Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

CASSR: Continuous A-Star Search through Reachability for real time footstep planning

Jiayi Wang, Steve Tonneau

TL;DR

Results show that CASSR enables fast, reliable, and real-time footstep planning for biped robots, and surpasses a commercial MIP solver.

Abstract

Footstep planning involves a challenging combinatorial search. Traditional A* approaches require discretising reachability constraints, while Mixed-Integer Programming (MIP) supports continuous formulations but quickly becomes intractable, especially when rotations are included. We present CASSR, a novel framework that recursively propagates convex, continuous formulations of a robot's kinematic constraints within an A* search. Combined with a new cost-to-go heuristic based on the EPA algorithm, CASSR efficiently plans contact sequences of up to 30 footsteps in under 125 ms. Experiments on biped locomotion tasks demonstrate that CASSR outperforms traditional discretised A* by up to a factor of 100, while also surpassing a commercial MIP solver. These results show that CASSR enables fast, reliable, and real-time footstep planning for biped robots.

CASSR: Continuous A-Star Search through Reachability for real time footstep planning

TL;DR

Results show that CASSR enables fast, reliable, and real-time footstep planning for biped robots, and surpasses a commercial MIP solver.

Abstract

Footstep planning involves a challenging combinatorial search. Traditional A* approaches require discretising reachability constraints, while Mixed-Integer Programming (MIP) supports continuous formulations but quickly becomes intractable, especially when rotations are included. We present CASSR, a novel framework that recursively propagates convex, continuous formulations of a robot's kinematic constraints within an A* search. Combined with a new cost-to-go heuristic based on the EPA algorithm, CASSR efficiently plans contact sequences of up to 30 footsteps in under 125 ms. Experiments on biped locomotion tasks demonstrate that CASSR outperforms traditional discretised A* by up to a factor of 100, while also surpassing a commercial MIP solver. These results show that CASSR enables fast, reliable, and real-time footstep planning for biped robots.
Paper Structure (29 sections, 6 equations, 3 figures, 1 table, 3 algorithms)

This paper contains 29 sections, 6 equations, 3 figures, 1 table, 3 algorithms.

Table of Contents

  1. INTRODUCTION
  2. State of the art
  3. Overview
  4. Assumptions and definitions
  5. Polytope-based representation of constraints
  6. Reminder on reachability computation
  7. 1-step reachability from a 3D foot position
  8. 1-step reachability from a polytope
  9. Rotations introduce a combinatorics
  10. $A^*$ implementation
  11. Reminder on the $A^*$ algorithm
  12. Specific implementation details
  13. Node structure
  14. expandNode()
  15. $\textrm{nodeAlreadyExpanded()}$
  16. ...and 14 more sections

Figures (3)

  • Figure 1: The pictures are taken from the back of the robot (the forward direction is towards the stepping stones). Given the position of a foot (grey rectangle) and assuming no rotation, we can compute the reachability constraints for the other foot as a convex polytope (red for constraints on the right foot, green for the left foot). The set of positions reachable from the red polygon (b) is obtained by computing the Minkowski sum of all reachability polytopes for all positions in the polygon (c, d, e). This is computed efficiently as the convex hull of the reachability polytopes of the extreme points (d). Not all extreme points are shown in (d) for readability. The union of the 3 green polygons in (f) is the 2 step-reachable set and corresponds to 3 nodes added to the $A^*$ graph.
  • Figure 2: Reachable set (red polytope) for the right foot with respect to the left foot (rectangle), seen from the back of the robot. To avoid leg crossing the right foot is constrained to be "on the right" of the left foot by construction of the set.
  • Figure 3: Scenarios and contact plans found by CASSR with (line 2) or without (line 1) rotation. Target is the yellow sphere and Talos is always at the start position. The blue (resp. red) rectangles indicate a contact placement for the left (resp. right).