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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.01549 |
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| _version_ | 1866913222572900352 |
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| author | Wu, Pengcheng Martinez, Sonia Chen, Jun |
| author_facet | Wu, Pengcheng Martinez, Sonia Chen, Jun |
| contents | This paper proposes a mechanism to fine-tune convex approximations of probabilistic reachable sets (PRS) of uncertain dynamic systems. We consider the case of unbounded uncertainties, for which it may be impossible to find a bounded reachable set of the system. Instead, we turn to find a PRS that bounds system states with high confidence. Our data-driven approach builds on a kernel density estimator (KDE) accelerated by a fast Fourier transform (FFT), which is customized to model the uncertainties and obtain the PRS efficiently. However, the non-convex shape of the PRS can make it impractical for subsequent optimal designs. Motivated by this, we formulate a mixed integer nonlinear programming (MINLP) problem whose solution result is an optimal $n$ sided convex polygon that approximates the PRS. Leveraging this formulation, we propose a heuristic algorithm to find this convex set efficiently while ensuring accuracy. The algorithm is tested on comprehensive case studies that demonstrate its near-optimality, accuracy, efficiency, and robustness. The benefits of this work pave the way for promising applications to safety-critical, real-time motion planning of uncertain dynamic systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_01549 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fine-Tuned Convex Approximations of Probabilistic Reachable Sets under Data-driven Uncertainties Wu, Pengcheng Martinez, Sonia Chen, Jun Robotics This paper proposes a mechanism to fine-tune convex approximations of probabilistic reachable sets (PRS) of uncertain dynamic systems. We consider the case of unbounded uncertainties, for which it may be impossible to find a bounded reachable set of the system. Instead, we turn to find a PRS that bounds system states with high confidence. Our data-driven approach builds on a kernel density estimator (KDE) accelerated by a fast Fourier transform (FFT), which is customized to model the uncertainties and obtain the PRS efficiently. However, the non-convex shape of the PRS can make it impractical for subsequent optimal designs. Motivated by this, we formulate a mixed integer nonlinear programming (MINLP) problem whose solution result is an optimal $n$ sided convex polygon that approximates the PRS. Leveraging this formulation, we propose a heuristic algorithm to find this convex set efficiently while ensuring accuracy. The algorithm is tested on comprehensive case studies that demonstrate its near-optimality, accuracy, efficiency, and robustness. The benefits of this work pave the way for promising applications to safety-critical, real-time motion planning of uncertain dynamic systems. |
| title | Fine-Tuned Convex Approximations of Probabilistic Reachable Sets under Data-driven Uncertainties |
| topic | Robotics |
| url | https://arxiv.org/abs/2303.01549 |