THE SQUARE-ROOT UNSCENTED INFORMATION FILTER
FOR STATE ESTIMATION AND SENSOR FUSION
Guoliang Liu, Florentin W¨org¨otter and Irene Markeli´c
III Physikalisches Institut - Biophysik, University of G¨ottingen, G¨ottingen, Germany
Keywords:
Nonlinear Estimation, Sensor Fusion, Unscented Information Filter, Square-root Unscented Information Filter.
Abstract:
This paper presents a new recursive Bayesian estimation method, which is the square-root unscented informa-
tion filter (SRUIF). The unscented information filter (UIF) has been introduced recently for nonlinear system
estimation and sensor fusion. In the UIF framework, a number of sigma points are sampled from the prob-
ability distribution of the prior state by the unscented transform and then propagated through the nonlinear
dynamic function and measurement function. The new state is estimated from the propagated sigma points. In
this way, the UIF can achieve higher estimation accuracies and faster convergence rates than the extended in-
formation filter (EIF). As the extension of the original UIF, we propose to use the square-root of the covariance
in the SRUIF instead of the full covariance in the UIF for estimation. The new SRUIF has better numerical
properties than the original UIF, e.g., improved numerical accuracy, double order precision and preservation
of symmetry.
1 INTRODUCTION
Recently, the information filter (IF), which is the
dual of the Kalman filter (KF) (Anderson and Moore,
1979), has attracted much attention for multiple sen-
sor fusion (Wang et al., 2010; Liu et al., 2011). Both
the IF and the KF represent distributions of random
state variables with Gaussians. However, in contrast
to moment parametrization as done in the KF, the IF
uses an information matrix and an information vector
to represent the Gaussians. This difference in param-
eterization makes the IF superior to the KF concern-
ing multiple sensor fusion, as computations are sim-
pler and no prior information of the system state is
required (Lee, 2008).
In the case of nonlinear estimation problems, an
extended version of the IF can be obtained using the
first order term of the Taylor series expansions of the
nonlinear functions, i.e., the dynamic and measure-
ment functions of the system, which is called ex-
tended information filter (EIF). This approximation
can introduce large errors when the system model is
highly nonlinear, and the higher order terms of Taylor
series are important (Van der Merwe and Wan, 2001).
To address this issue, the unscented information filter
(UIF) has been proposed by Kim et al. (Kim et al.,
2008) and Lee (Lee, 2008). Kim developed the UIF
by using minimum mean square error estimation. In
contrast, Lee’s UIF algorithm is derived by embed-
ding statistical linear error propagation into the EIF
architecture. Although their methods are different, re-
sults are essentially identical (Lee, 2008; Kim et al.,
2008). The UIF uses a number of deterministic sigma
points to capture the true information matrix and the
information vector, which can be accurate up to the
second order of any nonlinearity. These sigma points
are generated by unscented transform in the UIF. As
shown in (Lee, 2008), the UIF is superior to the EIF
not only in terms of estimation accuracy but also con-
cerning the convergence speed for nonlinear estima-
tion and multiple sensor fusion. The other way to
generate the sigma points is the Stirling’s interpola-
tion, which has similar performance to the unscented
transform, but with fewer predefined parameters (Liu
et al., 2011).
In this paper, we propose the square-root exten-
sion of the UIF. In the unscented transform, a square-
root of the prior covariance has to be calculated to
generate sigma points. This step is computationally
expensive and requires that the covariance matrix to
be positive definite. To save computational cost and
increase numerical robustness, a square-root form of
the covariance can be directly taken and updated in
the algorithm. The square-root forms achieve bet-
ter numerical characteristics than the regular ones,
e.g., improved numerical accuracy, double order pre-
169
Liu G., Wörgötter F. and Markeli
´
c I..
THE SQUARE-ROOT UNSCENTED INFORMATION FILTER FOR STATE ESTIMATION AND SENSOR FUSION.
DOI: 10.5220/0003839101690173
In Proceedings of the 1st International Conference on Sensor Networks (SENSORNETS-2012), pages 169-173
ISBN: 978-989-8565-01-3
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

