NAS: N-step computation of All Solutions to the footstep planning problem

TL;DR

NAS is an algorithm that efficiently computes all possible footstep solutions for humanoid robots, combining continuous and discrete aspects to produce a globally optimal policy in real time.

Contribution

It introduces NAS, the first algorithm to simultaneously consider continuous and discrete factors to compute all solutions and a globally optimal policy for footstep planning.

Findings

Reduces practical complexity to a bilinear form

Maintains completeness guarantees

Enables real-time GPU parallelisation

Abstract

How many ways are there to climb a staircase in a given number of steps? Infinitely many, if we focus on the continuous aspect of the problem. A finite, possibly large number if we consider the discrete aspect, \emph{i.e.} on which surface which effectors are going to step and in what order. We introduce NAS, an algorithm that considers both aspects simultaneously and computes \emph{all} the possible solutions to such a contact planning problem, under standard assumptions. To our knowledge NAS is the first algorithm to produce a globally optimal policy, efficiently queried in real time for planning the next footsteps of a humanoid robot. Our empirical results (in simulation and on the Talos platform) demonstrate that, despite the theoretical exponential complexity, optimisations reduce the practical complexity of NAS to a manageable bilinear form, maintaining completeness guarantees and enabling efficient GPU parallelisation. NAS is demonstrated in a variety of scenarios for the Talos robot, both in simulation and on the hardware platform. Future work will focus on further reducing computation times and extending the algorithm's applicability beyond gaited locomotion. Our video is available at \url{https://youtu.be/I5yFe0ez0sI}