# Performance Analysis and Comparison for High Maneuver Target Track Based on Different Jerk Models

Authors: Qinghai Meng; Bowen Hou; Dong Li; Zhangming He; Jiongqi Wang
Journal: Journal of Control Science and Engineering
 (2018)

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## Abstract

The Jerk model is widely used for the track of the maneuvering targets. Different Jerk model has its own state expression and is suitable to different track situation. In this paper, four Jerk models commonly used in the maneuvering target track are advanced. The performances of different Jerk models for target track with the state variables and the characters are compared. The corresponding limit conditions in the practical applications are also analyzed. Besides, the filter track is designed with UKF algorithm based on the four different models for the high-maneuvering target. The simplified dynamic model is used to gain the standard trajectory with Runge-Kutta numerical integration method. The mathematical simulations show that Jerk model with self-adaptive noise variance has the best robustness while other models may diverge when the initial error is much larger. If the process noise level is much lower, the track accuracy for four Jerk models is similar and stationary in the steady track situation, but it will be descended greatly in the much highly maneuvering situation.


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## Body

## 1. Introduction

Target tracking refers to estimating the motion parameters, such as the position and the velocity for a target through the noise-containing measurement data acquired by the measurement device in real time. Because of the uncertainty of the maneuvering target motion, the measurement process uncertainty, and the difficulty of estimating the nonlinear system, the maneuvering target tracking has always been a research focus for more than half a century and there are a lot of proposed algorithms. To sum up, the track algorithms study has been focused on two parts,i.e., maneuver target modeling and nonlinear filter design [1, 2].For the nonlinear filter design, there are three main methods, including Extended Kalman filter (EKF), Unscented Kalman filter (UKF), and Particle filter (PF) [3, 4]. Many articles have sufficiently studied the three algorithms in the nonlinear fitness, filter accuracy, filter stability, computational complexity, and other aspects [5, 6]. The conclusion is that UKF has the best performance in much practical applications [7].For the maneuver target modeling, there are lots of motion model describing the maneuver process of the target. The accuracy of the target modeling directly affects the tracking performance of maneuvering target, detection of the target [8, 9], and the fault diagnosis of the target [10, 11]. According to their modeling state dimensions, they can be divided into second-order model, three-order model and four-order model. Two-order models contain CV (Constant Velocity) model and CT (Constant Turn) model; three-order models contain CA (Constant Acceleration) model, Singer model, CS (Current Statistic) model, semi-Markov model, and so on [12, 13]. Four-order models contain Jerk (Jerk denotes the rate of the acceleration change) model and its corresponding improved version. The higher the order of maneuvering model is, the higher the order of the target is described. Jerk model extends the target maneuvering form via estimating the acceleration changing rate in real time. Theoretically, it can be applied on the highly maneuvering target tracking much better [14, 15].Therefore, in this paper, we mainly focused on the maneuvering target track with Jerk model and UKF filter algorithm.The references related to Jerk model has shown the good simulation results. However, due to the simple and the special simulation background, the trajectory is different from the true target. Therefore, the results cannot be convincing. This paper summarizes the various Jerk models in the references. Firstly, the advantages and disadvantages are analyzed in theory. Then every model is applied on the near-space high-speed maneuvering target, and the simulation results are compared. The track trajectory is generated through the integral of the simplified dynamic equation, which is close to the true target. Therefore, it is convictive to some extent.


## 2. Description for Different Jerk Model

Jerk denotes the acceleration changing rate. Jerk model is a model which describes the target Jerk mathematically. The models in the references mainly contain the following different kinds.


### 2.1. SJ Model

The earliest Jerk model is proposed by Mehrota, etc. [16]. They used Singer model for reference and modeled the Jerk model of the target as a zero-mean and first-order time related process. To distinguish other Jerk models, it is marked as SJ (Singer Jerk) model. Taking a one-dimension linear motion as an example, SJ model is expressed as(1)j˙t=-αjt+wtwhere j(t)=x⃛(t) denotes the target Jerk, the α denotes “Jerk” frequency (the reciprocal of the “Jerk” constant), w(t) denotes zero-mean Gaussian white noise, and the covariance is σw2=2ασj2, σj2 denotes the covariance of target Jerk.


