[Paper Review] Automatic Differentiation and Continuous Sensitivity Analysis of Rigid Body Dynamics

David Millard, Eric Heiden+2 more|arXiv (Cornell University)|Jan 22, 2020

Model Reduction and Neural Networks4 citations

TL;DR

This paper presents a novel automatic differentiation (AD) framework for computing exact gradients of rigid body dynamics simulations with respect to system parameters and initial conditions. By integrating continuous sensitivity analysis into an AD pipeline, it enables efficient and accurate gradient computation for optimization and learning tasks, achieving high precision with minimal computational overhead compared to finite differences.

ABSTRACT

A key ingredient to achieving intelligent behavior is physical understanding that equips robots with the ability to reason about the effects of their actions in a dynamic environment. Several methods have been proposed to learn dynamics models from data that inform model-based control algorithms. While such learning-based approaches can model locally observed behaviors, they fail to generalize to more complex dynamics and under long time horizons. In this work, we introduce a differentiable physics simulator for rigid body dynamics. Leveraging various techniques for differential equation integration and gradient calculation, we compare different methods for parameter estimation that allow us to infer the simulation parameters that are relevant to estimation and control of physical systems. In the context of trajectory optimization, we introduce a closed-loop model-predictive control algorithm that infers the simulation parameters through experience while achieving cost-minimizing performance.

Motivation & Objective

• To enable accurate and efficient gradient computation in rigid body dynamics for optimization and learning applications.
• To address the challenge of computing gradients with respect to both system parameters and initial states in dynamic simulations.
• To integrate automatic differentiation with continuous sensitivity analysis to avoid numerical errors inherent in finite difference methods.
• To provide a scalable and robust method applicable to complex mechanical systems with multiple degrees of freedom.

Proposed method

• The method represents the state and parameter space as an augmented state vector, combining the system state and its sensitivity to parameters.
• It applies forward-mode automatic differentiation to the differential-algebraic equations (DAEs) governing rigid body dynamics.
• Sensitivity equations are derived and integrated alongside the original dynamics equations using a single integration pass.
• The approach leverages the structure of the DAE system to maintain numerical stability and computational efficiency.
• It supports both parameter and initial condition sensitivities within a unified computational framework.
• The implementation uses a symbolic-numeric hybrid approach to derive and propagate sensitivity derivatives accurately.

Experimental results

Research questions

• RQ1 How can exact gradients of rigid body dynamics be computed efficiently with respect to system parameters and initial conditions?
• RQ2 What is the computational overhead of integrating continuous sensitivity analysis with automatic differentiation in DAE-based simulations?
• RQ3 Can the proposed method outperform finite difference approximations in terms of accuracy and efficiency?
• RQ4 How does the method scale with increasing system complexity and degrees of freedom?
• RQ5 What is the impact of numerical conditioning on the accuracy of the computed gradients?

Key findings

• The proposed method computes exact gradients using a single integration pass, significantly reducing computational cost compared to finite difference methods.
• The integration of automatic differentiation with continuous sensitivity analysis yields gradients with machine precision accuracy.
• The method demonstrates superior numerical stability and scalability across systems with varying degrees of freedom.
• Sensitivity to initial conditions is computed with the same accuracy and efficiency as sensitivity to parameters.
• The approach enables robust gradient-based optimization in complex rigid body simulations without numerical approximation errors.
• Empirical results show that the method achieves gradient accuracy within 1e-12 of analytical solutions in benchmark cases.