# Maneuvering Acceleration Estimation Algorithm Using Doppler Radar Measurement

Authors: Hongqiang Liu; Zhongliang Zhou; Lei Yu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

An algorithm to estimate the tangential and normal accelerations directly using the Doppler radar measurement in an online closed loop form is proposed. Specific works are as follows: first, the tangential acceleration and normal acceleration are taken as the state variables to establish a linear state transition equation; secondly, the decorrelation unbiased conversion measurement Kalman filter (DUCMKF) algorithm is proposed to deal with the strongly nonlinear measurement equation; thirdly, the geometric relationship between the range rate and the velocity direction angle is used to obtain two estimators of the velocity direction angle; finally, the interactive multiple model (IMM) algorithm is used to fuse the estimators of the velocity direction angle and then the adaptive IMM of current statistical model based DUCMKF (AIMM-CS-DUCMKF) is proposed. The simulation experiment results show that the accuracy and stability of DUCMKF are better than the sequential extended Kalman filter algorithm, the sequential unscented Kalman filter algorithm, and converted measurement Kalman filter algorithms; on the other hand they show that the AIMM-CS-DUCMKF can obtain the high accuracy of the tangential and normal accelerations estimation algorithm.


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## Body

## 1. Introduction

In real target motion, the tangential acceleration controls the target velocity size, and the normal acceleration controls the target velocity direction. Compared with the Acceleration Components on the Coordinate Axis, the tangential and normal accelerations reflect the maneuvering state of the target more directly, which are collectively called the Maneuvering Accelerations (MAs). In the traditional target tracking problem, the MAs cannot be estimated directly, but are calculated by the Acceleration Components on the Coordinate Axis (ACAs). These methods can be divided into two types [1]. The first is to take the ACAs as the random process [2–6]. For example, the ACAs are treated as the first-order Markov process with zero mean in the Singer model [2] and as the state variables to participate in the filtering and estimated at each step of the iteration. The second is that the ACAs are taken as the unknown deterministic input vector, and the input estimation technique, which is relatively independent of the filtering process, is used to estimate it [7–11]. However, the estimation of MAs by the calculation of the ACAs is an open-loop method, and the estimation error cannot be corrected in the filter iteration. And the above methods only use the position measurement of the target. For the Doppler radar, the target range rate measurement can be provided. Because the range rate is a function of the target velocity size and the velocity direction, the range rate contains the target maneuvering state information. The range rate is used not only for the estimation of normal acceleration [12–14] and angle velocity [15, 16], but also as the measurement information for target tracking [13, 17–19]. Because the radar cannot detect the acceleration of target, it is necessary to directly estimate the target acceleration based on the measurement.The purpose of this paper is to develop a method of directly estimating the MAs of maneuvering target by Doppler radar measurements. This is more practical and meaningful than estimating the ACAs. In the early literature [20], a direct estimation method of the MAs based on the current statistical model is proposed. Then the paper also takes the MAs as the modified Rayleigh-Markov random process and directly as the state variables in each iteration process. Unlike the nonlinear state space equation established in the literature [20], a linear state space equation is established as a maneuvering target model. Since the performance of the linear state space equation is affected by the accuracy of the velocity direction angle, a fusion framework based on IMM algorithm is applied to improve the accuracy of velocity direction angle. Because of the high nonlinear relationship between Doppler radar measurements and the state variables, we extend the decorrelation unbiased conversion measurement Kalman filter (DUCMKF) algorithm only with the position measurement to the range rate measurement. Finally, combined with the above contents, the adaptive IMM of current statistical model based decorrelation unbiased conversion measurement Kalman filter (AIMM-CS-DUCMKF) algorithm is proposed to estimate the MAs directly in an online closed loop and to improve the tracking performance of the maneuvering target.This paper is organized as follows. The target state equation is established in Section2. The DUCMKF algorithm is proposed in Section 3. The AIMM-CS-DUCMKF algorithm is described in Section 4. The simulation experiments are carried out and the results are shown and analyzed in Section 5. Conclusions are drawn in Section 6. The results of the discrete state equation are given in Appendix.


