**5.1 Dual momentum**

Dual momentum concept is introduced due to the acceleration is a dual pseudo-vector, that means that it can not be established as a dual vector.

$$\prescript{B}{}{\hat{H}}\_A = \int\_A \left( \prescript{B}{}{\vec{P}}\_p + \varepsilon^B \prescript{B}{}{\vec{H}}\_p \right) \tag{26}$$

The terms & *B B P H p p* are the linear and angular momentum of a particle "p" on a body "A" respectively, in terms of frame "B".

Kinematic and Dynamic Modelling of

From Table 2, the dual angles ˆ

ˆ 

11 1

<sup>2</sup> , <sup>2</sup> *d* ; for example if

<sup>1</sup> *<sup>d</sup>* <sup>2</sup> *<sup>d</sup>*

1

<sup>1</sup> *<sup>d</sup>* <sup>2</sup> *<sup>d</sup>*

1

Fig. 10. Possible topologies for different values of

1 Table 2. Denavith and Hartemberg parameters of 2C robotic arm.

*d* 22 2 ˆ 

 &

2

<sup>2</sup> *d*

i

1

2

<sup>1</sup> , <sup>1</sup> *d* ,

Serial Robotic Manipulators Using Dual Number Algebra 79

*<sup>i</sup> Si <sup>i</sup> a*

ˆ are constructed as:

It is observed that different topologies can be solved from the assigned values to

then nine different robots can be solved from the same aforementioned equations.

*d* 11 1 ˆ 

1 *d*

1 *d*

1 *d*

1 2 & .

<sup>2</sup> *d* <sup>2</sup> *d*

2

<sup>1</sup> <sup>1</sup> *d* <sup>1</sup>*l* 90°

 *a* 22 2 ˆ *a*

<sup>1</sup> is 0 the robot will change the original topology CC to PC

1

1

<sup>2</sup> <sup>2</sup> *d* <sup>2</sup>*l* 0°

*i*

2

2

2

$$\begin{aligned} \,^B P\_A &= m\_A \left[ \,^B V\_A^B \right] - \left[ \,^B S\_A^B \right] \left[ \,^B \alpha\_A^B \right] \\\\ \,^B H\_A &= \left[ \,^B S\_A^B \right] \left[ \,^B V\_A^B \right] + \left[ \,^B J\_A^B \right] \left[ \,^B \alpha\_A^B \right] \end{aligned} $$

#### **5.2 Dual inertial force**

According with (Pennock & Meehan, 2000) the dual inertial forces on a rigid body are the derivative of the dual momentum:

 *B B* <sup>ˆ</sup> <sup>ˆ</sup> *<sup>A</sup> <sup>A</sup> <sup>d</sup> f H dt* (27) ˆ ˆ ˆ 0 ˆ ˆ ˆ ˆ ˆˆ 0 ˆ ˆ 0 ˆ ˆ *B BB BB B Ai Ak Aj Ai B B BB BB B A Aj Ak Ai Aj BB BB <sup>B</sup> <sup>B</sup> Ak Aj Ai Ak H V V H f H V VH H V V H* ˆ ˆˆ ˆˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆ ˆ ˆˆ ˆˆ *B B B B B BB <sup>B</sup> Ai Ak Aj Aj Ak Ai B B B BB B B B Aj Aj Ai Ak Ak Ai B B B BB B BB Ak Ak Aj Ai Ai Aj f H VH VH f H VH VH f H VH VH* 

#### **5.3 Dynamic equilibrium**

Extending the D'Alembert principle to dual numbers for dynamic equilibrium

$${}^{B}\hat{M}\_{A}{}^{A}\hat{F}\_{A} - {}^{B}F\_{B} = {}^{B}\hat{f}\_{A} \tag{28}$$

#### **6. Example: Robot with cylindrical joints**

The robot shown in the Figure 9 is a clear example where the dual numbers can be employed:

Fig. 9. Assignment of reference systems and Denavit-Hartenberg parameters.

*B BB B B B <sup>B</sup> Pm V S <sup>A</sup> AA A A*

*B B B B B BB B B H SV J <sup>A</sup> AA AA*

According with (Pennock & Meehan, 2000) the dual inertial forces on a rigid body are the

*B B* <sup>ˆ</sup> <sup>ˆ</sup> *<sup>A</sup> <sup>A</sup>*

ˆ ˆ ˆ ˆˆ 0

*f H V VH*

ˆ ˆˆ ˆˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆ ˆ ˆˆ ˆˆ

Extending the D'Alembert principle to dual numbers for dynamic equilibrium

Fig. 9. Assignment of reference systems and Denavit-Hartenberg parameters.

