Application of interacting multiple model in integrated positioning system of vehicle

WEI Wen-jun， GAO Xue-ze， GE Li-min， GAO Zhong-jun

(1. School of Automaton and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China；2. School of Mechatronics Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China；3. China Academy of Railway Sciences, Beijing 100081, China)

Abstract： To solve low precision and poor stability of the extended Kalman filter (EKF) in the vehicle integrated positioning system owing to acceleration, deceleration and turning (hereinafter referred to as maneuvering), the paper presents an adaptive filter algorithm that combines interacting multiple model (IMM) and non-linear Kalman filter. The algorithm describes the motion mode of vehicle by using three state space models. At first, the parallel filter of each model is realized by using multiple nonlinear filters. Then the weight integration of filtering result is carried out by using the model matching likelihood function so as to get the system positioning information. The method has advantages of nonlinear system filter and overcomes disadvantages of single model of filtering algorithm that has poor effects on positioning the maneuvering target. At last, the paper uses IMM and EKF methods to simulate the global positioning system (GPS)/inertial navigation system (INS)/dead reckoning (DR) integrated positioning system, respectively. The results indicate that the IMM algorithm is obviously superior to EKF filter used in the integrated positioning system at present. Moreover, it can greatly enhance the stability and positioning precision of integrated positioning system.

Key words： vehicle; integrated positioning system; information fusion algorithm; extended Kalman filter (KEF); interacting multiple model (IMM)

0　Introduction

Reliable and high-precision position information guarantees the realization of vehicle-assisted driving and automatic parking, and technically relieves the traffic jam at the same time. To provide reliable and high-precision positioning information, the positioning system has developed from a single positioning mode to an integrated positioning system[1]. It has optimized various separate positioning systems and formed an integrated system with complementary performance so as to enhance the precision and reliability of the overall positioning system. In the process, the information integration algorithm plays a vital role in the final positioning precision of integrated positioning system. Ref.[2] proposed a WiFi positioning scheme based on a triangular algorithm. Moreover, it adjusted the weight coefficient of the positioning result by means of global positioning system (GPS) signal. Complicated and quickly changed vehicle motion environment makes the real-time weight adjustment become very difficult. Moreover, a simple weight combination is difficult to get optimal global estimation. Ref.[3] taked the federation Kalman algorithm as a information integration method in the WiFi/GPS integrated positioning system. Ref.[4] proposed a particle swarm optimization method combined with extended Kalman algorithm in the wireless sensor network positioning environment. The data integration algorithms used above have enhanced the positioning precision of vehicle to some extent. However, they do not have ideal effects on the positioning output of vehicle at maneuvering stage because the design of filter is usually based on a fixed model system and supposed as a constant parameter in the motion process of vehicle for the purpose of simplifying the problem analysis.

After multi-model estimation theory was born in 1970s, American scholar proposed an interacting multiple model (IMM) algorithm. The algorithm is widely used for tracking the high-speed maneuvering target[5-7]. To sum up, we propose an integrated GPS/inertial navigation system (INS)/dead reckoning (DR) vehicle positioning system based on the IMM Kalman filter. Against the integrated navigation system, we construct a multi-mode adaptive filter as well as a maneuvering model and a non-maneuvering model of vehicle. Moreover, we adopt different filtering algorithms against different models so as to enhance the positioning precision and stability of the system in the maneuvering operation of vehicle.

1　System structure

Based on the GPS/INS/DR integrated positioning system, by synthesizing the advantages of these three single positioning modes, the performance becomes complementary and thus the reliable operation of the positioning system is improved greatly. However, the integral action of gyro devices makes the INS error accumulate constantly as time goes[8]and the error of DR system decrease[9]. Therefore, in this paper, these two methods are synthesized and the attitude matrix of INS is introduced to the state computation matrix of DR so as to inhibit the error diffusion of INS by means of DR.

The combined positioning system is composed of INS/DR data integration treatment module, and the INS/DR positioning data and GPS positioning data integration treatment module. The system structure diagram is shown in Fig.1.

Fig.1　Integrated positioning system

Odometer, triaxial accelerometer and gyro compose the INS and DR system. The output positioning information and GPS positioning data are processed by multi-model adaptive filter to realize the valid combination of three single positioning modes and further track the vehicle in a maneuvering state and non-maneuvering state precisely.

