Natural frames and interacting particles in three dimensions

Details Natural frames and interacting particles in three dimensions Natural frames and interacting particles in three dimensions

2005-03-18

arxiv.org/abs/math/0503390v1

Mathematics

Optimization and Control

8 pages, submitted to 44th IEEE Conf. Decision and Control, 2005

Scientific paper

Motivated by the problem of formation control for vehicles moving at unit speed in three-dimensional space, we are led to models of gyroscopically interacting particles, which require the machinery of curves and frames to describe and analyze. A Lie group formulation arises naturally, and we discuss the general problem of determining (relative) equilibria for arbitrary G-invariant controls (where G = SE(3) is a symmetry group for the control law). We then present global convergence (and non-collision) results for specific two-vehicle interaction laws in three dimensions, which lead to specific formations (i.e., relative equilibria). Generalizations of the interaction laws to n vehicles is also discussed, and simulation results presented.

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