*f H VH VH f H VH VH f H VH VH*

*B B B B B BB <sup>B</sup> Ai Ak Aj Aj Ak Ai B B B BB B B B Aj Aj Ai Ak Ak Ai B B B BB B BB Ak Ak Aj Ai Ai Aj*

The robot shown in the Figure 9 is a clear example where the dual numbers can be

*B B BB BB B A Aj Ak Ai Aj BB BB <sup>B</sup> <sup>B</sup> Ak Aj Ai Ak*

 

ˆ ˆ ˆ 0 ˆ

*H V V H*

*B BB BB B Ai Ak Aj Ai*

ˆ ˆ 0 ˆ ˆ

*H V V H* 

**5.2 Dual inertial force** 

**5.3 Dynamic equilibrium** 

employed:

**6. Example: Robot with cylindrical joints** 

derivative of the dual momentum:

*<sup>d</sup> f H dt* (27)

*BA B B* ˆ ˆ <sup>ˆ</sup> *MA F F AB A <sup>f</sup>* (28)

1

<sup>1</sup> *<sup>d</sup>* <sup>2</sup> *<sup>d</sup>*

2


Table 2. Denavith and Hartemberg parameters of 2C robotic arm.

From Table 2, the dual angles ˆ &ˆ are constructed as:

$$
\hat{\theta}\_1 = \theta\_1 + \varepsilon d\_1 \qquad \qquad \hat{\theta}\_2 = \theta\_2 + \varepsilon d\_2 \qquad \qquad \hat{a}\_1 = a\_1 + \varepsilon a\_1 \qquad \qquad \hat{a}\_2 = a\_2 + \varepsilon a\_2
$$

It is observed that different topologies can be solved from the assigned values to <sup>1</sup> , <sup>1</sup> *d* , <sup>2</sup> , <sup>2</sup> *d* ; for example if <sup>1</sup> is 0 the robot will change the original topology CC to PC then nine different robots can be solved from the same aforementioned equations.

Fig. 10. Possible topologies for different values of 1 2 & .

Kinematic and Dynamic Modelling of

Dual velocities:

11 1 00

**6.1 Dynamic analysis** 

Derivating the above expressions:

1,0 0 1,0 11 1

*V MV v*

ˆ ˆˆ sin ( ) ˆ

Serial Robotic Manipulators Using Dual Number Algebra 81

0 2 0 1 1 1 2,0 1 2 1 2 1 2 2

Once obtained the velocities, the next step for solving the dynamic equations is, according

 1 11 1 11 1 11 1 11 0 1 1,0 1 1,0 1 1,0 1 1,0 *H mV S* <sup>ˆ</sup>

 2 22 2 22 2 22 2 22 1 2 2,0 2 2,0 2 2,0 2 2,0 *HmV S* <sup>ˆ</sup>

> 1 1 11 1 1 0 1 1,0 1 1,0

<sup>ˆ</sup> *d d Hm V S*

1 11 1 11 1 1,0 1 1,0

 

> *i i j j k k i kj jk j ik ki k ji ij*

*i k j k j j k j k*

*H H VP H VP*

 11 1 11 1 11 1 11 1 0 1,0 0 1,0 0 1,0 0 1,0 0 1 11 1 11 1 11 1 11 1 0 1,0 0 1,0 0 1,0 0 1,0 0 *j i k ik k i ki k j i j i i j i j*

*H H VP H VP*

 

*d d SV I dt dt*

1 1 0 0 1 1 0 0 1 1 0 0 1 11 1 11 1 0 1,0 0 1,0 0 1 11 1 11 1 0 1,0 0 1,0 0 1 11 1 11 1 0 1,0 0 1,0 0 1 11 1 11 1 11 1 11 1 0 1,0 0 1,0 0 1,0 0 1,0 0

 

*f t f t f t PPP PP P P PP*

 

*dt dt*

 

 

 

 

 

 

 

> 

*H VP H V P*

 

 

*T T*

22 2 11

*SV I*

*SV I*

 

0

2 2

*v*

2,0 1 2,0 22 2

*V MV v*

ˆ ˆˆ sin ( ) ˆ

22 2

 

 

*v*

0

cos ( ) ˆ

0 0

0

 

