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Galileo: A Pseudospectral Collocation Framework for Legged Robots

Ethan Chandler, Akshay Jaitly, Mahdi Agheli

TL;DR

A new transcription scheme is introduced that enables pseudospectral collocation to optimize directly on Lie Groups, such as SE(3) and quaternions without special normalization constraints without special normalization constraints.

Abstract

Dynamic maneuvers for legged robots present a difficult challenge due to the complex dynamics and contact constraints. This paper introduces a versatile trajectory optimization framework for continuous-time multi-phase problems. We introduce a new transcription scheme that enables pseudospectral collocation to optimize directly on Lie Groups, such as SE(3) and quaternions without special normalization constraints. The key insight is the change of variables - we choose to optimize over the history of the tangent vectors rather than the states themselves. Our approach uses a modified Legendre-Gauss-Radau (LGR) method to produce dynamic motions for various legged robots. We implement our approach as a Model Predictive Controller (MPC) and track the MPC output using a Quadratic Program (QP) based whole-body controller. Results on the Go1 Unitree and WPI HURON humanoid confirm the feasibility of the planned trajectories.

Galileo: A Pseudospectral Collocation Framework for Legged Robots

TL;DR

A new transcription scheme is introduced that enables pseudospectral collocation to optimize directly on Lie Groups, such as SE(3) and quaternions without special normalization constraints without special normalization constraints.

Abstract

Dynamic maneuvers for legged robots present a difficult challenge due to the complex dynamics and contact constraints. This paper introduces a versatile trajectory optimization framework for continuous-time multi-phase problems. We introduce a new transcription scheme that enables pseudospectral collocation to optimize directly on Lie Groups, such as SE(3) and quaternions without special normalization constraints. The key insight is the change of variables - we choose to optimize over the history of the tangent vectors rather than the states themselves. Our approach uses a modified Legendre-Gauss-Radau (LGR) method to produce dynamic motions for various legged robots. We implement our approach as a Model Predictive Controller (MPC) and track the MPC output using a Quadratic Program (QP) based whole-body controller. Results on the Go1 Unitree and WPI HURON humanoid confirm the feasibility of the planned trajectories.
Paper Structure (8 sections, 5 equations, 2 figures)

This paper contains 8 sections, 5 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Galileo
  3. Multi-phase Optimization Problem Formulation
  4. Optimizing with Deviations
  5. Galileo for Legged Robot Control
  6. Quaternion Parameterization for Legged Robot Pose
  7. Results
  8. Conclusions and Future Work

Figures (2)

  • Figure 1: Galileo MPC solving legged robot motions at 50Hz, simulated in Gazebo. Video results shown in galileo.
  • Figure 2: Visualization of the polynomial approximation used in pseudospectral collocation.