optimal feedback control for human gait with function electrical stimulation
DESCRIPTION
Presented at World Congress of Biomechanics 2010. Abstract:TRANSCRIPT
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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
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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.
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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
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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
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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
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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
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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
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“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
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Results and Discussion
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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
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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
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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
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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
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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
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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?
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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)
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Continuous walking with optimal combined feedback
• Why not stable, as predicted by MFE?• Limitations of Floquet analysis
– accuracy– linearization
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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
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Acknowledgments
• Programming: – Marko Ackermann
• U.S Department of Veterans Affairs– B4668R (Hardin)– B2933R (Triolo)
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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?
+