By Sébastien Briot, Wisama Khalil

This booklet starts off with a quick recapitulation on easy suggestions, universal to any sorts of robots (serial, tree constitution, parallel, etc.), which are additionally valuable for computation of the dynamic types of parallel robots. Then, as dynamics calls for using geometry and kinematics, the final equations of geometric and kinematic versions of parallel robots are given. After, it's defined that parallel robotic dynamic versions might be bought through decomposing the genuine robotic into digital platforms: a tree-structure robotic (equivalent to the robotic legs for which all joints will be actuated) plus a loose physique similar to 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 true inflexible parallel robotic is bought by way of last the loops by utilizing the Lagrange multipliers. the matter of the dynamic version degeneracy close to singularities is handled and optimum trajectory making plans for crossing singularities is proposed. finally, the process is prolonged to versatile parallel robots and the algorithms for computing their symbolic version within the such a lot 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**

**Example text**

20), defining rot(u, θ ) (Sect. 1), after rewriting its elements as a function of Q j . Thus, the orientation matrix is given as: ⎡ 2(Q 21 + Q 22 ) − 1 ⎣ R = 2(Q 2 Q 3 + Q 1 Q 4 ) 2(Q 2 Q 4 − Q 1 Q 3 ) ⎤ 2(Q 2 Q 3 − Q 1 Q 4 ) 2(Q 2 Q 4 + Q 1 Q 3 ) 2(Q 21 + Q 23 ) − 1 2(Q 3 Q 4 − Q 1 Q 2 )⎦ . 30) Inverse problem. 21). Equating the elements of the diagonals of the right sides of Eqs. 4 Parameterization of the General Matrices of Rotation 27 which is always positive. If we then subtract the second and third diagonal elements from the first diagonal element, we can write after simplifying: 4Q 22 = sx − n y − az + 1.

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 . . a(a(k)) a(k) k} a list containing the number of each intermediate frames separating the frame Fk from the frame F0 , ordered by 48 4 Kinematic Description of Multibody Systems successive frames.

In summary, the geometric description of a structure with closed loops is defined by an equivalent tree structure that is obtained by cutting each closed loop at one of its passive joints and by adding two aligned frames, but with different antecedent, at each cut joint. The total number of frames is equal to n + 2B and the geometric parameters of the last B frames are constant. Let us define a vector qT = [qaT qdT qcT ] of dimension n q in which: • qa is the vector containing the n a active joint variables; • qd is the vector containing the n d = n −n a passive joint variables of the equivalent tree structure; • qc is the vector containing the B variables of the cut joints.