Legendre Pseudo-Spectral Method for Missile Trajectory Optimization with Free Final Time

This paper presents a Legendre pseudo-spectral method for solving trajectory optimization problems with realistic state and input constraints. Fuel mass flow rate, rate of bank angle, amount of fuel, and angle of attack as considered as constraints. The final flight path angle and minimum final velocity constraints are added for a high probability of interception and maneuverability. Lift/drag coefficients and specific impulse which are functions of Mach number, angle of attack, and altitude, were handled with the lagging technique. These parameters are fixed at the current optimization iteration and updated at the next iteration. The original problem is converted to a free-final time problem using a time scale variable, which is an additional optimization variable. In the Pseudo-Spectral method, states and inputs are approximated with the Lagrange polynomial at every flipped Legendre–Gauss-Radau (LGR) collocation point. Dynamic constraints and other constraints are transcribed to algebraic equations at these LGR points. This discretized problem is solved with a MATLAB optimization solver, and values of states and inputs at every LGR collocation point are obtained. The Pseudo-Spectral method is proven to be effective in solving a trajectory optimization problem having realistic constraints, and is robust in solving trajectory optimization problems with a crude initial guess. A numerical simulation validates that this method produces accurate results with a small number of node points.