 

*v*

0

12 12 1

*cc cs s*

ˆ ˆ ˆ

*V sc ss c M M*

2 2

11 1

 

 

*v*

0

cos ( ) ˆ

with Fisher, to compute the dual momentum, being for each link:

1

*H*

*s c*

$${}^{0}\hat{M}\_{2} = \begin{bmatrix} \cos\hat{\theta}\_{1} & -\cos\hat{\alpha}\_{1}\sin\hat{\theta}\_{1} & \sin\hat{\alpha}\_{1}\sin\hat{\theta}\_{1} \\ \sin\hat{\theta}\_{1} & \cos\hat{\alpha}\_{1}\cos\hat{\theta}\_{1} & -\sin\hat{\alpha}\_{1}\cos\hat{\theta}\_{1} \\ 0 & \sin\hat{\alpha}\_{1} & \cos\hat{\alpha}\_{1} \end{bmatrix} \begin{bmatrix} \cos\hat{\theta}\_{2} & -\cos\hat{\alpha}\_{2}\sin\hat{\theta}\_{2} & \sin\hat{\alpha}\_{2}\sin\hat{\theta}\_{2} \\ \sin\hat{\theta}\_{2} & \cos\hat{\alpha}\_{2}\cos\hat{\theta}\_{2} & -\sin\hat{\alpha}\_{2}\cos\hat{\theta}\_{2} \\ 0 & \sin\hat{\alpha}\_{2} & \cos\hat{\alpha}\_{2} \end{bmatrix}$$

For the inverse solution:

$$\begin{aligned} ^0\hat{M}\_2 &= \begin{bmatrix} n\_x & o\_x & a\_x \\ n\_y & o\_y & a\_y \\ n\_z & o\_z & a\_z \end{bmatrix} \\ + \varepsilon \begin{bmatrix} -p\_z n\_y + p\_y n\_z & -p\_z o\_y + p\_y o\_z & -p\_z a\_y + p\_y a\_z \\ p\_z n\_x - p\_x n\_z & p\_z o\_x - p\_x o\_z & p\_x a\_x - p\_x a\_z \\ -p\_y n\_x + p\_x n\_y & -p\_y o\_x + p\_x o\_y & -p\_y a\_x + p\_z a\_y \end{bmatrix} \\ \end{aligned} $$

$$^0\hat{M}\_2 = \begin{bmatrix} c\_1 c\_2 & -c\_1 s\_2 & s\_1 \\ s\_1 c\_2 & -s\_1 s\_2 & -c\_1 \\ s\_2 & c\_2 & 0 \end{bmatrix} $$

$$+ \varepsilon \begin{bmatrix} 100.5\,\mathrm{s\_1}\,\mathrm{s\_2} - d\_1\,\mathrm{s\_1}\,\mathrm{c\_2} - d\_2\,\mathrm{c\_1}\,\mathrm{s\_2} & 70\,\mathrm{s\_1} + 100.5\,\mathrm{s\_1}\,\mathrm{c\_2} - d\_2\,\mathrm{c\_1}\,\mathrm{c\_2} + d\_1\,\mathrm{s\_1}\,\mathrm{s\_2} & \mathrm{c\_1}(d\_1 + 70\,\mathrm{s\_2}) \\ 100.5\,\mathrm{c\_1}\,\mathrm{s\_2} + d\_1\,\mathrm{c\_1}\,\mathrm{c\_2} - d\_2\,\mathrm{s\_1}\,\mathrm{s\_2} & 70\,\mathrm{c\_1} - 100.5\,\mathrm{s\_1}\,\mathrm{s\_2} & \mathrm{c\_2}(d\_1 + 70\,\mathrm{s\_2}) \\ d\_2\,\mathrm{c\_2} & -d\_2\,\mathrm{s\_2} & -70\,\mathrm{s\_2} \end{bmatrix} $$

Dividing elements (2,1) and (1,1) in both matrices:

$$\theta\_1 = \text{tg}^{-1}\left(\frac{n\_y}{n\_x}\right);\ \theta\_2 = \text{tg}^{-1}\left(\frac{\pm\sqrt{1-o\_z^2}}{o\_z^2}\right)$$