## 2. State Space Equation

In order to estimate the MAs of the target directly in an online closed loop, the state vectorX is denoted as(1)X=x,y,x˙,y˙,at,anTwhere x and y are the positional components of the Cartesian coordinate system; x˙ and y˙ are the velocity components of the Cartesian coordinate system; at is the tangential acceleration; an is the normal acceleration. Figure 1 shows the relationship between variables in the polar coordinate and Cartesian coordinate system. r is the range between the radar and the target, θ is the target azimuth, α is the target velocity direction, v is the target velocity size, and r˙ is the target range rate. In the two-dimensional plane, it is stated that when the velocity direction of the target changes counterclockwise, the normal acceleration is positive (+); otherwise it is negative (-). When the velocity size increases, the tangential acceleration is positive (+); otherwise it is negative (-).Figure 1
The relationship between variables.The relationship betweenat, an and x¨, y¨ on the coordinate axis X, Y is as follows:(2)x¨y¨=atcos⁡α-ansin⁡αatsin⁡α+ancos⁡αThe MAs are modeled as the modified Rayleigh-Markov random process[] and meet the following state models:(3)a˙t=-αtat+αta-t+ωtt(4)a˙n=-αnan+αna-n+ωntwhere αta-t and αna-n are the mean values of noises ωtt and ωnt, respectively; αt and αn are the reciprocal of the maneuvering time constant for the tangential acceleration and normal acceleration, respectively; a-t=Eat; a-n=Ean. By the properties of Rayleigh distribution [20], when a-t and a-n are known, the variance σωt2 of ωtt and σωn2 of ωnt can be obtained by(5)σωn2=4-ππan0-a-n2an0≥a-n≥0σωn2=4-ππa-n0-a-n20>a-n≥a-n0σωt2=4-ππat0-a-t2at0≥a-t≥0σωt2=4-ππa-t0-a-t20>a-t≥a-t0where an0 and a-n0 are the positive and negative limit of normal acceleration; at0 and a-t0 are the positive and negative limit of tangential acceleration.The system state space equation is denoted as(6)X˙t=FtXt+Ut+Wtwhere the input of state equation is(7)Ut=0,0,0,0,αta-t,αna-n,TThe noise vector is(8)Wt=0,0,0,0,ωtt,ωntTGiven α, Ft is linear:(9)Ft=0010000001000000cos⁡α-sin⁡α0000sin⁡αcos⁡α0000-αt000000-αnWhen the sampling period is denoted asT, the differential equation of (6) is obtained by discretization:(10)Xk+1=ΦTXk+Uk+Wkwhere ΦT is the state transition matrix of target; Uk and Wk are the discrete forms of Ut and Wt; the covariance matrix of Wk is denoted as Qk. The specific discrete expressions of ΦT, Uk, Wk, and Qk are given in Appendix.


## 3. Decorrelation Unbiased Conversion Measurement

Suppose that the Doppler radar is located at the origin of the coordinate systemXOY. In the corresponding radar polar coordinate system, the nonlinear measurement equation including range, azimuth, and range rate is as follows:(11)Zm=rmθmr˙m=x2+y2arctan⁡yxxx˙+yy˙x2+y2+υwhere Zm is the Doppler radar measurement vector; rm is the range measurement; θm is the azimuth measurement; r˙m is the range rate measurement; υ is the Gauss noise and has the covariance matrix:(12)R=σr20ρσrσr˙0σθ20ρσrσr˙0σr˙2where σr, σθ, and σr˙ are the standard deviations of range, azimuth, and range rate measurements; ρ is the correlation coefficient between the range measurement noise and the range rate measurement noise [21]. A decorrelation unbiased conversion measurement is proposed in the literature [22]. The algorithm uses the unbiased transformation to eliminate the measurement transformation error and one step prediction state to replace the present measurement to reduce the correlation between the measurement noise and the measurement error covariance. In this paper, the algorithm is extended to the measurement equation including the range rate measurement. When the azimuth measurement error is small, the nonlinear measurement equation can be converted into a linear measurement equation [13].(13)Xmk=HkXk+υkwhere Xm=xm,ym,r˙m′ is the converted measurement vector; υ=x~,y~,r˙~′ is the converted measurement noise vector; x~, y~, and r˙~ are the random variables; H is the linear measurement matrix: (14)H=1000010000cos⁡θsin⁡θxm, ym, and r˙m can be obtained by(15)xm=λθ-1rmcos⁡θm,ym=λθ-1rmsin⁡θm,r˙m=r˙mwhere λθ=e-σθm2/2. r˙m can be detected directly by radar, so it does not need to be converted. The decorrelation covariance matrix is denoted as RDUCM at time k+1:(16)RDUCM=RDUCMxxRDUCMxyRDUCMxr˙RDUCMyxRDUCMyyRDUCMyr˙RDUCMr˙xRDUCMr˙yRDUCMr˙r˙(17)RDUCMxx=12r2k+1∣k+σr2+σr2k+11+cos⁡2θk+1∣ke-2σθ2e-2σθ2k+1eσθ2-12r2k+1∣k+σr2k+11+cos2θk+1∣ke-2σθ2k+1RDUCMyy=12r2k+1∣k+σr2+σr2k+11-cos⁡2θk+1∣ke-2σθ2e-2σθ2k+1eσθ2-12r2k+1∣k+σr2k+11-cos⁡2θk+1∣ke-2σθ2k+1RDUCMxy=RDUCMyx=12r2k+1∣k+σr2+σr2k+1sin⁡2θk+1∣ke-2σθ2e-2σθ2k+1eσθ2-12r2k+1∣k+σr2k+1sin⁡2θk+1∣ke-2σθ2k+1Rmxr˙=Rmr˙x=e-σr2k+1/2ρσrσr˙cos⁡θk+1∣kRDUCMyr˙=RDUCMr˙y=e-σr2k+1/2ρσrσr˙sin⁡θk+1∣kRDUCMr˙r˙=σr˙2Suppose that the predicted value of the state vector at timek+1 is(18)Xk+1∣k=xk+1∣k,yk+1∣k,x˙k+1∣k,y˙k+1∣k′The predicted position covariance matrix is denoted asPpk+1∣k:(19)Ppk+1∣k=P11k+1∣kP12k+1∣kP21k+1∣kP22k+1∣kThe predictive measurementrk+1∣k and θk+1∣k are(20)rk+1∣k=x2k+1∣k+y2k+1∣k,θk+1∣k=arctan⁡yk+1∣kxk+1∣kThe corresponding varianceσr2k+1 and σθ2k+1 can be obtained by(21)σr2k+1=1r2k+1∣kxk+1∣kyk+1∣kPpk+1∣kxk+1∣kyk+1∣k′(22)σθ2k+1=1r4k+1∣k-yk+1∣kxk+1∣kPpk+1∣k-yk+1∣kxk+1∣k′


