SPECIAL ISSUE - REVIEW
Combined feedforward and feedback control of a redundant,
nonlinear, dynamic musculoskeletal system
Dimitra Blana Æ Robert F. Kirsch Æ
Edward K. Chadwick
Received: 6 June 2008 / Accepted: 11 March 2009 / Published online: 3 April 2009

International Federation for Medical and Biological Engineering 2009
Abstract A functional electrical stimulation controller is
presented that uses a combination of feedforward and
feedback for arm control in high-level injury. The feed-
forward controller generates the muscle activations
nominally required for desired movements, and the feed-
back controller corrects for errors caused by muscle fatigue
and external disturbances. The feedforward controller is an
artificial neural network (ANN) which approximates the
inverse dynamics of the arm. The feedback loop includes a
PID controller in series with a second ANN representing
the nonlinear properties and biomechanical interactions of
muscles and joints. The controller was designed and tested
using a two-joint musculoskeletal model of the arm that
includes four mono-articular and two bi-articular muscles.
Its performance during goal-oriented movements of vary-
ing amplitudes and durations showed a tracking error of
less than 4 in ideal conditions, and less than 10 even in
the case of considerable fatigue and external disturbances.
Keywords Feedback control
Musculoskeletal
modeling
Functional electrical stimulation
Shoulder

Elbow
1 Introduction
Functional electrical stimulation (FES) systems restore
function after spinal cord injury by stimulating paralyzed
muscles to contract in appropriately coordinated patterns.
Various FES systems have been developed that benefit
different populations, such as standing systems for people
with paraplegia [9], and hand grasp systems for people with
mid-cervical level spinal cord injuries [15]. We are inter-
ested in expanding the FES benefits to individuals with
high-level tetraplegia, whose entire upper extremity is
essentially paralyzed. This FES system aims to restore
shoulder and arm function to this population, and allow
them to move their arm throughout a functional workspace.
Current FES systems include highly developed
hardware (e.g., implanted stimulators and numerous elec-
trodes), but very basic control algorithms to calculate the
stimulation patterns needed for the desired function,
namely, predefined patterns, fixed for every task [15]. This
limits the functional benefit of the FES system to a set
number of tasks, and does not guarantee its performance in
the presence of muscle fatigue or unexpected perturbations.
A more sophisticated controller is needed for our upper-
extremity FES system, because unlike the cyclic move-
ments of the lower extremity, upper-extremity tasks are
goal directed [8]. This means that the amplitude, speed and
direction of the motions change continuously, so the con-
troller needs to continuously calculate the stimulation
patterns for a wide range of motions commanded by the
user.
There are different control strategies used in existing
FES systems. Feedforward control is commonly used in
clinical practice [13, 15]. The output of this type of con-
troller depends only on the user command, and not on the
system performance. The advantage of this design is that it
does not need sensors to measure the system output, but the
disadvantage is that it is unable to make corrections if the
actual movement deviates from the desired movement.
Feedback control uses sensors to monitor the system
D. Blana
R. F. Kirsch (&)
E. K. Chadwick
Department of Biomedical Engineering,
Case Western Reserve University, Cleveland, OH, USA
e-mail: rfk3@case.edu
D. Blana
e-mail: dimitra.blana@case.edu
123
Med Biol Eng Comput (2009) 47:533–542
DOI 10.1007/s11517-009-0479-3