The velocities in the cylindrical joints are given by:

$$\begin{aligned} \,^0\hat{V}\_{1,0}^0 = \begin{bmatrix} 0 \\ 0 \\ a\_1 \end{bmatrix} + \varepsilon \begin{bmatrix} 0 \\ 0 \\ v\_1 \end{bmatrix} \end{aligned} \qquad \qquad \begin{aligned} \,^1\hat{V}\_{2,1}^1 = \begin{bmatrix} 0 \\ 0 \\ a\_2 \end{bmatrix} + \varepsilon \begin{bmatrix} 0 \\ 0 \\ v\_2 \end{bmatrix} \end{aligned}$$

Computing the velocities on the end-effector:

$${}^{0}\hat{V}\_{2,0}^{2} = {}^{0}T\_{2}\left\{{}^{2}\hat{M}\_{0}\,{}^{0}\hat{V}\_{1,0}^{0} + {}^{2}\hat{M}\_{1}\,{}^{1}\hat{V}\_{2,1}^{1}\right\}$$

The above expression can be rewritten in terms of dual Jacobian matrix.

$$\begin{aligned} \,^0\hat{V}\_{2,0}^2 = \begin{bmatrix} \,^0\boldsymbol{\alpha}\_x + \varepsilon \boldsymbol{V}\_x\\ \,^0\boldsymbol{\alpha}\_y + \varepsilon \boldsymbol{V}\_y\\ \,^0\boldsymbol{\alpha}\_z + \varepsilon \boldsymbol{V}\_z \end{bmatrix} = \begin{bmatrix} \,^0T\_2 \end{bmatrix} \begin{bmatrix} \,^2\hat{M}\_0 & \vert & \,^2\hat{M}\_1 \end{bmatrix} \begin{bmatrix} \,^0\hat{V}\_{1,0}^0\\ \,^1\hat{V}\_{2,1}^1 \end{bmatrix} \end{aligned} $$

$$\begin{aligned}{}^{0}\hat{V}\_{2,0}^{2} = \begin{bmatrix} c\_{1}c\_{2} & -c\_{1}s\_{2} & s\_{1} \\ s\_{1}c\_{2} & -s\_{1}s\_{2} & -c\_{1} \\ s\_{2} & c\_{2} & 0 \end{bmatrix} \begin{bmatrix} ^{0}\hat{M}\_{2}^{T} & ^{1}\hat{M}\_{2}^{T} \end{bmatrix} \begin{bmatrix} o\_{1} + \varepsilon v\_{1} \\ 0 \\ 0 \\ 0 \\ o\_{2} + \varepsilon v\_{2} \end{bmatrix} \end{aligned}$$

Dual velocities:

80 Serial and Parallel Robot Manipulators – Kinematics, Dynamics, Control and Optimization

2 1 11 11 2 22 22

<sup>2</sup> <sup>ˆ</sup> *xxx yyy zzz z y y z z y y z z y y z z x x z zx xz zx xz y x x y y x x y y x x y*

*M noa*

 

 

0

0

ˆ

*d d*

1

1 1

The above expression can be rewritten in terms of dual Jacobian matrix.

*V*

*x x y y z z*

 

 

 

*v* 

0 0

ˆ*V* 0 0

 ;

*tg*

*y x n*

*n*

1

ˆ ˆ ˆˆ ˆ ˆ cos cos sin sin sin cos cos sin sin sin ˆˆ ˆˆ ˆ ˆ ˆ ˆˆ ˆ ˆ sin cos cos sin cos sin cos cos sin cos ˆˆ ˆˆ 0 sin ˆˆ ˆˆ cos 0 sin cos

*noa*

 

*noa p n pn po po pa pa p n pn po po pa pa p n pn po po pa pa*

12 12 1

*cc cs s M sc ss c s c*

12 112 212 1 12 212 112 1 1 2 12 112 212 1 12 112 212 1 1 2 2 2 2 2 2

*d d dd d d d dd d*

 

0

c s 70c 100.5

1 2 2

> 1 1 2,1

*<sup>V</sup> <sup>V</sup>*

 

ˆ

*tg*

2

2 2

*v* 

0 0

*V*

ˆ

2,1

0 0

*o*

1 *<sup>z</sup> z*

*o*

 

*V* 0 0

2 12 12 1 2 2

100.5s s s c c s 70s 100.5s c c c s s c 70s 100.5c s c c s s 70c 100.5c c c s s c s 70s

 02 0 2 00 2 11 2,0 2 0 1,0 1 2,1 *V T MV MV* ˆ ˆˆ ˆˆ

1,0 0 2 02 2 2,0 2 0 1 1 1

<sup>ˆ</sup> ˆ ˆ <sup>ˆ</sup>

*V V TM M*

 