cision and preservation of symmetry (Arasaratnam
and Haykin, 2009). In the literature, Potter (Potter
and Stern., 1963) introduced the first square-root filter
which was used in the Apollo manned mission. Since
then, many square-root extensions of conventional fil-
ters have been introduced. Recently, Van der Merwe
(Van der Merwe and Wan, 2001) has proposed square-
root forms of sigma-point Kalman filters which have
better numerical stability and less computational cost.
Here we employ a similar idea, and introduce the
square-root unscented information filter (SRUIF) for
solving nonlinear state estimation and sensor fusion
problems.
This paper is organized as follows: First, we first
show the basic UIF algorithm for nonlinear estimation
and multiple sensor fusion in Section 2, and then the
square-root UIF is proposed in Section 3. Simulation
results of target tracking are presented and discussed
in Section 4. Finally, the work is concluded in Section
5.
2 UNSCENTED INFORMATION
FILTER
The Unscented Information filter (UIF) employs the
unscented transform to generate sigma points, which
can be further used to estimate the mean and covari-
ance of the system state. The UIF algorithm is sum-
marized in Algorithm 1, where γ =
p
(λ+ L) is the
composite scaling parameter, λ = α
2
(L + κ) −L, α
and κ are scaling parameters that determine how far
the sigma points spread from the mean value (Van der
Merwe, 2004), L is the dimension of the state, R
v
and
R
n
are process noise covariance and observation noise
covariance respectively, w
(m)
i
and w
(c)
i
are weights
calculated by w
m
0
=
λ
L+λ
, w
c
0
=
λ
L+λ
+ (1 −α
2
+ β),
w
m
i
= w
c
i
=
1
2(L+λ)
, i = 1,··· , 2L.
2.1 Global Information Fusion
In case of multiple sensors N, where the measurement
noises between the sensors are uncorrelated, the mea-
surement update for information fusion is simply ex-
pressed as a linear combination of the local informa-
tion contribution terms:
y
k
= y
−
k
+
N
∑
i=1
φ
i,k
(1)
Y
k
= Y
−
k
+
N
∑
i=1
Φ
i,k
, (2)
where φ
i,k
and Φ
i,k
are defined in (14) and (15)
respectively.
Algorithm 1: UIF for state estimation.
• Initialization:
ˆx
0
= E(x
0
), P
x
0
= E

(x
0
− ˆx
0
)(x
0
− ˆx
0
)
T

.
• For k = 1,··· ,∞:
1. Generate sigma points for prediction:
ˆx
a
v
k−1
=

ˆx
k−1
v

, P
a
v
k−1
=

P
x
k−1
0
0 R
v

(3)
X
a
v
k−1
=
h
ˆx
a
v
k−1
ˆx
a
v
k−1
+ γ
q
P
a
v
k−1
ˆx
a
v
k−1
−γ
q
P
a
v
k−1
i
(4)
2. Prediction equations:
X
x
k|k−1
= f (X
x
k−1
,X
v
k−1
,u
k−1
) (5)
ˆx
−
k
=
2L
∑
i=0
w
(m)
i
X
x
i,k|k−1
(6)
P
−
x
k
=
2L
∑
i=0
w
(c)
i

X
x
i,k|k−1
− ˆx
−
k

X
x
i,k|k−1
− ˆx
−
k

T
(7)
Y
−
k
= (P
−
x
k
)
−1
(8)
ˆy
−
k
= Y
−
k
ˆx
−
k
(9)
3. Generate sigma points for measurement update:
X
k|k−1
=

