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

Transcription

1 Biomechanics MCE 493/593 & ECE 492/592 Prosthesis Design and Control September Antonie J. (Ton) van den Bogert Mechanical Engineering Cleveland State University 1

2 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

3 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

4 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

5 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? DEKA Arm 5

6 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? 6

7 Knee joint rotation in sagittal plane flexion extension range of motion: 160 degrees mostly controlled by muscles (active) rotation in transverse plane internal rotation external rotation range of motion about ± 10 degrees mostly controlled by ligaments (passive) neglected in prosthetic limb design 7

8 Knee joint as a 4-bar linkage instant center of rotation human knee 4-bar (polycentric) prosthetic knee 8

9 Polycentric knee vs. stance control polycentric knee $80 C-Leg ($50,000) microprocessor controlled hydraulic damper pro and con of each concept? Mauch SNS ($5,000) hydraulic damper with mechanical switching of damping coefficient 9

10 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

11 Relationship between knee torque and 1-DOF model q: knee flexion angle y: downward direction Velocity Jacobian Dynamics equation M( q) q C( q, q ) q external force g( q) Static case, neglect leg weight: F y L sin q 2 does minus sign make sense? J T F y F y L L q 11 y

12 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

13 structure of skeletal muscle fascicle: bundle of cells cell (muscle fiber): mm, 1-10 cm long myofibril: 1-2 mm diameter 13

14 sarcomere is the basic contractile unit 2 mm 14

15 mechanism of contraction actin filament ADP ATP myosin filament & myosin head (crossbridge) ATP is energy source of muscle contraction - needed for crossbridge to detach - rigor mortis: crossbridges remain attached 15

16 muscle activation: electricity Luigi Galvani ( ) 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

17 Effect of stimulation frequency on force rat gastrocnemius, supramaximal pulses (100 ms) on nerve cuff electrode Hz Hz fused tetanus Hz twitches Hz Roszek & Huijing,

18 Motor units a set of muscle fibers that are all innervated by the same motor neuron 10 fibers (eye muscle) to fibers (gastrocnemius) number of motor units decreases with age 18

19 Fiber types Type I or type S (slow) - Slow twitch, fatigueresistant (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

20 Motor unit recruitment CNS: smallest MU first Electrical (FES): largest first, therefore it is hard to control muscle force Walmsley et al. (1978) 20

21 Mechanical properties dependence of force on length in activated muscle tissue Frog semitendinosus, single fiber, fixed ends (isometric) activate muscle measure force turn off activation change length repeat Gordon, Huxley & Julian, J Physiol,

22 Mechanical properties contribution of passive and active properties to isometric force-length relationship (whole muscle) total force passive force optimal fiber length 22

23 Force-length relationship becomes a torque-angle relationship that can be measured Pincivero et al., J Biomech Maximum extensor torque as a function of knee flexion angle At which knee angle are the fibers shortened? 23

24 Non-isometric conditions Force-velocity property *quick release experiments 1. Stimulate the muscle fiber to isometric force F 0 at L Release the catch after the muscle force reaches steady state. 3. Isotonic contraction (constant force) 4. Steady-state 24

25 FORCE-VELOCITY RELATIONSHIP OF ACTIVE MUSCLE FIBER 1.5 eccentric isometric concentric force isometric force Hill s equation for concentric contraction: F Fiso V V max max V V a shortening velocity (fiber lengths per second) Vmax is about 10 fiber lengths per second (mixed fiber type muscle) a (the Hill constant) is about 0.25 maximal eccentric force is % larger than maximal isometric force AV Hill, Proc Royal Soc 1938; B Katz, J Physiol (Lond)

26 Power-velocity curve 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

27 The force-length-velocity relationship 27

28 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: max F PCSA max max N cm 250 kpa 28

29 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

30 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 m/s Electric motor (example) 200 W, 300 g (667 W/kg) efficiency 89% max speed depends on design and gearbox 30

31 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 pogostick effects in locomotion, with almost no muscle length change other advantages? 31

32 Three-element Hill muscle model neural stimulation CE bone bone SEE PEE Contractile Element (active tissue), Parallel Elastic Element, and Series Elastic Element (passive properties) Force response to dynamic length changes computer model experiment 32

33 mathematical model max.stim CE SEE motor controls muscle length L m (t) 1 cm 2.5 cm Force in contractile element: F CE a F max f ( L CE ) g( V CE ) a f ( L CE ) g dl dt CE isometric activation force-length relationship (between 0 and 1) (between zero and one) velocity-dependence (between 0 and 1.5)

34 mathematical model max.stim CE SEE motor controls muscle length L m (t) Force in series elastic element: F SEE F SEE k( L SEE 0 L slack ) 2 if if L L L L slack slack Because CE and SEE are in series F CE = F SEE L CE + L SEE = L m (t) L slack L SEE 34

35 F CE a F a F Muscle contraction dynamics F max max solve for dl dt CE SEE f ( L f ( L dl dt CE g CE CE : inv dl ) g dt CE dl ) g dt CE max.stim CE k( L k( L k( L m( t) LCE L a Fmax f ( L CE SEE m slack ) SEE L ( t) L ) 2 slack CE ) 2 L slack ) 2 motor controls muscle length L m (t) differential equation, solve for state variable L CE (t) and then compute muscle force 2 FSEE k( Lm ( t) LCE ( t) Lslack ) Matlab program available on course website 35

36 simulation of ramp stretch experiment max.stim CE SEE motor controls muscle length L m (t) 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)

37 Dynamic shortening against inertial load Galantis et al., J Physiol 2000 computational model muscle velocity endpoint velocity 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

38 Elastic tissue can generate catapult-like effects When muscle fibers alone are too slow 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 38

39 Natural oscillations in a horse forelimb simulated with 3-element Hill muscle models Wilson et al., Nature