Natural frames and the design of interaction laws in three dimensions

Eric Justh
University of Maryland

Joint work with P.S. Krishnaprasad

In recent work we have demonstrated the utility of curves and moving frames for describing the motion of interacting particles subject to gyroscopic forces (i.e., forces which change the direction of motion of the particles without changing their speed). Particle trajectories are described using natural Frenet frames with the corresponding natural curvatures as controls. We have also formulated interaction laws based on such models for scenarios of cooperation (e.g., formation control for unmanned aerial vehicles) and conflict (e.g., motion camouflage behavior by certain bats and flying insects). In this talk, we focus on the design of these interaction laws. Specifically, we discuss the symmetries we impose, the techniques we use in generalizing from the planar to the three-dimensional setting, and the interpretation of the resulting three-dimensional laws. The symmetries include dependence of the controls on shape (i.e.,
relative) quantities, and the freedom in the initial choice of natural Frenet frame normal vectors. In the planar setting, we have (in previous work) provided biologically plausible interpretations for our formation control and motion camouflage laws. The formation control laws have clearly identifiable heading-alignment terms and a separation-dependent term (which also prevents collisions). For motion camouflage, the goal (for a pursuer) is to approach an evader while always appearing at a constant bearing from the point of view of the evader. In generalizing from planar steering laws to three- dimensional laws, we are guided by the calculation of the time- derivative of a Lyapunov function (for formation-control), or a cost function (for motion camouflage), along with the symmetry constraints. Moreover, the resulting three-dimensional laws take an intuitively reasonable form, further supporting the effectiveness of natural Frenet frames for modeling interacting mobile collectives in both nature and engineering.

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