2　Design of multi-model adaptive filter

2.1　IMM introduction

IMM adaptive filter adopts multiple Kalman filters for parallel processing[10]. Each filter corresponds to a space model in a specific motion state. Different state space models describe different motion modes of a vehicle and each filter estimates the current position of the system independently according to its own model and input quantity. In this way, each filter makes different estimations against the target state. Then we compute the predicated value of the measured value according to the estimated value. After comparing it with the actually measured value, we get the residual and take it as the approximation degree indication of the filter model and actual system model. The smaller the residual is, the more it indicates that the filter model matches with the actual system model. The structural diagram of IMM adaptive filter is shown in Fig.2.

Fig.2　Structure of IMM adaptive filter

2.2　Multi-model integration estimation algorithm

The application process of IMM algorithm in the integrated positioning of vehicle is as follows.

Firstly, we set uprmeasuring noises, namely white noise model set with zero-mean Gaussian distribution and different sizes of covariance matrices. The covariance matrix of model isRj,j=1, …,r. Moreover, we set upR1as the measuring noise covariance matrix of the integrated navigation in the normal state. Then we make one model valid at current moment and thus gain the initial condition of filter matching by using such a specific model through the state estimation value of all the filters at the moment before mixing. After that, we suppose the vehicle hasrkinds of motion states, corresponding tormotion models, namelyrstate transition equations. We weight the integrated estimation of each gained model using model probability and finally get the target estimation：

Suppose the state equation of thejth model at the timek+1 is

Xj(k+1)=Φj(k)Xj(k)+Gj(k)Wj(k),

(1)

and measuring equation is

Z(k)=H(k)X(k)+V(k),

(2)

whereWj(k) is the white noise sequence with mean 0 and covarianceQj, and the value ofQjwill be given in Section 3 based on the settings of the special model;Gj(k) is the noise drive matrix;H(k) is the measurement matrix;V(k) is observation noise. The inter-model transition is determined by Markov probability transfer matrix. Supposing elementPijexpresses the probability of the vehicle transferring from modelito modelj, the transfer matrix is

(3)

IMM algorithm is carried out in the recursion form. Each recursion is mainly composed of input interacting model, model filter, model probability update and output interacting model[11-12].

Step 1: input interacting (modelj)

(4)

P0j(k-1|k-1)=

(5)

whereμij(k-1) is the mixing probability from modelito modelj. The computation formula is

(6)

Step 2: model filter

Step 3: Model probability updating

The computation of model probability update is realized by adopting likelihood function. Suppose the likelihood function of modeljis

(7)

Probability of modeljis

(8)

Step 4: Output interacting model

(9)

The estimation of total variance is

(10)

3　Establishment of system model

When the system model changes, the routine Kalman filtering algorithm is difficult to adapt which affects the precision and stability of filter. For the vehicle positioning system adopting the GPS/INS/DR integrated positioning, such phenomenon is mainly reflected in the maneuvering of vehicle. However, the basic Kalman filtering algorithm cannot get the ideal result. Thus we establish multi-model set of the system and adopt different filtering algorithms to adapt to different motions and enhance the positioning precision and stability of filtering.

The fine model can enhance the positioning precision of the model to some extent, but it may bring about dramatic increasing of computational complexity. In addition, it may cause model competition[13].

To simplify the computation, the paper divides the motion state of a vehicle into the three following models: constant-velocity linear motion (CV), constant-acceleration model (CA) and constant-velocity turning model (CT)[14]. Hereinto, CV is a non-maneuvering model, supposing the disturbance variance of its system isQ1=0; CA and CT are maneuvering models, supposing the system disturbance variance of these two models areQ2=0.001I2×2andQ3=0.014 4I2×2, andI2×2is a second-order unit.

Supposing the vehicle moves in a two-dimensional plane, the state equation is

X(k+1|k)=Φj(k)X(k)+G(k)W(k),

(11)

The state transfer matrix and noise drive matrix are expressed respectively by

(12)

(13)

(14)

whereωis the angular velocity when the vehicle turns, andTis the adopted period. Observer obtains error value in the planar coordinate of integrated navigation. Under the above three modes, the observation equation of system is

Z(k)=H(k)X(k)+V(k),

(15)

where

(16)

4　Simulation and analysis

To check the tracking precision and stability of the IMM Kalman filtering algorithm to maneuvering the target, we carry out simulation experiments in the two scenarios using EKF filtering method. For clarification, we set left turn as positive and right turn as negative in the motion process of vehicle.

