James Biggs

Department of Systems Engineering

University of Reading, England

**Hamiltonian Systems on Lie Groups**

In this lecture I will tackle the motion planning problem for underactuated control systems defined on matrix Lie groups. Examples of such problems include motion control of autonomous underwater vehicles and attitude control of satellites. This problem is formulated as one of optimal control which aims to steer the system along paths that minimize control energy. An application of Pontryagin’s Maximum Principle to this optimal control problem leads to the appropriate Hamiltonian and its corresponding Hamiltonian vector fields. In this context the motion planning problem becomes one of solving Hamiltonian systems defined on Lie groups. Finally, this methodology is applied in detail to the motion planning problem for an autonomous underwater vehicle. It will be illustrated that for an axially symmetric underwater vehicle the motion planning problem can be solved analytically.

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