Convergence properties of gradient-based numerical motion-optimizations for manipulator arms amid static or moving obstacles

Abstract

This paper demonstrates the convergence stability and the actual usefulness of the gradient-based motion optimizations for manipulator arms. An optimal motion-planning problem is converted into a finite-dimensional nonlinear programming problem that utilizes cubic or quintic B-splines as basis functions. This study shows that the numerically calculated gradient is a useful tool in finding minimum torque, minimum energy, minimum overload, and minimum time motions for manipulator arms in the presence of static or moving obstacles. A spatial 6-link manipulator is simulated without simplifying any of the kinematic, dynamic or geometric properties of the manipulator or obstacles.

Trajectory planning for optimal robot catching in real-time

“…The proposed optimization method is based on direct single shooting, with parameterization of the system independent states in time, as for e.g. in [2], i.e., θ = θ(t, p) with p ⊆ ℜ N , for N optimization parameters, as described in Section III-A. The control forces are then computed from the state transition Eq.…”

“…More midpoints of the grid were then also computed with the initial guess taken from the coarse grid, to produce a finer grid, of sufficient fineness for the subsequent learning process. The resulting data set contains 1825 (sample, target) pairs, with a spacing in the target velocity space of [0.17,0.25,0.125] m/s in the range [-2,-6], [2,5], [2.5,5.5] m/s. The total computation time of the data set on an Intel Xeon CPU W3520 2.67GHz machine is in the order of magnitude of 100 hours, which makes it unfeasible for online optimization.…”

“…The minimum energy problem for a non-redundant 6 DoF manipulator executing point-to-point maneuvers in configuration space was treated in [2], including collision avoidance. The resulting constrained boundary value problem was solved with direct single shooting.…”

“…Furthermore, the optimization problem considered here is different from those formulated, for e.g. in [2] and [3], in that: firstly, the final robot configuration is not given; secondly, the final time is not fixed; thirdly, there is a high kinematic redundancy resulting from using a 7 DoF robot and for the given task. The first point adds a set of nonlinear equality constraints, while the second and third add complexity, due to a resulting increase in local minima.…”

“…The mechanical energy is computed here as follows (similarly to [2] and [3]): (7) where the first term is the integral of the power for a robot model which neglects the joint motor inertias and the second term represents the motor kinetic energy, which is of comparable size to the first (not included in [2] and [3]). The symbol J motor = diag(J 1 N 1 , ..., J n N n ) expresses an (n × n) diagonal matrix with elements whose i th diagonal element is the i th motor inertia J i multiplied by the i th gear reduction ratio N i .…”