output, so it can correct for errors in the system trajectory [8].
Feedback is necessary in order to maintain good tracking
performance in the presence of fatigue and any external
disturbances encountered. However, inherent delays in the
response of the system can cause problems to the feedback
controller [1], especially in the case of fast movements
[23]. Therefore, a combination of feedforward and feed-
back control has the best results, and is the preferred
control method in several FES system designs [1, 6, 20],
including the one presented in this paper.
The feedforward controller is typically an inverse-
dynamic model of the controlled system [19]. Since an
actual inverse-dynamic model is usually not available, data
collected from the system itself are used to train an artifi-
cial neural network (ANN) to behave like the inverse-
dynamic model. An ANN that includes time-delayed inputs
can approximate any dynamic system based solely on the
knowledge of inputs and outputs [19]. In order to train the
network to represent the inverse dynamics of the system,
inputs and outputs are recorded from the system and used
as the outputs and inputs, respectively, of the ANN. This
has been used in many FES controllers of single-joint,
single-muscle systems [2, 3, 6, 20, 26]. However, in the
case of a system with multiple muscles crossing one joint,
the inverse of the system is not unique, and the ANN will
train on the average of possible solutions, which is not
always a correct solution [14].
For this reason, we have used the actual inverse-
dynamic model of the system to train the ANN in the
feedforward component of our controller. Optimization
was used to ensure uniqueness of the solution. In [10], the
inverse-dynamic model itself was used as a feedforward
controller, but a relatively simple model had to be designed
so that it was fast enough to run in real-time. An ANN,
however, is a system of simple processing elements, con-
nected into a network by a set of weights [19], so it is
computationally light. Off-line training of the ANN
according to the inverse-dynamic model allows us to
include all the complexities of the musculoskeletal system
in a real-time controller.
For the feedback part of the controller, the feasibility of
PID controllers for use in FES applications has been
examined [1, 6, 25]. However, the performance of these
controllers has been limited. As discussed in [20], PID
controllers are linear and have been incapable of providing
good control over highly nonlinear musculoskeletal sys-
tems [4]. Moreover, the PID controllers mentioned above
were all tested in single-joint, single-muscle systems, but
the complexity of musculoskeletal systems increases sub-
stantially when they include more degrees of freedom and
more muscles, some of them bi-articular, that cause
mechanical coupling between joints [3]. A multivariable
feedback controller that uses PI control is described by Lan
[17], but its performance was only evaluated under iso-
metric conditions. These authors suggest that nonlinear
control methods may be necessary for control of move-
ments of a multi-joint system.
In order to address these issues, we have created a
neuro-PID controller for the feedback loop, which com-
bines the attractive features of PID control (robustness and
ease of implementation) with the nonlinear nature of ANN.
The goal of the ANN is to model the nonlinear relation-
ships among muscles and joints, allowing the linear PID
controller to deal solely with the dynamic response.
The feedforward–feedback FES controller presented in
this paper was designed and tested in simulation. Model-
based evaluations were used to explore various control
strategies before implementing invasive, expensive FES
systems in human subjects. The model-based approach is
intended to allow initial development of the control system
to a point where a human implementation can be justified.
The model used includes two joints, and six muscles, two
of them bi-articular. This system is sufficiently complex to
include both the nonlinear properties of the muscles
themselves and the nonlinearities and complicated inter-
actions between multiple muscles and joints. The controller
was evaluated for a large set of goal-directed movements
that cover the range of both joints. Muscle fatigue and
external disturbances were also simulated to evaluate the
performance of the controller for realistic conditions.
2 Methods
2.1 The model
The model used to design and test the controller is a two-
dimensional model of the upper extremity, in the horizontal
plane (no gravity). It includes six muscles (anterior and
posterior deltoid, long head of the biceps, brachialis, long
and lateral head of the triceps) and two degrees of freedom
(shoulder flexion–extension and elbow flexion–extension).
The range of the shoulder angle is from -20 to 110, and
the range of the elbow angle is from 0 to 170. Four of the
muscles are mono-articular, and two (long head of biceps
and long head of triceps) are bi-articular. A schematic of
the model is shown in Fig. 1.
The muscle and joint parameters for the model were
obtained from cadaver studies by Klein-Breteler et al. [16].
These parameters include the position of joint centers,
inertial parameters for body segments, and the optimal fiber
length, origin and insertion, tendon slack length, and
physiological cross-sectional area of every muscle. The
muscle model is a Hill-type model that includes contraction
dynamics, force-length dependence and force-velocity
dependence. It was developed by McLean et al. [18].
534 Med Biol Eng Comput (2009) 47:533–542
123

