A DAE for Skiing Turns P. Kaps1), U. Filippi Oberegger1) M. Mössner2), D. Heinrich2), W. Nachbauer2), K. Schindelwig2) 1)Department

of Engineering Mathematics 2)Department of Sport Science University of Innsbruck

Outline of the talk Multibody model of the skier-ski system Equation of motion in descriptor form for a carved turn is a Differential Algebraic Equation (DAE) Balance problem => DAE with nonholonomic constraints Numerical solution with RADAU5 Result: Motion of the skier, forces and torques Model validation by video analysis of a turn

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Data collection • Video analysis of a run of an expert skier • Kistler force plate fixed on right ski • 3D object coordinates reconstructed from image coordinates by DLT (Direct Linear Transformation) • Smoothing of time history of Kistler forces and marker coordinates

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Necessity of smoothing resting point at origin xi

x-coordinate from frame i

errors: x1 = + 1 cm, x2 = - 1cm, x3 = + 1cm acceleration for

a

x1 2 x2 2 t

x3

t 10 4 10 4 10

2

s

2

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400 m/s

2

4

Film sequence of run

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Track of right ski Force plate on right ski => analysis of left turn => reference point for the path of the skier is a marker at toe piece of right binding

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Solution with Software • Model – skier, ski, snow – joints connecting segments – driving constraints

• Turn initiation by edging – inward leaning by bending the inner knee and extending the outer – hip angulation

• Solution with LMS Virtual.Lab Hairer60 - Genf, June 17 - 20, 2009

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Model of skier Skier Sledge

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Model of skier – 5 segments 2 shanks 2 thighs 1 trunk

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Monoskier - Joints rotational joint ankle joint knee joint

ball and socket joint

hip joint

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Driving constraints – Inverse Dynamics Equation of motion

Video analysis => positions, orientations, velocities, accelerations => reaction forces and torques mi vi

(0,0, mi g )

T

'  J i

' i

' i

' i

n'

J

' i

fi

fri

Ri

f

n' i

R

1 no. of segment i

7 fri

• Coulomb friction at ski segments fi • Prime for terms in body-fixed coordinates ' • Computation of i by rotation matrices Hairer60 - Genf, June 17 - 20, 2009

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Hanavan model Anthropometric measurements according to the Hanavan model => for each segment position of center of mass, position of joints, mass and inertia tensor

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Rotation matrices Ai • Transformation body fixed coordinates to global coordinates

ri

P

Ai

ri

' i i

As

for shank:

[ f i , gi , hi ]

• fi ,gi ,hi body-fixed orthonormal vectors • hi along segment axis, fi in ski direction • trunk: gi , hi computed from quadrilateral formed by 4 markers Hairer60 - Genf, June 17 - 20, 2009 13

Angular velocities wi • Obtained from rotation matrices Ai: ' ~ Matrix i ~' a

A Ai T i

' a z'

• Arrows indicate positive values x'

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y' 14

Results • Angular velocities of loaded outer ski (right) x':

longitudinal axis of ski z': normal to running surface of ski y': laterally inwards

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Reaction force of loaded outer ski Fitted load distribution of trunk (right/left ski): factors kr=0.9, kl=0.1 for t in [1.36, 2.74], afterwards linear load shift from right to left ski

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Forward dynamics • Mono-skier model: trunk, thigh, shank • Balance problem => nonholonomic constraints • Equation of motion in descriptor form solved as index 3 DAE by RADAU5 • Applied forces: gravity, Coulomb friction

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Model of mono-skier ri positions of centers of gravity rc contact point rC

r1

' 1 1

As

A1 [ f1 , g1 , h1 ]

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Newton–Euler equation of motion for an unconstrained rigid body mr F ~ ~ 'x ' x J '  ' n' ' J ' ' m

mass

r

position of the center of mass in the inertial frame

F

total external or applied force acting on the body

J‘

the inertia tensor with respect to the centroidal body fixed frame ‘

n‘

angular velocity of the body fixed frame relative to the inertial frame in body fixed coordinates torque of the external forces with respect to the origin of the body fixed frame

system of ODEs for r and is not integrable, i.e. it is not the time derivative of any vector Hairer60 - Genf, June 17 - 20, 2009

