A Spectral Patching Method for Direct Trajectory Optimization
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A Spectral Patching Method for Direct Trajectory Optimization Fariba Fahroo 1 and I. Michael Ross ' Abstract A new direct method for trajectory optimization is presented in this paper. This method is based on using global orthogonal polynomials instead of the traditional spline-integration methods to discretize the dynamical constraints. The states and controls are approximated by interpolating polynomials with unknown coefficients as the values of the functions at the collocation points. The collocation points are the Legendre-Gauss- Lobatto points which provide the smallest error in interpolation in the least-squares sense. They also provide an expression for the Lagrange interpolating polynomials used as trial functions in terms of Legendre polynomials. In order to achieve a greater freedom in the distribution of the collocation points, we present a patching method wherein the time domain may be arbitrarily divided into a few subintervals. The state equations over these subintervals are discretized by the spectral collocation method. Then, the solutions on these subintervals are "patched" by a continuity condition. In this patching method, by judiciously combining the ideas of the traditional collocation methods with global spectral methods, we achieve advantages of both methods while using neither. Results for the simplified Moon -landing problem (selected in honor of Professor Battin) show that the spectral patching method works well for just two subintervals.
Introduction Direct optimization methods have been successfully used to solve a variety of practical trajectory optimization problems arising in aeronautics and astronautics [1]. Of the direct methods, the collocation (transcription) methods are widely used and form the mathematical and algorithmic basis for much software. These direct collocation methods, that we shall call "traditional" collocation methods, use local piecewise continuous polynomials to satisfy the dynamic constraints at the socalled collocation points [2-4]. The spectral collocation method [5-8] differs from these traditional collocation methods in that we use global orthogonal polynomials. 'Assistant Professor, Department of Mathematics, Code Ma/Ff, Naval Postgraduate School, Monterey, CA 93943, E-mail: [email protected]. Senior Member AIAA. 2Visiting Associate Professor, Charles Stark Draper Laboratory, MS 70, 555 Technology Square, Cambridge, MA 02139, E-mail: [email protected]. Associate Fellow AIAA. Member, AAS.
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Spectral methods have been successfully applied to solve problems in fluid dynamics [9-11] and they appear to have some distinct advantages in solving optimal control problems. The advantages of using global orthogonal polynomials over local ones are many. For example, in the spectral method, the structure of the problem is preserved leading to an elegant relationship between the Karush-Kuhn-Tucker (KKT) multipliers associated with the nonlinear programming (NLP) problem and the costates associated with the optimal control problem [7]. Th
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