Biomechanics

MCE 493/593 & ECE 492/592 Prosthesis Design and Control

September 9. 2014

Antonie J. (Ton) van den Bogert Mechanical Engineering

Cleveland State University

1

Viewpoint

• Scientific background – physics, (comparative) anatomy

– & a little bit of neurophysiology

• Engineering background – mechanical engineering

– & a little bit of control engineering

• What does a prosthetic designer need to know? – design and control of human limbs

2

Components

• Skeleton and joints (mechanical linkage) – kinematics, statics

• Muscles – mechanical properties

– “motor” specifications

– how they are attached to the skeleton

• Neural control – sensors

– circuits

3

Joints

• Ball joints, 3 DOF

– hip, shoulder

• Hinge joints (revolute joints), 1 DOF

– elbow

• Universal joints, 2 DOF

– wrist, ankle

• Knee

– can be approximated as a hinge (but more later)

4

Kinematics

• Human arm has 7 degrees of freedom (excluding fingers) – one more than required for hand positioning

– is the extra DOF useful?

– are prosthetic arms designed the same way?

https://www.youtube.com/watch?v=R0_mLumx-6Y

DEKA Arm

5

Kinematics

• Human leg has approximately 6 degrees of freedom

– foot can have any position and orientation relative to pelvis

– why is this important?

• Are prosthetic legs designed the same way?

http://www.austpar.com/portals/prosthetics/c-leg.php

6

Knee joint

rotation in transverse plane internal rotation – external rotation range of motion about ± 10 degrees mostly controlled by ligaments (passive) neglected in prosthetic limb design

rotation in sagittal plane flexion – extension range of motion: 160 degrees mostly controlled by muscles (active)

7

Knee joint as a 4-bar linkage

human knee 4-bar (polycentric) prosthetic knee

instant center of rotation

8

Polycentric knee vs. stance control

C-Leg ($50,000) microprocessor controlled hydraulic damper

polycentric knee www.d-rev.org $80

Mauch SNS ($5,000) hydraulic damper with mechanical switching of damping coefficient

pro and con of each concept?

http://d-rev.org/media/?video=kamal-jaipurknee

9

Human leg operates close to kinematic singularity of the 2-link leg

• Robots and animals seem to avoid such postures

• Why do people do this? • Implications for prosthesis design

10

Relationship between knee torque and external force

• 1-DOF model

– q: knee flexion angle

– y: downward direction

• Velocity Jacobian

• Dynamics equation

• Static case, neglect leg weight:

q

y Fy

yT FJqgqqqCqqM )(),()(

2sin qLFy

does minus sign make sense?

L

L

11

Standing up from deep squat

• Is not done well by current prostheses (why?)

• Homework assignment (on the course website)

• What motor torque and motor speed are required?

• Find a commercially available motor with this capability

• Are size and weight appropriate for a prosthetic limb?

12

fascicle: bundle of cells cell (muscle fiber): 50-100 mm, 1-10 cm long myofibril: 1-2 mm diameter

structure of skeletal muscle

13

sarcomere is the basic contractile unit

2 mm

14

mechanism of contraction http://www.mrc-lmb.cam.ac.uk/myosin/motility/XBcycle.html

actin filament

myosin filament & myosin head (crossbridge)

ATP ADP

ATP is energy source of muscle contraction - needed for crossbridge to detach - rigor mortis: crossbridges remain attached

15

muscle activation: electricity

Luigi Galvani (1737-1798) action potential travels along fiber

release of Ca ions from sarcoplasmic reticulum

Ca ions bind to Troponin C in actin filament

myosin heads can now attach to actin filament

16

Effect of stimulation frequency on force rat gastrocnemius, supramaximal pulses (100 ms) on nerve cuff

electrode

Roszek & Huijing, 1997

Hz

Hz

Hz

Hz

twitches

fused tetanus

17

Motor units

• a set of muscle fibers that are all innervated by the same motor neuron

• 10 fibers (eye muscle) to 1000+ fibers (gastrocnemius)

• number of motor units decreases with age

18

Fiber types

• Type I or type S (slow) - Slow twitch, fatigue-resistant (smallest)

• Type IIa or type FR (fast, resistant) - Fast twitch, fatigue-resistant (larger)

• Type IIb or type FF (fast, fatigable) - Fast twitch, easily fatigable units (largest force)

19

Motor unit recruitment

Walmsley et al. (1978)

• CNS: smallest MU first

• Electrical (FES): largest first,

therefore it is hard to control

muscle force

http://nmrc.bu.edu/tutorials/motor_units/

20

• Frog semitendinosus,

single fiber, fixed ends

(isometric)

• activate muscle

• measure force

• turn off activation

• change length

• repeat

Mechanical properties dependence of force on length in activated muscle tissue

Gordon, Huxley & Julian, J Physiol, 1966 21

Mechanical properties contribution of passive and active properties to isometric force-length

relationship (whole muscle)

“optimal” fiber length

passive force

total force

22

Force-length relationship becomes a torque-angle relationship that can be measured

Pincivero et al., J Biomech 2004. Maximum extensor torque as a function of knee flexion angle At which knee angle are the fibers shortened?

