Direct Trajectory Optimization and Costate Estimation of General Optimal Control Problems Using a Radau Pseudospectral Method

Abstract

A method is presented for direct trajectory optimization and costate estimation using global collocation at Legendre-Gauss-Radau (LGR) points. The method is formulated first by casting the dynamics in integral form and computing the integral from the initial point to the interior LGR points and the terminal point. The resulting integration matrix is nonsin-gular and thus can be inverted so as to express the dynamics in inverse integral form. Then, by appropriate choice of the approximation for the state, a pseudospectral (i.e., differential) form that is equivalent to the inverse integral form is derived. As a result, the method presented in this paper can be thought of as either a global implicit integration method or a pseudospectral method. Moreover, the formulation derived in this paper enables solving general finite-horizon problems using global collocation at the LGR points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the non-linear programming problem (NLP) to the costates of the optimal control problem. Finally, * M.S. Student, it is shown that a previously developed Radau collocation method, which is restricted to infinite-horizon problems, is subsumed by the method presented in this paper. The results of this paper show that the use of LGR collocation as described in this paper leads to the ability to determine accurate primal and dual solutions to general finite-horizon optimal control problems.

Progress in reentry trajectory planning for hypersonic vehicle

“…One of the important advantages is that the RPM can accurately build a complete mapping between the costates of the OCP and the KKT multipliers of NLP [42,45]. Unlike the LPM, the approximation of the costate converges exponentially in the RPM.…”

“…To solve the infinite-horizon nonlinear OCP, the RPM was introduced by Fahroo et al [60], followed by a study of the direct trajectory optimization for the finite-horizon problem [42]. Numerical examples showed when the RPM would fail [61] and when it was proper to use the RPM for the OCP [31,56].…”

“…The popular pseudospectral methods include the Chebyshev pseudospectral method (CPM) [33,34], Legendre pseudospectral method (LPM) [35][36][37][38][39][40][41], Radau pseudospectral method (RPM) [42][43][44][45][46][47], and Gauss pseudospectral method (GPM) [48][49][50][51][52][53][54][55][56]. The features of the four methods are shown in Table 1.…”

Gauss Pseudospectral Method Based Trajectory Optimization for Hypersonic Glide Vehicles

“…The cordial acceleration term and the transported acceleration term after the dimensionless treatment are expressed as following 2 …”

“…The trajectory optimization for aircraft has always been a concern and a complex problem [1][2][3] . The most significant feature of glide trajectory optimization problem distinguished from other trajectory optimization problems is that hypersonic glide vehicles fly at high speed in the near-space for a long time, and the aerodynamic and thermal environment is very bad.…”