ˆx
−
k
ˆx
−
k
+ γ
q
P
−
x
k
ˆx
−
k
−γ
q
P
−
x
k

(10)
4. Measurement update equations:
Z
k|k−1
= h

X
k|k−1

(11)
ˆz
−
k
=
2L
∑
i=0
w
(m)
i
Z
i,k|k−1
(12)
P
x
k
z
k
=
2L
∑
i=0
w
(c)
i
h
X
i,k|k−1
− ˆx
−
k
][Z
i,k|k−1
−z
−
k
i
T
(13)
φ
k
= Y
−
k
P
x
k
z
k
R
−1
n
h
z
k
−z
−
k
+ (P
x
k
z
k
)
T
y
−
k
i
(14)
Φ
k
= Y
−
k
P
x
k
z
k
R
−1
n
(P
x
k
z
k
)
T
(Y
−
k
)
T
(15)
y
k
= y
−
k
+ φ
k
(16)
Y
k
= Y
−
k
+ Φ
k
(17)
3 SQUARE-ROOT UIF
The square-root UIF benefits from three powerful
matrix factorization techniques: QR decomposition,
Cholesky factor updating and efficient least squares.
In the following, we will use qr, chol, cholupdate
and ’\’ (backslash) to refer to the QR decomposi-
tion, Cholesky decomposition, Cholesky factor updat-
SENSORNETS 2012 - International Conference on Sensor Networks
170

ing and efficient least squares respectively
1
.
Algorithm 2: Square-root UIF for state estimation.
• Initialization:
ˆx
0
= E(x
0
), S
v
=
√
R
v
and S
n
=
√
R
n
, S
x0
=
chol

E

(x
0
− ˆx
0
)(x
0
− ˆx
0
)
T

.
• For k = 1, ··· ,∞:
1. Generate sigma points for prediction:
ˆx
a
v
k−1
=

ˆx
k−1
v

, S
a
v
k−1
=

S
x
k−1
0
0 S
v

(18)
X
a
v
k−1
=

ˆx
a
v
k−1
ˆx
a
v
k−1
+ γS
a
v
k−1
ˆx
a
v
k−1
−γS
a
v
k−1

(19)
2. Prediction equations:
X
x
k|k−1
= f (X
x
k−1
,X
v
k−1
,u
k−1
) (20)
ˆx
−
k
=
2L
∑
i=0
w
(m)
i
X
x
i,k|k−1
(21)
S
−
x
k
= qr

q
w
(c)
1

X
x
1:2L,k|k−1
− ˆx
−
k


(22)
C =
q
w
(c)
0

X
x
0
− ˆx
−
k

(23)
S
−
x
k
= cholupdate
n
S
−
x
k
,C,sign{w
(c)
0
}
o
(24)
ˆy
−
k
=

S
−
x
k

T
\

S
−
x
k
\ˆx
−
k

(25)
S
−
y
k
= qr

S
−
x
k
\I

(26)
3. Generate sigma points for measurement update:
X
k|k−1
=

ˆx
−
k
ˆx
−
k
+ γS
−
x
k
ˆx
−
k
−γS
−
x
k

(27)
4. Measurement update equations:
Z
k|k−1
= h

X
k|k−1

(28)
ˆz
−
k
=
2L
∑
i=0
w
(m)
i
Z
i,k|k−1
(29)
P
x
k
z
k
=
2L
∑
i=0
w
(c)
i
[X
i,k|k−1
− ˆx
−
k
][Z
i,k|k−1
−z
−
k
]
T
(30)
U =

S
−
x
k

T
\

S
−
x
k
\P
x
k
z
k

/S
T
n
(31)
y
k
= ˆy
−
k
+U/S
n
(z
k
−ˆz
−
k
+ P
T
x
k
z
k
ˆy
−
k
) (32)
S
y
k
= cholupdate{S
−
y
k
,U,+1} (33)
• QR Decomposition. In the UIF, the square-root
of the covariance matrix S is derived by Cholesky
decomposition on P: S = chol(P)
T
where S is a
lower triangular matrix and fulfills P = SS
T
. If
we know P = AA
T
, the square-root factor S can
be directly calculated from A by QR decomposi-
tion: S = qr(A)
T
. If the matrix A ∈ R
L×N
, then
the computational complexity of a QR decompo-
sition is O(NL
2
).
• Cholesky Factor Updating. If the original up-
date of the covariance matrix is P±uu
T
and S is
the Cholesky factor, then the rank 1 update of S
is S = cholupdate(S,u, ±) where u is the update
vector. If u is a matrix, we can update each col-
umn of u one by one in a loop. For each column
vector, the computational complexity is O(L
2
).
This procedure can alternatively be implemented
as S = qr([S ±u]
T
) using QR decomposition with-
out the loop updates.
• Efficient Least Squares. The least squares solution
for the linear equation Px = b can be solved effi-
ciently using forward and back substitution if the
Cholesky factor S is known and satisfies P = SS
T
.
For example, we can solve it by x = S
T
\(S\b)
where ’\’ is the backslash. This operation only
requires computational complexity O(L
2
).
The whole process is shown in Algorithm 2,
which looks similar to the general UIF algorithm in-
troduced in Section 2, except that the Cholesky factor
S
x
k
is used instead of the covariance P
k
. The square-
root UIF also comprises two steps, the first is the pre-
diction and the second is the measurement update.
For each step, a number of sigma points are generated
using unscented transform in (19) and (27). However,
the square root of covariance S
x
k
is directly used to
calculate the sigma points without the Cholesky fac-
torization.
In the prediction step, the Cholesky factor S
x
k
is updated using QR decomposition on the weighted
sigma points. This step replaces the P
k
update in (7)
and has complexity O(L
3
). The information vector
ˆy
−
k
= (P
−
k
)
−1
ˆx
−
x
k
=

