1. technology and computing

An Improved UKF Algorithm based on Minimum Sample Skewness and the Ratio Correction Factor
Xiuhua ZHANG, Lianjun ZHANG, Yongcheng JIANG
An Improved UKF Algorithm based on Minimum Sample Skewness and
the Ratio Correction Factor
*1, 2
1
Xiuhua ZHANG, 2Lianjun ZHANG, 2Yongcheng JIANG
Mechanical & Dynamic Engineering College, Harbin University of Science & Technology,
Harbin, China, [email protected]
2
College of Mechanical Engineering, Jiamusi University, Jiamusi, China
Abstract
In the positioning and navigation system of the mobile robot, it is necessary to be considered not
only the speed and accuracy in the calculation, but also the non-local effects. An improved algorithm
was presented by the strategy of the smallest proportional correction factor and the minimum degree of
sampling bias. In the method, the sampling strategy of the minimum degree was applied in the system
of positioning and navigation; the proportional correction factor was applied in the sampling strategy
of the minimum degree. The results show that the non-local effects were solved by the minimum degree
of partial sampling strategies in the filtering process. The theoretical foundation of the positioning and
navigation in the mobile robot system was proposed.
Keywords: Ratio Correction Factor, Minimum Sample Skewness, Improved UKF Algorithm, NonLocal Effect
1. Introduction
Kalman filter theory was put forward by the Hungary mathematician R.E.Kalman in 1960[1]. At
present, it has been widely used in the field of aeronautics and astronautics, navigation positioning,
target tracking, controlling, etc. Most of the practical systems are non-linear systems and Kalman
filtering theory is only applicable to linear systems, so Extended Kalman Filtering (EKF) was put
forward by Bucy and Sunahara. Kalman filter theory is applied further to the nonlinear field [2]. The
idea of EKF: the predictive model is acquired by Taylor series expansion for state equation or
observation equation of nonlinear and 1st order of approximation can be used. The linearization error
from the state equation or measurement equation is produced undergo the Taylor series expansion. To
solve the above problem, UKF algorithm was put forward by Julier and Uhlmann (British scholar) in
1995 [3]. Used nonlinear model for recursive estimation in this method to avoid the introduction of the
linearization error and calculate the Jacobian matrix [4, 20].
UKF (Unscented Kalman Filter) algorithm had been widely used in state estimation of nonlinear
systems [5, 6, 7]. When used the method to deal the state equation, first, the variables need to be
transformed by UT [8], then used the state variables through UT transform filter to estimates for
reducing the estimation error and improving the filtering accuracy. At present, UKF algorithm based
on the minimum sample skewness strategy is widely used in navigation systems and robotics
positioning systems in order to meet Real-time requirements [9, 10, 11, 12].
Li Dan and Liu Jian-ye (College of Automation Engineering, Nanjing University of Aeronautics
and Astronautics) proposed an improved UKF algorithm to meet the reliability, accuracy and real-time
requirements that the minimum sample strategy combined with UKF algorithm to improve the
calculation speed. The algorithm was applied in the magnetometer/Radar Altimeter Navigation System.
In [10], the minimum sample skewness strategy was applied in the mobile robot navigation system
when in real-time location algorithm and the application of the creation of the map. Through
experiments this algorithm was superior to other sampling strategies. In [11], Andersen M N and
Wheeler K., Pan Ping-jun and Feng Xin-xi (Air Force Engineering University) proposed UKF
algorithm based on the minimum single sample strategy was applied in dual-band infrared radiation of
IRST maneuvering target tracking algorithm to effectively track the target and the value of the
algorithm had been well represented in practical applications [17, 18, 19].
The sampling strategy had been applied well in navigation and positioning system. However, the
samples of non-local effects were also easy produced to result in nonlinear function. For this the
International Journal of Advancements in Computing Technology(IJACT)
Volume5,Number2,January 2013
doi:10.4156/ijact.vol5.issue2.92
747
An Improved UKF Algorithm based on Minimum Sample Skewness and the Ratio Correction Factor
Xiuhua ZHANG, Lianjun ZHANG, Yongcheng JIANG
correction factor is applied to the minimum ratio of skewness in the sampling strategy simplex to solve
the sampling problem of non-local effects.
2. Unscented transformation and minimum sample skewness strategy
UKF algorithm is an efficient method in nonlinear systems which are transformed by Unscented
Transformation (UT).
2.1. Unscented transformation
The Unscented Transformation is a method of calculating the statistical information of random
variable and it is a nonlinear transformation. The basic idea is as follows: a probability distribution
fitting is easier than an arbitrary nonlinear transformation fitting or function fitting. The basic principle
of unscented transformation is illustrated in Figure 1.
Figure 1. The Principle of the Unscented Transform
1) The points are selected (Sigma points) of sample mean is x and covariance is Pxx .
2) The new points are acquired from the nonlinear transformation and the statistical information of
which is y and Pyy .
2.2. Minimum sample skewness strategy
Selection of sampling strategy (number of sigma points, position and weight) for sigma points is
important in the unscented transformation. Currently, the sampling strategy: symmetric sampling, the
minimum skewness simplex sampling and sampling simplex hypersphere [13, 14].
Currently, symmetric sampling was usually used. In the symmetry, sigma number of points for
L  2n  1 and sigma points were transformed by UT. It resulted in the large calculation and the
differential real-time. In high real-time systems the sigma points were asked to be further reduced to
reduce the computing load. According to the analysis of [15], for an n-dimensional distribution of the
state space, n  1 points at least needed to be determined. In the sampling strategy of the single form,
number of sigma points is the L  n  2 (the center point is considered) [13] and only n  2  sigma
points were transformed by UT. The computational complexity of the minimum skewness simplex
sampling is much less than the computational complexity of the symmetric sampling.
3-order moments (skewness) were required minimum in the minimum sampling before the first two
moments were matched. The sigma points were obtained as follows [13]:
(1) Weight of center point
(1)
0  W0  1
(2) Other Sigma weights
1  W0
,
i  1,2;

