Riemannian Optimization Techniques on Lie Groups Describing Rigid Body Attitude and Pose

Brennan McCann () and Morad Nazari ()
Additional contact information
Brennan McCann: Johns Hopkins University Applied Physics Laboratory
Morad Nazari: Embry-Riddle Aeronautical University

Journal of Optimization Theory and Applications, 2025, vol. 207, issue 3, No 23, 46 pages

Abstract: Abstract This paper presents a series of unconstrained Riemannian optimization methods for optimizing over Lie groups describing rigid body attitude and pose, along with novel comparisons between the efficiency and accuracy of those methods across different groups. Requisite operations for Riemannian optimization over the Lie groups of attitude (i.e. the special orthogonal group $$\textsf{SO}(3)$$ SO ( 3 ) and the space of quaternions $$\mathbb {H}$$ H ), and the Lie groups of pose (i.e. the special Euclidean group $$\textsf{SE}(3)$$ SE ( 3 ) and the space of dual quaternions $$\mathcal {D}\mathbb {H}$$ D H ) are presented. Cost functions over all these spaces are proposed and analyzed. Riemannian optimization techniques, including unconstrained gradient descent and conjugate gradient algorithms, and several methods of step length determination, are presented and compared against each other. Furthermore, an analysis of the relative efficiency of the Riemannian techniques as compared with Euclidean optimizers constrained to obey structure-preserving criteria is conducted. Case studies, including a classical single shooting optimal controls problem and a pose equilibrium determination scenario, are presented to examine the efficacy of these techniques in a variety of problems. Results suggest that the unconstrained Riemannian optimization methods tend to outperform the conventional, constrained Euclidean optimizers and that certain algorithms may be better suited to specific Lie groups than others.

Keywords: Riemannian optimization; Quaternions; Dual quaternions; Rotation matrix; Special orthogonal group; Special Euclidean group; Manifold; 22E70; 49Q99; 53C25; 57R15 (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10957-025-02810-5 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:207:y:2025:i:3:d:10.1007_s10957-025-02810-5

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10957/PS2

DOI: 10.1007/s10957-025-02810-5

Access Statistics for this article

Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull

More articles in Journal of Optimization Theory and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().