PONTRYAGIN'S MAXIMUM PRINCIPLE AND OPTIMIZATION OF THE FLIGHT PHASE IN SKI JUMPING

摘要

There are several factors (the initial ski jumper's body position and its changes at the transition to the flight phase, the magnitude and the direction of the velocity vector of the jumper's center of mass, the magnitude of the aerodynamic drag and lift forces, etc.), which determine the trajectory of the jumper ski system along with the total distance of the jump. The objective of this paper is to bring out a method based on Pontryagin's maximum principle, which allows us to obtain a solution of the optimization problem for flight style control with three constrained control variables - the angle of attack (alpha), body ski angle (beta), and ski opening angle (V). The flight distance was used as the optimality criterion. The borrowed regression function was taken as the source of information about the dependence of the drag (D) and lift (L) area on control variables with tabulated regression coefficients. The trajectories of the reference and optimized jumps were compared with the K = 125 m jumping hill profile in Frenstat pod Radhostem (Czech Republic) and the appropriate lengths of the jumps, aerodynamic drag and lift forces, magnitudes of the ski jumper system's center of mass velocity vector and it's vertical and horizontal components were evaluated. Admissible control variables were taken at each time from the bounded set to respect the realistic posture of the ski jumper system in flight. It was found that a ski jumper should, within the bounded set of admissible control variables, minimize the angles alpha and beta, whereas angle V should be maximized. The length increment due to optimization is 17. For future work it is necessary to determine the dependence of the aerodynamic forces acting on the ski jumper system on the flight via regression analysis of the experimental data as well as the application of the control variables related to the ski jumper's mental and motor abilities.