By Sébastien Briot, Wisama Khalil

This booklet begins with a brief recapitulation on uncomplicated suggestions, universal to any kinds of robots (serial, tree constitution, parallel, etc.), which are additionally priceless for computation of the dynamic types of parallel robots. Then, as dynamics calls for using geometry and kinematics, the overall equations of geometric and kinematic versions of parallel robots are given. After, it's defined that parallel robotic dynamic types might be acquired through decomposing the true 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. therefore, the dynamics of inflexible tree-structure robots is analyzed and algorithms to procure their dynamic versions within the so much compact shape are given. The dynamic version of the genuine inflexible parallel robotic is acquired by means of remaining 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 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 proven via experiments.

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24) if sθ = 0. When sθ is small, the elements u x , u y and u z cannot be determined with good accuracy by this equation. )” indicates the sign function of the expression between brackets, thus sign(e) = +1 if e > 0, sign(e) = −1 if e < 0 and sign(e) = 0 if e = 0. 2 Quaternions The quaternions are also called Euler parameters or Olinde-Rodrigues parameters. This is another way of parameterizing the rotation of an angle θ (0 ≤ θ ≤ π ) about an axis u. In this representation, the orientation is expressed by four parameters.

2010). b The DualV from the LIRMM, France (van der Wijk et al. 2011): an actuation redundant PPM(4-RRR robot) can include larger reachable workspace, avoidance of kinematic singularities, and dexterity improvement (Ebrahimi et al. 2008). 2 Actuation Redundancy Actuation redundancy occurs when the number n a of actuators is greater than the number of robot DOF Ndo f . Mathematically speaking, we have n a > Ndo f . As a consequence, • we cannot independently choose the active joint variables as they are constrained by n c equations.

Each element of a matrix containing at least one mathematical operation is replaced by an intermediate variable. This variable is written in the output file that contains the customized model. The elements that do not contain any operation are kept without modification. We propagate the matrix obtained in the subsequent equations. Consequently, customizing eliminates multiplications by one and zero, and additions with zero. 2 Modified Denavit-Hartenberg Parameters 49 interesting to replace the corresponding product of matrices by a single matrix, which is calculated using Eq.