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

1 Willow, Inria, DI-ENS Paris, PSL University, 75006 Paris, France.
2 Division of Robotics, Automation and Mechatronics, Department of Mechanical Engineering, KU Leuven, and FlandersMake@KULeuven, 3000 Leuven, Belgium.

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 recently proposed PV-OSIM algorithm 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 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 the Delassus matrix is an $m×m$ sized matrix and its computation must consider all the n joints, PV-OSIMr's asymptotic computational complexity is optimal. We further benchmark our algorithm and find it to be often more efficient than the PV-OSIM and EFPA in practice.

Overview

Overview computational complexity of algorithms.

Selected Benchmarks

Operation count for shadow hand.
Operation count for shadow hand attached to Atlas.

BibTeX

@ARTICLE{sathya_pvosimr_2024,
      author={Sathya, Ajay Suresha and Decré, Wilm and Swevers, Jan},
      journal={IEEE Robotics and Automation Letters}, 
      title={PV-OSIMr: A Lowest Order Complexity Algorithm for Computing the Delassus Matrix}, 
      year={2024},
      volume={9},
      number={11},
      pages={10224-10231},
      keywords={Heuristic algorithms;Force;Kinematics;Aerospace electronics;Robot kinematics;End effectors;Computational complexity;Computational efficiency;Dynamics;multi-contact whole-body motion planning and control;whole-body motion planning and control},
      doi={10.1109/LRA.2024.3469829}}