**5. Dynamic analysis: Dual force**

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

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

, ,

From dual velocities theorem, the vector of dual velocities in the end of *n* link in terms of

0 0 1 1 ,0 1 ,1 1 ˆ ˆˆ *<sup>n</sup> n n ii n n i ii i V TMV*

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 0 2,1 ,0 012 1

*z z n n*

<sup>ˆ</sup> <sup>ˆ</sup> ˆˆˆ ˆ ...

*<sup>V</sup> V V TMMM M*

*n nnn n n yy n n*

*QP Q PP* ˆ ˆ *V TV <sup>j</sup> <sup>i</sup> p j <sup>i</sup>* (20)

,,, ˆˆˆ *VVV k i <sup>j</sup> i k <sup>j</sup>* (21)

(22)

*V*

ˆ

1 1 , 1

*n n*

(23)

ˆ

 

*V*

*Vk i* in dual form is established as:

Fig. 6. Dual velocity scheme.

the *n* frame can be found as:

Where <sup>0</sup>

The relative velocity of a link *<sup>k</sup>* with respect to link *<sup>i</sup>* , <sup>ˆ</sup>

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

*x x*

 

 

 

*V*

*V*

*Tn* is the primary component of the dual matrix(19).

frame *R* is done:

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

In dual algebra, if a force and a momentum act with respect a coordinate system, they can be represented in an expression called dual force:

$$
\hat{F} = \vec{F} + \mathcal{E}\vec{\tau} \tag{24}
$$

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

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

$${}^{AB}\hat{F}\_A = {}^{B}T\_A {}^{A}\vec{F}\_A + \mathfrak{s} {}^{B}T\_A {}^{A}\vec{\tau}\_A \tag{25}$$
