By Sébastien Briot, Wisama Khalil

ISBN-10: 3319197878

ISBN-13: 9783319197876

ISBN-10: 3319197886

ISBN-13: 9783319197883

This publication starts off with a quick recapitulation on simple strategies, universal to any varieties of robots (serial, tree constitution, parallel, etc.), which are additionally precious for computation of the dynamic versions of parallel robots. Then, as dynamics calls for using geometry and kinematics, the final equations of geometric and kinematic types of parallel robots are given. After, it really is defined that parallel robotic dynamic types should be bought through decomposing the true robotic into digital structures: a tree-structure robotic (equivalent to the robotic legs for which all joints will be actuated) plus a loose physique resembling the platform. hence, the dynamics of inflexible tree-structure robots is analyzed and algorithms to acquire their dynamic types within the such a lot compact shape are given. The dynamic version of the genuine inflexible parallel robotic is received by way of ultimate the loops by using the Lagrange multipliers. the matter of the dynamic version degeneracy close to singularities is taken care of and optimum trajectory making plans for crossing singularities is proposed. finally, the strategy is prolonged to versatile parallel robots and the algorithms for computing their symbolic version within the so much compact shape are given. All theoretical advancements are verified via experiments.

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**Extra resources for Dynamics of Parallel Robots: From Rigid Bodies to Flexible Elements**

**Sample text**

2 Modified Denavit-Hartenberg Parameters 47 Fig. 1 Example of MDH parameters for a closed loop Let us consider the single planar closed loop shown in Fig. 7, which is composed of four revolute joints and one prismatic joint. The system’s mobility is equal to 2 (see Appendix A). We suppose that the active joints are the joints 1 and 3 which are fixed on the base. According to the above mentioned notations, it is possible to assign the frames of the system in the way presented in Fig. 7. 2. 3 Computation of the Homogeneous Transformation Matrix Representing the Location of the Frame Fk with Respect to the Frame F i Let us denote as ak = {0 .

35) Q4 = 1 sign(s y − n x ) −sx − n y + az + 1. 36) Contrary to Euler angles, roll-pitch-yaw angles and T&T angles (see next sections), quaternion representation is free of singularity. For more information on the algebra of quaternions, the reader can refer to (de Casteljau 1987). 3 Euler Angles The orientation of frame Fk expressed in frame Fi can be determined by specifying three angles, φ, θ and ψ corresponding to a sequence of three rotations (Fig. 5). Let us consider two intermediate frames F j and F j defined by F j (Oi , x j , y j , z j ) and F j (O j , x j , y j , z j ) and characterized by: • z j ≡ zi and y j is the intersection between the two planes (Oi , xi , yi ) and (Oi , xk , yk ), • y j ≡ y j and z j ≡ zk .

A(a(k)) a(k) k} a sub-list of ak with a length n k containing the number of each intermediate frames separating the frame Fk from the frame Fi , ordered by successive frames. 8) Ta (i:k) ( j) j=2 where a (i:k) ( j) denotes the jth component of the list a(i:k) . • Case 2: i ∈ / ak ; in that case, we create a new list ai = {0 . . a(a(i)) a(i) i} containing the number of each intermediate frames separating the frame Fi from the frame F0 , ordered by successive frames. Let us denote as rc the highest number common to both lists.

### Dynamics of Parallel Robots: From Rigid Bodies to Flexible Elements by Sébastien Briot, Wisama Khalil

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