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Chapter 4 / Target Tracking
_ _
The innovation (∆Z) is the difference between the actual and predicted
measurements,
∆
− Zˆ Z ~
⎛
⎜
⎝
∂ h
⋅ ∆X
+ 2
∂ X
T
Z : = : =
⋅ ∆X
2
4-12
−1
2
∂ h ⎞
⋅ ⋅ ∆X
+ L ⎟
+ υ
∂ X ⎠
Equation 4.4-15
The measurement Jacobian (H), and Hessian [G], are computed using the
state at time (tk+1) - in the EKF the Hessian is ignored. The measurement
innovation is then apportioned between the states by the Kalman gains so as
to minimise the trace of the covariance matrix, Gelb [G.12] (p186),
K
k+
1
: =
X ˆ
H
+
k+
1
k+
1
: =
C
⋅ C
−
k+
1
−
k+
1
X ˆ
−
k+
1
⋅ H
⋅ H
T
k+
1
T
k+
1
+ K
+ R
k+
1
k+
1
⋅ ∆Z
k+
1
≡
C
−
k+
1
⋅ H
A
The covariance matrix is update using the simple expression,
C
+
k+
1
−
( I − K ⋅ H ) ⋅ C
= 3 k+
1 k+
1 k+
1
Equation 4.4-16
k+
1
T
k+
1
Equation 4.4-17
Equation 4.4-18
Nishimara [N.3] showed that Joseph's form of this equation yields the correct
covariance whether the Kalman gains are optimal or not. This form should
always be used in the IEKF, Kerr [K.4] ,
−
T
T
( I − K ⋅ H ) ⋅ C ⋅ ( I − K ⋅ H ) + K ⋅ R ⋅ K
+
C k+
1 : =
k+
1 k+
1 k+
1
k+
1 k+
1 k+
1 k+
1 k+
1
Equation 4.4-19
Measurements should be processed serially to avoid matrix inversion, and to
accommodate reduced order and asynchronous measurement sets. The
effect of process model and linearisation errors on [A], [H], [Φ] and (∆Z) is
quantified by Hanlon [H.8] .
4.4.2 Iterated EKF
EKF measurement linearisation can be performed about any trajectory. If
the filter is to be well conditioned the trajectory about which linearisation is
performed should ideally be close to the reference trajectory. If the state
estimates become too inaccurate, as might be the case after a long period of
prediction, transformed measurements are often substituted. The IEKF
iterates process and measurement linearisation, a process that if taken to