Presentation Outline 1. Introduction 1. Introduction

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May 23, 2010 - Dynamics of Mechanical Systems. Presentation Outline. 1. Introduction. 2. Foot anatomy and Biomechanics. 3. Mathematical Foot Model. 4.
23-05-2010

Presentation Outline Dynamics of Mechanical Systems

Dynamics of Mechanical Systems

A BIOMECHANICAL MULTIBODY FOOT MODEL FOR FORWARD DYNAMIC ANALYSIS

Dynamics of Mechanical Systems

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

1. Introduction Dynamics of Mechanical Systems

• Biomechanical Foot Models

• 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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1. Introduction Dynamics of Mechanical Systems

1. Introduction Dynamics of Mechanical Systems

• Biomechanical Foot Models

• Biomechanical Foot Models

• 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).

1. Introduction Dynamics of Mechanical Systems

foot

Dynamics of Mechanical Systems

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: ƒ 1. Definition of the geometric properties; ƒ 2. Development of a methodology for contact detection; ƒ 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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2. Foot Anatomy and Biomechanics

Dynamics of Mechanical Systems

• 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.

2. Foot Anatomy and Biomechanics

Dynamics of Mechanical Systems

Dynamics of Mechanical Systems

2. Foot Anatomy and Biomechanics

• Gait Cycle:

Dynamics of Mechanical Systems

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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3. Mathematical Foot Model

Dynamics of Mechanical Systems

3. Mathematical Foot Model

Dynamics of Mechanical Systems

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

Dynamics of Mechanical Systems

(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

Dynamics of Mechanical Systems

(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

Dynamics of Mechanical Systems

5. Foot Model Application to Human Gait

Dynamics of Mechanical Systems

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]

5. Foot Model Application to Human Gait

Coulomb Friction

-4 -4.5

0

Dynamics of Mechanical Systems

Viscous Friction

-3.5

0

0.1

0.2

0.3

0.4

0.5

Time [s]

5. Foot Model Application to Human Gait

Dynamics of Mechanical Systems

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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5. Foot Model Application to Human Gait

Dynamics of Mechanical Systems

5. Foot Model Application to Human Gait

Dynamics of Mechanical Systems

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

Dynamics of Mechanical Systems

5. Foot Model Application to Human Gait

Dynamics of Mechanical Systems

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

Experimental Results

-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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5. Foot Model Application to Human Gait

Dynamics of Mechanical Systems

5. Foot Model Application to Human Gait

Dynamics of Mechanical Systems

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 (%)

Dynamics of Mechanical Systems

Sphere 1

Numerical Results

-100

Experimental Results

0

Experimental Results

-150 0

20

40 60 Gait Cycle %

80

100

5. Foot Model Application to Human Gait

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

Vertical Ground Reaction Force (N)

Anterior/Posterior Ground Reaction Force (N)

Relative Penetration (mm)

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

5. Foot Model Application to Human Gait

Dynamics of Mechanical Systems

Vertical Ground Reaction Force (N)

Anterior/Posterior Ground Reaction Force (N)

Relative Penetration (mm)

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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5. Foot Model Application to Human Gait

Dynamics of Mechanical Systems

5. Foot Model Application to Human Gait

Dynamics of Mechanical Systems

• 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

Vertical Ground Reaction Force (N)

Anterior/Posterior Ground Reaction Force (N)

Relative Penetration (mm)

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 Penetration (mm)

5. Foot Model Application to Human Gait

• 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

Dynamics of Mechanical Systems

0

20

40

60

Gait Cycle %

80

100

Verical Ground Reaction Force (N)

Dynamics of Mechanical Systems

800 700 600 500 400 300 200 Numericall Results

100

Experimental Results

0 0

Cavus Foot

Flat Foot

20

40

60

80

100

Gait Cycle %

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Dynamics of Mechanical Systems

6. Conclusions and Future Perspectives

• 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.

Dynamics of Mechanical Systems

6. Conclusions and Future Perspectives

• 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

Dynamics of Mechanical Systems

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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