Advanced Topics of Tensor Analysis

**Chapter 6**

**Abstract**

solution.

⊗ *<sup>L</sup>*, <sup>1</sup>

**87**

**1. Introduction**

separated schemes restricted to scheme X.

<sup>1</sup> , is a Lefschetz motive ð Þ<sup>1</sup> , [1].

determination of the tensor triangulated category DM‐

*Francisco Bulnes*

Derived Tensor Products

In this research we studied **t**he tensor product on derived categories of Étale sheaves with transfers considering as fundamental, the tensor product of categories *X* ⊗ *Y* ¼ *X* � *Y*, on the category Cor*k*, (finite correspondences category) by understanding it to be the product of the underlying schemes on *k*. Although, to this is required to build a total tensor product on the category PST(*k*), where this

construction will be useful to obtain generalizations on derived categories using pre-sheaves and contravariant and covariant functors on additive categories to define the exactness of infinite sequences and resolution of spectral sequences. Some concrete applications are given through a result on field equations

**Keywords:** algebraic variety, additive pre-sheaves, derived categories, derived tensor products, finite correspondences category, schemes, singularities, varieties

This study is focused on the derived tensor product whose functors have images as cohomology groups that are representations of integrals of sheaves represented for its pre-sheaves in an order modulo *k*. This study is remounted to the *K*-theory on the sheaves cohomologies constructed through pre-sheaves defined by the tensor product on commutative rings. The intention of this study is to establish a methodology through commutative rings and their construction of a total tensor product

on the category PST(*k*), considering extensions of the tensor products ⊗ *<sup>R</sup>*ð Þ <sup>A</sup> , to obtain resolution in the projective sense of infinite sequences of modules of Étale sheaves. These sheaves are pre-sheaves of Abelian groups on the category of smooth

Likewise, the immediate application of the derived tensor products will be the

to be equivalent to the derived category of discrete *=m*‐ modules over the Galois

́ettð Þ *k*, *=m* , of Étale motives

**2010 Mathematics Subject classification**: 13D09, 18D20, 13D15

and Their Applications

#### **Chapter 6**

## Derived Tensor Products and Their Applications

*Francisco Bulnes*

#### **Abstract**

In this research we studied **t**he tensor product on derived categories of Étale sheaves with transfers considering as fundamental, the tensor product of categories *X* ⊗ *Y* ¼ *X* � *Y*, on the category Cor*k*, (finite correspondences category) by understanding it to be the product of the underlying schemes on *k*. Although, to this is required to build a total tensor product on the category PST(*k*), where this construction will be useful to obtain generalizations on derived categories using pre-sheaves and contravariant and covariant functors on additive categories to define the exactness of infinite sequences and resolution of spectral sequences. Some concrete applications are given through a result on field equations solution.

**Keywords:** algebraic variety, additive pre-sheaves, derived categories, derived tensor products, finite correspondences category, schemes, singularities, varieties **2010 Mathematics Subject classification**: 13D09, 18D20, 13D15

#### **1. Introduction**

This study is focused on the derived tensor product whose functors have images as cohomology groups that are representations of integrals of sheaves represented for its pre-sheaves in an order modulo *k*. This study is remounted to the *K*-theory on the sheaves cohomologies constructed through pre-sheaves defined by the tensor product on commutative rings. The intention of this study is to establish a methodology through commutative rings and their construction of a total tensor product ⊗ *<sup>L</sup>*, <sup>1</sup> on the category PST(*k*), considering extensions of the tensor products ⊗ *<sup>R</sup>*ð Þ <sup>A</sup> , to obtain resolution in the projective sense of infinite sequences of modules of Étale sheaves. These sheaves are pre-sheaves of Abelian groups on the category of smooth separated schemes restricted to scheme X.

Likewise, the immediate application of the derived tensor products will be the determination of the tensor triangulated category DM‐ ́ettð Þ *k*, *=m* , of Étale motives to be equivalent to the derived category of discrete *=m*‐ modules over the Galois

<sup>1</sup> , is a Lefschetz motive ð Þ<sup>1</sup> , [1].

group *<sup>G</sup>* <sup>¼</sup> Gal *ksep=<sup>k</sup>* , which says on the equivalence of functors of tensor triangulated categories<sup>2</sup> .

Then the mean result of derived tensor products will be in tensor triangulated category DMeff,� *Nis* ð Þ *k*, *R* , of effective motives and their subcategory of effective geometric motives DMeff,� *gm* ð Þ *<sup>k</sup>*, *<sup>R</sup> :* Likewise, the motive *<sup>M</sup>*(X*)*, of a scheme *<sup>X</sup>*, is an object of DMeff,� *Nis* ð Þ *<sup>k</sup>*, *<sup>R</sup>* , and belongs to DMeff,� *gm* ð Þ *<sup>k</sup>*, *<sup>R</sup>* , if *<sup>X</sup>*, is smooth. However, this requires the use of cohomological properties of sheaves associated with homotopy invariant pre-sheaves with transfers for Zariski topology, Nisnevich and cdh topologies.

Finally, all this treatment goes in-walked to develop a motivic cohomology to establish a resolution in the field theory incorporating singularities in the complex Riemannian manifolds where singularities can be studied with deformation theory through operads, motives, and deformation quantization.

#### **2. Fundaments of derived tensor products**

We consider the Abelian category Ab, which is conformed by all functor images that are contravariant additive functors *F* : A ! Ab, on small category of ð Þ A *:* Likewise, ð Þ A , is the category of all additive pre-sheaves on A. Likewise, we can define this category as of points space:

$$\mathbb{Z}(\mathcal{A}) = \{ F|F: \mathcal{A} \to \mathbf{Ab} \}, \tag{1}$$

Likewise, we have the Yoneda embedding as the mapping<sup>3</sup> :

$$h: \mathcal{A} \to \mathbb{Z}(\mathcal{A}),\tag{2}$$

*h*<sup>X</sup> : *A* ↦*R* ⊗ HomAð Þ *A*, X , (6)

Hom*R*ð Þ <sup>A</sup> ð Þffi *h*X, *F F*ð Þ X , (7)

*<sup>F</sup>* <sup>¼</sup> *<sup>h</sup>*X*=*<sup>A</sup> <sup>⊕</sup> , (8)

x : ⊕ *h*<sup>X</sup> ! *F*, (9)

A ¼ *Cork*, (11)

ð Þ *F* ⊗ *RG* ð Þ¼ X *F*ð Þ X ⊗ *RG*ð Þ X , (13)

ð Þ *M* ⊗ *N<sup>α</sup>* , (12)

*h*X, (10)

■

Likewise, the following lemma introduces the representable pre-sheaves and functors and their role to construct pre-sheaves ⊗ *<sup>R</sup>*ð Þ <sup>A</sup> , that can be extended to presheaves <sup>⊗</sup> *<sup>L</sup>*, first using the projective objects of *<sup>R</sup>*ð Þ <sup>A</sup> , and define the projective

**Lemma 1.1.** Every representable pre-sheaf *h*X, is a projective object of *R*ð Þ A , every projective object of *R*ð Þ A is a direct summand of a direct sum of representable

Then each object *h*X, is a projective object in *R*ð Þ A *:* Likewise, each *F* ∈*R*ð Þ A , is a

Then from the additive category until functional additive category modulus

<sup>⊕</sup> <sup>x</sup><sup>∈</sup> *<sup>F</sup>*ð Þ <sup>X</sup> x 6¼ 0

Now suppose that A, with an additive symmetric monoidal structure ⊗ , is such

ffi ⊕ *α*∈ *A*

We extend <sup>⊗</sup> , on <sup>A</sup> <sup>⊕</sup> , in the same way, and this extends to tensor product of corresponding projectives. Then ⊗ , can be extended to a tensor product on

Likewise, if *F*, *G* ∈ *R*ð Þ A , then we have a pre-sheaf tensor product in the

However, this does not correspond to *R*ð Þ A , since *F* ⊗ *<sup>R</sup> G*, is not additive. However, this could be additive when one component *<sup>F</sup>*, or *<sup>G</sup>*, is element of <sup>A</sup> <sup>⊕</sup> . But if we want to get a tensor product on *R*ð Þ A , we need a more complicated or specialized construction. For this, we consider X, Y∈ A, then *h*<sup>X</sup> ⊗ *h*Y, of their

This means that ⊗ , commutes with direct sum. Let *Nα*, *α* ∈ *A*, and *M*, be

*N<sup>α</sup>* 

which is representable of the *R*-mod.

*Derived Tensor Products and Their Applications DOI: http://dx.doi.org/10.5772/intechopen.92869*

resolution to infinite complexes sequence.

then there exist a surjection x, such that

quotient

that

all of *R*ð Þ A .

**89**

following way:

<sup>A</sup> <sup>⊕</sup> <sup>⊂</sup>ð Þ <sup>A</sup> , we have:

which proves the lemma.

*R*-modules; then is clear that:

functors, and every *F*, in *R*ð Þ A , has a projective resolution.

*Proof*. We consider an analogue to (6) in the functor context:

*F* ¼ ⊕ X∈ A

*<sup>M</sup>* ⊗ ⊕*<sup>α</sup>*<sup>∈</sup> *<sup>A</sup>*

which has correspondence rule

$$\mathbf{X} \mapsto \oplus \mathbf{X}\_i,\tag{3}$$

or

$$h\_{\mathbf{X}} = \oplus h\_{\mathbf{X}\_i},\tag{4}$$

We need a generalization of the before categories and functors, therefore we give a ring *R*, originating the ring structure Að Þ *R* , to be the Abelian category of the additive functors

$$F: \mathcal{A} \to \mathcal{R}-\text{mod},\tag{5}$$

being *R*-mod, the category of the modules that originate the ring structure. Then *hX*, is the functor

$$\operatorname{D}^{-}(\operatorname{G},\mathbb{Z}/m) \xrightarrow{\pi^{\*}} \mathcal{L} \to \operatorname{D}(\mathcal{W}\_{A}^{\cdot 1}) = \operatorname{DM}\_{\operatorname{Er}}^{\operatorname{eff}\_{\*}-}(k,\mathbb{Z}/m),$$

until the category D�ð Þ *Sh*́*et*ð Þ *Cork*, *=m :*

<sup>2</sup> **Theorem.** If 1*=*<sup>m</sup> <sup>∈</sup>*k*, the space ð Þ <sup>L</sup>, <sup>⊗</sup> <sup>L</sup> , is a tensor triangulated category and the functors

<sup>3</sup> The obtained image by the Yoneda embedding has the pre-sheaf <sup>A</sup> <sup>⊕</sup> <sup>⊂</sup>ð Þ <sup>A</sup> *:*

*Derived Tensor Products and Their Applications DOI: http://dx.doi.org/10.5772/intechopen.92869*

group *<sup>G</sup>* <sup>¼</sup> Gal *ksep=<sup>k</sup>* , which says on the equivalence of functors of tensor

this requires the use of cohomological properties of sheaves associated with

Then the mean result of derived tensor products will be in tensor triangulated

geometric motives DMeff,� *gm* ð Þ *<sup>k</sup>*, *<sup>R</sup> :* Likewise, the motive *<sup>M</sup>*(X*)*, of a scheme *<sup>X</sup>*, is an

homotopy invariant pre-sheaves with transfers for Zariski topology, Nisnevich and

Finally, all this treatment goes in-walked to develop a motivic cohomology to establish a resolution in the field theory incorporating singularities in the complex Riemannian manifolds where singularities can be studied with deformation theory

We consider the Abelian category Ab, which is conformed by all functor images that are contravariant additive functors *F* : A ! Ab, on small category of ð Þ A *:* Likewise, ð Þ A , is the category of all additive pre-sheaves on A. Likewise, we can

*h*<sup>X</sup> ¼ ⊕ *h*<sup>X</sup>*<sup>i</sup>*

We need a generalization of the before categories and functors, therefore we give a ring *R*, originating the ring structure Að Þ *R* , to be the Abelian category of the

being *R*-mod, the category of the modules that originate the ring structure. Then

*A* <sup>¼</sup> DMeff,�

<sup>2</sup> **Theorem.** If 1*=*<sup>m</sup> <sup>∈</sup>*k*, the space ð Þ <sup>L</sup>, <sup>⊗</sup> <sup>L</sup> , is a tensor triangulated category and the functors

<sup>D</sup>�ð Þ! *<sup>G</sup>*, *=<sup>m</sup> <sup>π</sup>* <sup>∗</sup> L ! <sup>D</sup> *<sup>W</sup>*‐<sup>1</sup>

<sup>3</sup> The obtained image by the Yoneda embedding has the pre-sheaf <sup>A</sup> <sup>⊕</sup> <sup>⊂</sup>ð Þ <sup>A</sup> *:*

ð Þ¼ A f g *F F*j :A ! Ab , (1)

*h* : A ! ð Þ A , (2)

X ↦ ⊕ X*i*, (3)

*F* : A ! *R* � mod, (5)

́*Et* ð Þ *k*,*=m* ,

:

, (4)

*Nis* ð Þ *k*, *R* , of effective motives and their subcategory of effective

*Nis* ð Þ *<sup>k</sup>*, *<sup>R</sup>* , and belongs to DMeff,� *gm* ð Þ *<sup>k</sup>*, *<sup>R</sup>* , if *<sup>X</sup>*, is smooth. However,

triangulated categories<sup>2</sup>

category DMeff,�

object of DMeff,�

cdh topologies.

or

additive functors

*hX*, is the functor

**88**

until the category D�ð Þ *Sh*́*et*ð Þ *Cork*, *=m :*

.

*Advances on Tensor Analysis and Their Applications*

through operads, motives, and deformation quantization.

Likewise, we have the Yoneda embedding as the mapping<sup>3</sup>

**2. Fundaments of derived tensor products**

define this category as of points space:

which has correspondence rule

$$h\_{\mathbf{X}}: A \mapsto \mathbf{R} \otimes\_{\mathbb{Z}} \mathbf{Hom}\_{\mathcal{A}}(A, \mathbf{X}), \tag{6}$$

which is representable of the *R*-mod.

Likewise, the following lemma introduces the representable pre-sheaves and functors and their role to construct pre-sheaves ⊗ *<sup>R</sup>*ð Þ <sup>A</sup> , that can be extended to presheaves <sup>⊗</sup> *<sup>L</sup>*, first using the projective objects of *<sup>R</sup>*ð Þ <sup>A</sup> , and define the projective resolution to infinite complexes sequence.

**Lemma 1.1.** Every representable pre-sheaf *h*X, is a projective object of *R*ð Þ A , every projective object of *R*ð Þ A is a direct summand of a direct sum of representable functors, and every *F*, in *R*ð Þ A , has a projective resolution.

*Proof*. We consider an analogue to (6) in the functor context:

$$\mathbf{Hom}\_{\mathcal{R}(\mathcal{A})}(h\_{\mathbf{X}}, F) \cong F(\mathbf{X}),\tag{7}$$

Then each object *h*X, is a projective object in *R*ð Þ A *:* Likewise, each *F* ∈*R*ð Þ A , is a quotient

$$F = h\_{\mathcal{X}} / \mathcal{A}^{\oplus},\tag{8}$$

then there exist a surjection x, such that

$$\mathbf{x}: \ \oplus h\_{\mathbf{X}} \to F,\tag{9}$$

Then from the additive category until functional additive category modulus <sup>A</sup> <sup>⊕</sup> <sup>⊂</sup>ð Þ <sup>A</sup> , we have:

$$F = \bigoplus\_{\mathbf{X} \in \mathcal{A}} \bigoplus\_{\mathbf{x} \in F(\mathbf{X})} h\_{\mathbf{X}},\tag{10}$$
 
$$\mathbf{x} \neq \mathbf{0}$$

which proves the lemma.

■ Now suppose that A, with an additive symmetric monoidal structure ⊗ , is such that

$$\mathcal{A}\_- = \text{Cor}\_k,\tag{11}$$

This means that ⊗ , commutes with direct sum. Let *Nα*, *α* ∈ *A*, and *M*, be *R*-modules; then is clear that:

$$M \otimes \left( \bigoplus\_{a \in A} N\_a \right) \cong \bigoplus\_{a \in A} (M \otimes N\_a), \tag{12}$$

We extend <sup>⊗</sup> , on <sup>A</sup> <sup>⊕</sup> , in the same way, and this extends to tensor product of corresponding projectives. Then ⊗ , can be extended to a tensor product on all of *R*ð Þ A .

Likewise, if *F*, *G* ∈ *R*ð Þ A , then we have a pre-sheaf tensor product in the following way:

$$(F \otimes\_R G)(\mathbf{X}) = F(\mathbf{X}) \otimes\_R G(\mathbf{X}),\tag{13}$$

However, this does not correspond to *R*ð Þ A , since *F* ⊗ *<sup>R</sup> G*, is not additive. However, this could be additive when one component *<sup>F</sup>*, or *<sup>G</sup>*, is element of <sup>A</sup> <sup>⊕</sup> . But if we want to get a tensor product on *R*ð Þ A , we need a more complicated or specialized construction. For this, we consider X, Y∈ A, then *h*<sup>X</sup> ⊗ *h*Y, of their

representable pre-sheaves should be represented by X ⊗ Y. As a first step, we can extend ⊗ , to a tensor product

$$\otimes : \mathcal{A}^{\oplus} \times \mathcal{A}^{\oplus} \to \mathcal{A}^{\oplus},\tag{14}$$

Then in (20) we have:

We consider the following examples.

*Derived Tensor Products and Their Applications DOI: http://dx.doi.org/10.5772/intechopen.92869*

Kähler 1-differentials module, is *M* ⊗ <sup>L</sup>

compatible with the monoidal pairing

*Proof*. We have Hom, as defined in (18):

If *F*1, *G*1, *F*2, *G*<sup>2</sup> ∈ *R*ð Þ A *:*, then

Here, for any two modules *A*, *B*∈*R*ð Þ A , we have:

**Example 2.1.** Let A, be the category of *R*-modules *M*, such that:

*<sup>R</sup>*ð Þ <sup>A</sup> . Here the category is of the cotangent complexes of *<sup>R</sup>*.

But also,

Hom*R*, and ⊗ *<sup>R</sup>:*

and ⊗ tor*:*

and

**91**

⊗ *<sup>L</sup>*

Hom*R*ð Þ <sup>A</sup> *<sup>h</sup>*X, Hom*<sup>R</sup>*ð Þ <sup>A</sup> ð Þ *<sup>h</sup>*<sup>X</sup> <sup>⊗</sup> *<sup>h</sup>*Y, *<sup>G</sup>* <sup>¼</sup> *<sup>G</sup>*ð Þ <sup>X</sup> <sup>⊗</sup> <sup>Y</sup> , (23)

where the lemma is proved. ■

**Example 1.1.** We consider the category A, of free *R*-modules over a commutative ring *R*ð Þ A . This category is equivalent to the category of all *R*-modules where pre-sheaf associated to *M*, is *M* ⊗ *<sup>R</sup>*, and Hom, and ⊗ , are the familiar

where *K*, is a fraction field<sup>6</sup> and *M*tor, is the torsion submodule of *M:* Then associated to *M*, is 1 ⊗ *RM*, which is pre-sheaf. Here Hom*:* and ⊗ , are Hom*R*,

**Example 3.1.** Let *R*, be a simplicial commutative ring and *R*ð Þ!A A , be a category cofibrant replacement. Here, the pre-sheaf associated to *M*, which is the

**Proposition 1.1.** If *Fi*, and *Gi*, are in *R*ð Þ A , then there is a natural mapping

HomAð Þ *U* � *A*1, X1 ⊗ HomAð Þ! *U* � *A*2, X2 HomAð Þ *U* � *U* � *A*<sup>1</sup> � *A*2, X1 � X2

<sup>6</sup> The field of fractions of an integral domain is the smallest field in which this domain can be embedded.

Homð Þ *F*1, *G*<sup>1</sup> ⊗ Homð Þ! *F*2, *G*<sup>2</sup> Homð Þ *F*<sup>1</sup> ⊗ *G*1, *F*<sup>2</sup> ⊗ *G*<sup>2</sup> , (27)

Homð Þ *F*1, *G*<sup>1</sup> : X1 ! Hom*<sup>R</sup>*ð Þ <sup>A</sup> *F*<sup>1</sup> ⊗ *h*X1 ð Þ , *G*<sup>1</sup> , (29)

Homð Þ *F*2, *G*<sup>2</sup> : X2 ! Hom*<sup>R</sup>*ð Þ <sup>A</sup> *F*<sup>2</sup> ⊗ *h*X2 ð Þ , *G*<sup>2</sup> , (30)

*G*ð Þ¼ X ⊗ Y Hom*R*ð Þ <sup>A</sup> ð Þ *h*<sup>X</sup> ⊗ *h*Y, G , (24)

*A* ⊗ *RB* : *A* ⊗ *B* ¼ *B* ⊗ *A*, (25)

*K* ⊗ *RM* ffi *K* ⊗ *<sup>R</sup>*ð Þ *M=M*tor , (26)

*<sup>R</sup>*ð Þ <sup>A</sup> , and here Hom, and <sup>⊗</sup> , are Hom*R*, and

! HomAð Þ *U* � *A*<sup>1</sup> � *A*2, X1 � X2 ,

(28)

commuting with <sup>⊕</sup> . Thus if *<sup>L</sup>*1, *<sup>L</sup>*<sup>2</sup> <sup>∈</sup>Ch� <sup>A</sup> <sup>⊕</sup> , of the above co-chain complexes as follows:

$$\dots \to F^{\mathfrak{n}} \to \mathbf{0} \to \dots,\tag{15}$$

the chain complex *L*<sup>1</sup> ⊗ *L*2, is defined as the total complex of the double complex *L*<sup>∗</sup> <sup>1</sup> ⊗ *L*<sup>∗</sup> 2 .

Then we can define a legitimate tensor product between two categories *F*, *G* ∈*R*ð Þ A , as follows:

**Definition 1.1.** Let be *F*, *G* ∈ *R*ð Þ A , choosing projective resolutions

$$P\_\* \to F, \quad Q\_\* \to G,\tag{16}$$

we define *F* ⊗ *<sup>L</sup> G*, <sup>4</sup> to be *P* ⊗ *Q*, which means that the tensor product is total having that Tot *P*<sup>∗</sup> ⊗ *Q* <sup>∗</sup> ð Þ*:* Then the tensor product to these pre-sheaves and the Hom, pre-sheaves is defined as:

$$F \otimes G = H\_0(F \otimes^L G),\tag{17}$$

and

$$\operatorname{Hom}(F,G): \mathbf{X} \mapsto \operatorname{Hom}\_{\mathbb{R}(\mathcal{A})}(F \otimes h\_{\mathbf{X}}, G),\tag{18}$$

The relation (17) means the chain homotopy equivalent of the *F* ⊗ *<sup>L</sup> G*, is well defined up to chain homotopy equivalence, and analogous for Homð Þ *F*, *G :*

In particular, given that *h*X, and *h*Y, are projective, we have

$$\begin{aligned} h\_{\mathbf{X}} \otimes^L h\_{\mathbf{Y}} &= h\_{\mathbf{X}} \otimes h\_{\mathbf{Y}} = h\_{\mathbf{X} \otimes \mathbf{Y}}, \\ \forall \mathbf{X}, \mathbf{Y} \in \mathcal{A}^{\oplus}. \end{aligned} \tag{19}$$

Likewise, the ring *R*ð Þ A , is an additive symmetric monoidal category. We consider the following lemma.

Lemma 1.2. The functor Homð Þ *F*,• , is right adjoint to *F* ⊗ •*:* In particular Homð Þ *F*,• , is left exact and *F* ⊗ •, is right exact.

*Proof*. Let be

$$F = \text{Hom}(h\_{\mathcal{Y}}, \mathcal{G}), \tag{20}$$

Then<sup>5</sup>

$$\operatorname{Hom}\_{\mathcal{R}(\mathcal{A})}(h\_{\mathbf{X}}, F) = \operatorname{Hom}\_{\mathcal{R}(\mathcal{A})}(h\_{\mathbf{X}}, \operatorname{Hom}(h\_{\mathbf{Y}}, G)),\tag{21}$$

We consider

$$\operatorname{Hom}(h\_{\mathbf{Y}}, \mathbf{G}) = \operatorname{Hom}\_{\mathbb{R}(\mathcal{A})}(h\_{\mathbf{X}} \otimes h\_{\mathbf{Y}}, \mathbf{G}),\tag{22}$$

<sup>4</sup> ⊗ *<sup>L</sup>*, is a total tensor product.

<sup>5</sup> Homð Þ *<sup>h</sup>*Y, *<sup>G</sup>* : <sup>Y</sup> ! Hom*<sup>R</sup>*ð Þ <sup>A</sup> ð Þ *<sup>F</sup>* <sup>⊗</sup> *<sup>h</sup>*Y, G *:*

Then in (20) we have:

$$\mathbf{Hom}\_{\mathbb{R}(\mathcal{A})} \left( h\_{\mathbf{X}}, \mathbf{Hom}\_{\mathbb{R}(\mathcal{A})} (h\_{\mathbf{X}} \otimes h\_{\mathbf{Y}}, \mathbf{G}) \right) = \mathbf{G} (\mathbf{X} \otimes \mathbf{Y}), \tag{23}$$

But also,

representable pre-sheaves should be represented by X ⊗ Y. As a first step, we can

commuting with <sup>⊕</sup> . Thus if *<sup>L</sup>*1, *<sup>L</sup>*<sup>2</sup> <sup>∈</sup>Ch� <sup>A</sup> <sup>⊕</sup> , of the above co-chain com-

the chain complex *L*<sup>1</sup> ⊗ *L*2, is defined as the total complex of the double com-

Then we can define a legitimate tensor product between two categories

having that Tot *P*<sup>∗</sup> ⊗ *Q* <sup>∗</sup> ð Þ*:* Then the tensor product to these pre-sheaves and the

The relation (17) means the chain homotopy equivalent of the *F* ⊗ *<sup>L</sup> G*, is well

*<sup>h</sup>*<sup>X</sup> <sup>⊗</sup> *Lh*<sup>Y</sup> <sup>¼</sup> *<sup>h</sup>*<sup>X</sup> <sup>⊗</sup> *<sup>h</sup>*<sup>Y</sup> <sup>¼</sup> *<sup>h</sup>*<sup>X</sup> <sup>⊗</sup> Y,

defined up to chain homotopy equivalence, and analogous for Homð Þ *F*, *G :* In particular, given that *h*X, and *h*Y, are projective, we have

Likewise, the ring *R*ð Þ A , is an additive symmetric monoidal category.

Lemma 1.2. The functor Homð Þ *F*,• , is right adjoint to *F* ⊗ •*:* In particular

**Definition 1.1.** Let be *F*, *G* ∈ *R*ð Þ A , choosing projective resolutions

<sup>⊗</sup> : <sup>A</sup> <sup>⊕</sup> � A <sup>⊕</sup> ! A <sup>⊕</sup> , (14)

… ! *<sup>F</sup><sup>n</sup>* ! <sup>0</sup> ! … , (15)

*P*<sup>∗</sup> ! *F*, *Q* <sup>∗</sup> ! *G*, (16)

*<sup>F</sup>* <sup>⊗</sup> *<sup>G</sup>* <sup>¼</sup> *H0 <sup>F</sup>* <sup>⊗</sup> *LG* , (17)

to be *P* ⊗ *Q*, which means that the tensor product is total

Homð Þ *F*, *G* : X ↦Hom*<sup>R</sup>*ð Þ <sup>A</sup> ð Þ *F* ⊗ *h*X, *G* , (18)

<sup>∀</sup>X, Y<sup>∈</sup> <sup>A</sup> <sup>⊕</sup> *:* (19)

*F* ¼ Homð Þ *h*Y, *G* , (20)

Hom*<sup>R</sup>*ð Þ <sup>A</sup> ð Þ¼ *h*X, *F* Hom*<sup>R</sup>*ð Þ <sup>A</sup> ð Þ *h*X, Homð Þ *h*Y, *G* , (21)

Homð Þ¼ *h*Y, *G* HomRð Þ <sup>A</sup> ð Þ *h*<sup>X</sup> ⊗ *h*Y, *G* , (22)

extend ⊗ , to a tensor product

*Advances on Tensor Analysis and Their Applications*

plexes as follows:

<sup>1</sup> ⊗ *L*<sup>∗</sup> 2 .

*F*, *G* ∈*R*ð Þ A , as follows:

we define *F* ⊗ *<sup>L</sup> G*, <sup>4</sup>

Hom, pre-sheaves is defined as:

We consider the following lemma.

*Proof*. Let be

We consider

<sup>4</sup> ⊗ *<sup>L</sup>*, is a total tensor product.

<sup>5</sup> Homð Þ *<sup>h</sup>*Y, *<sup>G</sup>* : <sup>Y</sup> ! Hom*<sup>R</sup>*ð Þ <sup>A</sup> ð Þ *<sup>F</sup>* <sup>⊗</sup> *<sup>h</sup>*Y, G *:*

Then<sup>5</sup>

**90**

Homð Þ *F*,• , is left exact and *F* ⊗ •, is right exact.

plex *L*<sup>∗</sup>

and

$$G(\mathbf{X}\otimes\mathbf{Y}) = \mathbf{Hom}\_{\mathbb{R}(\mathcal{A})}(h\_{\mathbf{X}}\otimes h\_{\mathbf{Y}}, \mathbf{G}),\tag{24}$$

where the lemma is proved. ■

We consider the following examples.

**Example 1.1.** We consider the category A, of free *R*-modules over a commutative ring *R*ð Þ A . This category is equivalent to the category of all *R*-modules where pre-sheaf associated to *M*, is *M* ⊗ *<sup>R</sup>*, and Hom, and ⊗ , are the familiar Hom*R*, and ⊗ *<sup>R</sup>:*

Here, for any two modules *A*, *B*∈*R*ð Þ A , we have:

$$A \otimes\_R B : A \otimes B = B \otimes A,\tag{25}$$

**Example 2.1.** Let A, be the category of *R*-modules *M*, such that:

$$K \otimes\_R M \cong K \otimes\_R (M/M\_{\text{tor}}),\tag{26}$$

where *K*, is a fraction field<sup>6</sup> and *M*tor, is the torsion submodule of *M:* Then associated to *M*, is 1 ⊗ *RM*, which is pre-sheaf. Here Hom*:* and ⊗ , are Hom*R*, and ⊗ tor*:*

**Example 3.1.** Let *R*, be a simplicial commutative ring and *R*ð Þ!A A , be a category cofibrant replacement. Here, the pre-sheaf associated to *M*, which is the Kähler 1-differentials module, is *M* ⊗ <sup>L</sup> *<sup>R</sup>*ð Þ <sup>A</sup> , and here Hom, and <sup>⊗</sup> , are Hom*R*, and ⊗ *<sup>L</sup> <sup>R</sup>*ð Þ <sup>A</sup> . Here the category is of the cotangent complexes of *<sup>R</sup>*.

**Proposition 1.1.** If *Fi*, and *Gi*, are in *R*ð Þ A , then there is a natural mapping

$$\operatorname{Hom}(F\_1, G\_1) \otimes \operatorname{Hom}(F\_2, G\_2) \to \operatorname{Hom}(F\_1 \otimes G\_1, F\_2 \otimes G\_2),\tag{27}$$

compatible with the monoidal pairing

$$\begin{aligned} \operatorname{Hom}\_{\mathcal{A}}(U \times A\_1, \mathbf{X}\_1) \otimes \operatorname{Hom}\_{\mathcal{A}}(U \times A\_2, \mathbf{X}\_2) &\to \operatorname{Hom}\_{\mathcal{A}}(U \times U \times A\_1 \times A\_2, \mathbf{X}\_1 \times \mathbf{X}\_2) \\ &\to \operatorname{Hom}\_{\mathcal{A}}(U \times A\_1 \times A\_2, \mathbf{X}\_1 \times \mathbf{X}\_2), \end{aligned} \tag{28}$$

*Proof*. We have Hom, as defined in (18):

$$\operatorname{Hom}(F\_1, G\_1) : \mathbf{X}\_1 \to \operatorname{Hom}\_{\mathcal{R}(\mathcal{A})}(F\_1 \otimes h\_{\mathbf{X}\_1}, G\_1), \tag{29}$$

and

$$\operatorname{Hom}(F\_2, G\_2) : \mathbf{X}\_2 \to \operatorname{Hom}\_{\mathcal{R}(\mathcal{A})}(F\_2 \otimes h\_{\mathbf{X}\_2}, G\_2), \tag{30}$$

If *F*1, *G*1, *F*2, *G*<sup>2</sup> ∈ *R*ð Þ A *:*, then

<sup>6</sup> The field of fractions of an integral domain is the smallest field in which this domain can be embedded.

$$\begin{aligned} \operatorname{Hom}(F\_1, G\_1) \otimes \operatorname{Hom}(F\_2, G\_2) &= \operatorname{Hom}\_{\mathcal{R}(\mathcal{A})}(F\_1 \otimes h\_{\mathcal{X}\_1} \otimes F\_2 \otimes h\_{\mathcal{X}\_2}, G\_1 \otimes G\_2) \\ &= \operatorname{Hom}\_{\mathcal{R}(\mathcal{A})}(F\_1 \otimes F\_2 \otimes h\_{\mathcal{X} \otimes \mathcal{Y}\_1}, G\_1 \otimes G\_2) \\ &= \operatorname{Hom}(F\_1 \otimes F\_2, G\_1 \otimes G\_2), \end{aligned} \tag{31}$$

We consider the Universal mapping which is commutative:

$$F \times G \xrightarrow{\sim} F \otimes G$$

$$\mathfrak{S}$$

Then (31) is compatible with the monoidal pairing. ■

If the (projective) objects *hX*, are flat, that is to say, *h*<sup>X</sup> ⊗ •, is an exact functor then ⊗ , is called a balanced functor [2]. Here *F* ⊗ *<sup>L</sup> G*, agrees with the usual left derived functor *L F*ð Þ ⊗ • *G:* But here we do not know when the *hX*, are flat. This is true in Example 1.1. But it is not true in PSTð Þ¼ *k* ð Þ *Cork :* Then we need to extend <sup>⊗</sup> *<sup>L</sup>*, to a total tensor product on the category Ch�*R*ð Þ <sup>A</sup> , of bounded above cochain complexes (15). This would be the usual derived functor if ⊗ , were balanced [2], and our construction is parallel. Likewise, if *C*, is a complex in Ch�*R*ð Þ A , there is a quasi-isomorphism *P*ffi! C, with *P*, a complex of projective objects. Any such complex *P*, is called a projective resolution of *C*, and any other projective resolution of *C*, is chain homotopic to *P* [3].

Likewise, if *D*, is any complex in Ch�*R*ð Þ A , and

$$Q \xrightarrow{\cong} D,\tag{32}$$

We consider the following lemma to obtain in the extension (36) a derived triangulated category that will be useful in the context of derived tensor categories

category of spectra or more generally, the homotopy category of a stable ∞-category. In both cases is carried a structure of triangulated category.

**3. Derived triangulated categories with structure by pre-sheaves ⊗** *<sup>L</sup>***,**

**Proposition 3.1.** The derived category D�*R*ð Þ <sup>A</sup> , equipped with <sup>⊗</sup> *<sup>L</sup>*‐ structure is

*Proof*. We consider a projective object X∈P, where P, is a projective category

P ¼ f g X∈*R*ð ÞA A j is addative with ⊗ *<sup>R</sup>* � structure , (37)

*<sup>H</sup>*<sup>0</sup> <sup>Λ</sup> <sup>⊗</sup> *LQ* <sup>¼</sup> Homð Þ *<sup>h</sup>*X, Homð Þ <sup>Λ</sup> <sup>⊗</sup> *<sup>h</sup>*Y, *<sup>G</sup>* , (39)

Λð Þ¼ X Hom*<sup>R</sup>*ð Þ <sup>A</sup> ð Þ *h*X,Λ ∈ D�*R*ð Þ A , (40)

X ! Homð Þ *h*X, Homð Þ Λ ⊗ *h*Y, *G* , (42)

which is risked from <sup>⊗</sup> *<sup>L</sup>*� structure when <sup>⊗</sup> ffi <sup>⊗</sup> *<sup>L</sup>*, in <sup>P</sup>, which then is true from the lemma 2.1. ■ Now, for bounded complexes of pre-sheaves we can give the following

**Definition 3.1**. Let *C*, and *D*, be bounded complexes of pre-sheaves. There is a

which was foresee in the Definition 1.1. By right exactness of ⊗ *<sup>R</sup>*, and ⊗ , given

in Lemma 1.1, it suffices to construct a natural mapping of pre-sheaves

D�*R*ð Þffi A K�ð Þ P , (41)

*C* ⊗ *<sup>R</sup> D* ! *C* ⊗ *D*, (43)

Λ : P ! K�ð Þ P , (38)

The importance of a triangulated category together with the additional structure as the given by pre-sheaves ⊗ *<sup>L</sup>*, lies in obtaining distinguished triangles of categories that generate the long exact sequences of homology that can be described through of short exact sequences of Abelian categories. Likewise, the immediate examples are the derived categories of Abelian category and the stable homotopy

whose pre-sheaves are Étale pre-sheaves.

*Derived Tensor Products and Their Applications DOI: http://dx.doi.org/10.5772/intechopen.92869*

We enounce the following proposition.

via the chain homotopy. For other side

We consider the application Λ, defined by the mapping:

where the objects Λð Þ X , are those that are determined by

**and ⊗** *tr*

or

definitions.

**93**

canonical mapping:

Then we have

*L***,***ét*

defined as the points set

a tensor-triangulated category.

is a projective resolution, we define

$$\mathbb{C}\otimes^L D = P \otimes Q,\tag{33}$$

Now, how do we understand the extensions of these tensor products in chain homotopy equivalence?

