Optimal trajectory planning via chebyshev-based state parameterization

Two state parameterization methods based on finite-term Chebyshev representations are developed to determine the optimal state and control trajectories of unconstrained linear timeinvariant dynamic systems with quadratic performance indices. In one method, each state variable of a dynamic system is approximated by a shifted Chebyshev series. In the second method, each state variable is represented by the superposition of a shifted Chebyshev series and a third order polynomial. In both approaches, the necessary and sufficient condition of optimality is derived as a system of linear algebraic equations. The results of simulation studies demonstrate that the Chebyshev-pluspolynomial method offers computational advantages relative to the direct Chebyshev method, a previous Chebyshev method and a state-transition approach.