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

2. 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

3. Optimal landing
Workshop Ibk 05

4. Landing in backward position
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5. Workshop Ibk 05

6. Workshop Ibk 05

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

8. Control points at Russi jump
Workshop Ibk 05

9. Camera position at Russi jump
Workshop Ibk 05

10. Video frame on PC
Workshop Ibk 05

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

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

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

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

15. 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

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

17. 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

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

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

20. 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

21. Force of seriell-elastic elements
Force of parallel-elastic elements
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22. Force-length-relation
Fmax maximal
isometric force
isometric vCE = 0
maximal activation q = 1
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23. Force-velocity relation
vCE = d/dt LCE
maximal activation q = 1
optimal muscle length LCE = LCEopt
Workshop Ibk 05

24. Hill equation (1938) Force-velocity relation
concentric
contraction
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25. Activation model (Hatze 1981) muscle activation
LCE length of the contractile elements
calcium-ion concentration
value of the non activated muscle
Workshop Ibk 05

26. LCEopt optimal length of contractile elements
Workshop Ibk 05

27. 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

28. 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

29. State of a muscle three state variables actual muscle length
length of the contractile element
calcium-ion conzentration
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30. 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

31. Comparison measured ( ) and simulated ( ) landing movement
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32. Techn. University, Vienna
Turns in Alpine skiing
Simulation with DADS
Peter Lugner
Franz Bruck
Techn. University, Vienna
Workshop Ibk 05

33. 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

34. 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

35. 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

36. 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

37. 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

38. Results truncated values more exact values
[ , ] 1= (cdA)1=0.9094
[0.1777, ] 2= (cdA)2=0.9070
[0.5834, ] 1= (cdA)1=0.5534
Workshop Ibk 05

39. Comparison more exact values
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40. Comparison truncated values
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41. 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

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