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Adaptive Step Duration for Accurate Foot Placement: Achieving Robust Bipedal Locomotion on Terrains with Restricted Footholds

Zhaoyang Xiang, Victor Paredes, Guillermo A. Castillo, Ayonga Hereid

TL;DR

This paper introduces a novel multi-step preview foot placement planning algorithm based on the step-to-step discrete evolution of the Divergent Component of Motion of walking robots, which adaptively changes the step duration and the swing foot trajectory for optimal foot placement under constraints, thereby enhancing the long-term stability of the robot and significantly improving its ability to navigate environments with tight constraints on viable footholds.

Abstract

Traditional one-step preview planning algorithms for bipedal locomotion struggle to generate viable gaits when walking across terrains with restricted footholds, such as stepping stones. To overcome such limitations, this paper introduces a novel multi-step preview foot placement planning algorithm based on the step-to-step discrete evolution of the Divergent Component of Motion (DCM) of walking robots. Our proposed approach adaptively changes the step duration and the swing foot trajectory for optimal foot placement under constraints, thereby enhancing the long-term stability of the robot and significantly improving its ability to navigate environments with tight constraints on viable footholds. We demonstrate its effectiveness through various simulation scenarios with complex stepping-stone configurations and external perturbations. These tests underscore its improved performance for navigating foothold-restricted terrains, even with external disturbances.

Adaptive Step Duration for Accurate Foot Placement: Achieving Robust Bipedal Locomotion on Terrains with Restricted Footholds

TL;DR

This paper introduces a novel multi-step preview foot placement planning algorithm based on the step-to-step discrete evolution of the Divergent Component of Motion of walking robots, which adaptively changes the step duration and the swing foot trajectory for optimal foot placement under constraints, thereby enhancing the long-term stability of the robot and significantly improving its ability to navigate environments with tight constraints on viable footholds.

Abstract

Traditional one-step preview planning algorithms for bipedal locomotion struggle to generate viable gaits when walking across terrains with restricted footholds, such as stepping stones. To overcome such limitations, this paper introduces a novel multi-step preview foot placement planning algorithm based on the step-to-step discrete evolution of the Divergent Component of Motion (DCM) of walking robots. Our proposed approach adaptively changes the step duration and the swing foot trajectory for optimal foot placement under constraints, thereby enhancing the long-term stability of the robot and significantly improving its ability to navigate environments with tight constraints on viable footholds. We demonstrate its effectiveness through various simulation scenarios with complex stepping-stone configurations and external perturbations. These tests underscore its improved performance for navigating foothold-restricted terrains, even with external disturbances.
Paper Structure (17 sections, 1 theorem, 14 equations, 7 figures, 2 tables)

This paper contains 17 sections, 1 theorem, 14 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Divergent Component of Motion Analysis of Bipedal Locomotion
  3. Center of Mass Dynamics of Bipedal Robots
  4. Discrete Dynamics of the initial DCM
  5. Swing Foot Placement Planning via Discrete Model Predictive Control
  6. Step Position
  7. Step Duration
  8. Initial DCM
  9. Discrete MPC Formulation
  10. Adaptation of Variable Step Duration in the Low-level Controller
  11. Simulation Results
  12. Robust Walking on Stepping Stones
  13. Profile I
  14. Profile II
  15. Profile III and IV
  16. ...and 2 more sections

Key Result

Proposition 1

Given appropriate allowable sets $\mathcal{D}$, $\mathcal{T} \subset \mathbb{R}$ of the $k$-th step, for any initial DCM $\xi^x_{0,k}$ in the corresponding $\mathcal{X}$ that evolves once into $\xi^x_{0,k+1}$ using eq:dcm_reset_map, there exists at least one allowable foot placement $(L_k,T_k)$ such

Figures (7)

  • Figure 1: Digit robot stably walks across four challenging stepping stone scenarios in MuJoCo simulation by dynamically adjusting both step duration and stepping locations.
  • Figure 2: Schematic view of the DCM evolution with step positions and footprints. The blue arrows mark the initial DCM at the beginning of steps and the orange ones mark the final DCM. During the $k$-th step, the DCM evolves from $\boldsymbol{\xi}_{0,k}$ to $\boldsymbol{\xi}_{T,k}$ and is reset to $\boldsymbol{\xi}_{0,k+1}$ in the new contact frame after the touchdown.
  • Figure 3: Illustration of updating $\tau(t)$ with respect to $(T_{1,\mathrm{tgt}})_i$. For instance, when $(T_{1,\mathrm{tgt}})_2$ is updated at $t_2$, $\tau(t)$ transitions linearly from $\tau_2$ to $1$ if $t$ reaches $(T_{1,\mathrm{tgt}})_2$, until $t$ reaches $t_3$ when the 3rd update takes place.
  • Figure 4: Comparison of footprints in Profile II test. The black triangles mark the actual step positions, with the black frames illustrating the flat feet of the Digit, while the green rectangles mark the step position bounds on the gray stepping stones. (Left, DCM-MPC) The robot walks stably through the random stones with the bounded DCM; (Right, DCM-QP) The robot falls after the 24th step (marked as red frames) due to a non-viable foot placement of the 23rd step.
  • Figure 5: Comparison of step duration and initial DCM in the test of Profile II. The vertical gray lines mark the touchdown moments of walking steps. (Left, DCM-MPC) The step duration is adjusted to bound the DCM with smoother changes; (Right, DCM-QP) DCM-QP fails with a boundary step duration that indicates the non-viable DCM.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Proposition
  • proof
  • Remark