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x.(k +1 | k) = E[x(k + 1) | Zk]= F (k).x(k | k)+ G(k)u(k), (B-6)
where the term involving the noise drops out because E[v(k)] = 0 by de.nition. Note that in the prediction the measurement at time k +1 is not used. The state prediction covariance simpli.es to
P (k +1 | k)= F (k)P (k | k)F (k)T + B(k)Q(k)B(k)T , (B-7)
where P (k | k) is the covariance of the posterior state estimate. One could predict what the measurement at time k + 1 would be by simply transforming the predicted state estimate to arrive at
z.(k +1 | k) = E[z(k + 1) | Zk]= H(k + 1).x(k +1 | k), (B-8)
where
the
zero-mean
noise
is
again
not
present.
Similar
to
Eq.
(B-7),
the
measurement
prediction
covariance is given by
S(k +1) = H(k + 1)P (k +1 | k)H(k + 1)T + R(k + 1), (B-9)
which is equivalent to Pzz described above. Calculating the cross-covariance between the state and measurement, Pxz, one arrives at the equation for the .lter gain of the Kalman .lter:
W (k +1) = P (k +1 | k)H(k + 1)T S(k + 1).1 . (B-10)
In the update step, the predicted state and covariance are updated using the measurement. Using
Eq.
(B-4),
the
updated
state
estimate
at
time
k + 1 is given by
x.(k +1 | k +1) = .x(k +1 | k)+ W (k + 1)ν(k + 1), (B-11)
where ν(k +1) ≡ z(k +1). z.(k +1 | k) is the measurement residual or innovation. The quantity S can
be
viewed
as
the
innovation
covariance.
The
updated
covariance
is
obtained
by
using
Eq.
(B-5)
and making the necessary substitutions:
P (k +1 | k +1) = P (k +1 | k) . W (k + 1)S(k + 1)W (k + 1)T . (B-12)
Figure
B-1
is
a
high-level
overview
representing
one
cycle
of
the
Kalman
.lter
that
summarizes
the relevent equations presented above. Note that the covariance is independent of the state and measurements
and
can
be
calculated
o.ine,
as
Eq.
(B-12)
implies.
Evolution of the system
Estimation of the state State covariance computation
B.2 UNSCENTED KALMAN FILTER
The unscented Kalman .lter is an extension to the Kalman .lter that uses the unscented transform to
allow
for
nonlinear
dynamic
systems
and
measurement
models
[53].
The
unscented
transform
approximates the distribution of a random variable y that results from an arbitrary nonlinear transformation y = f(x), where the mean xˉand covariance Px of x ∈ Rnx are known. It does this by deterministically choosing sample points, also referred to as sigma points, de.ned by
κ
χi =ˉxWi = i =0,
L nx+κ 1
χi =ˉx + ( (nx + κ)Px Wi = 2(nx+κ) i =1,...,nx, (B-13)
( Li
χi =ˉx . (nx + κ)Px Wi = 1 i = nx +1,..., 2nx,
i.nx 2(nx+κ)
( L
where κ is a scaling parameter and (nx + κ)Px is the ith row of the matrix square root of
i
(nx + κ)Px. These sigma points capture the true mean and covariance of x. The .rst sigma point is called the central sigma point. They are then propagated through the function f to obtain the points βi = f(χi), i =0,..., 2nx. The .rst two moments of y are recovered as
2nx
yˉ= Wiβi, (B-14)
i=0
2nx
Py = Wi(βi . yˉ)(βi . yˉ)T . (B-15)
i=0
Figure
B-2
illustrates
how
the
unscented
transform
works
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