Since *P*, and *Q*, are bounded above, each

$$(P \otimes Q)^{n} = \bigoplus\_{i+j=n} P^{i} \otimes Q^{j},\tag{34}$$

is a finite sum, and *C* ⊗ *<sup>L</sup> D*, is bounded above. Then, since *P*, and *Q*, are defined up to chain homotopy, the complex *C* ⊗ *<sup>L</sup> D*, is independent (up to chain homotopy equivalence) of the choice of *P*, and *Q*. Then there exists a mapping

$$\mathbf{C} \otimes^L \mathbf{D} \to \mathbf{C} \otimes \mathbf{D},\tag{35}$$

which extends the mapping

$$F \otimes^L G \to F \otimes G,\tag{36}$$

of Definition 1.1.

*Derived Tensor Products and Their Applications DOI: http://dx.doi.org/10.5772/intechopen.92869*

We consider the following lemma to obtain in the extension (36) a derived triangulated category that will be useful in the context of derived tensor categories whose pre-sheaves are Étale pre-sheaves.

The importance of a triangulated category together with the additional structure as the given by pre-sheaves ⊗ *<sup>L</sup>*, lies in obtaining distinguished triangles of categories that generate the long exact sequences of homology that can be described through of short exact sequences of Abelian categories. Likewise, the immediate examples are the derived categories of Abelian category and the stable homotopy category of spectra or more generally, the homotopy category of a stable ∞-category. In both cases is carried a structure of triangulated category.

#### **3. Derived triangulated categories with structure by pre-sheaves ⊗** *<sup>L</sup>***, and ⊗** *tr L***,***ét*

We enounce the following proposition.

**Proposition 3.1.** The derived category D�*R*ð Þ <sup>A</sup> , equipped with <sup>⊗</sup> *<sup>L</sup>*‐ structure is a tensor-triangulated category.

*Proof*. We consider a projective object X∈P, where P, is a projective category defined as the points set

$$\mathcal{P} = \{ \mathbf{X} \in \mathcal{R}(\mathcal{A}) | \mathcal{A} \text{ is additive with } \otimes\_{\mathcal{R}} - \text{structure} \}, \tag{37}$$

We consider the application Λ, defined by the mapping:

$$
\Lambda : \mathcal{P} \to \mathbb{K}^-(\mathcal{P}),
\tag{38}
$$

where the objects Λð Þ X , are those that are determined by

$$H\_0(\Lambda \otimes^L \mathbf{Q}) = \text{Hom}(h\_\mathbf{X}, \text{Hom}(\Lambda \otimes h\_\mathbf{Y}, \mathbf{G})),\tag{39}$$

or

Homð Þ *F*1, *G*<sup>1</sup> ⊗ Homð Þ¼ *F*2, *G*<sup>2</sup> Hom*R*ð Þ <sup>A</sup> *F*<sup>1</sup> ⊗ *h*X1 ⊗ *F*<sup>2</sup> ⊗ *h*X2 ð Þ , *G*<sup>1</sup> ⊗ *G*<sup>2</sup>

Then (31) is compatible with the monoidal pairing. ■ If the (projective) objects *hX*, are flat, that is to say, *h*<sup>X</sup> ⊗ •, is an exact functor then ⊗ , is called a balanced functor [2]. Here *F* ⊗ *<sup>L</sup> G*, agrees with the usual left derived functor *L F*ð Þ ⊗ • *G:* But here we do not know when the *hX*, are flat. This is true in Example 1.1. But it is not true in PSTð Þ¼ *k* ð Þ *Cork :* Then we need to extend <sup>⊗</sup> *<sup>L</sup>*, to a total tensor product on the category Ch�*R*ð Þ <sup>A</sup> , of bounded above cochain complexes (15). This would be the usual derived functor if ⊗ , were balanced [2], and our construction is parallel. Likewise, if *C*, is a complex in Ch�*R*ð Þ A , there is a quasi-isomorphism *P*ffi! C, with *P*, a complex of projective objects. Any such complex *P*, is called a projective resolution of *C*, and any other projective resolution

*Q* !

Now, how do we understand the extensions of these tensor products in chain

*i*þ*j*¼*n*

is a finite sum, and *C* ⊗ *<sup>L</sup> D*, is bounded above. Then, since *P*, and *Q*, are defined up to chain homotopy, the complex *C* ⊗ *<sup>L</sup> D*, is independent (up to chain homotopy equivalence) of the choice of *P*, and *Q*. Then there exists a mapping

*Pi* ⊗ *Q <sup>j</sup>*

ð Þ *<sup>P</sup>* <sup>⊗</sup> *<sup>Q</sup> <sup>n</sup>* <sup>¼</sup> <sup>⊕</sup>

We consider the Universal mapping which is commutative:

*Advances on Tensor Analysis and Their Applications*

of *C*, is chain homotopic to *P* [3].

homotopy equivalence?

which extends the mapping

of Definition 1.1.

**92**

is a projective resolution, we define

Since *P*, and *Q*, are bounded above, each

Likewise, if *D*, is any complex in Ch�*R*ð Þ A , and

¼ Hom*R*ð Þ <sup>A</sup> ð Þ *F*<sup>1</sup> ⊗ *F*<sup>2</sup> ⊗ *h*<sup>X</sup> <sup>⊗</sup> Y, *G*<sup>1</sup> ⊗ *G*<sup>2</sup>

¼ Homð Þ *F*<sup>1</sup> ⊗ *F*2, *G*<sup>1</sup> ⊗ *G*<sup>2</sup> , (31)

ffi *D*, (32)

, (34)

<sup>C</sup> <sup>⊗</sup> *<sup>L</sup> <sup>D</sup>* <sup>¼</sup> *<sup>P</sup>* <sup>⊗</sup> *<sup>Q</sup>*, (33)

*<sup>C</sup>* <sup>⊗</sup> *<sup>L</sup> <sup>D</sup>* ! *<sup>C</sup>* <sup>⊗</sup> *<sup>D</sup>*, (35)

*<sup>F</sup>* <sup>⊗</sup> *<sup>L</sup> <sup>G</sup>* ! *<sup>F</sup>* <sup>⊗</sup> *<sup>G</sup>*, (36)

$$\Lambda(\mathbf{X}) = \mathbf{Hom}\_{\mathcal{R}(\mathcal{A})}(h\_{\mathbf{X}}, \Lambda) \in \mathbf{D}^- \mathcal{R}(\mathcal{A}),\tag{40}$$

Then we have

$$\mathbf{D}^{-}\mathcal{R}(\mathcal{A}) \cong \mathbf{K}^{-}(\mathcal{P}),\tag{41}$$

via the chain homotopy. For other side

$$\mathbf{X} \to \mathbf{Hom}(h\_{\mathbf{X}}, \mathbf{Hom}(\Lambda \otimes h\_{\mathbf{Y}}, \mathbf{G})),\tag{42}$$

which is risked from <sup>⊗</sup> *<sup>L</sup>*� structure when <sup>⊗</sup> ffi <sup>⊗</sup> *<sup>L</sup>*, in <sup>P</sup>, which then is true from the lemma 2.1. ■

Now, for bounded complexes of pre-sheaves we can give the following definitions.

**Definition 3.1**. Let *C*, and *D*, be bounded complexes of pre-sheaves. There is a canonical mapping:

$$C \otimes\_R D \to C \otimes D,\tag{43}$$

which was foresee in the Definition 1.1. By right exactness of ⊗ *<sup>R</sup>*, and ⊗ , given in Lemma 1.1, it suffices to construct a natural mapping of pre-sheaves

*Advances on Tensor Analysis and Their Applications*

$$\eta: h\_{\mathbf{X}} \otimes\_{R} h\_{\mathbf{Y}} \to h\_{\mathbf{X} \otimes \mathbf{Y}},\tag{44}$$

$$\forall \mathbf{X}, \mathbf{Y} \in \mathcal{A}^{\oplus}.$$

The above can be generalized through the following lemma.

*tr*ð Þ! X*<sup>i</sup>* ⊕

! <sup>⊕</sup>

with transfer and whose morphisms are natural transformations.

*i*

*tr* X1 � … � X*<sup>j</sup>* � … � X*<sup>j</sup>* � … � X*<sup>n</sup>*

! *tr*ðX1 � ⋯ � X*n*Þ ! *tr*ð Þ! X1∧⋯∧X*<sup>n</sup>* 0,

*Derived Tensor Products and Their Applications DOI: http://dx.doi.org/10.5772/intechopen.92869*

Then, it is sufficient to demonstrate that ⊗ *tr*

Likewise, we have the definition.

the Étale sheaf associated to *F* ⊗ *trG:*

and *Q* ffi *D:* Then there is a natural mapping

*é t <sup>D</sup>*, for ð Þ *<sup>C</sup>* <sup>⊗</sup> *trD é t*, and

*C* ⊗ *tr*

*C* ⊗ *tr*

*C* ⊗ *tr*

0 ! *tr*ðX1 � ⋯ � X*n*Þ ! ⊕

! ⊕ *i*, *j*

we write

to ⊗ *tré t:*

shall write *C* ⊗ *tr*

induced by

**95**

**Lemma 3.2.** If *F*, and *F*<sup>0</sup>

locally constant, the mapping:

! Δ0

**Lemma 3.1.** The pre-sheaf *tr*ð Þ ð Þ X1, x1 ∧⋯∧ð Þ X*n*, x*<sup>n</sup>* , is a direct summand of *tr*ð Þ X1 � ⋯ � X*<sup>n</sup>* . In particular, it is projective object of PST. Likewise, for the following sequence of pre-sheaves with transfers, the exactness is explicit<sup>9</sup>

*i*, *j*

**Definition 3.2.** A pre-sheaf with transfers is a contravariant additive functor:

to describe the functor category on the field *k*, whose objects are pre-sheaves

Likewise, analogously we can define to the tensor product ⊗ *tr*, their extension

**Definition 3.3.** If *F*, and *G*, are pre-sheaves of *R*-modules with transfers, we write:

*<sup>F</sup>* <sup>⊗</sup> *tr* ð Þ *<sup>G</sup> é t* ! *<sup>F</sup>* <sup>⊗</sup> *tr*

If *C*, and *D*, are bounded above complexes of pre-sheaves with transfers, we

*<sup>L</sup> <sup>D</sup>* ffi *<sup>P</sup>* <sup>⊗</sup> *tr*

*<sup>L</sup>*,*é t <sup>D</sup>* ! *<sup>C</sup>* <sup>⊗</sup> *tr*

*<sup>h</sup>*Xð Þ *<sup>U</sup>* <sup>⊗</sup> *<sup>R</sup> <sup>h</sup>*Yð Þ¼ *<sup>U</sup>* HomAð Þ *<sup>U</sup>*, X <sup>⊗</sup> HomAð Þ! *<sup>U</sup>*, Y <sup>⊗</sup> HomAð Þ *<sup>U</sup>* <sup>⊗</sup> *<sup>U</sup>*, X <sup>⊗</sup> <sup>Y</sup>

<sup>9</sup> *tr*ð Þffi <sup>X</sup> *tr* <sup>⊕</sup> *tr*ð Þ X, x , *tr*ð Þffi X1 � X2 *tr* <sup>⊕</sup> *tr*ð Þ X1, x1 <sup>⊕</sup> *tr*ð Þ X2, x2 <sup>⊕</sup> *tr*ð Þ X1∧X2 *:*

where *P*, and *Q*, are complexes of representable sheaves with transfers, *P* ffi *C*,

*tr* X*<sup>i</sup>* � X*<sup>j</sup>*

! …

i

:

(55)

*tr* X1 � … � <sup>X</sup>^ *<sup>j</sup>* � … � <sup>X</sup>*<sup>n</sup>* 

*<sup>L</sup>*,*é t*, preserve quasi-isomorphisms.

*é tG*, (58)

*é tQ*, (59)

*é tD*, (60)

*<sup>L</sup>*, *<sup>D</sup>* ! *<sup>C</sup>* <sup>⊗</sup> *trD*, (61)

, are Étale sheaves of *R*-modules with transfers, and *F*, is

HomAð Þ¼ *U*, X ⊗ Y *h*<sup>X</sup> <sup>⊗</sup> <sup>Y</sup>ð Þ *U* , (62)

*F* ¼ *Cork* ! Ab, (56)

Pre*Sh Cor* ð Þ!*<sup>k</sup>* PSTð Þ¼ *k* PST, (57)

For *U*, in A, *ηU*, is the monoidal product in A, followed by the diagonal mapping of triangle:

$$
\eta: U \to U \otimes U,\tag{45}
$$

that is to say,

$$h\_{\mathbf{X}}(U) \otimes\_R h\_{\mathbf{Y}}(U) \to \operatorname{Hom}\_{\mathcal{A}}(U, \mathbf{X}) \otimes\_R \operatorname{Hom}\_{\mathcal{A}}(U, \mathbf{Y}),\tag{46}$$

satisfies the triangle<sup>7</sup> :

$$\begin{aligned} \operatorname{Hom}\_{\mathcal{A}}(U, \mathbf{X}) \otimes\_{\mathbb{R}} \operatorname{Hom}\_{\mathcal{A}}(U, \mathbf{Y}) & \xrightarrow{\otimes} \operatorname{Hom}\_{\mathcal{A}}(U \otimes U, \mathbf{X} \otimes \mathbf{Y})\\ \Delta \rhd & & \downarrow \Delta \\ & & \operatorname{Hom}\_{\mathcal{A}}(U, \mathbf{X} \otimes \mathbf{Y}) \end{aligned} \tag{47}$$

where

$$\operatorname{Hom}\_{\mathcal{A}}(U, \mathbf{X} \otimes \mathbf{Y}) = h\_{\mathbf{X} \otimes \mathbf{Y}}(U), \tag{48}$$

With all these dispositions and generalities, now we can specialize to the case when<sup>8</sup>

$$\mathcal{A} = \mathbf{C}r\_k,\tag{49}$$

and ⊗ , is the tensor product

$$\mathbf{X} \otimes \mathbf{Y} = \mathbf{X} \times \mathbf{Y}',\tag{50}$$

Then we have the Yoneda embedding:

$$\text{PST}(k) \subset \text{Cor}\_k \oplus \text{Cor}\_k,\tag{51}$$

We denote as <sup>⊗</sup> *tr*, for the tensor product on PSTð Þ¼ *<sup>k</sup>* ð Þ *Cork* , or

$$\text{PST}(k, R) = R(Cor\_k), \tag{52}$$

and ⊗ *trL*, for ⊗ *<sup>L</sup>*. Then there is a natural mapping

$$\mathbf{C} \otimes\_{L}^{tr} \mathbf{D} \to \mathbf{C} \otimes^{tr} \mathbf{D},\tag{53}$$

Here <sup>⊗</sup> *trL*, is the tensor product induced to ð Þ *Cork :* But, before we will keep using the product ⊗ *tr*, which we can define as:

$$\mathbf{X} \otimes \mathbf{Y} = \mathbf{X} \times \mathbf{Y}.$$

$$R\_{tr}(\mathbf{X}) \otimes^{tr} R\_{tr}(\mathbf{Y}) = R\_{tr}(\mathbf{X}, \mathbf{Y}), \tag{54}$$

being *<sup>h</sup>*<sup>X</sup> <sup>¼</sup> *<sup>R</sup>*trð Þ <sup>X</sup> , <sup>∀</sup>*h*<sup>X</sup> <sup>∈</sup> Hom, <sup>∀</sup>X<sup>∈</sup> <sup>A</sup> <sup>⊕</sup> *:*

$$\mathbf{x} \otimes \mathbf{y} = \mathbf{x} \times \mathbf{y}.$$

<sup>7</sup> *<sup>η</sup>U*∘<sup>Δ</sup> <sup>¼</sup> <sup>Δ</sup><sup>0</sup> .

<sup>8</sup> Def. If X, Y∈*Cork*, their tensor product X ⊗ Y, is defined to be the product underlying schemes over *k*,

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The above can be generalized through the following lemma.

**Lemma 3.1.** The pre-sheaf *tr*ð Þ ð Þ X1, x1 ∧⋯∧ð Þ X*n*, x*<sup>n</sup>* , is a direct summand of *tr*ð Þ X1 � ⋯ � X*<sup>n</sup>* . In particular, it is projective object of PST. Likewise, for the following sequence of pre-sheaves with transfers, the exactness is explicit<sup>9</sup> :

$$\begin{split} \mathbf{0} \to \mathbb{Z}\_{\text{tr}}(\mathbf{X}\_{1} \times \cdots \times \mathbf{X}\_{\text{n}}) \to \bigoplus\_{i} \mathbb{Z}\_{\text{tr}}(\mathbf{X}\_{i}) \to \bigoplus\_{i,j} \mathbb{Z}\_{\text{tr}}(\mathbf{X}\_{i} \times \mathbf{X}\_{j}) \to \dots \\ \to \bigoplus\_{i,j} \mathbb{Z}\_{\text{tr}}(\mathbf{X}\_{1} \times \ldots \times \mathbf{X}\_{j} \times \ldots \times \mathbf{X}\_{j} \times \ldots \times \mathbf{X}\_{\text{n}}) \to \bigoplus\_{i} \mathbb{Z}\_{\text{tr}}(\mathbf{X}\_{1} \times \ldots \times \hat{\mathbf{X}}\_{j} \times \ldots \times \mathbf{X}\_{\text{n}}) \to \\ \to \mathbb{Z}\_{\text{tr}}(\mathbf{X}\_{1} \times \cdots \times \mathbf{X}\_{\text{n}}) \to \mathbb{Z}\_{\text{tr}}(\mathbf{X}\_{1} \wedge \cdots \wedge \mathbf{X}\_{\text{n}}) \to \mathbf{0}, \end{split} \tag{55}$$

Then, it is sufficient to demonstrate that ⊗ *tr <sup>L</sup>*,*é t*, preserve quasi-isomorphisms. **Definition 3.2.** A pre-sheaf with transfers is a contravariant additive functor:

$$F = Cor\_k \to \mathbf{Ab},\tag{56}$$

we write

*η* : *h*<sup>X</sup> ⊗ *Rh*<sup>Y</sup> ! *h*<sup>X</sup> <sup>⊗</sup> Y, (44)

*η* : *U* ! *U* ⊗ *U*, (45)

(47)

*h*Xð Þ *U* ⊗ *Rh*Yð Þ! *U* HomAð Þ *U*, X ⊗ *<sup>R</sup>*HomAð Þ *U*, Y , (46)

<sup>∀</sup>X, Y <sup>∈</sup> <sup>A</sup> <sup>⊕</sup> *:*

HomAð Þ *<sup>U</sup>*, X <sup>⊗</sup> *<sup>R</sup>*HomAð Þ! *<sup>U</sup>*, Y <sup>⊗</sup> HomAð Þ *<sup>U</sup>* <sup>⊗</sup> *<sup>U</sup>*, X <sup>⊗</sup> <sup>Y</sup>

HomAð Þ *U*, X ⊗ Y

With all these dispositions and generalities, now we can specialize to the case when<sup>8</sup>

X ⊗ Y ¼ X � Y<sup>0</sup>

Here <sup>⊗</sup> *trL*, is the tensor product induced to ð Þ *Cork :* But, before we will keep

X ⊗ Y ¼ X � Y*:*

<sup>8</sup> Def. If X, Y∈*Cork*, their tensor product X ⊗ Y, is defined to be the product underlying schemes over *k*,

X ⊗ Y ¼ X � Y*:*

PSTð Þ*k* ⊂*Cork*

We denote as <sup>⊗</sup> *tr*, for the tensor product on PSTð Þ¼ *<sup>k</sup>* ð Þ *Cork* , or

*C* ⊗ *tr*

and ⊗ *trL*, for ⊗ *<sup>L</sup>*. Then there is a natural mapping

HomAð Þ¼ *U*, X ⊗ Y *h*<sup>X</sup> <sup>⊗</sup> <sup>Y</sup>ð Þ *U* , (48)

A ¼ *Cork*, (49)

, (50)

<sup>⊕</sup> ⊂*Cork*, (51)

PSTð Þ¼ *k*, *R R Cor* ð Þ*<sup>k</sup>* , (52)

*Rtr*ð Þ <sup>X</sup> <sup>⊗</sup> *tr Rtr*ð Þ¼ <sup>Y</sup> *Rtr*ð Þ X, Y , (54)

*<sup>L</sup> <sup>D</sup>* ! *<sup>C</sup>* <sup>⊗</sup> *trD*, (53)

Δ↘ ↓Δ

For *U*, in A, *ηU*, is the monoidal product in A, followed by the diagonal

mapping of triangle:

that is to say,

where

<sup>7</sup> *<sup>η</sup>U*∘<sup>Δ</sup> <sup>¼</sup> <sup>Δ</sup><sup>0</sup>

**94**

.

satisfies the triangle<sup>7</sup>

and ⊗ , is the tensor product

Then we have the Yoneda embedding:

using the product ⊗ *tr*, which we can define as:

being *<sup>h</sup>*<sup>X</sup> <sup>¼</sup> *<sup>R</sup>*trð Þ <sup>X</sup> , <sup>∀</sup>*h*<sup>X</sup> <sup>∈</sup> Hom, <sup>∀</sup>X<sup>∈</sup> <sup>A</sup> <sup>⊕</sup> *:*

:

*Advances on Tensor Analysis and Their Applications*

$$\text{PreSh}(\text{Cor}\_k) \to \text{PST}(k) = \text{PST},\tag{57}$$

to describe the functor category on the field *k*, whose objects are pre-sheaves with transfer and whose morphisms are natural transformations.

Likewise, analogously we can define to the tensor product ⊗ *tr*, their extension to ⊗ *tré t:*

Likewise, we have the definition.

**Definition 3.3.** If *F*, and *G*, are pre-sheaves of *R*-modules with transfers, we write:

$$(F \otimes^{tr} G)\_{\acute{e}t} \to F \otimes^{tr}\_{\acute{e}t} G,\tag{58}$$

the Étale sheaf associated to *F* ⊗ *trG:*

If *C*, and *D*, are bounded above complexes of pre-sheaves with transfers, we shall write *C* ⊗ *tr é t <sup>D</sup>*, for ð Þ *<sup>C</sup>* <sup>⊗</sup> *trD é t*, and

$$(\mathcal{C}\otimes\_L^{tr}D)\cong P\otimes\_{\acute{e}t}^{tr}Q,\tag{59}$$

where *P*, and *Q*, are complexes of representable sheaves with transfers, *P* ffi *C*, and *Q* ffi *D:* Then there is a natural mapping

$$\left(\mathbb{C}\otimes\_{L,\acute{e}t}^{tr}D\right)\to\mathbb{C}\otimes\_{\acute{e}t}^{tr}D,\tag{60}$$

induced by

$$(\mathcal{C}\otimes\_{L,}^{tr}D)\to\mathcal{C}\otimes^{tr}D,\tag{61}$$

**Lemma 3.2.** If *F*, and *F*<sup>0</sup> , are Étale sheaves of *R*-modules with transfers, and *F*, is locally constant, the mapping:

$$h\_{\mathbf{X}}(U)\otimes\_{R}h\_{\mathbf{Y}}(U) = \operatorname{Hom}\_{\mathcal{A}}(U, \mathbf{X})\otimes\operatorname{Hom}\_{\mathcal{A}}(U, \mathbf{Y}) \stackrel{\otimes}{\to} \operatorname{Hom}\_{\mathcal{A}}(U\otimes U, \mathbf{X}\otimes\mathbf{Y})$$

$$\stackrel{\Delta'}{\to} \operatorname{Hom}\_{\mathcal{A}}(U, \mathbf{X}\otimes\mathbf{Y}) = h\_{\mathbf{X}\otimes\mathbf{Y}}(U),\tag{62}$$

<sup>9</sup> *tr*ð Þffi <sup>X</sup> *tr* <sup>⊕</sup> *tr*ð Þ X, x , *tr*ð Þffi X1 � X2 *tr* <sup>⊕</sup> *tr*ð Þ X1, x1 <sup>⊕</sup> *tr*ð Þ X2, x2 <sup>⊕</sup> *tr*ð Þ X1∧X2 *:*

induces an isomorphism

$$F \otimes\_{\text{et}} F' \stackrel{\rightharpoonup}{\to} F \otimes\_{\acute{e}t}^{tr} F',\tag{63}$$

Through the characterization of connections for derived tensor products, we search precisely generalize the connections through pre-sheaves with certain special

*<sup>L</sup>*,*é t*, induces a tensor-triangulated structure to a

*<sup>p</sup>*,*<sup>q</sup>* <sup>¼</sup> *Lp*<sup>Φ</sup> *<sup>H</sup>*q*<sup>C</sup>* , (67)

*Lp*þ*<sup>q</sup>*Φð Þ¼ *C* 0, (68)

Chð Þ *B* <sup>≥</sup>*<sup>B</sup>* \$ *s*CRing*<sup>A</sup>==<sup>B</sup>*, (69)

ð Þ *C* , of *C*, (for an integer *i*) is calculated as follows: take a quasi-isomorphism *ψ* :

*é t* ,�

ð Þ *k*, *=m* ,

gmð Þ *k*, *R* .

*é tF*, from the

*<sup>L</sup>*,*é tF:*

ffi Mod*B:*

ð Þ *C* , of *C*, is then the

Remember we want to generalize the field theory on spaces that admit decomposing into components that can be manageable in the complex manifolds whose complex varieties can be part of those components called motives, creating a decomposition in the derived category of its spectrum considering the functor Spec, and where solutions of the field equations are defined in a hypercohomology.<sup>11</sup> Likewise, this goes focused to obtain a good integrals theory (solutions) in the hypercohomology context considering the knowledge of spectral theory of the cycle sequences in motivic theory that searches the solution of the field equations even

properties, as can be the Étale sheaves.

*Derived Tensor Products and Their Applications DOI: http://dx.doi.org/10.5772/intechopen.92869*

We can demonstrate that ⊗ *tr*

category DMeff

then

Then is had the result.

functor is *B* ↦ΩX*=<sup>A</sup>:*

adjunction complies:

hypercohomology *H<sup>i</sup>*

cohomology *H<sup>i</sup>*

**97**

*é t* ,�

with singularities of the complex Riemann manifold.

derived category more general than D�*R*ð Þ <sup>A</sup> , as for example, DMeff

which is our objective. In this case, we want geometrical motives, where this last

category PSTð Þ *k*, *R* , of pre-sheaves of *R*-modules with transfers to the category of the Étale sheaves of *R*-modules and transfers. Likewise, their left functors *LP*Φð Þ *F* , are the homology sheaves of the total left derived functor <sup>Φ</sup>ð Þ¼ *<sup>F</sup> Rtr*ð Þ <sup>Y</sup> <sup>⊗</sup> *tr*

We consider *A*, and *B*∈ A, where A, is a category as has been defined before.

Then a hypercohomology as given to *d*da ¼ 0, can be obtained through double functor work *A* ! *B* ! *B*, through an inclusion of a category Mod*B*, in CRing*<sup>A</sup>==<sup>B</sup>:*

**Theorem 4.2.** The character for an adjoint lifts for a homotopically meaningful

<sup>11</sup> **Definition.** A hyperhomology or hypercohomology of a complex of objects of an abelian category is

hypercohomology is suppose that A, is an abelian category with enough injectives and Φ, a left exact functor to another abelian category B. If *C*, is a complex of objects of A, bounded on the left, the

an extension of the usual homology of an object to complexes. The mechanism to give a

*<sup>C</sup>* ! *<sup>I</sup>*, where *<sup>I</sup>*, is a complex of injective elements of <sup>A</sup>. The hypercohomology *<sup>H</sup><sup>i</sup>*

ð Þ Φð Þ*I* , of the complex Φð Þ*I :*

**Theorem 4.1.** The left adjoint to the inclusion functor Mod*B*, CRing*<sup>A</sup>==<sup>B</sup>:* is defined by X ↦ΩX*=<sup>A</sup>* ⊗ <sup>X</sup>*B:* In particular, the image of *A* ! *B* ! *B*, under this

**Proposition 4.1.** There is equivalence between categories Ab CRing*<sup>A</sup>==<sup>B</sup>*

We consider and fix *<sup>Y</sup>*, and the right exact functor <sup>Φ</sup>ð Þ¼ *<sup>F</sup> Rtr*ð Þ <sup>Y</sup> <sup>⊗</sup> *tr*

Considering a chain complex *C*, the hypercohomology spectral sequence is:

E2

Then the corresponding infinite sequence is exact.

The derived tensor product is a regular tensor product.

We have the following proposition.

ð Þ *k*, *=m* , can be identified for the derived category DM�

Remember that a pre-sheaf is defined as:

**Definition 3.4.** A pre-sheaf *F*, of Abelian groups on *Sm=k*, is an Étale sheaf if it restricts to an Étale sheaf on each X, in *Sm=k*, , that is if:

i. The sequence

$$\mathbf{0} \to F(\mathbf{X}) \stackrel{\text{diag}}{\to} F(U) \stackrel{(+,\text{-})}{\to} F(U \times\_{\mathbf{X}} U),\tag{64}$$

is exact for every surjective Étale morphism of smooth schemes,

$$U \to \mathbf{X},\tag{65}$$

ii. *F*ð Þ¼ X∪Y *F*ð Þ X ⊕ *F*ð Þ Y , ∀X, Y, schemes.

We demonstrate Lemma 3.2.

*Proof*. We want the tensor product ⊗ *tr <sup>L</sup>*,*é t*, which induces to tensor triangulated structure on the derived category of Étale sheaves of *R*-modules with transfers<sup>10</sup> defined in other expositions [4]. Considering Proposition 3.1, we have:

$$\left(\mathbb{C}\otimes\_{L,\acute{e}t}^{tr}D\right)\to D\otimes\_{L,\acute{e}t}^{tr}\mathbb{C},\tag{66}$$

Then, it is sufficient to demonstrate that ⊗ *tr <sup>L</sup>*,*é t*, preserve quasi-isomorphisms. The details can be found in [5].

Then the tensor product ⊗ *tr é t*, as pre-sheaf to Étale sheaves can have a homology space of zero dimension that vanishes in certain component right exact functor <sup>Φ</sup>ð Þ¼ *<sup>F</sup> Rtr*ð Þ <sup>Y</sup> <sup>⊗</sup> *tr é tF*, from the category PSTð Þ *k*, *R* , of pre-sheaves of *R*-modules with transfers to the category of the Étale sheaves of *R*-modules and transfers. Then every derived functor *Ln*Φ, vanishes on *H*<sup>0</sup> *C*~ , to certain complex of Étale.

Then, all right exact functors *Rtr*ð Þ <sup>Y</sup> <sup>⊗</sup> *tr é tF*, are acyclic. This is the machinery to demonstrate the functor exactness and resolution in modules through of induce from ⊗ *tr <sup>L</sup>*,*é t*, a tensor-triangulated structure to a derived category more general that D�*R*ð Þ A *:*

Also we have:

**Lemma 3.3.** Fix *<sup>Y</sup>*, and set <sup>Φ</sup> <sup>¼</sup> *Rtr*ð Þ *<sup>Y</sup>* <sup>⊗</sup> *tr é t:* If *F*, is a pre-sheaf of *R*-modules with transfers such that *Fé t* ¼ 0, then *Ln*Φð Þ¼ *F* 0, ∀*n:*

#### **4. Some considerations to mathematical physics**

Remember that in the derived geometry we work with structures that must support *R*-modules with characterizations that should be most general to the case of singularities, where it is necessary to use irregular connections, if it is the case, for example in field theory in mathematical physics when studying the quantum field equations on a complex Riemann manifold with singularities.

<sup>10</sup> **Definition.** A pre-sheaf with transfers is a contravariant additive functor from the category *Cork*, to the category of abelian groups Ab.

*Derived Tensor Products and Their Applications DOI: http://dx.doi.org/10.5772/intechopen.92869*

Through the characterization of connections for derived tensor products, we search precisely generalize the connections through pre-sheaves with certain special properties, as can be the Étale sheaves.

Remember we want to generalize the field theory on spaces that admit decomposing into components that can be manageable in the complex manifolds whose complex varieties can be part of those components called motives, creating a decomposition in the derived category of its spectrum considering the functor Spec, and where solutions of the field equations are defined in a hypercohomology.<sup>11</sup> Likewise, this goes focused to obtain a good integrals theory (solutions) in the hypercohomology context considering the knowledge of spectral theory of the cycle sequences in motivic theory that searches the solution of the field equations even with singularities of the complex Riemann manifold.

We can demonstrate that ⊗ *tr <sup>L</sup>*,*é t*, induces a tensor-triangulated structure to a derived category more general than D�*R*ð Þ <sup>A</sup> , as for example, DMeff *é t* ,� ð Þ *k*,*=m* , which is our objective. In this case, we want geometrical motives, where this last category DMeff *é t* ,� ð Þ *k*, *=m* , can be identified for the derived category DM� gmð Þ *k*, *R* .

We consider and fix *<sup>Y</sup>*, and the right exact functor <sup>Φ</sup>ð Þ¼ *<sup>F</sup> Rtr*ð Þ <sup>Y</sup> <sup>⊗</sup> *tr é tF*, from the category PSTð Þ *k*, *R* , of pre-sheaves of *R*-modules with transfers to the category of the Étale sheaves of *R*-modules and transfers. Likewise, their left functors *LP*Φð Þ *F* , are the homology sheaves of the total left derived functor <sup>Φ</sup>ð Þ¼ *<sup>F</sup> Rtr*ð Þ <sup>Y</sup> <sup>⊗</sup> *tr <sup>L</sup>*,*é tF:* Considering a chain complex *C*, the hypercohomology spectral sequence is:

$$\mathbf{E}\_{p,q}^2 = L\_p \Phi(H\_\mathbf{q} \mathbf{C}),\tag{67}$$

then

induces an isomorphism

i. The sequence

Remember that a pre-sheaf is defined as:

*Advances on Tensor Analysis and Their Applications*

restricts to an Étale sheaf on each X, in *Sm=k*, , that is if:

ii. *F*ð Þ¼ X∪Y *F*ð Þ X ⊕ *F*ð Þ Y , ∀X, Y, schemes.

Then, it is sufficient to demonstrate that ⊗ *tr*

every derived functor *Ln*Φ, vanishes on *H*<sup>0</sup> *C*~

Then, all right exact functors *Rtr*ð Þ <sup>Y</sup> <sup>⊗</sup> *tr*

**Lemma 3.3.** Fix *<sup>Y</sup>*, and set <sup>Φ</sup> <sup>¼</sup> *Rtr*ð Þ *<sup>Y</sup>* <sup>⊗</sup> *tr*

transfers such that *Fé t* ¼ 0, then *Ln*Φð Þ¼ *F* 0, ∀*n:*

**4. Some considerations to mathematical physics**

equations on a complex Riemann manifold with singularities.

*Proof*. We want the tensor product ⊗ *tr*

We demonstrate Lemma 3.2.

The details can be found in [5]. Then the tensor product ⊗ *tr*

<sup>Φ</sup>ð Þ¼ *<sup>F</sup> Rtr*ð Þ <sup>Y</sup> <sup>⊗</sup> *tr*

Also we have:

the category of abelian groups Ab.

**96**

from ⊗ *tr*

D�*R*ð Þ A *:*

<sup>0</sup> ! *<sup>F</sup>*ð Þ! <sup>X</sup> diag

is exact for every surjective Étale morphism of smooth schemes,

*F* ⊗ ́et*F*<sup>0</sup> !

ffi *F* ⊗ *tr é tF*<sup>0</sup>

**Definition 3.4.** A pre-sheaf *F*, of Abelian groups on *Sm=k*, is an Étale sheaf if it

*F U*ð Þ!ð Þ <sup>þ</sup>, ‐

structure on the derived category of Étale sheaves of *R*-modules with transfers<sup>10</sup>

*<sup>L</sup>*,*é t <sup>D</sup>* ! *<sup>D</sup>* <sup>⊗</sup> *tr*

space of zero dimension that vanishes in certain component right exact functor

demonstrate the functor exactness and resolution in modules through of induce

Remember that in the derived geometry we work with structures that must support *R*-modules with characterizations that should be most general to the case of singularities, where it is necessary to use irregular connections, if it is the case, for example in field theory in mathematical physics when studying the quantum field

<sup>10</sup> **Definition.** A pre-sheaf with transfers is a contravariant additive functor from the category *Cork*, to

with transfers to the category of the Étale sheaves of *R*-modules and transfers. Then

*é tF*, from the category PSTð Þ *k*, *R* , of pre-sheaves of *R*-modules

*<sup>L</sup>*,*é t*, a tensor-triangulated structure to a derived category more general that

defined in other expositions [4]. Considering Proposition 3.1, we have:

*C* ⊗ *tr*

, (63)

*F U*ð Þ �X*U* , (64)

*U* ! X, (65)

*<sup>L</sup>*,*é t*, which induces to tensor triangulated

*<sup>L</sup>*,*é tC*, (66)

*<sup>L</sup>*,*é t*, preserve quasi-isomorphisms.

*é t*, as pre-sheaf to Étale sheaves can have a homology

, to certain complex of Étale.

*é tF*, are acyclic. This is the machinery to

*é t:* If *F*, is a pre-sheaf of *R*-modules with

$$L\_{p+q} \Phi(\mathcal{C}) = \mathbf{0},\tag{68}$$

Then the corresponding infinite sequence is exact.

We consider *A*, and *B*∈ A, where A, is a category as has been defined before. We have the following proposition.

**Proposition 4.1.** There is equivalence between categories Ab CRing*<sup>A</sup>==<sup>B</sup>* ffi Mod*B:* Then a hypercohomology as given to *d*da ¼ 0, can be obtained through double functor work *A* ! *B* ! *B*, through an inclusion of a category Mod*B*, in CRing*<sup>A</sup>==<sup>B</sup>:* Then is had the result.

**Theorem 4.1.** The left adjoint to the inclusion functor Mod*B*, CRing*<sup>A</sup>==<sup>B</sup>:* is defined by X ↦ΩX*=<sup>A</sup>* ⊗ <sup>X</sup>*B:* In particular, the image of *A* ! *B* ! *B*, under this functor is *B* ↦ΩX*=<sup>A</sup>:*

The derived tensor product is a regular tensor product.