### 2.2. CSJ Model

Qiao [15] used the analyzing method which is also applied on Singer model tracking accuracy in [17] to analyze the SJ model. He proposed that SJ model shows a steady-state deterministic error in the tracking step Jerk signals. Therefore, the Jerk model with nonzero mean and first-order time correlation is built using CS model for reference. According to the same analyzing process, the new model has eliminated the steady-state deterministic error. The model is marked as CSJ (Current Statistic Jerk) model. Taking the one-dimension model as an example, CSJ model is expressed as(2)x⃛t=j-CSJt+jCSJtj˙CSJt=-αjCSJt+wtwhere j-CSJ(t) denotes the nonzero mean of Jerk. jCSJ(t) denotes the covariance of the zero-mean colored Jerk noise, and w(t) denotes zero-mean Gaussian white noise. Other parameters are defined as SJ model.The model should setα and σj2 previously before the practical application. Therefore, Pan proposed a novel CSJ algorithm to describe the probability density of Jerk according to truncated normal distribution and construct the connection between σj2 and current Jerk estimation. It is(3)σj2k=Jmax-x⃛^k∣k29In this way, the probable extreme Jerk can be predefined as Jmax. That is the covariance of the state noise can be self-adapted according to the Jerk estimation during the filter process to adapt to different maneuvering situations. The model is marked as MCSJ (Modified CSJ) model.


### 2.3.αJ Model

The Jerk models above cannot avoid the problem that the Jerk frequencyα should be predefined. However, α cannot be directly measured and it is constantly changing in the target practical motion process. For this reason, Luo [7] considered α in SJ model as an estimated parameter and took it as the extension variable. Therefore, it can be estimated in real time during the filter process. α is modeled as nonzero-mean Gaussian white noise and the derivative of it is zero mean Gaussian white noise which can be considered as(4)α˙=εtwhere ε(t) denotes the zero-mean input noise, whose variance is σε2. This is called αJ (Alpha Jerk) model. αJ model can estimate α in real time, but σj2 and σε2 need to be predefined carefully. If the values of them are not proper, the estimation accuracy of α will be decreased severely and even cause the divergence of the filter.Take the one-dimension motion as an example. (Three-dimension situation has the same principle as the one-dimension motion.) The state variable, state equation, and the characteristic of the four models are shown in Table1. The specific form of the matrices in Table 1 can be found in the related references [14, 16].Table 1
Description for four different Jerk models.
Model State Variable Continuous State Equation Discrete State Equation Characteristic SJ [ x , x ˙ , x ¨ , x ⃛ ] T X ˙ ( t ) = F S X ( t ) + B w ( t ) X k + 1 = Φ S X k + W k  Qk=2ασj2Q0 Tracking step Jerk signal shows deterministic error.α and σj2 need to be predefined. Parameters are constant.  CSJ [ x , x ˙ , x ¨ , x ⃛ ] T X ˙ ( t ) = F S X ( t ) + A j - C S J ( t ) + B w ( t ) X k + 1 = Φ S X k + U j - k + 1 + W k  j-k+1=x⃛^k  Qk=2ασj2Q0 Tracking step Jerk signal does not show steady-state deterministic error.α and σj2 need to be predefined. Parameters are constant.  MCSJ [ x , x ˙ , x ¨ , x ⃛ ] T Ditto Ditto, butσj2 is self-adaption Tracking step Jerk signal does not show steady-state deterministic error.α and Jmax need to be predefined. σj2 is self-adaption.  αJ [ x , x ˙ , x ¨ , x ⃛ , α ] T X ˙ a ( t ) = F a X a ( t ) + B a w a ( t )  Fa=diag⁡FS,0  wa(t)=wt,εt′ X a ( k + 1 ) = Φ a X a ( k ) + W a ( k )  Φa=diag⁡(ΦS,1)  Qk=diag⁡(2αkσj2Q0,Tσε2) Tracking step Jerk signal shows steady-state deterministic error.σj2 and σε2 need to be predefined. α needs to be estimated in real time.


## 2.1. SJ Model

The earliest Jerk model is proposed by Mehrota, etc. [16]. They used Singer model for reference and modeled the Jerk model of the target as a zero-mean and first-order time related process. To distinguish other Jerk models, it is marked as SJ (Singer Jerk) model. Taking a one-dimension linear motion as an example, SJ model is expressed as(1)j˙t=-αjt+wtwhere j(t)=x⃛(t) denotes the target Jerk, the α denotes “Jerk” frequency (the reciprocal of the “Jerk” constant), w(t) denotes zero-mean Gaussian white noise, and the covariance is σw2=2ασj2, σj2 denotes the covariance of target Jerk.


