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Numerical Approaches for Constrained and Unconstrained, Static Optimization on the Special Euclidean Group SE(3)

Author

Listed:
  • Brennan McCann

    (Johns Hopkins University Applied Physics Laboratory)

  • Morad Nazari

    (Embry-Riddle Aeronautical University)

  • Christopher Petersen

    (University of Florida)

Abstract

In this paper, rigid body static optimization is investigated on the Riemannian manifold of rigid body motion groups. This manifold, which is also a matrix manifold, provides a framework to formulate translational and rotational motions of the body, while considering any coupling between those motions, and uses members of the special orthogonal group $$\textsf{SO}(3)$$ SO ( 3 ) to represent the rotation. Hence, it is called the special Euclidean group $$\textsf{SE}(3)$$ SE ( 3 ) . Formalism of rigid body motion on $$\textsf{SE}(3)$$ SE ( 3 ) does not fall victim to singularity or non-uniqueness issues associated with attitude parameterization sets. Benefiting from Riemannian matrix manifolds and their metrics, a generic framework for unconstrained static optimization and a customizable framework for constrained static optimization are proposed that build a foundation for dynamic optimization of rigid body motions on $$\textsf{SE}(3)$$ SE ( 3 ) and its tangent bundle. The study of Riemannian manifolds from the perspective of rigid body motion introduced here provides an accurate tool for optimization of rigid body motions, avoiding any biases that could otherwise occur in rotational motion representation if attitude parameterization sets were used.

Suggested Citation

  • Brennan McCann & Morad Nazari & Christopher Petersen, 2024. "Numerical Approaches for Constrained and Unconstrained, Static Optimization on the Special Euclidean Group SE(3)," Journal of Optimization Theory and Applications, Springer, vol. 201(3), pages 1116-1150, June.
    • Handle: RePEc:spr:joptap:v:201:y:2024:i:3:d:10.1007_s10957-024-02431-4
    DOI: 10.1007/s10957-024-02431-4
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    References listed on IDEAS

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    1. Du Nguyen, 2023. "Operator-Valued Formulas for Riemannian Gradient and Hessian and Families of Tractable Metrics in Riemannian Optimization," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 135-164, July.
    2. Xiaojing Zhu & Hiroyuki Sato, 2020. "Riemannian conjugate gradient methods with inverse retraction," Computational Optimization and Applications, Springer, vol. 77(3), pages 779-810, December.
    3. Yan, Zheping & Zhang, Jinzhong & Tang, Jialing, 2021. "Path planning for autonomous underwater vehicle based on an enhanced water wave optimization algorithm," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 181(C), pages 192-241.
    4. Yuya Yamakawa & Hiroyuki Sato, 2022. "Sequential optimality conditions for nonlinear optimization on Riemannian manifolds and a globally convergent augmented Lagrangian method," Computational Optimization and Applications, Springer, vol. 81(2), pages 397-421, March.
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