On the Fundamental Importance of Gauss-Newton in Motion Optimization

TL;DR

This paper demonstrates that the Gauss-Newton approximation closely matches the true Hessian in motion optimization, enabling more efficient computation and improved convergence in high-degree-of-freedom systems.

Contribution

It provides a theoretical proof that Gauss-Newton Hessian approximations are nearly optimal for many objective terms, introduces a novel kinetic energy reformulation, and presents practical techniques for efficient curvature computation.

Findings

Gauss-Newton Hessian approximates the true Hessian well as discretization becomes small.

The new kinetic energy reformulation captures curvature efficiently with minimal computation.

Experimental results validate the theoretical insights on real-world motion optimization.

Abstract

Hessian information speeds convergence substantially in motion optimization. The better the Hessian approximation the better the convergence. But how good is a given approximation theoretically? How much are we losing? This paper addresses that question and proves that for a particularly popular and empirically strong approximation known as the Gauss-Newton approximation, we actually lose very little--for a large class of highly expressive objective terms, the true Hessian actually limits to the Gauss-Newton Hessian quickly as the trajectory's time discretization becomes small. This result both motivates it's use and offers insight into computationally efficient design. For instance, traditional representations of kinetic energy exploit the generalized inertia matrix whose derivatives are usually difficult to compute. We introduce here a novel reformulation of rigid body kinetic energy designed explicitly for fast and accurate curvature calculation. Our theorem proves that the Gauss-Newton Hessian under this formulation efficiently captures the kinetic energy curvature, but requires only as much computation as a single evaluation of the traditional representation. Additionally, we introduce a technique that exploits these ideas implicitly using Cholesky decompositions for some cases when similar objective terms reformulations exist but may be difficult to find. Our experiments validate these findings and demonstrate their use on a real-world motion optimization system for high-dof motion generation.