Time-Optimal Control via Heaviside Step-Function Approximation

Least-squares programming is a popular tool in robotics due to its simplicity and availability of open-source solvers. However, certain problems like sparse programming in the $\ell_{0}$- or $\ell_{0}-\mathbf{norm}$ for time-optimal control are not equivalently solvable. In this work, we propose a non-linear hierarchical least-squares programming (NL-HLSP) for time-optimal control of non-linear discrete dynamic systems. We use a continuous approximation of the heaviside step function with an additional term that avoids vanishing gradients. We use a simple discretization method by keeping states and controls piece-wise constant between discretization steps. This way, we obtain a comparatively easily implementable NL-HLSP in contrast to direct transcription approaches of optimal control. We show that the NL-HLSP indeed recovers the discrete time-optimal control in the limit for resting goal points. We confirm the results in simulation for linear and non-linear control scenarios.