Constraints’ resolution by optimal trajectory planning for anholonom devices

Most of the wheeled vehicles are anholonom systems the position and the rotational pose of which cannot arbitrarily be determined. This problem typically is solved by iterative, small, ldquoback and forthrdquo type movements while leaving a place/occupying a vacancy in a crowded parking place and also needs simple solution in the tracking control of a smooth path of considerable velocity. For this purpose convenient solution can be the prescription of kinematically not exactly realizable position and pose data that can only be approximated by using optimal control finding compromise between the requirements in contradiction instead of inventing realizable nominal trajectories that mathematically may be cumbersome since it normally requires the use of Frenet frames. Instead applying the ldquoorthodoxrdquo method by Pontryagin expressing the kinematic restrictions as constraints with associated Lagrange multipliers in the proposed solution these restrictions are explicitly built in so the simplest form of the Gradient Descent Method becomes applicable. Simulation calculations using MS EXCELpsilas SOLVER package and Visual Basic taking into account the dynamics of the steering wheel and that of the vehicle in the longitudinal direction are presented to show the applicability of this concept.