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Chapter 4 / Target Tracking
_ _
S
X ˆ
+
− ⎛ 1 ⎛
= X +
⎜ ∑ ⎜
⎝
2 , j ⎝
ˆ :
[ K ] ⋅ ⎜ ∆Z
− ⋅ ⎜ C ( i , j )
4-14
∂ h
⋅
∂ X ⋅ ∂ X
i i j
P ⋅ H
K : = − T
−1
H ⋅ P ⋅ H + R + 2 ⋅ S
( i , j ) : = ⎜
⋅ C ( m , n ) ⋅ C ( i , j )
−
T
2
⎞
⎟
⎠
⎞
⎟
⎟
⎠
Equation 4.4-25
Equation 4.4-26
∑ ⎟ ⎛ 2
2
∂ h
∂ h ⎞
⋅
⎜
m,
n,
j ⎝
∂ Xi
⋅ ∂ Xm
∂ Xn
⋅ ∂ X
⎠
i, j
Equation 4.4-27
The EKF projects the covariance surface linearly along the gradient to the
estimated state trajectory. Although the SOF replaces this with parabolic
projection, in highly non-linear systems even this can result in deviation
from the true covariance. A crude approach is to increase the system noise
thereby expanding the uncertainties to ensure that they encompass the true
errors at the expense of observability. A promising technique presented by
Julier [J.2] propagates defining points on the covariance surface using (f) and
re-computes the correct mean and covariance statistics after projection,.
Although the method is only accurate to 2 nd order it is shown to improve the
3 rd and 4 th moments that by definition are zero in the SOR.
4.4.4 Converted Measurement Kalman Filter
In the Converted Measurement KF (CMKF) the measurements and their
uncertainties are converted into the state space before performing the
measurement update. Park [P.6] showed that the CMKF is more accurate than
the conventional EKF formulation if the bearing error is < 1.5º. Bar-
Shalom [B.5] also obtained better results from the CMKF for measurement
errors > 0.5º albeit using a de-biasing transformation algorithm. Nearoptimal
results for bearing errors up to 10° are possible providing the
angular uncertainty is know accurately, Longbin [L.6] and Lerro [L7] . In this
application the measurement errors are relatively small, certainly less than
6 mrad, and the technique was not considered.
4.4.5 Modified Gain EKF
Universal linearisation implies that (h) can be expressed as, Speyer [S.13] ,
( ) X X ˆ
X : g Z ,
ˆ X h ⎛ - ⎞ = ⎛ -
−
⎞ ⋅
h ⎜ ⎟ ⎜ ⎟ ∆
⎝ ⎠ ⎝ ⎠
Equation 4.4-28