Presentation Outline 1. Introduction 1. Introduction

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Presentation Outline Dynamics of Mechanical Systems

Dynamics of Mechanical Systems

A BIOMECHANICAL MULTIBODY FOOT MODEL FOR FORWARD DYNAMIC ANALYSIS


1. Introduction 2. Foot anatomy and Biomechanics

Pedro Moreira1 | Miguel Silva2 | Paulo Flores1

3. Mathematical Foot Model 4. Constitutive Equations for Contact Forces Universidade do Minho Escola de Engenharia

1University of Minho Mechanical Engineering Department Campus de Azurém 4800-058 Guimarães Portugal

2IDMEC/IST Technical University of Lisbon Av. Rovisco Pais 1 1049-001 Lisboa Portugal

5. Foot Model Application to Human Gait 6. Conclusions and Future Perspectives

1. Introduction Dynamics of Mechanical Systems


• Biomechanical Foot Models


• The first models consider the foot as single rigid body with the ankle joint directly connected to the ground, disregarding the existence of the foot.

• Gilchrist and Winter (1996) developed a three-dimensional foot model that describes the foot-ground contact during the stance phase of gait using viscoelastic elements.

• Meglan (1992) introduced the first biomechanical foot model that does not include any kinematic conditions to the movement of the feet.

• Two segment foot model with one revolute joint. • 9 vertically oriented spring – damper components

• Meglan, D.A.: A 3D passive mechanical model of human foot for use in locomotion synthesis. Proc. NACOB II: The Second North American Congress on Biomechanics, Chicago (1992).

• Gilchrist, L.A. and Winter, D.A.: A two-part, viscoelastic foot model for use in gait simulations, Journal of Biomechanics, 29(6), 795-798 (1996).

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• Barbosa et al. (2005) presented a new idea to study the foot-ground interaction by introducing ellipsoids to represent the contact between the foot and the ground.

• Millard et al. (2007) used a multibody mechanical model with a contact model.

• The normal ground reaction force is calculated using the relative pseudopenetration.

• Barbosa, I.C.J., 2008, Identification of a foot model for human gait analysis using optimization methodologies. PhD Dissertation, Universidade Técnica de Lisboa, Instituto Superior Técnico, Lisboa, Portugal.

• Two-sphere single segmented rigid foot model, Three-sphere two segment foot contact model and Four-sphere two segment foot contact model.

• Millard, M. McPhee, J. and Kubica, E.: Multi-Step Forward Dynamic Gait Simulation. In Proceedings of Multibody Dynamics 2007, ECCOMAS Thematic Conference, (Carlo L. Bottasso eds.), Milan, Italy, June 25-28 (2007).


foot


2. Foot Anatomy and Biomechanics

• Scope and Objectives:

• Human Foot:

• To develop a three-dimensional computational foot contact model.

• Sereig and Arvikar (1989) defined the foot as a multi-segmented and highly ligamentous structure, articulated at numerous joints and assisted by a meticulous arrangement of extrinsic and intrinsic muscles that provide a diverse range of motions and functions.

• To characterize the interaction between the foot and floor. floor Any contact problem in multibody systems can be divided into three main steps: 1. Definition of the geometric properties; 2. Development of a methodology for contact detection; 3. Application of appropriate constitutive laws for contact; • To apply the formulation developed to the analysis and simulation of human gait analysis.

• Sereig, A. and Arvikar, R.: Biomechanical analysis of the musculoskeletal structure for medicine and sports, Hemisphere Publishing Corporation (1989). • Tortora, G. and Grabowsky S., 2004, Introduction to Human Body: essentials ofanatomy and physiology, 6th edition John Willey & Sons, New York.

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• Foot Biomechanics:

• Foot Biomechanics:

Inversion / Eversion

Supination / Pronation

• Abboud, R.J., 2002, Relevant foot biomechanics, Current Orthopaedics, 16(3), 165-179.

Inversion / Eversion

Supination / Pronation

• Abboud, R.J., 2002, Relevant foot biomechanics, Current Orthopaedics, 16(3), 165-179.





• Gait Cycle:


3. Mathematical Foot Model

• Methods and Methodologies:

A typical normal gait cycle illustrating the events of gait

• Foot Pathologies:

Normal Foot

Flat Foot

Cavus Foot

• Winter, D.A.: Biomechanics and Motor Control of Human Movement, J. Wiley & Sons, Toronto, Canada, 3rd edition (1990). • Altay Scientific Foot Models, 2009.

