[Paper Review] First Steps Towards Full Model Based Motion Planning and Control of Quadrupeds: A Hybrid Zero Dynamics Approach

Wen-Loong Ma, Kaveh Akbari Hamed+1 more|arXiv (Cornell University)|Sep 17, 2019

Robotic Locomotion and Control29 references4 citations

TL;DR

This paper presents a full-model-based motion planning and control framework for quadrupedal robots using Hybrid Zero Dynamics (HZD), enabling stable gait generation and exponential stabilization for walking, ambling, and trotting. It combines direct collocation-based nonlinear programming for trajectory optimization with iterative BMI-based controller synthesis via Poincaré return map analysis, successfully demonstrating robust ambling on the 36-state, 12-actuator Vision 60 robot in simulation and real-world experiments.

ABSTRACT

The hybrid zero dynamics (HZD) approach has become a powerful tool for the gait planning and control of bipedal robots. This paper aims to extend the HZD methods to address walking, ambling and trotting behaviors on a quadrupedal robot. We present a framework that systematically generates a wide range of optimal trajectories and then provably stabilizes them for the full-order, nonlinear and hybrid dynamical models of quadrupedal locomotion. The gait planning is addressed through a scalable nonlinear programming using direct collocation and HZD. The controller synthesis for the exponential stability is then achieved through the Poincaré sections analysis. In particular, we employ an iterative optimization algorithm involving linear and bilinear matrix inequalities (LMIs and BMIs) to design HZD-based controllers that guarantee the exponential stability of the fixed points for the Poincaré return map. The power of the framework is demonstrated through gait generation and HZD-based controller synthesis for an advanced quadruped robot, ---Vision 60, with 36 state variables and 12 control inputs. The numerical simulations as well as real world experiments confirm the validity of the proposed framework.

Motivation & Objective

• To extend the Hybrid Zero Dynamics (HZD) framework—proven effective in bipedal robots—toward full-order, nonlinear, hybrid models of quadrupedal locomotion.
• To address the lack of formal stability guarantees in existing quadruped control methods that rely on model reduction and simplified dynamics.
• To develop a scalable, systematic framework for generating optimal, dynamically stable gaits (walking, ambling, trotting) using direct collocation and HZD optimization.
• To synthesize HZD-based controllers that provably ensure exponential stability of periodic orbits via Poincaré return map analysis and bilinear matrix inequality (BMI) optimization.
• To validate the framework experimentally on the Vision 60 quadruped robot with 36 state variables and 12 control inputs, demonstrating stable ambling at 0.3 m/s.

Proposed method

• Model the quadruped as a hybrid dynamical system with 13 links and 18 degrees of freedom, using a 36-state, 12-input model of the Vision 60 robot.
• Formulate gait planning as a nonlinear programming (NLP) problem using direct collocation to generate optimal periodic trajectories for walking, ambling, and trotting.
• Apply the HZD framework to define a controlled invariant manifold, ensuring that the desired gait corresponds to a periodic solution of the hybrid system.
• Use Poincaré return map analysis to assess local stability of the periodic orbit, with the Jacobian's dominant eigenvalues indicating stability margins.
• Design HZD-based controllers via iterative optimization involving linear and bilinear matrix inequalities (LMIs and BMIs), parameterizing output functions as $ y_a(q,\xi_v) = H_0(\xi_v)q $ to stabilize the fixed point.
• Implement a time-based PD controller for real-time torque tracking of optimized trajectories, with domain switching triggered by time-based event functions.

Experimental results

Research questions

• RQ1 Can the HZD framework be systematically extended from bipedal to quadrupedal robots to achieve full-model-based motion planning and control?
• RQ2 How can optimal, dynamically stable gaits (walking, ambling, trotting) be generated for a full-order, nonlinear, hybrid dynamical model of a quadruped?
• RQ3 What controller synthesis method ensures exponential stability of the periodic gait orbits in the presence of hybrid dynamics and high-dimensional state spaces?
• RQ4 Can BMI-based optimization effectively stabilize complex quadrupedal gaits by adjusting controller parameters to shift the dominant eigenvalues of the Poincaré map?
• RQ5 To what extent can the proposed framework bridge the gap between simulation and real-world implementation on a complex, high-DOF quadruped robot?

Key findings

• The HZD-based framework successfully generated stable, optimal trajectories for walking, ambling, and trotting gaits using direct collocation on the full 36-state, 12-input Vision 60 model.
• The BMI optimization algorithm stabilized a 4-domain trotting gait by shifting the dominant eigenvalues of the Poincaré map from {-1.1062, 1.000, -0.4201+0.7278i} to {0.8877, -0.1507±0.8661i, -0.8830±0.0994i}, ensuring local exponential stability.
• Real-world experiments confirmed stable ambling at 0.3 m/s on the Vision 60 robot, with phase portraits showing convergence to the desired orbit.
• The average torque inputs were 7.73 N·m (hip roll), 9.46 N·m (hip pitch), and 16.17 N·m (knee), indicating feasible actuator loads.
• The time-based PD implementation achieved robust tracking without additional feedback layers, demonstrating the feasibility of direct model-based control.
• Each BMI iteration took approximately 30 minutes on a standard high-end machine, showing the method's computational feasibility despite complexity.