By Bruckmann, Tobias; Pott, Andreas
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Additional info for Cable-Driven Parallel Robots
In most of the motions optimization has focused on minimizing motion time of manipulators, the control inputs are physically unrealizable due to the typical discontinuities at the switching times . Previous efforts for solving the energy optimization problem include a dynamic programming search in the state space for point to point motions  and for motions along specified paths [8, 9]. Cable driven manipulator motions are investigated in several dynamical criteria . Much of this prior work aims to minimize the execution time of a desired trajectory, which often adversely affects energy efficiency [11–13].
Since the CDDR is underconstrained, S is a square matrix of order two and the computation of the cable tensions is straightforward; • fS is the force exerted by the passive serial support on the end-effector, which varies both in magnitude and direction during the motion; • pE is the weight force vector applied to the end-effector; • M is the Cartesian mass matrix of the end-effector; • x¨ are the Cartesian accelerations of the end-effector (at the TCP). In  it has been proved that when a two-link serial support is employed, it exerts a reaction force f S which takes the following form: f S = IS x¨ + NS (JS−1 x˙ )2 + pS (2) where: • x˙ and x¨ are the end-effector (TCP) Cartesian velocities and accelerations; • the elements of the matrices IS and NS depend on the inertial and geometrical properties of the serial support and on the positions of its links (and hence, in the end, on the Cartesian position x of the TCP); • the elements of matrix JS only depend on the lengths and the positions of the links; • the elements of vector pS account for the position-dependent gravitational effect introduced by the serial support.
Quintic polynomial trajectories are adopted for point-to-point planning. Quintic polynomial trajectories are suitable to CDDR trajectory planning because they are characterized by continuous velocity, acceleration and jerk profiles. Hence they are smooth enough not to excite vibrational phenomena induced by cable elasticity. Additionally, minimum-time trajectories can be easily computed. Clearly, the time computed is not the absolute minimum for a given point-to-point motion . Let us express the path coordinate l through the following polynomial: l(t) = b0 + b1 t + b2 t 2 + b3 t 3 + b4 t 4 + b5 t 5 (9) It holds 0 ≤ l ≤ L t , where L t is the path length.