## 2.2. CSJ Model

Qiao [15] used the analyzing method which is also applied on Singer model tracking accuracy in [17] to analyze the SJ model. He proposed that SJ model shows a steady-state deterministic error in the tracking step Jerk signals. Therefore, the Jerk model with nonzero mean and first-order time correlation is built using CS model for reference. According to the same analyzing process, the new model has eliminated the steady-state deterministic error. The model is marked as CSJ (Current Statistic Jerk) model. Taking the one-dimension model as an example, CSJ model is expressed as(2)x⃛t=j-CSJt+jCSJtj˙CSJt=-αjCSJt+wtwhere j-CSJ(t) denotes the nonzero mean of Jerk. jCSJ(t) denotes the covariance of the zero-mean colored Jerk noise, and w(t) denotes zero-mean Gaussian white noise. Other parameters are defined as SJ model.The model should setα and σj2 previously before the practical application. Therefore, Pan proposed a novel CSJ algorithm to describe the probability density of Jerk according to truncated normal distribution and construct the connection between σj2 and current Jerk estimation. It is(3)σj2k=Jmax-x⃛^k∣k29In this way, the probable extreme Jerk can be predefined as Jmax. That is the covariance of the state noise can be self-adapted according to the Jerk estimation during the filter process to adapt to different maneuvering situations. The model is marked as MCSJ (Modified CSJ) model.


## 2.3.αJ Model

The Jerk models above cannot avoid the problem that the Jerk frequencyα should be predefined. However, α cannot be directly measured and it is constantly changing in the target practical motion process. For this reason, Luo [7] considered α in SJ model as an estimated parameter and took it as the extension variable. Therefore, it can be estimated in real time during the filter process. α is modeled as nonzero-mean Gaussian white noise and the derivative of it is zero mean Gaussian white noise which can be considered as(4)α˙=εtwhere ε(t) denotes the zero-mean input noise, whose variance is σε2. This is called αJ (Alpha Jerk) model. αJ model can estimate α in real time, but σj2 and σε2 need to be predefined carefully. If the values of them are not proper, the estimation accuracy of α will be decreased severely and even cause the divergence of the filter.Take the one-dimension motion as an example. (Three-dimension situation has the same principle as the one-dimension motion.) The state variable, state equation, and the characteristic of the four models are shown in Table1. The specific form of the matrices in Table 1 can be found in the related references [14, 16].Table 1
Description for four different Jerk models.
Model State Variable Continuous State Equation Discrete State Equation Characteristic SJ [ x , x ˙ , x ¨ , x ⃛ ] T X ˙ ( t ) = F S X ( t ) + B w ( t ) X k + 1 = Φ S X k + W k  Qk=2ασj2Q0 Tracking step Jerk signal shows deterministic error.α and σj2 need to be predefined. Parameters are constant.  CSJ [ x , x ˙ , x ¨ , x ⃛ ] T X ˙ ( t ) = F S X ( t ) + A j - C S J ( t ) + B w ( t ) X k + 1 = Φ S X k + U j - k + 1 + W k  j-k+1=x⃛^k  Qk=2ασj2Q0 Tracking step Jerk signal does not show steady-state deterministic error.α and σj2 need to be predefined. Parameters are constant.  MCSJ [ x , x ˙ , x ¨ , x ⃛ ] T Ditto Ditto, butσj2 is self-adaption Tracking step Jerk signal does not show steady-state deterministic error.α and Jmax need to be predefined. σj2 is self-adaption.  αJ [ x , x ˙ , x ¨ , x ⃛ , α ] T X ˙ a ( t ) = F a X a ( t ) + B a w a ( t )  Fa=diag⁡FS,0  wa(t)=wt,εt′ X a ( k + 1 ) = Φ a X a ( k ) + W a ( k )  Φa=diag⁡(ΦS,1)  Qk=diag⁡(2αkσj2Q0,Tσε2) Tracking step Jerk signal shows steady-state deterministic error.σj2 and σε2 need to be predefined. α needs to be estimated in real time.


