Optimal Paths in Still Air for a Sailplane with a Quadratic Glide Polar

Abstract

This paper considers the optimal control problem of minimizing the altitude loss of a glider maneuvering instill air to a nearby position and heading angle under shallow bank angle assumptions. The glider’s motion,as viewed from above, is modeled as a kinematic car with control inputs of forward speed and turn rate. (Inpractice, a glider can adjust its flight path angle and bank angle to achieve a desired forward speed and turnrate.) The speed control is strictly positive, varying from the stall speed to the maximum speed, and the turnrate is symmetrically bounded about zero. The sink rate of the glider is assumed to be a quadratic function ofthe forward speed, approximating the “glide polar”. Necessary conditions derived from the Minimum Principleare used to characterize the extremal controls. Further, suboptimality conditions are identified geometricallyto arrive at a finite and sufficient set of candidate optimal controls. The extremal paths are shown to consist of(i) straight line segments flown at the glider’s “best glide” speed and (ii) maximum rate turns with either: (a)a heading dependent speed input, (b) the stall speed, or (c) the minimum sink speed. A synthesis procedure isproposed to solve for the optimal path. These results may be applicable to autonomous sailplanes or mannedaircraft experiencing loss of thrust (under autopilot control).