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

• Published in: IEEE robotics and automation letters Vol. 9; no. 11; pp. 10224 - 10231
• Main Authors: Sathya, Ajay Suresha, Decre, Wilm, Swevers, Jan
• Format: Journal Article
• Language: English
• Published: Piscataway IEEE 01-11-2024; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Aerospace electronics; Algorithms; Complexity; Computation; Computational complexity; Computational efficiency; Dynamics; End effectors; Force; Heuristic algorithms; Kinematics; multi-contact whole-body motion planning and control; Optimization; Robot kinematics; Robots; Vectors; whole-body motion planning and control
• ISSN: 2377-3766; 2377-3766
• DOI: 10.1109/LRA.2024.3469829

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 <inline-formula><tex-math notation="LaTeX">O(n+m^{2})</tex-math></inline-formula> compared to <inline-formula><tex-math notation="LaTeX"> O(n + m^{2}d)</tex-math></inline-formula> of the PV-OSIM algorithm and <inline-formula><tex-math notation="LaTeX">O(n+md +m^{2})</tex-math></inline-formula> of the extended force propagator algorithm (EFPA), where <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula> is the number of joints, <inline-formula><tex-math notation="LaTeX">m</tex-math></inline-formula> is the number of constraints and <inline-formula><tex-math notation="LaTeX">d</tex-math></inline-formula> is the depth of the kinematic tree. Since the Delassus matrix is an <inline-formula><tex-math notation="LaTeX">m \times m</tex-math></inline-formula> sized matrix and its computation must consider all the <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula> 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.