Biomechanics (part 2) MCE 493/593 & ECE 492/592 Prosthesis Design and Control

September 11, 2014

Antonie J. (Ton) van den Bogert

Mechanical Engineering

Cleveland State University

1

Today

• Coupling between muscles and skeleton

• Neural control

Joint Moments

FJqgqqqCqqM T )(),()(

joint moments forces and moments applied from outside

In the human body, joint moments are generated by 1. muscles 2. ligaments (mostly at end of range of motion) 3. prosthetic / orthotic devices

q, , F are vectors

Terminology: “moment” or “torque”?

Mechanical power output of muscle: Power output of joint torque: So, a joint torque is equivalent to a muscle force F if:

Muscle length Lm is a function of joint angle q Cosine rule:

Joint moment generated by muscle

q

muscle

a

b

qabbaL

qabbaL

m

m

cos2

cos2

22

222

mLFP

qP

Fdq

dL

qqdq

dLF

qLF

m

m

m

dq

dLmis the “moment arm” or “mechanical advantage” of the muscle with respect to coordinate q

In this example: m

m

L

qab

qabba

qab

dq

dL sin

cos2

sin

22

Anatomist’s definition of moment arm

• “distance between muscle and joint”

• engineer and anatomist agree on this one

q

muscle

a

b

d

mL

qabd

sin

In general (many DOFs and muscles)

extT

m

T

m FJFdq

dLqgqqqCqqM

other)(),()(

joint moments from muscles

forces and moments applied from outside

q, , Fm , Lm , Fext are vectors

dq

dLmis a matrix of moment arms matrix element i,j is the moment arm of muscle i with respect to coordinate j

(and you do it the same way with linear actuators in a prosthetic device!)

Muscle path does not have to be a straight line

wrapping points

wrapping surface

If your model allows you to calculate muscle length as a function of q, and take the derivative, you have the moment arms. Opensim software http://opensim.stanford.edu/

Muscle does not have to cross the joint to have a moment arm

Fourbar linkage (for example: closed chain exercise)

q 0dq

dLm

an anatomist would have some trouble with this idea is the “moment arm” in this example positive or negative?

Muscle can cross more than one joint

• And have a counterintuitive moment arm

Andrews JG (1985) A general method for

determining the functional role of a muscle.

ASME J Biomech Eng 107:348-353

1 DOF: knee flexion angle q

Hamstrings length is a function of q: Lm(q)

moment arm is negative

so hamstrings are a knee extensor! (when you have

these constraints on foot and pelvis)

dq

dLm

Moment arms can be measured

• Distance to joint (not the best idea)

• Move the joints, measure muscle length change Lm(q), and take partial derivatives – “tendon travel method”

joint angle [radians!!]

muscle length

L

multiple joints: partial derivatives

Two-joint muscles and energy transfer

• Gastrocnemius crosses knee and ankle

• Coordinates:

– knee flexion angle (q4), ankle plantarflexion angle (q5)

• During push-off phase of gait

– knee is extending

– ankle is plantarflexing

• Gastrocnemius produces

– knee flexion moment

– ankle plantarflexion moment

0

0

5

4

q

q

0

0

5

4

)(generates 0

(absorbs) 0

5

4

P

P

Power:

Analysis of normal gait

Farris & Sawicki, J Royal Soc Interface 2012

This analysis does not consider muscles (joint torques only) It shows that half of the required ankle power could have been transferred from the knee by the Gastrocnemius muscle. Quadriceps moves the ankle! May improve control! This transfer could be done by a fully passive elastic structure (but then you can’t turn it off)

I used to think that…

• Robotic systems needed elastic structures and actuators crossing more than one joint

– to match human control and efficiency

• But:

– elasticity and energy transfer can be achieved in the electric domain

– and controlled also

Closed loop system

spinal cord

muscles

muscle stimulation “efferent nerves”

brain

sensory signals “afferent nerves”

movement

skeleton

muscle forces

tissue mechanics

sensory organs

Neural activation

Action potential: a polarity reversal that travels along the axon Each neuron activates one motor unit with varying firing rate Action potentials travel along muscle fibers EMG (electromyography) is the resultant of all these action potentials as seen at the electrode

neuron

axon neuromuscular junction

Time delays

• Multiple conversions between chemical and electrical signals – especially in spinal circuit with multiple neurons

– “polysynaptic reflexes”

• Action potentials travel at 80-120 m/s – From spinal cord to leg: 10 ms

• Muscle activation is a chemical process – 20 ms to peak force, 40 ms to decay (depends on fiber type)

muscle twitches at 15 Hz

Spinal circuits

• Central Pattern Generator (CPG) – neural oscillator, sends rhythmic stimulation to muscles – “feedforward control”