(2)
Wi   2 n
i 1
 2 W1 , i  3,, L.
(3) Initial iteration vector
748
An Improved UKF Algorithm based on Minimum Sample Skewness and the Ratio Correction Factor
Xiuhua ZHANG, Lianjun ZHANG, Yongcheng JIANG
  01  0, i  0





 11   1 

2W1 



 1  1 

 2  

 2W1 
For the dimension of variable are j  2,  , n , the iterative formula:






j
i  






  0j 1 

,
 0 








(3)
i0


,
2W j 1 

0 
1 ,
2W j 1 

 i j 1
i  1,  , j
1
(4)
i  j 1
Mean and covariance of x were added to the generated sigma points.
 x


Pxx 
(5)
Sampling Eq. (2) ~Eq. (5) show that distribution of sigma points were not subject to Centro
symmetric in the minimum sampling skewness simplex but were subject to axial symmetry. Sigma
points weight of the formation of low-dimensional augmented were larger than the high-dimensional
point. Corresponding to the distance from the center will also be longer. Non-cumulative local effects
were easy to be produced, result to the higher error of the higher order terms [16].
To reduce these effects, change in the proportion of UT was proposed by Julie that it was applied in
sampling process of the UKF algorithm. Non-local effect was solved by adjusting the value of
parameter  . The prediction covariance matrix semi-positive definite was ensured by introduced
parameter  and  . In the sampling strategy of symmetric it has been proved [17, 18], but in other
sampling strategy it had not been applied [14]. The transformation of UT ratio was applied to the
minimum sample skewness strategy in the paper. In the positioning system of the mobile robot the
problem of non-local effects not only had been solved, but also the real-time of positioning had been
improved.
3. An improved ukf algorithm
UKF algorithm of the minimum sample skewness strategy based on the ratio correction coefficient
is as follows.
The nonlinear model is considered as follows.
State equation:
(6)
xk  f k 1 xk 1   wk 1
Measurement equation:
z k  hk xk   vk
(7)
In the formula, xk  R n is system status, f k is vector function of n -dimension, hk is vector
function of m -dimension, wk is random process noise of n -dimension, vk is random measurement
noise of m -dimension. Before filtered the followings are supposed: process noise and measurement
noise are zero mean uncorrelated white noise, Qk is covariance of process noise wk . Rk is covariance
of measurement noise vk .
749
An Improved UKF Algorithm based on Minimum Sample Skewness and the Ratio Correction Factor
Xiuhua ZHANG, Lianjun ZHANG, Yongcheng JIANG
(1) Initialize parameters of UKF algorithm
x0 (Initial state) is independent with all the noise. The priori mean and covariance matrix are as
follows.
The priori mean matrix:
(8)
E x0   x0  xˆ0|0
The priori covariance matrix:


covx0   E x0  x x0  x   P0
(9)
(2) The sigma points of n  2 and their weights are computed.
Transform of ratio UT is applied to the minimum percentage of UT skewness simplex sampling
strategy. As follows:
T
 i'   0    i   0      P  i j
(10)
Above formula:

n

P   wi  i'    i'  

