Second-order DMOC using projection

Discrete mechanics and optimal control (DMOC) is a recent development in optimal control of mechanical systems that takes advantage of the variational structure of mechanics when discretizing the optimal control problem. Typically, the discrete Euler-Lagrange equations are used as constraints on the feasible set of solutions, and then the objective function is minimized using a constrained optimization algorithm, such as sequential quadratic programming (SQP). In contrast, this paper illustrates that by reducing dimensionality by projecting onto the feasible subspace and then performing optimization, one can obtain significant improvements in convergence, going from superlinear to quadratic convergence. Moreover, whereas numerical SQP can run into machine precision problems before terminating, the projection-based technique converges easily. Double and single pendulum examples are used to illustrate the technique.