Geoffrey T. Huntington

Corrigendum: Algorithm 902: GPOPS, a MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method

An algorithm is described to solve multiple-phase optimal control problems using a recently developed numerical method called the Gauss pseudospectral method. The algorithm is well suited for use in modern vectorized programming languages such as FORTRAN 95 and MATLAB. The algorithm discretizes the cost functional and the differential-algebraic equations in each phase of the optimal control problem. The phases are then connected using linkage conditions on the state and time. A large-scale nonlinear programming problem (NLP) arises from the discretization and the significant features of the NLP are described in detail. A particular reusable MATLAB implementation of the algorithm, called GPOPS, is applied to three classical optimal control problems to demonstrate its utility. The algorithm described in this article will provide researchers and engineers a useful software tool and a reference when it is desired to implement the Gauss pseudospectral method in other programming languages.

Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method

A pseudospectral method, called the Gauss pseudospectral method, for solving nonlinear optimal control problems is presented. In the method presented here, orthogonal collocation of the dynamics is performed at the Legendre-Gauss points. This form of orthogonal collocation leads a nonlinear programming problem (NLP) whose Karush-Kuhn-Tucker (KKT) multipliers can be mapped to the costates of the continuous-time optimal control problem. In particular the Legendre-Gauss collocation leads to a costate mapping at the boundary points. The method is demonstrated on an example problem where it is shown that highly accurate costates are obtained. The results presented in this paper show that the Gauss pseudospectral method is a viable apprach for direct trajectory optimization and costate estimation.

Brief paper: A unified framework for the numerical solution of optimal control problems using pseudospectral methods

A unified framework is presented for the numerical solution of optimal control problems using collocation at Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and Legendre-Gauss-Lobatto (LGL) points. It is shown that the LG and LGR differentiation matrices are rectangular and full rank whereas the LGL differentiation matrix is square and singular. Consequently, the LG and LGR schemes can be expressed equivalently in either differential or integral form, while the LGL differential and integral forms are not equivalent. Transformations are developed that relate the Lagrange multipliers of the discrete nonlinear programming problem to the costates of the continuous optimal control problem. The LG and LGR discrete costate systems are full rank while the LGL discrete costate system is rank-deficient. The LGL costate approximation is found to have an error that oscillates about the true solution and this error is shown by example to be due to the null space in the LGL discrete costate system. An example is considered to assess the accuracy and features of each collocation scheme.

Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method

A method is presented for direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using global collocation at Legendre-Gauss-Radau (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 to the costates of the optimal control problem. More precisely, it is shown that the dual multipliers for the discrete scheme correspond to a pseudospectral approximation of the adjoint equation using polynomials one degree smaller than that used for the state equation. The relationship between the coefficients of the pseudospectral scheme for the state equation and for the adjoint equation is established. Also, it is shown that the inverse of the pseudospectral LGR differentiation matrix is precisely the matrix associated with an implicit LGR integration scheme. Hence, the method presented in this paper can be thought of as either a global implicit integration method or a pseudospectral method. Numerical results show that the use of LGR collocation as described in this paper leads to the ability to determine accurate primal and dual solutions for both finite and infinite-horizon optimal control problems.

Optimal Reconfiguration of Spacecraft Formations Using the Gauss Pseudospectral Method

This paper addresses the problem of how to reconfigure a tetrahedral formation in a fuel-optimal manner. The objective of this research is to formulate and solve an optimal control problem that results in a single-orbit, minimum-fuel reconfiguration strategy such that, after reconfiguration, the four spacecraft are in an orbit that creates an acceptable tetrahedral configuration throughout a portion of the orbit. In particular, an acceptable tetrahedron is one that meets specified size and shape requirements. The reconfiguration problem is posed as a multiple-phase nonlinear optimal control problem and is solved via direct transcription using the Gauss pseudospectral method. The results obtained in this study provide insight into the structure of the optimal mission design and demonstrate the generality, computational efficiency, and accuracy of the Gauss pseudospectral method.

A Comparison of Accuracy and Computational Efficiency of Three Pseudospectral Methods

A comparison is made between three pseudospectral methods used to numerically solve optimal control problems. In particular, the accuracy of the state, control, and costate obtained using the Legendre, Radau, and Gauss pseudospectral methods is compared. Three examples with different degrees of complexity are used to identify key differences between the three methods. The results of this study indicate that the Radau and Gauss methods are very similar in accuracy, while both significantly outperform the Legendre method with respect to costate accuracy. Furthermore, it is found that the computational efficiency of the three methods is comparable. Based on these results and a detailed analysis of the mathematics of each method, a rationale is created to determine when each method should be implemented to solve optimal control problems.