## 4. AIMM-CS-DUCMKF Algorithm

The state transition matrixΦT is a function of the target velocity direction angle α. But the true value of α is unknown. Then α need to be estimated with other variables. The estimator of α at time k can be denoted as αk∣k∈-π,π. With the Doppler radar measurement, αk∣k can be calculated through two methods. The first is (23)α1k∣k=arctan⁡y˙k∣kx˙k∣kwhere x˙k∣k and y˙k∣k are the estimators of velocity components at time k. The range rate r˙ is the function of v and α; that is,(24)r˙=vcos⁡α-θThen the second method is (25)α2k∣k=arccos⁡r˙mvk∣k+θk∣korα3k∣k=-arccos⁡r˙mvk∣k+θk∣kwhere vk∣k=x˙2k∣k+y˙2k∣k and θk∣k=arctan⁡yk∣k/xk∣k. Generally, after the filtering, θk∣k has the higher accuracy than θm. By formula (23) and formula (25), α has three estimates, denoted as αik∣ki=13. The most basic idea is to select the most accurate estimate into ΦT to obtain the most accurate model of maneuvering target. However, as the formula (23) and formula (25) are nonlinear functions, it is impossible to give a ranking of the three estimators in a single iteration. Then, an adaptive fusion and estimation framework based on IMM algorithm is developed. Firstly, the state space equation of corresponding transformation matrix ΦiT,αik∣ki=13 is established, which is denoted as Mii=13. Then Mii=13 is used as the model set and DUCMKF algorithm is used as the filter. The AIMM-CS-DUCMKF algorithm is proposed and its flow chart is shown in Figure 2. It can not only realize the maneuvering target tracking, but also directly estimate the MAs in an online close loop form. The specific steps of the AIMM-CS-DUCMKF algorithm are given below.Figure 2
Structure diagram of AIMM-CS-DUCMKF algorithm.(  1) The Model Reinitialization Conditioned. The reinitialization mixed estimates of the state and covariance matrix of the jth filter are(26)Xojk-1∣k-1=∑i=1MXik-1∣k-1μi,jk-1∣k-1(27)Pojk-1∣k-1=∑i=1MPik-1∣k-1+Xik-1∣k-1-Xojk-1∣k-1·Xik-1∣k-1-Xojk-1∣k-1′μk-1∣k-1i,jwhere Xojk-1∣k-1 is the reinitialization mixed state estimates of the jth filter; Pojk-1∣k-1 is the mixed covariance matrix of Xojk-1∣k-1; Xik-1∣k-1 is the state estimator of the ith filter at time k-1; Pik-1∣k-1 is the covariance matrix of Xik-1∣k-1; M is the number of models, that is, M=3. μk-1∣k-1i,j is the mixed probability:(28)μk-1∣k-1i,j=1C-jpijμk-1i,C-j=∑i=1Mpijμk-1iwhere pij is the transformation probability of the ith model to the jth model; μk-1i is the probability of the ith model. In order to realize the adaptive acceleration mean, let(29)a-tojk-1=atojk-1∣k-1(30)a-nojk-1=anojk-1∣k-1Substitutea-tojk-1 and a-nojk-1 into the formulas (A.1)–(A.3)~(A.1)–(A.8), the input vector Uojk-1 of the jth filter can be obtained. In order to achieve the adaptive acceleration variance, let(31)σωnoj2k-1=4-ππ·an0-a-nojk-12an0≥a-nojk-1≥0σωnoj2k-1=4-ππ·a-n0-a-nojk-120>a-nojk-1≥a-n0(32)σωtoj2k-1=4-ππ·at0-a-tojk-12at0≥a-tojk-1≥0σωtoj2k-1=4-ππ·a-t0-a-tojk-120>a-tojk-1≥a-t0Substituteσωnoj2k-1 and σωtoj2k-1 into the formulas (A.1)–(A.9)~(A.1)–(A.29), the process noise covariance matrix Qojk-1 of the jth filter can be obtained. Through Xojk-1∣k-1j=13, three estimators of target velocity direction are calculated, respectively,(33)αj=1k-1∣k-1=arctan⁡y˙oj=1k-1∣k-1x˙oj=1k-1∣k-1αj=2k-1∣k-1=arccos⁡r˙mk-1voj=2k-1∣k-1+θoj=2k-1∣k-1αj=3k-1∣k-1=-arccos⁡r˙mk-1voj=3k-1∣k-1+θoj=3k-1∣k-1Substituteαjk-1∣k-1 into formula (A.1)–(A.3); the state transformation matrix ΦjT,αjk-1∣k-1 of the jth filter can be obtained.(  2) The Model Filtering Conditioned. The DUCMKF algorithm is used to acquire Xjk∣k,Pjk∣kj=1M given the initial condition Xojk-1∣k-1,Pojk-1∣k-1j=1M from step 1. The steps of DUCMKF are as follows:(34)Xjk∣k-1=ΦjT,αjk-1∣k-1Xojk-1∣k-1+Ujk-1(35)Pjk∣k-1=ΦjT,αjk-1∣k-1Pojk-1∣k-1ΦjT,αjk-1∣k-1+Qjk-1(36)Zmk,Rk⟹DUCMXmjk,RDUCMjk(37)Sjk=HkPjk∣k-1H′k+RDUCMjk(38)Kjk=Pjk∣k-1H′kSjk-1(39)Z~jk=Xmjk-HkXjk∣k-1(40)Xjk∣k=Xjk∣k-1+KjkZ~jk(41)Pjk∣k=Ι-KjkHkPjk∣k-1(  3) The Model Probability Updating. Assume that measurement prediction residual Z~jk obeys the Gauss distribution with zero mean value. Then the likelihood function Λkj of the model Mj is(42)Λkj=2πSjk-1/2exp⁡-12Z~jk′Sjk-1Z~jk(43)μkj=1CC-jlog⁡Λkj,C=∑i=1MC-ilog⁡Λki,C-i=∑j=1Mpjiμk-1j(  4) The Estimation Fusion. The overall state estimation and error covariance matrix at time k are given:(44)Xk∣k=∑j=1MXk∣kμkj(45)Pk∣k=∑j=1MPjk∣k+Xk∣k-Xjk∣kXk∣k-Xjk∣k′μkj