The model can run both forward and inverse-dynamic
simulations. In the forward-dynamic mode, the inputs are the
activations of the six muscles, and the outputs are the angles
and angular velocities of the two joints. In the inverse-
dynamic mode, the inputs are the desired angles and angular
velocities of the two joints, and the outputs are the required
muscle activations. Both modes are used in the controller
design and testing, as described in the next section. In the
case of inverse-dynamic simulations, an optimization rou-
tine is needed to distribute the muscle forces based on the
required joint torques, since there are more muscles than
degrees of freedom. The objective function currently used
was proposed by Praagman [21] to minimize energy
consumption:
Em ¼ Ef þ Ea ¼ m c1
Fm
PCSA
þ c2
Fm
Fmax

2( )
ð1Þ
where Ef and Ea are the muscle energy consumption due to
the detachment of cross bridges and re-uptake of calcium,
respectively, m is the muscle mass, Fm is the muscle force,
PCSA is the physiological cross-sectional area, Fmax is the
maximum muscle force, and c1 and c2 are two constants
chosen such that 50–50 contribution from the linear and
nonlinear terms at 50% activation is reached [21].
2.2 The controller
The controller consists of a feedforward and a feedback
part (Fig. 2). The feedforward part is an ANN trained to
behave like the inverse-dynamic model of the arm, so that
it can generate the muscle activations required for a
desired movement, based on knowledge of system
dynamics. This ANN is a two-layer network with a sig-
moidal transfer function in the hidden layer, and a linear
transfer function in the output layer. The inputs are the
shoulder and elbow angles, and four past values of the
angles, each delayed by 80 ms, used to estimate angular
velocity. The outputs are activation levels for the six
muscles.
Training of this ANN was done using data from inverse-
dynamic simulations using the model. Forty movements
with a duration of 60 s each were used for training. The
movements were trajectories covering the range of both
shoulder and elbow angles, with bell-shaped velocity pro-
files. The bell-shaped curves had random amplitudes,
within the range of the angles, and random maximum
velocities, up to 4 rad/s. This is the type of trajectory seen
in goal-directed movements performed by able-bodied
subjects [12], and it covers a wide range of amplitudes and
frequencies, which is a desired characteristic for ANN
learning [23]. In order to improve learning, Gaussian white
noise with mean zero and standard deviation equal to 5% of
the maximum angle was added to the input data, as
described in [20]. The ability of the network to approxi-
mate the inverse-dynamic model was quantified using the
root mean squared error between the original model-based
activations (i.e., the outputs of the inverse-dynamic simu-
lations) and ANN-predicted muscle activations (i.e., the
output of the ANN that approximates the inverse-dynamic
model) as follows:
RMSE ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN
i¼1 ðyANNi  ymodeli Þ
2
N
s
ð2Þ
Networks with different numbers of neurons in the
hidden layer were trained, and it was found that the
network with 15 neurons had the smallest testing RMS
error: 0.07 (no units, this is normalized muscle activation
ranging from 0 to 1). Consequently, this ANN was chosen
as the feedforward part of the controller.
The feedback part of the controller was designed as a
traditional PID controller in series with a static ANN. The
PID controller is described by:
uiðtÞ ¼ KPieiðtÞ þ KDi
deiðtÞ
dt
þ KIi
Z t
0
eiðtÞ; i ¼ 1; 2 ð3Þ
where KPi, KDi and KIi are the proportional, derivative and
integral gains for the shoulder (i = 1) and the elbow
(i = 2). The outputs of the PID controller (u1 and u2) then
serve as the inputs to the static ANN. The PID controller
gains were tuned using the Ziegler–Nichols step response
method, which correlates the controller parameters to fea-
tures of the step response [5], with additional manual fine-
tuning. The values found after tuning were: KP = 2.5,
anterior
elbow
wrist
shoulder
1
2
3
4
5
6
shoulder angle
elbow angle
y
x
Fig. 1 The two-dimensional model of the human arm used in this
study. It has two degrees of freedom (shoulder flexion–extension and
elbow flexion–extension), and six muscles numbered in the figure: 1
anterior deltoid, 2 posterior deltoid, 3 brachialis, 4 lateral triceps, 5 long
head of biceps, and 6 long head of triceps. 1 and 2 cross the shoulder
joint, 3 and 4 cross the elbow joint, and 5 and 6 cross both joints
Med Biol Eng Comput (2009) 47:533–542 535
123