## 3. Performance Comparison

### 3.1. Trajectory State Equation

Tracking target is the near-space and high-speed target whose true trajectory comes from the integral of the dynamic equation. To simplify the dynamic equation of the target, the earth is assumed as an irrational ball and the influence on the center of mass motion of the target of the controlling force is neglected. The target has nonliteral movement and the simple dynamic equation in the launching frame is expressed as(5)v˙xv˙y=ρVS2m-Cxvx-CyvyCyvx-Cxvy+g0R02r3x-R0xy-R0ywhere ρ denotes the atmosphere density which is used as American 1976 standard atmosphere model. V denotes the resultant velocity; S denotes the aerodynamic reference area of the target; m denotes the mass of the target; Cx and Cy denote the resistance coefficient and the lift coefficient, respectively (they are the function of the attack angle the Mach number Ma); g0 denotes the gravitational acceleration on the sea level; R0 denotes the radius of the earth; R0x and R0y denote the coordinates of the earth center in the launching frame; r denotes the geocentric distance of the target.The target parameterS and m and the aerodynamic parameters Cx and Cy use the public values in American high-performance general air vehicle (CAV-H). The longitude, the latitude, the height, and the launching angle are 0°, 0°, 0m, and 90°. The initial longitude, the latitude, and the height of the target are 0°, 0°, and 30km, respectively. The initial Mach number is 10. The local velocity angle is -1°, and the velocity drift angle is 0°. The controlled quantity is the attacking angle α and is constant with 15°. The integral method of the trajectory is fourth-order Runge-Kutta numerical integration. The height and the voyage are shown in Figure 1.Figure 1
Target trajectory.


### 3.2. Measurement Setting Condition

The measurement device is a pulse radar and the measurement elements are the distance R, the azimuth angle A, the pitch angle E, and the radial velocity V. The measure error standard deviations of the distance, the angle, and the velocity are 10m, 0.5mrad, and 1m/s, respectively. The longitude, the latitude, and the height of the radar station are 14°E, 1°N, and 0m, respectively. The sampling period is 0.05s and the tracking period is 70~790s. To compare the tracking performance of the models (including the tracking accuracy, the filter robustness, and the computation complexity), the simulation results cannot be compared as the general references setting a parameter. Mehrotra proposed that “Jerk” frequencyα has a good robustness in [7] via the simulation. That is, the filter performance can maintain stable when α changes in a large range. Therefore, the paper focuses on the standard deviation of Jerk and the robustness of the filter initial value error. The initial position and velocity in filter are equal to the true value adding the random error. The standard deviations of the position and the velocity are σp0 and σv0, respectively. The acceleration and the Jerk are 0. αJ model needs to set the initial value of α and the standard deviation σε of the input noise ε(t). MCSJ model needs to set the extreme Jerk value Jmax. The parameter sets of the three filters are expressed as P1, P2, and P3, respectively. The specific parameter values are shown in Table 2. The Monte Carlo simulation number is 100.Table 2
Filter parameters.
Parameters P1 P2 P3 α 0.1  α 0 α  σ j 1 1 0.1  J m a x 3 σ j  σ p 0 1000 100 100  σ v 0 100 10 10  σ ε 0.1


### 3.3. Simulation Results

The filter root mean square error (RMSE) of the position and the velocity under the parameter sets is shown in Figures2, 3, and 4, respectively. Because of the different values of P1 and P2, the initial value robustness of the four models can be compared. Comparing Figures 2 and 3, when the filter initial value is large, only the self-adaption state noise of MCSJ model can maintain the convergence and the initial value robustness is much better. But the other three models will present filter divergence performance during the 100 Monte Carlo steps and the convergence is influenced by the initial value which is not proper for the practical tracking application. On the other hand, because the only difference between P2 and P3 is σj. The parameter robustness of the four models can be compared. When comparing Figures 3 and 4, when σj is small, the tracking accuracy of the four models in the steady period of the target motion can be improved. However, the filter error has a large jump when the target has a severe maneuvering motion. The computation complexity ratio of SJ, CSJ, MCSJ, and αJ is 8:8:9:12.Figure 2
Filtering RMSE for four models in P1 parameter.
(a)
RMSE of position (b)
RMSE of velocityFigure 3
Filtering RMSE for four models in P2 parameter.
(a)
RMSE of position (b)
RMSE of velocityFigure 4
Filtering RMSE for four models in P3 parameter.
(a)
RMSE of position (b)
RMSE of velocity