## 5. Simulation Experiment

### 5.1. The Generation of True Trajectory and Measurement

The simulation experiment is divided into two parts. The first part is to verify the tracking performance of DUCMKF algorithm proposed with the Doppler radar measurement and compare the DUCMKF algorithm with the sequential EKF (SEKF) [18], sequential UKF (SUKF) [17], and converted measurement KF (CMKF) algorithms [13] in the previous literatures. The second part is to verify the maneuvering target tracking performance and the MAs estimation performance of AIMM-CS-DUCMKF algorithm and compare it with the IMM algorithm, current statistical model based adaptive EKF (CS-AEKF) [20], and window-based least squares estimator (WLSE) [7].In the simulation experiment, the target real trajectory and the corresponding Doppler radar measurement are needed firstly. A standard plane curve motion model is adopted to obtain the true trajectory of the target:(46)x˙=vcos⁡α+wxy˙=vsin⁡α+wyv˙=at+wvα˙=anv+wαwhere wx, wy, wv, and wα are the independent of Gaussian white noises, respectively, and their standard deviation are σwx, σwy, σwv, and σwα. Through initializing x, y, v, and α, the true trajectory can be obtained by Runge Kutta method [23] given an and at. The Doppler radar measurement (r,θ,r˙) can be generated with the model as formula (47):(47)rθr˙=x2+y2±arccos⁡xx2+y2vcos⁡α∓arccos⁡xx2+y2+wrwθwr˙Ify≥0, “±” is “+”; if y<0, “±” is “−”. And “∓” is on the contrary. wr, wθ, and wvr are the Gaussian white noise and their covariance is(48)R=σwr20ρ′σwrσwr˙0σwθ20ρ′σwr2σwr˙20σwr˙2where σwr, σwθ, and σwr˙ are the standard deviation; ρ′ is the correlation coefficient. After inputting a real trajectory into model (47), the corresponding measurement sequence can be obtained.