**Theorem 4.2.** The character for an adjoint lifts for a homotopically meaningful adjunction complies:

$$\text{Ch}(B)\_{\geq\_{B}B} \longleftrightarrow \mathfrak{s}\text{CRing}\_{\text{A}//B},\tag{69}$$

<sup>11</sup> **Definition.** A hyperhomology or hypercohomology of a complex of objects of an abelian category is an extension of the usual homology of an object to complexes. The mechanism to give a

hypercohomology is suppose that A, is an abelian category with enough injectives and Φ, a left exact functor to another abelian category B. If *C*, is a complex of objects of A, bounded on the left, the hypercohomology *H<sup>i</sup>* ð Þ *C* , of *C*, (for an integer *i*) is calculated as follows: take a quasi-isomorphism *ψ* : *<sup>C</sup>* ! *<sup>I</sup>*, where *<sup>I</sup>*, is a complex of injective elements of <sup>A</sup>. The hypercohomology *<sup>H</sup><sup>i</sup>* ð Þ *C* , of *C*, is then the cohomology *H<sup>i</sup>* ð Þ Φð Þ*I* , of the complex Φð Þ*I :*

Meaning that, it is an adjunction of categories, which induces an adjunction to level of homotopy categories.

We define the cotangent complex required in derived geometry and QFT.

**Definition 4.1.** The cotangent complex *LA=B*, is the image of functor *A* ! *B* ! *B*, under the left functor of the Kahler differentials module *M* ⊗ *<sup>L</sup> <sup>R</sup>*ð Þ <sup>A</sup> , . Likewise, if *P*• ! *B*, be a free resolution then

$$L\_{A/B} = \Omega\_{\mathcal{P}\_{\bullet}/A} \otimes\_{P\_{\bullet}} B,\tag{70}$$

Then the functor *LA*<sup>1</sup> , induces a tensor operation on D�

� ð Þ*k* , this gives us the functor

*Derived Tensor Products and Their Applications DOI: http://dx.doi.org/10.5772/intechopen.92869*

where we have the formula

rical stacks in mathematical physics.

The category DMeff

with Galois group GalMT*:*

**99**

The inverting of the objects � ⊗ *:*ð Þ1 *:*

*<sup>A</sup>*<sup>1</sup> ShNisð Þ *Cor k*ð Þ also a triangulated tensor category. Likewise, explicitly in

*<sup>m</sup>* : *Smk* ! DMeff

If we consider the embedding theorem, then we can establish the following

gmð Þ*k*

which has implications in the geometrical motives applied to bundle of geomet-

**Theorem 5.1 (F. Bulnes).** Suppose that *M*, is a complex Riemannian manifold with singularities. Let *X*, and *Y*, be smooth projective varieties in *M* 12. We know that solutions of the field equations *d*da ¼ 0, are given in a category Specð Þ *Sm*<sup>k</sup> , (see Example 4). Context Solutions of the quantum field equations for *d*da ¼ 0, are defined in hyper-cohomology on ‐ coefficients from the category *Smk*, defined on

Then the following triangulated tensor category scheme is true and commutative:

Triangulated category of geometrical motives DMgmð Þ *k*, *R* , or written simply as DMgmð Þ*<sup>k</sup>* , is defined formally inverting the functor of the Tate objects<sup>13</sup> (are objects of a motivic category called Tannakian category) ð Þ1 , to be image of the complex

<sup>12</sup> Singular projective varieties useful in quantization process of the complex Riemannian manifold. The quantization condition compact quantizable Käehler manifolds can be embedded into projective space. <sup>13</sup> Let MTð Þ , denote the category of mixed Tate motives unramified over . It is a Tannakian category

Remember that a scheme is a mathematical structure that enlarges the concept of algebraic variety in several forms, such as taking account of multiplicities. The schemes can to be of a same algebraic variety different and allowing "varieties" defined over any commutative ring. In many cases, the family of all

varieties of a type can be viewed as a variety or scheme, known as a moduli space.

gmð Þ *k*, *R* , has a tensor structure and the tensor product of its

DQFT *i*↙ ↘ *F* MDgm ðÞ \$ MDð Þ O<sup>Y</sup>

*m* ↘ ↕ Id DMeff gmð Þ*k*

*Smk* ! DMeff

a numerical field *k*, considering the derived tensor product ⊗ *tr*

motives is as defined in (75) *m*ð Þ X ⊗ *m*ð Þ¼ Y *m*ð Þ X � Y .

that D�

defined by

triangulated scheme

DMeff

*<sup>A</sup>*<sup>1</sup> ShNisð Þ *Cor k*ð Þ , making

(77)

(78)

*é t*, of pre-sheaves.

� ð Þ*k* , (74)

*<sup>m</sup>*ð Þ <sup>X</sup> <sup>≔</sup> *<sup>C</sup>*Susð Þ <sup>Z</sup>*tr*ð Þ <sup>X</sup> , (75)

*m*ð Þ¼ X�*k*Y *m*ð Þ X ⊗ *m*ð Þ Y , (76)

The cotangent complex as defined in (69) lives in the derived category Mod*B:* We observe that choosing the particular resolution of *B*, then Ω*P*•*=A*, is a co-fibrant object in the derived category Mod*P*• , which no exist distinction between the derived tensor product and the usual tensor product. Then to any representation automorphic of *G A*ð Þ, the *G F*ð Þ*=G A*ð Þ, can be decomposed as the tensor product ⊗ *<sup>n</sup> <sup>i</sup>*¼*<sup>1</sup>πI:* This last fall in the geometrical Langlands ramifications.

**Example 4.1.** (66) in the context of solution of field equations as *d*da ¼ 0, has solution in the hypercohomology of a spectral sequence of D�*R*ð Þ A , (established on the infinite sequence … ! *<sup>F</sup><sup>n</sup>* ! <sup>0</sup> ! … *:* [6]) when its functors whose image <sup>Ω</sup>*<sup>B</sup>=<sup>A</sup>*, have as its cotangent complex the image under of the functor *LA=<sup>B</sup>*, which is the functor image *A* ! *B* ! *B*, under the left derived functor of Kahler differentials.

To demonstrate this, it is necessary to define an equivalence between derived categories in the level of derived categories *<sup>D</sup> <sup>L</sup>*Bun, <sup>D</sup> , and *<sup>D</sup> LLoc*, <sup>O</sup> , where geometrical motives can be risked with the corresponding moduli stack to holomorphic bundles. The integrals are those whose functors image will be in Spec*H*SymT OP ð Þ *LG* ð Þ *D* , which is the variety of opers on the formal disk *D*, or neighborhood of all points in a surface Σ, in a complex Riemannian manifold [6].

#### **5. Applications**

As was shown, the geometrical motives required in our research are a result of embedding the derived category DM� gmð Þ *k*, *R* , (geometrical motives category) in the DM*eff é t* ,� ð Þ *k*,*=m* , considering the category of smooth schemes on the field *k*.

We consider the following functors. For each *<sup>F</sup>* <sup>∈</sup> <sup>D</sup>� ShNisð Þ *Cor k*ð Þ , there is *LA*<sup>1</sup> *F* ∈ Deff � ð Þ*k* , the resulting functor is:

$$L^{A^1} : \mathbf{D}^- \left( \text{Sh}^{\text{Nis}}(\text{Cor}(k)) \right) \to \mathbf{D}\_-^{\text{eff}}(k),\tag{71}$$

which is exact and left-adjoint to the inclusion

$$\mathbf{D}\_{-}^{\mathrm{eff}}(\boldsymbol{k}) \to \mathbf{D}^{-} \left( \mathbf{S} \mathbf{h}^{\mathrm{Nis}}(\mathrm{Cor}(\boldsymbol{k})) \right), \tag{72}$$

Also the functor (70) descends to an equivalence of triangulated categories. This is very useful to make Deff � ð Þ*k* , into a tensor category as follows. We consider the Nisnevich sheaf *tr*ð Þ*k* , with transfer *tr* : Y ! *c*ð Þ Y, X *:* We define

$$\mathbb{Z}\_{tr}(k)\otimes\mathbb{Z}\_{tr}(k) := \mathbb{Z}\_{tr}(\mathbf{X}\times\_k \mathbf{Y}),\tag{73}$$

Then it can be demonstrated that the operation realized in (70) can be extended to give D� ShNisð Þ *Cor k*ð Þ , with the structure of a triangulated tensor category.

Then the functor *LA*<sup>1</sup> , induces a tensor operation on D� *<sup>A</sup>*<sup>1</sup> ShNisð Þ *Cor k*ð Þ , making that D� *<sup>A</sup>*<sup>1</sup> ShNisð Þ *Cor k*ð Þ also a triangulated tensor category. Likewise, explicitly in DMeff � ð Þ*k* , this gives us the functor

$$m: \text{Sm}\_k \to \text{DM}\_-^{\text{eff}}(k),\tag{74}$$

defined by

Meaning that, it is an adjunction of categories, which induces an adjunction to

The cotangent complex as defined in (69) lives in the derived category Mod*B:* We observe that choosing the particular resolution of *B*, then Ω*P*•*=A*, is a co-fibrant object in the derived category Mod*P*• , which no exist distinction between the derived tensor product and the usual tensor product. Then to any representation automorphic of *G A*ð Þ, the *G F*ð Þ*=G A*ð Þ, can be decomposed as the tensor product

**Example 4.1.** (66) in the context of solution of field equations as *d*da ¼ 0, has solution in the hypercohomology of a spectral sequence of D�*R*ð Þ A , (established on the infinite sequence … ! *<sup>F</sup><sup>n</sup>* ! <sup>0</sup> ! … *:* [6]) when its functors whose image <sup>Ω</sup>*<sup>B</sup>=<sup>A</sup>*, have as its cotangent complex the image under of the functor *LA=<sup>B</sup>*, which is the functor image *A* ! *B* ! *B*, under the left derived functor of Kahler differentials. To demonstrate this, it is necessary to define an equivalence between derived categories in the level of derived categories *<sup>D</sup> <sup>L</sup>*Bun, <sup>D</sup> , and *<sup>D</sup> LLoc*, <sup>O</sup> , where geometrical motives can be risked with the corresponding moduli stack to holomorphic bundles. The integrals are those whose functors image will be in Spec*H*SymT OP ð Þ *LG* ð Þ *D* , which is the variety of opers on the formal disk *D*, or neighborhood of all points in a surface Σ, in a complex Riemannian manifold [6].

As was shown, the geometrical motives required in our research are a result of

ð Þ *k*,*=m* , considering the category of smooth schemes on the field *k*.

� ð Þ! *<sup>k</sup>* <sup>D</sup>� ShNisð Þ *Cor k*ð Þ

Also the functor (70) descends to an equivalence of triangulated categories. This

Then it can be demonstrated that the operation realized in (70) can be extended

We consider the following functors. For each *<sup>F</sup>* <sup>∈</sup> <sup>D</sup>� ShNisð Þ *Cor k*ð Þ

: <sup>D</sup>� ShNisð Þ *Cor k*ð Þ 

*LA=<sup>B</sup>* ¼ Ω*P*•*=<sup>A</sup>* ⊗ *<sup>P</sup>*•*B*, (70)

gmð Þ *k*, *R* , (geometrical motives category) in the

! <sup>D</sup>eff

� ð Þ*k* , into a tensor category as follows. We consider the

*tr*ð Þ*k* ⊗ *tr*ð Þ*k* ≔ *tr*ð Þ X�*k*Y , (73)

, with the structure of a triangulated tensor category.

� ð Þ*k* , (71)

, (72)

, there is

*<sup>R</sup>*ð Þ <sup>A</sup> , . Likewise, if

We define the cotangent complex required in derived geometry and QFT. **Definition 4.1.** The cotangent complex *LA=B*, is the image of functor *A* ! *B* !

*B*, under the left functor of the Kahler differentials module *M* ⊗ *<sup>L</sup>*

*<sup>i</sup>*¼*<sup>1</sup>πI:* This last fall in the geometrical Langlands ramifications.

level of homotopy categories.

*Advances on Tensor Analysis and Their Applications*

*P*• ! *B*, be a free resolution then

⊗ *<sup>n</sup>*

**5. Applications**

DM*eff é t* ,�

*LA*<sup>1</sup>

**98**

*F* ∈ Deff

is very useful to make Deff

to give D� ShNisð Þ *Cor k*ð Þ

embedding the derived category DM�

� ð Þ*k* , the resulting functor is:

*LA*<sup>1</sup>

which is exact and left-adjoint to the inclusion

Deff

Nisnevich sheaf *tr*ð Þ*k* , with transfer *tr* : Y ! *c*ð Þ Y, X *:* We define

$$m(\mathbf{X}) \coloneqq \mathbf{C}^{\mathrm{Sus}}(\mathbf{Z}\_{tr}(\mathbf{X})),\tag{75}$$

where we have the formula

$$m(\mathbf{X} \times\_k \mathbf{Y}) = m(\mathbf{X}) \otimes m(\mathbf{Y}),\tag{76}$$

If we consider the embedding theorem, then we can establish the following triangulated scheme

$$\begin{aligned} \text{Sym}\_k &\to \text{DM}\_{\text{gm}}^{\text{eff}}(k) \\ m &\searrow \text{fd} \\ \text{DM}\_{\text{gm}}^{\text{eff}}(k) \end{aligned} \tag{77}$$

which has implications in the geometrical motives applied to bundle of geometrical stacks in mathematical physics.

**Theorem 5.1 (F. Bulnes).** Suppose that *M*, is a complex Riemannian manifold with singularities. Let *X*, and *Y*, be smooth projective varieties in *M* 12. We know that solutions of the field equations *d*da ¼ 0, are given in a category Specð Þ *Sm*<sup>k</sup> , (see Example 4). Context Solutions of the quantum field equations for *d*da ¼ 0, are defined in hyper-cohomology on ‐ coefficients from the category *Smk*, defined on a numerical field *k*, considering the derived tensor product ⊗ *tr é t*, of pre-sheaves. Then the following triangulated tensor category scheme is true and commutative:

$$\begin{array}{cccc} & \text{DQFT} & & \\ & i\angle & & \searrow F & \\ \text{MD}\_{\text{gm}}(\mathbb{Q}) & \longleftrightarrow & \text{MD}(\mathcal{O}\_{\text{Y}}) \end{array} \tag{78}$$

The category DMeff gmð Þ *k*, *R* , has a tensor structure and the tensor product of its motives is as defined in (75) *m*ð Þ X ⊗ *m*ð Þ¼ Y *m*ð Þ X � Y .

Triangulated category of geometrical motives DMgmð Þ *k*, *R* , or written simply as DMgmð Þ*<sup>k</sup>* , is defined formally inverting the functor of the Tate objects<sup>13</sup> (are objects of a motivic category called Tannakian category) ð Þ1 , to be image of the complex

<sup>12</sup> Singular projective varieties useful in quantization process of the complex Riemannian manifold. The quantization condition compact quantizable Käehler manifolds can be embedded into projective space.

<sup>13</sup> Let MTð Þ , denote the category of mixed Tate motives unramified over . It is a Tannakian category with Galois group GalMT*:*

The inverting of the objects � ⊗ *:*ð Þ1 *:*

Remember that a scheme is a mathematical structure that enlarges the concept of algebraic variety in several forms, such as taking account of multiplicities. The schemes can to be of a same algebraic variety different and allowing "varieties" defined over any commutative ring. In many cases, the family of all varieties of a type can be viewed as a variety or scheme, known as a moduli space.

*<sup>ℙ</sup>*<sup>1</sup> ! ½ � Specð Þ*<sup>k</sup>* , where the motive in degree *<sup>p</sup>* <sup>¼</sup> 2, 3, will be *m p*ð Þ¼ *<sup>m</sup>* <sup>⊗</sup> ð Þ<sup>1</sup> <sup>⊗</sup> <sup>p</sup> , or to any motive *m* ∈ DMeff gmð Þ*k* , ∀*p* ∈ *ℕ:*

Likewise, the important fact is that the canonical functor DMeff gmð Þ*k* , !

DMgmð Þ*k* , is full embedding [7]. Therefore we work in the category DMgmð Þ*k* . Likewise, for *X*, and *Y*, smooth projective varieties and for any integer *i*, there exists an isomorphism:

$$\operatorname{Hom}\_{\operatorname{DM}^{\operatorname{aff}}\_{\text{gm}}(k)}(m(\mathbf{X}), m(\mathbf{Y})(i)[\mathbf{2}i]) \cong A^{m+i}(\mathbf{X} \times \mathbf{Y}), \qquad m = \dim \mathbf{Y},\tag{79}$$

We demonstrate the Theorem 5.1.

*Proof*. ∀ X∈ *Smk*, the category *Smk*, extends to a pseudo-tensor equivalence of cohomological categories over motives on *k*, that is to say, MMð Þ*k* , is the image of functors

$$\text{DM}^{\text{eff}}(k) \to \text{DM}\_{\text{gm}}(k), \tag{80}$$

**Acknowledgements**

**Author details**

Francisco Bulnes

Mexico

**101**

Research COMECYT-077/111/21.

*Derived Tensor Products and Their Applications DOI: http://dx.doi.org/10.5772/intechopen.92869*

Posdoctoral research was supported by State of Mexico Council of Scientific

Research Department in Mathematics and Engineering, TESCHA, IINAMEI (Investigación Internacional en Matemáticas Avanzadas e Ingeniería), Chalco,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: francisco.bulnes@tesch.edu.mx

provided the original work is properly cited.

which is an equivalence of the underlying triangulated tensor categories. On the other hand, the category DQFT can be defined for the motives in a hypercohomology from the category *Smk*, defined as:

$$\operatorname{Hom}\_{\operatorname{DM}^{\operatorname{eff}}\_{\text{pr}}(k)}(m(\mathbf{X}), \mathbb{Q}(q)[\mathbf{p}]) \cong H^{\star}\_{\operatorname{Ni}}(\mathbf{X}, \mathbb{Q}, (q)) = \mathbb{H}^{p,q}(\mathbf{X}\_{\operatorname{Ni}}, \mathbb{Q}, (q)),\tag{81}$$

which comes from the hypercohomology

$$H\_L^{p,q}(\mathbf{X}, \mathbb{Q}) = \mathbb{H}^{p,q}(\mathbf{X}, \mathbb{Q}),\tag{82}$$

We observe that if a Zariski sheaf of ‐modules with transfers *<sup>F</sup>*, is such that *<sup>F</sup>* <sup>¼</sup> *<sup>H</sup>qC*, for all *<sup>C</sup>*, a complex defined on ð Þ*<sup>q</sup>* ‐modules (being a special case when *<sup>C</sup>* <sup>¼</sup> ð Þ*<sup>q</sup>* ), where the cohomology groups of the isomorphism *<sup>H</sup><sup>p</sup> é t*ð Þffi X, *Fé t Hp Nis*ð Þ X, *FNis* , can be vanished for *p* >dim Yð Þ.

Then survives a hypercohomology *<sup>q</sup>* ð Þ X, *:* If we consider Specð Þ *Sm*<sup>k</sup> , we can to have the quantum version of this hyper-cohomology with an additional work on moduli stacks of the category Mod*B*, in a study on equivalence between derived categories in the level of derived categories *<sup>D</sup> <sup>L</sup>*Bun, <sup>D</sup> , and *<sup>D</sup> LLoc*, <sup>O</sup> , where geometrical motives can be risked with the corresponding moduli stack to holomorphic bundles14.

For other way, with other detailed work of quasi-coherent sheaves [6] we can to obtain the category MOOð Þ Y *:* The functors are constructed using the Mukai-Fourier transforms. ■

<sup>14</sup> We consider the functor *F*, defined as:

where <sup>K</sup> *<sup>F</sup><sup>r</sup>* ð Þ, the kernel space of the functor *<sup>F</sup><sup>r</sup>* , is the functor that induces the equivalence Mod*T*ð Þffi *<sup>D</sup>*ð Þ <sup>X</sup>�YX <sup>⊥</sup><sup>K</sup> *<sup>F</sup><sup>r</sup>* ð Þ, and the operator *<sup>T</sup>* <sup>¼</sup> *<sup>F</sup><sup>r</sup>* ∘*F*, acting on category *D*ð Þ X�YX .

*Derived Tensor Products and Their Applications DOI: http://dx.doi.org/10.5772/intechopen.92869*

## **Acknowledgements**

*<sup>ℙ</sup>*<sup>1</sup> ! ½ � Specð Þ*<sup>k</sup>* , where the motive in degree *<sup>p</sup>* <sup>¼</sup> 2, 3, will be *m p*ð Þ¼ *<sup>m</sup>* <sup>⊗</sup> ð Þ<sup>1</sup> <sup>⊗</sup> <sup>p</sup> ,

*Proof*. ∀ X∈ *Smk*, the category *Smk*, extends to a pseudo-tensor equivalence of cohomological categories over motives on *k*, that is to say, MMð Þ*k* , is the image of

which is an equivalence of the underlying triangulated tensor categories. On the other hand, the category DQFT can be defined for the motives in a

*<sup>L</sup>* ð Þ¼ X, *<sup>p</sup>*,*<sup>q</sup>*

We observe that if a Zariski sheaf of ‐modules with transfers *<sup>F</sup>*, is such that *<sup>F</sup>* <sup>¼</sup> *<sup>H</sup>qC*, for all *<sup>C</sup>*, a complex defined on ð Þ*<sup>q</sup>* ‐modules (being a special case when

to have the quantum version of this hyper-cohomology with an additional work on moduli stacks of the category Mod*B*, in a study on equivalence between derived categories in the level of derived categories *<sup>D</sup> <sup>L</sup>*Bun, <sup>D</sup> , and *<sup>D</sup> LLoc*, <sup>O</sup> , where geometrical motives can be risked with the corresponding moduli stack to

For other way, with other detailed work of quasi-coherent sheaves [6] we can to obtain the category MOOð Þ Y *:* The functors are constructed using the Mukai-Fourier transforms. ■

DMgmð Þ*k* , is full embedding [7]. Therefore we work in the category DMgmð Þ*k* . Likewise, for *X*, and *Y*, smooth projective varieties and for any integer *i*, there

gmð Þ*k* , !

ð Þ X � Y , *m* ¼ dimY, (79)

DMeffð Þ! *<sup>k</sup>* DMgmð Þ*<sup>k</sup>* , (80)

*Nis*ð Þ¼ X, ,ð Þ*<sup>q</sup> <sup>p</sup>*,*<sup>q</sup>* <sup>X</sup>*Nis* ð Þ , ,ð Þ*<sup>q</sup>* , (81)

ð Þ X, , (82)

ð Þ X, *:* If we consider Specð Þ *Sm*<sup>k</sup> , we can

, is the functor that induces the equivalence

∘*F*, acting on category *D*ð Þ X�YX .

*é t*ð Þffi X, *Fé t*

gmð Þ*k* , ∀*p* ∈ *ℕ:* Likewise, the important fact is that the canonical functor DMeff

gmð Þ*<sup>k</sup>* <sup>ð</sup>*m*ð Þ <sup>X</sup> , *<sup>m</sup>*ð Þ <sup>Y</sup> ð Þ*<sup>i</sup>* ½ � <sup>2</sup>*<sup>i</sup>* Þ ffi *Am*þ*<sup>i</sup>*

hypercohomology from the category *Smk*, defined as:

gmð Þ*<sup>k</sup>* <sup>ð</sup>*m*ð Þ <sup>X</sup> , ð Þ*<sup>q</sup>* ½ � <sup>p</sup> Þ ffi *<sup>H</sup>*•

*H<sup>p</sup>*,*<sup>q</sup>*

*<sup>C</sup>* <sup>¼</sup> ð Þ*<sup>q</sup>* ), where the cohomology groups of the isomorphism *<sup>H</sup><sup>p</sup>*

which comes from the hypercohomology

*Nis*ð Þ X, *FNis* , can be vanished for *p* >dim Yð Þ. Then survives a hypercohomology *<sup>q</sup>*

or to any motive *m* ∈ DMeff

exists an isomorphism:

HomDMeff

HomDMeff

holomorphic bundles14.

<sup>14</sup> We consider the functor *F*, defined as: where <sup>K</sup> *<sup>F</sup><sup>r</sup>* ð Þ, the kernel space of the functor *<sup>F</sup><sup>r</sup>*

Mod*T*ð Þffi *<sup>D</sup>*ð Þ <sup>X</sup>�YX <sup>⊥</sup><sup>K</sup> *<sup>F</sup><sup>r</sup>* ð Þ, and the operator *<sup>T</sup>* <sup>¼</sup> *<sup>F</sup><sup>r</sup>*

functors

*Hp*

**100**

We demonstrate the Theorem 5.1.

*Advances on Tensor Analysis and Their Applications*

Posdoctoral research was supported by State of Mexico Council of Scientific Research COMECYT-077/111/21.

## **Author details**

Francisco Bulnes Research Department in Mathematics and Engineering, TESCHA, IINAMEI (Investigación Internacional en Matemáticas Avanzadas e Ingeniería), Chalco, Mexico

\*Address all correspondence to: francisco.bulnes@tesch.edu.mx

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Chapter 7**

**Abstract**

**1. Introduction**

**103**

Lie Algebra

*Daniel Condurache*

Higher-Order Kinematics in Dual

In this chapter, using the ring properties of dual number algebra, vector and tensor calculus, a computing method for the higher-order acceleration vector field properties in general rigid body motion is proposed. The higher-order acceleration field of a rigid body in a general motion is uniquely determined by higher-order time derivative of a dual twist. For the relative kinematics of rigid body motion, equations that allow the determination of the higher-order acceleration vector field are given, using an exponential Brockett-like formula in the dual Lie algebra. In particular cases, the properties for velocity, acceleration, jerk, and jounce fields are given. This approach uses the isomorphism between the Lie algebra of the rigid displacements *se*(3), of the Special Euclidean group, *S*3, and the Lie algebra of dual

The kinematic analysis of multibody systems has been traditionally considered as the determination of the positions, velocities, accelerations, jerks and jounces of their constitutive members. This is an old field with a long history, which has attracted the attention of mathematicians and engineers. Michel Chasles discovered (1834) that any rigid body displacement is equivalent to a screw displacement [1]. Screw theory is an efficient mathematical tool for the study of spatial kinematics. The pioneering work of Ball [2], the treatises of Hunt [3], and Phillips [4] and the multitude of contributions appearing in the literature are evidence of this. The isomorphism between screw theory and the Lie algebra, se(3), of the Special Euclidean group, *S*3, provide with a wealth of results and techniques from modern

A kinematic mapping relates the motion of a rigid body to the joint motions of a kinematic chain. Its time derivatives yield the twist, acceleration, jerk and jounce etc. of the body. Time derivatives of the twists of members in a kinematic chain and derivatives of screws are essential operations in kinematics. Recognizing the Lie group nature of rigid body motions, and correspondingly the Lie algebra nature of screws, Karger [5], Rico et al. [6], Lerbet [7] and Müller [8, 9] derived closed form

In this chapter, using the tensor calculus and the dual numbers algebra, a new computing method for studying the higher-order accelerations field properties is proposed in the case of the general rigid body motion. For the spatial kinematic

vectors. The results are coordinate free and in a closed form.

**Keywords:** higher-order kinematics, dual algebra, lie group

differential geometry and Lie group theory [5–9].

expressions of higher-order time derivatives of twist.

## **References**

[1] Adams JF. Stable homotopy and generalised homology. In: Chicago Lectures in Mathematics, MR 53 #6534. Chicago, IL: University of Chicago Press; 1974

[2] Mazza C, Voevodsky V, Weibel C, editors. Lecture Notes on Motivic Cohomology. Vol. 2. Cambridge, MA, USA: AMS Clay Mathematics Institute; 2006

[3] Aubry M. Homotopy Theory and Models. Boston, MA: Birkhäuser; 1995

[4] Milne JS. Étale Cohomology. Princeton, NJ: Princeton University Press; 1980. ISBN: 0-691-08238-3

[5] Bosch S. Algebraic Geometry and Commutative Algebra. NY, USA: Springer; 2013

[6] Bulnes F. Extended d-cohomology and integral transforms in derived geometry to QFT-equations solutions using Langlands correspondences. Theoretical Mathematics and Applications. 2017;**7**(2):51-62

[7] Voevodsky V. Triangulated categories of motives over a field. In: Cycles, Transfers, and Motivic Homology Theories, Vol. 143 of Annals of Mathematics Studies. USA: Princeton University Press; 2000. pp. 188-238

**Chapter 7**

**References**

Press; 1974

2006

Springer; 2013

**102**

[1] Adams JF. Stable homotopy and generalised homology. In: Chicago Lectures in Mathematics, MR 53 #6534. Chicago, IL: University of Chicago

*Advances on Tensor Analysis and Their Applications*

[2] Mazza C, Voevodsky V, Weibel C, editors. Lecture Notes on Motivic Cohomology. Vol. 2. Cambridge, MA, USA: AMS Clay Mathematics Institute;

[3] Aubry M. Homotopy Theory and Models. Boston, MA: Birkhäuser; 1995

[5] Bosch S. Algebraic Geometry and Commutative Algebra. NY, USA:

[6] Bulnes F. Extended d-cohomology and integral transforms in derived geometry to QFT-equations solutions using Langlands correspondences. Theoretical Mathematics and Applications. 2017;**7**(2):51-62

[7] Voevodsky V. Triangulated categories of motives over a field. In: Cycles, Transfers, and Motivic

Homology Theories, Vol. 143 of Annals of Mathematics Studies. USA: Princeton University Press; 2000. pp. 188-238

[4] Milne JS. Étale Cohomology. Princeton, NJ: Princeton University Press; 1980. ISBN: 0-691-08238-3

## Higher-Order Kinematics in Dual Lie Algebra

*Daniel Condurache*

#### **Abstract**

In this chapter, using the ring properties of dual number algebra, vector and tensor calculus, a computing method for the higher-order acceleration vector field properties in general rigid body motion is proposed. The higher-order acceleration field of a rigid body in a general motion is uniquely determined by higher-order time derivative of a dual twist. For the relative kinematics of rigid body motion, equations that allow the determination of the higher-order acceleration vector field are given, using an exponential Brockett-like formula in the dual Lie algebra. In particular cases, the properties for velocity, acceleration, jerk, and jounce fields are given. This approach uses the isomorphism between the Lie algebra of the rigid displacements *se*(3), of the Special Euclidean group, *S*3, and the Lie algebra of dual vectors. The results are coordinate free and in a closed form.

**Keywords:** higher-order kinematics, dual algebra, lie group

#### **1. Introduction**

The kinematic analysis of multibody systems has been traditionally considered as the determination of the positions, velocities, accelerations, jerks and jounces of their constitutive members. This is an old field with a long history, which has attracted the attention of mathematicians and engineers. Michel Chasles discovered (1834) that any rigid body displacement is equivalent to a screw displacement [1]. Screw theory is an efficient mathematical tool for the study of spatial kinematics. The pioneering work of Ball [2], the treatises of Hunt [3], and Phillips [4] and the multitude of contributions appearing in the literature are evidence of this. The isomorphism between screw theory and the Lie algebra, se(3), of the Special Euclidean group, *S*3, provide with a wealth of results and techniques from modern differential geometry and Lie group theory [5–9].

A kinematic mapping relates the motion of a rigid body to the joint motions of a kinematic chain. Its time derivatives yield the twist, acceleration, jerk and jounce etc. of the body. Time derivatives of the twists of members in a kinematic chain and derivatives of screws are essential operations in kinematics. Recognizing the Lie group nature of rigid body motions, and correspondingly the Lie algebra nature of screws, Karger [5], Rico et al. [6], Lerbet [7] and Müller [8, 9] derived closed form expressions of higher-order time derivatives of twist.

In this chapter, using the tensor calculus and the dual numbers algebra, a new computing method for studying the higher-order accelerations field properties is proposed in the case of the general rigid body motion. For the spatial kinematic

chains, equations that allow the determination of the nth order accelerations field are given, using a Brockett-like formula. The crucial observation is that the nth order time derivative of twist of the terminal body in a kinematic chain can be determined by propagating the kth order time derivative of twists of the bodies in the chain, for *<sup>k</sup>* <sup>¼</sup> 0,�*n*. The results are coordinate-free and in a closed form.

#### **2. Theoretical consideration on rigid body motion**

The general framework of this chapter is a rigid body that moves with respect to a fixed reference frame <sup>R</sup><sup>0</sup> � �. Consider another reference frame f g <sup>R</sup> originated in a point *Q* that moves together with the rigid body. Let **ρ***<sup>Q</sup>* denote the position vector of point *Q* with respect to frame R<sup>0</sup> � �, **v***<sup>Q</sup>* its absolute velocity and **a***<sup>Q</sup>* its absolute acceleration.

Then the vector parametric equation of motion is:

$$
\boldsymbol{\mathfrak{p}} = \boldsymbol{\mathfrak{p}}\_{\boldsymbol{Q}} + \mathbf{R} \mathbf{r} \tag{1}
$$

**2.1 The velocity field in rigid body motion**

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

tensor **Φ**<sup>1</sup> and the velocity invariant **a**1:

**2.2 The acceleration field in rigid body motion**

is invertible and its inverse is (see [10]):

**Φ**�<sup>1</sup>

<sup>2</sup> <sup>¼</sup> <sup>1</sup>

**3. The vector field of the** *nth* **order accelerations**

ð Þ **ω** � **ε**

**v** � **v***<sup>Q</sup>* ¼ **Φ**<sup>1</sup> **ρ** � **ρ***<sup>Q</sup>*

The instantaneous angular velocity **ω** of the rigid body may be determined as **ω** ¼ vect**Φ**1. The major property that may be highlighted from Eq. (4) is that the velocity of a given point of the rigid may be computed when knowing the velocity

**a** � **a***<sup>Q</sup>* ¼ **Φ**<sup>2</sup> **ρ** � **ρ***<sup>Q</sup>*

The absolute acceleration of a given point of the rigid body may be computed

The instantaneous angular acceleration of the rigid body may be determined as:

<sup>2</sup> **ε** ⊗ **ε** þ ð Þ **ω** ⊗ **ω**

It follows that if tensor **Φ**<sup>2</sup> is non-singular, then for an arbitrary given acceleration **a** we may find a point of the rigid that has this acceleration. Its absolute position

Particularly, if **Φ**<sup>2</sup> is non-singular, then there exists a point *G* of zero accelera-

**<sup>ρ</sup>** <sup>¼</sup> **<sup>Φ</sup>**�<sup>1</sup>

tion, named the **acceleration center**. Its absolute position vector is given by:

**<sup>ρ</sup>***<sup>G</sup>* ¼ �**Φ**�<sup>1</sup>

This section extends some of the previous considerations to the case of the *nth*

order accelerations. We define the *nth* order acceleration of a point as:

when knowing the acceleration tensor **Φ**<sup>2</sup> and the acceleration invariant **a**2:

The determinant of tensor **Φ**<sup>2</sup> is (see [10]): det**Φ**<sup>2</sup> ¼ �ð Þ **ω** � **ε**

� � (5)

� � (7)

**v** ¼ **a**<sup>1</sup> þ **Φ**1**ρ** (6)

**a** ¼ **a**<sup>2</sup> þ **Φ**2**ρ** (8)

**ε** ¼ vect**Φ**<sup>2</sup> (9)

**<sup>2</sup>** � **<sup>ω</sup>** g~ **<sup>2</sup> ε** h i (10)

<sup>2</sup> ð Þ **a** � **a**<sup>2</sup> (11)

<sup>2</sup> **a**<sup>2</sup> (12)

2

. It follows that if

It is described by:

It is described by

**ω** � **ε** 6¼ **0**, then tensor **Φ**2*:*

is given by (see also Eq. (8)):

**105**

where **ρ** represents the absolute position of a generic point *P* of the rigid body with respect to <sup>R</sup><sup>0</sup> � � and **<sup>R</sup>** <sup>¼</sup> **<sup>R</sup>**ð Þ*<sup>t</sup>* is an orthogonal proper tensorial function in **SO** <sup>3</sup> . Vector **r** is constant and it represents the relative position vector of the arbitrary point *P* with respect to f g R .

The results of this section succinctly present the velocity and acceleration vector field in rigid body motion. These results lead to the generalization presented in the next section.

With the denotations that were introduced, the vector fields of velocities and accelerations are described by:

$$\begin{cases} \mathbf{v} - \mathbf{v}\_Q = \dot{\mathbf{R}} \mathbf{R}^T (\boldsymbol{\rho} - \boldsymbol{\rho}\_Q) \\\\ \mathbf{a} - \mathbf{a}\_Q = \ddot{\mathbf{R}} \mathbf{R}^T (\boldsymbol{\rho} - \boldsymbol{\rho}\_Q) \end{cases} \tag{2}$$

Tensors:

$$\begin{cases} \boldsymbol{\Phi}\_1 = \dot{\mathbf{R}} \mathbf{R}^T \\\\ \boldsymbol{\Phi}\_2 = \ddot{\mathbf{R}} \mathbf{R}^T \end{cases} \tag{3}$$

represent the **velocity tensor** respectively the **acceleration tensor**. Tensor **<sup>Φ</sup>**<sup>1</sup> <sup>¼</sup> **<sup>ω</sup>**<sup>~</sup> <sup>∈</sup>**so** <sup>3</sup> is the skew-symmetric tensor associated to the instantaneous angular velocity **ω** ∈ **V** <sup>3</sup> . Tensor **<sup>Φ</sup>**<sup>2</sup> <sup>¼</sup> **<sup>ω</sup>**~<sup>2</sup> <sup>þ</sup> <sup>~</sup>**ε**, where **<sup>ε</sup>** <sup>¼</sup> **<sup>ω</sup>**\_ is the instantaneous angular acceleration of the rigid body. One may remark that vectors:

$$\begin{cases} \mathbf{a}\_1 = \mathbf{v} - \Phi\_1 \mathfrak{p} = \mathbf{v}\_Q - \Phi\_1 \mathfrak{p}\_Q \\\\ \mathbf{a}\_2 = \mathbf{a} - \Phi\_2 \mathfrak{p} = \mathbf{a}\_Q - \Phi\_2 \mathfrak{p}\_Q \end{cases} \tag{4}$$

do not depend on the choice of point *P* of the rigid body. They are called the **velocity invariant** respectively the **acceleration invariant** (at a given moment of time).

