Computational Intelligence - May 2014 - 44
t
x ( t)
dE of error function with respect to i using the directly-learning style [34], the optimal identified parameters i * are
i
)
= D @ x,
It is worth noting here that Kalman filter with forgetting
factor for predicting muscular torque has to be with a properly-chosen fixed forgetting factor, which usually distributes
within the range [0.9, 1] [32]. Otherwise, Kalman filter may
encounter instability and unsatisfactory output results since the
forgetting factor may sensitively affect its stability. Compared
with Kalman filter with forgetting factor, the proposed
NARX-RNN model with its directly-learning estimator, may
be regarded as a more robust alternative.
where notation D @ is the pseudoinverse of matrix D . This estimation equation provides a more straightforward way to obtain
the accurate parameters for identifying NARX-RNN models.
In the procedures above, we can observe that many zeros
appear in the aforementioned matrices and vectors, which
implies that computational time consumption for solving pseudoinverse of matrix D could be rather low. That is to say, the
time cost for identifying the parameters of NARX-RNN
model (1) describing the relationship between FES-induced
torque and eEMG can be low. This is important for online predictive FES control of muscle to establish the predictive model
instantly after the identification phase.
As the torque sensors may not be available in practice for
patient's daily life, so as to perform better predictive control of
muscle activity, we need to predict the muscle torque based
only on eEMG. In this case, real-time identification of muscle
dynamics is switched off after time instant t id and thus muscle
model updating is stopped. We use the latest identified parameters i * till t = t id to predict the muscle torque, and measured
torque is never considered afterwards. According to equation
(1), the predicted torque is as follows.
by RNN Update
Normalized Torque
Parameters
by Kalman Filtering
Normalized Torque
V. Verification Based on Experimental Data
In this section, we focus on the verification of the proposed
NARX-RNN model with data from five SCI subjects. For
the purposes of comparison, Kalman filter with forgetting
factor [32] is also applied for the muscular torque identification and prediction, based on the NARX-type muscle model
used in [32].
As in Fig. 5, identification is terminated at time instant t id
and then prediction is started afterwards with the latest estimated
parameters from time instant t id and ended at time instant t pr .
The process cycle is repeated periodically. During the first time
period [t id, t pr ], the NARX-RNN model shows similar prediction results as those of Kalman filter. However, as observed from
Identification Off
Identification Off
3
= i *T [xt (t - 1), g, xt (t - l ), u (t - 1), g,
u n (t - m), u (t - m) xt (t - 1), u (t - 1) xt (t - l )] T . (4)
Identification Off
2
1
0
v1(t)
w11(t)
-1
-2
0
10
20
30
40
50
60
70
80
90
a(t)
b(t)
100
Measured
Estimated
eEMG Based Prediction
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
70
80
90
100
Measured
Estimated
eEMG Based Prediction
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
Time (s)
60
70
80
90
100
Figure 5 Performance of NARX-RNN model and Kalman filter with forgetting factor with eEMG for periodic torque estimation. Identification was
switched off for 18s (with blue background color in middle and lower plots). From top to bottom: updated parameters v 1 (t), w 11 (t), a (t), and
b (t) of NARX-RNN model; identification and prediction synthesized by NARX-RNN; identification and prediction results synthesized by Kalman
filter with forgetting factor m = 0.965.
44
IEEE ComputatIonal IntEllIgEnCE magazInE | may 2014
Table of Contents for the Digital Edition of Computational Intelligence - May 2014
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