• Example: furnace with timer • can be modulated by brain input

– exists in fish, insects, cats, infants – existence in adult humans is controversial

• Sensory inputs – needed for “feedback control”

• Example: furnace with thermostat -> “equilibrium point hypothesis” • Neural transmission delay in lower extremity: 50-100 ms

– can “entrain” the CPG – can assist learning of feedforward strategies – reflex pathways have been well studied

Sensory organs

• Vision, hearing:

– not essential for low-level control of walking and standing

• Essential sensors:

– cutaneous (skin pressure) sensors

– muscle receptors (force, stretch)

– balance sensors

• Sensors send action potentials to the spinal cord

– afferent neurons

Muscle sensors

Muscle spindles: respond to stretch & stretch velocity

Golgi tendon organ: responds to force

Stretch reflex

Skin pressure receptors

Cutaneous (skin) reflexes

• Negative feedback: withdrawal reflex – muscle action removes

stimulus

– helps prevent stumbling

• Positive feedback – muscle action increases

stimulus

– cats

– stance control

• Sign of reflex gain can depend on phase of movement! Zehr & Stein, 1999

Balance

linear and angular “accelerometers”

Vestibulo-ocular reflex

Reflex-based control

Geyer & Herr, IEEE Trans Neural Syst Rehabil Eng, 2010

Herr invented the Rheo knee and the BIOM foot BIOM uses reflexes in some way

No CPG, no clock! System is autonomous

xuxfx

tuxfx

uxfx

,

,

,

open loop

reflex based

Simulation results

You probably need realistic muscle mechanical properties to make this work The controller is a bit too “hand-crafted” for my taste optimization?

My work in 2009 • Open loop optimal control solution xO(t), uO(t) • Feedback controller:

– u = uO(t) + G·[ s – s(xO(t)) ]

• Gain matrix (16 x 30)

• Gains – Signs fixed, positive (●) or negative (●) – Same gain magnitude within each sensor type

• Model will follow trajectory xO(t) until perturbed

G =

feet ang.vel angles muscle spindles

right side muscles

left side muscles

Formal stability analysis of limit cycle

• Linearization: (xk+1 – x*) = A·(xk – x*) • Matrix A calculated from model • Eigenvalues of A: Floquet multipliers λ (50) • Floquet exponents: μ = log(λ)/T units: s-1 • Movement is stable when

– Maximum Floquet Exponent (MFE) < 0 δx(t)~eμt

Dingwell & Kang, J Biomech Eng 2007.

Floquet analysis

Quantify the growth/damping

of perturbations from one gait

cycle to the next

“Anecdotal” stability analysis

• Perturb forward velocity by 2%

– Equivalent to impulsive force

• Simulate half a gait cycle

• By how much has the trunk fallen?

– Vertical Trunk Excursion (VTE)

initial state final state

VTE

Open loop optimal control solution

-10

0

10

20

30Hip Angle

[degre

es]

0

20

40

60

Knee Angle

70

80

90

100Ankle Angle

File name: ./result100half.mat

Number of nodes: 100

Initial guess: ../007result.mat

Model used: ../../Legs2dMEX/CCFmodel

Gait data tracked: ../wintergaitdata.mat

Weffort: 1

Norm of constraints: 0.00092369

Cost function value: 0.029958

0

0.2

0.4

0.6

0.8

1

1.2 GRF Y

[BW

]

0 50 100

-0.2

-0.1

0

0.1

0.2GRF X

[BW

]

Time [% of gait cycle]

0

400 Muscle Forces

Ilio

0

400

Glu

0

600

Ham

0

150

RF

0

600

Vas

0

1500

Gas

0

1000

Sol

0 50 1000

800

TA

0

1

Ilio Muscle Activations

0

1

Glu

0

1

Ham

0

1R

F

0

1

Vas

0

1

Gas

0

1

Sol

0 50 1000

1

TA

Can be done with

• subject-specific model

• subject-specific gait data

Stability analysis of open loop controlled model

7921.8152

9.3582

2.7494

1.7277

1.2257

1.0000

0.0449

0.0449

0.0140

0.0140

0.0049

0.0011

0.0001

0.0001

0.0000

14.0537

3.5008

1.5833

0.8560

0.3186

0.0000

-4.8597

-4.8597

-6.6858

-6.6858

-8.3317

-10.7149

-15.4383

-15.4383

-18.3008

Floquet multipliers

λ

Floquet exponents (s-1)

μ = log(λ)/T

Floquet multiplier 1.0000

• Eigenvector = 1

0

0

0

0

0

0

.

.

.

.

.

.

