Download T - Numerical Analysis

Simulation in Alpine Skiing
Peter Kaps
Werner Nachbauer
University of Innsbruck, Austria
Workshop Ibk 05
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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
b1 X  b2Y  b3Z  b4
b5 X  b6Y  b7 Z  b8
y
x
b9 X  b10Y  b11Z 1
b9 X  b10Y  b11Z 1
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
mx  f x
my  f y
mz  f z
My  f
(x,y,z)T
Rigid body
center of gravity: y=(x,y,z)T
J  n    J
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Constrained
equation of motion in 2D
mi xi  f xi  rxi
mi zi  f zi  rzi
Iii  fi  ri
z
( xi, zi )T
αi
unconstrained r=0
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x
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Constrained Newton-Euler
equation of motion
My  f  r
f applied forces
r reaction forces
g ( y, t )  0
geometric constraint
G  gy
r  G 
T
d‘Alembert‘s principle
My  G   f
T
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DAE
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Constrained Newton-Euler
equation of motion
My  G   f
T
DAE
g ( y, t )  0
index 3
position level
Gy  g
1
index 2
velocity level
Gy  g
2
index 1
acceler. level
G  gy
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Equation of motion
I
0

0

0
0
I
0
0
0
0
0
0

0  u


0 v

0  w
 
0  
 
T(u,t)v
 
w

T
  Mw  G  
 
1
  Gv  g



f


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
Fz a| z| (1 bz)
p
Force between snow and ski
normal to snow surface
Fz,Ende
Fy,Ende
Fz,Mitte
Fy,Mitte
Fz,Vorne
Schneeoberfläche
Skispitze
3 nonlinear viscoelastic contact elements
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Musculo-skeletal model of a skier
m. iliopsoas
mm. glutei
m. rectus femoris
mm. vasti
mm. ischiocrurales
m. gastrocnemius
m. tibialis anterior
m. soleus
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
LCE
LPEE
LSEE
CE
SEE
PEE
contractile element
seriell elastic element
parallel elastic element
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Force of seriell-elastic elements
FSEE  f ( LSEE )  f ( L LCE )
FSEE
LSEE0
LSEE
Force of parallel-elastic elements
FPEE  f ( LPEE )  f ( LCE )
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Force-length-relation
FCE
Fmax
Fmax maximal
isometric force
LCEopt
LCE
2wLCEopt
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
bFmax  avCE
FCE 
b  vCE
concentric
contraction
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Activation model (Hatze 1981)
muscle activation
FCE  qFmax
q0 [ ρ( LCE )γ]2
q
1[ ρ( LCE )γ]2
LCE length of the contractile elements
γ
q0  0.005
calcium-ion concentration
value of the non activated muscle
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 ( LCE) 66200
1.90
LCEopt
2.90

1
LCE
LCEopt
optimal length of contractile elements
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Activation model (Hatze 1981)
Ordinary differential equation for
the calcium-ion concentration 
dγ m(cη γ ), γ(0) γ
0
dt
Control parameter: relative stimulation rate
η f
fmax
0 η1
f
stimulation rate,
fmax maximum stimulation rate
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Equilibrium
FCEFSEEFPEE  f ( L, LCE )
LCE
LPEE
LSEE
LSEELLCE, LPEELCE
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
L, LCE , γ
L
actual muscle length
LCE length of the contractile element
γ
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 (
) and
simulated (
) 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
mx  f  r
g ( x, t )  0
f
r
ODE
algebraic equation
applied forces
reaction forces
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r = -gxT 
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Applied forces
gravity snow friction
drag
 0 


1
f   0   Nt  cd A v 2 t
2
  mg 


v
unit vector in tangential direction t 
|| v ||

friction coefficient
N
normal force N = ||r||
cd A drag area

density
v
velocity
t
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Snow friction and drag area
piecewice constant values
i
ti 1  t  ti
cd Ai
determination of i , cd Ai , ti
by a least squares argument

x ( ti )  xi
x(ti)
xi
2
minimum
DAE-solution at time ti
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-Version by Ch. Engstler
Optimization problem
Nelder-Mead simplex algorithmus
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Results
truncated values
t<0.58
0.40
0.90
μ
cdA
t>0.58
0.10
0.55
more exact values
[0
, 0.1777]
[0.1777, 0.5834]
[0.5834, 1.9200]
1=0.4064
2=0.4041
1=0.1008
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(cdA)1=0.9094
(cdA)2=0.9070
(cdA)1=0.5534
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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
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