KI = 3 and KD = 1 for the shoulder angle, and KP = 2.5,
KI = 5 and KD = 1 for the elbow angle.
The feedback ANN has the same architecture as the
feedforward ANN (two layers, sigmoidal and linear transfer
function in the hidden and output layer, respectively).
However, this ANN is static, since it uses only the present
angle values (and no past values) as inputs. In order to obtain
training data, a 100 by 100 grid of shoulder and elbow angles
was created, and inverse-dynamic simulations were run with
these angles as constant inputs. From each simulation, we
recorded the set of muscle activations calculated by the
model in the steady state at each of the 10,000 positions. The
angles and the corresponding muscle activations were used
to train networks with different numbers of neurons in the
hidden layer, and the network with 10 neurons was chosen
because its RMS error of 0.09 was the lowest.
The controller was then tested using forty 60-s trajec-
tories, similar to the trials used for training (i.e., goal-
directed movements with bell-shaped velocity profiles of
different amplitudes and maximum velocities) but not the
same. Its performance was quantified using the RMS error
between the desired and actual trajectory (Eq. 2).
In order to investigate the benefits of each component of
the controller, the feedforward-only (FF) component,
feedback-only (FB) component, and combination of feed-
forward and feedback (FF ? FB) components were tested
separately.
To test the performance of the controller for more
realistic, non-ideal conditions, a simple model of muscle
fatigue was incorporated into the model. For this fatigue
model, the maximum force (Fj) of each muscle was line-
arly reduced according to the equation:
FjðtÞ ¼ F0jðtÞð1  0:005  tÞ; j ¼ 1; . . .; 6 ð4Þ
where F0j is the maximum isometric force for muscle j.
Using a ‘‘fatigue rate’’ of 0.005 resulted in a 50% force
reduction over 100 s, which has also been used by [1]
and [20]. The performance of the controller in the
presence of fatigue was tested using forty 160-s goal-
oriented movements, similar to the ones described ear-
lier, and ten cyclical 160-s movements, with repeated
reaching between two randomly chosen goals. The
cyclical movements were included because they can
clearly show the effects of increasing muscle fatigue on
the model trajectory. The performance of the controller
in the presence of fatigue was quantified using the RMS
error between the desired and actual trajectory (Eq. 2)
for all 50 trials.
To test the ability of the controller to resist external
disturbances, forces of different amplitudes were applied
to the wrist of the model, in both the x (medial-lateral)
and y (anterior–posterior) directions. The forces were
applied during goal-oriented movements similar to those
described earlier, and tended to displace the arm from
the desired trajectory. They had bell-shaped amplitudes,
and lasted between 1 s (simulating, for example, a sud-
den push) and 10 s (e.g., carrying an object along the
trajectory). The maximum amplitude of the perturbations
was drawn from a normal distribution with standard
deviation 1 N and four different means: 1, 5, 10 and
15 N, to simulate smaller or larger disturbances. Ten 60-s
goal-oriented movements were used for testing: first they
were performed without perturbations, and then they
were performed again for each case of force amplitude
(about 1, 5, 10 and 15 N). The mean RMSE between the
desired and actual trajectory was calculated for the ten
trials in each case.
Finally, the trajectory errors were also expressed in
terms of endpoint (ep) position, since the hand location is
more functionally meaningful than the individual joint
angles. This was calculated as the RMS of the Euclidean
distance between the desired and actual endpoint position:
Reference
trajectory
dynamic
inverse model
ANN
PID
steady-state
inverse model
ANN
2 DOF,
6 muscle
arm model
Output
trajectory
error
muscle
activations
feedforward
feedback
+
+
+
-
Fig. 2 The combined
feedforward and feedback
(FF ? FB) controller scheme,
shown controlling the 2 DOF
model from Fig. 1. The
‘‘reference trajectory’’ includes
the desired joint angles
(shoulder and elbow) and
desired joint angular velocities
(shoulder and elbow). The
output trajectories indicate the
same quantities as generated by
the overall system (controller
plus arm model)
536 Med Biol Eng Comput (2009) 47:533–542
123