0

*M*

For the inverse solution:

 

Dividing elements (2,1) and (1,1) in both matrices:

The velocities in the cylindrical joints are given by:

0 0 1,0

Computing the velocities on the end-effector:

1 11 11 2 22 22

 

 

11 2 2

 

   

 

 

11 1 00 1,0 0 1,0 11 1 11 1 0 ˆ ˆˆ sin ( ) ˆ cos ( ) ˆ *V MV v v* 22 2 11 2,0 1 2,0 22 2 22 2 0 ˆ ˆˆ sin ( ) ˆ cos ( ) ˆ *V MV v v* 

#### **6.1 Dynamic analysis**

Once obtained the velocities, the next step for solving the dynamic equations is, according with Fisher, to compute the dual momentum, being for each link:

$$\begin{aligned} \hat{H}\_0 &= m\_1 \Big[\,^1\vec{V}\_{1,0}^1\Big] - \,^1S\_1\left[\,^1\vec{\phi}\_{1,0}^1\right] + \mathcal{E}\left\{\left[\,^1S\_1\right]\left[\,^1\vec{V}\_{1,0}^1\right] + \left[\,^1I\_1\right]\left[\,^1\vec{\phi}\_{1,0}^1\right]\right\} \\\\ \hat{H}\_1 &= m\_2 \Big[\,^2\vec{V}\_{2,0}^2\Big] - \,^2S\_2\left[\,^2\vec{\phi}\_{2,0}^2\right] + \mathcal{E}\left\{\left[\,^2S\_2\right]\left[\,^2\vec{V}\_{2,0}^2\right] + \left[\,^2I\_2\right]\left[\,^2\vec{\phi}\_{2,0}^2\right]\right\} \end{aligned}$$

Derivating the above expressions:

 1 1 11 1 1 0 1 1,0 1 1,0 1 11 1 11 1 1,0 1 1,0 <sup>ˆ</sup> *d d Hm V S dt dt d d SV I dt dt* 1 1 0 0 1 1 0 0 1 1 0 0 1 11 1 11 1 0 1,0 0 1,0 0 1 11 1 11 1 0 1,0 0 1,0 0 1 11 1 11 1 0 1,0 0 1,0 0 1 11 1 11 1 11 1 11 1 0 1,0 0 1,0 0 1,0 0 1,0 0 1 *i i j j k k i kj jk j ik ki k ji ij i k j k j j k j k f t f t f t PPP PP P P PP H H VP H VP H* 11 1 11 1 11 1 11 1 0 1,0 0 1,0 0 1,0 0 1,0 0 1 11 1 11 1 11 1 11 1 0 1,0 0 1,0 0 1,0 0 1,0 0 *j i k ik k i ki k j i j i i j i j H VP H V P H H VP H VP* 

Kinematic and Dynamic Modelling of

**8. Acknowledgments** 

robots.

**9. References** 

for their valuable comments and suggestions.

Serial Robotic Manipulators Using Dual Number Algebra 83

The dynamic model is treated by using the dual momentum, wherein the inertial forces are computed by means of a set of linear equations, thus a 6 *n* vector of forces is calculated, and in consequence one obtains a complete description of the robotic manipulator. An appropriate way of dual numbers programming will yield a suitable software alternative to

The authors gratefully acknowledge the support of CONACYT, IPN and ICyTDF for research projects and scholarships. They also would like to thank the anonymous reviewers

Al-Widyan, K.; Qing Ma, Xiao & Angeles, J. (2011). The robust design of parallel spherical

Bandyopadhyay, S. (2004). Analytical determination of principal twists in serial, parallel and

Bandyopadhyay, S. (2006). *Analysis and Design of Spatial Manipulators: An Exact Algebraic* 

Bayro-Corrochano, E. & Kähler, D. (2000). Motor algebra approach for computing the

Bayro-Corrochano, E. & Kähler, D. (2001), Kinematics of Robot Manipulators in the Motor

Institute of Computer Science and Applied Mathematics, University of Kiel. Brodsky, V. &Shoham, M. (1998). *Derivation of dual forces in robot manipulators*. Journal of

Cecchini, E., Pennestri, E., & Vergata, T. (2004). A dual number approach to the Kinematic

Cheng, H.H. (1994). Programming with dual numbers and its applications in mechanisms

Fischer, I. (2003). Velocity analysis of mechanisms with ball joints. *Journal of Mechanics* 

Fisher, I. S. (1998). *Dual Number Methods in Kinematics, Statics and dynamics*. (1st Edition),

Fisher, I. S. (1998). The dual angle and axis of a screw motion. *Journal of Mechanisms and* 

Fisher, I. S. (2000). Numerical analysis of displacements in spatial mechanisms with ball

*Machine Theory*, Vol. 39, (2004), pp. (1289-1305), ISSN: 0094-114X.

of Mechanica Engineeriring, Indian Institute of Science.