#### **2.1 The velocity field in rigid body motion**

It is described by:

chains, equations that allow the determination of the nth order accelerations field are given, using a Brockett-like formula. The crucial observation is that the nth order time derivative of twist of the terminal body in a kinematic chain can be determined by propagating the kth order time derivative of twists of the bodies in the chain, for

The general framework of this chapter is a rigid body that moves with respect to a fixed reference frame <sup>R</sup><sup>0</sup> � �. Consider another reference frame f g <sup>R</sup> originated in a point *Q* that moves together with the rigid body. Let **ρ***<sup>Q</sup>* denote the position vector of point *Q* with respect to frame R<sup>0</sup> � �, **v***<sup>Q</sup>* its absolute velocity and **a***<sup>Q</sup>* its absolute

where **ρ** represents the absolute position of a generic point *P* of the rigid body with respect to <sup>R</sup><sup>0</sup> � � and **<sup>R</sup>** <sup>¼</sup> **<sup>R</sup>**ð Þ*<sup>t</sup>* is an orthogonal proper tensorial function in

The results of this section succinctly present the velocity and acceleration vector field in rigid body motion. These results lead to the generalization presented in the

With the denotations that were introduced, the vector fields of velocities and

**<sup>v</sup>** � **<sup>v</sup>***<sup>Q</sup>* <sup>¼</sup> **RR**\_ *<sup>T</sup>* **<sup>ρ</sup>** � **<sup>ρ</sup>***<sup>Q</sup>*

**<sup>a</sup>** � **<sup>a</sup>***<sup>Q</sup>* <sup>¼</sup> **RR**€ *<sup>T</sup>* **<sup>ρ</sup>** � **<sup>ρ</sup>***<sup>Q</sup>*

**<sup>Φ</sup>**<sup>1</sup> <sup>¼</sup> **RR**\_ *<sup>T</sup>* **<sup>Φ</sup>**<sup>2</sup> <sup>¼</sup> **RR**€ *<sup>T</sup>*

represent the **velocity tensor** respectively the **acceleration tensor**. Tensor

**a**<sup>1</sup> ¼ **v** � **Φ**1**ρ** ¼ **v***<sup>Q</sup>* � **Φ**1**ρ***<sup>Q</sup>* **a**<sup>2</sup> ¼ **a** � **Φ**2**ρ** ¼ **a***<sup>Q</sup>* � **Φ**2**ρ***<sup>Q</sup>*

do not depend on the choice of point *P* of the rigid body. They are called the **velocity invariant** respectively the **acceleration invariant** (at a given moment of

(

� �

� �

<sup>3</sup> is the skew-symmetric tensor associated to the instantaneous angular

<sup>3</sup> . Tensor **<sup>Φ</sup>**<sup>2</sup> <sup>¼</sup> **<sup>ω</sup>**~<sup>2</sup> <sup>þ</sup> <sup>~</sup>**ε**, where **<sup>ε</sup>** <sup>¼</sup> **<sup>ω</sup>**\_ is the instantaneous angular

<sup>3</sup> . Vector **r** is constant and it represents the relative position vector of the

**ρ** ¼ **ρ***<sup>Q</sup>* þ **Rr** (1)

(2)

(3)

(4)

*<sup>k</sup>* <sup>¼</sup> 0,�*n*. The results are coordinate-free and in a closed form.

*Advances on Tensor Analysis and Their Applications*

**2. Theoretical consideration on rigid body motion**

Then the vector parametric equation of motion is:

8 < :

acceleration of the rigid body. One may remark that vectors:

(

arbitrary point *P* with respect to f g R .

accelerations are described by:

acceleration.

next section.

Tensors:

**<sup>Φ</sup>**<sup>1</sup> <sup>¼</sup> **<sup>ω</sup>**<sup>~</sup> <sup>∈</sup>**so**

time).

**104**

velocity **ω** ∈ **V**

**SO**

$$\mathbf{v} - \mathbf{v}\_Q = \Phi\_1(\mathfrak{p} - \mathfrak{p}\_Q) \tag{5}$$

The instantaneous angular velocity **ω** of the rigid body may be determined as **ω** ¼ vect**Φ**1. The major property that may be highlighted from Eq. (4) is that the velocity of a given point of the rigid may be computed when knowing the velocity tensor **Φ**<sup>1</sup> and the velocity invariant **a**1:

$$\mathbf{v} = \mathbf{a}\_1 + \Phi\_1 \mathbf{p} \tag{6}$$

#### **2.2 The acceleration field in rigid body motion**

It is described by

$$\mathbf{a} - \mathbf{a}\_{Q} = \Phi\_{2}(\mathfrak{p} - \mathfrak{p}\_{Q}) \tag{7}$$

The absolute acceleration of a given point of the rigid body may be computed when knowing the acceleration tensor **Φ**<sup>2</sup> and the acceleration invariant **a**2:

$$\mathbf{a} = \mathbf{a}\_2 + \Phi\_2 \mathbf{\hat{p}} \tag{8}$$

The instantaneous angular acceleration of the rigid body may be determined as:

$$e = \text{vect}\Phi\_2 \tag{9}$$

The determinant of tensor **Φ**<sup>2</sup> is (see [10]): det**Φ**<sup>2</sup> ¼ �ð Þ **ω** � **ε** 2 . It follows that if **ω** � **ε** 6¼ **0**, then tensor **Φ**2*:*

is invertible and its inverse is (see [10]):

$$\boldsymbol{\Phi}\_{2}^{-1} = \frac{1}{\left(\boldsymbol{\mathfrak{o}} \times \boldsymbol{\mathfrak{e}}\right)^{2}} \left[\boldsymbol{\mathfrak{e}} \otimes \boldsymbol{\mathfrak{e}} + \left(\boldsymbol{\mathfrak{o}} \otimes \boldsymbol{\mathfrak{o}}\right)^{2} - \widehat{\boldsymbol{\mathfrak{o}}}^{2} \boldsymbol{\overline{\mathfrak{e}}}\right] \tag{10}$$

It follows that if tensor **Φ**<sup>2</sup> is non-singular, then for an arbitrary given acceleration **a** we may find a point of the rigid that has this acceleration. Its absolute position is given by (see also Eq. (8)):

$$
\mathfrak{p} = \Phi\_2^{-1}(\mathbf{a} - \mathbf{a}\_2) \tag{11}
$$

Particularly, if **Φ**<sup>2</sup> is non-singular, then there exists a point *G* of zero acceleration, named the **acceleration center**. Its absolute position vector is given by:

$$\mathfrak{p}\_G = -\mathfrak{d}\_2^{-1}\mathfrak{a}\_2\tag{12}$$

#### **3. The vector field of the** *nth* **order accelerations**

This section extends some of the previous considerations to the case of the *nth* order accelerations. We define the *nth* order acceleration of a point as:

*Advances on Tensor Analysis and Their Applications*

$$\mathbf{a}\_{\mathfrak{p}}^{[n]} \stackrel{\text{def}}{=} \frac{d^n}{dt^n} \mathfrak{p}, n \ge 1 \tag{13}$$

**Remark 4.** *By defining the nth order instantaneous nth order angular acceleration of*

*k*¼1

The set of affine maps, *g* : **V**<sup>3</sup> ! **V**3, *g*ð Þ¼ **u Ru** þ **w**, where **R** is an orthogonal proper tensor and **w** a vector in **V**<sup>3</sup> is a group under composition and it is called *the group of direct affine isometries* or *rigid motions* and it is denoted *S*3. Any rigid finite motion may be described by such a map. Tensor **R** models the rotation of the considered rigid body and vector **w** its translation. An affine map from *S*<sup>3</sup> may be

> *<sup>g</sup>* <sup>¼</sup> **R w 0** 1

> > ¼

**R w**

" #�<sup>1</sup>

**0** 1

and the range *S*3. The parametric vector equation of the rigid body motion (1) may be rewritten with the help of a homogenous matrix function in *S*

> <sup>¼</sup> **<sup>R</sup> <sup>ρ</sup>***<sup>Q</sup>* **0** 1 � � **r**

<sup>¼</sup> **<sup>R</sup>**\_ **<sup>ρ</sup>**\_ *<sup>Q</sup>* **0** 0

and by making the computations and taking into account Eqs. (3) and (4) it

<sup>¼</sup> **<sup>Φ</sup>**<sup>1</sup> **<sup>a</sup>**<sup>1</sup> **0** 0 � � **ρ**

By using the previous considerations, it follows that Eq. (25) may be extended

1

" # **<sup>R</sup>***<sup>T</sup>* �**R***<sup>T</sup>***ρ***<sup>Q</sup>*

1

**0** 1

" # **<sup>ρ</sup>**

**3.1 Homogenous matrix approach to the field of** *nth* **order accelerations**

( ) " #

*dk*

*dtn*�<sup>1</sup> **ω***, it follows from Eq. (21) that its associated skew-symmetric*

*dtn*�<sup>1</sup> **<sup>Φ</sup>**<sup>1</sup> *. The expression of the instantaneous nth order*

� � (23)

**R**1**R**<sup>2</sup> **R**1**w**<sup>2</sup> þ **w**<sup>1</sup>

<sup>¼</sup> **<sup>R</sup>***<sup>T</sup>* �**R***<sup>T</sup>***<sup>w</sup> 0** 1

<sup>3</sup> , the set of the functions with the domain ℝ

� � (25)

1

� � (27)

� � (26)

" #

" #

**0** 1

, *n*≥3 (22)

(24)

<sup>3</sup> like it

*dtk* ð Þ **<sup>Φ</sup>***<sup>n</sup>*�*k*�<sup>1</sup>**Φ**<sup>1</sup>

*the rigid body* **ε**½ � *<sup>n</sup>* ≝ *dn*�<sup>1</sup>

*angular acceleration is:*

*tensor may be expressed as* <sup>~</sup>**ε**½ � *<sup>n</sup>* <sup>¼</sup> *<sup>d</sup>n*�<sup>1</sup>

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

represented with a 4 � 4 square matrix:

8 >>>>>><

>>>>>>:

We may extend now *S*<sup>3</sup> to *S*

From Eq. (25), it follows that:

<sup>¼</sup> **<sup>R</sup>**\_ **<sup>ρ</sup>**\_ *<sup>Q</sup>* **0** 0 " # **r**

**ρ**\_ 0 � �

follows:

follows that:

like: **107**

**<sup>ε</sup>**½ � *<sup>n</sup>* <sup>¼</sup> *vect* **<sup>Φ</sup>***<sup>n</sup>* � **<sup>Φ</sup>***<sup>n</sup>*�<sup>1</sup>**Φ**<sup>1</sup> �X*n*�<sup>2</sup>

One may remark that the following relations hold true:

**0** 1

**ρ** 1 � �

> 1 � �

**ρ**\_ 0 � �

" #

" # **R**<sup>2</sup> **w**<sup>2</sup>

**R**<sup>1</sup> **w**<sup>1</sup>

**0** 1

For *n* ¼ 1, it represents the velocity, and for *n* ¼ 2, the acceleration. By derivation with respect to time successively in Eq. (2), it follows that:

$$\mathbf{a}\_{\mathfrak{p}}^{[n]} - \mathbf{a}\_{Q}^{[n]} = \mathbf{R}^{(n)} \mathbf{R}^{T} (\mathfrak{p} - \mathfrak{p}\_{Q}), \text{where } \mathbf{R}^{(n)} \stackrel{\text{def}}{=} \frac{d^{n}}{dt^{n}} \mathbf{R} \tag{14}$$

We define:

$$\boldsymbol{\Phi}\_n \stackrel{\text{def}}{=} \mathbf{R}^{(n)} \mathbf{R}^T \tag{15}$$

the *nth* **order acceleration tensor** in rigid body motion. A vector invariant is immediately highlighted from Eq. (14) with the denotation (15). Vector:

$$\mathbf{a}\_{n} = \mathbf{a}\_{\rho}^{[n]} - \boldsymbol{\Phi}\_{n}\boldsymbol{\mathfrak{p}} = \mathbf{a}\_{Q}^{[n]} - \boldsymbol{\Phi}\_{n}\boldsymbol{\mathfrak{p}}\_{Q} \tag{16}$$

does not depend on the choice of the point of the rigid body for which the acceleration **a**½ � *<sup>n</sup>* is computed. Vector **a***<sup>n</sup>* is named the **invariant vector** of the *nth* **order accelerations**. Then Eq. (7) may be generalized as it follows:

$$\mathbf{a}\_{\boldsymbol{\rho}}^{[n]} - \mathbf{a}\_{Q}^{[n]} = \boldsymbol{\Phi}\_{n} (\boldsymbol{\rho} - \boldsymbol{\rho}\_{Q}) \tag{17}$$

The next Theorem gives the fundamental properties of the vector field of the *nth* order accelerations.

**Theorem 1.** *In the rigid body motion, at a moment of time t, there exist tensor* **Φ***<sup>n</sup> defined by Eq. (15) and vector* **a***<sup>n</sup> such as:*

$$\begin{aligned} \mathbf{a}\_{\boldsymbol{\mathfrak{p}}}^{[n]} - \mathbf{a}\_{Q}^{[n]} &= \boldsymbol{\Phi}\_{n} (\boldsymbol{\mathfrak{p}} - \boldsymbol{\mathfrak{p}}\_{Q}) \\ \mathbf{a}\_{\boldsymbol{n}} &= \mathbf{a}\_{\boldsymbol{\mathfrak{p}}}^{[n]} - \boldsymbol{\Phi}\_{n} \boldsymbol{\mathfrak{p}} = \mathbf{a}\_{Q}^{[n]} - \boldsymbol{\Phi}\_{n} \boldsymbol{\mathfrak{p}}\_{Q} \end{aligned} \tag{18}$$

for any point *P* of the rigid body with the absolute position defined by vector **ρ***.* **Remark 1.** *Given the absolute position of a point of the rigid body and knowing* **Φ***<sup>n</sup> and* **a***n, its acceleration is computed from:*

$$\mathbf{a}\_{\rho}^{[n]} = \mathbf{a}\_{n} + \boldsymbol{\Phi}\_{n}\boldsymbol{\mathfrak{p}} \tag{19}$$

**Remark 2.** *Tensor* **Φ***<sup>n</sup> and vector* **a***<sup>n</sup> generalize the notions of velocity/acceleration tensor respectively velocity/acceleration invariant. They are fundamental in the study of the vector field of the nth order accelerations. The recursive formulas for computing* **Φ***<sup>n</sup> and* **a***<sup>n</sup> are:*

$$\begin{cases} \Phi\_{n+1} = \dot{\Phi}\_n + \Phi\_n \Phi\_1\\ \mathbf{a}\_{n+1} = \dot{\mathbf{a}}\_n + \Phi\_n \mathbf{a}\_1 \end{cases}, n \ge 1, where \ \Phi\_1 = \ddot{\mathbf{o}}, \mathbf{a}\_1 = \mathbf{v}\_Q - \Phi\_1 \mathbf{o}\_Q \tag{20}$$

**Remark 3.** *One may remark that from Eq. (20) it follows by direct computation:*

$$\begin{cases} \Phi\_n = \Phi\_{n-1}\Phi\_1 + \left(\frac{d^{n-1}}{dt^{n-1}}\Phi\_1\right) + \sum\_{k=1}^{n-2} \left[\frac{d^k}{dt^{n-1}}(\Phi\_{n-k-1}\Phi\_1)\right] \\\\ \mathbf{a}\_n = \Phi\_{n-1}\mathbf{a}\_1 + \left(\frac{d^{n-1}}{dt^{n-1}}\mathbf{a}\_1\right) + \sum\_{k=1}^{n-2} \left[\frac{d^k}{dt^{n-1}}(\Phi\_{n-k-1}\mathbf{a}\_1)\right] \end{cases}, n \ge 3 \tag{21}$$

**106**

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

**a**½ � *<sup>n</sup>* **ρ**

derivation with respect to time successively in Eq. (2), it follows that:

*<sup>Q</sup>* <sup>¼</sup> **<sup>R</sup>**ð Þ *<sup>n</sup>* **<sup>R</sup>***<sup>T</sup>* **<sup>ρ</sup>** � **<sup>ρ</sup>***<sup>Q</sup>*

immediately highlighted from Eq. (14) with the denotation (15). Vector:

**<sup>a</sup>***<sup>n</sup>* <sup>¼</sup> **<sup>a</sup>**½ � *<sup>n</sup>*

**order accelerations**. Then Eq. (7) may be generalized as it follows:

**a**½ � *<sup>n</sup>* **<sup>ρ</sup>** � **a** ½ � *n*

**a** ½ � *n* **<sup>ρ</sup>** � **a** ½ � *n*

**a***<sup>n</sup>* ¼ **a** ½ � *n*

*dn*�<sup>1</sup> *dtn*�<sup>1</sup> **<sup>Φ</sup>**<sup>1</sup> !

> *dn*�<sup>1</sup> *dtn*�<sup>1</sup> **<sup>a</sup>**<sup>1</sup> !

**a**½ � *<sup>n</sup>* **<sup>ρ</sup>** � **a** ½ � *n*

*Advances on Tensor Analysis and Their Applications*

We define:

order accelerations.

(

8 >>>>><

>>>>>:

**106**

*defined by Eq. (15) and vector* **a***<sup>n</sup> such as:*

*and* **a***n, its acceleration is computed from:*

**<sup>Φ</sup>***<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> **<sup>Φ</sup>**\_ *<sup>n</sup>* <sup>þ</sup> **<sup>Φ</sup>***n***Φ**<sup>1</sup> **a***<sup>n</sup>*þ<sup>1</sup> ¼ **a**\_ *<sup>n</sup>* þ **Φ***n***a**<sup>1</sup>

**Φ***<sup>n</sup>* ¼ **Φ***<sup>n</sup>*�<sup>1</sup>**Φ**<sup>1</sup> þ

**a***<sup>n</sup>* ¼ **Φ***<sup>n</sup>*�<sup>1</sup>**a**<sup>1</sup> þ

<sup>≝</sup> *dn*

For *n* ¼ 1, it represents the velocity, and for *n* ¼ 2, the acceleration. By

the *nth* **order acceleration tensor** in rigid body motion. A vector invariant is

½ � *n*

*<sup>Q</sup>* ¼ **Φ***<sup>n</sup>* **ρ** � **ρ***<sup>Q</sup>*

*<sup>Q</sup>* ¼ **Φ***<sup>n</sup>* **ρ** � **ρ***<sup>Q</sup>*

for any point *P* of the rigid body with the absolute position defined by vector **ρ***.* **Remark 1.** *Given the absolute position of a point of the rigid body and knowing* **Φ***<sup>n</sup>*

**Remark 2.** *Tensor* **Φ***<sup>n</sup> and vector* **a***<sup>n</sup> generalize the notions of velocity/acceleration tensor respectively velocity/acceleration invariant. They are fundamental in the study of the vector field of the nth order accelerations. The recursive formulas for computing* **Φ***<sup>n</sup> and* **a***<sup>n</sup> are:*

**Remark 3.** *One may remark that from Eq. (20) it follows by direct computation:*

þX*<sup>n</sup>*�<sup>2</sup> *k*¼1

> þX*<sup>n</sup>*�<sup>2</sup> *k*¼1

*dk*

*dk*

� �

½ � *n <sup>Q</sup>* � **Φ***n***ρ***<sup>Q</sup>*

The next Theorem gives the fundamental properties of the vector field of the *nth*

**Theorem 1.** *In the rigid body motion, at a moment of time t, there exist tensor* **Φ***<sup>n</sup>*

**<sup>ρ</sup>** � **Φ***n***ρ** ¼ **a**

**a**½ � *<sup>n</sup>*

**<sup>ρ</sup>** � **Φ***n***ρ** ¼ **a**

does not depend on the choice of the point of the rigid body for which the acceleration **a**½ � *<sup>n</sup>* is computed. Vector **a***<sup>n</sup>* is named the **invariant vector** of the *nth*

� �, where **<sup>R</sup>**ð Þ *<sup>n</sup>* <sup>≝</sup> *dn*

*dtn* **<sup>ρ</sup>**, *<sup>n</sup>*<sup>≥</sup> <sup>1</sup> (13)

**Φ***<sup>n</sup>* ≝ **R**ð Þ *<sup>n</sup>* **R***<sup>T</sup>* (15)

*<sup>Q</sup>* � **Φ***n***ρ***<sup>Q</sup>* (16)

� � (17)

**<sup>ρ</sup>** ¼ **a***<sup>n</sup>* þ **Φ***n***ρ** (19)

, *n* ≥1, *where* **Φ**<sup>1</sup> ¼ **ω**~, **a**<sup>1</sup> ¼ **v***<sup>Q</sup>* � **Φ**1**ρ***<sup>Q</sup>* (20)

*dtn*�<sup>1</sup> ð Þ **<sup>Φ</sup>***<sup>n</sup>*�*k*�<sup>1</sup>**Φ**<sup>1</sup> " #

*dtn*�<sup>1</sup> ð Þ **<sup>Φ</sup>***<sup>n</sup>*�*k*�<sup>1</sup>**a**<sup>1</sup> " # (18)

, *n* ≥3 (21)

*dt<sup>n</sup>* **<sup>R</sup>** (14)

**Remark 4.** *By defining the nth order instantaneous nth order angular acceleration of the rigid body* **ε**½ � *<sup>n</sup>* ≝ *dn*�<sup>1</sup> *dtn*�<sup>1</sup> **ω***, it follows from Eq. (21) that its associated skew-symmetric tensor may be expressed as* <sup>~</sup>**ε**½ � *<sup>n</sup>* <sup>¼</sup> *<sup>d</sup>n*�<sup>1</sup> *dtn*�<sup>1</sup> **<sup>Φ</sup>**<sup>1</sup> *. The expression of the instantaneous nth order angular acceleration is:*

$$e^{[n]} = \text{vect}\left\{\Phi\_n - \Phi\_{n-1}\Phi\_1 - \sum\_{k=1}^{n-2} \left[\frac{d^k}{dt^k}(\Phi\_{n-k-1}\Phi\_1)\right] \right\}, n \ge 3 \tag{22}$$

#### **3.1 Homogenous matrix approach to the field of** *nth* **order accelerations**

The set of affine maps, *g* : **V**<sup>3</sup> ! **V**3, *g*ð Þ¼ **u Ru** þ **w**, where **R** is an orthogonal proper tensor and **w** a vector in **V**<sup>3</sup> is a group under composition and it is called *the group of direct affine isometries* or *rigid motions* and it is denoted *S*3. Any rigid finite motion may be described by such a map. Tensor **R** models the rotation of the considered rigid body and vector **w** its translation. An affine map from *S*<sup>3</sup> may be represented with a 4 � 4 square matrix:

$$\mathbf{g} = \begin{bmatrix} \mathbf{R} & \mathbf{w} \\ \mathbf{0} & \mathbf{1} \end{bmatrix} \tag{23}$$

One may remark that the following relations hold true:

$$\begin{cases} \begin{bmatrix} \mathbf{R}\_1 & \mathbf{w}\_1 \\ \mathbf{0} & \mathbf{1} \end{bmatrix} \begin{bmatrix} \mathbf{R}\_2 & \mathbf{w}\_2 \\ \mathbf{0} & \mathbf{1} \end{bmatrix} = \begin{bmatrix} \mathbf{R}\_1 \mathbf{R}\_2 & \mathbf{R}\_1 \mathbf{w}\_2 + \mathbf{w}\_1 \\ \mathbf{0} & \mathbf{1} \end{bmatrix} \\\\ \begin{bmatrix} \mathbf{R} & \mathbf{w} \\ \mathbf{0} & \mathbf{1} \end{bmatrix}^{-1} = \begin{bmatrix} \mathbf{R}^T & -\mathbf{R}^T \mathbf{w} \\\\ \mathbf{0} & \mathbf{1} \end{bmatrix} \end{cases} \tag{24}$$

We may extend now *S*<sup>3</sup> to *S* <sup>3</sup> , the set of the functions with the domain ℝ and the range *S*3. The parametric vector equation of the rigid body motion (1) may be rewritten with the help of a homogenous matrix function in *S* <sup>3</sup> like it follows:

$$
\begin{bmatrix} \mathbf{p} \\ \mathbf{1} \end{bmatrix} = \begin{bmatrix} \mathbf{R} & \mathbf{p}\_Q \\ \mathbf{0} & \mathbf{1} \end{bmatrix} \begin{bmatrix} \mathbf{r} \\ \mathbf{1} \end{bmatrix} \tag{25}
$$

From Eq. (25), it follows that:

$$
\begin{bmatrix} \dot{\boldsymbol{\Phi}} \\ \mathbf{0} \end{bmatrix} = \begin{bmatrix} \dot{\mathbf{R}} & \dot{\boldsymbol{\rho}}\_{Q} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{r} \\ \mathbf{1} \end{bmatrix} = \begin{bmatrix} \dot{\mathbf{R}} & \dot{\boldsymbol{\rho}}\_{Q} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{R}^{T} & -\mathbf{R}^{T}\boldsymbol{\rho}\_{Q} \\ \mathbf{0} & \mathbf{1} \end{bmatrix} \begin{bmatrix} \mathbf{\dot{\rho}} \\ \mathbf{1} \end{bmatrix} \tag{26}
$$

and by making the computations and taking into account Eqs. (3) and (4) it follows that:

$$
\begin{bmatrix} \dot{\boldsymbol{\rho}} \\ \mathbf{0} \end{bmatrix} = \begin{bmatrix} \boldsymbol{\Phi}\_1 & \mathbf{a}\_1 \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \boldsymbol{\rho} \\ \mathbf{1} \end{bmatrix} \tag{27}
$$

By using the previous considerations, it follows that Eq. (25) may be extended like:

*Advances on Tensor Analysis and Their Applications*

$$
\begin{bmatrix} \mathbf{a}\_{\rho}^{[n]} \\ \mathbf{0} \end{bmatrix} = \begin{bmatrix} \Phi\_n & \mathbf{a}\_n \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \rho \\ \mathbf{1} \end{bmatrix} \tag{28}
$$

*where* **<sup>D</sup>** <sup>¼</sup> *<sup>d</sup>*

tionship of recurrence for **P***n*ð Þ **D** :

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

**0** 0

*nth order accelerations will be written as*

*where* **P***<sup>n</sup> fulfills the relationship of recurrence*

(

Since **<sup>Ψ</sup>**<sup>1</sup> <sup>¼</sup> **<sup>ω</sup>**<sup>~</sup> **<sup>v</sup>**

It follows

Thus, it follows:

• the velocity field invariants

• the acceleration field invariants

• hyper-jerk (jounce) field invariants

• jerk field invariants

**109**

*dt is the operator of time derivative*.

(

*Proof*: Taking into account Eqs. (36) and (33) we will have the following rela-

**P***<sup>n</sup>*þ<sup>1</sup> ¼ **DP***<sup>n</sup>* þ **P***n*ð Þ **Ψ**<sup>1</sup>

(37)

(39)

(40)

(41)

(42)

(43)

**P**<sup>0</sup> ¼ **I**

**Theorem 3.** *There is a unique polynomial with the coefficients in the noncommutative ring* **L V**ð Þ 3, **V**<sup>3</sup> *such that the vector respectively the tensor invariants of the*

**a***<sup>n</sup>* ¼ **P***n***v**

**P**<sup>2</sup> ¼ **D** þ **ω**~ **<sup>P</sup>**<sup>3</sup> <sup>¼</sup> **<sup>D</sup>**<sup>2</sup> <sup>þ</sup> **<sup>ω</sup>**~**<sup>D</sup>** <sup>þ</sup> <sup>2</sup>**ω**~\_ <sup>þ</sup> **<sup>ω</sup>**~<sup>2</sup> **<sup>P</sup>**<sup>4</sup> <sup>¼</sup> **<sup>D</sup>**<sup>3</sup> <sup>þ</sup> **<sup>ω</sup>**~**D**<sup>2</sup> <sup>þ</sup> <sup>3</sup>**ω**~\_ <sup>þ</sup> **<sup>ω</sup>**~<sup>2</sup> � �**<sup>D</sup>** <sup>þ</sup> €3**ω**<sup>~</sup> <sup>þ</sup> <sup>3</sup>**ω**~\_ **<sup>ω</sup>**<sup>~</sup> <sup>þ</sup> <sup>2</sup>**ω**~**ω**~\_ <sup>þ</sup> **<sup>ω</sup>**~<sup>3</sup>

(

(

8 < :

**<sup>P</sup>***<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> **DP***<sup>n</sup>* <sup>þ</sup> **<sup>P</sup>***n*ð Þ **<sup>ω</sup>**<sup>~</sup> , *<sup>n</sup>*<sup>∈</sup> <sup>∗</sup>

**a**<sup>1</sup> ¼ **v Φ**<sup>1</sup> ¼ **ω**~

**a**<sup>2</sup> ¼ **v**\_ þ **ω**~**v <sup>Φ</sup>**<sup>2</sup> <sup>¼</sup> **<sup>ω</sup>**~\_ <sup>þ</sup> **<sup>ω</sup>**~<sup>2</sup>

**<sup>a</sup>**<sup>3</sup> <sup>¼</sup> **<sup>v</sup>**€ <sup>þ</sup> **<sup>ω</sup>**~**<sup>v</sup>** <sup>þ</sup> <sup>2</sup>**ω**~\_ **<sup>v</sup>** <sup>þ</sup> **<sup>ω</sup>**~<sup>2</sup>

**<sup>Φ</sup>**<sup>3</sup> <sup>¼</sup> **<sup>ω</sup>**€~ <sup>þ</sup> **<sup>ω</sup>**~**ω**~\_ <sup>þ</sup> <sup>2</sup>**ω**~\_ **<sup>ω</sup>**<sup>~</sup> <sup>þ</sup> **<sup>ω</sup>**~<sup>3</sup>

**v**

**<sup>Φ</sup>***<sup>n</sup>* <sup>¼</sup> **<sup>P</sup>***n***ω**<sup>~</sup> , *<sup>n</sup>* <sup>∈</sup> <sup>∗</sup> (38)

� � it follows the next outcome.

**P**<sup>1</sup> ¼ **I**

Eq. (28) represents a unified form of describing the vector field of the *nth* order accelerations in rigid body motion. The matrix:

$$
\Psi\_n = \begin{bmatrix}
\Phi\_n & \mathbf{a}\_n \\
\mathbf{0} & \mathbf{0}
\end{bmatrix} \tag{29}
$$

contains both the *nth* order acceleration tensor **Φ***<sup>n</sup>* and the vector invariant **a***n*. Eqs. (20) may be put in a compact form:

$$
\Psi\_{n+1} = \dot{\Psi}\_n + \Psi\_n \Psi\_1, n \ge 1 \tag{30}
$$

If follows that **Ψ***<sup>n</sup>* may be written as:

$$
\Psi \Psi\_n = \Psi\_{n-1} \Psi\_1 + \left(\frac{d^{n-1}}{dt^{n-1}} \Psi\_1\right) + \sum\_{k=1}^{n-2} \left[\frac{d^k}{dt^k} (\Psi\_n \Psi\_1)\right], n \ge 3 \tag{31}
$$

#### **4. Symbolic calculus of higher-order kinematics invariants**

We will present a method for the symbolic calculation of higher-order kinematics invariants for rigid motion.

Let be **a***<sup>n</sup>* and **Φ***n*, *n* ∈ vector invariant, respectively, tensor invariant for the *nth* order accelerations fields. We denote by

$$
\Psi\_n = \begin{bmatrix}
\Phi\_n & \mathbf{a}\_n \\
\mathbf{0} & \mathbf{0}
\end{bmatrix} \tag{32}
$$

and we have the following relationship of recurrence:

$$\begin{aligned} \Psi\_{n+1} &= \dot{\Psi}\_n + \Psi\_n \Psi\_1, n \in \mathbb{N} \\ \Psi\_1 &= \begin{bmatrix} \ddot{\mathbf{o}} & \mathbf{v} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \end{aligned} \tag{33}$$

The pair of vectors ð Þ **ω**, **v** is also known as *the spatial twist of rigid body.* Let be **A** the matrix ring

$$\mathcal{A} = \left\{ \mathbf{A} \in \mathcal{M}\_{3 \times 3}(\mathbb{R}) \, \middle| \, \mathbf{A} = \begin{bmatrix} \boldsymbol{\Phi} & \mathbf{a} \\ \mathbf{0} & \mathbf{0} \end{bmatrix}; \boldsymbol{\Phi} \in \mathrm{L}(\mathbf{V}\_3, \mathbf{V}\_3), \mathbf{a} \in \mathbf{V}\_3 \right\} \tag{34}$$

and **A**½ � X the set of polynomials with coefficients in the non-commutative ring **A**. A generic element of **A**½ � X has the form

$$\mathbf{P(X)} = \mathbf{A}\_0 \mathbf{X}^m + \mathbf{A}\_1 \mathbf{X}^{m-1} + \dots + \mathbf{A}\_{m-1} \mathbf{X} + \mathbf{A}\_m, \mathbf{A}\_k \in \mathcal{A}, k = \overline{0, m} \tag{35}$$

**Theorem 2.** *There is a unique polynomial* **P***<sup>n</sup>* ∈ **A**½ � *X such that* **Ψ**<sup>n</sup> *will be written as*

$$\Psi\_n = \mathbf{P}\_n(\mathbf{D})\Psi\_1, n \in \mathbb{N} \tag{36}$$

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

*where* **<sup>D</sup>** <sup>¼</sup> *<sup>d</sup> dt is the operator of time derivative*.

*Proof*: Taking into account Eqs. (36) and (33) we will have the following relationship of recurrence for **P***n*ð Þ **D** :

$$\begin{cases} \mathbf{P}\_{n+1} = \mathbf{D} \mathbf{P}\_n + \mathbf{P}\_n(\boldsymbol{\Psi}\_1) \\\\ \mathbf{P}\_0 = \mathbf{I} \end{cases} \tag{37}$$

Since **<sup>Ψ</sup>**<sup>1</sup> <sup>¼</sup> **<sup>ω</sup>**<sup>~</sup> **<sup>v</sup> 0** 0 � � it follows the next outcome.

**Theorem 3.** *There is a unique polynomial with the coefficients in the noncommutative ring* **L V**ð Þ 3, **V**<sup>3</sup> *such that the vector respectively the tensor invariants of the nth order accelerations will be written as*

$$\begin{aligned} \mathbf{a}\_{n} &= \mathbf{P}\_{n} \mathbf{v} \\ \boldsymbol{\Phi}\_{n} &= \mathbf{P}\_{n} \boldsymbol{\tilde{\mathbf{o}}} \end{aligned} , n \in \mathbb{N}^\* \tag{38}$$

*where* **P***<sup>n</sup> fulfills the relationship of recurrence*

$$\begin{cases} \mathbf{P}\_{n+1} = \mathbf{D} \mathbf{P}\_n + \mathbf{P}\_n(\tilde{\mathbf{u}}), n \in \mathbb{N}^\* \\\\ \mathbf{P}\_1 = \mathbf{I} \end{cases} \tag{39}$$

It follows

**a** ½ � *n* **ρ** 0

accelerations in rigid body motion. The matrix:

*Advances on Tensor Analysis and Their Applications*

Eqs. (20) may be put in a compact form:

If follows that **Ψ***<sup>n</sup>* may be written as:

**Ψ***<sup>n</sup>* ¼ **Ψ***<sup>n</sup>*�<sup>1</sup>**Ψ**<sup>1</sup> þ

ics invariants for rigid motion.

Let be **A** the matrix ring

**108**

order accelerations fields. We denote by

" #

<sup>¼</sup> **<sup>Φ</sup>***<sup>n</sup>* **<sup>a</sup>***<sup>n</sup>* **0** 0 � � **ρ**

Eq. (28) represents a unified form of describing the vector field of the *nth* order

**<sup>Ψ</sup>***<sup>n</sup>* <sup>¼</sup> **<sup>Φ</sup>***<sup>n</sup>* **<sup>a</sup>***<sup>n</sup>* **0** 0 � �

contains both the *nth* order acceleration tensor **Φ***<sup>n</sup>* and the vector invariant **a***n*.

þX*<sup>n</sup>*�<sup>2</sup> *k*¼1

We will present a method for the symbolic calculation of higher-order kinemat-

Let be **a***<sup>n</sup>* and **Φ***n*, *n* ∈ vector invariant, respectively, tensor invariant for the *nth*

**<sup>Ψ</sup>***<sup>n</sup>* <sup>¼</sup> **<sup>Φ</sup>***<sup>n</sup>* **<sup>a</sup>***<sup>n</sup>* **0** 0 � �

**<sup>Ψ</sup>***<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> **<sup>Ψ</sup>**\_ *<sup>n</sup>* <sup>þ</sup> **<sup>Ψ</sup>***n***Ψ**1, *<sup>n</sup>* <sup>∈</sup>

**<sup>Ψ</sup>**<sup>1</sup> <sup>¼</sup> **<sup>ω</sup>**<sup>~</sup> **<sup>v</sup>**

The pair of vectors ð Þ **ω**, **v** is also known as *the spatial twist of rigid body.*

**0** 0 � �

� �

and **A**½ � X the set of polynomials with coefficients in the non-commutative ring

P Xð Þ¼ **<sup>A</sup>**0X*<sup>m</sup>* <sup>þ</sup> **<sup>A</sup>**1X*<sup>m</sup>*�<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> **<sup>A</sup>***<sup>m</sup>*�1X <sup>þ</sup> **<sup>A</sup>***m*, **<sup>A</sup>***<sup>k</sup>* <sup>∈</sup> **<sup>A</sup>**, *<sup>k</sup>* <sup>¼</sup> 0, *<sup>m</sup>* (35)

**Theorem 2.** *There is a unique polynomial* **P***<sup>n</sup>* ∈ **A**½ � *X such that* **Ψ**<sup>n</sup> *will be written as*

**0** 0

*dn*�<sup>1</sup> *dt<sup>n</sup>*�<sup>1</sup> **<sup>Ψ</sup>**<sup>1</sup> !

