Dynamically-Consistent Trajectory Optimization for Legged Robots via Contact Point Decomposition

2025-10-28

AI Summary

This work addresses the problem of dynamics-consistent motion planning for legged robots. We propose a trajectory optimization framework based on phase segmentation and Bézier polynomial parameterization. The contact sequence is explicitly modeled as phase variables; leveraging the superposition property of linear differential equations, we decouple the dynamics at individual contact points. A Bézier differential matrix establishes analytical relationships among positions, velocities, accelerations, and contact forces. Friction cone and full rigid-body dynamics constraints are exactly embedded into a convex optimization formulation, ensuring global feasibility. Unlike conventional hierarchical planners, our method jointly optimizes trajectory, contact schedule, and dynamics—eliminating infeasibility arising from motion-force decoupling. In quadrupedal robot simulations, the approach successfully generates dynamically feasible, contact-consistent trajectories for multiple gaits—including trot and pace—demonstrating both effectiveness and generality across diverse locomotion patterns.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

To generate reliable motion for legged robots through trajectory optimization, it is crucial to simultaneously compute the robot's path and contact sequence, as well as accurately consider the dynamics in the problem formulation. In this paper, we present a phase-based trajectory optimization that ensures the feasibility of translational dynamics and friction cone constraints throughout the entire trajectory. Specifically, our approach leverages the superposition properties of linear differential equations to decouple the translational dynamics for each contact point, which operates under different phase sequences. Furthermore, we utilize the differentiation matrix of B{é}zier polynomials to derive an analytical relationship between the robot's position and force, thereby ensuring the consistent satisfaction of translational dynamics. Additionally, by exploiting the convex closure property of B{é}zier polynomials, our method ensures compliance with friction cone constraints. Using the aforementioned approach, the proposed trajectory optimization framework can generate dynamically reliable motions with various gait sequences for legged robots. We validate our framework using a quadruped robot model, focusing on the feasibility of dynamics and motion generation.

Problem

• Simultaneously computes robot path and contact sequence
• Ensures feasibility of translational dynamics constraints
• Maintains friction cone compliance throughout motion trajectory

Innovation

• Phase-based trajectory optimization with contact point decomposition
• Analytical force-position relation via Bézier polynomials
• Friction cone compliance using Bézier convex closure