Scenario 1: Supposing the target moves in a two-dimensional plane in the form of weak maneuvering, the initial value of the state variable isX=[1 000,25,1 000,20,0,0]T, the observed value is the position information of target, observation time interval isT=1 s, and total simulation time is 100 s. System noise and observation noise are mutually independent Gaussian white noise, which meet the noise requirements of Kalman filter. Supposing the target moves at a constant acceleration of 0.75 m/s2within 10-20 s, turns at a constant velocity of 3 °/s within 20-40 s, and turns at a constant velocity of -3 °/s within 60-80 s. The positioning algorithm carries out the integration computation for CV, CA and CT, supposing the initial probability of CV, CA and CT are 0.8, 0.1 and 0.1, respectively; and the initial values of transfer probability matrix of Markov chain transferred by the control model are

After Monte Carlo simulation for 100 times, we compare the IMM Kalman algorithm and the EKF Kalman algorithm to evaluate the performance of algorithm using position and its root-mean square error (RMSE). The simulation results are shown in Figs.3 and 4.

Fig.3　Position tracking error contrast in scenario 1

Fig.4　Position tracking RMSE contrast in scenario 1

The comparison of mean and variance of RMSE in Scenario 1 with two algorithms is shown in Table 1.

Table 1　Contrast values of RMSE in scenario 1

According to Fig.3, it can be seen that the error of these two integration algorithms is small at CV stage within 0-10 s, the errors of EKF algorithm inXdirection andYdirection increase obviously at CA stage, and the errors inXdirection andYdirection using EKF algorithm increase further at CT stage. After returning to the CV motion, the filter error of EKF decreases gradually. For IMM adaptive Kalman filtering algorithm, the large error mainly focuses on the moment before and after the model transfering, and the error is obviously smaller than that of EKF algorithm. Thus, the positioning precision of IMM Kalman filtering algorithm in the weak maneuvering process is obviously superior to that of the EKF algorithm. Table 1 presents the comparison of RMSE mean and variance of these two algorithms. Hereinto, the mean item further proves such conclusion in the numerical perspective. RMSE variance is commonly used as a criterion of measuring stability[15]. From the variance items in Fig.4 and Table 1, it can be seen that compared with EKF filtering algorithm, IMM Kalman filtering algorithm has better stability.

Scenario 2: Supposing the target moves in a two-dimensional plane in the form of strong maneuvering, the simulation duration is 100 s, the target turns at -5 °/s within 20-40 s, and at 5 °/s within 60-80 s. Other conditions are the same as that in scenario 1. Simulation results are shown in Figs.5 and 6.

Fig.5　Position tracking error contrast in scenario 2

Fig.6　Position tracking RMSE contrast in scenario 2

The mean and variance comparison of RMSE with two algorithms in this scenario is shown in Table 2.

Table 2　Contrast values of RMSE in scene II

According to Fig.5, it can be seen that for EKF algorithm, the positioning errors at two sharp CT stages inXdirection andYdirection increase greatly than that in scenario 1, and the errors of IMM algorithm before and after the moment of the model transfering increase dramatically, but it is still superior to that of EKF algorithm. After the completion of turning, the error in EKF algorithm decreases slowly, while the error in IMM algorithm enters and maintains a low range rapidly after the end of the model transfer. Mean item in Table 2 further proves the conclusion in the numerical perspective. The variance item in Fig.6 and Table 2 also indicates that the stability of the IMM Kalman filtering is superior to that of the EKF algorithm in the strong maneuvering motion state.

To sum up, when IMM Kalman filtering algorithm is taken as the data integration algorithm of the integrated positioning system, it can enhance the positioning precision and stability of the system to the maneuvering target no matter in a weak or strong maneuvering state than EKF algorithm.

5　Conclusions

1) According to the problem existing in the integrated positioning system of vehicle with EKF algorithm for data integration, the paper proposes an interactive multi-model Kalman filtering algorithm, as which is taken as a new way in integration algorithm of positioning system. This enhances the positioning precision and stability of a vehicle at the maneuvering stage.

2) The simulation experiment indicates, compared with EKF algorithm, such algorithm enhances the positioning precision and stability of a vehicle at the maneuvering stage.

3) The above research indicates that the algorithm proposed in the paper can be deemed as a new method in vehicle integrated positioning system data integration algorithm of a vehicle. With the constant development of an unmanned vehicle technology, the integrated positioning based on the interactive multi-model filtering algorithm will reflect preferable performance.