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Perceptive Variable-Timing Footstep Planning for Humanoid Locomotion on Disconnected Footholds

Zhaoyang Xiang, Upama Pant, Ayonga Hereid

TL;DR

An onboard, perceptive mixed-integer model predictive control framework that jointly plans foot placement and step duration using step-to-step Divergent Component of Motion (DCM) dynamics, and embeds capturability bounds in the DCM space.

Abstract

Many real-world walking scenarios contain obstacles and unsafe ground patches (e.g., slippery or cluttered areas), leaving a disconnected set of admissible footholds that can be modeled as stepping-stone-like regions. We propose an onboard, perceptive mixed-integer model predictive control framework that jointly plans foot placement and step duration using step-to-step Divergent Component of Motion (DCM) dynamics. Ego-centric depth images are fused into a probabilistic local heightmap, from which we extract a union of convex steppable regions. Region membership is enforced with binary variables in a mixed-integer quadratic program (MIQP). To keep the optimization tractable while certifying safety, we embed capturability bounds in the DCM space: a lateral one-step condition (preventing leg crossing) and a sagittal infinite-step bound that limits unstable growth. We further re-plan within the step by back-propagating the measured instantaneous DCM to update the initial DCM, improving robustness to model mismatch and external disturbances. We evaluate the approach in simulation on Digit on randomized stepping-stone fields, including external pushes. The planner generates terrain-aware, dynamically consistent footstep sequences with adaptive timing and millisecond-level solve times.

Perceptive Variable-Timing Footstep Planning for Humanoid Locomotion on Disconnected Footholds

TL;DR

An onboard, perceptive mixed-integer model predictive control framework that jointly plans foot placement and step duration using step-to-step Divergent Component of Motion (DCM) dynamics, and embeds capturability bounds in the DCM space.

Abstract

Many real-world walking scenarios contain obstacles and unsafe ground patches (e.g., slippery or cluttered areas), leaving a disconnected set of admissible footholds that can be modeled as stepping-stone-like regions. We propose an onboard, perceptive mixed-integer model predictive control framework that jointly plans foot placement and step duration using step-to-step Divergent Component of Motion (DCM) dynamics. Ego-centric depth images are fused into a probabilistic local heightmap, from which we extract a union of convex steppable regions. Region membership is enforced with binary variables in a mixed-integer quadratic program (MIQP). To keep the optimization tractable while certifying safety, we embed capturability bounds in the DCM space: a lateral one-step condition (preventing leg crossing) and a sagittal infinite-step bound that limits unstable growth. We further re-plan within the step by back-propagating the measured instantaneous DCM to update the initial DCM, improving robustness to model mismatch and external disturbances. We evaluate the approach in simulation on Digit on randomized stepping-stone fields, including external pushes. The planner generates terrain-aware, dynamically consistent footstep sequences with adaptive timing and millisecond-level solve times.
Paper Structure (8 sections, 22 equations, 5 figures, 1 table, 3 algorithms)

This paper contains 8 sections, 22 equations, 5 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Probabilistic Ego-centric Terrain Perception
  3. Depth Image Processing
  4. Probabilistic Local Heightmap
  5. Convex Region Constraints
  6. Variable Time Foot Placement via MIQP
  7. Simulation Results
  8. Conclusion

Figures (5)

  • Figure 1: (Left) Snapshots of the simulation terrain for all ablation tests. The blue rectangles show the planned step position for the previewed steps, and the red bar shows the goal position in the sagittal direction. (Top Right) 2.5D local heightmap colored by heights; (Bottom Right) Local heightmap mask with gray masks showing all the steppable regions and green polygons marking the selected regions.
  • Figure 2: Overall control architecture for locomotion on restricted footholds. The perception module processes raw depth images into linear region constraints representing feasible stepping areas. These constraints, together with a goal position, are passed to the MIQP planner that optimizes step position and duration using step-to-step DCM dynamics. The task-space controller tracks the planned step parameters and generates joint torques, which are executed on the Digit robot.
  • Figure 3: Convex regions extraction from the local heightmap as in Alg. \ref{['alg:region_extraction']}.
  • Figure 4: Initial DCM comparison of: (A) The proposed framework; (B) Planning with fixed step duration; (C) Planning with reduced number of previewed steps as $N=2$; (D) Planning without viability constraints on DCM evolution. (V) CoM velocity of the proposed framework. The $x$-velocity was roughly maintained around $1.0\,\mathrm{m/s}$ primarily due to the goal position tracking under the nominal stride length and step duration. The $y$-velocity was also roughly regulated to periodic oscillations. (T) Step duration of the proposed framework. It was being actively adjusted to produce viable DCM evolution through random footholds.
  • Figure 5: Top-down view of stance footprints and CoM and DCM trajectories of the proposed framework. With the goal position in the forward direction, the robot was able to find a viable path through all random footholds.