Determination of Trajectories for a Gliding Parachute System

The problem of the automatic guidance of a parachute subject to a constant wind is considered. A trajectory is to be determined, through variations of bank angle, to land on an intended target and to approach the target in the upwind direction. Optimal control, involving a minimum expenditure of control energy, requires the solution of a nonlinear two-point boundary value problem. However, the structure of the optimal control law is derived in terms of three unknown parameters. Nonlinear algebraic equations for these parameters involving elliptic integrals can be written, but their solution is tedious. A method for computing the parameters is developed through the numerical minimization of a terminal error function. This method requires the integration of a system of twelve differential equations at each iteration. A non-optimal guidance scheme is also given, which prescribes a given geometrical path parameterized by three constants as a trajectory.