T
(11)
i 1
1 

 1  2 



1  W0
2n   2




Wi m   




W0
i0
2
i  1,2
2i 2  W1
(12)
i  3,4,  , n  1
2


m 
2
 W0  1    
Wi c   
m 
Wi
i0
(13)
i0
In the formula Wi is a weight of sampling point i in the minimum sample skewness simplex. Wi m 
is the weight of mean and Wi c  is the weight of covariance. All other symbols are the same meaning as
the symbols of proportional sampling strategy.
Parameter  is needed to be selected a very small proportional parameters to reduce the impact of
higher order terms. Parameter  should to satisfy the following properties: first choice of the
prediction of variance should be guaranteed; followed the second order accuracy of the mean and
variance should be ensured.
(3) Computation of time updates equation
(14)
yi'  f xi'
 
n
yi'   wi' yi'
(15)
i 0
n



  1   y
Py'   wi' yi'  yi' yi'  yi'

T
i 0

n
  wi' yi'  y yi'  y
T
 Qk
2
i 0
'
0

 y y0'  y
(4) Calculation of observation vector updates equation and filter gain
zi'  h yi'
 

T
 Qk
(16)
(17)
n
zi'   wi' zi'
(18)
i 0
n






Pz'   wi' zi'  zi' zi'  zi'
i 0
T
 Rk
 Py'  1   2 y0  yi' y0  yi'

T
(19)
750
An Improved UKF Algorithm based on Minimum Sample Skewness and the Ratio Correction Factor
Xiuhua ZHANG, Lianjun ZHANG, Yongcheng JIANG
n


Pyz'   wi' yi'  yi' zi'  zi'

