PV-OSIMr: A Lowest Order Complexity Algorithm for Computing the Delassus Matrix

TL;DR

PV-OSIMr is a novel, highly efficient algorithm for computing the Delassus matrix with minimal computational complexity, outperforming existing methods in both theory and practice for kinematic trees.

Contribution

It introduces PV-OSIMr, the lowest order complexity algorithm for Delassus matrix computation, optimizing the PV solver for better efficiency.

Findings

PV-OSIMr has a computational complexity of O(n + m^2).

It outperforms PV-OSIM and EFPA in practical benchmarks.

The algorithm is optimal in asymptotic complexity for the problem.

Abstract

We present PV-OSIMr, an efficient algorithm for computing the Delassus matrix (also known as the inverse operational space inertia matrix) for a kinematic tree, with the lowest order computational complexity known in literature. PV-OSIMr is derived by optimizing the Popov-Vereshchagin (PV) solver computations using the compositionality of the force and motion propagators. It has a computational complexity of O(n + m^2 ) compared to O(n + m^2d) of the original PV-OSIM algorithm and O(n+md+m^2 ) of the extended force propagator algorithm (EFPA), where n is the number of joints, m is the number of constraints and d is the depth of the kinematic tree. Since Delassus matrix computation requires constructing an m x m sized matrix and must consider all the n joints at least once, the asymptotic computational complexity of PV-OSIMr is optimal. We further benchmark our algorithm and find it to be often more efficient than the PV-OSIM and EFPA in practice.