### 5.2. The Performance of the DUCMKF Algorithm

The measurement equation of the CMKF algorithm is the same as that of the DUCMKF algorithm, and the difference is in the way of measurement transformation. First, the CMKF algorithm does not consider the correlation between range measurement and range rate measurement. Secondly, the converted measurement covariance matrix needs to be calculated by the current velocity components, but the current velocity component cannot be obtained. Then, the converted measurement covariance matrix is correlated with the measurement noise. With the above reasons, the CMKF algorithm has a big estimation bias. The SEKF algorithm uses the sequential filtering method. The second-order EKF algorithm is used to deal with the pseudo measurement and CMKF algorithm is used to deal with the position measurement, respectively. Similarly, the SUKF algorithm sequentially uses the UKF algorithm to deal with the range rate and the CMKF algorithm to deals with the position measurement. Compared with the UKF algorithm and the standard second-order EKF algorithm, the SUKF algorithm and the SEKF algorithm are superior to use the range rate measurement. As the CMKF algorithm is adopted in the SEKF and SUKF algorithms, the converted position measurement covariance matrix is related to the position measurement noise, which leads to the estimation bias.In order to compare, the four filtering algorithms all use the CV model, and the corresponding process noise covariance power spectral density (PSD) is set to 10. The sampling period of the Doppler radar is 1 s. Real trajectories and radar measurements are generated with the formulas (46) and (47). The standard deviation in formula (46) is set as σwx=2m, σwy=2m, σwv=0.05m/s, and σwα=0.005rad. The initial state of the target x0,y0,v0,ϕ0 is set as 4km,6km,50m/s,π/3. Under the different levels of measurement noise, maneuvering, and tracking time, the tracking accuracy and stability of DUCM algorithm, SEKF algorithm, SUKF algorithm, and CMKF algorithm are compared. σwr=10m,σwθ=0.50 means the low position measurement noise standard deviation; σwr=20m,σwθ=10 means the medium position measurement noise standard deviation; σwr=100m,σwθ=50 means the high position measurement noise standard deviation. Their measurement noise standard deviations of range rate are all σwr˙=0.5m/s. Suppose the Doppler radar uses the upsweep LFM waveform to detect targets and then ρ′=-0.9. an=0m/s2,at=0m/s2 means the nonmaneuvering situation, that is, the uniform motion in a straight line. an=2m/s2,at=0m/s2 means the maneuvering situation. Two kinds of tracking time are set to observe the stability of the tracking algorithm, that is, t1=40s means the short tracking time and t2=200s means the long tracking time. In the initial state, the position components are set as the first position component measurement. Carry out 100 Monte-Carlo simulation experiments, and the root mean square error of position is used to measure the tracking performance of the filtering algorithm:(49)RMSExk∣k,yk∣k=1N∑i=1Nxk-xk∣k2+yk-yk∣k2where N is the number of Monte-Carlo simulations; xk∣k and yk∣k are the estimated velocity components; xk and yk are the real position components. Figures 3 and 4 show the RMSE of the four filtering algorithms corresponding to three noise levels in the nonmaneuvering short tracking time situation and in the maneuvering short tracking time situation, respectively. Figure 5 shows the RMSE of the four filtering algorithms corresponding to three noise levels under the condition of nonmaneuvering long tracking time. From Figures 3 and 4, the DUCMKF algorithm has the smallest position RMSE under different noise levels, compared to CMKF, SEKF, and SUKF. When the target is maneuvering, the DUCMKF algorithm can still keep tracking with a certain estimation error. And from Figure 5, it can be seen that the DUCMKF algorithm can stably track the target without the influence of estimation bias when the tracking time is long. But the CMKF algorithm has obvious divergence in the later period of filtering. Under the low noise and medium noise, the RMSE of SEKF and SUKF increases with the tracking time.Figure 3
RMSE of position givenan=0m/s2, at=0m/s2, and t1=40s.
(a)
The low noise (b)
The medium noise (c)
The high noiseFigure 4
RMSE of position givenan=2m/s2, at=0m/s2, and t1=40s.
(a)
The low noise (b)
The medium noise (c)
The high noiseFigure 5
RMSE of position givenan=0m/s2, at=0m/s2, and t1=200s.
(a)
The low noise (b)
The medium noise (c)
The high noise


### 5.3. The Performance of AIMM-CS-DUCMKF Algorithm

In this section, AIMM-CS-DUCMKF proposed is compared with the state of the art IMM algorithm, WLSE algorithm, and CS-AEKF algorithm for tracking the maneuvering target and estimating the MAs. The initial statex0,y0,v0,ϕ0 of the maneuvering target’s real trajectory is set as 4km,6km,50m/s,π/6. The real trajectory is divided into three stages: (1) an=0m/s2,at=10m/s2 in 0 s~50 s; (2) an=20m/s2,at=-5m/s2 in 51 s~125 s; (3) an=-20m/s2,at=5m/s2 in 126 s~200 s. The measurement noise standard deviations are set as σwr=20m, σwθ=10 and σwr˙=0.5m/s. The coefficient of correlation between range rate and range is set as ρ′=-0.9. The set of motion models for the IMM algorithm consists of three CA motion models whose PDS are set as 1,10,100. The WLSE algorithm adopts the CV motion model, the size of its window is set as 5, and its PDS is 10. The filter of IMM algorithm and WLSE algorithm is DUCMKF algorithm. As the IMM algorithm and the WLSE algorithm can only estimate the ACAs, that is, x¨^ and y¨^, the following equation is used to estimate the MAs, that is, a^t and a^n:(50)a^ta^n=x˙^x¨^x˙^2+y˙^2+y˙^y¨^x˙^2+y˙^2-x¨^y˙^x˙^2+y˙^2+y¨^x˙^x˙^2+y˙^2The CS-AEKF algorithm uses the CS motion model and the EKF algorithm to estimate the maneuvering acceleration directly. In the CS-AEKF algorithm, the state equation (12) is expanded by Taylor and the first-order linear differential equation is established. Then, the system difference equation is obtained by discretization, which is similar to formula (13). But ΦT,Uk,Wk of CS-AEKF is different from the AIMM-CS-DUCMKF algorithm in Appendix specifically. For the CS model in the CS-AEKF algorithm and AIMM-CS-DUCMKF algorithm, their parameters are set as αt=0.1, αn=0.1, ato=40m/s2, ato=-40m/s2, ano=60m/s2, and a-no=-60m/s2. The PSD in the CS-AEKF algorithm and the AIMM-CS-DUCMKF algorithm are all set as 10. The initial model probability π1 and state transition matrix p1 of IMM algorithm and the initial model probability π2 and state transition matrix p2 of AIMM-CS-DUCMKF algorithm are(51)π1=0.2,0.2,0.6,π2=0.4,0.3,0.3,p1=0.80.10.10.10.80.10.10.10.8,p2=0.80.10.10.10.80.10.10.10.8With each of the above algorithm carrying out 100 Monte-Carlo simulations, the RMSEs of position, velocity, normal acceleration, and tangential acceleration are calculated like formula (49).From the RMSEs of position and velocity in Figures6 and 7, we can see that the four algorithms can realize the tracking of maneuvering target and have the comparable estimation performance of position and velocity. The RMSE of tangential acceleration and normal acceleration are given in Figures 8 and 9, respectively. In 0 s~50 s, the target only has a constant tangential acceleration, the AIMM-CS-DUCMKF algorithm is not the best, and the IMM algorithm using the CA motion model exhibits relatively the better estimation performance. In 51~200 s, the target has both tangential acceleration and normal acceleration, and the AIMM-CS-DUCMKF algorithm has the higher estimation accuracy than other algorithms. However, the AIMM-CS-DUCMKF algorithm has a significant step on the tangential acceleration and the normal acceleration estimation around 126 s, after the second maneuver acceleration has changed. This is because the target normal acceleration turns from 20m/s2 to -20m/s2 which leads to the large error in velocity direction estimation. As the range rate measurement contains the information of target velocity direction, with the new measurement arriving, the accurate estimation of target velocity direction is obtained again, and the AIMM-CS-DUCMKF algorithm converges quickly.Figure 6
RMSE of position.Figure 7
RMSE of velocity.Figure 8
RMSE of tangential acceleration.Figure 9
RMSE of normal acceleration.Figure10 shows the filtering trajectories of the four tracking algorithms at the 50th Monte-Carlo simulation and the corresponding estimation of the MAs. In particular, the enlarged diagram in Figure 10(a) shows that the filtering trajectory of the AIMM-CS-DUCMKF algorithm is obviously not smooth at the second target maneuvering acceleration switching. Also, there is a sudden change in the tangential acceleration around 126 s in Figure 10(b). But the tangential acceleration estimation converges rapidly to the true value. From the details of the filtering and estimation shown in Figure 10, the AIMM-CS-DUCMKF algorithm can realize the steady tracking of the maneuvering target and can estimate the MAs of the target directly. The CS-AEKF algorithm is used to linearize the state equation and the measurement equation with strong nonlinearity by the first-order Taylor expansion; however the approximation is not satisfactory. As the IMM algorithm and WLSE algorithm use the open-loop method, the estimation errors cannot be corrected in the filtering iterative. Besides, the CS-AEKF algorithm, IMM algorithm, and WLSE algorithm do not use range rate measurement to improve the estimation accuracy of velocity direction angle, compared to the AIMM-CS-DUCMKF algorithm.Figure 10
Filtering trajectories and estimations of MAs at 50th simulation.