0

forward translation

Floquet multiplier 7921.8 (Floquet exponent 14.1 s-1)

• Eigenvector =

0.0004

-0.0001

0.0006

-0.0013

0.0004

-0.0003

-0.0219

0.0411

-0.0084

generalized

coordinates

0.0075

-0.0015

0.0098

-0.0226

-0.0047

0.0247

-0.4608

0.8657

-0.1841

generalized

velocities

0.0099

-0.0062

-0.0002

-0.0142

-0.0175

0.0027

-0.0060

0.0032

0.0006

-0.0004

-0.0006

0.0002

-0.0001

-0.0000

-0.0001

0.0001

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

muscle CE

lengths

muscle active

states

stance leg

joint angles

Floquet multiplier 9.36 (Floquet exponent 3.50 s-1)

• Eigenvector =

generalized

coordinates

generalized

velocities

muscle CE

lengths

muscle active

states

-0.0019

-0.0009

0.0389

-0.0374

-0.0045

-0.0003

-0.0643

0.0303

-0.0090

-0.0045

-0.0126

0.1487

-0.1505

-0.1604

0.3743

-0.4743

0.6346

-0.3877

0.0306

-0.0193

-0.0181

0.0079

-0.0129

0.0012

-0.0062

0.0033

0.0177

-0.0111

-0.0258

0.0121

0.0016

-0.0030

0.0000

0.0000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Response of open loop controlled model to forward pull

final state

VTE = 16.6 cm

with feedback control added?

Muscle spindle feedback

0 1 2 30

5

10

15

20

Spindle gain (m-1 s)

Max F

loquet

Exponent

(s-1

)

0 1 2 30

0.05

0.1

0.15

0.2

Spindle gain (m-1 s)

Vert

ical T

runk E

xcurs

ion (

m)Floquet VTE

gain = 1.96 m-1 s gain = 0

0 0.5 1 1.5 24

6

8

10

12

14

16

angle gain (rad-1)

Max F

loquet

Exponent

(s-1

)

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

angle gain (rad-1)

Vert

ical T

runk E

xcurs

ion (

m)

Joint angle feedback

Floquet VTE

gain = 0.7 rad-1 gain = 0

Joint angular velocity feedback

0 0.1 0.2 0.3 0.4 0.50

5

10

15

angular velocity gain (rad-1 s)

Max F

loquet

Exponent

(s-1

)

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

angular velocity gain (rad-1 s)

Vert

ical T

runk E

xcurs

ion (

m)

Floquet VTE

gain = 0.22 rad-1 s gain = 0

0 1 2 3

x 10-3

10

15

20

25

30

35

40

GRF gain (N-1)

Max F

loquet

Exponent

(s-1

)

0 1 2 3

x 10-3

0

0.05

0.1

0.15

0.2

GRF gain (N-1)

Vert

ical T

runk E

xcurs

ion (

m)

Forefoot pressure feedback Floquet VTE

gain = 0.00138 N-1 gain = 0

Effect of simple feedback

• Feedback from each type of sensor could improve stability

• Agreement between Floquet analysis and finite perturbation response

• An optimal feedback gain always existed

• Stability (MFE<0) was not yet achieved

– Feedback from combination of sensor types?

00.1

0.20.3

0.4

0

1

2-5

0

5

10

angular velocity gain (rad-1 s)angle gain (rad-1)

Max.

Flo

quet

Exponent

(s-1

)

Combined feedback

• Lowest MFE: −0.1482 s-1

– Angle gain 1.40 rad-1

– Angular velocity gain 0.12 rad-1 s

MFE (s-1)

Continuous walking with optimal combined feedback?

• Why not stable, as predicted by MFE? • Limitations of Floquet analysis

– Accuracy – Linearization around optimal trajectory (small foot clearance!)

Limitations of control system

• Sensors

– All sensors in one group had same gain

– Only some sensor combinations were tested

– Missing sensors • Vestibular, etc.

• Physiological feedback is not always linear

– Threshold effects

– Reflex modulation

– Stumble response

Human stability tests Able bodied subject

Impulsive force 10% BW for 20 ms

(Δv ≈ 0.02 m/s)

Human stability test Gait analysis data (Trial 25, 0.85 m/s)

3 4 5 6 7 8 9 10-1000

0

1000

2000

3 4 5 6 7 8 9 100

0.2

0.4

3 4 5 6 7 8 9 100

0.2

0.4

3 4 5 6 7 8 9 100

0.05

0.1

3 4 5 6 7 8 9 100

0.05

0.1

3 4 5 6 7 8 9 10-100

0

100

200

R GRF

L GRF

R GAS

L GAS

R VL

L VL

R AnkleMoment

L AnkleMoment

GRF

EMG R.Gastroc

EMG L.Gastroc

EMG R. Vastus Lateralis

EMG L. Vastus Lateralis

Ankle moments