**4. Symbolic calculus of higher-order kinematics invariants**

and we have the following relationship of recurrence:

**<sup>A</sup>** <sup>¼</sup> **<sup>A</sup>** <sup>∈</sup>M<sup>3</sup>�<sup>3</sup>ð Þ **<sup>A</sup>** <sup>¼</sup> **<sup>Φ</sup> <sup>a</sup>**

**A**. A generic element of **A**½ � X has the form

� � � �

1 � �

**<sup>Ψ</sup>***<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> **<sup>Ψ</sup>**\_ *<sup>n</sup>* <sup>þ</sup> **<sup>Ψ</sup>***n***Ψ**1, *<sup>n</sup>* <sup>≥</sup><sup>1</sup> (30)

*dtk* ð Þ **<sup>Ψ</sup>***n***Ψ**<sup>1</sup> " #

" # (33)

; **Φ** ∈Lð Þ **V**3, **V**<sup>3</sup> , **a**∈ **V**<sup>3</sup>

**Ψ***<sup>n</sup>* ¼ **P***n*ð Þ **D Ψ**1, *n*∈ (36)

*dk*

(28)

(29)

(32)

(34)

, *n* ≥3 (31)

$$\begin{aligned} \mathbf{P\_2} &= \mathbf{D} + \ddot{\mathbf{o}} \\ \mathbf{P\_3} &= \mathbf{D^2} + \ddot{\mathbf{o}} \mathbf{D} + 2\dot{\mathbf{o}} + \ddot{\mathbf{o}}^2 \end{aligned} \tag{40}$$

$$\mathbf{P\_4} = \mathbf{D^3} + \ddot{\mathbf{o}} \mathbf{D^2} + \left( 3\dot{\ddot{\mathbf{o}}} + \ddot{\mathbf{o}}^2 \right) \mathbf{D} + 3\ddot{\ddot{\mathbf{o}}} + 3\dot{\ddot{\mathbf{o}}} \ddot{\mathbf{o}} + 2\ddot{\mathbf{o}} \dot{\ddot{\mathbf{o}}} + \ddot{\mathbf{o}}^3$$

Thus, it follows:

• the velocity field invariants

$$\begin{cases} \mathbf{a}\_1 = \mathbf{v} \\ \clubsuit \mathbf{0}\_1 = \mathbf{\tilde{o}} \end{cases} \tag{41}$$

• the acceleration field invariants

$$\begin{cases} \mathbf{a}\_2 = \dot{\mathbf{v}} + \ddot{\mathbf{o}} \mathbf{v} \\\\ \boldsymbol{\Phi}\_2 = \dot{\ddot{\mathbf{o}}} + \ddot{\mathbf{o}}^2 \end{cases} \tag{42}$$

• jerk field invariants

$$\begin{cases} \mathbf{a}\_3 = \ddot{\mathbf{v}} + \ddot{\mathbf{o}}\mathbf{v} + 2\dot{\ddot{\mathbf{o}}}\mathbf{v} + \ddot{\mathbf{o}}^2 \mathbf{v} \\\\ \boldsymbol{\Phi}\_3 = \ddot{\ddot{\mathbf{o}}} + \ddot{\mathbf{o}}\dot{\ddot{\mathbf{o}}} + 2\dot{\ddot{\mathbf{o}}}\ddot{\mathbf{o}} + \ddot{\mathbf{o}}^3 \end{cases} \tag{43}$$

• hyper-jerk (jounce) field invariants

$$\begin{cases} \mathbf{a}\_{4} = \ddot{\mathbf{v}} + \ddot{\mathbf{o}}\ddot{\mathbf{v}} + \left( 3\dot{\ddot{\mathbf{o}}} + \ddot{\mathbf{o}}^{2} \right) \dot{\mathbf{v}} + 3\ddot{\ddot{\mathbf{o}}} \mathbf{v} + 3\dot{\ddot{\mathbf{o}}} \ddot{\mathbf{o}} \mathbf{v} + 2\ddot{\mathbf{o}} \dot{\ddot{\mathbf{o}}} \mathbf{v} + \ddot{\mathbf{o}}^{3} \mathbf{v} \\\\ \boldsymbol{\Phi}\_{4} = \ddot{\ddot{\mathbf{o}}} + \ddot{\mathbf{o}} \ddot{\ddot{\mathbf{o}}} + \left( 3\dot{\ddot{\mathbf{o}}} + \ddot{\mathbf{o}}^{2} \right) \dot{\ddot{\mathbf{o}}} + 3\ddot{\ddot{\mathbf{o}}} \ddot{\mathbf{o}} + 3\dot{\ddot{\mathbf{o}}} \ddot{\mathbf{o}}^{2} + 2\ddot{\mathbf{o}} \dot{\ddot{\mathbf{o}}} \ddot{\mathbf{o}} + \ddot{\mathbf{o}}^{4} \end{cases} \tag{44}$$

**Remark 5.** *The higher-order time derivative of spatial twist solve completely the problem of determining the field of the nth order acceleration of rigid motion*.

#### **4.1 Higher-order acceleration center and vector invariants of rigid body motion**

Equation (16) may be written as

$$\mathbf{a}\_{\rho}^{[n]} - \phi\_n \mathfrak{p} = \mathbf{a}\_Q^{[n]} - \Phi\_n \mathfrak{p}\_Q, n \in \mathbb{N}^\*.\tag{45}$$

This shows us that the vector function

$$\mathbf{I}\_{\mathbf{n}} = \mathbf{a}\_{\rho}^{[\mathbf{n}]} - \boldsymbol{\Phi}\_{\mathbf{n}} \boldsymbol{\rho}, \mathbf{n} \in \mathbb{N}^\* \tag{46}$$

knows, still an open problem in theoretical kinematics field. We will propose a method based on the tensors algebra that will give a closed form, coordinate- free

*div a*½ � *<sup>n</sup>*

*curl a*½ � *<sup>n</sup>*

**Theorem 4.** *The adjugate tensor and determinant of the tensor* **Φ** *is*:

Let **<sup>Φ</sup>**<sup>n</sup> the n-th order acceleration tensor, **<sup>Φ</sup>**<sup>n</sup> <sup>¼</sup> <sup>~</sup>**t***<sup>n</sup>* <sup>þ</sup> **<sup>S</sup>***n:*

**<sup>t</sup>**<sup>n</sup>þ<sup>1</sup> <sup>¼</sup> **<sup>t</sup>**\_<sup>n</sup> <sup>þ</sup>

8 ><

>:

• Velocity field: **Φ**<sup>1</sup> ¼ **ω**~, **t**<sup>1</sup> ¼ **ω**, **S1** ¼ **0**

**Φ**<sup>1</sup> is singular for any **ω**. In this case,

• Acceleration field: **<sup>Φ</sup>**<sup>2</sup> <sup>¼</sup> **<sup>ω</sup>**~<sup>2</sup> <sup>þ</sup> **<sup>ω</sup>**~\_ ,**t**<sup>2</sup> <sup>¼</sup> **<sup>ω</sup>**\_ , **<sup>S</sup>**<sup>2</sup> <sup>¼</sup> **<sup>ω</sup>**~<sup>2</sup>

**Φ**<sup>∗</sup>

**Φ**�<sup>1</sup> <sup>2</sup> <sup>¼</sup> **<sup>ω</sup>**<sup>~</sup> g<sup>2</sup>

**Φ**<sup>2</sup> is nonsingular if and only if **ω** � **ω**\_ 6¼ **0**. In this case

**t**<sup>1</sup> ¼ **ω**

(

The vector field of the higher-order acceleration is a non-stationary vector field. Differential operator *div* and *curl* is expressed, taking into account Eq. (47), through

*<sup>ϱ</sup>* ¼ *trace* **Φ***<sup>n</sup>*

Let **Φ ∈ L V**ð Þ **<sup>3</sup>**, **V3** a tensor and we note **t** ¼ vect**Φ** and **S** ¼ sym**Φ**. The below

**<sup>Φ</sup>**<sup>∗</sup> <sup>¼</sup> **<sup>S</sup>**<sup>∗</sup> � **St**<sup>~</sup> <sup>þ</sup> **<sup>t</sup> <sup>⊗</sup> <sup>t</sup>**

The vectors **t***<sup>n</sup>* and the symmetric tensors **S***n*, *n* ∈ <sup>∗</sup> can be obtained with the

**<sup>S</sup>**<sup>n</sup>þ<sup>1</sup> <sup>¼</sup> **<sup>S</sup>**\_ <sup>n</sup> <sup>þ</sup> symð Þ **<sup>Φ</sup>**n**ω**<sup>~</sup>

<sup>1</sup> ¼ **ω ⊗ ω**

<sup>2</sup> � **<sup>ω</sup>**<sup>~</sup> g<sup>2</sup>

**ω**\_ � ð Þ **ω ⊗ ω**

**ω**\_ þ **ω**\_ ⊗ **ω**\_

<sup>2</sup> � **<sup>ω</sup>**\_ <sup>⊗</sup> **<sup>ω</sup>**\_

det**Φ**<sup>2</sup> ¼ �ð Þ **<sup>ω</sup>** � **<sup>ω</sup>**\_ <sup>2</sup> (58)

ð Þ **<sup>ω</sup>** � **<sup>ω</sup>**\_ <sup>2</sup> (59)

<sup>2</sup> ð Þ trace**Φ**<sup>n</sup> **<sup>I</sup>** � **<sup>Φ</sup>**<sup>T</sup>

� �**ω**

1

**S1** ¼ **0**

**Φ**<sup>∗</sup>

div **a** ½ � 1 <sup>ϱ</sup> ¼ 0

curl **a** ½ � 1 <sup>ϱ</sup> ¼ 2**ω**

<sup>2</sup> ¼ ð Þ **ω ⊗ ω**

*<sup>ϱ</sup>* ¼ 2*vect* **Φ***<sup>n</sup>* (52)

det**<sup>Φ</sup>** <sup>¼</sup> det**<sup>S</sup>** <sup>þ</sup> **tSt** (53)

**n**

det**Φ**<sup>1</sup> <sup>¼</sup> <sup>0</sup> (56)

(54)

(55)

(57)

solution, dependent to the time derivative of spatial twist.

the linear invariants of the tensor **Φ***n*, as below:

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

theorem takes place.

below recurrence relation:

It follows that:

**111**

has the same value in every point of the rigid body under the general spatial motion, at a given moment of time *t*. It represents a **vector invariant** of the n-th order acceleration field.

The invariant value of vector **I**<sup>n</sup> is obtained for **ρ** ¼ **0** and it is the n-th order acceleration of the point of the rigid body that passes the origin of the fixed reference frame at a given moment of time: **I**<sup>n</sup> ¼ **a** ½ � **n** <sup>0</sup> ≝**a**n. Eq. (46) becomes:

$$\mathbf{a}\_{\mathfrak{p}}^{[\mathfrak{n}]} = \mathbf{a}\_{\mathfrak{n}} + \phi\_{\mathfrak{n}} \mathfrak{p}. \tag{47}$$

Let be **Φ<sup>∗</sup> <sup>n</sup>** be the adjugate tensor of **Φ<sup>n</sup>** uniquely defined by:**ΦnΦ<sup>∗</sup> <sup>n</sup>** ¼ ð Þ det**Φ<sup>n</sup> I***:* From Eq. (46), results another invariant

$$\mathbf{J}\_{\mathbf{n}} = \boldsymbol{\Phi}\_{\mathbf{n}}^{\*} \mathbf{a}\_{\boldsymbol{\rho}}^{[\mathbf{n}]} - (\det \boldsymbol{\Phi}\_{\mathbf{n}}) \boldsymbol{\rho}, \mathbf{n} \in \mathbb{N}^{\*}.\tag{48}$$

The value of this invariant is **<sup>J</sup>**<sup>n</sup> <sup>¼</sup> **<sup>Φ</sup><sup>∗</sup>** <sup>n</sup> **a**n*:*

In the specific case when tensor **Φ<sup>n</sup>** is non-singular (det**Φ<sup>n</sup>** 6¼ 0), from (47) results the position vector having an imposed n-th order acceleration **a <sup>∗</sup>**:

$$\boldsymbol{\mathfrak{p}}^{\*} = \boldsymbol{\Phi}\_{\mathrm{n}}^{-1}(\mathbf{a}^{\*} - \mathbf{a}\_{\mathrm{n}}), \mathbf{n} \in \mathbb{N}^{\*}.\tag{49}$$

In a particular case of the **n-th order acceleration center** Gn (i.e. the point that have **a <sup>∗</sup>** ¼ **0**) on obtain:

$$\boldsymbol{\mathfrak{p}}\_{\mathcal{G}\_{\mathbf{n}}} = -\boldsymbol{\Phi}\_{\mathbf{n}}^{-1} \mathbf{a}\_{\mathbf{n}} \tag{50}$$

Assuming that the tensor **Φ<sup>n</sup>** is non-singular, the previous relations lead to a new vector invariant that characterize the accelerations of n-th and m-th order (n, m **∈ <sup>∗</sup>** ):

$$\mathbf{K}\_{\mathbf{m},\mathbf{n}} = \mathbf{a}\_{\rho}^{[\mathbf{m}]} - \boldsymbol{\Phi}\_{\mathbf{m}} \boldsymbol{\Phi}\_{\mathbf{n}}^{-1} \mathbf{a}\_{\rho}^{[\mathbf{n}]}, \mathbf{m}, \mathbf{n} \in \mathbb{N}^\*. \tag{51}$$

The value of this invariant is **<sup>K</sup>**m,n <sup>¼</sup> **<sup>a</sup>**<sup>m</sup> � **<sup>Φ</sup>**m**Φ**�<sup>1</sup> <sup>n</sup> **a**n.

The problem of the determination the adjugate tensor of the n-th acceleration tensor and the conditions in which these tensors are inversable is, as the author

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

**<sup>a</sup>**<sup>4</sup> <sup>¼</sup> **<sup>v</sup>**€ <sup>þ</sup> **<sup>ω</sup>**~**v**€ <sup>þ</sup> <sup>3</sup>**ω**~\_ <sup>þ</sup> **<sup>ω</sup>**~<sup>2</sup> � �

*Advances on Tensor Analysis and Their Applications*

8 ><

>:

**<sup>Φ</sup>**<sup>4</sup> <sup>¼</sup> **<sup>ω</sup>**€~ <sup>þ</sup> **<sup>ω</sup>**~**ω**€~ <sup>þ</sup> <sup>3</sup>**ω**~\_ <sup>þ</sup> **<sup>ω</sup>**~<sup>2</sup> � �

Equation (16) may be written as

order acceleration field.

Let be **Φ<sup>∗</sup>**

have **a <sup>∗</sup>** ¼ **0**) on obtain:

):

(n, m **∈ <sup>∗</sup>**

**110**

This shows us that the vector function

reference frame at a given moment of time: **I**<sup>n</sup> ¼ **a**

From Eq. (46), results another invariant

The value of this invariant is **<sup>J</sup>**<sup>n</sup> <sup>¼</sup> **<sup>Φ</sup><sup>∗</sup>**

**<sup>J</sup>**<sup>n</sup> <sup>¼</sup> **<sup>Φ</sup>**<sup>∗</sup>

**a**½ � *<sup>n</sup>*

*<sup>ρ</sup>* � ϕ*n***ρ** ¼ **a**

**<sup>I</sup>**<sup>n</sup> <sup>¼</sup> **<sup>a</sup>**½ � **<sup>n</sup>**

**a**½ � *<sup>n</sup>*

<sup>n</sup> **a**½ � <sup>n</sup>

results the position vector having an imposed n-th order acceleration **a <sup>∗</sup>**:

**<sup>ρ</sup>**<sup>∗</sup> <sup>¼</sup> **<sup>Φ</sup>**�<sup>1</sup>

**<sup>K</sup>**m,n <sup>¼</sup> **<sup>a</sup>**½ � <sup>m</sup>

The value of this invariant is **<sup>K</sup>**m,n <sup>¼</sup> **<sup>a</sup>**<sup>m</sup> � **<sup>Φ</sup>**m**Φ**�<sup>1</sup>

**<sup>v</sup>**\_ <sup>þ</sup> <sup>3</sup>**ω**€~**<sup>v</sup>** <sup>þ</sup> <sup>3</sup>**ω**~\_ **<sup>ω</sup>**~**<sup>v</sup>** <sup>þ</sup> <sup>2</sup>**ω**~**ω**~\_ **<sup>v</sup>** <sup>þ</sup> **<sup>ω</sup>**~<sup>3</sup>

**<sup>ω</sup>**~\_ <sup>þ</sup> <sup>3</sup>**ω**€~**ω**<sup>~</sup> <sup>þ</sup> <sup>3</sup>**ω**~\_ **<sup>ω</sup>**~<sup>2</sup> <sup>þ</sup> <sup>2</sup>**ω**~**ω**~\_ **<sup>ω</sup>**<sup>~</sup> <sup>þ</sup> **<sup>ω</sup>**<sup>~</sup> <sup>4</sup>

*<sup>Q</sup>* � **<sup>Φ</sup>***n***ρ**<sup>Q</sup> , *<sup>n</sup>* <sup>∈</sup> <sup>∗</sup> *:* (45)

**<sup>ρ</sup>** � **<sup>Φ</sup>nρ**, n<sup>∈</sup> <sup>∗</sup> (46)

**<sup>ρ</sup>** ¼ **a**<sup>n</sup> þ ϕ*n***ρ***:* (47)

**<sup>ρ</sup>** � ð Þ det**Φ**<sup>n</sup> **<sup>ρ</sup>**, n<sup>∈</sup> <sup>∗</sup> *:* (48)

<sup>n</sup> **<sup>a</sup>** <sup>∗</sup> ð Þ � **<sup>a</sup>**<sup>n</sup> , n<sup>∈</sup> <sup>∗</sup> *:* (49)

<sup>n</sup> **a**<sup>n</sup> (50)

**<sup>ρ</sup>** , m, n<sup>∈</sup> <sup>∗</sup> *:* (51)

<sup>0</sup> ≝**a**n. Eq. (46) becomes:

**<sup>n</sup>** ¼ ð Þ det**Φ<sup>n</sup> I***:*

**Remark 5.** *The higher-order time derivative of spatial twist solve completely the*

**4.1 Higher-order acceleration center and vector invariants of rigid body motion**

½ � *n*

has the same value in every point of the rigid body under the general spatial motion, at a given moment of time *t*. It represents a **vector invariant** of the n-th

The invariant value of vector **I**<sup>n</sup> is obtained for **ρ** ¼ **0** and it is the n-th order acceleration of the point of the rigid body that passes the origin of the fixed

**<sup>n</sup>** be the adjugate tensor of **Φ<sup>n</sup>** uniquely defined by:**ΦnΦ<sup>∗</sup>**

<sup>n</sup> **a**n*:* In the specific case when tensor **Φ<sup>n</sup>** is non-singular (det**Φ<sup>n</sup>** 6¼ 0), from (47)

In a particular case of the **n-th order acceleration center** Gn (i.e. the point that

Assuming that the tensor **Φ<sup>n</sup>** is non-singular, the previous relations lead to a new

<sup>n</sup> **a**½ � <sup>n</sup>

<sup>n</sup> **a**n.

**<sup>ρ</sup>**Gn ¼ �**Φ**�<sup>1</sup>

vector invariant that characterize the accelerations of n-th and m-th order

**<sup>ρ</sup>** � **<sup>Φ</sup>**m**Φ**�<sup>1</sup>

The problem of the determination the adjugate tensor of the n-th acceleration tensor and the conditions in which these tensors are inversable is, as the author

½ � **n**

*problem of determining the field of the nth order acceleration of rigid motion*.

**v**

(44)

knows, still an open problem in theoretical kinematics field. We will propose a method based on the tensors algebra that will give a closed form, coordinate- free solution, dependent to the time derivative of spatial twist.

The vector field of the higher-order acceleration is a non-stationary vector field. Differential operator *div* and *curl* is expressed, taking into account Eq. (47), through the linear invariants of the tensor **Φ***n*, as below:

$$\operatorname{div}\mathfrak{a}\_{\emptyset}^{[n]} = \operatorname{trace}\,\Phi\_n$$

$$\operatorname{curl}\mathfrak{a}\_{\emptyset}^{[n]} = 2\operatorname{vect}\,\Phi\_n\tag{52}$$

Let **Φ ∈ L V**ð Þ **<sup>3</sup>**, **V3** a tensor and we note **t** ¼ vect**Φ** and **S** ¼ sym**Φ**. The below theorem takes place.

**Theorem 4.** *The adjugate tensor and determinant of the tensor* **Φ** *is*:

$$\begin{aligned} \boldsymbol{\Phi}^\* &= \mathbf{S}^\* - \tilde{\mathbf{S}}\mathbf{t} + \mathbf{t} \otimes \mathbf{t} \\ \text{det}\boldsymbol{\Phi} &= \text{det}\mathbf{S} + \mathbf{t}\mathbf{S}\mathbf{t} \end{aligned} \tag{53}$$

Let **<sup>Φ</sup>**<sup>n</sup> the n-th order acceleration tensor, **<sup>Φ</sup>**<sup>n</sup> <sup>¼</sup> <sup>~</sup>**t***<sup>n</sup>* <sup>þ</sup> **<sup>S</sup>***n:*

The vectors **t***<sup>n</sup>* and the symmetric tensors **S***n*, *n* ∈ <sup>∗</sup> can be obtained with the below recurrence relation:

$$\begin{cases} \mathbf{t}\_{\mathbf{n}+1} = \dot{\mathbf{t}}\_{\mathbf{n}} + \frac{1}{2} \left[ (\text{trace} \boldsymbol{\Phi}\_{\mathbf{n}}) \mathbf{I} - \boldsymbol{\Phi}\_{\mathbf{n}}^{T} \right] \boldsymbol{\Phi} \\\\ \mathbf{t}\_{1} = \boldsymbol{\Phi} \end{cases} \tag{54}$$

$$\begin{cases} \mathbf{S\_{n+1}} = \dot{\mathbf{S\_n}} + \text{sym}(\boldsymbol{\Phi\_n}\tilde{\boldsymbol{\phi}}) \\ \mathbf{S\_I} = \mathbf{0} \end{cases} \tag{55}$$

It follows that:

• Velocity field: **Φ**<sup>1</sup> ¼ **ω**~, **t**<sup>1</sup> ¼ **ω**, **S1** ¼ **0**

$$\begin{aligned} \clubsuit \spadesuit^\* &= \spadesuit \oplus \spadesuit\\ \clubsuit \spadesuit &= \mathbf{0} \end{aligned} \tag{56}$$

**Φ**<sup>1</sup> is singular for any **ω**. In this case,

$$\begin{aligned} \operatorname{div} \mathbf{a}\_{\mathbb{Q}}^{[1]} &= \mathbf{0} \\ \operatorname{curl} \mathbf{a}\_{\mathbb{Q}}^{[1]} &= \mathbf{2} \mathbf{0} \end{aligned} \tag{57}$$

• Acceleration field: **<sup>Φ</sup>**<sup>2</sup> <sup>¼</sup> **<sup>ω</sup>**~<sup>2</sup> <sup>þ</sup> **<sup>ω</sup>**~\_ ,**t**<sup>2</sup> <sup>¼</sup> **<sup>ω</sup>**\_ , **<sup>S</sup>**<sup>2</sup> <sup>¼</sup> **<sup>ω</sup>**~<sup>2</sup>

$$\begin{aligned} \boldsymbol{\Phi}\_{2}^{\*} &= \left(\boldsymbol{\mathfrak{o}} \otimes \boldsymbol{\mathfrak{o}}\right)^{2} - \widehat{\boldsymbol{\mathfrak{o}}}^{2} \boldsymbol{\breve{\mathfrak{o}}} + \boldsymbol{\mathring{\mathfrak{o}}} \otimes \boldsymbol{\mathring{\mathfrak{o}}} \\ &\quad \det \boldsymbol{\Phi}\_{2} = -\left(\boldsymbol{\mathfrak{o}} \times \boldsymbol{\mathring{\mathfrak{o}}}\right)^{2} \end{aligned} \tag{58}$$

**Φ**<sup>2</sup> is nonsingular if and only if **ω** � **ω**\_ 6¼ **0**. In this case

$$\boldsymbol{\Phi}\_{2}^{-1} = \frac{\overbrace{\boldsymbol{\Phi}^{2}\boldsymbol{\phi}} - \left(\boldsymbol{\alpha}\otimes\boldsymbol{\alpha}\right)^{2} - \boldsymbol{\phi}\otimes\boldsymbol{\phi}}{\left(\boldsymbol{\alpha}\times\boldsymbol{\phi}\right)^{2}}\tag{59}$$

*Advances on Tensor Analysis and Their Applications*

$$\operatorname{div}\mathbf{a}\_{\mathbf{q}}^{[2]} = -\mathbf{2}\mathbf{o}^{2}$$

$$\operatorname{curl}\mathbf{a}\_{\mathbf{q}}^{[2]} = \mathbf{2}\mathbf{\dot{o}}.\tag{60}$$

In the hypothesis (68), there is jounce center, determined by

div **a**½ � <sup>4</sup>

vectors and dual tensors. More details can be found in [10–25].

Let the set of real dual numbers to be denoted by

curl **a**½ � <sup>4</sup>

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

**5. Dual algebra in rigid body kinematics**

**5.1 Dual numbers**

S⊂ such that:

**5.2 Dual vectors**

called unit dual vector.

**5.3 Dual tensors**

**113**

**<sup>ρ</sup>**G4 ¼ �**Φ**�<sup>1</sup>

<sup>4</sup> **a**<sup>4</sup>

<sup>ϱ</sup> <sup>¼</sup> <sup>2</sup>**ω**€ <sup>þ</sup> <sup>2</sup>**ω**€ � **<sup>ω</sup>** � <sup>4</sup>**ω<sup>2</sup>ω**\_ � <sup>8</sup>ð Þ **<sup>ω</sup>** � **<sup>ω</sup>**\_ **<sup>ω</sup>**

In this section, we will present some algebraic properties for dual numbers, dual

<sup>¼</sup> <sup>a</sup> <sup>þ</sup> <sup>ε</sup>a0 a, a0 <sup>∈</sup> , <sup>ε</sup><sup>2</sup> <sup>¼</sup> 0, <sup>ε</sup> 6¼ <sup>0</sup> �

f : S⊂ ! ; fð Þ¼ a f að Þþ εa0f

following expressions: cos a ¼ cosa � εa0sina; sin a ¼ sina þ εa0cosa;

denoted by **V**3. The ensemble of dual vectors is defined as:

**a** ¼

8 ><

>:

Based on the previous property, two of the most important functions have the

In the Euclidean space, the linear space of free vectors with dimension 3 will be

where **a** ¼ Re ð Þ**a** is the real part of **a** and **a**<sup>0</sup> ¼ Duð Þ**a** the dual part. For any three dual vectors **a**, **b**, **c**, the following notations will be used for the basic products: **a** � **b—**scalar product, **a** � **b—**cross product and h i **a**, **b**, **c** ¼ **a** � ð Þ **b** � **c** —triple scalar product. Regarding algebraic structure, **V**<sup>3</sup> ð Þ , þ , � is a free -module [13, 14]. The magnitude of j j **a** , denoted by j j **a** , is the dual number computed from

> **a**<sup>0</sup> � **a** k k**a**

where k k*:* is the Euclidean norm. For any dual vector **a**∈ **V**3, if j j **a** ¼ 1 then **a** is

An -linear application of **V**<sup>3</sup> into **V**<sup>3</sup> is called an Euclidean dual tensor:

k k**a** þ ε

**<sup>V</sup>**<sup>3</sup> <sup>¼</sup> **<sup>a</sup>** <sup>þ</sup> <sup>ε</sup>**a**0; **<sup>a</sup>**, **<sup>a</sup>**<sup>0</sup> **<sup>∈</sup> <sup>V</sup>**3, <sup>ε</sup><sup>2</sup> <sup>¼</sup> 0, <sup>ε</sup> 6¼ <sup>0</sup> � � (73)

, Re ð Þ**a** 6¼ **0**

εk k **a**<sup>0</sup> , Re ð Þ¼ **a 0**

where a ¼ Re ð Þa is the real part of a and a0 ¼ Duð Þa the dual part. The sum and product between dual numbers generate a ring with zero divisors structure for . Any differentiable function f : S⊂ ! , f ¼ f að Þ can be completely defined on

<sup>ϱ</sup> ¼ �2 4**<sup>ω</sup>** � **<sup>ω</sup>**€ <sup>þ</sup> <sup>3</sup>**ω**\_ **<sup>2</sup>** <sup>þ</sup> **<sup>ω</sup><sup>4</sup>** � � (70)

� � � (71)

0

ð Þa (72)

(74)

\* Jerk féld:  $\Phi\_{3} = \ddot{\mathbf{\ddot{o}}} + 2\dot{\mathbf{o}}\,\ddot{\mathbf{o}} + \dot{\mathbf{o}}\,\ddot{\mathbf{o}}\,\ddot{\mathbf{o}} + \dot{\mathbf{o}}\,^{3},$   $\mathbf{t}\_{3} = \ddot{\mathbf{o}} + \frac{1}{2}\dot{\mathbf{o}} \times \boldsymbol{\alpha} - \mathbf{o}^{2}\boldsymbol{\alpha},$   $\mathbf{S}\_{3} = \frac{3}{2}\left[\ddot{\mathbf{o}}\,\ddot{\mathbf{o}}\,\ddot{\mathbf{o}} + \dot{\mathbf{o}}\,\ddot{\mathbf{o}}\,\ddot{\mathbf{o}}\right],$  
$$\boldsymbol{\Phi}\_{3}^{\*} = \frac{9}{4}\left[(\boldsymbol{\alpha}\otimes\dot{\boldsymbol{\alpha}})^{2} + (\dot{\mathbf{o}}\otimes\boldsymbol{\alpha})^{2} + (\boldsymbol{\alpha}\times\dot{\boldsymbol{\alpha}})\otimes(\dot{\boldsymbol{\alpha}}\times\boldsymbol{\alpha})\right] - \widehat{\mathbf{S}\_{3}\mathbf{t}\_{3}} + \mathbf{t}\_{3}\otimes\mathbf{t}\_{3}$$

$$\det \Phi\_3 = \frac{\mathbf{1}\mathbf{2}(\mathbf{t}\_3 \times \dot{\mathbf{o}})(\mathbf{o} \times \mathbf{t}\_3) + \mathbf{27}\mathbf{o} \cdot \dot{\mathbf{o}}(\mathbf{o} \times \dot{\mathbf{o}})}{4}.\tag{61}$$

**<sup>Φ</sup>**<sup>3</sup> is nonsingular if and only if 4ð Þ **<sup>ω</sup>**\_ � **<sup>t</sup>**<sup>3</sup> ð Þ **<sup>ω</sup>** � **<sup>t</sup>**<sup>3</sup> 6¼ <sup>9</sup>**<sup>ω</sup>** � **ω ω** \_ ð Þ � **<sup>ω</sup>**\_ <sup>2</sup> . In this case

$$\boldsymbol{\Phi}\_{3}^{-1} = -\frac{\mathcal{B}\left[\left(\boldsymbol{\alpha}\otimes\dot{\boldsymbol{\alpha}}\right)^{2} + \left(\dot{\boldsymbol{\alpha}}\otimes\boldsymbol{\alpha}\right)^{2} + \left(\boldsymbol{\alpha}\times\dot{\boldsymbol{\alpha}}\right)\otimes\left(\dot{\boldsymbol{\alpha}}\times\boldsymbol{\alpha}\right)\right] - 4\boldsymbol{\widetilde{\mathcal{S}}\_{3}\mathbf{t}\_{3}} + 4\mathbf{t}\_{3}\otimes\mathbf{t}\_{3}}{\mathbf{1}2\left(\mathbf{t}\_{3}\times\dot{\boldsymbol{\alpha}}\right)\left(\boldsymbol{\alpha}\times\mathbf{t}\_{3}\right) + 2\boldsymbol{\mathcal{T}}\boldsymbol{\alpha}\cdot\dot{\boldsymbol{\alpha}}\left(\boldsymbol{\alpha}\times\dot{\boldsymbol{\alpha}}\right)^{2}}$$

$$\text{div }\mathbf{a}\_{\boldsymbol{\alpha}}^{[3]} = -\boldsymbol{\delta}\cdot\boldsymbol{\alpha}\cdot\dot{\boldsymbol{\alpha}}$$

$$\text{curl }\mathbf{a}\_{\boldsymbol{\alpha}}^{[3]} = 2\ddot{\boldsymbol{\alpha}} + \dot{\boldsymbol{\alpha}}\times\boldsymbol{\alpha} - 2\boldsymbol{\alpha}^{2}\boldsymbol{\alpha}. \tag{62}$$

• Jounce field:

$$\mathbf{O}\_4 = \ddot{\mathbf{\ddot{o}}} + \ddot{\mathbf{o}}\ddot{\mathbf{o}}\ddot{\mathbf{o}} + \left(3\dot{\mathbf{o}} + \dot{\mathbf{o}}^2\right)\dot{\mathbf{o}} + 3\ddot{\mathbf{o}}\mathbf{o}\mathbf{o} + 3\dot{\mathbf{o}}\mathbf{o}^2 + 2\ddot{\mathbf{o}}\dot{\mathbf{o}}\mathbf{o}\mathbf{o} + \ddot{\mathbf{o}}^4$$

$$\mathbf{t}\_4 = \ddot{\mathbf{o}}\mathbf{i} + \ddot{\mathbf{o}}\times\mathbf{o} - 2\mathbf{o}^2\dot{\mathbf{o}} - 4(\mathbf{o}\cdot\dot{\mathbf{o}})\mathbf{o}$$

$$\mathbf{S}\_4 = 2\text{sym}\left(2\ddot{\mathbf{o}}\ddot{\mathbf{o}} + \dot{\mathbf{o}}\dot{\mathbf{o}}^2\right) + 3\dot{\mathbf{o}}^2 + \ddot{\mathbf{o}}^4\tag{63}$$

$$\mathbf{S}\_4^\* = 2 \text{sym} \left[ \mathbf{\hat{z}} \dot{\mathbf{o}} (\mathbf{o} \otimes \mathbf{w}) \dot{\mathbf{o}} - \mathbf{o} \mathbf{o} \otimes \mathbf{w} \right] - \ddot{\mathbf{o}} (\mathbf{w} \otimes \mathbf{w}) \ddot{\mathbf{o}} - \mathbf{ja} \dot{\mathbf{o}} \otimes \dot{\mathbf{o}} + \mathbf{a} \mathbf{o}^4 \mathbf{I} \tag{64}$$

$$
\Phi\_4^\* = \mathbf{S}\_4^\* - \widehat{\mathbf{S}\_4}\overline{\mathbf{t}\_4} + \mathbf{t}\_4 \otimes \mathbf{t}\_4 \tag{65}
$$

$$\det \Phi\_4 = \alpha \left[ \mathsf{6} (\mathsf{a} \times \mathsf{do}) \cdot (\mathbf{w} \times \mathsf{do}) - (\mathsf{a} \times \mathbf{w})^2 + 2\mathsf{ao} \mathbf{o} \cdot \mathbf{w} + 3\mathsf{ao} \mathsf{b}^2 + \mathsf{a}^2 \right] - \mathsf{3} (\mathsf{do}, \mathsf{o}, \mathbf{w})^2 \tag{66}$$

In Eqs. (65) and (66), the following notation has been used:

$$\begin{aligned} \mathbf{w} &= \boldsymbol{\bullet} - \boldsymbol{\bullet} \times \boldsymbol{\bullet} - \boldsymbol{\bullet}^2 \boldsymbol{\bullet} \\ \boldsymbol{\alpha} &= \boldsymbol{\bullet}^4 - 2\boldsymbol{\bullet}^2 - 2\boldsymbol{\bullet} \cdot \boldsymbol{\bullet} \end{aligned} \tag{67}$$

If

$$\mathcal{B}(\mathsf{do}, \mathsf{o}, \mathbf{w})^2 \neq \mathfrak{a} \left[ \mathsf{6} (\mathsf{o} \times \mathsf{do}) \cdot (\mathbf{w} \times \mathsf{do}) - (\mathsf{o} \times \mathbf{w})^2 + 2\mathsf{o}\mathsf{o}\mathsf{o} \cdot \mathbf{w} + 2\mathsf{o}\mathsf{o}^2 + \mathsf{a}^2 \right] \tag{68}$$

then **Φ**<sup>4</sup> is inversible and

$$\Phi\_4^{-1} = \frac{\Phi\_4^\*}{\det \Phi\_4} \tag{69}$$

In the hypothesis (68), there is jounce center, determined by

$$\mathfrak{p}\_{\mathbf{G}\_{4}} = -\mathfrak{D}\_{4}^{-1}\mathbf{a}\_{4}$$

$$\operatorname{div}\,\mathbf{a}\_{\mathbf{q}}^{[4]} = -2\left(4\mathfrak{o}\cdot\mathring{\mathfrak{o}} + 3\mathring{\mathfrak{o}}^{2} + \mathfrak{o}^{4}\right)\tag{70}$$

$$\operatorname{curl}\,\mathbf{a}\_{\mathbf{q}}^{[4]} = 2\ddot{\mathfrak{o}} + 2\ddot{\mathfrak{o}}\times\mathfrak{o} - 4\mathfrak{o}^{2}\dot{\mathfrak{o}} - 8(\mathfrak{o}\cdot\mathring{\mathfrak{o}})\mathfrak{o}$$

#### **5. Dual algebra in rigid body kinematics**

In this section, we will present some algebraic properties for dual numbers, dual vectors and dual tensors. More details can be found in [10–25].