23

Non-isometric conditions

• Force-velocity property

*quick release experiments

1. Stimulate the muscle fiber to isometric force F0 at L0.

2. Release the catch after the muscle force reaches steady state.

3. Isotonic contraction (constant force)

4. Steady-state

24

shortening velocity (fiber lengths per second)

force

isometric force

-10 -5 0 5 10 0

0.5

1

1.5

isometric eccentric concentric

• Vmax is about 10 fiber lengths per second (mixed fiber type muscle)

• a (the Hill constant) is about 0.25

• maximal eccentric force is 20-100% larger than maximal isometric force

AV Hill, Proc Royal Soc 1938; B Katz, J Physiol (Lond) 1939

FORCE-VELOCITY RELATIONSHIP OF ACTIVE MUSCLE FIBER

aVV

VV

F

F

iso

max

max

Hill’s equation for concentric contraction:

25

Power-velocity curve

-10 -5 0 5 10 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

shortening velocity (fiber lengths per second)

• power (Watts) =

force (Newton) * velocity (m/s)

• peak power output occurs at

about 33% of maximal

shortening velocity • in bicycling, choose a gear

that causes the main

muscles to operate at a

shortening speed of 3-4 fiber

lengths per second

26

The force-length-velocity relationship

27

Muscle architecture

parallel-fibered pennate

with same muscle volume, more force (more fibers in parallel) but shorter fibers and therefore a smaller shortening range

Physiological Cross-Sectional Area (PCSA): muscle volume divided by average fiber length. Muscle strength is determined by PCSA:

kPa 250cm N 25 -2max

PCSAF maxmax

28

Peak power per kg muscle mass

mass = 1; % assume a 1 kg muscle density = 1000; % density, in kg per cubic meter volume = mass/density; fiberlength = 0.08; % typical for human leg muscle PCSA = volume/fiberlength; Fmax = 250e3*PCSA % maximal isometric force Vmax = 10*fiberlength; V = 0:0.01:1; % look at speeds up to 1 m/s a = 0.25; % Hill constant F = Fmax.*(Vmax-V)./(Vmax+V/a); % Hill equation F = max(F,0); % only positive forces are possible P = F.*V; % compute power plot(V,P); xlabel('contraction velocity (m/s) ') ylabel('power output at full activation (W) ');

29

Muscle vs. motor

• Muscle – average power 100 W/kg

• muscle is not always at optimal length

• muscle is not always at optimal velocity

– efficiency 25% (75% heat)

– max contraction speed 0.5 – 1.0 m/s

• Electric motor (example) – 200 W, 300 g (667 W/kg)

– efficiency 89%

– max speed depends on design and gearbox

30

Series Elasticity

• Contractile force is transmitted through an elastic aponeurosis and (often a long) tendon

• How does such a combined muscle-tendon unit function?

• Advantages – allows efficient “pogo-

stick” effects in locomotion, with almost no muscle length change

– other advantages?

31

Three-element Hill muscle model

• Contractile Element (active tissue), Parallel Elastic Element, and Series Elastic Element (passive properties)

• Force response to dynamic length changes

– computer model

– experiment

bone bone

neural stimulation

PEE

CE

SEE

32

mathematical model

1 cm 2.5 cm

motor controls muscle length Lm(t)

max.stim

CE SEE

Force in contractile element:

dt

dLgLfaVgLfFaF CE

CECECECE )()()(max

activation (between zero and one)

isometric force-length relationship (between 0 and 1)

velocity-dependence (between 0 and 1.5)

-10 -5 0 5 10 0

0.5

1

1.5

33

mathematical model

motor controls muscle length Lm(t)

max.stim

CE SEE

Force in series elastic element:

slack2

slack

slack

if)(

if0

LLLLk

LLF

SEESEE

Lslack

LSEE

FSEE

Because CE and SEE are in series • FCE = FSEE • LCE + LSEE = Lm(t)

34

Muscle contraction dynamics

motor controls muscle length Lm(t)

max.stim

CE SEE

)(

))((

: for solve

))(()(

)()(

max

2slack

2slackmax

2slackmax

CE

CEminvCE

CE

CEmCE

CE

SEECE

CE

SEECE

LfFa

LLtLkg

dt

dL

dt

dL

LLtLkdt

dLgLfFa

LLkdt

dLgLfFa

FF

differential equation, solve for state variable LCE(t) and then compute muscle force 2

slack ))()(( LtLtLkF CEmSEE

Matlab program available on course website 35

simulation of ramp stretch experiment

motor controls muscle length Lm(t)

max.stim

CE

SEE

• Muscle has high “short-range stiffness”

• May play a role in control and stability

of movement

• Theoretical models can predict this

behavior

• Hill model

• crossbridge models (chemical kinetics)

Dynamic shortening against inertial load Galantis et al., J Physiol 2000

computational model

muscle velocity

endpoint velocity

http://www.rvc.ac.uk/Research/Video/ccspringmasslinear.dcr

“power amplification”

• Series elasticity allows muscle to work at a shortening velocity

that is different than the endpoint velocity

• Peak power output of muscle-tendon complex can be about 40% higher

than if the muscle acted alone (jumping, throwing)

• Also measured in human experiments

37

Elastic tissue can generate catapult-like effects

• When muscle fibers alone are too slow

38

Kurokawa et al., 2001 human jumping (homework assignment) Wilson et al., Nature 2003

horse galloping BE: elastic energy stored in biceps KE,PE: kinetic/potential energy of limb

Natural oscillations in a horse forelimb simulated with 3-element Hill muscle models

Wilson et al., Nature 2001

39