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We propose a method to analyze the three-dimensional nonholonomic system known as the Brockett integrator and to derive the (energy) optimal trajectories between two given points. For systems with nonholonomic constraint, it is well-known that the energy optimal trajectories corresponds to sub-Riemannian geodesics under a proper sub-Riemannian metric. Our method uses symmetry reduction and an analysis of the quotient space associated with the action of a (symmetry) group on R^3. By lifting the Riemannian geodesics with respect to an appropriate metric from the quotient space back to the original space R^3, we derive the optimal trajectories of the original problem.
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HV geometry for signal comparison
In order to compare and interpolate signals, we investigate a Riemannian geometry on the space of signals. The metric allows discontinuous signals and measures both horizontal (thus providing many benefits of the Wasserstein metric) and vertical deformations. Moreover, it allows for signed signals, which overcomes the main deficiency of optimal transportation-based metrics in signal processing. We characterize the metric properties of the space of signals and establish the regularity and stability of geodesics. Furthermore, we introduce an efficient numerical scheme to compute the geodesics and present several experiments which highlight the nature of the metric.
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- Award ID(s):
- 2206069
- PAR ID:
- 10519787
- Publisher / Repository:
- Americal Mathematical Society
- Date Published:
- Journal Name:
- Quarterly of Applied Mathematics
- Volume:
- 82
- Issue:
- 2
- ISSN:
- 0033-569X
- Page Range / eLocation ID:
- 391 to 430
- Subject(s) / Keyword(s):
- signal comparison, geometry on the space of functions
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/qam/1672
