A common approach to optimal robot trajectory planning divides the problem into two phases: selection of a geometric path and time scheduling along the chosen path. For a given parametrized path, the minimum transversal time problem under maximum torque constraints has been already solved by the same authors, taking into account the full nonlinear robot dynamics.
This paper addresses the problem of how to select among the infinitely many geometric paths joining two points, the one which yields the global minimum transversal time. The general exact solution is hard to find. However, several interesting approximate results are obtained.
For a gravity- and friction-free robot with bounds only on the total supplied power, geodesics in the manipulator inertia space are shown to be optimal. The concept of a super-manipulator is introduced, a robot with limits only on the maximum kinetic energy and on the torque components tangential to the path. Based on energy methods, the minimum achievable transversal time can be derived for this case. Since such a manipulator is capable of doing at least everything as the original one, a lower bound on transversal time for the general case is thus obtained.
Minimization of this lower bound is attained if the path to be traveled is a geodesics in the inertia space. Thus, geodesics in the inertia space are chosen as near-minimum time paths in any case. For interesting comments on this point [see also: Y. C. Chen, ibid., AC-32, 11, 1027-1028 (1987)]. To actually construct such paths, a two-point boundary value problem has to be solved. Indeed, due to simpler boundary conditions, the computational burden is smaller than by straight application of Pontryagin’s minimum principle to the point-to-point optimal motion problem.