Indirect Optimization of Multi-Phase Trajectories Involving Arbitrary Discrete Logic

Abstract: Multi-phase trajectories of aerospace vehicle systems involve multiple flight segments whose transitions may be triggered by boolean logic in continuous state variables, control and time. When the boolean logic is represented using only states and/or time, such systems are termed autonomously switched hybrid systems. The relaxed autonomously switched hybrid system approach (RASHS) was previously introduced to simplify the trajectory optimization process of such systems in the indirect framework when the boolean logic is solely represented using AND operations. This investigation enables cases involving arbitrary discrete logic. The new approach is termed the Generalized Relaxed Autonomously Switched Hybrid System (GRASHS) approach. Similar to the RASHS approach, the outcome of the GRASHS approach is the transformation of the necessary conditions of optimality from a multi-point boundary value problem to a two-point boundary value problem, which is simpler to handle. This is accomplished by converting the arbitrary boolean logic to the disjunctive normal form and applying smoothing using sigmoid and hyperbolic tangent functions. The GRASHS approach is demonstrated by optimizing a Mars entry, descent, and landing trajectory, where the parachute descent segment is active when the velocity is below the parachute deployment velocity or the altitude is below the parachute deployment altitude, and the altitude is above the powered descent initiation altitude. This set of conditions represents a combination of AND and OR logic. The previously introduced RASHS approach is not designed to handle such problems. The proposed GRASHS approach aims to fill this gap.