#### **5.1 Dual numbers**

div **a**½ � <sup>2</sup>

• Jerk field: **<sup>Φ</sup>**<sup>3</sup> <sup>¼</sup> **<sup>ω</sup>**€~ <sup>þ</sup> <sup>2</sup>**ω**~\_ **<sup>ω</sup>**<sup>~</sup> <sup>þ</sup> **<sup>ω</sup>**~**ω**~\_ <sup>þ</sup> **<sup>ω</sup>**~<sup>3</sup>

*Advances on Tensor Analysis and Their Applications*

<sup>4</sup> ð Þ **<sup>ω</sup> <sup>⊗</sup> <sup>ω</sup>**\_ <sup>2</sup> <sup>þ</sup> ð Þ **<sup>ω</sup>**\_ <sup>⊗</sup> **<sup>ω</sup>**

<sup>9</sup> ð Þ **<sup>ω</sup> <sup>⊗</sup> <sup>ω</sup>**\_ <sup>2</sup> <sup>þ</sup> ð Þ **<sup>ω</sup>**\_ <sup>⊗</sup> **<sup>ω</sup>**

,

**3**

**<sup>2</sup> <sup>ω</sup>**~**ω**~\_ <sup>þ</sup> **<sup>ω</sup>**~\_ **<sup>ω</sup>**<sup>~</sup> h i

> **Φ**<sup>∗</sup> <sup>3</sup> <sup>¼</sup> <sup>9</sup>

**Φ**�<sup>1</sup> <sup>3</sup> ¼¼

• Jounce field:

**S**∗

If

**112**

3ð Þ **ω**\_ , **ω**, **w**

then **Φ**<sup>4</sup> is inversible and

**Φ**<sup>4</sup> ¼**ω**~ *:::*

curl **a**½ � <sup>2</sup>

h i

**<sup>Φ</sup>**<sup>3</sup> is nonsingular if and only if 4ð Þ **<sup>ω</sup>**\_ � **<sup>t</sup>**<sup>3</sup> ð Þ **<sup>ω</sup>** � **<sup>t</sup>**<sup>3</sup> 6¼ <sup>9</sup>**<sup>ω</sup>** � **ω ω** \_ ð Þ � **<sup>ω</sup>**\_ <sup>2</sup>

h i

div **a**½ � <sup>3</sup>

<sup>þ</sup>**ω**€ � **<sup>ω</sup>** � <sup>2</sup>**ω**<sup>2</sup>

<sup>4</sup> � **S**

<sup>2</sup> <sup>þ</sup> <sup>2</sup>α**<sup>ω</sup>** � **<sup>w</sup>** <sup>þ</sup> <sup>3</sup>α**ω**\_ <sup>2</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> h i

**<sup>w</sup>** <sup>¼</sup> **<sup>ω</sup>**€ � **<sup>ω</sup>** � **<sup>ω</sup>**\_ � **<sup>ω</sup><sup>2</sup><sup>ω</sup>**

�<sup>1</sup> <sup>¼</sup> **<sup>Φ</sup>**<sup>∗</sup> 4 det**Φ**<sup>4</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>α**<sup>ω</sup>** � **<sup>w</sup>** <sup>þ</sup> <sup>2</sup>α**ω**\_ <sup>2</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> h i

**<sup>S</sup>**<sup>4</sup> <sup>¼</sup> 2sym 2**ω**€~**ω**<sup>~</sup> <sup>þ</sup> **<sup>ω</sup>**~\_ **<sup>ω</sup>**~<sup>2</sup> � �

curl **a**½ � <sup>3</sup>

<sup>þ</sup>**ω**~**ω**€~ <sup>þ</sup> <sup>3</sup>**ω**~\_ <sup>þ</sup> **<sup>ω</sup>**~<sup>2</sup> � �

**Φ**<sup>∗</sup> <sup>4</sup> <sup>¼</sup> **<sup>S</sup>**<sup>∗</sup>

In Eqs. (65) and (66), the following notation has been used:

<sup>2</sup> 6¼ <sup>α</sup> <sup>6</sup>ð Þ� **<sup>ω</sup>** � **<sup>ω</sup>**\_ ð Þ� **<sup>w</sup>** � **<sup>ω</sup>**\_ ð Þ **<sup>ω</sup>** � **<sup>w</sup>**

**Φ**<sup>4</sup>

**t**<sup>4</sup> ¼**ω** *:::*

h i

<sup>4</sup> <sup>¼</sup> 2sym 3**ω ω** ~\_ ð Þ **<sup>⊗</sup> <sup>w</sup> <sup>ω</sup>**~\_ � <sup>α</sup>**<sup>ω</sup> <sup>⊗</sup> <sup>w</sup>**

det **Φ**<sup>4</sup> ¼ α 6ð Þ� **ω** � **ω**\_ ð Þ� **w** � **ω**\_ ð Þ **ω** � **w**

det**Φ**<sup>3</sup> <sup>¼</sup> <sup>12</sup>ð Þ **<sup>t</sup>**<sup>3</sup> � **<sup>ω</sup>**\_ ð Þþ **<sup>ω</sup>** � **<sup>t</sup>**<sup>3</sup> <sup>27</sup>**<sup>ω</sup>** � **ω ω** \_ ð Þ � **<sup>ω</sup>**\_ <sup>2</sup>

<sup>ϱ</sup> ¼ �2**ω**<sup>2</sup>

,**t**<sup>3</sup> <sup>¼</sup> **<sup>ω</sup>**€ <sup>þ</sup> <sup>1</sup>

<sup>2</sup> <sup>þ</sup> ð Þ **<sup>ω</sup>** � **<sup>ω</sup>**\_ **<sup>⊗</sup>** ð Þ **<sup>ω</sup>**\_ � **<sup>ω</sup>**

<sup>2</sup> <sup>þ</sup> ð Þ **<sup>ω</sup>** � **<sup>ω</sup>**\_ **<sup>⊗</sup>** ð Þ **<sup>ω</sup>**\_ � **<sup>ω</sup>**

<sup>12</sup>ð Þ **<sup>t</sup>**<sup>3</sup> � **<sup>ω</sup>**\_ ð Þþ **<sup>ω</sup>** � **<sup>t</sup>**<sup>3</sup> <sup>27</sup>**<sup>ω</sup>** � **ω ω** \_ ð Þ � **<sup>ω</sup>**\_ <sup>2</sup>

<sup>ϱ</sup> ¼ �6 **ω** � **ω**\_

<sup>ϱ</sup> ¼ 2**ω**\_ *:* (60)

<sup>2</sup> **<sup>ω</sup>**\_ � **<sup>ω</sup>** � **<sup>ω</sup><sup>2</sup>ω**, **S3** <sup>¼</sup>

� **S**

<sup>4</sup> *:* (61)

� 4**S**

<sup>ϱ</sup> <sup>¼</sup> <sup>2</sup>**ω**€ <sup>þ</sup> **<sup>ω</sup>**\_ � **<sup>ω</sup>** � **<sup>2</sup>ω2ω***:* (62)

**<sup>ω</sup>**~\_ <sup>þ</sup> €3**ω**~**ω**<sup>~</sup> <sup>þ</sup> <sup>3</sup>**ω**~\_ **<sup>ω</sup>**~<sup>2</sup> <sup>þ</sup> <sup>2</sup>**ω**~**ω**~\_ **<sup>ω</sup>**<sup>~</sup> <sup>þ</sup> **<sup>ω</sup>**<sup>~</sup> <sup>4</sup>

<sup>α</sup> <sup>¼</sup> **<sup>ω</sup>**<sup>4</sup> � <sup>2</sup>**ω**\_ <sup>2</sup> � <sup>2</sup>**<sup>ω</sup>** � **<sup>ω</sup>**\_ (67)

� **<sup>ω</sup>**~ð Þ **<sup>w</sup>** <sup>⊗</sup> **<sup>w</sup> <sup>ω</sup>**<sup>~</sup> � <sup>3</sup>α**ω**\_ <sup>⊗</sup> **<sup>ω</sup>**\_ <sup>þ</sup> <sup>α</sup>**ω**<sup>4</sup>**<sup>I</sup>** (64)

g4**t**<sup>4</sup> þ **t**<sup>4</sup> ⊗ **t**<sup>4</sup> (65)

**ω**\_ � 4ð Þ **ω** � **ω**\_ **ω**

<sup>þ</sup> <sup>3</sup>**ω**~\_ <sup>2</sup>

g3**t**<sup>3</sup> þ **t**<sup>3</sup> ⊗ **t**<sup>3</sup>

g**3t**<sup>3</sup> þ **4t**<sup>3</sup> **⊗ t**<sup>3</sup>

<sup>þ</sup> **<sup>ω</sup>**<sup>~</sup> <sup>4</sup> (63)

� 3ð Þ **ω**\_ , **ω**, **w**

2

(66)

(68)

(69)

. In this case

Let the set of real dual numbers to be denoted by

$$\underline{\mathbb{R}} = \left\{ \mathbf{a} + \mathbf{e}\mathbf{a}\_0 \, \middle| \, \mathbf{a}, \mathbf{a}\_0 \in \mathbb{R}, \mathbf{e}^2 = \mathbf{0}, \mathbf{e} \neq \mathbf{0} \right\} \tag{71}$$

where a ¼ Re ð Þa is the real part of a and a0 ¼ Duð Þa the dual part. The sum and product between dual numbers generate a ring with zero divisors structure for .

Any differentiable function f : S⊂ ! , f ¼ f að Þ can be completely defined on S⊂ such that:

$$\mathbf{f}: \underline{\sf S} \subset \underline{\sf R} \to \underline{\sf R}; \mathbf{f}(\underline{\sf a}) = \mathbf{f}(\mathbf{a}) + e \mathbf{a}\_0 \mathbf{f}'(\mathbf{a})\tag{72}$$

Based on the previous property, two of the most important functions have the following expressions: cos a ¼ cosa � εa0sina; sin a ¼ sina þ εa0cosa;

#### **5.2 Dual vectors**

In the Euclidean space, the linear space of free vectors with dimension 3 will be denoted by **V**3. The ensemble of dual vectors is defined as:

$$\underline{\mathbf{V}\_3} = \left\{ \mathbf{a} + \varepsilon \mathbf{a}\_0; \mathbf{a}, \mathbf{a}\_0 \in \mathbf{V}\_3, \varepsilon^2 = \mathbf{0}, \mathbf{e} \neq \mathbf{0} \right\} \tag{73}$$

where **a** ¼ Re ð Þ**a** is the real part of **a** and **a**<sup>0</sup> ¼ Duð Þ**a** the dual part. For any three dual vectors **a**, **b**, **c**, the following notations will be used for the basic products: **a** � **b—**scalar product, **a** � **b—**cross product and h i **a**, **b**, **c** ¼ **a** � ð Þ **b** � **c** —triple scalar product. Regarding algebraic structure, **V**<sup>3</sup> ð Þ , þ , � is a free -module [13, 14].

The magnitude of j j **a** , denoted by j j **a** , is the dual number computed from

$$\underline{\mathbf{a}} = \begin{cases} \|\mathbf{a}\| + e \frac{\mathbf{a}\_0 \cdot \mathbf{a}}{\|\mathbf{a}\|}, \text{Re}\left(\underline{\mathbf{a}}\right) \neq \mathbf{0} \\\\ e \|\mathbf{a}\_0\|, \text{Re}\left(\underline{\mathbf{a}}\right) = \mathbf{0} \end{cases} \tag{74}$$

where k k*:* is the Euclidean norm. For any dual vector **a**∈ **V**3, if j j **a** ¼ 1 then **a** is called unit dual vector.

#### **5.3 Dual tensors**

An -linear application of **V**<sup>3</sup> into **V**<sup>3</sup> is called an Euclidean dual tensor:

$$\underline{\mathbf{T}(\underline{\boldsymbol{\lambda}},\underline{\mathbf{v}}\_{1}+\underline{\boldsymbol{\lambda}},\underline{\mathbf{v}}\_{2})} = \underline{\boldsymbol{\lambda}}\_{1}\underline{\mathbf{T}(\underline{\mathbf{v}}\_{1})} + \underline{\boldsymbol{\lambda}}\_{2}\underline{\mathbf{T}(\underline{\mathbf{v}}\_{2})}, \forall \underline{\boldsymbol{\lambda}}\_{1}, \underline{\boldsymbol{\lambda}}\_{2} \in \underline{\mathbb{R}}, \forall \underline{\mathbf{v}}\_{1}, \underline{\mathbf{v}}\_{2} \in \underline{\mathbf{V}}\_{3} \tag{75}$$

Let **L V3**, **V3** ð Þ be the set of dual tensors, then any dual tensor **T** ∈ **L V3**, **V3** ð Þ can be decomposed as **T** ¼ **T** þ ε**T**0, where **T**, **T0** ∈**L V**ð Þ 3, **V**<sup>3</sup> are real tensors. Also, the dual transposed tensor, denoted by **T**T, is defined by

$$
\underline{\mathbf{v}}\_1 \cdot (\underline{\mathbf{T}} \underline{\mathbf{v}}\_1) = \underline{\mathbf{v}}\_2 \cdot (\underline{\mathbf{T}}^T \underline{\mathbf{v}}\_1), \forall \underline{\mathbf{v}}\_1, \underline{\mathbf{v}}\_2 \in \underline{\mathbf{V}}\_3 \tag{76}
$$

Based on Theorems 6 and 7, for any orthogonal dual tensor **R**, a dual vector **ψ** ¼ α **u** ¼ **ψ** þ ε**ψ**<sup>0</sup> can be computed and represents the **screw dual vector**, which

The form of **ψ** implies that **ψ**~ ∈ log **R**. The types of rigid displacements that can

**Theorem 8.** *The natural invariants* α ¼ α þ εd, **u** ¼ **u** þ ε**u**<sup>0</sup> *can be used to directly*

**<sup>R</sup>** <sup>¼</sup> <sup>I</sup> <sup>þ</sup> sinα**u**<sup>~</sup> <sup>þ</sup> ð Þ <sup>1</sup> � cos<sup>α</sup> **<sup>u</sup>**~<sup>2</sup> **ρ** ¼ d**u** þ sinα**u**<sup>0</sup> þ ð Þ 1 � cosα **u** � **u**<sup>0</sup>

**Theorem 9 (isomorphism theorem).** *The special Euclidean group S*ð Þ 3, � *and*

Φ : S<sup>3</sup> ! S<sup>3</sup>

Being the rigid body motion given by the following parametric equation in a

**ρ** ¼ **ρ**ð Þt ∈ **V**<sup>3</sup> **R** ¼ **R**ð Þt ∈ **S**<sup>3</sup>

The dual orthogonal tensor that describes the rigid body motion is [13, 24]:

In relation (87), the skew symmetric tensor associated to the vector **ρ** is denoted

ð Þ¼ **R**

� � � � �

Φð Þ¼ g ð Þ **I** þ ε**ρ**~ **R** (84)

**R ρ 0** 1

**R** ¼ ð Þ **I** þ ε**ρ**~ **R** (87)

� � (85)

� � � � � �∈ε;

� � � � � �<sup>∈</sup> . �<sup>∈</sup> and **<sup>ψ</sup>** � � � � � � <sup>∉</sup> f g <sup>ε</sup> ;

(83)

(86)

embeds the screw axis and screw parameters.

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

• roto-translation if **ψ** 6¼ 0, **ψ**<sup>0</sup> 6¼ 0 and **ψ** � **ψ**<sup>0</sup> 6¼ 0 ⟺ if **ψ**

• pure translation if if **ψ** ¼ 0 and **ψ**<sup>0</sup> 6¼ 0 ⟺ if **ψ**

• pure rotation if **ψ** 6¼ 0 and **ψ** � **ψ**<sup>0</sup> ¼ 0 ⟺ if **ψ**

*recover the structural invariants* **R** *and* **ρ** *from Eq. (79):*

*S*<sup>3</sup> ð Þ , � *are connected via the isomorphism of the Lie groups*

� �, **<sup>R</sup>** <sup>∈</sup>S3, **<sup>ρ</sup>** <sup>∈</sup> **<sup>V</sup>**3.

*where* **<sup>R</sup>** <sup>¼</sup> Re ð Þ **<sup>R</sup>** , **<sup>ρ</sup>** <sup>¼</sup> vect Duð Þ� **<sup>R</sup> <sup>R</sup>**<sup>T</sup> � � .

**6. Higher-order kinematics in dual Lie algebra**

**<sup>Φ</sup>**�<sup>1</sup> : <sup>S</sup><sup>3</sup> \$ <sup>S</sup>3; **<sup>Φ</sup>**�<sup>1</sup>

(

be parameterized by **ψ** are:

*where* <sup>g</sup> <sup>¼</sup> **<sup>R</sup> <sup>ρ</sup>**

given reference frame:

by **ρ**~.

**115**

**0** 1

**Remark 6.** *The inverse of* **Φ** is

with t∈ **I**⊆ is time variable.

while ∀**v**1, **v**2, **v**<sup>3</sup> ∈ **V**3, Re **v**1, **v**2, **v**<sup>3</sup> � � 6¼ 0 the determinant is

$$
\langle \underline{\mathbf{T}\mathbf{v}\_1}, \underline{\mathbf{T}\mathbf{v}\_2}, \underline{\mathbf{T}\mathbf{v}\_3} \rangle = \det \underline{\mathbf{T}} \langle \underline{\mathbf{v}}\_1, \underline{\mathbf{v}}\_2, \underline{\mathbf{v}}\_3 \rangle. \tag{77}
$$

Orthogonal dual tensor maps are a powerful instrument in the study of the rigid motion with respect to an inertial and non-inertial reference frames.

Let the orthogonal dual tensor set be denoted by:

$$\underline{\mathbf{S}}\underline{\mathbf{Q}}\_3 = \left\{ \underline{\mathbf{R}} \in \mathbf{L}(\underline{\mathbf{V}}\_3, \underline{\mathbf{V}}\_3) \Big| \underline{\mathbf{R}} \underline{\mathbf{R}}^T = \underline{\mathbf{I}}, \det \underline{\mathbf{R}} = \mathbf{1} \right\} \tag{78}$$

where S<sup>3</sup> is the set of special orthogonal dual tensors and **I** is the unit orthogonal dual tensor.

**Theorem 5 (Structure Theorem).** *For any R*∈*S*<sup>3</sup> *a unique decomposition is viable*

$$\underline{\mathbf{R}} = (\mathbf{I} + e\tilde{\boldsymbol{\rho}})\mathbf{R} \tag{79}$$

*where R*∈*S*<sup>3</sup> *and ρ*∈ *V***<sup>3</sup>** *are called structural invariants.*

Taking into account the Lie group structure of S<sup>3</sup> and the result presented in previous theorem, it can be concluded that any orthogonal dual tensor **R** ∈S<sup>3</sup> can be used globally parameterize displacements of rigid bodies.

**Theorem 6 (Representation Theorem).** *For any orthogonal dual tensor* **R** *defined as in Eq. (79), a dual number α* ¼ α þ εd *and a dual unit vector* **u** ¼ **u** þ ε**u**<sup>0</sup> *can be computed to have the following equation*, [13–15]:

$$\underline{\mathbf{R}}(\underline{\mathbf{a}}, \underline{\mathbf{u}}) = \mathbf{I} + \sin \underline{\mathbf{a}} \,\underline{\mathbf{u}} + (1 - \cos \underline{\mathbf{a}}) \underline{\mathbf{u}}^2 = \exp \left( \underline{a} \widetilde{\underline{\mathbf{u}}} \right) \tag{80}$$

The parameters α and **u** are called the **natural invariants** of **R.** The unit dual vector **u** gives the Plücker representation of the Mozzi-Chasles axis [13, 14], while the dual angle α ¼ α þ εd contains the rotation angle α and the translated distance d.

The Lie algebra of the Lie group S<sup>3</sup> is the skew-symmetric dual tensor set denoted by <sup>s</sup><sup>3</sup> <sup>¼</sup> **<sup>α</sup>**<sup>~</sup> <sup>∈</sup>**<sup>L</sup> <sup>V</sup>**3, **<sup>V</sup>**<sup>3</sup> ð Þ **<sup>α</sup>**<sup>~</sup> ¼ �**α**~<sup>T</sup> � � � �, where the internal mapping is **<sup>α</sup>**~1, **<sup>α</sup>**~<sup>2</sup> h i <sup>¼</sup> **<sup>α</sup>**<sup>~</sup> <sup>~</sup> <sup>1</sup>**α**2.

The link between the Lie algebra s3, the Lie group *S*3, and the exponential map is given by the following.

**Theorem 7.** *The mapping*

$$\exp\colon \underline{\mathbf{s}\mathbf{o}}\_3 \to \underline{\mathbf{S}\mathbf{\mathbb{O}}}\_3,\\\exp\left(\underline{\tilde{a}}\right) = e^{\tilde{\underline{a}}} = \sum\_{k=0}^{\infty} \frac{\tilde{\underline{a}}^k}{k!} \tag{81}$$

is well defined and onto.

$$\log \colon \underline{\mathtt{S}} \mathbb{Q}\_3 \to \underline{\mathtt{s}} \mathbb{q}\_3,\\ \log \underline{\mathtt{R}} = \left\{ \underline{\tilde{\boldsymbol{\mu}}} \in \underline{\mathtt{s}} \mathbb{q}\_3 \, \middle| \, \exp \left( \underline{\tilde{\boldsymbol{\mu}}} \right) = \underline{\mathtt{R}} \right\} \tag{82}$$

and is the inverse of Eq. (81).

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

**T** λ1**v**<sup>1</sup> þ λ2**v**<sup>2</sup> ð Þ¼ λ1**T v**<sup>1</sup> ð Þþ λ2**T v**<sup>2</sup> ð Þ, ∀λ1, λ<sup>2</sup> ∈ , ∀**v**1, **v**<sup>2</sup> ∈ **V**<sup>3</sup> (75)

� � 6¼ 0 the determinant is

� �, <sup>∀</sup>**v**1, **<sup>v</sup>**<sup>2</sup> <sup>∈</sup> **<sup>V</sup>**<sup>3</sup> (76)

� � � (78)

**R** ¼ ð Þ **I** þ ε**ρ**~ **R** (79)

� �*:* (77)

Let **L V3**, **V3** ð Þ be the set of dual tensors, then any dual tensor **T** ∈ **L V3**, **V3** ð Þ can be decomposed as **T** ¼ **T** þ ε**T**0, where **T**, **T0** ∈**L V**ð Þ 3, **V**<sup>3</sup> are real tensors. Also, the

� � <sup>¼</sup> det**<sup>T</sup> <sup>v</sup>**1, **<sup>v</sup>**2, **<sup>v</sup>**<sup>3</sup>

<sup>S</sup><sup>3</sup> <sup>¼</sup> **<sup>R</sup>** <sup>∈</sup>**<sup>L</sup> <sup>V</sup>**3, **<sup>V</sup>**<sup>3</sup> ð Þ **RR**<sup>T</sup> <sup>¼</sup> **<sup>I</sup>**, det**<sup>R</sup>** <sup>¼</sup> <sup>1</sup> �

where S<sup>3</sup> is the set of special orthogonal dual tensors and **I** is the unit orthog-

**Theorem 5 (Structure Theorem).** *For any R*∈*S*<sup>3</sup> *a unique decomposition is*

Taking into account the Lie group structure of S<sup>3</sup> and the result presented in previous theorem, it can be concluded that any orthogonal dual tensor **R** ∈S<sup>3</sup> can

**Theorem 6 (Representation Theorem).** *For any orthogonal dual tensor* **R** *defined as in Eq. (79), a dual number α* ¼ α þ εd *and a dual unit vector* **u** ¼ **u** þ ε**u**<sup>0</sup> *can be*

The parameters α and **u** are called the **natural invariants** of **R.** The unit dual vector **u** gives the Plücker representation of the Mozzi-Chasles axis [13, 14], while the dual angle α ¼ α þ εd contains the rotation angle α and the translated distance d. The Lie algebra of the Lie group S<sup>3</sup> is the skew-symmetric dual tensor set

The link between the Lie algebra s3, the Lie group *S*3, and the exponential

*exp*: s<sup>3</sup> ! *S*3, *exp* ð Þ¼ *α*~ *e*

log: S<sup>3</sup> ! s3, log **R** ¼ **ψ**~ ∈s<sup>3</sup> exp **ψ**~

**<sup>R</sup>**ð Þ¼ <sup>α</sup>, **<sup>u</sup> <sup>I</sup>** <sup>þ</sup> sin <sup>α</sup>**u**<sup>~</sup> <sup>þ</sup> ð Þ <sup>1</sup> � cos <sup>α</sup> **<sup>u</sup>**~<sup>2</sup> <sup>¼</sup> exp ð Þ <sup>α</sup>**u**<sup>~</sup> (80)

� � �, where the internal mapping is

*<sup>α</sup>*<sup>~</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup> *k*¼0

� � � n o

*α*~*k*

� �

¼ **R**

*<sup>k</sup>*! (81)

(82)

Orthogonal dual tensor maps are a powerful instrument in the study of the rigid

dual transposed tensor, denoted by **T**T, is defined by

while ∀**v**1, **v**2, **v**<sup>3</sup> ∈ **V**3, Re **v**1, **v**2, **v**<sup>3</sup>

*Advances on Tensor Analysis and Their Applications*

onal dual tensor.

**<sup>α</sup>**~1, **<sup>α</sup>**~<sup>2</sup> h i <sup>¼</sup> **<sup>α</sup>**<sup>~</sup> <sup>~</sup>

**114**

<sup>1</sup>**α**2.

map is given by the following. **Theorem 7.** *The mapping*

is well defined and onto.

and is the inverse of Eq. (81).

*viable*

**<sup>v</sup>**<sup>1</sup> � **Tv**<sup>1</sup> ð Þ¼ **<sup>v</sup>**<sup>2</sup> � **<sup>T</sup>**<sup>T</sup>**v**<sup>1</sup>

**Tv**1, **Tv**2, **Tv**<sup>3</sup>

*where R*∈*S*<sup>3</sup> *and ρ*∈ *V***<sup>3</sup>** *are called structural invariants.*

be used globally parameterize displacements of rigid bodies.

*computed to have the following equation*, [13–15]:

denoted by <sup>s</sup><sup>3</sup> <sup>¼</sup> **<sup>α</sup>**<sup>~</sup> <sup>∈</sup>**<sup>L</sup> <sup>V</sup>**3, **<sup>V</sup>**<sup>3</sup> ð Þ **<sup>α</sup>**<sup>~</sup> ¼ �**α**~<sup>T</sup> �

Let the orthogonal dual tensor set be denoted by:

motion with respect to an inertial and non-inertial reference frames.

Based on Theorems 6 and 7, for any orthogonal dual tensor **R**, a dual vector **ψ** ¼ α **u** ¼ **ψ** þ ε**ψ**<sup>0</sup> can be computed and represents the **screw dual vector**, which embeds the screw axis and screw parameters.

The form of **ψ** implies that **ψ**~ ∈ log **R**. The types of rigid displacements that can be parameterized by **ψ** are:


**Theorem 8.** *The natural invariants* α ¼ α þ εd, **u** ¼ **u** þ ε**u**<sup>0</sup> *can be used to directly recover the structural invariants* **R** *and* **ρ** *from Eq. (79):*

$$\begin{aligned} \mathbf{R} &= \mathbf{I} + \sin \alpha \ddot{\mathbf{u}} + (\mathbf{1} - \cos \alpha) \ddot{\mathbf{u}}^2 \\ \rho &= \mathbf{d} \mathbf{u} + \sin \alpha \mathbf{u}\_0 + (\mathbf{1} - \cos \alpha) \mathbf{u} \times \mathbf{u}\_0 \end{aligned} \tag{83}$$

**Theorem 9 (isomorphism theorem).** *The special Euclidean group S*ð Þ 3, � *and S*<sup>3</sup> ð Þ , � *are connected via the isomorphism of the Lie groups*

$$\Phi: \mathbb{SE}\_3 \to \underline{\mathbb{SD}}\_3$$

$$\Phi(\mathbf{g}) = (\mathbf{I} + \varepsilon \tilde{\mathfrak{p}}) \mathbf{R} \tag{84}$$

*where* <sup>g</sup> <sup>¼</sup> **<sup>R</sup> <sup>ρ</sup> 0** 1 � �, **<sup>R</sup>** <sup>∈</sup>S3, **<sup>ρ</sup>** <sup>∈</sup> **<sup>V</sup>**3. **Remark 6.** *The inverse of* **Φ** is

$$\boldsymbol{\Phi}^{-1} : \underline{\mathbb{S}} \underline{\mathbb{S}}\_3 \hookrightarrow \operatorname{SE}\_3; \boldsymbol{\Phi}^{-1}(\underline{\mathbf{R}}) = \begin{bmatrix} \mathbf{R} & \boldsymbol{\rho} \\ \mathbf{0} & 1 \end{bmatrix} \tag{85}$$

*where* **<sup>R</sup>** <sup>¼</sup> Re ð Þ **<sup>R</sup>** , **<sup>ρ</sup>** <sup>¼</sup> vect Duð Þ� **<sup>R</sup> <sup>R</sup>**<sup>T</sup> � � .

#### **6. Higher-order kinematics in dual Lie algebra**

Being the rigid body motion given by the following parametric equation in a given reference frame:

$$\begin{cases} \mathfrak{p} = \mathfrak{p}(\mathfrak{t}) \in \mathbf{V}\_3 \\ \mathbf{R} = \mathbf{R}(\mathfrak{t}) \in \mathbf{S} \mathbb{O}\_3 \end{cases} \tag{86}$$

with t∈ **I**⊆ is time variable.

The dual orthogonal tensor that describes the rigid body motion is [13, 24]:

$$\underline{\mathbf{R}} = (\mathbf{I} + e\tilde{\boldsymbol{\rho}})\mathbf{R} \tag{87}$$

In relation (87), the skew symmetric tensor associated to the vector **ρ** is denoted by **ρ**~.

It can be easily demonstrated [14, 15], that:

$$\underline{\underline{\mathbf{R}}\underline{\mathbf{R}}^{\mathrm{T}} = \underline{\mathrm{I}}}$$

$$\det \underline{\mathbf{R}} = 1$$

$$\underline{\mathbf{a}} \cdot \underline{\mathbf{b}} = (\underline{\mathbf{R}} \underline{\mathbf{a}}) \cdot (\underline{\mathbf{R}} \underline{\mathbf{b}}), \forall \underline{\mathbf{a}}, \underline{\mathbf{b}} \in \underline{\mathbf{V}}\_{3}$$

$$\underline{\underline{\mathbf{R}}(\underline{\mathbf{a}} \times \underline{\mathbf{b}})} = \underline{\underline{\mathbf{R}}} \left( \underline{\mathbf{a}} \right) \times \underline{\underline{\mathbf{R}}} \left( \underline{\underline{\mathbf{b}}} \right), \forall \underline{\mathbf{a}}, \underline{\mathbf{b}} \in \underline{\mathbf{V}}\_{3} \tag{88}$$

The tensor **R** transports the dual vectors from the body frame in the space frame with the conservation of the dual angles and the relative orientation of lines that corresponds to the dual vectors **a** and **b**.

The dual angular velocity for the rigid body motion (86) is given by (87):

$$
\underline{\mathbf{o}} = \mathbf{vect} \dot{\underline{\mathbf{R}}}^{\mathrm{T}} \tag{89}
$$

From (97), results:

with PQ ¼ **ρ**<sup>Q</sup> � **ρ**P.

where we denoted

In (100), **aρ**, **j**

min **H<sup>ρ</sup>** � � �

ations, for *n*∈ :

*dn***ω**

**117**

Relation (97) is true for any P and Q.

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

Analogue with Eq. (95), the following invariants take place:

*e ε***ρ**~

> *e ε***ρ**~ **J**

*e ε***ρ**~

hyper-jerk (jounce), in a point given by the position vector **ρ**:

**J<sup>ρ</sup>** ¼ **j**

**H<sup>ρ</sup>** ¼ **h<sup>ρ</sup>** � **ω** � **j**

Analogue with Eq. (97) the following equations take place:

**J**

vectors **Aρ**,**Jρ**, **H<sup>ρ</sup>** have the minimum module value. Supplementary,

min **<sup>ρ</sup>**<sup>∈</sup> V3 **A<sup>ρ</sup>** � � �

min **ρ**∈ V3 **Jρ** � � �

min **<sup>ρ</sup>** <sup>∈</sup> V3 **H<sup>ρ</sup>** � � �

Interesting is the fact that for the plane motion min **A<sup>ρ</sup>**

� ¼ 0 because Duj j **ω**\_ ¼ Duj j **ω**€ ¼ Du **ω**

jerk (jounce), in a point given by the position vector **ρ***:*

The lines corresponding to the dual vectors **ω**\_ , **ω**€, **ω**

**J**

**A<sup>ρ</sup>** ¼ **ω**\_ , ∀**ρ**∈ **V**<sup>3</sup>

**<sup>ρ</sup>** <sup>¼</sup> **<sup>ω</sup>**€, <sup>∀</sup>**<sup>ρ</sup>** <sup>∈</sup> **<sup>V</sup>**<sup>3</sup>

**H<sup>ρ</sup>** ¼ **ω**€, ∀**ρ**∈ **V**<sup>3</sup>

**A<sup>ρ</sup>** ¼ **ω**\_ þ ε**A<sup>ρ</sup>**

**<sup>ρ</sup>** <sup>¼</sup> **<sup>ω</sup>**€ <sup>þ</sup> <sup>ε</sup>**J<sup>ρ</sup> H<sup>ρ</sup>** ¼ **ω**€ þ ε**H<sup>ρ</sup>**

with **Aρ**,**Jρ**, **H<sup>ρ</sup>** the reduced acceleration, reduced jerk, respectively the reduced

**A<sup>ρ</sup>** ¼ **a<sup>ρ</sup>** � **ω** � **v<sup>ρ</sup>**

**<sup>A</sup>**<sup>P</sup> <sup>¼</sup> <sup>e</sup><sup>ε</sup>PQ<sup>e</sup> **<sup>A</sup>**<sup>Q</sup>

<sup>P</sup> <sup>¼</sup> <sup>e</sup><sup>ε</sup>PQ<sup>e</sup> **<sup>J</sup>**

**<sup>H</sup>**<sup>P</sup> <sup>¼</sup> <sup>e</sup><sup>ε</sup>PQ<sup>e</sup> **<sup>H</sup>**<sup>Q</sup>

Q

� ¼ j j Duj j **ω**\_

� ¼ j j Duj j **ω**€

� ¼ Du **ω** *:::* � � � � � � � �

> *:::* � � � � ¼ 0

All properties are extended for higher-order accelerations. The vector **<sup>ω</sup>**ð Þ *<sup>n</sup>* <sup>¼</sup>

*dtn* , *n* ∈ describes completely the helicoidally field of the *n* order reduced acceler-

� � �

**<sup>ρ</sup>** � **ω** � **a<sup>ρ</sup>** � 2**ω**\_ � **v<sup>ρ</sup>**

**<sup>ρ</sup>** � 3**ω**\_ � **a<sup>ρ</sup>** � 3**ω**€ � **v<sup>ρ</sup>**

**<sup>ρ</sup>** and **h<sup>ρ</sup>** are, respectively, the acceleration, the jerk, and the hyper-

, (101)

*:::* represent the loci, where the

� ¼ min **J<sup>ρ</sup>** � � � � ¼

**<sup>V</sup>**<sup>P</sup> <sup>¼</sup> <sup>e</sup><sup>ε</sup>PQ<sup>~</sup> **<sup>V</sup>**<sup>Q</sup> (97)

(98)

(99)

(100)

(102)

It can be demonstrated that:

$$
\underline{\mathbf{o}} = \mathbf{o} + \varepsilon \mathbf{v} \tag{90}
$$

where

$$\mathbf{u} = \mathbf{v}\mathbf{c}\mathbf{t}\mathbf{R}^{\mathrm{T}} \tag{91}$$

is the instantaneous angular velocity of the rigid body and

$$\mathbf{v} = \dot{\mathfrak{p}} - \mathfrak{w} \times \mathfrak{p} \tag{92}$$

is the linear velocity of a point of the rigid body that coincides with the origin of the reference frame at that given moment.

The dual angular velocity **ω** completely characterizes the distribution of the velocity field of the rigid body. The pair (**ω**, **v**) is called "the twist of the rigid body motion" [13, 14].

Being:

$$
\underline{\mathbf{V}}\_{\mathfrak{\rho}} = \mathfrak{\mathfrak{o}} + \varepsilon \mathbf{v}\_{\mathfrak{\mathfrak{\rho}}} \tag{93}
$$

the dual velocity for a point localized in the reference frame by the position vector **ρ***.*

In (93), **ω** is the instantaneous angular velocity of the rigid body and **v<sup>ρ</sup>** is the linear velocity of the point. Using the next equation,

$$\mathbf{e}^{\varepsilon\tilde{\mathbf{p}}} = \mathbf{I} + \varepsilon\tilde{\mathbf{p}} \tag{94}$$

from (90), (92)–(94), results:

$$\mathbf{e}^{\epsilon\bar{\rho}} \underline{\mathbf{V}}\_{\rho} = \underline{\mathbf{o}} \tag{95}$$

Consequently, e<sup>ε</sup>~**<sup>r</sup> V**<sup>r</sup> is an invariant having the same value for any **r**.

Writing this invariant in two different points of the rigid body (noted with P and Q ), results that:

$$\mathbf{e}^{\varepsilon \vec{\tau}\_{\mathsf{P}}} \underline{\mathbf{V}}\_{\mathsf{P}} = \mathbf{e}^{\varepsilon \vec{\tau}\_{\mathsf{Q}}} \underline{\mathbf{V}}\_{\mathsf{Q}} \tag{96}$$

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

From (97), results:

It can be easily demonstrated [14, 15], that:

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corresponds to the dual vectors **a** and **b**.