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Euler parameters u

uT u 1

rotation axis rotation angle

e0

cos

p

[e 0 , e ]

T

p p

cos

2

, e u sin

2

T

2

u u sin

2

2

2

1

Euler parameter normalization constraint Hairer60 - Genf, June 17 - 20, 2009

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Derivatives with help of Euler parameters ~ e e0 I]

G [ e

p

1 T G 2

1 2

p

e1

e0

e3

e2 e3

e3 e2

e0 e1

e2 e1 e0

time derivative of Euler parameters

G

T

"derivative" with respect to virtual rotations Hairer60 - Genf, June 17 - 20, 2009

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Newton-Euler equation of motion first order system for

r, v, p, ω r v 1 T p G 2 mv F ~  J n J Hairer60 - Genf, June 17 - 20, 2009

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Newton-Euler equation of motion for a constrained multibody system T 1

T nb

r [ r , , r ] T 1

T nb

p [ p , , p ] K

( r , p, t )

0

kinematic constraints

( r , p, t )

0

driving constraints

( r , p, t )

0

constraint equation

D

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Equation of motion for a constrained multibody system

Mr J

T r

F n

~J

constraint on

( r , p, t )   rr r

r



0

position level

t

r

velocity level acceleration level

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Balance problem Force components determining inward lean: Fr Fn Fla Fc – Gravitational force component normal to hill Fn – Downhill lateral force Fla Fg sin cos q – Traverse angle

Fg cos

Fc fall line track

Fn

Fla

Φ Fr

Inward lean angle FR n, FR

cos( ) • Hill normal n • Resultant force FR

FN

FLA

FC

– Gravitational component normal to hill FN – Gravitational component on hill lateral to travel direction FLA – Centrifugal force FC Hairer60 - Genf, June 17 - 20, 2009

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Constraints at ankle joint 1 2 3 4 5

2 C

r

3 C

r

2 1

p

3 1

p

4 1

p

1 C

rC

s (r ) 1 C

' 1 1

r1 A s

2 C

h( r , r ) 2 C

p

r1

y

p1

3 C

p

p

4 C

i

full row rank

y Hairer60 - Genf, June 17 - 20, 2009

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Nonholonomic constraints solution as index-3 problem with RADAU5 works constraints at velocity level – not full row rank

pC (v) 4( v ) 5( v ) 3

h

pC ( ( v )) h y

i

v

v v

row rank = 1 Hairer60 - Genf, June 17 - 20, 2009

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Comparison FD - measured forces

Difference between curves may help in estimating load on inner ski

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Discussion Balance problem in forward dynamics • Inward lean angle such that resultant force FR of weight and centrifugal force points onto base of support of skier • Leads to nonholonomic constraints – centrifugal force depends on velocity – index 3 problem works with RADAU5 but needs theoretical investigation Hairer60 - Genf, June 17 - 20, 2009

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Conclusion • After smoothing the data are sufficiently accurate for inverse dynamics. Computation of reaction forces/moments is important for medical investigations like rupture or damage of ligaments or tendons. • It is possible to derive the equation of motion in descriptor form without multibody system software. Formula manipulation programs like Maple are necessary. Hairer60 - Genf, June 17 - 20, 2009

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Thank you for your attention This work was supported by the Austrian Science Foundation (FWF) under the project no. P20870.

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musculo-skeletal model of 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 Hairer60 - Genf, June 17 - 20, 2009

References DLT: Nachbauer et al.: J. Appl. Biomech. 12, 104-115 (1996) Jump with musculo-skeletal model (2D case): Gerritsen et al.: J. Biomechanics 29, 845-854 (1996) Kaps et al.: Spectrum der Sportwissenschaften 12, 6-26 (2000)

Sledge model: Mössner et al.: J. of ASTM International 5 (2008) Paper ID: JAI101387 Hairer60 - Genf, June 17 - 20, 2009

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