Biomedical Engineering Reference
In-Depth Information
Once the system parameters are determined using least squares on the training data, the
model obtained (
a
,
C
,
U
) can be used in the Kalman filter to generate estimates of the hand posi-
tions from neuronal firing measurements. Essentially, this model assumes a linear dynamical rela-
tionship between current and future trajectory states.
The Kalman filter is an adaptive state estimator (observer) where the observer gain is op-
timized to minimize the state estimation error variance. In real-time operation, the Kalman gain
matrix
K
(
5.19
), is updated using the projection of the error covariance in (
5.18
) and the error co-
variance update in (
5.21
). During model testing, the Kalman gain correction is a powerful method
for decreasing estimation error. The state in (
5.20
) is updated by adjusting the current state value by
the error multiplied with the Kalman gain.
-
T
(5.18)
P
(
t
+
1 =
)
AP A
( )
t
+
U
T
T
-
1
(5.19)
+
-
-
+
K
(
t
1
)
=
P
(
t
+
1
)
C CP
(
(
t
1
)
C
)
+
(5.20)
�
x
(
t
+
1
)
=
Ax
�
( )
t
+
K
(
t
+
1
)(
Z
(
t
1
)
-
CAx
�
( ))
t
-
(5.21)
+
=
-
+
+
P
(
t
1
)
(
I K
(
t
1
)
C P
)
(
t
1
)
where the notation
P
-
(
t
+ 1) means an intermediate (prior) value of
P
at time
t
+ 1. Using the propaga-
tion equations above, the Kalman filter approach provides a recursive and on-line estimation of hand
kinematics from the firing rate, which is more realistic than the traditional linear filtering techniques
and potentially better. The testing outputs for the Kalman BMI are presented in Figure
5.4
. The Kal-
man filter performs better than the linear filter in peak accuracy (CC = 0.78 ± 0.20) but suffers in
80
60
40
20
0
-20
-40
0
50
100
150
200
250
300
350
400
450
500
Time (100ms)
FIgURE 5.4:
Testing performance for a Kalman filter for a reaching task. Here, the red curves are the
desired
x
,
y
, and
z
coordinates of the hand trajectory, whereas the blue curves are the model outputs.