It can be demonstrated that:

the reference frame at that given moment.

linear velocity of the point. Using the next equation,

from (90), (92)–(94), results:

Consequently, e<sup>ε</sup>~**<sup>r</sup>**

and Q ), results that:

**116**

where

motion" [13, 14]. Being:

vector **ρ***.*

**RR**<sup>T</sup> <sup>¼</sup> **<sup>I</sup>** det**R** ¼ 1 **a** � **b** ¼ ð Þ� **Ra** ð Þ **Rb** , ∀**a**, **b**∈ **V**<sup>3</sup>

The tensor **R** transports the dual vectors from the body frame in the space frame with the conservation of the dual angles and the relative orientation of lines that

is the linear velocity of a point of the rigid body that coincides with the origin of

The dual angular velocity **ω** completely characterizes the distribution of the velocity field of the rigid body. The pair (**ω**, **v**) is called "the twist of the rigid body

the dual velocity for a point localized in the reference frame by the position

In (93), **ω** is the instantaneous angular velocity of the rigid body and **v<sup>ρ</sup>** is the

**V**<sup>r</sup> is an invariant having the same value for any **r**.

Writing this invariant in two different points of the rigid body (noted with P

eε**ρ**~

The dual angular velocity for the rigid body motion (86) is given by (87):

is the instantaneous angular velocity of the rigid body and

**R**ð Þ¼ **a** � **b R** ð Þ� **a R** ð**b**Þ, ∀**a**, **b**∈ **V**<sup>3</sup> (88)

**<sup>ω</sup>** <sup>¼</sup> vect**R**\_ **<sup>R</sup>**<sup>T</sup> (89)

**ω** ¼ **ω** þ ε**v** (90)

**<sup>ω</sup>** <sup>¼</sup> vect**RR**\_ <sup>T</sup> (91)

**v** ¼ **ρ**\_ � **ω** � **ρ** (92)

**V<sup>ρ</sup>** ¼ **ω** þ ε**v<sup>ρ</sup>** (93)

<sup>e</sup><sup>ε</sup>**ρ**<sup>~</sup> <sup>¼</sup> **<sup>I</sup>** <sup>þ</sup> <sup>ε</sup>**ρ**<sup>~</sup> (94)

**V<sup>ρ</sup>** ¼ **ω** (95)

<sup>e</sup><sup>ε</sup>~r*P***V**<sup>P</sup> <sup>¼</sup> <sup>e</sup><sup>ε</sup>~r*Q***V**<sup>Q</sup> (96)

$$
\underline{\mathbf{V}}\_{\mathcal{P}} = \mathbf{e}^{\mathrm{e\bar{P}Q}} \underline{\mathbf{V}}\_{Q} \tag{97}
$$

with PQ ¼ **ρ**<sup>Q</sup> � **ρ**P. Relation (97) is true for any P and Q. Analogue with Eq. (95), the following invariants take place:

$$\begin{aligned} \epsilon^{r\bar{\rho}} \underline{\mathbf{A}}\_{\rho} &= \underline{\mathbf{\dot{o}}}, \forall \mathfrak{\rho} \in \mathbf{V}\_3 \\ \epsilon^{r\bar{\rho}} \underline{\mathbf{J}}\_{\rho} &= \underline{\mathbf{\ddot{o}}}, \forall \mathfrak{\rho} \in \mathbf{V}\_3 \\ \epsilon^{r\bar{\rho}} \underline{\mathbf{H}}\_{\rho} &= \underline{\mathbf{\dot{o}}}, \forall \mathfrak{\rho} \in \mathbf{V}\_3 \end{aligned} \tag{98}$$

where we denoted

$$\begin{aligned} \underline{\mathbf{A}}\_{\mathfrak{\rho}} &= \dot{\mathfrak{o}} + \varepsilon \mathbf{A}\_{\mathfrak{\rho}} \\ \underline{\mathbf{J}}\_{\mathfrak{\rho}} &= \dot{\mathfrak{o}} + \varepsilon \mathbf{J}\_{\mathfrak{\rho}} \\ \underline{\mathbf{H}}\_{\mathfrak{\rho}} &= \ddot{\mathfrak{o}} + \varepsilon \mathbf{H}\_{\mathfrak{\rho}} \end{aligned} \tag{99}$$

with **Aρ**,**Jρ**, **H<sup>ρ</sup>** the reduced acceleration, reduced jerk, respectively the reduced hyper-jerk (jounce), in a point given by the position vector **ρ**:

$$\mathbf{A}\_{\rho} = \mathbf{a}\_{\rho} - \mathbf{o} \times \mathbf{v}\_{\rho}$$

$$\mathbf{J}\_{\rho} = \mathbf{j}\_{\rho} - \mathbf{o} \times \mathbf{a}\_{\rho} - 2\dot{\mathbf{o}} \times \mathbf{v}\_{\rho} \tag{100}$$

$$\mathbf{H}\_{\rho} = \mathbf{h}\_{\rho} - \mathbf{o} \times \mathbf{j}\_{\rho} - 3\dot{\mathbf{o}} \times \mathbf{a}\_{\rho} - 3\dot{\mathbf{o}} \times \mathbf{v}\_{\rho}$$

In (100), **aρ**, **j <sup>ρ</sup>** and **h<sup>ρ</sup>** are, respectively, the acceleration, the jerk, and the hyperjerk (jounce), in a point given by the position vector **ρ***:*

Analogue with Eq. (97) the following equations take place:

$$\begin{aligned} \mathbf{A}\_{\mathrm{P}} &= \mathbf{e}^{\mathrm{e}\widetilde{\mathbf{Q}}} \mathbf{A}\_{\mathrm{Q}} \\ \mathbf{J}\_{\mathrm{P}} &= \mathbf{e}^{\mathrm{e}\widetilde{\mathbf{P}}\mathbf{Q}} \mathbf{I}\_{\mathrm{Q}} \quad , \\ \mathbf{\underline{H}}\_{\mathrm{P}} &= \mathbf{e}^{\mathrm{e}\widetilde{\mathbf{P}}\mathbf{Q}} \mathbf{H}\_{\mathrm{Q}} \end{aligned} \tag{101}$$

The lines corresponding to the dual vectors **ω**\_ , **ω**€, **ω** *:::* represent the loci, where the vectors **Aρ**,**Jρ**, **H<sup>ρ</sup>** have the minimum module value. Supplementary,

$$\begin{aligned} \min\_{\boldsymbol{\rho} \in \mathcal{V}\_{\mathcal{I}}} & ||\mathbf{A}\_{\boldsymbol{\rho}}|| = |\mathbf{D}\mathbf{u}| \underline{\hat{\mathbf{u}}}||\\ \min\_{\boldsymbol{\rho} \in \mathcal{V}\_{\mathcal{I}}} & ||\mathbf{J}\_{\boldsymbol{\rho}}|| = |\mathbf{D}\mathbf{u}| \ddot{\underline{\mathbf{u}}}||\\ \min\_{\boldsymbol{\rho} \in \mathcal{V}\_{\mathcal{I}}} & ||\mathbf{H}\_{\boldsymbol{\rho}}|| = |\mathbf{D}\mathbf{u}| \ddot{\underline{\mathbf{u}}}|| \end{aligned} \tag{102}$$

Interesting is the fact that for the plane motion min **A<sup>ρ</sup>** � � � � ¼ min **J<sup>ρ</sup>** � � � � ¼ min **H<sup>ρ</sup>** � � � � ¼ 0 because Duj j **ω**\_ ¼ Duj j **ω**€ ¼ Du **ω** *:::* � � � � ¼ 0

All properties are extended for higher-order accelerations. The vector **<sup>ω</sup>**ð Þ *<sup>n</sup>* <sup>¼</sup> *dn***ω** *dtn* , *n* ∈ describes completely the helicoidally field of the *n* order reduced accelerations, for *n*∈ :

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$$
\varepsilon^{\epsilon\tilde{\rho}} \underline{\mathbf{A}}\_{\mathfrak{p}}^{[n]} = \underline{\mathbf{o}}^{(n)} \tag{103}
$$

it follows successively

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

> **Ψ**<sup>1</sup> **Ψ**<sup>2</sup> **Ψ**<sup>3</sup> **Ψ**<sup>4</sup>

**Figure 1.**

**119**

*Higher-order time derivative of dual twist.*

**P**<sup>1</sup> ¼ **ω**~ **P**<sup>2</sup> ¼ **D** þ **ω**~ **<sup>P</sup>**<sup>3</sup> <sup>¼</sup> **<sup>D</sup>**<sup>2</sup> <sup>þ</sup> **<sup>ω</sup>**~**<sup>D</sup>** <sup>þ</sup> <sup>2</sup>**ω**~\_ <sup>þ</sup> **<sup>ω</sup>**~<sup>2</sup> **<sup>P</sup>**<sup>4</sup> <sup>¼</sup> **<sup>D</sup>**<sup>3</sup> <sup>þ</sup> **<sup>ω</sup>**~**D2** <sup>þ</sup> <sup>3</sup>**ω**~\_ <sup>þ</sup> **<sup>ω</sup>**~<sup>2</sup> � �**<sup>D</sup>** <sup>þ</sup> <sup>3</sup>**ω**€~ <sup>þ</sup> <sup>2</sup>**ω**~**ω**~\_ <sup>þ</sup> <sup>3</sup>**ω**~\_ **<sup>ω</sup>**<sup>~</sup> <sup>þ</sup> **<sup>ω</sup>**~<sup>3</sup>

If we denote **<sup>T</sup>** <sup>¼</sup> ð Þ **<sup>v</sup>**, **<sup>ω</sup>**<sup>~</sup> and by **<sup>Ψ</sup>***<sup>n</sup>* <sup>¼</sup> ð Þ **<sup>a</sup>***n*, **<sup>Φ</sup>***<sup>n</sup>* , *<sup>n</sup>*<sup>∈</sup> <sup>∗</sup> , for the case of the

<sup>3</sup>**ω**€~ <sup>þ</sup> <sup>2</sup>**ω**~**ω**~\_ <sup>þ</sup> <sup>3</sup>**ω**~\_ **<sup>ω</sup>**<sup>~</sup> <sup>þ</sup> **<sup>ω</sup>**~<sup>3</sup> **<sup>3</sup>ω**~\_ <sup>þ</sup> **<sup>ω</sup>**<sup>~</sup> **<sup>2</sup> <sup>ω</sup>**<sup>~</sup> **<sup>I</sup>**

k¼1

**Theorem 10.** *The nth order accelerations field of a rigid body in a general motion is*

*uniquely determined by the kth order time derivative of a dual twist* **<sup>ω</sup>**, *<sup>k</sup>* <sup>¼</sup> 0, *<sup>n</sup>* � 1.

**C**k

<sup>n</sup>�**1ω**~ð Þ <sup>n</sup>�1�<sup>k</sup> **<sup>Ψ</sup>**<sup>k</sup>

**<sup>Ψ</sup>**<sup>n</sup> <sup>¼</sup> **<sup>P</sup>**n**T**, n<sup>∈</sup> <sup>∗</sup> (116)

**<sup>T</sup>**ð Þ <sup>n</sup>�**<sup>1</sup>** <sup>¼</sup> **<sup>Ψ</sup>**<sup>n</sup> �X<sup>n</sup>�<sup>1</sup>

**I 0 00** �**ω**~ **I 00** �2**ω**~\_ �**ω**<sup>~</sup> **I 0** �3**ω**€~ �3**ω**~\_ �**ω**<sup>~</sup> **<sup>I</sup>**

**I 0 00 ω**~ **I 00** <sup>2</sup>**ω**~\_ <sup>þ</sup> **<sup>ω</sup>**~<sup>2</sup> **<sup>ω</sup>**<sup>~</sup> **I 0**

**Ψ**<sup>1</sup> **Ψ**<sup>2</sup> **Ψ**<sup>3</sup> **Ψ**<sup>4</sup>

**T T**\_ **T**€ **T** *:::*

velocities, accelerations, jerks and jounces, on obtain (**Figure 1**):

**T T**\_ **T**€ **T**€

(113)

(114)

(115)

In Eq. (103) **A**½ � *<sup>n</sup>* **<sup>ρ</sup>** denote the *nth* order of the dual reduced acceleration in a given point by the position vector **ρ.**

It follows that the dual part of the *nth* order differentiation of **ω**ð Þ *<sup>n</sup>*

$$
\underline{\mathbf{o}}^{(n)} = \mathbf{o}^{(n)} + \mathbf{e} \mathbf{v}^{(n)} \tag{104}
$$

is the *nth* order reduced acceleration of that point of the rigid body that at the given time pass by the origin of the reference frame.

From equation

$$\mathbf{v} = \dot{\mathfrak{p}} - \mathfrak{w} \times \mathfrak{p} \tag{105}$$

it follows that

$$\mathbf{v}^{(n)} = \mathfrak{p}^{(n+1)} - \sum\_{k=0}^{n} \mathbf{C}\_{n}^{k} \mathbf{o}^{(k)} \times \mathfrak{p}^{(n-k)}, n \in \mathbb{N} \tag{106}$$

with the following notations

$$\mathbf{a}\_{\boldsymbol{\rho}}^{[n]} \triangleq \boldsymbol{\mathfrak{p}}^{(n+1)}, n \in \mathbb{N} \tag{107}$$

for the *n* ∈ order acceleration of the point given by the position vector **ρ** and

$$\mathbf{A}\_{\mathfrak{p}}^{[n]} \triangleq \mathbf{a}\_{\mathfrak{p}}^{[n]} - \sum\_{k=0}^{n-1} \mathbf{C}\_{n}^{k} \mathbf{o}^{(n-k)} \mathbf{a}\_{\mathfrak{p}}^{[k]} \tag{108}$$

for the *nth* order reduced acceleration of the same point the equation:

$$\mathbf{A}\_{\mathfrak{p}}^{[n]} = \mathbf{v}^{(n)} + \mathfrak{o}^{(n)} \times \mathfrak{p} \tag{109}$$

which proves the character of the helicoidally field of the *nth* order reduced accelerations field.

For **<sup>ρ</sup>** <sup>¼</sup> 0, the relations between the *nth* order reduced acceleration and the n order acceleration from point O, the origin of the reference frame, are written

$$\mathbf{A}\_{\mathbf{0}}^{[n]} = \mathbf{v}^{(n)} = \mathbf{a}\_{n} - \sum\_{k=1}^{n-1} \mathbf{C}\_{n}^{k} \mathbf{o}^{(n-k)} \mathbf{a}\_{k}, n \in \mathbb{N} \tag{110}$$

The invert of previous equation is written:

$$\mathbf{a}\_n = \mathbf{P}\_n(\mathbf{v}), n \in \mathbb{N} \tag{111}$$

where **P***<sup>n</sup>* is the polynomial with the coefficients in the ring of the second order Euclidean tensors and the polynomials **P***n*½ � **D** follow the recurrence equation:

$$\begin{cases} \mathbf{P}\_{n+1} = \mathbf{D} \mathbf{P}\_n + \mathbf{P}\_n(\ddot{\mathbf{u}}) \\\\ \mathbf{P}\_0 = \mathbf{I} \end{cases} \tag{112}$$

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

it follows successively

*e ε***ρ**~ **A**½ � *<sup>n</sup>*

It follows that the dual part of the *nth* order differentiation of **ω**ð Þ *<sup>n</sup>*

given time pass by the origin of the reference frame.

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**<sup>v</sup>**ð Þ *<sup>n</sup>* <sup>¼</sup> **<sup>ρ</sup>**ð Þ *<sup>n</sup>*þ<sup>1</sup> �X*<sup>n</sup>*

**A**½ � *<sup>n</sup>* **ρ** ≜**a**½ � *<sup>n</sup>* **<sup>ρ</sup>** �X*<sup>n</sup>*�<sup>1</sup> *k*¼0 C*k*

**A**½ � *<sup>n</sup>*

**<sup>0</sup>** <sup>¼</sup> **<sup>v</sup>**ð Þ *<sup>n</sup>* <sup>¼</sup> **<sup>a</sup>***<sup>n</sup>* �X*<sup>n</sup>*�<sup>1</sup>

**A**½ � *<sup>n</sup>*

The invert of previous equation is written:

*k*¼0 C*k*

for the *nth* order reduced acceleration of the same point the equation:

which proves the character of the helicoidally field of the *nth* order reduced

For **<sup>ρ</sup>** <sup>¼</sup> 0, the relations between the *nth* order reduced acceleration and the n order acceleration from point O, the origin of the reference frame, are written

> *k*¼1 C*k*

where **P***<sup>n</sup>* is the polynomial with the coefficients in the ring of the second order

**P***<sup>n</sup>*þ<sup>1</sup> ¼ **DP***<sup>n</sup>* þ **P***n*ð Þ **ω**~

Euclidean tensors and the polynomials **P***n*½ � **D** follow the recurrence equation:

**P**<sup>0</sup> ¼ **I**

(

for the *n* ∈ order acceleration of the point given by the position vector **ρ** and

*n***ω**ð Þ *<sup>n</sup>*�*<sup>k</sup>* **a**½ � *<sup>k</sup>*

**a**½ � *<sup>n</sup>* **ρ**

is the *nth* order reduced acceleration of that point of the rigid body that at the

In Eq. (103) **A**½ � *<sup>n</sup>*

From equation

it follows that

accelerations field.

**118**

with the following notations

point by the position vector **ρ.**

**<sup>ρ</sup>** <sup>¼</sup> **<sup>ω</sup>**ð Þ *<sup>n</sup>* (103)

**<sup>ω</sup>**ð Þ *<sup>n</sup>* <sup>¼</sup> **<sup>ω</sup>**ð Þ *<sup>n</sup>* <sup>þ</sup> **<sup>ε</sup>v**ð Þ *<sup>n</sup>* (104)

**v** ¼ **ρ**\_ � **ω** � **ρ** (105)

*<sup>n</sup>***ω**ð Þ*<sup>k</sup>* � **<sup>ρ</sup>**ð Þ *<sup>n</sup>*�*<sup>k</sup>* , *<sup>n</sup>*<sup>∈</sup> (106)

**<sup>ρ</sup>** (108)

*<sup>n</sup>***ω**ð Þ *<sup>n</sup>*�*<sup>k</sup>* **a***k*, *n* ∈ (110)

(112)

≜**ρ**ð Þ *<sup>n</sup>*þ<sup>1</sup> , *n*∈ (107)

**<sup>ρ</sup>** <sup>¼</sup> **<sup>v</sup>**ð Þ *<sup>n</sup>* <sup>þ</sup> **<sup>ω</sup>**ð Þ *<sup>n</sup>* � **<sup>ρ</sup>** (109)

**a***<sup>n</sup>* ¼ **P***n*ð Þ **v** , *n*∈ (111)

**<sup>ρ</sup>** denote the *nth* order of the dual reduced acceleration in a given

$$\begin{aligned} \mathbf{P\_1} &= \ddot{\mathbf{o}} \\ \mathbf{P\_2} &= \mathbf{D} + \ddot{\mathbf{o}} \end{aligned} $$
 
$$\mathbf{P\_3} = \mathbf{D^2} + \ddot{\mathbf{o}}\mathbf{D} + 2\dot{\mathbf{o}} + \ddot{\mathbf{o}}^2 \tag{113}$$
 
$$\mathbf{P\_4} = \mathbf{D^3} + \ddot{\mathbf{o}}\mathbf{D^2} + \left( 3\dot{\mathbf{o}} + \ddot{\mathbf{o}}^2 \right) \mathbf{D} + 3\ddot{\mathbf{o}} + 2\ddot{\mathbf{o}}\dot{\mathbf{o}} \dot{\mathbf{o}} + 3\dot{\mathbf{o}}\dot{\mathbf{o}}\mathbf{d} + \ddot{\mathbf{o}}^3$$

If we denote **<sup>T</sup>** <sup>¼</sup> ð Þ **<sup>v</sup>**, **<sup>ω</sup>**<sup>~</sup> and by **<sup>Ψ</sup>***<sup>n</sup>* <sup>¼</sup> ð Þ **<sup>a</sup>***n*, **<sup>Φ</sup>***<sup>n</sup>* , *<sup>n</sup>*<sup>∈</sup> <sup>∗</sup> , for the case of the velocities, accelerations, jerks and jounces, on obtain (**Figure 1**):

$$
\begin{bmatrix} \mathbf{T} \\ \dot{\mathbf{T}} \\ \ddot{\mathbf{T}} \\ \ddot{\mathbf{T}} \end{bmatrix} = \begin{bmatrix} \mathbf{I} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ -\ddot{\mathbf{o}} & \mathbf{I} & \mathbf{0} & \mathbf{0} \\ -2\dot{\mathbf{o}} & -\ddot{\mathbf{o}} & \mathbf{I} & \mathbf{0} \\ -3\ddot{\mathbf{o}} & -3\dot{\mathbf{o}} & -\ddot{\mathbf{o}} & \mathbf{I} \end{bmatrix} \begin{bmatrix} \Psi\_1 \\ \Psi\_2 \\ \Psi\_3 \\ \Psi\_4 \end{bmatrix} \tag{114}
$$

$$
\begin{bmatrix}
\Psi\_1 \\
\Psi\_2 \\
\Psi\_3 \\
\Psi\_4
\end{bmatrix} = \begin{bmatrix}
\mathbf{I} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\ddot{\mathbf{0}} & \mathbf{I} & \mathbf{0} & \mathbf{0} \\
2\dot{\ddot{\mathbf{0}}} + \ddot{\mathbf{0}}^2 & \ddot{\mathbf{0}} & \mathbf{I} & \mathbf{0} \\
3\ddot{\ddot{\mathbf{0}}} + 2\ddot{\mathbf{o}}\dot{\ddot{\mathbf{o}}} + 3\dot{\ddot{\mathbf{o}}}\ddot{\mathbf{o}} + \ddot{\mathbf{o}}^3 & 3\dot{\ddot{\mathbf{o}}} + \ddot{\mathbf{o}}^2 & \ddot{\mathbf{o}} & \mathbf{I}
\end{bmatrix} \begin{bmatrix}
\mathbf{T} \\
\ddot{\mathbf{T}} \\
\ddot{\mathbf{T}} \\
\dddot{\mathbf{T}}
\end{bmatrix} \tag{115}
$$

$$\mathbf{T}^{(\mathbf{n}-1)} = \boldsymbol{\Psi}\_{\mathbf{n}} - \sum\_{\mathbf{k}=1}^{\mathbf{n}-1} \mathbf{C}\_{\mathbf{n}-1}^{\mathbf{k}} \tilde{\mathbf{o}}^{(\mathbf{n}-1-\mathbf{k})} \boldsymbol{\Psi}\_{\mathbf{k}}$$

$$\boldsymbol{\Psi}\_{\mathbf{n}} = \mathbf{P}\_{\mathbf{n}} \mathbf{T}, \mathbf{n} \in \mathbb{N}^\* \tag{116}$$

**Theorem 10.** *The nth order accelerations field of a rigid body in a general motion is uniquely determined by the kth order time derivative of a dual twist* **<sup>ω</sup>**, *<sup>k</sup>* <sup>¼</sup> 0, *<sup>n</sup>* � 1.

**Figure 1.** *Higher-order time derivative of dual twist.*

#### **7. Higher-order kinematics of spatial chain using dual Lie algebra**

Consider a spatial kinematic chain of the bodies *Ck*, *k* ¼ 0, *m* where the relative motion of the rigid body *Ck* with respect to *Ck*�<sup>1</sup> is given by the proper orthogonal tensor *<sup>k</sup>*�<sup>1</sup> **R***<sup>k</sup>* ∈**SO** <sup>3</sup> . The relative motion properties of the body *Cm* with respect to *C*<sup>0</sup> are described by the orthogonal dual tensor (**Figure 2**):

$$\underline{\mathbf{R}} = \,^0 \underline{\mathbf{R}}\_1 \cdot \,^1 \underline{\mathbf{R}}\_2 \dots \,^{m-1} \underline{\mathbf{R}}\_m \tag{117}$$

where **ω***<sup>k</sup>* is the dual twist of the relative motion of the body *Ck* in relation to the

To determine the field of the *nth* order accelerations of a rigid body *Cm* we have

In order to determine the nth order accelerations field of a rigid body *Cm*, we

<sup>3</sup> *and* **ω**p, **Ω**<sup>p</sup> ∈ **V**

*<sup>p</sup>* the nth order derivative of the relative

<sup>3</sup> , *then*

<sup>n</sup>**Φ**<sup>n</sup>�<sup>k</sup>**D**½ � <sup>k</sup> , (124)

<sup>p</sup> *and* **Φ**<sup>p</sup> *are dual tensors*

<sup>3</sup> , p ¼ 0, n, (125)

*<sup>n</sup>***Φ***<sup>n</sup>*�*<sup>k</sup>***D**½ � *<sup>k</sup>* (128)

<sup>p</sup> and **Φ***<sup>p</sup> are dual tensors*

<sup>3</sup> , *p* ¼ 0, *n* (129)

*:* (126)

*dt*, *with coefficients in*

dt, *with coefficients in*

<sup>p</sup> ¼ **p**nð Þ **ω ω**p, p ¼ 1, n (123)

**Remark 7.** *For m* ¼ 2, <sup>0</sup>**ω**<sup>2</sup> ¼ **ω**<sup>1</sup> þ **ω**2, *we will obtain the space replica of Aronhold-Kennedy Theorem: the instantaneous screw axis for the three relative rigid body motions has in every moment a common perpendicular, at any given time. The*

**<sup>R</sup>***<sup>p</sup>*�<sup>1</sup>**Ω**ð Þ *<sup>n</sup>*

*<sup>m</sup>* , *n*∈ <sup>∗</sup> we will use the following

*common perpendicular is line that corresponds to the dual vector* **ω**<sup>1</sup> � **ω**2*.*

body *Ck*�<sup>1</sup> observed from the body *C*0.

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

*<sup>m</sup>* , *n*∈ .

dual twist **Ω***p*, resolved in the body frame of *C*0.

*the non-commutative ring of Euclidian dual tensors.*

1 **R**<sup>2</sup> … *<sup>p</sup>*�<sup>2</sup>

*<sup>m</sup>* , *n* ∈ <sup>∗</sup> .

**ω**ð Þ <sup>n</sup>

*<sup>n</sup> is the binomial coefficient*, **<sup>D</sup>**½ � <sup>k</sup> **<sup>ω</sup>**<sup>p</sup> <sup>¼</sup> **<sup>ω</sup>**½ � <sup>k</sup>

(

**Theorem 11.** *The following equation takes place*

*the non-commutative ring of Euclidian dual tensors*

*where* **<sup>p</sup>**nð Þ **<sup>ω</sup>** *are polynomials of the differential operator* **<sup>D</sup>** <sup>¼</sup> <sup>d</sup>

**<sup>p</sup>**nð Þ¼ **<sup>ω</sup>** <sup>X</sup><sup>n</sup>

**<sup>Φ</sup>**<sup>p</sup> <sup>¼</sup> **<sup>R</sup>**ð Þ <sup>p</sup> **<sup>R</sup>**T, **<sup>R</sup>** <sup>∈</sup>**SO**

**Φ**<sup>0</sup> ¼ **I**

*where* **<sup>p</sup>***n*ð Þ **<sup>ω</sup>** *are polynomials of the derivative operator* **<sup>D</sup>** <sup>¼</sup> *<sup>d</sup>*

*<sup>n</sup> is the binomial coefficient*, **<sup>D</sup>**½ � <sup>k</sup> **<sup>ω</sup>**<sup>p</sup> <sup>¼</sup> **<sup>ω</sup>**½ � <sup>k</sup>

**<sup>p</sup>***n*ð Þ¼ **<sup>ω</sup>** <sup>X</sup>*<sup>n</sup>*

**<sup>Φ</sup>***<sup>p</sup>* <sup>¼</sup> **<sup>R</sup>**ð Þ *<sup>p</sup>* � **<sup>R</sup>***<sup>T</sup>*, **<sup>R</sup>** <sup>∈</sup>**SO**

*k*¼0 C*k*

**<sup>Φ</sup>**<sup>p</sup>þ<sup>1</sup> <sup>¼</sup> **<sup>Φ</sup>**\_ <sup>p</sup> <sup>þ</sup> **<sup>Φ</sup>**p**ω**<sup>~</sup>

, p∈

<sup>þ</sup> **<sup>p</sup>***<sup>n</sup>* **<sup>ω</sup>**<sup>1</sup> <sup>þ</sup> **<sup>ω</sup>**<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> **<sup>ω</sup>***<sup>m</sup>*�<sup>1</sup> ð Þ**ω***m*; <sup>∀</sup>*<sup>n</sup>* <sup>∈</sup> (127)

ð Þ *<sup>n</sup>* <sup>þ</sup> **<sup>p</sup>***<sup>n</sup>* **<sup>ω</sup>**<sup>1</sup> ð Þ**ω**<sup>2</sup> <sup>þ</sup> **<sup>p</sup>***<sup>n</sup>* **<sup>ω</sup>**<sup>1</sup> <sup>þ</sup> **<sup>ω</sup>**<sup>2</sup> ð Þ**ω**<sup>3</sup> <sup>þ</sup> …

k¼0 Ck

*<sup>p</sup>* <sup>¼</sup> <sup>0</sup>**R**<sup>1</sup>

**Lemma**: *If* **<sup>ω</sup>**<sup>p</sup> <sup>¼</sup> **<sup>R</sup> <sup>Ω</sup>**<sup>p</sup> *with* **<sup>R</sup>** <sup>∈</sup>**SO**

*which follow the recurrence equation*:

0**ω**ð Þ *<sup>n</sup> <sup>m</sup>* ¼ **ω**<sup>1</sup>

*which follow the recurrence equation*:

to determine the <sup>0</sup>**ω**ð Þ *<sup>n</sup>*

We denote **ω**½ � *<sup>n</sup>*

have to determine the <sup>0</sup>**ω**ð Þ *<sup>n</sup>*

To compute <sup>0</sup>**ω**ð Þ *<sup>n</sup>*

*where* C*<sup>k</sup>*

*where* C*<sup>k</sup>*

**121**

Instantaneous dual angular velocity (dual twist) of the rigid body in relation to the reference frame it will be given by the equation

$$\mathbf{a}\_0 \underline{\mathbf{a}}\_n = \text{vect} \underline{\dot{\mathbf{R}} \mathbf{R}^T} \tag{118}$$

It follows from (110) and (111) that:

$$\mathbf{R}\_0 \underline{\mathbf{o}}\_m = \underline{\mathbf{O}}\_1 + {}^0 \mathbf{R}\_1 \underline{\mathbf{O}}\_2 + \dots + {}^0 \mathbf{R}\_1 \cdot {}^1 \mathbf{R}\_2 \dots {}^{m-2} \mathbf{R}\_{m-1} \underline{\mathbf{O}}\_m \tag{119}$$

where

$$\underline{\mathbf{Q}}\_{k} = \mathbf{vect}^{k-1} \dot{\underline{\mathbf{R}}}\_{k}^{k-1} \underline{\mathbf{R}}\_{k}^{T} \tag{120}$$

Using the denotation

$$
\underline{\mathbf{u}}\_{k} = {}^{0}\underline{\mathbf{R}}\_{1} \cdot {}^{1}\underline{\mathbf{R}}\_{2} \dots {}^{k-2}\underline{\mathbf{R}}\_{k-1}\underline{\mathbf{Q}}\_{k} \tag{121}
$$

Eq. (118) will be written

$$
\mathfrak{a}\_0 \underline{\mathfrak{a}}\_m = \underline{\mathfrak{a}}\_1 + \underline{\mathfrak{a}}\_2 + \dots + \underline{\mathfrak{a}}\_m \tag{122}
$$

**Figure 2.** *Orthogonal dual tensors of relative rigid body motion.*

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

**7. Higher-order kinematics of spatial chain using dual Lie algebra**

**<sup>R</sup>** <sup>¼</sup> <sup>0</sup>**R1** � <sup>1</sup>

<sup>0</sup>**ω***<sup>m</sup>* <sup>¼</sup> **<sup>Ω</sup>**<sup>1</sup> <sup>þ</sup> <sup>0</sup>**R1Ω**<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> <sup>0</sup>**R1** � <sup>1</sup>

**Ω***<sup>k</sup>* ¼ vect

**<sup>ω</sup>***<sup>k</sup>* <sup>¼</sup> <sup>0</sup>**R1** � <sup>1</sup>

*C*<sup>0</sup> are described by the orthogonal dual tensor (**Figure 2**):

the reference frame it will be given by the equation

*Advances on Tensor Analysis and Their Applications*

It follows from (110) and (111) that:

tensor *<sup>k</sup>*�<sup>1</sup>

where

**Figure 2.**

**120**

*Orthogonal dual tensors of relative rigid body motion.*

Using the denotation

Eq. (118) will be written

**R***<sup>k</sup>* ∈**SO**

Consider a spatial kinematic chain of the bodies *Ck*, *k* ¼ 0, *m* where the relative motion of the rigid body *Ck* with respect to *Ck*�<sup>1</sup> is given by the proper orthogonal

**R2** … *<sup>m</sup>*�<sup>1</sup>

Instantaneous dual angular velocity (dual twist) of the rigid body in relation to

*k*�1 **R**\_ *k k*�1 **R***<sup>T</sup>*

**R2** … *<sup>k</sup>*�<sup>2</sup>

<sup>3</sup> . The relative motion properties of the body *Cm* with respect to

**R***<sup>m</sup>* (117)

**R***m*�**1Ω***<sup>m</sup>* (119)

*<sup>k</sup>* (120)

**R***k*�**1Ω***<sup>k</sup>* (121)

<sup>0</sup>**ω***<sup>m</sup>* <sup>¼</sup> vect**R**\_ **<sup>R</sup>***<sup>T</sup>* (118)

**R2** … *<sup>m</sup>*�<sup>2</sup>

<sup>0</sup>**ω***<sup>m</sup>* ¼ **ω**<sup>1</sup> þ **ω**<sup>2</sup> þ … þ **ω***<sup>m</sup>* (122)

where **ω***<sup>k</sup>* is the dual twist of the relative motion of the body *Ck* in relation to the body *Ck*�<sup>1</sup> observed from the body *C*0.

**Remark 7.** *For m* ¼ 2, <sup>0</sup>**ω**<sup>2</sup> ¼ **ω**<sup>1</sup> þ **ω**2, *we will obtain the space replica of Aronhold-Kennedy Theorem: the instantaneous screw axis for the three relative rigid body motions has in every moment a common perpendicular, at any given time. The common perpendicular is line that corresponds to the dual vector* **ω**<sup>1</sup> � **ω**2*.*

To determine the field of the *nth* order accelerations of a rigid body *Cm* we have to determine the <sup>0</sup>**ω**ð Þ *<sup>n</sup> <sup>m</sup>* , *n*∈ .

We denote **ω**½ � *<sup>n</sup> <sup>p</sup>* <sup>¼</sup> <sup>0</sup>**R**<sup>1</sup> 1 **R**<sup>2</sup> … *<sup>p</sup>*�<sup>2</sup> **<sup>R</sup>***<sup>p</sup>*�<sup>1</sup>**Ω**ð Þ *<sup>n</sup> <sup>p</sup>* the nth order derivative of the relative dual twist **Ω***p*, resolved in the body frame of *C*0.

In order to determine the nth order accelerations field of a rigid body *Cm*, we have to determine the <sup>0</sup>**ω**ð Þ *<sup>n</sup> <sup>m</sup>* , *n* ∈ <sup>∗</sup> .