Comparison of Global and Local Collocation Methods for Optimal Control

O PTIMAL control problems are often solved numerically via direct methods [1]. In recent years, considerable attention has been focused on a class of direct transcription methods called pseudospectral [2–4] or orthogonal collocation [5,6] methods. In a pseudospectral method, a finite basis of global interpolating polynomials is used to approximate the state at a set of discretization points. The time derivative of the state is approximated by differentiating the interpolating polynomial and constraining the derivative to be equal to the vector field at a finite set of collocation points. Although any set of unique collocation points can be chosen, generally speaking, an orthogonal collocation is chosen, i.e., the collocation points are the roots of an orthogonal polynomial (or linear combinations of such polynomials and their derivatives). Because pseudospectral methods are commonly implemented via orthogonal collocation, the terms pseudospectral and orthogonal collocation are interchangeable (thus researchers in onefieldmay use the term pseudospectral [3], whereas others may use the term orthogonal collocation [5]). One advantage to pseudospectral methods is that for smooth problems, pseudospectral methods typically have faster convergence rates than other methods, exhibiting “spectral accuracy” [4]. Pseudospectral methods were traditionally used to solve fluid dynamics problems [3], whereas orthogonal collocation methods were first established in the chemical engineering community [6]. Seminal work in the mathematics of orthogonal collocation methods for optimal control dates back to 1979 [7] and, in recent years, the following orthogonal collocationmethods have risen to prominence: the Legendre pseudospectral method [8], the Chebyshev pseudospectral method [9], the Radau orthogonal collocation method [10,11], and the Gauss pseudospectral method [12]. Within the class of pseudospectral methods, there are two very different and widely used implementation strategies that can be best described as local and global approaches. In a local approach, the time interval is divided into a large number of subintervals called segments or finite elements [6] and a small number of collocation points are usedwithin each segment. The segments are then linked via continuity conditions on the state, the independent variable, and possibly the control. The rationale for using local collocation is that a local method provides so-called local support [13] (i.e., the discretization points are located so that they support the local behavior of the dynamics) and is both computationally simple and efficient. Although local methods have a long history in solving optimal control problems, much of the recent work has shown great success in the application of global collocation [14,15] (i.e., collocation using a global polynomial across the entire time interval). Researchers are also looking into higher-order local methods [16,17], which lie in between global and local methods. In light of the recent results that promote global orthogonal collocation and the long history of the use local collocation, it is important to gain a better understanding as to how these two different philosophies compare in practice. This Note provides a comparison between global and local orthogonal collocation solutions for two optimal control problems.A recently developed orthogonal collocation method called the Gauss pseudospectral method (GPM) [12] is employed in both a global format and a local format. In the global approach, the GPM is implemented on a single segment and the number of collocation points is varied. As a local approach, the GPM is implemented such that a number of equal width segments are varied, while a small fixed number of collocation points are used in each segment. We note that the local application of the GPM is similar to the approaches of [6,10,18]. The results obtained in this research suggest that, except in special circumstances, global orthogonal collocation is preferable to local orthogonal collocation. For the smooth example in this study, the global GPM is much more accurate than the local GPM for a given number of total collocation points. Furthermore, for a desired accuracy, the global approach is computationally more efficient than the local approach in smooth problems. For nonsmooth problems, as in the second example, the local and global approach are quite similar in terms of accuracy.

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

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 integralfrom the initialpoint 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,

A Comparison between Global and Local Orthogonal Collocation Methods for Solving Optimal Control Problems

A comparison is made between local and global orthogonal collocation methods for solving optimal control problems. The study is performed using an orthogonal collocation method called the Gauss pseudospectral method. In particular, this method is employed in two different ways. In the first approach, the method is applied globally and solutions are computed for various numbers of collocation points. In the second approach, the Gauss pseudospectral method is applied locally, meaning the problem is segmented, with each segment containing a small, fixed number of collocation points. Solutions are then computed for various numbers of equal-width segments. These two different approaches are then compared in terms of their accuracy and computational efficiency. The results of this study indicate that, for a given number of collocation points, global collocation is much more accurate than local collocation for smooth problems and can provide comparable results for problems with discontinuities. Furthermore, the results obtained in this paper indicate that for a desired accuracy on smooth problems the global approach is computationally more efficient. In order to substantiate the analysis, the comparison is performed on two examples with vastly different characteristics.