Simulation in Alpine Skiing Peter Kaps Werner Nachbauer University of Innsbruck, Austria Workshop Ibk 05

Data Collection Trajectory of body points Landing movement after jumps in Alpine downhill skiing, Lillehammer (Carved turns, Lech) Turn, World Cup race, Streif, Kitzbühel Workshop Ibk 05

Optimal landing Workshop Ibk 05

Landing in backward position Workshop Ibk 05

Workshop Ibk 05

Workshop Ibk 05

Direct linear transformation x,y image coordinates X,Y,Z object coordinates bi DLT-parameters Z y Y x X Workshop Ibk 05

Control points at Russi jump Workshop Ibk 05

Camera position at Russi jump Workshop Ibk 05

Video frame on PC Workshop Ibk 05

Unconstrained Newton-Euler equation of motion (x,y,z)T Rigid body center of gravity: y=(x,y,z)T Workshop Ibk 05

Constrained equation of motion in 2D unconstrained r = 0 Workshop Ibk 05

Constrained Newton-Euler equation of motion f applied forces r reaction forces geometric constraint d‘Alembert‘s principle DAE Workshop Ibk 05

Constrained Newton-Euler equation of motion DAE index 3 position level index 2 velocity level index 1 acceler. level Workshop Ibk 05

Equation of motion T(u,t)v MATLAB-version of Ch. Engstler Index-2-DAE Solved with RADAU 5 (Hairer-Wanner) MATLAB-version of Ch. Engstler Workshop Ibk 05

Ton van den Bogert Karin Gerritsen Kurt Schindelwig Jumps in Alpine skiing Ton van den Bogert Karin Gerritsen Kurt Schindelwig Workshop Ibk 05

Force between snow and ski normal to snow surface Fy,Ende Fy,Mitte Fz,Ende Fz,Mitte Fz,Vorne Skispitze Schneeoberfläche 3 nonlinear viscoelastic contact elements Workshop Ibk 05

Musculo-skeletal model of a skier muscle model van Soest, Bobbert 1993 Workshop Ibk 05

Muscle force production of force – contractile element ligaments - seriell elastic element connective tissue - parallel elastic element Workshop Ibk 05

Muscle model of Hill total length L = LCE + LSEE LCE LSEE LPEE CE contractile element SEE seriell elastic element PEE parallel elastic element Workshop Ibk 05

Force of seriell-elastic elements Force of parallel-elastic elements Workshop Ibk 05

Force-length-relation Fmax maximal isometric force isometric vCE = 0 maximal activation q = 1 Workshop Ibk 05

Force-velocity relation vCE = d/dt LCE maximal activation q = 1 optimal muscle length LCE = LCEopt Workshop Ibk 05

Hill equation (1938) Force-velocity relation concentric contraction Workshop Ibk 05

Activation model (Hatze 1981) muscle activation LCE length of the contractile elements calcium-ion concentration value of the non activated muscle Workshop Ibk 05

LCEopt optimal length of contractile elements Workshop Ibk 05

Activation model (Hatze 1981) Ordinary differential equation for the calcium-ion concentration  Control parameter: relative stimulation rate f stimulation rate, fmax maximum stimulation rate Workshop Ibk 05

Equilibrium FCE(L,vCE,q) = f(L,LCE) Solving for vCE LSEE LCE LPEE FCE(L,vCE,q) = f(L,LCE) Solving for vCE vCE = d/dt LCE = fH(L,LCE,q(,LCE)) Workshop Ibk 05

State of a muscle three state variables actual muscle length length of the contractile element calcium-ion conzentration Workshop Ibk 05

Force of muscle-ligament complex according to Hill-Modell Input: L, LCE,  compute equivalent torque muscle force times lever arm Dk for joint k Dk constant Workshop Ibk 05

Comparison measured ( ) and simulated ( ) landing movement Workshop Ibk 05

Techn. University, Vienna Turns in Alpine skiing Simulation with DADS Peter Lugner Franz Bruck Techn. University, Vienna Workshop Ibk 05

Trajectory of a ski racer x(t)=(X(t), Y(t), Z(t))T position as a function of time Mean value between the toe pieces of the left and right binding Track Position constraint Z-h(X,Y)=0 Y-s(X)=0 g(x,t)=0 Workshop Ibk 05

Equation of Motion Skier modelled as a mass point descriptor form dependent coordinates x Differential-Algebraic Equation DAE ODE algebraic equation f applied forces r reaction forces r = -gxT  Workshop Ibk 05

Applied forces gravity snow friction drag t unit vector in tangential direction  friction coefficient N normal force N = ||r|| cd A drag area  density v velocity Workshop Ibk 05

Snow friction and drag area piecewice constant values determination of i , , ti by a least squares argument minimum x(ti) DAE-solution at time ti xi smoothed DLT-result at time ti Workshop Ibk 05

Software for Computation Computations were performed in MATLAB DAE-solver RADAU5 of Hairer-Wanner MATLAB-Version by Ch. Engstler Optimization problem Nelder-Mead simplex algorithmus Workshop Ibk 05

Results truncated values more exact values [0 , 0.1777] 1=0.4064 (cdA)1=0.9094 [0.1777, 0.5834] 2=0.4041 (cdA)2=0.9070 [0.5834, 1.9200] 1=0.1008 (cdA)1=0.5534 Workshop Ibk 05

Comparison more exact values Workshop Ibk 05

Comparison truncated values Workshop Ibk 05

Conclusions In Alpine skiing biomechanical studies under race conditions are possible. The results are reasonable, although circumstances for data collection are not optimal: no markers, position of control points must not disturb the racers, difficulties with commercial rights Results like loading of the anterior cruciate ligament (ACL) as function of velocity or inclination of the slope during landing or the possibility of a rupture of the ACL without falling are interesting applications in medicine. Informations on snow friction and drag in race conditions are interesting results, but a video analysis is expensive (digitizing the data, geodetic surveying). Workshop Ibk 05

Applications Determination of an optimal trajectory Virtual skiing, with vibration devices, in analogy to flight simulators Workshop Ibk 05