Relaxed Autonomously Switched Hybrid System Approach to Indirect Multiphase Aerospace Trajectory Optimization

This investigation presents a methodology that simplifies the design of multiphase trajectories for aerospace vehicles using indirect methods. Such systems can be viewed as autonomously switched hybrid systems. The trajectories for such systems are subject to piecewise continuous dynamic equations. Moreover, each phase can be associated with a unique cost functional. Therefore, the cost functional of the overall trajectory design problem is also piecewise continuous. Consequently, the necessary conditions of optimality result in a multipoint boundary value problem in a system of differential-algebraic equations. These equations can be difficult to solve because the convergence of existing numerical algorithms is contingent upon supplying an initial guess that is close to the solution, which is not straightforward. The proposed method addresses this limitation, in part, by reducing the problem to a two-point boundary value problem by introducing sigmoid functions to obtain a single system of dynamic equations and a cost functional that are continuous and differentiable throughout the multiphase trajectory. The resultant reduced problem is solved by employing a continuation scheme, wherein the solution approach begins with solving a trivial problem, which is then evolved iteratively to the problem of interest. The proposed method is demonstrated using two examples: 1) time-switched Atlas V 4H launch to circular orbit; and 2) state-switched Mars entry, descent, and landing. The resultant trajectories are found to approximate the solutions of the respective original multipoint boundary value problems very well.