3 Results
Figure 3 shows the performance of the FF, FB, and
FF ? FB controller for a 20-s segment of one of the testing
movements. Panel (a) shows the shoulder angle, and panel (b)
shows the elbow angle. The RMS error for the three con-
trollers was 8.3, 4.6, and 3.3, respectively, for the shoulder
angle, and 8.4, 4.0, and 2.7, respectively, for the elbow
angle. The muscle activations corresponding to that
movement are shown in panels (c), (d) and (e) for the FF,
FB and FF ? FB controllers, respectively. Note that the
feedforward-only muscle activations were much smoother
and lower in maximum magnitude than the feedback-only
activations, with the overall feedforward–feedback activa-
tions showing a combination of smooth, moderate
amplitude activations when the desired angles changed
slowly and fast adjustments when the desired angles
changed rapidly.
The mean tracking performance of the three controllers
across all forty 60-s testing movements, calculated as the
mean RMS error (Eq. 2), is summarized in Fig. 4. The
combined feedforward–feedback controller had the best
performance, with an RMS error of less than 4 for both the
shoulder and the elbow. The feedback-only controller had
an RMS error of 5–6, and the feedforward-only controller
had the worst tracking performance, with an RMS error of
15–20. The three controllers were significantly different
from each other, for both the shoulder and the elbow
(paired t test, P \ 0.001).
Figure 5 shows an example of a cyclical movement that
was performed while the maximum forces of all six mus-
cles were continuously decreased according to our fatigue
model (Eq. 4). The movement is continued beyond the
160 s used to quantify the controller performance, to
illustrate the model operation in the presence of extreme
muscle fatigue: since the maximum muscle force was
0
50
100
(a)
sh
ou
ld
er
(d
eg
)

desired
feedforward
feedback
feedforward−feedback
0
50
100
(b)
el
bo
w
(d
eg
)
0
0.5
1
(c)
FF
−o
nl
y
ac
tiv
at
io
ns


anterior deltoid
posterior deltoid
brachialis
triceps short
triceps long
biceps
0
0.5
1
(d)
FB
−o
nl
y
ac
tiv
at
io
ns
0 2 4 6 8 10 12 14 16 18 20
0
0.5
1
time (sec)
(e)
FF
+F
B
ac
tiv
at
io
ns
Fig. 3 An example of the
performance of the FF, FB and
FF ? FB controllers for a 20-s
segment of a goal-oriented
movement. Panel a illustrates
the shoulder angles for the
various controller configurations
and panel b illustrates the
corresponding elbow angles.
Panel c shows the activations
calculated by the FF-only
controller, panel d by the FB-
only controller, and panel e for
the FF ? FB controller
RMSE ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PN
i¼1 ðdesired epx  actual epxÞ
2 þ ðdesired epy  actual epyÞ
2
h i
N
vuut
Med Biol Eng Comput (2009) 47:533–542 537
123