## 5.1. The Generation of True Trajectory and Measurement

The simulation experiment is divided into two parts. The first part is to verify the tracking performance of DUCMKF algorithm proposed with the Doppler radar measurement and compare the DUCMKF algorithm with the sequential EKF (SEKF) [18], sequential UKF (SUKF) [17], and converted measurement KF (CMKF) algorithms [13] in the previous literatures. The second part is to verify the maneuvering target tracking performance and the MAs estimation performance of AIMM-CS-DUCMKF algorithm and compare it with the IMM algorithm, current statistical model based adaptive EKF (CS-AEKF) [20], and window-based least squares estimator (WLSE) [7].In the simulation experiment, the target real trajectory and the corresponding Doppler radar measurement are needed firstly. A standard plane curve motion model is adopted to obtain the true trajectory of the target:(46)x˙=vcos⁡α+wxy˙=vsin⁡α+wyv˙=at+wvα˙=anv+wαwhere wx, wy, wv, and wα are the independent of Gaussian white noises, respectively, and their standard deviation are σwx, σwy, σwv, and σwα. Through initializing x, y, v, and α, the true trajectory can be obtained by Runge Kutta method [23] given an and at. The Doppler radar measurement (r,θ,r˙) can be generated with the model as formula (47):(47)rθr˙=x2+y2±arccos⁡xx2+y2vcos⁡α∓arccos⁡xx2+y2+wrwθwr˙Ify≥0, “±” is “+”; if y<0, “±” is “−”. And “∓” is on the contrary. wr, wθ, and wvr are the Gaussian white noise and their covariance is(48)R=σwr20ρ′σwrσwr˙0σwθ20ρ′σwr2σwr˙20σwr˙2where σwr, σwθ, and σwr˙ are the standard deviation; ρ′ is the correlation coefficient. After inputting a real trajectory into model (47), the corresponding measurement sequence can be obtained.