S
−
x
k

T
\

S
−
x
k
\ˆx
−
k

is derived us-
ing efficient least squares in (25). Because
ˆ
S
−
x
k
is
a square and triangular matrix, we can directly use
back-substitution for solving ˆy
−
k
without the need for
matrix inversion. The back substitution only requires
O(L
2
). Next is the calculation of the square-root in-
formation matrix S
−
y
k
in (26). This step requires a QR
decomposition since S
−
y
k
is a upper triangular matrix,
and meets

S
−
y
k

T
S
−
y
k
=

S
−
x
k

−T

S
−
x
k

−1
. As S
−
x
k
is a
lower triangular matrix, QR decomposition is used to
solve the Cholesky factor S
−
y
k
of the information ma-
1
The abbreviations qr, chol, cholupdate and ’\’ (back-
slash) are in accordance with the function names for QR de-
composition, Cholesky decomposition, Cholesky factor up-
dating and efficient least squares in Matlab.
THE SQUARE-ROOT UNSCENTED INFORMATION FILTER FOR STATE ESTIMATION AND SENSOR FUSION
171

trix Y
k|k−1
. To avoid the inversion, here we use effi-
cient least squares to solve

S
−
x
k

−1
as S
−
x
k
\I, where I
is an identity matrix.
In the measurementupdate step, the updated infor-
mation vector y
k
is derived by efficient least squares
in (32). If the observation dimension is M, the up-
dated square-root information matrix S
y
k
is calculated
in (33) by applying an M-sequential Cholesky update
to S
−
y
k
. The columns of matrix U are update vectors.
This requires O(L
2
M) and replaces the measurement
update of Y
k
in (17).
3.1 Square-root UIF for Multiple
Sensor Fusion
In the case where information from multiple sensors
is available, i.e., N > 1, we can fuse this using the
square-root UIF. For the i
th
sensor, the information
contribution for the information vector is
φ
i,k
= U/S
T
n
(z
k
− ˆz
−
k
+ P
T
x
k
z
k
ˆy
−
k
) (34)
where U is defined in (31). The information contribu-
tion for the square-root information matrix is
S
i,φ
k
= U. (35)
The final estimated result is derived by:
y
k
= ˆy
−
k
+
N
∑
i=0
φ
i,k
(36)
S
y
k
= cholupdate{S
−
y
k
,[S
1,φ
k
S
2,φ
k
··· S
N,φ
k
],+1}.
(37)
4 EXPERIMENTS
In this section, we consider a nonlinear bearing-only
tracking (BOT) problem using the UIF and SRUIF
and compare their performances. The bearing-only
tracking problem has become an important bench-
mark for different probability inference methods
(Bar-Shalom et al., 2001; Sadhu et al., 2004). Har-
tikainen and S¨arkk¨a (Hartikainen and S¨arkk¨a, 2008)
have developed a toolbox which includes the compar-
ison between the UKF, the EKF, and their smoothers
by solving the BOT problem with static sensors.
Here we use the same system model as in (Har-
tikainen and S¨arkk¨a, 2008). A moving target object
is tracked by two static angular sensors as shown in
Fig. 1. The discrete time update of the dynamic ob-
ject on time step k is
x
k
=