decreased by 0.5% per second, by the end of the 190-s
movement the maximum force was 5% of the initial
maximum force. The maximum force, as a percentage of
the initial maximum force, is shown in panel (a). Panel (b)
presents the resulting shoulder motions and panel (c) pre-
sents the resulting elbow motions. Panels (d) and (e) are the
corresponding joint angle errors (absolute difference
between desired and output angles) for the shoulder and
elbow, respectively. Panel (f) shows the muscle activations
calculated by the FF ? FB controller. Note that the error
for both the shoulder and elbow does not exceed 5 until
after 160-s of movement, which corresponds to a maximum
muscle force of just 20% of the ‘‘pre-fatigue’’ levels. For
further decreases in maximum muscle forces, most muscle
activation levels saturate at their maximum value of 1
(panel f) and the desired arm trajectory can no longer be
accurately tracked.
Figure 6 shows the mean RMS errors across all fifty
160-s trials with fatigue as a function of time. The trials
were split into 40-s intervals, and the RMS error was cal-
culated for each one. The maximum force as a percentage
of the initial maximum force is also shown for each
interval. For both the shoulder and the elbow, the RMS
error increased as the fatigue level increased. Towards the
end of each trial, which corresponds to muscle forces that
have decreased to levels of 40–20%, the RMS error was
10–15. However, the RMS error was less than 8 for the
first half of the trials, which corresponds to muscle force as
low as 50%.
Figure 7 shows an example of the controller perfor-
mance during an unperturbed movement (panels a, b, c),
and for the same movement in the presence of two sets of
random external disturbances at the wrist: small forces
(panels d, e, f) and large forces (panels g, h, i). The forces
are shown on the top panels: zero force for the unperturbed
movement (panel a), forces around 1 N in panel (d) and
forces around 15 N in panel (g). Panels (b), (e) and (h)
show the shoulder angles for the three cases, and (c), (f)
and (i) show the elbow angles. For the unperturbed
movement, the RMS errors were 2.7 (shoulder) and 2.5
(elbow). In the case of small perturbations, the RMS errors
were 3.0 (shoulder) and 2.7 (elbow). Finally, in the case
of large perturbations, the RMS errors were 5.8 (shoulder)
and 3.4 (elbow).
The mean RMS error for the shoulder and elbow angles,
for ten 60-s trials performed under five different conditions
(unperturbed, external forces of about 1, 5, 10 and 15 N)
are shown in Fig. 8. The error increased with the force
amplitude, but in all cases it was below 10.
Finally, Fig. 9 shows the mean RMS errors for the
endpoint position in four of the cases presented above,
using ten trials for each case: ideal conditions (no fatigue or
perturbations), large external forces of about 15 N, mod-
erate fatigue that has reduced the muscle forces to 60–80%
of their maximum output, and extreme fatigue that has
reduced the muscle forces to 20–40%. The mean RMS
errors for the four cases were 2.9, 4.8, 3.8 and 17.6 cm,
respectively.
4 Discussion
A feedforward–feedback controller has been developed
for a two-joint, six-muscle arm model. An inverse-
dynamic model of the arm was used to train the ANN-
based feedforward component so that it accounted for
the complex interactions between muscles and joints,
while solving the muscle redundancy problem by dis-
tributing the muscle forces according to a minimum
energy consumption criterion. The inverse-dynamic
model in the steady state was also used to train a sep-
arate static ANN that was used in conjunction with a
linear PID controller in the feedback loop to create a
neuro-PID feedback controller capable of handling the
highly nonlinear and dynamic nature of the musculo-
skeletal system. The controller showed excellent tracking
performance during goal-oriented movements, with less
than 4 joint error (3-cm endpoint error) in ideal con-
ditions and less than 10 (6 cm) even in the case of
considerable fatigue and large external disturbances.
Using the actual inverse-dynamic model for ANN
training solved the muscle redundancy problem caused by
the greater number of muscles (six) than joints (two) in the
model. In addition, the muscle activations calculated by
the controller were almost always at low levels (except for
the case of extreme fatigue), since the criterion used to
select the unique solution of the muscle force distribution
minimized energy consumption. As shown in Figs. 3 and 5,
feedforward feedback feedforward+feedback
0
5
10
15
20
25
30
R
M
S
er
ro
r (
de
gr
ee
s)
shoulder angle
elbow angle
Fig. 4 The mean RMS errors for shoulder angles (filled bars) and
elbow angles (open bars) across all forty 60-s trials tested using the
three controllers: FF, FB, and FF ? FB. The paired t test showed
significant differences among the three controllers, for both the
shoulder and the elbow
538 Med Biol Eng Comput (2009) 47:533–542
123

