Trajectory Optimization for DDE Models of Supercavitating Underwater Vehicles

摘要

In this study we first develop a flight mechanics model for supercavitating vehicles, whichnis formulated to account for the dependence of the cavity shape from the past history ofnthe system. This mathematical model is governed by a particular class of delay differen-ntial equations, featuring time delays on the states of the system. Next, flight trajectoriesnand maneuvering strategies for supercavitating vehicles are obtained by solving an op-ntimal control problem, whose solution, given a cost function and general constraints andnbounds on states and controls, yields the control time histories that maneuver the vehiclenaccording to a desired strategy, together with the associated flight path. The optimalncontrol problem is solved using a novel direct multiple shooting approach, which isnformulated to properly handle conditions dictated by the delay differential equation for-nmulation governing the dynamic behavior of the vehicle. Specifically, the new formulationnenforces the state continuity line conditions in a least-squares sense using local interpo-nlations, which supports local time stepping and drastically reduces the number of opti-nmization unknowns. Examples of maneuvers and resulting trajectories demonstrate theneffectiveness of the proposed methodology and the generality of the formulation. Thenresults are also compared with those obtained from a previously developed model gov-nerned by ordinary differential equations to highlight the differences and demonstrate thenneed for the current formulation.