Energy-Optimal Paths for a Glider with Speed and Load Factor Controls

摘要

This work investigated the problem of steering a glider to a desired planar position and heading angle, while minimizing changes in potential energy. A realistic turn rate and sink rate model with speed and load factor controls was developed. The hodograph analysis was made possible by a change of variables (from time to arclength) so that the control inputs only appeared in one state rate and the cost rate equation. Thus, the hodograph could be easily visualized and used to geometrically interpret the minimization operation of the minimum principle. This analysis provided useful insight into the structure of the extremal controls. A rigorous analysis of the necessary conditions of the minimum principle further characterized the extremals and gave the flight conditions associated with each extremal arc. Knowledge of these relevant flight conditions, for optimal gliding turning motions, may inform future glider designs as well as piloting and control techniques. The path synthesis problem was solved using a commercially available pseudospectral optimal control solver. However, the geometric and analytical results give a much deeper understanding of the extremal controls than could be obtained using the numerical solver alone. The analytical results also provide a means to aid and refine the results of the numerical solver, for example, by constructing an analytical expression a posteriori for each extremal arc returned by the solver. The motivation for this work was control of underwater glider motion, but the results are equally applicable to sailplane flight or to emergency operation of powered aircraft that have lost thrust.