Example of a mechanical system described with fully Cartesian coordinates: rigid bodies are defined using the Cartesian coordinates of a set of points and vectors that are located at the joints and extremities of elements. Assembled mechanical system with a revolute joint created sharing point i and vector u.

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Shank

Revolute joint Main foot part

Toes

Revolute joint

• Three rigid bodies that represent the shank, the main foot part and the toes. • The ground is the fourth rigid body, which is rigid and flat.

• The main foot part and the toes are represented by spheres. • The number of spheres, their radius and their locations can be adjusted by the user.

• The foot parts are constrained by kinematic revolute joints.

4. Constitutive Equations for Contact Forces


(i)

Z

G dn

G d

Y X

G r jP

G dt

G n

Pj

•

• Mathematical methodology to deal with the contact detection between the ground and spheres that represent the plantar foot surface and toes.

Ci Ri

G ri C

4. Constitutive Equations for Contact Forces


(j)

caused by a linear damping function similar to the one in the Kelvin-Voigt model:

• d = r j − ri (distance vector d between points Ci and Pj ) P

We assume that this energy loss is

•

A f n

C

A hysteresis form for the damping coefficient was proposed by Hunt and

c

K Zoom

• δ = R i − d n (geometric condition used to check if the sphere and plane are in contact)

• δ = v n (pseudo-velocity of penetration)

Crossley:

μc μd

f = ( K δ m + cδ )n n

ft

•

c = χδ m

The constitutive law for the tangential force is expressed as:

ft = −( μc f n + μd vt )t

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5. Foot Model Application to Human Gait




Contact model with viscous component and maximum Coulomb friction

0.5

0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4

Sphere 1 X Velocity [m/s]

Contact model without viscous component and maximum Coulomb friction

Sphere 1 X Displacement [m]

Point A

Viscous Friction Coulomb Friction

0 -0.5 -1 -1.5 -2 -2.5 -3

0.1

0.2

0.3

0.4

0.5

Time [s]


Coulomb Friction

-4 -4.5

0


Viscous Friction

-3.5

0

0.1

0.2

0.3

0.4

0.5

Time [s]



Knee trajectory

ζ1

• 5 vectors, 22 points and 81 natural coordinates

η1

• Male 1,70 m and 70 kg

ξ1

Revolute joint

Shank

ζ2

η2 ξ2

Z

Y Revolute joint X

• 6 functional degrees of freedom

Main foot part Torsional spring-damper

ζ3

η3 ξ3

Toes

• Forward Dynamic Analysis with full prescribed kinematics ( knee trajectory, knee rotation and ankle rotation)

Knee Z Trajectory (m)

0.48

• 4 rigid bodies (shank, main foot part, toes and floor)

Thigh

Raw Signal

0.47

First Filtered Signal

0.46 0.45 0.44 0.43 0.42 0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

• Low pass Butterworth 2nd order filter • Cut-off frequency 2.5Hz • Laboratory of Biomechanics of Motion of Technical University of Lisbon • Silva, M.: Human motion analysis using multibody dynamics and optimization tools, PhD Dissertation, Instituto Superior Técnico, Lisboa (2003).

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Thigh

Sphere number

ξ [mm]

η [mm]

ζ [mm]

Radius [mm]

1 2 3 4 5 6 7 8 9

-94.70 -94.70 -75.20 0.00 26.97 56.09 -22.70 0.00 21.602

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-15.34 -15.89 -22.89 -25.89 -32.36 -34.52 -15.10 -17.26 -12.94

30.81 39.53 31.10 15.10 12.94 15.10 12.94 10.78 10.79

Knee trajectory

ζ1

η1 ξ1 Shank

Revolute joint

Main foot part

ζ2

η2 ξ2

Z

Torsional spring-damper

ζ3

η3

ξ3

Toes

Y Revolute joint

Segment

Description

Mass [kg]

1 2 3

Shank Main foot part Toes

4.76 1.33 0.35

Moment of inertia [kgm2] Iηη Iζζ Iξξ 8.230 8.230 8.230 2.250 2.250 2.250 0.471 0.471 0.471

Animation Sequence of the Human Gait cycle of the Biomechanical Model

X

5. Foot Model Application to Human Gait Analysis




l

Fn = ∑ f nk 150 Anterior/Posterior Groun nd Reaction Force (N)

