Practical direct collocation methods for computational optimal control

The development of numerical methods for optimal control and, specifically, trajectory optimisation, has been correlated with advances in the fields of space exploration and digital computing. Space exploration presented scientists and engineers with challenging optimal control problems. Specialised numerical methods implemented in software that runs on digital computers provided the means for solving these problems. This chapter gives an introduction to direct collocation methods for computational optimal control. In a direct collocation method, the state is approximated using a set of basis functions, and the dynamics are collocated at a given set of points along the time interval of the problem, resulting in a sparse nonlinear programming problem. This chapter concentrates on local direct collocation methods,which are based on low-order basis functions employed to discretise the state variables over a time segment. This chapter includes sections that discuss important practical issues such as multi-phase problems, sparse nonlinear programming solvers, efficient differentiation, measures of accuracy of the discretisation, mesh refinement, and potential pitfalls. A space relevant example is given related to a four-phase vehicle launch problem.