To compute <sup>0</sup>**ω**ð Þ *<sup>n</sup> <sup>m</sup>* , *n*∈ <sup>∗</sup> we will use the following **Lemma**: *If* **<sup>ω</sup>**<sup>p</sup> <sup>¼</sup> **<sup>R</sup> <sup>Ω</sup>**<sup>p</sup> *with* **<sup>R</sup>** <sup>∈</sup>**SO** <sup>3</sup> *and* **ω**p, **Ω**<sup>p</sup> ∈ **V** <sup>3</sup> , *then*

$$
\underline{\mathbf{a}}\_{\mathbf{p}}^{(\mathbf{n})} = \mathbf{p}\_{\mathbf{n}}(\underline{\mathbf{a}}) \underline{\mathbf{a}}\_{\mathbf{p}}, \mathbf{p} = \overline{\mathbf{1}, \mathbf{n}} \tag{123}
$$

*where* **<sup>p</sup>**nð Þ **<sup>ω</sup>** *are polynomials of the differential operator* **<sup>D</sup>** <sup>¼</sup> <sup>d</sup> dt, *with coefficients in the non-commutative ring of Euclidian dual tensors.*

$$\mathbf{p}\_{\mathbf{n}}(\underline{\mathbf{a}}) = \sum\_{\mathbf{k}=0}^{\mathrm{n}} \mathbf{C}\_{\mathbf{n}}^{\mathrm{k}} \underline{\mathbf{\underline{\mathbf{d}}}}\_{\mathbf{n}-\mathbf{k}} \mathbf{D}^{[\mathbf{k}]},\tag{124}$$

*where* C*<sup>k</sup> <sup>n</sup> is the binomial coefficient*, **<sup>D</sup>**½ � <sup>k</sup> **<sup>ω</sup>**<sup>p</sup> <sup>¼</sup> **<sup>ω</sup>**½ � <sup>k</sup> <sup>p</sup> *and* **Φ**<sup>p</sup> *are dual tensors*

$$
\underline{\boldsymbol{\Phi}}\_{\mathcal{P}} = \underline{\mathbf{R}}^{(p)} \underline{\mathbf{R}}^{\mathrm{T}}, \underline{\mathbf{R}} \in \underline{\mathbf{SO}}\_{\mathcal{I}}^{\mathbb{R}}, \mathbf{p} = \overline{\mathbf{0}, \mathbf{n}}, \tag{125}
$$

*which follow the recurrence equation*:

$$\begin{cases} \underline{\Phi}\_{\mathsf{p}+1} = \dot{\underline{\Phi}}\_{\mathsf{p}} + \underline{\Phi}\_{\mathsf{p}} \ddot{\underline{\Phi}}\\ \underline{\Phi}\_{0} = \underline{\mathsf{I}} \end{cases}, \mathsf{p} \in \mathbb{N}. \tag{126}$$

**Theorem 11.** *The following equation takes place*

$$\begin{array}{c} \mathbf{0}\_{0}\underline{\mathbf{o}}\_{m}^{(n)} = \underline{\mathbf{o}}\_{1}^{(n)} + \mathbf{p}\_{n}(\underline{\mathbf{o}}\_{1})\underline{\mathbf{o}}\_{2} + \mathbf{p}\_{n}(\underline{\mathbf{o}}\_{1} + \underline{\mathbf{o}}\_{2})\underline{\mathbf{o}}\_{3} + \dots \\ + \mathbf{p}\_{n}(\underline{\mathbf{o}}\_{1} + \underline{\mathbf{o}}\_{2} + \dots + \underline{\mathbf{o}}\_{m-1})\underline{\mathbf{o}}\_{m}; \forall n \in \mathbb{N} \end{array} \tag{127}$$

*where* **<sup>p</sup>***n*ð Þ **<sup>ω</sup>** *are polynomials of the derivative operator* **<sup>D</sup>** <sup>¼</sup> *<sup>d</sup> dt*, *with coefficients in the non-commutative ring of Euclidian dual tensors*

$$\mathbf{p}\_n(\underline{\mathbf{a}}) = \sum\_{k=0}^n \mathbf{C}\_n^k \underline{\mathbf{\mathbf{Q}}}\_{n-k} \mathbf{D}^{[k]} \tag{128}$$

*where* C*<sup>k</sup> <sup>n</sup> is the binomial coefficient*, **<sup>D</sup>**½ � <sup>k</sup> **<sup>ω</sup>**<sup>p</sup> <sup>¼</sup> **<sup>ω</sup>**½ � <sup>k</sup> <sup>p</sup> and **Φ***<sup>p</sup> are dual tensors*

$$\underline{\mathbf{OP}}\_p = \underline{\mathbf{R}}^{(p)} \cdot \underline{\mathbf{R}}^T, \underline{\mathbf{R}} \in \underline{\mathbf{SO}}\_3^{\mathbb{R}}, p = \overline{0, n} \tag{129}$$

*which follow the recurrence equation*:

*Advances on Tensor Analysis and Their Applications*

$$\begin{cases} \underline{\underline{\Phi}}\_{p+1} = \dot{\underline{\Phi}}\_{p} + \underline{\underline{\Phi}}\_{p} \ddot{\underline{\Phi}}\\ \underline{\underline{\Phi}}\_{0} = \underline{\underline{\mathbf{I}}} \end{cases}, p \in \mathbb{N}. \tag{130}$$

Other equivalent forms of Eq. (127) are the following recursive formulas (**Figures 3** and **4**):

$$\underline{\mathbf{a}}\_{m}^{(n)} = \underline{\mathbf{a}}\_{1}^{(n)} + \mathbf{p}\_{n} (\_{0}\underline{\mathbf{a}}\_{1})\underline{\mathbf{a}}\_{2} + \mathbf{p}\_{n} (\_{0}\underline{\mathbf{a}}\_{2})\underline{\mathbf{a}}\_{3} + \dots + \mathbf{p}\_{n} (\_{0}\underline{\mathbf{a}}\_{m-1})\underline{\mathbf{a}}\_{m}, \forall n \in \mathbb{N} \tag{131}$$

The previous equations are valid in the most general situation where there are no kinematic links between the rigid bodies *C*1,*C*2, … ,*Cm*.

The following identity can be proved:

$$\underline{\Phi}\_{k}(\underline{\bf{\bf{o}}}\_{1}+\underline{\bf{o}}\_{2}+\ldots,+\underline{\bf{o}}\_{p-1}) = \sum\_{k\_{1}+k\_{2}+\ldots+k\_{p-1}=k} \mathsf{C}\_{\mathtt{n}}^{k\_{1},\ldots,k\_{p-1}} \underline{\Phi}\_{k\_{1}}(\underline{\bf{o}}\_{1}) \underline{\Phi}\_{k\_{2}}(\underline{\bf{o}}\_{2}) \ldots \underline{\Phi}\_{k\_{p-1}}(\underline{\bf{o}}\_{p-1}) \tag{132}$$

where Ck1, … ,kp�<sup>1</sup> <sup>n</sup> <sup>¼</sup> *<sup>n</sup>*! *<sup>k</sup>*1! … *kp*�1! is the multinomial coefficient.

From Eq. (131), on obtain the closed form non-recursive coordinate-free formula:

$$\begin{aligned} \mathbf{^0\underline{\mathbf{a}}}\_{\mathbf{m}}^{(\mathbf{n})} &= \underline{\mathbf{a}}\_{\mathbf{1}}^{[\mathbf{n}]} + \underline{\mathbf{a}}\_{\mathbf{2}}^{[\mathbf{n}]} + \dots + \underline{\mathbf{a}}\_{\mathbf{m}}^{[\mathbf{n}]} + \\ + \sum\_{p=2}^{m} \sum\_{k=1}^{n} \mathbf{C}\_{n}^{k} \sum\_{\mathbf{k}\_{1}+\dots+\mathbf{k}\_{p-1}=\mathbf{k}} \mathbf{C}\_{\mathbf{n}}^{\mathbf{k}\_{1},\dots,\mathbf{k}\_{p-1}} \underline{\mathbf{a}}\_{\mathbf{k}\_{1}} (\underline{\mathbf{a}}\_{1}) \dots \underline{\mathbf{a}}\_{\mathbf{k}\_{p-1}} (\underline{\mathbf{a}}\_{\mathbf{p}-1}) \underline{\mathbf{a}}\_{\mathbf{p}}^{[\mathbf{n}-\mathbf{k}]}, \end{aligned} \tag{133}$$

where

$$\begin{aligned} \underline{\underline{\Phi}}\_{0}(\underline{\underline{a}}) &= \underline{\underline{I}} \\ \underline{\underline{\Phi}}\_{1}(\underline{\underline{a}}) &= \underline{\underline{\Phi}} \end{aligned} \tag{134}$$

**<sup>Φ</sup>**2ð Þ¼ **<sup>ω</sup> <sup>ω</sup>**~½ � <sup>1</sup> <sup>þ</sup> **<sup>ω</sup>**~<sup>2</sup> **<sup>Φ</sup>**3ð Þ¼ **<sup>ω</sup> <sup>ω</sup>**~½ � <sup>2</sup> <sup>þ</sup> **<sup>ω</sup>**~**ω**~½ � <sup>1</sup> <sup>þ</sup> <sup>2</sup>**ω**~½ � <sup>1</sup> **<sup>ω</sup>**<sup>~</sup> <sup>þ</sup> **<sup>ω</sup>**~<sup>3</sup> **<sup>Φ</sup>**4ð Þ¼ **<sup>ω</sup> <sup>ω</sup>**~½ � <sup>3</sup> <sup>þ</sup> **<sup>ω</sup>**<sup>~</sup> **<sup>ω</sup>**~½ � <sup>2</sup> <sup>þ</sup> <sup>3</sup>**ω**~½ � <sup>1</sup> <sup>þ</sup> **<sup>ω</sup>**~<sup>2</sup> **<sup>ω</sup>**~½ � <sup>1</sup> <sup>þ</sup> <sup>3</sup>**ω**~½ � <sup>2</sup> **<sup>ω</sup>**<sup>~</sup> <sup>þ</sup> <sup>3</sup>**ω**~½ � <sup>1</sup> **<sup>ω</sup>**~<sup>2</sup> <sup>þ</sup> <sup>2</sup>**ω**<sup>~</sup> **<sup>ω</sup>**~½ � <sup>1</sup> **<sup>ω</sup>**<sup>~</sup> <sup>þ</sup> **<sup>ω</sup>**<sup>~</sup> <sup>4</sup>

We'll apply the general results obtained in the previous chapter for the particular case of four degrees of freedom 2C general manipulator. In this case the relative motions of three bodies *C*0,*C*1,*C*<sup>2</sup> are given, the spatial motion of the terminal body

1

<sup>0</sup>**R**<sup>2</sup> <sup>¼</sup> <sup>0</sup>**R**<sup>1</sup>

<sup>0</sup>**R**<sup>1</sup> <sup>¼</sup> *<sup>e</sup>*

In Eqs. (138) and (139), the dual angles *α*1ð Þ*t* and *α*2ð Þ*t* are four times differentiable functions, and unit dual vectors <sup>0</sup>**u**<sup>1</sup> and <sup>1</sup>**u**<sup>2</sup> being constant. To simplify the

1 **R**<sup>2</sup> ¼ *e*

**<sup>Φ</sup>**nð Þ¼ **<sup>ω</sup> <sup>P</sup>**n**ω**~, n<sup>∈</sup> <sup>∗</sup> (135)

**R**<sup>2</sup> (136)

*<sup>α</sup>*1ð Þ*<sup>t</sup>* <sup>0</sup>**u**~<sup>1</sup> (137)

*<sup>α</sup>*2ð Þ*<sup>t</sup>* <sup>1</sup>**u**~<sup>2</sup> (138)

…

**8. Higher-order kinematics for general 2C manipulator**

*Higher-order time derivative of dual twist of relative motion on terminal body.*

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

*C*<sup>2</sup> been described by dual orthogonal tensor as it follows:

where

**123**

**Figure 4.**

writing, we will denote:

**Figure 3.** *Higher-order time derivative of dual twist of relative motion.*

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

**Figure 4.**

**<sup>Φ</sup>***<sup>p</sup>*þ<sup>1</sup> <sup>¼</sup> **<sup>Φ</sup>**\_ *<sup>p</sup>* <sup>þ</sup> **<sup>Φ</sup>***p***ω**<sup>~</sup>

Other equivalent forms of Eq. (127) are the following recursive formulas

(

*Advances on Tensor Analysis and Their Applications*

� �**ω**<sup>2</sup> <sup>þ</sup> **<sup>p</sup>***<sup>n</sup>* <sup>0</sup>**ω**<sup>2</sup>

kinematic links between the rigid bodies *C*1,*C*2, … ,*Cm*.

<sup>¼</sup> <sup>X</sup> *k*1þ*k*2þ … þ*kp*�1¼*k*

0**ω**ð Þ <sup>n</sup>

X k1þ … þkp�1¼k

*Higher-order time derivative of dual twist of relative motion.*

<sup>m</sup> <sup>¼</sup> **<sup>ω</sup>**½ � <sup>n</sup>

Ck1, … ,kp�<sup>1</sup>

(**Figures 3** and **4**):

**<sup>Φ</sup>***<sup>k</sup>* **<sup>ω</sup>**<sup>1</sup> <sup>þ</sup> **<sup>ω</sup>**<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> **<sup>ω</sup>***p*�<sup>1</sup> � �

where Ck1, … ,kp�<sup>1</sup>

X*n k*¼1 C*k n*

formula:

þ X*m p*¼2

where

**Figure 3.**

**122**

<sup>1</sup> þ **p***<sup>n</sup>* <sup>0</sup>**ω**<sup>1</sup>

<sup>n</sup> <sup>¼</sup> *<sup>n</sup>*!

The following identity can be proved:

**ω**ð Þ *<sup>n</sup> <sup>m</sup>* <sup>¼</sup> **<sup>ω</sup>**ð Þ *<sup>n</sup>* **Φ**<sup>0</sup> ¼ **I**

� �**ω**<sup>3</sup> <sup>þ</sup> … <sup>þ</sup> **<sup>p</sup>***<sup>n</sup>* <sup>0</sup>**ω***<sup>m</sup>*�<sup>1</sup>

The previous equations are valid in the most general situation where there are no

Ck1, … ,kp�<sup>1</sup>

*<sup>k</sup>*1! … *kp*�1! is the multinomial coefficient. From Eq. (131), on obtain the closed form non-recursive coordinate-free

<sup>1</sup> <sup>þ</sup> **<sup>ω</sup>**½ � <sup>n</sup>

<sup>n</sup> **Φ**k1 **ω**<sup>1</sup> ð Þ … **Φ**kp�<sup>1</sup>

**Φ**0ð Þ¼ **ω I**

<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> **<sup>ω</sup>**½ � <sup>n</sup>

, *p*∈

<sup>n</sup> **Φ***k*<sup>1</sup> **ω**<sup>1</sup> ð Þ**Φ***k*<sup>2</sup> **ω**<sup>2</sup> ð Þ … **Φ***kp*�<sup>1</sup>

<sup>m</sup> þ

� **ω**<sup>p</sup>�<sup>1</sup> � **ω**½ � <sup>n</sup>�<sup>k</sup>

**<sup>Φ</sup>**1ð Þ¼ **<sup>ω</sup> <sup>ω</sup>**<sup>~</sup> (134)

*:* (130)

� �**ω***m*, <sup>∀</sup>*n*<sup>∈</sup> (131)

� **ω***p*�<sup>1</sup>

<sup>p</sup> , (133)

� (132)

*Higher-order time derivative of dual twist of relative motion on terminal body.*

$$\underline{\underline{\Phi}}\_2(\underline{\underline{\alpha}}) = \underline{\underline{\tilde{\alpha}}}^{[1]} + \underline{\underline{\tilde{\alpha}}}^2$$

$$\underline{\underline{\Phi}}\_3(\underline{\underline{\alpha}}) = \underline{\underline{\tilde{\alpha}}}^{[2]} + \underline{\underline{\tilde{\alpha}}} \underline{\underline{\tilde{\alpha}}}^{[1]} + 2 \underline{\underline{\tilde{\alpha}}}^{[1]} \underline{\underline{\tilde{\alpha}}} + \underline{\underline{\tilde{\alpha}}}^3$$

$$\underline{\underline{\Phi}}\_4(\underline{\underline{\alpha}}) = \underline{\underline{\tilde{\alpha}}}^{[3]} + \underline{\underline{\tilde{\alpha}}} \underline{\underline{\tilde{\alpha}}}^{[2]} + \left( 3 \underline{\underline{\tilde{\alpha}}}^{[1]} + \underline{\underline{\tilde{\alpha}}}^2 \right) \underline{\underline{\tilde{\alpha}}}^{[1]} + 3 \underline{\underline{\tilde{\alpha}}}^{[2]} \cdot \underline{\underline{\tilde{\alpha}}} + 3 \underline{\underline{\tilde{\alpha}}}^{[1]} \cdot \underline{\underline{\tilde{\alpha}}}^2 + 2 \underline{\underline{\tilde{\alpha}}} \cdot \underline{\underline{\tilde{\alpha}}}^{[1]} \cdot \underline{\underline{\tilde{\alpha}}} + \underline{\underline{\tilde{\alpha}}}^4$$

$$\dots$$

$$\underline{\underline{\Phi}}\_n(\underline{\omega}) = \mathbf{P}\_n \underline{\underline{\tilde{\alpha}}}, \mathbf{n} \in \mathbb{N}^\* \tag{135}$$

#### **8. Higher-order kinematics for general 2C manipulator**

We'll apply the general results obtained in the previous chapter for the particular case of four degrees of freedom 2C general manipulator. In this case the relative motions of three bodies *C*0,*C*1,*C*<sup>2</sup> are given, the spatial motion of the terminal body *C*<sup>2</sup> been described by dual orthogonal tensor as it follows:

$$\mathbf{^0\underline{R}}\_2 = \mathbf{^0\underline{R}}\_1 \mathbf{^1\underline{R}}\_2 \tag{136}$$

where

$$\mathbf{^0\underline{R}\_1} = \mathbf{c^{\underline{a}\_t(t)} \mathbf{^0\underline{a}\_t}} \tag{137}$$

$$\mathbf{^1\underline{R}}\_2 = e^{\underline{a}\_2(t)\_1 \underline{\hat{u}}\_2} \tag{138}$$

In Eqs. (138) and (139), the dual angles *α*1ð Þ*t* and *α*2ð Þ*t* are four times differentiable functions, and unit dual vectors <sup>0</sup>**u**<sup>1</sup> and <sup>1</sup>**u**<sup>2</sup> being constant. To simplify the writing, we will denote:

$$\mathbf{u}\_0 \underline{\mathbf{u}}\_1 = \cdot \underline{\mathbf{u}}\_1 \tag{139}$$

$$\mathbf{u}\_{0}\underline{\mathbf{u}}\_{2} = \left(I + \sin\underline{a}\_{10}\bar{\mathbf{u}}\_{1} + (1 - \cos\underline{a}\_{1})\_{0}\bar{\mathbf{u}}\_{1}^{2}\right)\underline{\mathbf{u}}\_{2} = \cdot\underline{\mathbf{u}}\_{2} \tag{140}$$

$$
\underline{\alpha}\_1 = \dot{a}\_1(t) + \epsilon \dot{d}\_1(t) \tag{141}
$$

$$
\underline{\alpha}\_2 = \dot{a}\_2(t) + \epsilon \dot{d}\_2(t) \tag{142}
$$

According to the observations from Section 6, the vector field of the velocity, the acceleration, the jerk, the jounce is uniquely determined by the dual vectors **ω**, **ω**\_ , **ω**€, **ω** *:::*. Taking into account Eq. (133), we will have:

$$
\underline{\mathbf{o}} = \underline{\mathbf{o}}\_1 \underline{\mathbf{u}}\_1 + \underline{\mathbf{o}}\_2 \underline{\mathbf{u}}\_2 \tag{143}
$$

**Author details**

**125**

Daniel Condurache

Technical University of Iasi, Iasi, Romania

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

provided the original work is properly cited.

\*Address all correspondence to: daniel.condurache@tuiasi.ro

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

$$
\underline{\dot{\mathbf{u}}} = \dot{\underline{\omega}}\_1 \underline{\mathbf{u}}\_1 + \dot{\underline{\omega}}\_2 \underline{\mathbf{u}}\_2 + \underline{\omega}\_1 \underline{\omega}\_2 \underline{\mathbf{u}}\_1 \times \underline{\mathbf{u}}\_2 \tag{144}
$$

$$
\underline{\ddot{\mathbf{u}}} = \ddot{\underline{\omega}}\_1 \underline{\mathbf{u}}\_1 + \ddot{\underline{\omega}}\_2 \underline{\mathbf{u}}\_2 + (2\underline{\alpha}\_1 \dot{\underline{\omega}}\_2 + \dot{\underline{\omega}}\_1 \underline{\alpha}\_2) \underline{\mathbf{u}}\_1 \times \underline{\mathbf{u}}\_2 + \underline{\alpha}\_1^2 \underline{\alpha}\_2 \underline{\mathbf{u}}\_1 \times (\underline{\mathbf{u}}\_1 \times \underline{\mathbf{u}}\_2) \tag{145}
$$

$$\underline{\ddot{\mathbf{o}}} = \underline{\ddot{\mathbf{u}}}\_1 \underline{\mathbf{u}}\_1 + \underline{\ddot{\mathbf{o}}}\_2 \underline{\mathbf{u}}\_2 + \left(\underline{\ddot{\mathbf{o}}}\_1 \underline{\mathbf{o}}\_2 + \mathbf{3} \underline{\dot{\mathbf{o}}}\_2 \underline{\dot{\mathbf{o}}}\_2 + \mathbf{3} \underline{\mathbf{o}}\_1 \underline{\dot{\mathbf{o}}}\_2 - \underline{\mathbf{u}}\_1^3 \underline{\mathbf{o}}\_2\right) \underline{\mathbf{u}}\_1 \times \underline{\mathbf{u}}\_2 + \mathbf{3} \left(\underline{\underline{\dot{\mathbf{o}}}}\_2 \underline{\dot{\mathbf{o}}}\_2 + \underline{\dot{\mathbf{o}}}\_2 \underline{\mathbf{o}}\_2 \underline{\mathbf{o}}\_2\right) \underline{\mathbf{u}}\_1 \times \left(\underline{\mathbf{u}}\_1 \times \underline{\mathbf{u}}\_2\right) \tag{146}$$

$$\begin{array}{l} \ddot{\underline{\mathbf{u}}} = \left[ \ddot{\underline{\mathbf{u}}}\_{1} + \mathfrak{Z} \big( \underline{\mathbf{u}}\_{1}^{2} \dot{\underline{\mathbf{u}}}\_{2} + \dot{\underline{\mathbf{u}}}\_{1} \underline{\mathbf{u}}\_{1} \underline{\mathbf{u}}\_{2} \big) \underline{\mathbf{u}}\_{1} \cdot \underline{\mathbf{u}}\_{2} \right] \underline{\mathbf{u}}\_{1} + \left[ \ddot{\underline{\mathbf{u}}}\_{2} - \mathfrak{Z} \big( \underline{\mathbf{u}}\_{1}^{2} \dot{\underline{\mathbf{u}}}\_{2} + \dot{\underline{\mathbf{u}}}\_{1} \underline{\mathbf{u}}\_{1} \underline{\mathbf{u}}\_{2} \big) \right] \underline{\mathbf{u}}\_{2} \\\ + \left( \ddot{\underline{\mathbf{u}}}\_{1} \underline{\mathbf{u}}\_{2} + \mathfrak{Z} \dot{\underline{\mathbf{u}}}\_{1} \dot{\underline{\mathbf{u}}}\_{2} + \dot{\underline{\mathbf{u}}}\_{2} \ddot{\underline{\mathbf{u}}}\_{1} \dot{\underline{\mathbf{u}}}\_{2} - \underline{\mathbf{u}}\_{1}^{3} \underline{\mathbf{u}}\_{2} \right) \underline{\mathbf{u}}\_{1} \times \underline{\mathbf{u}}\_{2} \end{array} \tag{147}$$

Similarly, the results for six degrees of freedom general 3 C manipulator can be obtained, the calculus being a little longer.

#### **9. Conclusions**

The higher-order kinematics properties of rigid body in general motion had been deeply studied. Using the isomorphism between the Lie group of the rigid displacements *S*<sup>3</sup> and the Lie group of the orthogonal dual tensors *S*3, a general method for the study of the field of arbitrary higher-order accelerations is described. It is proved that all information regarding the properties of the distribution of highorder accelerations are contained in the n-th order derivatives of the dual twist of the rigid body. These derivatives belong to the Lie algebra associated to the Lie group *S*3*:*

For the case of the spatial relative kinematics, equations that allow the determination of the n-th order field accelerations are given, using a Brockett-like formulas specific to the dual algebra. In particular cases the properties for velocity, acceleration, jerk, hyper-jerk (jounce) fields are given.

The obtained results interest the theoretical kinematics, jerk and jounce analysis in the case of parallel manipulations, control theory and multibody kinematics.

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

<sup>0</sup>**u**<sup>1</sup> ¼ �**u**<sup>1</sup> (139)

<sup>1</sup>**u**<sup>2</sup> ¼ �**u**<sup>2</sup> (140)

*d*1ð Þ*t* (141)

*d*2ð Þ*t* (142)

<sup>1</sup>ω2**u**<sup>1</sup> � **u**<sup>1</sup> � **u**<sup>2</sup> ð Þ (145)

**<sup>u</sup>**<sup>1</sup> � **<sup>u</sup>**<sup>1</sup> � **<sup>u</sup>**<sup>2</sup> ð Þ (146)

1

**ω** ¼ ω1**u**<sup>1</sup> þ ω2**u**<sup>2</sup> (143)

<sup>1</sup>ω\_ <sup>2</sup> þ ω\_ <sup>1</sup>ω1ω<sup>2</sup>

<sup>1</sup>ω\_ <sup>2</sup> þ ω\_ <sup>1</sup>ω1ω<sup>2</sup> **<sup>u</sup>**<sup>2</sup>

**ω**\_ ¼ ω\_ <sup>1</sup>**u**<sup>1</sup> þ ω\_ <sup>2</sup>**u**<sup>2</sup> þ ω1ω2**u**<sup>1</sup> � **u**<sup>2</sup> (144)

<sup>2</sup> � <sup>3</sup> <sup>ω</sup><sup>2</sup>

<sup>0</sup>**u**<sup>2</sup> <sup>¼</sup> *<sup>I</sup>* <sup>þ</sup> *sin <sup>α</sup>*10**u**~<sup>1</sup> <sup>þ</sup> <sup>1</sup> � *cos <sup>α</sup>*<sup>1</sup> ð Þ0**u**~<sup>2</sup>

<sup>ω</sup><sup>1</sup> <sup>¼</sup> *<sup>α</sup>*\_ <sup>1</sup>ðÞþ*<sup>t</sup> <sup>ε</sup>* \_

<sup>ω</sup><sup>2</sup> <sup>¼</sup> *<sup>α</sup>*\_ <sup>2</sup>ðÞþ*<sup>t</sup> <sup>ε</sup>* \_

acceleration, the jerk, the jounce is uniquely determined by the dual vectors

<sup>1</sup>ω<sup>2</sup> **<sup>u</sup>**<sup>1</sup> � **<sup>u</sup>**<sup>2</sup> <sup>þ</sup> <sup>3</sup> <sup>ω</sup><sup>2</sup>

<sup>1</sup>ω<sup>2</sup>

The higher-order kinematics properties of rigid body in general motion had been deeply studied. Using the isomorphism between the Lie group of the rigid displacements *S*<sup>3</sup> and the Lie group of the orthogonal dual tensors *S*3, a general method for the study of the field of arbitrary higher-order accelerations is described. It is proved that all information regarding the properties of the distribution of highorder accelerations are contained in the n-th order derivatives of the dual twist of the rigid body. These derivatives belong to the Lie algebra associated to the Lie

For the case of the spatial relative kinematics, equations that allow the determination of the n-th order field accelerations are given, using a Brockett-like formulas specific to the dual algebra. In particular cases the properties for velocity, accelera-

The obtained results interest the theoretical kinematics, jerk and jounce analysis

in the case of parallel manipulations, control theory and multibody kinematics.

Similarly, the results for six degrees of freedom general 3 C manipulator can be

*:::*

**<sup>u</sup>**<sup>1</sup> � **<sup>u</sup>**<sup>2</sup> (147)

. Taking into account Eq. (133), we will have:

**<sup>ω</sup>**€ <sup>¼</sup> <sup>ω</sup>€1**u**<sup>1</sup> <sup>þ</sup> <sup>ω</sup>€2**u**<sup>2</sup> <sup>þ</sup> <sup>2</sup>ω1ω\_ <sup>2</sup> <sup>þ</sup> <sup>ω</sup>\_ <sup>1</sup>ω<sup>2</sup> ð Þ**u**<sup>1</sup> � **<sup>u</sup>**<sup>2</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup>

<sup>2</sup>**u**<sup>2</sup> <sup>þ</sup> <sup>ω</sup>€1ω<sup>2</sup> <sup>þ</sup> <sup>3</sup>ω\_ <sup>1</sup>ω\_ <sup>2</sup> <sup>þ</sup> <sup>3</sup>ω1ω€<sup>2</sup> � <sup>ω</sup><sup>3</sup>

<sup>1</sup>ω\_ <sup>2</sup> þ ω\_ <sup>1</sup>ω1ω<sup>2</sup> **<sup>u</sup>**<sup>1</sup> � **<sup>u</sup>**<sup>2</sup> **<sup>u</sup>**<sup>1</sup> <sup>þ</sup> <sup>ω</sup>

<sup>þ</sup> <sup>ω</sup>€1ω<sup>2</sup> <sup>þ</sup> <sup>3</sup>ω\_ <sup>1</sup>ω\_ <sup>2</sup> <sup>þ</sup> <sup>3</sup>ω1ω€<sup>2</sup> � <sup>ω</sup><sup>3</sup>

*Advances on Tensor Analysis and Their Applications*

obtained, the calculus being a little longer.

tion, jerk, hyper-jerk (jounce) fields are given.

**ω**, **ω**\_ , **ω**€, **ω** *:::*

> **ω** *:::* ¼ ω *:::* <sup>1</sup>**u**<sup>1</sup> þ ω *:::*

> > **ω** *:::* ¼ ω *:::*

**9. Conclusions**

group *S*3*:*

**124**

<sup>1</sup> <sup>þ</sup> <sup>3</sup> <sup>ω</sup><sup>2</sup>

According to the observations from Section 6, the vector field of the velocity, the

#### **Author details**

Daniel Condurache Technical University of Iasi, Iasi, Romania

\*Address all correspondence to: daniel.condurache@tuiasi.ro

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

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[2] Ball RS. Theory of Screws. London: Cambridge University Press Warehouse; 1900

[3] Hunt KH. Kinematic Geometry of Mechanisms. New York: Oxford University Press; 1978

[4] Phillips J. Freedom in Machinery: Introducing Screw Theory. Vol. 1. Cambridge: Cambridge University Press; 1984

[5] Karger A. Singularity analysis of serial robot-manipulators. ASME Journal of Mechanical Design. 1996; **118**(4):520-525

[6] Rico JM, Gallardo J, Duffy J. Screw theory and higher order kinematic analysis of open serial and closed chains. Mechanism and Machine Theory. 1999; **34**(4):559-586

[7] Lerbet J. Analytic geometry and singularities of mechanisms. Zeitschrift für Angewandte Mathematik und Mechanik. 1999;**78**(10):687-694

[8] Müller A. Higher derivatives of the kinematic mapping and some applications. Mechanism and Machine Theory. 2014;**76**:70-85

[9] Müller A. An overview of formulae for the higher-order kinematics of lower-pair chains with applications in robotics and mechanism theory. Mechanism and Machine Theory. 2019;**142**

[10] Condurache D, Matcovschi M. Computation of angular velocity and acceleration tensors by direct measurements. Acta Mechanica. 2002; **153**(3–4):147-167

[11] Angeles J. The application of dual algebra to kinematic analysis, computational methods. Mechanical Systems. 1998;**161**:3-31

[19] Condurache D. Higher-order kinematics of rigid bodies. A tensors algebra approach. In: Kecskeméthy A, Flores FG, Carrera E, Elias DA, editors. Interdisciplinary Applications of Kinematics. Mechanisms and Machine Science. Vol. 71. Cham: Springer; 2019.

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

[20] Condurache D. Higher-order relative kinematics of rigid body

[21] Condurache D. Higher-order acceleration centers and kinematic invariants of rigid body. In: The 5th Joint International Conference on Multibody System Dynamics; Lisbon;

[22] Condurache D. Higher-order Rodrigues dual vectors. Kinematic equations and tangent operator. In: The 5th Joint International Conference on Multibody System Dynamics; Lisbon;

[23] Pennestrı E, Valentini PP. Linear dual algebra algorithms and their application to kinematics. In: Multibody Dynamics, Computational Methods and Applications. Dordrecht: Springer;

[24] Samuel A, McAree P, Hunt K. Unifying screw geometry and matrix transformations. The International Journal of Robotics Research. 1991;

[25] Veldkamp GR. Canonical systems and instantaneous invariants in spatial kinematics. Journal of Mechanisms.

motions: A dual lie algebra approach. In: Lenarcic J, Parenti-Castelli V, editors. Advances in Robot Kinematics. Vol. 8.

pp. 215-225

2018. pp. 83-91

2018

2018

2009. pp. 207-229

**10**(5):454-472

1967;**2**(3):329-388

**127**

[12] Bokelberg EH, Hunt KH, Ridley PR. Spatial motion-I: Points of inflection and the differential geometry of screws Raumliche bewegung-I wendepunkte und die differentiale schraubengeometrie. Mechanism and Machine Theory. 1992; **27**(1):1-15

[13] Condurache D, Burlacu A. Orthogonal dual tensor method for solving the AX=XB sensor calibration problem. Mechanism and Machine Theory. 2016;**104**:382-404

[14] Condurache D, Burlacu A. Dual tensors based solutions for rigid body motion parameterization. Mechanism and Machine Theory. 2014;**74**:390-412

[15] Condurache D, Burlacu A. Dual lie algebra representations of the rigid body motion. In: AIAA/AAS Astrodynamics Specialist Conference; San Diego, CA; 2014

[16] Condurache D. A Davenport dual angles approach for minimal parameterization of the rigid body displacement and motion. Mechanism and Machine Theory. 2019; **140**:104-122

[17] Condurache D. General rigid body motion parameterization using modified Cayley transform for dual tensors and dual quaternions. In: The 4th Joint International Conference on Multibody System Dynamics; Montreal; 2016

[18] Condurache D. Higher-order accelerations on rigid bodies motions. A tensors and dual lie algebra approach. Acta Technica Napocensis – Series: Applied Mathematics, Mechanics and Engineering. 2018;**61**(1):29-38

*Higher-Order Kinematics in Dual Lie Algebra DOI: http://dx.doi.org/10.5772/intechopen.91779*

**References**

33-66

[1] Everett JD. On the kinematics of a rigid body. The Quarterly Journal of Pure and Applied Mathematics. 1875;**13**:

*Advances on Tensor Analysis and Their Applications*

[11] Angeles J. The application of dual

[12] Bokelberg EH, Hunt KH, Ridley PR. Spatial motion-I: Points of inflection and the differential geometry of screws Raumliche bewegung-I wendepunkte und die differentiale schraubengeometrie. Mechanism and Machine Theory. 1992;

algebra to kinematic analysis, computational methods. Mechanical

[13] Condurache D, Burlacu A. Orthogonal dual tensor method for solving the AX=XB sensor calibration problem. Mechanism and Machine

Theory. 2016;**104**:382-404

[14] Condurache D, Burlacu A. Dual tensors based solutions for rigid body motion parameterization. Mechanism and Machine Theory. 2014;**74**:390-412

[15] Condurache D, Burlacu A. Dual lie algebra representations of the rigid body motion. In: AIAA/AAS Astrodynamics Specialist Conference; San Diego, CA;

[16] Condurache D. A Davenport dual

Mechanism and Machine Theory. 2019;

[17] Condurache D. General rigid body motion parameterization using modified Cayley transform for dual tensors and dual quaternions. In: The 4th Joint International Conference on

Multibody System Dynamics; Montreal;

[18] Condurache D. Higher-order accelerations on rigid bodies motions. A tensors and dual lie algebra approach. Acta Technica Napocensis – Series: Applied Mathematics, Mechanics and Engineering. 2018;**61**(1):29-38

angles approach for minimal parameterization of the rigid body

displacement and motion.

Systems. 1998;**161**:3-31

**27**(1):1-15

2014

**140**:104-122

2016

[2] Ball RS. Theory of Screws. London:

[3] Hunt KH. Kinematic Geometry of Mechanisms. New York: Oxford

[4] Phillips J. Freedom in Machinery: Introducing Screw Theory. Vol. 1. Cambridge: Cambridge University

[5] Karger A. Singularity analysis of serial robot-manipulators. ASME Journal of Mechanical Design. 1996;

[6] Rico JM, Gallardo J, Duffy J. Screw theory and higher order kinematic analysis of open serial and closed chains. Mechanism and Machine Theory. 1999;

[7] Lerbet J. Analytic geometry and singularities of mechanisms. Zeitschrift für Angewandte Mathematik und Mechanik. 1999;**78**(10):687-694

[8] Müller A. Higher derivatives of the

applications. Mechanism and Machine

[9] Müller A. An overview of formulae for the higher-order kinematics of lower-pair chains with applications in robotics and mechanism theory. Mechanism and Machine Theory.

[10] Condurache D, Matcovschi M. Computation of angular velocity and

measurements. Acta Mechanica. 2002;

acceleration tensors by direct

**153**(3–4):147-167

kinematic mapping and some

Theory. 2014;**76**:70-85

2019;**142**

**126**

Cambridge University Press

Warehouse; 1900

Press; 1984

**118**(4):520-525

**34**(4):559-586

University Press; 1978

[19] Condurache D. Higher-order kinematics of rigid bodies. A tensors algebra approach. In: Kecskeméthy A, Flores FG, Carrera E, Elias DA, editors. Interdisciplinary Applications of Kinematics. Mechanisms and Machine Science. Vol. 71. Cham: Springer; 2019. pp. 215-225

[20] Condurache D. Higher-order relative kinematics of rigid body motions: A dual lie algebra approach. In: Lenarcic J, Parenti-Castelli V, editors. Advances in Robot Kinematics. Vol. 8. 2018. pp. 83-91

[21] Condurache D. Higher-order acceleration centers and kinematic invariants of rigid body. In: The 5th Joint International Conference on Multibody System Dynamics; Lisbon; 2018

[22] Condurache D. Higher-order Rodrigues dual vectors. Kinematic equations and tangent operator. In: The 5th Joint International Conference on Multibody System Dynamics; Lisbon; 2018

[23] Pennestrı E, Valentini PP. Linear dual algebra algorithms and their application to kinematics. In: Multibody Dynamics, Computational Methods and Applications. Dordrecht: Springer; 2009. pp. 207-229

[24] Samuel A, McAree P, Hunt K. Unifying screw geometry and matrix transformations. The International Journal of Robotics Research. 1991; **10**(5):454-472

[25] Veldkamp GR. Canonical systems and instantaneous invariants in spatial kinematics. Journal of Mechanisms. 1967;**2**(3):329-388

## *Edited by Francisco Bulnes*

This book brings together recent advances in tensor analysis and studies of its invariants such as twistors, spinors, kinematic tensors and others belonging to tensor algebras with extended structures to Lie algebras, Kac-Moody algebras, and enveloping algebras, among others. Chapters cover such topics as classical tensors and bilinear forms, tensors for exploring space–time, tensor applications in geometry and continuum media, and advanced topics in tensor analysis such as invariant theory, derived categories, hypercohomologies, k-modules, extensions of kinematic tensors, infinite dimensional operators, and more.

Published in London, UK © 2020 IntechOpen © Goettingen / iStock

Advances on Tensor Analysis and their Applications

Advances on Tensor Analysis

and their Applications

*Edited by Francisco Bulnes*