A Global Coordinate-Free Approach to Invariant Contraction on Homogeneous Manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916814057897984 |
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| author | Harapanahalli, Akash Coogan, Samuel |
| author_facet | Harapanahalli, Akash Coogan, Samuel |
| contents | In this work, we provide a global condition for contraction with respect to an invariant Riemannian metric on reductive homogeneous spaces. Using left-invariant frames, vector fields on the manifold are horizontally lifted to the ambient Lie group, where the Levi-Civita connection is globally characterized as a real matrix multiplication. By linearizing in these left-invariant frames, we characterize contraction using matrix measures on real square matrices, avoiding the use of local charts. Applying this global condition, we provide a necessary condition for a prescribed subset of the manifold to possibly admit a contracting system, which accounts for the underlying geometry of the invariant metric. Applied to the sphere, this condition implies that no great circle can be contained in a contraction region. Finally, we apply our results to compute reachable sets for an attitude control problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_15441 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Global Coordinate-Free Approach to Invariant Contraction on Homogeneous Manifolds Harapanahalli, Akash Coogan, Samuel Systems and Control Optimization and Control In this work, we provide a global condition for contraction with respect to an invariant Riemannian metric on reductive homogeneous spaces. Using left-invariant frames, vector fields on the manifold are horizontally lifted to the ambient Lie group, where the Levi-Civita connection is globally characterized as a real matrix multiplication. By linearizing in these left-invariant frames, we characterize contraction using matrix measures on real square matrices, avoiding the use of local charts. Applying this global condition, we provide a necessary condition for a prescribed subset of the manifold to possibly admit a contracting system, which accounts for the underlying geometry of the invariant metric. Applied to the sphere, this condition implies that no great circle can be contained in a contraction region. Finally, we apply our results to compute reachable sets for an attitude control problem. |
| title | A Global Coordinate-Free Approach to Invariant Contraction on Homogeneous Manifolds |
| topic | Systems and Control Optimization and Control |
| url | https://arxiv.org/abs/2410.15441 |