2000), pp. (495-516), DOI: 10.1002/1097-4563

Mechanism and Machine Theory, 33: 1241-1248.

*Engineering Conference*, Salt Lake City, Utah, USA.

design. *Engineering with Computers*, Springer.

CRC Press, ISBN: 9780849391156, U. S. A.

(1623 - 1640), doi:10.1016/S0094-114X(99)00058-0

10.1016/S0093-6413(02)00350-6

114X(97)00039-6

hybrid manipulators using dual vectors and matrices*. Journal of Mechanism and* 

*Approach using Dual Numbers and Symbolic Computation*. Ph. D. Thesis, Department

kinematics of robot manipulators. *Journal of Robotic Systems*, Vol. 17, 9, (September

Algebra, In: *Geometric computing with Clifford algebras*, Springer, ISBN: 3-540-41198-4

analysis of spatial linkages with dimensional and geometric tolerances. *Proceedings of Design Engineering Technical Conferences and Computers and Information in* 

*Research Communications*, Vol. 30, 1, January-February 2003, pp. (69-78), doi:

*Machine Theory,* Vol. 33, 3, (April 1998), pp. (331 - 340), DOI: 10.1016/S0094-

joints. *Journal of Mechanisms and Machine Theory,* Vol. 35, 11, (November 2000), pp.

simulate and analyze different serial robotic manipulators topologies.

2 2 1 1 2 2 1 1 2 2 1 1 2 22 2 22 2 1 2,0 1 2,0 1 2 22 2 22 2 1 2,0 1 2,0 1 2 22 2 22 2 1 2,0 1 2,0 1 2 22 2 22 2 22 2 22 2 1 2,0 1 2,0 1 2,0 1 2,0 1 2 *i i j j k k i kj jk j ik ki k ji ij i k j k j j k j k f t f t f t PPP PP P P PP H H VP H VP H* 22 2 22 2 22 2 22 2 1 2,0 1 2,0 1 2,0 1 2,0 1 2 22 2 22 2 22 2 22 2 1 2,0 1 2,0 1 2,0 1 2,0 1 *j i k ik k i ki k j i j i i j i j H VP H V P H H VP H VP* 

From dynamic equilibrium:

$$\begin{aligned} \,^B M\_A \,^A \vec{F}\_A - \,^B \vec{F}\_B - \,^B \vec{f}\_A + \varepsilon \left( \,^B M\_A \,^A \vec{T}\_A + \,^B D\_A \,^A \vec{F}\_A - \,^B \vec{T}\_B - \,^B \vec{f}\_A \right) = 0 \\\\ \,^B M\_A \,^A \vec{F}\_A - \,^B \vec{F}\_B - \,^B \vec{f}\_A = 0 \\\\ \,^B M\_A \,^A \vec{T}\_A + \,^B D\_A \,^A \vec{F}\_A - \,^B \vec{T}\_B - \,^B \vec{f}\_A = 0 \\\\ \,^2 \vec{F}\_2 = \left[ \,^2 M\_1 \right]^{-1} \left\{ \,^2 \vec{f}\_1 \right\} \\\\ \,^1 \vec{F}\_1 = \left[ \,^1 M\_0 \right]^{-1} \left\{ \,^2 \vec{F}\_2 + \,^1 \vec{f}\_0 \right\} \\\\ \,^2 \vec{T}\_2 = \left[ \,^2 M\_1 \right]^{-1} \left\{ \,^2 \vec{t}\_1 - \,^2 D\_1 \,^2 \vec{F}\_2 \right\} \\\\ \,^1 \vec{T}\_1 = \left[ \,^1 M\_0 \right]^{-1} \left\{ \,^2 \vec{T}\_2 + \,^1 \vec{t}\_0 - \,^1 D\_0 \,^1 \vec{F}\_1 \right\} \end{aligned} $$
