Convex subspaces for time-optimal control of robotic systems on prescribed paths

For robotic manipulators designed to follow a prescribed maneuver, this paper solves the classic time-optimal control problem using a convex optimization approach. By assuming a Bernstein polynomial basis for the path velocity, and enforcing torque constraints at discrete points along the path, the time-optimal control problem becomes a finite-dimensional optimization problem with non-convex equality constraints. However, a rank 2 spectral decomposition of the torque constraints yields equivalent semi-definite quadratic constraints which allow us to solve the problem as a sequence of convex subproblems guaranteed to converge to the global minimum. An advantage of this approach is that bang-bang control solutions are replaced by smooth motor commands, even though the higher-order derivatives of the system are not explicitly constrained. Numerical examples illustrate the algorithm.