CDS Invited Lecture: James Biggs, "Hamiltonian Systems on Lie Groups"
Thursday, October 25, 2007
2460 A.V. Williams Building
301 405 6602
Control and Dynamical Systems Invited Lecture Series
Some Hamiltonian Systems on Lie Groups
School of Systems Engineering
University of Reading, England
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 Pontryagins 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.