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

Jiayi Wang, Saeid Samadi, Hefan Wang, Pierre Fernbach, Olivier Stasse, Sethu Vijayakumar, Steve Tonneau

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}

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

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}
Paper Structure (41 sections, 5 equations, 4 figures, 3 algorithms)

This paper contains 41 sections, 5 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. State of the art
  3. Combinatorics models for gaited locomotion
  4. Combinatorics models for multi-contact locomotion
  5. Towards combinatorics-free contact locomotion?
  6. definitions, notations and problem statement
  7. Problem statement
  8. Surfaces, reachable areas, and kinematic constraints
  9. Overview
  10. 1-step feasibility
  11. Reachability from a given position (Fig. \ref{['fig:overview']}.1-2)
  12. Reachability from a set of positions (Fig. \ref{['fig:overview']}.4-7)
  13. Computing the 1-step feasible set (Fig. \ref{['fig:overview']}.7-9)
  14. The NAS algorithm
  15. Tree description and initialisation
  16. ...and 26 more sections

Figures (4)

  • Figure 1: A 2-step plan for the left foot to reach the red target.
  • Figure 2: Two-step feasibility computation. 1. Red: The goal for this contact planning problem is to find a feasible contact sequence such that the left foot reaches the red position $\mathbf{p}_l$, thus we define $\mathcal{G} = \{\mathbf{p}_l\}$. Green: The environment $\mathcal{S}$ is the union of 2 convex polygons. 2. All positions of the right foot such that $\mathbf{p}_l$ can be reached by the left foot in one step are bounded by the blue polytope ${^{\mathcal{G}}\mathcal{R}}$. It is obtained by translating the antecedent polytope $^{l}\mathcal{A}_{r}$ by $\mathbf{p}_l$. 3. One-step feasible set for the right foot ${^{\mathcal{G}}\mathcal{F}}$. 4. Reachability polytope $^{r}\mathcal{A}_{l}$ for the position of the right foot indicated by the arrow. 5.$^{r}\mathcal{A}_{l}$ translated by each extreme point of ${^{\mathcal{G}}\mathcal{F}}$. 6. & 7. Minkowski sum of $^{r}\mathcal{A}_{l}$ and ${^{\mathcal{G}}\mathcal{F}}$. 8. & 9. Two-step feasible set for the right foot, composed of 2 convex polygons.
  • Figure 3: Exponential growth of nodes without node merging.
  • Figure 4: In all instances, node merging allows the number of nodes and generation time for $\mathcal{T}{}$ to grow as $O(m*n)$.