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| Autori principali: | , , , |
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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2511.18800 |
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| _version_ | 1866912726482157568 |
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| author | Hampsey, Matthew van Goor, Pieter Banavar, Ravi Mahony, Robert |
| author_facet | Hampsey, Matthew van Goor, Pieter Banavar, Ravi Mahony, Robert |
| contents | Mechanical control systems such as aerial, marine, space, and terrestrial robots often naturally admit a state-space that has the structure of a Lie group. The kinetic energy of such systems is commonly invariant to the induced action by the Lie group, and the system dynamics can be written as a coupled ordinary differential equation on the group and the dual space of its Lie algebra, termed a Lie-Poisson system. In this paper, we show that Lie-Poisson systems can also be written as a left-invariant system on a semi-direct Lie group structure placed on the trivialised cotangent bundle of the symmetry group. The authors do not know of a prior reference for this observation and we are confident the insight has never been exploited in the context of tracking control. We use this representation to build a right-invariant tracking error for the full state of a Lie-Poisson mechanical system and show that the error dynamics for this error are themselves of Lie-Poisson structure, albeit with time-varying inertia. This allows us to tackle the general trajectory tracking problem using an energy shaping design metholodology. To demonstrate the approach, we apply the proposed design methodology to a simple attitude tracking control. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18800 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equivariant Tracking Control for Fully Actuated Mechanical Systems on Matrix Lie Groups Hampsey, Matthew van Goor, Pieter Banavar, Ravi Mahony, Robert Systems and Control Mechanical control systems such as aerial, marine, space, and terrestrial robots often naturally admit a state-space that has the structure of a Lie group. The kinetic energy of such systems is commonly invariant to the induced action by the Lie group, and the system dynamics can be written as a coupled ordinary differential equation on the group and the dual space of its Lie algebra, termed a Lie-Poisson system. In this paper, we show that Lie-Poisson systems can also be written as a left-invariant system on a semi-direct Lie group structure placed on the trivialised cotangent bundle of the symmetry group. The authors do not know of a prior reference for this observation and we are confident the insight has never been exploited in the context of tracking control. We use this representation to build a right-invariant tracking error for the full state of a Lie-Poisson mechanical system and show that the error dynamics for this error are themselves of Lie-Poisson structure, albeit with time-varying inertia. This allows us to tackle the general trajectory tracking problem using an energy shaping design metholodology. To demonstrate the approach, we apply the proposed design methodology to a simple attitude tracking control. |
| title | Equivariant Tracking Control for Fully Actuated Mechanical Systems on Matrix Lie Groups |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2511.18800 |