Simulation in Alpine Skiing

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Simulation in Alpine Skiing
Peter Kaps Werner Nachbauer University of
Innsbruck, Austria
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Data Collection

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

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Optimal landing
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Landing in backward position
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Direct linear transformation
x,y image coordinates X,Y,Z object
coordinates bi DLT-parameters
Z
y
Y
X
x
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Control points at Russi jump
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Camera position at Russi jump
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Video frame on PC
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Unconstrained Newton-Euler equation of motion
(x,y,z)T
Rigid body center of gravity y(x,y,z)T
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Constrained equation of motion in 2D
unconstrained r 0
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Constrained Newton-Euler equation of
motion
f applied forces
r reaction forces
geometric constraint
dAlemberts principle
DAE
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Constrained Newton-Euler equation of motion
DAE
index 3 position level
index 2 velocity level
index 1 acceler. level
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Equation of motion
T(u,t)v
Index-2-DAE
Solved with RADAU 5 (Hairer-Wanner)
MATLAB-version of Ch. Engstler
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Jumps in Alpine skiing

• Ton van den Bogert
• Karin Gerritsen
• Kurt Schindelwig

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Force between snow and ski
Force between snow and ski normal to snow surface
3 nonlinear viscoelastic contact elements
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Musculo-skeletal model of a skier
muscle model van Soest, Bobbert 1993
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Muscle force
production of force contractile element
ligaments - seriell elastic element
connective tissue - parallel elastic element
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Muscle model of Hill
total length
L LCE LSEE
CE contractile element
SEE seriell elastic element
PEE parallel elastic element
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Force of seriell-elastic elements
Force of parallel-elastic elements
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Force-length-relation
Fmax maximal isometric force
isometric vCE 0
maximal activation q 1
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Force-velocity relation
vCE d/dt LCE
maximal activation q 1
optimal muscle length LCE LCEopt
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Hill equation (1938) Force-velocity relation
concentric contraction
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Activation model (Hatze 1981)muscle activation
LCE length of the contractile elements
calcium-ion concentration
value of the non activated muscle
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LCEopt optimal length of contractile elements
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Activation model (Hatze 1981)
Ordinary differential equation for the
calcium-ion concentration ?
Control parameter
relative stimulation rate
f stimulation rate, fmax maximum
stimulation rate
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Equilibrium
FCE(L,vCE,q) f(L,LCE)
Solving for vCE
vCE d/dt LCE fH(L,LCE,q(?,LCE))
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State of a muscle
three state variables
actual muscle length
length of the contractile element
calcium-ion conzentration
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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
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Comparison measured ( ) andsimulated (
) landing movement
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Turns in Alpine skiing
Simulation with DADS Peter Lugner Franz
Bruck Techn. University, Vienna
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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
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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 ?
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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
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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
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Software for Computation
Computations were performed in MATLAB
DAE-solver RADAU5 of Hairer-Wanner MATLAB-Versio
n by Ch. Engstler
Optimization problem Nelder-Mead simplex
algorithmus
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Results
truncated values
more exact values
0 , 0.1777 ?10.4064 (cdA)10.9094 0.17
77, 0.5834 ?20.4041 (cdA)20.9070 0.5834,
1.9200 ?10.1008 (cdA)10.5534
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Comparison more exact values
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Comparison truncated values
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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).
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Applications
Determination of an optimal trajectory Virtual
skiing, with vibration devices, in analogy to
flight simulators