**4. Dual Jacobian matrix**

If a point P on a body j is moving with respect to a body I (Fig. 6), the velocity can be expressed in terms of inertial frame R , *R P*ˆ*Vj <sup>i</sup>* .

Fig. 6. Dual velocity scheme.

When the dual velocity needs to be represented in terms of frame *Q* , a rotation from frame *R* is done:

$${}^{Q}\hat{V}\_{j,i}^{P} = {}^{Q}T\_p {}^{P}\hat{V}\_{j,i}^{P} \tag{20}$$

Kinematic and Dynamic Modelling of

**5. Dynamic analysis: Dual force** 

around to screw.

Fig. 8. Example of dual force

**5.1 Dual momentum** 

The terms & *B B P H p p*

respectively, in terms of frame "B".

"A" in terms of the frame "B" is given by:

means that it can not be established as a dual vector.

represented in an expression called dual force:

Serial Robotic Manipulators Using Dual Number Algebra 77

One of the most important features of dual number formulation is the capability of generalization for a great variety of robot topologies, without modifying the main program, this is an advantage when compared to typical homogenous matrices wherein is required to

In dual algebra, if a force and a momentum act with respect a coordinate system, they can be

A clear example would be a screwdriver where is necessary to apply a force axially and

If a dual force is applied on a point "B" different to the origin point "A", the effect on the point "B" will be determined by a coordinate transformation. Then a dual force applied on

> *B BA BA* ˆ *F TF T <sup>A</sup> AA AA*

Dual momentum concept is introduced due to the acceleration is a dual pseudo-vector, that

 *B BB* <sup>ˆ</sup> *A pp A H PH*

are the linear and angular momentum of a particle "p" on a body "A"

(25)

(26)

(24)

ˆ *F F*

specify in dynamical model whether a joint is rotational or prismatic.

The block matrix 012 1 ˆˆˆ ˆ ... *nnn n MMM Mn* is called Dual Jacobian matrix (Brodsky).

The relative velocity of a link *<sup>k</sup>* with respect to link *<sup>i</sup>* , <sup>ˆ</sup> *Vk i* in dual form is established as:

$$
\hat{V}\_{k,i} = \hat{V}\_{j,i} + \hat{V}\_{k,j} \tag{21}
$$

Fig. 7. Relative dual velocity theorem in a kinematic chain.

From dual velocities theorem, the vector of dual velocities in the end of *n* link in terms of the *n* frame can be found as:

$${}^{0}\hat{V}\_{n,0}^{n} = {}^{0}T\_{n} \sum\_{i=1}^{n} {}^{n}\hat{M}\_{i-1} \, {}^{i-1}\hat{V}\_{i,i-1}^{i-1} \tag{22}$$

Where <sup>0</sup> *Tn* is the primary component of the dual matrix(19).

The generalized dual Jacobian matrix is obtained by applying the relative velocity theorem in dual form. The differential motions, whether axial or radial, are expressed in a matrix formed by the dual homogenous matrices, in contrast with the conventional Jacobian matrix that is obtained from specific columns of homogeneous transformation matrix (Sai-Kai, 2000).

$${}^{0}\hat{V}\_{n,0}^{n} = \begin{bmatrix} \alpha\_{x} + \varepsilon V\_{x} \\ \alpha\_{y} + \varepsilon V\_{y} \\ \alpha\_{z} + \varepsilon V\_{z} \end{bmatrix} = \begin{bmatrix} {}^{0}T\_{n} \\ \end{bmatrix} \begin{bmatrix} {}^{n}\hat{M}\_{0} & {}^{n}\hat{M}\_{1} & {}^{n}\hat{M}\_{2} \dots & {}^{n}\hat{M}\_{n-1} \end{bmatrix} \begin{bmatrix} \frac{{}^{0}\hat{V}\_{1,0}^{0}}{} \\ \frac{{}^{1}\hat{V}\_{2,1}^{1}}{} \\ \vdots \\ {}^{n-1}\hat{V}\_{n,n-1}^{n-1} \end{bmatrix} \tag{23}$$

The block matrix 012 1 ˆˆˆ ˆ ... *nnn n MMM Mn* is called Dual Jacobian matrix (Brodsky).