700

(i)

600 500

Fti

400 300

Ftj

Z 50

Y M yi

Numerical Results

100

0

20

40

60

80

100

Fni + Fnj

Ft = ∑ f tk

(j)

M yj

YCOP = −

k =1

M xi + M xj Fni + Fnj

X

0

Numerical Results

-100

Experimental Results

0

Fnj

l

Fni

-50

200

M yi + M yj

M xj

M xi

100

X-coordinate of the center of pressure (m)

Vertical Ground Raection n Force (N)

800

X COP = −

k =1


-150 0

20

40 60 Gait Cycle %

80

100

Gait Cycle (%)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

• http://www.liv.ac.uk/premog/images/Hpp.jpg

20

40

60

80

100

Gait Cycle %

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Anterior/Posterior Groun nd Reaction Force (N)

Vertical Ground Raection n Force (N)

150 800

100

700 600 500 400 300

50 0

-50

200

Numerical Results

100 0

20

40

60

80

100

Gait Cycle (%)


Sphere 1

Numerical Results

-100


0


-150 0

20

40 60 Gait Cycle %

80

100


Vertical Ground Reaction Force (N)

Anterior/Posterior Ground Reaction Force (N)

Relative Penetration (mm)

192.4

-12.24

28.49

Sphere 2

281.2

-12.22

36.69

Sphere 3

203.0

-12.18

29.52

• Spheres 1-3 in contact with the ground • 27% of stance phase • End of heel contact




98.42

-12.06

18.21

181.8

-12.04

27.41

59.56

-12.23

13.02

Sphere 1 Sphere 2 Sphere 3

• Spheres 1-3 in contact with the ground • 10% of stance phase. • Initial heel contact






Sphere 1

115.6

-1.527

20.31

Sphere 2

203.1

-1.493

29.58

Sphere 3

158.8

-1.187

25.09

Sphere 4

0.4764

-1.51

0.5216

Sphere 5

0.6367

-1.289

0.6326

Sphere 6

0.3986

-1.351

0.4628

Sphere 7

7.837

-1.327

3.371

Sphere 8

3.006

-1.280

1.779

• Spheres 1-8 in contact with the ground • 49% of stance phase • Mid Stance (Plantar support)

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• Dissipative curve of Sphere 6 • 11 mm maximum penetration • Hunt and Crossley diagram shape 0.03

Damping Force (N)

0.025 0.02 0.015 0.01 0.005

Sphere 1

0

Sphere 2 0

2

4

6

8

10

12

Sphere 3




30.00

-3.400

8.2690

21.89

-3.356

6.7010

1.103

-3.744

0.9137

• Spheres 7.9 in contact with the ground • 68% of stance phase • Toe off



• Relative Stiffness, Coordinates.

800 Vertical Gro ound Raection Force (N)

Vertical Gro ound Raection Force (N)

900 800 700 600 500 400 300 Normal Foot

200 100

Cavus Foot

0

700

20

40

60

Gait Cycle %

80

100

Damping

Coefficient,

Radius

and

Local

600 500

• Methodology was based on the trial and error procedure.

400

900

300 200

Normal Foot

100

Flat Foot

0 0

6. Conclusions and Future Perspectives


0

20

40

60

Gait Cycle %

80

100

Verical Ground Reaction Force (N)


800 700 600 500 400 300 200 Numericall Results

100


0 0

Cavus Foot

Flat Foot

20

40

60

80

100

Gait Cycle %

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• Reliable Reproduction of Contact between Foot and Floor. • Viscous component of the friction force proved to be important as it is responsible for preventing the foot from sliding.



• Better identification of the material properties of the contact surfaces is required. • Better parameter identification optimization procedures.

should

be

carried

out

using

• Performs well either in forward and inverse dynamics analysis. • Implementation of medial/lateral ground reaction force. • Allows an automatic synchronization between kinematics and kinetics.

• Incorporation of the model in already existing human biomechanical models, allowing its use for foot and gait pathologies analysis


THANK YOU Dynamics of Mechanical Systems

Acknowledgements ● The first author would like to thank the Portuguese Foundation for Science and Technology (FCT) for the support given through: ● PhD Grant SFRH/BD/64477/2009.

[email protected] ℡ +351 253 510 220 +351 253 516 007

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