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Multi-contact Stochastic Predictive Control for Legged Robots with Contact Locations Uncertainty

Ahmad Gazar, Majid Khadiv, Andrea Del Prete, Ludovic Righetti

TL;DR

This work advances safe legged locomotion under contact-location uncertainty by formulating a stochastic kino-dynamic MPC (SNMPC) that enforces contact-location chance constraints through covariance-informed back-offs. By linearizing joint chance-constraints, decomposing surface polytopes, and propagating the state covariance under Gaussian disturbances, the authors derive a deterministic SNMPC on the mean that preserves NMPC-like computational costs. Key contributions include tractable joint chance-constraints, per-half-space back-offs tied to uncertainty, and a real-time SQP-based solution that delivers robust safety with a $100\%$ success rate in dynamic simulations, outperforming NMPC and heuristic margins. The practical impact lies in enabling safer, agile quadruped locomotion on uncertain terrains, with future work extending to contact-mode uncertainty and real-world experiments.

Abstract

Trajectory optimization under uncertainties is a challenging problem for robots in contact with the environment. Such uncertainties are inevitable due to estimation errors, control imperfections, and model mismatches between planning models used for control and the real robot dynamics. This induces control policies that could violate the contact location constraints by making contact at unintended locations, and as a consequence leading to unsafe motion plans. This work addresses the problem of robust kino-dynamic whole-body trajectory optimization using stochastic nonlinear model predictive control (SNMPC) by considering additive uncertainties on the model dynamics subject to contact location chance-constraints as a function of robot's full kinematics. We demonstrate the benefit of using SNMPC over classic nonlinear MPC (NMPC) for whole-body trajectory optimization in terms of contact location constraint satisfaction (safety). We run extensive Monte-Carlo simulations for a quadruped robot performing agile trotting and bounding motions over small stepping stones, where contact location satisfaction becomes critical. Our results show that SNMPC is able to perform all motions safely with 100% success rate, while NMPC failed 48.3% of all motions.

Multi-contact Stochastic Predictive Control for Legged Robots with Contact Locations Uncertainty

TL;DR

This work advances safe legged locomotion under contact-location uncertainty by formulating a stochastic kino-dynamic MPC (SNMPC) that enforces contact-location chance constraints through covariance-informed back-offs. By linearizing joint chance-constraints, decomposing surface polytopes, and propagating the state covariance under Gaussian disturbances, the authors derive a deterministic SNMPC on the mean that preserves NMPC-like computational costs. Key contributions include tractable joint chance-constraints, per-half-space back-offs tied to uncertainty, and a real-time SQP-based solution that delivers robust safety with a success rate in dynamic simulations, outperforming NMPC and heuristic margins. The practical impact lies in enabling safer, agile quadruped locomotion on uncertain terrains, with future work extending to contact-mode uncertainty and real-world experiments.

Abstract

Trajectory optimization under uncertainties is a challenging problem for robots in contact with the environment. Such uncertainties are inevitable due to estimation errors, control imperfections, and model mismatches between planning models used for control and the real robot dynamics. This induces control policies that could violate the contact location constraints by making contact at unintended locations, and as a consequence leading to unsafe motion plans. This work addresses the problem of robust kino-dynamic whole-body trajectory optimization using stochastic nonlinear model predictive control (SNMPC) by considering additive uncertainties on the model dynamics subject to contact location chance-constraints as a function of robot's full kinematics. We demonstrate the benefit of using SNMPC over classic nonlinear MPC (NMPC) for whole-body trajectory optimization in terms of contact location constraint satisfaction (safety). We run extensive Monte-Carlo simulations for a quadruped robot performing agile trotting and bounding motions over small stepping stones, where contact location satisfaction becomes critical. Our results show that SNMPC is able to perform all motions safely with 100% success rate, while NMPC failed 48.3% of all motions.
Paper Structure (12 sections, 11 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 12 sections, 11 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Multi-Contact Robot Dynamics
  4. Cost function and constraint penalties
  5. Stochastic Optimal Control for Kinodynamic Trajectory Optimization
  6. Tractable formulation of Joint Chance-constraints
  7. Deterministic Reformulation of Individual Contact location Chance-Constraints
  8. Deterministic Reformulation of SNMPC
  9. Simulations Results
  10. Kino-dynamic Monte-Carlo Simulations
  11. Whole-body Simulation
  12. Discussion and Conclusions

Figures (4)

  • Figure 1: Effect of equally distributed back-offs design of the linearized contact location chance-constraints.
  • Figure 2: Norm of the contact location deviations from the contact surface center using NMPC, HNMPC, and SNMPC.
  • Figure 3: Snapshots of a trot motion on non-coplanar stones.
  • Figure 4: Snapshots of a bound motion on non-coplanar stones.

Theorems & Definitions (1)

  • Remark 1