A Linear-Time Variational Integrator for Multibody Systems

Abstract

We present an efficient variational integrator for simulating multibody systems. Variational integrators reformulate the equations of motion for multibody systems as discrete Euler-Lagrange (DEL) equation, transforming forward integration into a root-finding problem for the DEL equation. Variational integrators have been shown to be more robust and accurate in preserving fundamental properties of systems, such as momentum and energy, than many frequently used numerical integrators. However, state-of-the-art algorithms suffer from O(n 3 ) complexity, which is prohibitive for articulated multibody systems with a large number of degrees of freedom, n, in generalized coordinates. Our key contribution is to derive a quasi-Newton algorithm that solves the root-finding problem for the DEL equation in O(n), which scales up well for complex multibody systems such as humanoid robots. Our key insight is that the evaluation of DEL equation can be cast into a discrete inverse dynamic problem while the approximation of inverse Jacobian can be cast into a continuous forward dynamic problem. Inspired by Recursive Newton-Euler Algorithm (RNEA) and Articulated Body Algorithm (ABA), we formulate the DEL equation individually for each body rather than for the entire system, such that both inverse and forward dynamic problems can be solved efficiently in O(n). We demonstrate scalability and efficiency of the variational integrator through several case studies.

Time Integrating Articulated Body Dynamics Using Position-Based Collocation Method

“…Collocation methods approximate the locus of configuration using high-order polynomials. Unlike these general-purpose integrators, special integrators such as Lie-group integrators [15] and variational integrators [17] can be developed to respect the Lie group structure of articulated bodies, resulting in desirable conservative properties in linear/angular momentum and the Hamiltonian.…”

“…First, numerical dissipation cannot totally be avoided, although we can reduce it using smaller timestep sizes or highorder collocation methods. Second, to recast the articulated body dynamics as an optimization problem and avoid high-order derivatives, we discretize the velocities in a Euclidean workspace, instead of using a Lie-Group structure [17]. As a result, our PBAD method can be less accurate compared with Lie-Group integrators.…”