1 0 ∆t 0
0 1 0 ∆t
0 0 1 0
0 0 0 1



x
k−1
+ v
k−1
(38)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
X (m)
Y (m)
Real trajectory
UIFa estimate
SRUIFa estimate
UIFb estimate
SRUIFb estimate
Positions of sensors
Figure 1: The ground truth and estimated trajectories for
the bearing-only tracking. In the figure, the estimated tra-
jectories from UIFa and SRUIFa are overlapped, and the es-
timated trajectories from UIFb and SRUIFb are overlapped
too.
where the system state is x
k
= (x
k
,y
k
, ˙x
k
, ˙y
k
)
T
, which
includes the target position (x
k
,y
k
) and velocity
( ˙x
k
, ˙y
k
). ∆t is the time interval between time step k
and k−1, which is set to ∆t = 0.01 in the simulation.
v
k−1
is Gaussian noise with zero mean and the covari-
ance is
R
v
=




1
3
∆t
3
0
1
2
∆t
2
0
0
1
3
∆t
3
0
1
2
∆t
2
1
2
∆t
2
0 ∆t 0
0
1
2
∆t
2
0 ∆t




ξ (39)
where ξ is the spectral density of the noise (Har-
tikainen and S¨arkk¨a, 2008) and set to ξ = 0.1 in our
experiment. The target is tracked by sensors located
at (x
s
,y
s
), where s = 1, 2 in the case of two sensors.
The measurement model of the s
th
sensor is defined
as
θ
s
= tan
−1

y
k
−y
s
x
k
−x
s

+ e
θ,s
(40)
where e
θ,s
∼ N (0,R
n,s
) is the measurement noise of
the s
th
sensor. The sensors are located at (x
1
,y
1
) =
(−1,− 2) and (x
2
,y
2
) = (1, 1), and their measurement
noise variances are R
n,1
= R
n,2
= 0.05
2
. The initial
prior state
ˆ
x
0
and the covariance
ˆ
P
0
are given by:
ˆ
x
0
= [0, 0, 1, 0]
T
(41)
ˆ
P
0
= diag(0.1,0.1,10,10), (42)
where diag means the diagonal matrix.
The estimated results from different filters are
summarized in Table 1. We also show the estimated
SENSORNETS 2012 - International Conference on Sensor Networks
172

trajectories in the Fig. 1. It can be seen that the UIF
and SRUIF have equal accuracy for nonlinear estima-
tion and sensor fusion, but the SRUIF is slightly faster
than the UIF. Van der Merwe (Van der Merwe and
Wan, 2001) shown that the square-root UKF can be
20% faster than the UKF, but in our case the SRUIF
only achieves 3% faster than the UIF.
Table 1: Means (E) and standard deviations (STD) of
RMSE values of the position and average run time (T) in
100 Monte Carlo runs of the bearing-only tracking. UIFa
and SRUIFa use one sensor, whereas UIFb and SRUIFb use
two sensors.
Method E STD T (s)
UIFa 0.6794 0.1725 0.5102
SRUIFa 0.6794 0.1725 0.4944
UIFb 0.1145 0.0283 0.8154
SRUIFb 0.1145 0.0283 0.7763
5 CONCLUSIONS
In this paper, we present the square-root UIF for non-
linear estimation and sensor fusion. It has the same
computational complexity as the original UIF, i.e.,
O(L
3
). However, the SRUIF has better numerical
properties, such as the improved numerical accuracy,
double order precision and preservation of symme-
try. In addition since the square-root of the covari-
ance matrix is directly available, the SRUIF can save
computational costs in the step of sigma-points calcu-
lation. The experimental results show that the SRUIF
runs slightly faster than the original UIF. In the future,
we plan to investigate their performances with differ-
ent sensor network architectures (Lee, 2008), and fur-
ther improve the estimation accuracies, e.g., by com-
bining the proposed filters with the adaptive consen-
sus algorithm (Casbeer and Beard, 2009).
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