Dynamic Programming Trajectory Optimization by Piecewise Linear Approximation

Abstract

The rapid increase of computational capability due to high-performance processors and parallel processing, as well as continual decreases of hardware costs, have revived interest in previously overlooked numerical methods. The practical application of Dynamic Programming, one such method for trajectory optimization, has been prevented by the "menace of the expanding grid," otherwise known as the dimensional difference problem. This problem occurs in cases where the number of control variables is fewer than that of state variables. The present paper proposes a promising method that overcomes the problem by piecewise linear approximation to obtain the optimum return function. The accuracy achieved by the method is illustrated with a simple example in which the exact solution is provided analytically. Furthermore, the method may be applied to aircraft longitudinal flight optimization, where it generates the most efficient and practically applicable reference flight trajectory in real time. NomenclatureB A, = system matrix of the continuous time system equation C = constraints D = drag f = function vector g = gravity acceleration H = geopotential altitude J = objective function L = Lagrangian, lift m = dimension of the control variable, mass of aircraft M = Mach number n = dimension of the state variable, number of divisions P = solution of the Riccati equation Q = weight matrix for state variables in an objective function r = weight for control variables in an objective function, dimension of constraints R = weight matrix for control variables in an objective function 0 R = radius of the Earth R = set of real numbers S = wing area t = time T = thrust 1 Graduate student, School of Engineering,AIAA SciTech u = control variable V = velocity x = state vector, down range γ = flight path angle η = cross range θ = longitude µ = fuel flow ξ = down range ρ = air density φ = latitude, objective function at terminal ψ = constraint function at terminal Subscripts f , 0 = initial, final c = continuous-time system d = discrete-time system j i, = element numbers for vector and matrix k = time stage number in the discrete time system max = maximum min = minimum MO = maximum operating opt = optimal Superscripts  = zeroth-order hold

A Modified Dynamic Programming Approach for 4D Minimum Fuel and Emissions Trajectory Optimization

“…Miyazawa et al proposed a Moving Search Space Dynamic Programming (MS-DP) to reduce the computation time of traditional DP and applied it to the generation of a conflictfree and minimum-fuel 4D optimal trajectory [28]. Harada et al proposed a method by using the piecewise linear approximation to overcome the limitation of the menace of the expanding grid problem of DP [29].…”