Trajectory optimization on a parallel processor

The design of a trajectory for an aerospace vehicle involves choosing a set of variables to optimally shape the path of the vehicle. Typically the trajectory is simulated by numerically solving the differential equations describing the dynamics of the vehicle. The optimal trajectory is usually determined by using a nonlinear programming (parameter optimization) algorithm to select the variables. Problems that require choosing control functions are usually reduced to choosing a finite set of parameters. The computational expense of a trajectory optimization is dominated by two factors: the cost of simulating a trajectory and the cost of computing gradient information for the optimization algorithm. This paper presents a technique for using a parallel processor to reduce the cost of these calculations. The trajectory is broken into phases, which can be simulated in parallel, thereby reducing the cost of an individual trajectory. This multiple shooting technique has been suggested by a number of authors. The nonlinear optimization problem that results from this formulation produces a Jacobian matrix that is sparse. The Jacobian is computed using sparse finite differencing, which is also performed in parallel, thereby reducing the cost of obtaining gradient information for the optimization algorithm. This paper describes the application of sparse finite differencing to a multiple shooting formulation of the two-point boundary-value problem, in a manner suitable for implementation on a parallel processor. Computational experience with the algorithm as implemented on the BBN GPIOOO (Butterfly) parallel processing computer is described.