rapidly approached the maximum. Addition of the FF
component smoothed the activation and its maximum did
not increase above 0.7.
The fatigue model used here was fairly simplistic, since
it did not include activation level and stimulation frequency
dependence, or a recovery with rest component [22].
However, our goal was to test the controller in the case of
reduced muscle force output, and the fatigue model we
chose created the right conditions for that. As shown in
Figs. 5 and 6, under moderate fatigue conditions the
tracking performance of the controller was as good as the
non-fatigued case. During normal functional use (i.e.,
occasional goal-directed movements), it is very unlikely
that all the muscles will fatigue below 50% without the
chance for recovery.
The perturbations added to the wrist simulated obstacles
or objects that are picked up and held during the move-
ment. They had a much smaller effect on the controller
performance than fatigue, with an average error of less than
10 for forces up to 15 N.
Figure 9 shows that the generally small joint angle
errors translate into small endpoint errors: in ideal condi-
tions the endpoint error is less than 3 cm, and even in the
presence of the largest perturbations tested, the error does
not exceed 6 cm. An error at that level is functionally
trivial, since it would certainly allow the FES user to eat, or
0
5
10
15
20
25
R
M
S
er
ro
r (
de
gr
ee
s)
0−40sec 40−80sec 80−120sec 120−160sec
maxF: 100−80% 80−60% 60−40% 40−20%
shoulder angle
elbow angle
Fig. 6 Mean controller performance during progressive fatigue. The
mean RMS errors across all fifty 160-s trials with fatigue are plotted
for different intervals
−16
−8
0
8
16
(a)
pe
rtu
rb
at
io
n
(N
)
(d)


force x
force y
(g)
10
60
110
(b)
sh
ou
ld
er
(d
eg
)
(e)


desired angle
output angle
(h)
0 5 10 15 20
0
50
100
(c)
el
bo
w
(d
eg
)
0 5 10 15 20
(f)
time(sec)
0 5 10 15 20
(i)
Fig. 7 Example of the FF ? FB controller performance during a
goal-oriented movement without perturbations (left panels), and in the
presence of two sets of random external forces at the wrist: small
forces (middle panels) and large forces (right panels). The forces are
shown in panels a, d and g. Panels b, e and h illustrate the shoulder
angles, and panels c, f and i illustrate the elbow angles
540 Med Biol Eng Comput (2009) 47:533–542
123

reach for various objects. The endpoint error approaches
20 cm only during extreme fatigue, when there is insuffi-
cient muscle force for trajectory tracking.
The controller evaluated here was tested using a two-
joint, six-muscle system, but it could easily be extended for
use with a full arm model. If the inverse-dynamic model is
available, the ANN could be trained with more inputs and
outputs, although a larger number of neurons in the hidden
layer may be needed. One important practical benefit of
using PID in series with an ANN instead of a purely PID
feedback loop is that the number of gains that need to be
tuned is reduced to 6 in the case of our model (propor-
tional, integral and derivative for each angle) instead of 36
(the previous 6 PID gains for each of the 6 muscles). This
will become a major advantage when the number of mus-
cles increases.
For the controller to be implemented in the real FES
system, the first step will be customizing the arm model to
reflect the FES user’s arm as closely as possible, by
including information about possible voluntary muscle
forces, denervation, as well as which muscles are the tar-
gets for FES stimulation. A controller for this customized
arm model will be built, but because of the differences
between the model and the FES user’s arm, it is unlikely
that the model-based ANN parameters will result in opti-
mal performance. This controller will instead be used as a
starting point for adaptation, using data collected from the
FES-driven arm, as shown previously in [1]. While the FES
system is in use, the stimulation levels and resulting
movements will be recorded for either online or off-line
adaptation of the ANN parameters. As both the feedfor-
ward and feedback ANN are adapted to the arm dynamics
of the specific user, the controller performance will con-
tinue to improve, produce more accurate movements, and
provide more functional benefits to the user.
Acknowledgments The authors would like to thank Dr. Antonie
van den Bogert for his help with the muscle model implementation.
This study was funded by NIH/NINDS contract N01-NS-5-2365.
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0
5
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R
M
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