Symbolic–Numeric Indirect Method for Solving Optimal Control Problems for Large Multibody SystemsThe Time-Optimal Racing Vehicle Example

This work presents a methodological framework, based on an indirect approach, for the automatic generation and numerical solution of Optimal Control Problems (OCP) for mechatronic systems, described by a system of Differential Algebraic Equations (DAEs). The equations of the necessary condition for optimality were derived exploiting the DAEs structure, according to the Calculus of Variation Theory. A collection of symbolic procedures was developed within general-purpose Computer Algebra Software. Those procedures are general and make it possible to generate both OCP equations and their jacobians, once any DAE mathematical model, objective function, boundary conditions and constraints are given. Particular attention has been given to the correct definition of the boundary conditions especially for models described with set of dependent coordinates. The non-linear symbolic equations, their jacobians with the sparsity patterns, generated by the procedures above mentioned, are translated into a C++ source code. A numerical code, based on a Newton Affine Invariant scheme, was also developed to solve the Boundary Value Problems (BVPs) generated by such procedures. The software and methodological framework here presented were successfully applied to the solution of the minimum-lap time problem of a racing motorcycle.