This paper describes a computational method for solving optimal control problems involving large-scale, nonlinear, dynamical systems. Central to the approach is the idea that any optimal control problem can be converted into a standard nonlinear programming problem by parameterizing each control history using a set of nodal points, which then become the variables in the resulting parameter optimization problem. A key feature of the method is that it dispenses with the need to solve the two-point, boundary-value problem derived from the necessary conditions of optimal control theory. Gradient-based methods for solving such problems do not always converge due to computational errors introduced by the highly nonlinear characteristics of the costate variables. Instead, by converting the optimal control problem into a parameter optimization problem, any number of well-developed and proven nonlinear programming algorithms can be used to compute the near-optimal control trajectories. The utility of the parameter optimization approach for solving general optimal control problems for human movement is demonstrated by applying it to a detailed optimal control model for maximum-height human jumping. The validity of the near-optimal control solution is established by comparing it to a solution of the two-point, boundary-value problem derived on the basis of a bang-bang optimal control algorithm. Quantitative comparisons between model and experiment further show that the parameter optimization solution reproduces the major features of a maximum-height, countermovement jump (i.e., trajectories of body-segmental displacements, vertical and fore-aft ground reaction forces, displacement, velocity, and acceleration of the whole-body center of mass, pattern of lower-extremity muscular activity, jump height, and total ground contact time).
A two-level decomposition-aggregation approach for large-scale optimal control problems
