By Sébastien Briot, Wisama Khalil
This publication begins with a brief recapitulation on easy suggestions, universal to any forms of robots (serial, tree constitution, parallel, etc.), which are additionally beneficial 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 really is defined that parallel robotic dynamic versions may be received by way of 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 unfastened physique similar to the platform. hence, the dynamics of inflexible tree-structure robots is analyzed and algorithms to procure their dynamic types within the so much compact shape are given. The dynamic version of the true inflexible parallel robotic is acquired by way of ultimate the loops by using 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 technique 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 established via experiments.
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Extra info for Dynamics of Parallel Robots: From Rigid Bodies to Flexible Elements
11) where i rˆ j = i rˆ Oi O j Since j v j = j Ri i v j and j ω j = j Ri i ω j , Eq. 12) where j Ti is the (6 × 6) transformation matrix between screws: j Ti = jR i 03 − j Ri i rˆ j . 13) The transformation matrices between screws have the following properties: Property 1 Product: j 0 Tj = k−1 k=1 Tk = 1 T2 2 T3 · · · j−1 T j . 14) 36 3 Representation of Velocities and Forces/Acceleration of a Body Property 2 Inverse: j −1 Ti i R i rˆ i R j j j 03 i R j = = iTj. 15), can be rewritten as j T w j = i T j i wi .
6, the basic dynamic principles used in this book are introduced (Lagrange equations, Newton-Euler principle, principle of virtual works). • Part II deals with the dynamic modeling of rigid parallel robots: – In Chap. 7, the generic computation of the geometric and kinematic models of PKM is detailed. Moreover, the problem of the singularity of parallel robots is introduced. – In Chap. 8, the computation of the inverse and direct dynamic model of parallel robots (with and without redundancy) is treated.
33) The parameter Q 1 being always positive, the sign of Q 2 is that of (n z −a y ), which allows us to write: Q2 = 1 sign(n z − a y ) sx − n y − az + 1. 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.