Quickest descent line during alpine ski racing

Abstract

Time differences between medalists at Olympic or World Cup alpine ski races are often less than 0.01 s. One factor that could affect these small differences is the line taken between the numerous gates passed through while speeding down the ski slope. The determination of the ‘quickest line’ is therefore critical to winning races. In this study the quickest lines are calculated by direct optimal control theory which converts an optimal control problem into a parameter optimization problem that is solved using a nonlinear programming method. Specifically, the problem is described in terms of an objective function in which state and control variables are implicitly involved. The objective function is the time between the starting point and finishing gate, while state variables are positions of the ski-skier systems on a ski slope, rotational angles of skis, velocities, and rotational velocity at a discrete time, i.e., a node. The control variable at each node is the skier-controlled edging angle between the ski sole and snow surface. Equations of motion of the ski-skier system on a ski slope are numerically satisfied at the midpoint between neighbouring nodes, and the original problem is converted into a nonlinear programming problem with equality and inequality constraints. The problem is solved by the sequential quadratic programming method in which numerical calculations are carried out using the MATLAB Optimization Toolbox. Numerical calculations are presented to determine the quickest lines of an uphill and a downhill ski turn with a starting point, first gate, and second gate (finish line) having been successfully carried out. The quickest line through four gates could not be calculated due to numerical difficulty. Instead, the descent line was respectively calculated for an uphill and downhill turn and simply added, giving a resultant time that represents an upper bound.