Gauss Pseudospectral Method Research Articles

Computing short-time aircraft maneuvers using direct methods

This paper analyzes the applicability of direct methods to design optimal short-term spatial maneuvers for an unmanned vehicle in a faster than real-time scale. It starts by introducing different basic control schemes, which employ online trajectory generation. Next, it presents and analyzes the results obtained through two recently developed direct transcription (collocation) methods: the Gauss pseudospec-tral method and the Legendre-Gauss-Lobatto pseudosp ectral method. The achieved results are further compared with those found through the Pontryagin’s Maximum (Minimum) Principle, and the paper continues by providing another set of direct method simulations incorporating more realistic boundary conditions. Finally, the results obtained using the third direct method, based on inverse dynamics in the virtual domain, are presented and discussed.

Algorithm 902

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.

Solution of nonlinear delay optimal control problems using a composite pseudospectral collocation method

We develop a composite collocation approximation scheme for thenumerical solution of nonlinear delay optimal control problems.For this purpose, we present an extension and also modificationfor the Gauss pseudospectral method using the hybrid ofblock-pulse functions and Lagrange polynomials based on theLegendre-Gauss points. In this respect, we derive thecorresponding operational matrix of derivative according to theweak representation of derivative operator. In order todemonstrate the applicability, efficiency and accuracy of theproposed method, we examine two illustrative examples.

Relative Motion Integrated Coupled Control for Spacecraft Formation Using Gauss Pseudospectral Method

This paper addresses the problem of how to control spacecraft formation with coupled translational and rotational dynamics with optimized fuel usage using a single body fixed thruster and reaction wheels. The orientation of the thrust vector is constrained by the attitude and angular velocity. In particular, the star camera cannot point at the sun during formation maneuvering. The formation coupled control, which is a difficult problem with nonlinear and nonconvex attitude dynamics, is posed as a nonlinear optimal control problem and solved via direct transcription using the Gauss pseudospectral method. Examples demonstrate application of the proposed approach with simulated trajectory optimization of a dual-spacecraft formation initialization and a three-spacecraft reconfiguration problem.

Two-stage path planning approach for solving multiple spacecraft reconfiguration maneuvers

The paper presents a two-stage approach for designing optimal reconfiguration maneuvers for multiple spacecraft in close proximity. These maneuvers involve well-coordinated and highly-coupled motions of the entire fleet of spacecraft while satisfying an arbitrary number of constraints. This problem is complicated by the nonlinearity of the attitude dynamics, the non-convexity of some of the constraints, and the coupling that exists in some of the constraints between the positions and attitudes of all spacecraft. While there has been significant research to solve for the translation and/or rotation trajectories for the multiple spacecraft reconfiguration problem, the approach presented in this paper is more general and on a larger scale than the problems considered previously. The essential feature of the solution approach is the separation into two stages, the first using a simplified planning approach to obtain a feasible solution, which is then significantly improved using a smoothing stage. The first stage is solved using a bidirectional Rapidly-exploring Random Tree (RRT) planner. Then the second step optimizes the trajectories by solving an optimal control problem using the Gauss pseudospectral method (GPM). Several examples are presented to demonstrate the effectiveness of the approach for designing spacecraft reconfiguration maneuvers.

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.

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.

Optimal configuration of tetrahedral spacecraft formations

The problem of determining minimum-fuel maneuver sequences for a four-spacecraft formation is considered. The objective of this paper is to find fuel-optimal spacecraft trajectories that transfer four spacecraft from an initial parking orbit to a desired terminal reference orbit while satisfying a set of constraints on the formation at the terminal time. Trajectories involving both one and two allowable maneuvers per spacecraft are considered. The resulting nonlinear optimal control problem is solved numerically using a recently developed direct transcription method called the Gauss pseudospectral method. The results presented in this paper highlight interesting features of the fuel-optimal formation and control. Furthermore, by showing that the discretized first-order optimality conditions from an indirect formulation are satisfied, a post-optimality analysis of the results demonstrates the accuracy and usefulness of the Gauss pseudospectral method.

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.