M.P. Hennessey and C. Shakiban (USA)

brachistochrone, optimal control, calculus of variations, and singular control theory

The brachistochrone for a steerable particle moving on a 2D curved surface in a gravity ﬁeld is solved using an op timal control formulation with state feedback and extends previous work by the authors on a 1D curved surface. The process begins with a derivation of a fourth-order open loop plant model with the system input being the body yaw rate. Solving for the minimum-time control law entails introducing four co-states and solving the Euler-Lagrange equations with the Hamiltonian being stationary with re spect to the control. Since the system is autonomous, the Hamiltonian must be zero. A decoupled two-point bound ary value problem results with a transversality condition and its solution requires iteration of the initial bearing an gle so the integrated trajectory runs through the ﬁnal point. The k = 1 generalized Legendre-Clebsch necessary con dition from singular control theory is satisﬁed for all nu merical simulations performed and optimality is assured. Analytical formulations and/or simulations in MATLAB Re xercise the theory developed and illustrate application, minimizing travel time over a: (1) skewed plane, (2) in clined half-pipe, and (3) a bivariate quadratic surface when starting essentially from rest or a noticeable initial speed. Lastly, a control law singularity in particle speed is over come numerically when the particle is initially at rest.

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