T
(20)
i 0
K k  Pz' Pyz'
(21)
4. Simulated and analyzed to the system
In order to verify the application results of UKF algorithm based on the minimum degree of partial
sampling strategy simplex used the standard UKF algorithm and the UKF algorithm based on the
minimum degree of partial sampling strategy simplex to filter for the linear one-dimensional linear
uniform motion. Results were compared to the filtering.
1) Select the system model
Equation of state:
X K 1  X K  QK
(22)
Equation of measurement:
Z K 1  X K  RK
(23)
Parameter Wk is state noise and parameter Vk is measurement noise that mean subject to Gaussian
distribution. Parameter Qk is state covariance and parameter Rk is measurement covariance. System
state equation and measurement equation are nonlinear equations.
Initialize the state:
X 0  1;0;0.1 ;
(24)
Coefficients of UT transform:   0.01 ,   2 ;
Variables of state:
T
X  x1
x2
x3 
(25)
In above formula: position is parameter x1 , speed is parameter x2 and acceleration is parameter x3 .
Used two different non-linear filters to filter based on the initial parameters.
① Use the standard UKF algorithm.
② Use the UKF algorithm based on the minimum degree of partial sampling strategy simplex.
They are numbered UKF1 and UKF2.
Programmed and simulated in Windows XP system and MATLAB programming system.
2) Results and analysis
Simulated to the system and get the numerical simulation of the state variables in MATLAB system.
The results are showed as follows. The three parts are the results that the three state variables X are
simulated through 50 cycles.
Figure 2. The Filtering Results of UKF1 Algorithm
751
An Improved UKF Algorithm based on Minimum Sample Skewness and the Ratio Correction Factor
Xiuhua ZHANG, Lianjun ZHANG, Yongcheng JIANG
Figure 3. The Filtering Results of UKF2 Algorithm
(a)
(b)
(c)
Figure 4. Comparison Results
As above shown, use the samplings to calculate the 50-steps in the simulation. The results of
calculation are showed in the three maps by the method of two filtering (the standard UKF algorithm
and the correction coefficient based on the minimum sampling strategy skewness UKF algorithm).
Table 1. Comparison of the running time of algorithm
algorithm
UKF1
The running time (s)
1.36
UKF2
1.08
752
An Improved UKF Algorithm based on Minimum Sample Skewness and the Ratio Correction Factor
Xiuhua ZHANG, Lianjun ZHANG, Yongcheng JIANG
Figure 2, 3 and 4 show that the accuracy by UKF1 algorithm was more accurate than UKF2
algorithm in the filtering accuracy. Table 1 shows that the running time save the 21% times by UKF1
algorithm than UKF2 algorithm.
5. Conclusions
UKF algorithm based on the minimum degree of partial sampling strategy simplex is proposed to
meet the requirements of high real-time in robot autonomous positioning and navigation system. The
results of simulation show that the improved UKF algorithm saves the time 21% than the standard
UKF algorithm in the running time, so it can meet the requirements of real-time positioning and
navigation. The simulation results show that the method is feasibility. For the system of autonomous
positioning and navigation provide a theoretical basis.
6. Acknowledgements
This research was funded by grants of National High Technology Research and Development
Program (863 Program) (2007aa04Z255) and National Natural Science Foundation of China
(61002004).The authors gratefully acknowledge these supports.
7. References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
R.E. Kalman “A new approach to linear filtering and Prediction Problem”, Transactions of the
ASME-Journal of Basicengineering, vol. 2, no. 82, pp. 35-45, 1960.
R.S. Bucy, K.D. Senne, “Digital synthesis of nonlinear filters”, Defense Technical Information
Center. vol. 7, no. 10, 1970.
S.J.Julier, J.K.Uhlmann, H.F.Duxrant-Whyte, “A New Method for Nonlinear Transformation of
Means and Covariances in Filters and Estimators”, IEEE Transactions Automatic Control, vol.
45, no. 3, pp. 477-482, 2000.
YU Guoqiang, Yang Jianye, Zhang Hexin, “A Comparison of Several Widely Used Nonlinear
Filtering Approaches”, Electron Optics & Control. vol. 16, no. 12, pp. 48-52, 2009.
MU Jing, CAI Yuan-li, “Iterated cubature Kalman filter and its application”, Systems
Engineering and Electronics. vol. 33, no. 7, pp. 1454-1457, 2011.
YANG Wen-bo, LI Shao-yuan, “Autonomous navigation filtering algorithm for spacecraft based
on strong tracking UKF”, Systems Engineering and Electronics.vol. 33, no. 11, pp. 2485-2491,
2011.
DUAN Xiao-ju, XUE Xiao-zhong, “Comparison between Extended and Unscented Kalman
Filtering Applied to Ultra-tight GPS/INS Integration”, Fire Control & Command Control, vol.
35, no. 6, pp. 61-63, 2010.
Julier S J, Uhlmann J K.Durrant-Whyte H F, “A new approach for filtering nonlinear systems”,
In Proceeding (s) of American Control Conference, pp. 1628-1632, 1995.
Li Dan, Liu Jian-ye, “Root Unscented Kalman Filter for Satellite Autonomous Navigation
System Based on Minimal Skew Simplex Transformation”, Journal of Nanjing University of
Aeronautics & Astronautis, vol. 41, no. 1, pp. 54-58, 2009.
Julier S J, “The spherical simplex unscented transformation”, In Proceeding (s) of American
Control Conference on Denver Colorado, no. 3, pp. 2430-2434, 2003.
Andersen M N, Wheeler K, “Filtering in hybrid dynamic Bayesian networks”, In Proceeding (s)
of IEEE International Conference on Acoustics, pp. 773-776, 2004.
PAN Ping-jun, FENG Xin-xi, “An algorithm of maneuvering target tracking with IRSTS based
on infrared radiation information of double bands”, Journal of Naval University of Engineering,
vol. 20, no. 3, 2008.
Julier S J, Uhlmann J K, “Reduced sigma point filters for the propagation of means and
covariance through nonlinear transformations”, In Proceeding (s) of the American Control
Conference on Anchorage Alaska, no. 2, pp. 887-892, 2002.
PAN Quan, YANG Feng, YE Liang, “Survey of a kind of nonlinear filters-UKF”, Control and
Decision, vol. 20, no. 5, pp. 481-489, 2002.
753
An Improved UKF Algorithm based on Minimum Sample Skewness and the Ratio Correction Factor
Xiuhua ZHANG, Lianjun ZHANG, Yongcheng JIANG
[15] QIN Yong-yuan, ZHANG Hong-yue, “Kalman filter and Navigation Principles”, Northwestern
Polytechnical University Press, no. 3, 2010.
[16] Simon J.Julier, “The Scaled Unscented Transformation”, In proceeding (s) of the American
Control Conference on Anchorage AK, vol. 8-10, no. 5, pp. 4555-4559, 2002.
[17] Liu Xue-peng, ZHANG You-qun, BAO Kuo, “Improved UKF Algorithm for Pseudolite
Positioning System”, Journal of Geomatics Science and Technology, vol. 25, no. 2, pp. 108-111,
2008.
[18] DIAO Peng, XIE Nie, WU Xun-zhong, “The Low cost MINS/GPS Integrated Navigation
System’s Design and Application”, Chinese Journal of Sensors and Actuators, vol. 22, no. 9, pp.
1366-1370, 2009.
[19] Joseph Gallart Suarez, Manuel Tupia Anticona, "Two GRASP Metaheuristic for the Capacitated
Vehicle Routing Problem Considering Split Delivery and Solving the Three Dimensional Bin
Packing Problem", AISS: Advances in Information Sciences and Service Sciences, vol. 2, no. 2,
pp. 42-50, 2010
[20] Saleh Alqatan, Dalbir Singh, Kamsuriah Ahmad, "A Theoretic Discussion of Tourism Mcommerce", JCIT: Journal of Convergence Information Technology, Vol. 6, No. 12, pp. 100 ~
106, 2011
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