Optimal Feedback Control for Human Gait with Functional Electrical Stimulation

Ton van den BogertOrchard Kinetics LLC, Cleveland OH

Elizabeth HardinCleveland FES Center

Cleveland VA Medical Center

Functional Electrical Stimulation (FES) for gait

• Open loop stimulation patterns• Stability achieved via:

– upper body support– passive constraints on joint motion

• Long term goal: feedback control

Hardin, et al, J Rehabil Res Dev 44(3), 2007.

Model-based approach

• Musculoskeletal model• Make it walk with open loop control• Add feedback

– Muscle spindles (for comparison)– Joint angles– Joint angular velocities– Forefoot pressure

• Evaluate stability as function of– Feedback type– Feedback gain

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Methods

Generic musculoskeletal model

• 2D, 7 segments, 9 degrees of freedom• 16 Hill-based muscles• 50 state variables x(t)

– 9 generalized coordinates– 9 generalized velocities– 16 muscle active states– 16 muscle contractile states

• 16 muscle stimulations u(t)• Dynamic model:

glutei

iliopsoas

hamstrings

rectus femoris

vasti

gastrocnemius

soleus

tibialis anterior

u)f(x,x

Open loop optimal control

• Make model walk like a human– Track joint angles & ground reaction forces– Minimal effort

• Find x(t),u(t) such that– Objective function is minimized:

and constraints are satisfied• Dynamics:• Periodicity:

– Solved via direct collocation method• (Ackermann & van den Bogert, J Biomech 2010)

u)f(x,x vTT )()( 0xx

N

i

N

i

M

jjieffort

V

j j

jiji uMN

Wms

NVJ

1 1 1

2

1

2

11

tracking effort

Sensors for feedback

• 30 sensor signals s(t)– 2 forefoot pressures– 16 spindle signals d/dt(fiber length)– 6 joint angles– 6 joint angular velocities

• Sensor signals are a function of system state:

s(t) = s(x(t)) sensor model

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Model with feedback• Open loop optimal control solution xO(t), uO(t)• Feedback controller:

– u = uO(t) + G·[ s – s(xO(t)) ]• Gain matrix (16 x 30)

• Magnitude of gains was varied– Signs fixed, positive (●) or negative (●)

G =

feet ang.velanglesspindles

right side muscles

left side muscles

Formal stability analysis

• 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

– Maximum Floquet Exponent: MFE (s-1) (stable: <0)

Dingwell & Kang, J Biomech Eng 2007.

Floquet analysisQuantify the growth/damping of perturbations from one gait cycle to the next

“Anecdotal” stability analysis

• Perturb forward velocity by 2%• Simulate half a gait cycle• By how much has the trunk fallen?

– Vertical Trunk Excursion (VTE)

initial state final state

VTE

Results and Discussion

Open loop optimal control solution

-10

0

10

20

30Hip Angle

[degre

es]

0

20

40

60Knee 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]

0400 Muscle Forces

Ilio

0

400

Glu

0

600

Ham

0150

RF

0

600

Vas

0

1500

Gas

01000

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

Muscle spindle feedback

0 1 2 30

5

10

15

20

Spindle gain (m-1 s)

Max

Flo

quet

Exp

onen

t (s

-1)

0 1 2 30

0.05

0.1

0.15

0.2

Spindle gain (m-1 s)

Ver

tical

Tru

nk E

xcur

sion

(m

)Floquet VTE

gain = 1.96 m-1 sgain = 0

0 0.5 1 1.5 24

6

8

10

12

14

16

angle gain (rad-1)

Max

Flo

quet

Exp

onen

t (s

-1)

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

angle gain (rad-1)

Ver

tical

Tru

nk E

xcur

sion

(m

)

Joint angle feedback

Floquet VTE

gain = 0.7 rad-1gain = 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

Flo

quet

Exp

onen

t (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)

Ver

tical

Tru

nk E

xcur

sion

(m

)

Floquet VTE

gain = 0.22 rad-1 sgain = 0

0 1 2 3

x 10-3

10

15

20

25

30

35

40

GRF gain (N-1)

Max

Flo

quet

Exp

onen

t (s

-1)

0 1 2 3

x 10-3

0

0.05

0.1

0.15

0.2

GRF gain (N-1)

Ver

tical

Tru

nk E

xcur

sion

(m

)

Forefoot pressure feedbackFloquet VTE

gain = 0.00138 N-1gain = 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

. F

loqu

et E

xpon

ent

(s-1

)

Combined feedback

• Lowest MFE: −0.1482 s-1

– Angle gain 1.40 rad-1

– Angular velocity gain 0.12 rad-1 s

angle gain (rad -1) angular velocity gain (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

Limitations of control system

• Sensors– All sensors in one group had same gain– Limited sensor combinations were tested– Missing sensors

• Vestibular, etc.

• Physiological feedback is not always linear– Threshold effects– Reflex modulation– Stumble response

Acknowledgments

• Programming: – Marko Ackermann

• U.S Department of Veterans Affairs– B4668R (Hardin)– B2933R (Triolo)

Forefoot pressure

• Can possibly:– help control timing of push off– help stabilize against forward fall

• Evidence in cats and humans– Pratt, J Neurophysiol 1995; Nurse & Nigg, Clin Biomech 2001

• Theoretically useful in control of posture and hopping– Prochazka et al, J Neurophysiol 1997; Geyer et al., Proc R Soc Lond 2003

• Gait?

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