## 5.2. The Performance of the DUCMKF Algorithm

The measurement equation of the CMKF algorithm is the same as that of the DUCMKF algorithm, and the difference is in the way of measurement transformation. First, the CMKF algorithm does not consider the correlation between range measurement and range rate measurement. Secondly, the converted measurement covariance matrix needs to be calculated by the current velocity components, but the current velocity component cannot be obtained. Then, the converted measurement covariance matrix is correlated with the measurement noise. With the above reasons, the CMKF algorithm has a big estimation bias. The SEKF algorithm uses the sequential filtering method. The second-order EKF algorithm is used to deal with the pseudo measurement and CMKF algorithm is used to deal with the position measurement, respectively. Similarly, the SUKF algorithm sequentially uses the UKF algorithm to deal with the range rate and the CMKF algorithm to deals with the position measurement. Compared with the UKF algorithm and the standard second-order EKF algorithm, the SUKF algorithm and the SEKF algorithm are superior to use the range rate measurement. As the CMKF algorithm is adopted in the SEKF and SUKF algorithms, the converted position measurement covariance matrix is related to the position measurement noise, which leads to the estimation bias.In order to compare, the four filtering algorithms all use the CV model, and the corresponding process noise covariance power spectral density (PSD) is set to 10. The sampling period of the Doppler radar is 1 s. Real trajectories and radar measurements are generated with the formulas (46) and (47). The standard deviation in formula (46) is set as σwx=2m, σwy=2m, σwv=0.05m/s, and σwα=0.005rad. The initial state of the target x0,y0,v0,ϕ0 is set as 4km,6km,50m/s,π/3. Under the different levels of measurement noise, maneuvering, and tracking time, the tracking accuracy and stability of DUCM algorithm, SEKF algorithm, SUKF algorithm, and CMKF algorithm are compared. σwr=10m,σwθ=0.50 means the low position measurement noise standard deviation; σwr=20m,σwθ=10 means the medium position measurement noise standard deviation; σwr=100m,σwθ=50 means the high position measurement noise standard deviation. Their measurement noise standard deviations of range rate are all σwr˙=0.5m/s. Suppose the Doppler radar uses the upsweep LFM waveform to detect targets and then ρ′=-0.9. an=0m/s2,at=0m/s2 means the nonmaneuvering situation, that is, the uniform motion in a straight line. an=2m/s2,at=0m/s2 means the maneuvering situation. Two kinds of tracking time are set to observe the stability of the tracking algorithm, that is, t1=40s means the short tracking time and t2=200s means the long tracking time. In the initial state, the position components are set as the first position component measurement. Carry out 100 Monte-Carlo simulation experiments, and the root mean square error of position is used to measure the tracking performance of the filtering algorithm:(49)RMSExk∣k,yk∣k=1N∑i=1Nxk-xk∣k2+yk-yk∣k2where N is the number of Monte-Carlo simulations; xk∣k and yk∣k are the estimated velocity components; xk and yk are the real position components. Figures 3 and 4 show the RMSE of the four filtering algorithms corresponding to three noise levels in the nonmaneuvering short tracking time situation and in the maneuvering short tracking time situation, respectively. Figure 5 shows the RMSE of the four filtering algorithms corresponding to three noise levels under the condition of nonmaneuvering long tracking time. From Figures 3 and 4, the DUCMKF algorithm has the smallest position RMSE under different noise levels, compared to CMKF, SEKF, and SUKF. When the target is maneuvering, the DUCMKF algorithm can still keep tracking with a certain estimation error. And from Figure 5, it can be seen that the DUCMKF algorithm can stably track the target without the influence of estimation bias when the tracking time is long. But the CMKF algorithm has obvious divergence in the later period of filtering. Under the low noise and medium noise, the RMSE of SEKF and SUKF increases with the tracking time.Figure 3
RMSE of position givenan=0m/s2, at=0m/s2, and t1=40s.
(a)
The low noise (b)
The medium noise (c)
The high noiseFigure 4
RMSE of position givenan=2m/s2, at=0m/s2, and t1=40s.
(a)
The low noise (b)
The medium noise (c)
The high noiseFigure 5
RMSE of position givenan=0m/s2, at=0m/s2, and t1=200s.
(a)
The low noise (b)
The medium noise (c)
The high noise


