Single-Stage Optimization of Open-loop Stable Limit Cycles with Smooth, Symbolic Derivatives
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914031588081664 |
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| author | Hassan, Muhammad Saud Ul Hubicki, Christian |
| author_facet | Hassan, Muhammad Saud Ul Hubicki, Christian |
| contents | Open-loop stable limit cycles are foundational to legged robotics, providing inherent self-stabilization that minimizes the need for computationally intensive feedback-based gait correction. While previous methods have primarily targeted specific robotic models, this paper introduces a general framework for rapidly generating limit cycles across various dynamical systems, with the flexibility to impose arbitrarily tight stability bounds. We formulate the problem as a single-stage constrained optimization problem and use Direct Collocation to transcribe it into a nonlinear program with closed-form expressions for constraints, objectives, and their gradients.
Our method supports multiple stability formulations. In particular, we tested two popular formulations for limit cycle stability in robotics: (1) based on the spectral radius of a discrete return map, and (2) based on the spectral radius of the monodromy matrix, and tested five different constraint-satisfaction formulations of the eigenvalue problem to bound the spectral radius. We compare the performance and solution quality of the various formulations on a robotic swing-leg model, highlighting the Schur decomposition of the monodromy matrix as a method with broader applicability due to weaker assumptions and stronger numerical convergence properties.
As a case study, we apply our method on a hopping robot model, generating open-loop stable gaits in under 2 seconds on an Intel Core i7-6700K, while simultaneously minimizing energy consumption even under tight stability constraints. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10647 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Single-Stage Optimization of Open-loop Stable Limit Cycles with Smooth, Symbolic Derivatives Hassan, Muhammad Saud Ul Hubicki, Christian Robotics Systems and Control Open-loop stable limit cycles are foundational to legged robotics, providing inherent self-stabilization that minimizes the need for computationally intensive feedback-based gait correction. While previous methods have primarily targeted specific robotic models, this paper introduces a general framework for rapidly generating limit cycles across various dynamical systems, with the flexibility to impose arbitrarily tight stability bounds. We formulate the problem as a single-stage constrained optimization problem and use Direct Collocation to transcribe it into a nonlinear program with closed-form expressions for constraints, objectives, and their gradients. Our method supports multiple stability formulations. In particular, we tested two popular formulations for limit cycle stability in robotics: (1) based on the spectral radius of a discrete return map, and (2) based on the spectral radius of the monodromy matrix, and tested five different constraint-satisfaction formulations of the eigenvalue problem to bound the spectral radius. We compare the performance and solution quality of the various formulations on a robotic swing-leg model, highlighting the Schur decomposition of the monodromy matrix as a method with broader applicability due to weaker assumptions and stronger numerical convergence properties. As a case study, we apply our method on a hopping robot model, generating open-loop stable gaits in under 2 seconds on an Intel Core i7-6700K, while simultaneously minimizing energy consumption even under tight stability constraints. |
| title | Single-Stage Optimization of Open-loop Stable Limit Cycles with Smooth, Symbolic Derivatives |
| topic | Robotics Systems and Control |
| url | https://arxiv.org/abs/2312.10647 |