Presented at World Congress of Biomechanics 2010. Abstract:

1. Optimal Feedback Control for Human Gait
with Functional Electrical Stimulation
Ton van den Bogert
Orchard Kinetics LLC, Cleveland OH
Elizabeth Hardin
Cleveland FES Center
Cleveland VA Medical Center

2. 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.

3. 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
+

4. Methods

5. 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 

6. 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
j
jieffort
V
j j
jiji
u
MN
W
ms
NV
J
1 1 1
2
1
2
11

tracking effort

7. 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
+

8. 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

9. 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 analysis
Quantify the growth/damping
of perturbations from one gait
cycle to the next

10. “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

11. Results and Discussion

12. Open loop optimal control solution
-10
0
10
20
30
Hip Angle
[degrees]
0
20
40
60
Knee Angle
70
80
90
100
Ankle 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
0.2
GRF 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 50 100
0
800
TA
0
1
Ilio
Muscle Activations
0
1
Glu
0
1
Ham
0
1
RF
0
1
Vas
0
1
Gas
0
1
Sol
0 50 100
0
1
TA

13. Muscle spindle feedback
0 1 2 3
0
5
10
15
20
Spindle gain (m-1
s)
MaxFloquetExponent(s-1)
0 1 2 3
0
0.05
0.1
0.15
0.2
Spindle gain (m-1
s)
VerticalTrunkExcursion(m)
Floquet VTE
gain = 1.96 m-1 sgain = 0

14. 0 0.5 1 1.5 2
4
6
8
10
12
14
16
angle gain (rad-1
)
MaxFloquetExponent(s-1)
0 0.5 1 1.5 2
0
0.05
0.1
0.15
0.2
angle gain (rad-1
)
VerticalTrunkExcursion(m)
Joint angle feedback
Floquet VTE
gain = 0.7 rad-1gain = 0

15. Joint angular velocity feedback
0 0.1 0.2 0.3 0.4 0.5
0
5
10
15
angular velocity gain (rad-1
s)
MaxFloquetExponent(s-1)
0 0.1 0.2 0.3 0.4 0.5
0
0.05
0.1
0.15
0.2
angular velocity gain (rad-1
s)
VerticalTrunkExcursion(m)
Floquet VTE
gain = 0.22 rad-1 sgain = 0

16. 0 1 2 3
x 10
-3
10
15
20
25
30
35
40
GRF gain (N-1
)
MaxFloquetExponent(s-1)
0 1 2 3
x 10
-3
0
0.05
0.1
0.15
0.2
GRF gain (N-1
)
VerticalTrunkExcursion(m)
Forefoot pressure feedback
Floquet VTE
gain = 0.00138 N-1gain = 0

17. 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?

18. 0
0.1
0.2
0.3
0.4
0
1
2
-5
0
5
10
angular velocity gain (rad-1
s)angle gain (rad-1
)
Max.FloquetExponent(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)

19. Continuous walking with optimal
combined feedback
• Why not stable, as predicted by MFE?
• Limitations of Floquet analysis
– accuracy
– linearization

20. 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

21. Acknowledgments
• Programming:
– Marko Ackermann
• U.S Department of Veterans Affairs
– B4668R (Hardin)
– B2933R (Triolo)

22. 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?
+