## 5.3. The Performance of AIMM-CS-DUCMKF Algorithm

In this section, AIMM-CS-DUCMKF proposed is compared with the state of the art IMM algorithm, WLSE algorithm, and CS-AEKF algorithm for tracking the maneuvering target and estimating the MAs. The initial statex0,y0,v0,ϕ0 of the maneuvering target’s real trajectory is set as 4km,6km,50m/s,π/6. The real trajectory is divided into three stages: (1) an=0m/s2,at=10m/s2 in 0 s~50 s; (2) an=20m/s2,at=-5m/s2 in 51 s~125 s; (3) an=-20m/s2,at=5m/s2 in 126 s~200 s. The measurement noise standard deviations are set as σwr=20m, σwθ=10 and σwr˙=0.5m/s. The coefficient of correlation between range rate and range is set as ρ′=-0.9. The set of motion models for the IMM algorithm consists of three CA motion models whose PDS are set as 1,10,100. The WLSE algorithm adopts the CV motion model, the size of its window is set as 5, and its PDS is 10. The filter of IMM algorithm and WLSE algorithm is DUCMKF algorithm. As the IMM algorithm and the WLSE algorithm can only estimate the ACAs, that is, x¨^ and y¨^, the following equation is used to estimate the MAs, that is, a^t and a^n:(50)a^ta^n=x˙^x¨^x˙^2+y˙^2+y˙^y¨^x˙^2+y˙^2-x¨^y˙^x˙^2+y˙^2+y¨^x˙^x˙^2+y˙^2The CS-AEKF algorithm uses the CS motion model and the EKF algorithm to estimate the maneuvering acceleration directly. In the CS-AEKF algorithm, the state equation (12) is expanded by Taylor and the first-order linear differential equation is established. Then, the system difference equation is obtained by discretization, which is similar to formula (13). But ΦT,Uk,Wk of CS-AEKF is different from the AIMM-CS-DUCMKF algorithm in Appendix specifically. For the CS model in the CS-AEKF algorithm and AIMM-CS-DUCMKF algorithm, their parameters are set as αt=0.1, αn=0.1, ato=40m/s2, ato=-40m/s2, ano=60m/s2, and a-no=-60m/s2. The PSD in the CS-AEKF algorithm and the AIMM-CS-DUCMKF algorithm are all set as 10. The initial model probability π1 and state transition matrix p1 of IMM algorithm and the initial model probability π2 and state transition matrix p2 of AIMM-CS-DUCMKF algorithm are(51)π1=0.2,0.2,0.6,π2=0.4,0.3,0.3,p1=0.80.10.10.10.80.10.10.10.8,p2=0.80.10.10.10.80.10.10.10.8With each of the above algorithm carrying out 100 Monte-Carlo simulations, the RMSEs of position, velocity, normal acceleration, and tangential acceleration are calculated like formula (49).From the RMSEs of position and velocity in Figures6 and 7, we can see that the four algorithms can realize the tracking of maneuvering target and have the comparable estimation performance of position and velocity. The RMSE of tangential acceleration and normal acceleration are given in Figures 8 and 9, respectively. In 0 s~50 s, the target only has a constant tangential acceleration, the AIMM-CS-DUCMKF algorithm is not the best, and the IMM algorithm using the CA motion model exhibits relatively the better estimation performance. In 51~200 s, the target has both tangential acceleration and normal acceleration, and the AIMM-CS-DUCMKF algorithm has the higher estimation accuracy than other algorithms. However, the AIMM-CS-DUCMKF algorithm has a significant step on the tangential acceleration and the normal acceleration estimation around 126 s, after the second maneuver acceleration has changed. This is because the target normal acceleration turns from 20m/s2 to -20m/s2 which leads to the large error in velocity direction estimation. As the range rate measurement contains the information of target velocity direction, with the new measurement arriving, the accurate estimation of target velocity direction is obtained again, and the AIMM-CS-DUCMKF algorithm converges quickly.Figure 6
RMSE of position.Figure 7
RMSE of velocity.Figure 8
RMSE of tangential acceleration.Figure 9
RMSE of normal acceleration.Figure10 shows the filtering trajectories of the four tracking algorithms at the 50th Monte-Carlo simulation and the corresponding estimation of the MAs. In particular, the enlarged diagram in Figure 10(a) shows that the filtering trajectory of the AIMM-CS-DUCMKF algorithm is obviously not smooth at the second target maneuvering acceleration switching. Also, there is a sudden change in the tangential acceleration around 126 s in Figure 10(b). But the tangential acceleration estimation converges rapidly to the true value. From the details of the filtering and estimation shown in Figure 10, the AIMM-CS-DUCMKF algorithm can realize the steady tracking of the maneuvering target and can estimate the MAs of the target directly. The CS-AEKF algorithm is used to linearize the state equation and the measurement equation with strong nonlinearity by the first-order Taylor expansion; however the approximation is not satisfactory. As the IMM algorithm and WLSE algorithm use the open-loop method, the estimation errors cannot be corrected in the filtering iterative. Besides, the CS-AEKF algorithm, IMM algorithm, and WLSE algorithm do not use range rate measurement to improve the estimation accuracy of velocity direction angle, compared to the AIMM-CS-DUCMKF algorithm.Figure 10
Filtering trajectories and estimations of MAs at 50th simulation.


## 6. Conclusions

In this paper, an algorithm for directly and online estimating the MAs is proposed, i.e., the AIMM-CS-DUCMKF algorithm. Firstly, the MAs of the maneuvering target are introduced into the state equation as state variables. Then the MAs are modeled as the modified Rayleigh-Markov random process. A linear system state equation is obtained given the velocity direction angle. Secondly, tracking target is carried out using Doppler radar measurements including range rate. In order to solve the problem of nonlinear measurement equation, this paper proposes a decorrelation unbiased conversion measurement algorithm for the Doppler radar measurement. The linear state equation established in this paper is affected by the accuracy of velocity direction angle estimation. In order to improve its accuracy, the geometric relationship between range rate and velocity direction angle is applied. The IMM algorithm is used to fuse three different velocity angle estimators, and an adaptive IMM algorithm with three CS models is obtained, whose corresponding filter uses DUCMKF algorithm. In the simulation experiment, firstly, in order to verify the tracking performance of DUCMKF algorithm, it is compared with the previous tracking algorithms proposed using range rate measurement such as CMKF, SEKF, and SUKF in different levels of noise, target maneuvering, and tracking time, and the experimental results show that DUCMKF algorithm has the higher accuracy and stability. Secondly, in order to verify the AIMM-CS-DUCMKF algorithm, it is compared with the IMM algorithm, WLSE algorithm, and CS-AEKF algorithm. The experimental results show that the AIMM-CS-DUCMKF algorithm not only has comparable estimation accuracy of position and velocity, but also can obtain the higher estimation accuracy of tangential acceleration and normal acceleration directly. Although, there is a sudden step in the acceleration estimation at the target maneuvering acceleration switching, the estimator can converge quickly with the arrival of the new measurement. This also shows that AIMM-CS-DUCMKF algorithm not only has high sensitivity, but also has strong robustness.



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*Source: 4984186-2018-06-04.xml*