Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon 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 nonsingular 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 nonlinear programming problem (NLP) to the costates of the optimal control problem. Finally, * M.S. Student, Dept. 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.

Direct Trajectory Optimization and Costate Estimation of State Inequality Path-Constrained Optimal Control Problems Using a Radau Pseudospectral Method

“…[18][19][20] In the Radau pseudospectral method, the state and its time derivative are approximated, respectively, in each mesh interval S k , (k = 1, . .…”

“…In even more recent years, a great deal of research has been done on the class of direct collocation pseudospectral methods. 3,12,16,[18][19][20][21] In a pseudospectral method, the state is approximated using a basis of either Lagrange of Chebyshev polynomials and the dynamics are collocated at points associated with a Gaussian quadrature. The most common collocation points, which are the roots of a linear combination of Legendre polynomials or derivatives of Legendre polynomials, are Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and Legendre-Gauss-Lobatto (LGL) points.…”

“…The method developed in this paper utilizes a modified version of the Radau pseudospectral method. [18][19][20] Specifically, the optimal control problem is divided into a mesh such that the times of the mesh points are included as variables in the optimal control problem. This leads to an optimal control problem where it is desired to determine not only the optimal state and control in each mesh interval, but it is also desired to determine the optimal values of mesh point times.…”

A Method for Computing Derivatives in MATLAB

“…, N ) in Eqs. (3) and (5). The total number of LGR points, N = N k K, is obtained by dividing the problem into K mesh intervals using N k LGR points in each mesh interval (for details see Ref.…”

A Preliminary Analysis of Small Spacecraft Finite-Thrust Aeroassisted Orbital Transfer

“…The optimal aeroassisted orbital transfer problem is posed as a nonlinear multiple-phase optimal control problem, and the optimal control problem is solved via direct collocation using the Radau collocation hp-adaptive 24 version of the open-source optimal control software GPOPS. [25][26][27] The overall performance of the vehicle is analyzed as a function of the number of atmospheric passes, required inclination change, and maximum allowable heating rate. The results obtained in this study are also compared against a finitethrust all-propulsive orbital transfer.…”