The computations of the high-order partial derivatives in a given problem are often cumbersome or not accurate. To combat such shortcomings, a new method for calculating exact high-order sensitivities using multicomplex numbers is presented. Inspired by the recent complex step method that is only valid for firstorder sensitivities, the new multicomplex approach is valid to arbitrary order. The mathematical theory behind this approach is revealed, and an efficient procedure for the automatic implementation of the method is described. Several applications are presented to validate and demonstrate the accuracy and efficiency of the algorithm. The results are compared to conventional approaches such as finite differencing, the complex step method, and two separate automatic differentiation tools. The multicomplex method performs favorably in the preliminary comparisons and is therefore expected to be useful for a variety of algorithms that exploit higher order derivatives.
A new algorithm is presented to solve constrained nonlinear optimal control problems, with an emphasis on highly nonlinear dynamical systems. The algorithm, called HDDP, is a hybrid variant of differential dynamic programming, a proven second-order technique that relies on Bellman's Principle of Optimality and successive minimization of quadratic approximations. The new hybrid method incorporates nonlinear mathematical programming techniques to increase efficiency: quadratic programming subproblems are solved via trust region and range-space active set methods, an augmented Lagrangian cost function is utilized, and a multiphase structure is implemented. In addition, the algorithm decouples the optimization from the dynamics using first-and second-order state transition matrices. A comprehensive theoretical description of the algorithm is provided in this first part of the two paper series. Practical implementation and numerical evaluation of the algorithm is presented in Part 2.
Perturbed two-body problems play a special role in Celestial Mechanics as they capture the dominant dynamics for a broad range of natural and artificial satellites. In this paper, we investigate the classic Stark problem, corresponding to motion in a Newtonian gravitational field subjected to an additional uniform force of constant magnitude and direction. For both the two-dimensional and three-dimensional cases, the integrals of motion are determined, and the resulting quadratures are analytically integrated. A complete list of exact, closed-form solutions is deduced in terms of elliptic functions. It is found that all expressions rely on only seven fundamental solution forms. Particular attention is given to ensure that the expressions are well-behaved for very small perturbations. A comprehensive study of the phase space is also made using a boundary diagram to describe the domains of the general types of possible motion. Numerical examples are presented to validate the solutions.
Low-thrust propulsion is becoming increasingly considered for future space missions, but optimization of the resulting trajectories is very challenging. To solve such complex problems, differential dynamic programming is a proven technique based on Bellman's Principle of Optimality and successive minimization of quadratic approximations. In this paper, we build upon previous and existing optimization strategies to present an alternative hybrid variant of differential dynamic programming for robust low-thrust optimization. It uses first-and second-order state transition matrices to take advantage of an efficient discretization scheme and obtain the partial derivatives needed to perform the minimization. Unlike the traditional formulation, the state transition approach provides valuable constraint sensitivities and furthermore is naturally amenable to parallel computation. The method includes also a smoothing strategy to improve robustness of convergence when starting far from the optimum, as well as the capability to handle efficiently both soft and hard constraints. Procedures to drastically reduce the computation cost are mentioned. Preliminary numerical results are presented and compared to existing algorithms to illustrate the performance and the accuracy of our approach.
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