1 American Institute of Aeronautics and Astronautics Higher Order Sensitivities for Solving Nonlinear Two-Point Boundary-Value Problems D. Todd Griffith * Texas A&M University, College Station, Texas 77843-3141 James D. Turner † Amdyn Systems, White GA 30184 S. R. Vadali ‡ and John. L. Junkins § Texas A&M University, College Station, Texas 77843-3141 In this paper, we consider new computational approaches for solving nonlinear Two- Point Boundary-Value Problems. The sensitivity calculations required in the solution utilize the automatic differentiation tool OCEA (Object Oriented Coordinate Embedding Method). OCEA has broad potential in this area and many other areas since the partial derivative calculations required for solving these problems are automatically computed and evaluated freeing the analyst from deriving and coding them. In this paper, we demonstrate solving nonlinear Two-Point Boundary Value Problems by shooting and direct methods using automatic differentiation. We demonstrate standard first-order algorithms and higher- order extensions. Additionally, automatic generation of co-state differential equations and second- and higher-order midcourse corrections are considered. Optimization of a sample Low-thrust, Mars-Earth trajectory is considered as an example. Computational issues related to domain of convergence and rate of convergence will be detailed. I. Introduction PPLICATIONS which require the solution to a Two-Point Boundary-Value Problem (TPBVP) occur in many engineering disciplines. The solution of the TPBVP determines states or functions of the states at two points, usually the initial and final times, which satisfy the boundary conditions for a given mathematical problem. For linear problems, analytical solutions exist for determining the unknown initial or final conditions; however, for nonlinear problems we must resort to iterative methods. Many methods have been developed to solve nonlinear TPBVPs dealing with a wide range of issues, including stiffness of differential equations of motion, reduction in sensitivity, and speed of convergence. One particular solution is obtained by the method of differential corrections 1,2 . Here a guess is made for the unknown states, the state differential equations are integrated until the final time, and the boundary conditions are checked to see if they are satisfied. When the constraint conditions are not satisfied, corrections to the unknown states are computed and the process is repeated until convergence criteria are met. Typically, these corrections are computed by using first order sensitivity calculations or at best, second order calculations using an approximate Hessian. In this paper, we study the automatic generation of sensitivity equations for solving nonlinear TPBVPs. The Fortran 90 extension OCEA (Object Oriented Coordinate Embedding Method) will be used to perform the sensitivity calculations 3-4 . OCEA has been used to solve problems in optimization 3-4 , estimation 5 , and generation of equations of motion 6 . OCEA has broad potential in many areas since it is extremely useful for computing partial derivatives of scalar, vector, matrix, and higher order tensor functions. Optimization of a sample Low-thrust, Mars-Earth trajectory will be considered as one example. Direct optimization using Differential Inclusions 7 will be considered in a second example demonstrating OCEA’s * Graduate Research Assistant, Department of Aerospace Engineering, 3141 TAMU, Student Member AIAA. † Adjunct Faculty, Department of Aerospace Engineering, and President Amdyn Systems. ‡ Stewart & Stevenson-I Professor, Department of Aerospace Engineering, 3141 TAMU, Member AIAA. § George Eppright Chair, Distinguished Professor, Department of Aerospace Engineering, 3141 TAMU, Fellow AIAA. A AIAA/AAS Astrodynamics Specialist Conference and Exhibit 16 - 19 August 2004, Providence, Rhode Island AIAA 2004-5404 Copyright © 2004 by D. Todd Griffith. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.