## 3.1. Trajectory State Equation

Tracking target is the near-space and high-speed target whose true trajectory comes from the integral of the dynamic equation. To simplify the dynamic equation of the target, the earth is assumed as an irrational ball and the influence on the center of mass motion of the target of the controlling force is neglected. The target has nonliteral movement and the simple dynamic equation in the launching frame is expressed as(5)v˙xv˙y=ρVS2m-Cxvx-CyvyCyvx-Cxvy+g0R02r3x-R0xy-R0ywhere ρ denotes the atmosphere density which is used as American 1976 standard atmosphere model. V denotes the resultant velocity; S denotes the aerodynamic reference area of the target; m denotes the mass of the target; Cx and Cy denote the resistance coefficient and the lift coefficient, respectively (they are the function of the attack angle the Mach number Ma); g0 denotes the gravitational acceleration on the sea level; R0 denotes the radius of the earth; R0x and R0y denote the coordinates of the earth center in the launching frame; r denotes the geocentric distance of the target.The target parameterS and m and the aerodynamic parameters Cx and Cy use the public values in American high-performance general air vehicle (CAV-H). The longitude, the latitude, the height, and the launching angle are 0°, 0°, 0m, and 90°. The initial longitude, the latitude, and the height of the target are 0°, 0°, and 30km, respectively. The initial Mach number is 10. The local velocity angle is -1°, and the velocity drift angle is 0°. The controlled quantity is the attacking angle α and is constant with 15°. The integral method of the trajectory is fourth-order Runge-Kutta numerical integration. The height and the voyage are shown in Figure 1.Figure 1
Target trajectory.


## 3.2. Measurement Setting Condition

The measurement device is a pulse radar and the measurement elements are the distance R, the azimuth angle A, the pitch angle E, and the radial velocity V. The measure error standard deviations of the distance, the angle, and the velocity are 10m, 0.5mrad, and 1m/s, respectively. The longitude, the latitude, and the height of the radar station are 14°E, 1°N, and 0m, respectively. The sampling period is 0.05s and the tracking period is 70~790s. To compare the tracking performance of the models (including the tracking accuracy, the filter robustness, and the computation complexity), the simulation results cannot be compared as the general references setting a parameter. Mehrotra proposed that “Jerk” frequencyα has a good robustness in [7] via the simulation. That is, the filter performance can maintain stable when α changes in a large range. Therefore, the paper focuses on the standard deviation of Jerk and the robustness of the filter initial value error. The initial position and velocity in filter are equal to the true value adding the random error. The standard deviations of the position and the velocity are σp0 and σv0, respectively. The acceleration and the Jerk are 0. αJ model needs to set the initial value of α and the standard deviation σε of the input noise ε(t). MCSJ model needs to set the extreme Jerk value Jmax. The parameter sets of the three filters are expressed as P1, P2, and P3, respectively. The specific parameter values are shown in Table 2. The Monte Carlo simulation number is 100.Table 2
Filter parameters.
Parameters P1 P2 P3 α 0.1  α 0 α  σ j 1 1 0.1  J m a x 3 σ j  σ p 0 1000 100 100  σ v 0 100 10 10  σ ε 0.1


## 3.3. Simulation Results

The filter root mean square error (RMSE) of the position and the velocity under the parameter sets is shown in Figures2, 3, and 4, respectively. Because of the different values of P1 and P2, the initial value robustness of the four models can be compared. Comparing Figures 2 and 3, when the filter initial value is large, only the self-adaption state noise of MCSJ model can maintain the convergence and the initial value robustness is much better. But the other three models will present filter divergence performance during the 100 Monte Carlo steps and the convergence is influenced by the initial value which is not proper for the practical tracking application. On the other hand, because the only difference between P2 and P3 is σj. The parameter robustness of the four models can be compared. When comparing Figures 3 and 4, when σj is small, the tracking accuracy of the four models in the steady period of the target motion can be improved. However, the filter error has a large jump when the target has a severe maneuvering motion. The computation complexity ratio of SJ, CSJ, MCSJ, and αJ is 8:8:9:12.Figure 2
Filtering RMSE for four models in P1 parameter.
(a)
RMSE of position (b)
RMSE of velocityFigure 3
Filtering RMSE for four models in P2 parameter.
(a)
RMSE of position (b)
RMSE of velocityFigure 4
Filtering RMSE for four models in P3 parameter.
(a)
RMSE of position (b)
RMSE of velocity


## 4. Conclusion

The paper summarizes the proposed Jerk models which are applied on the high-maneuvering target tracking. The tracking accuracy, the filter robustness, and the computation complexity of the models are compared in the theoretical analysis and the tracking simulation. The tracking simulation trajectory is created by the integral of the dynamic equation which is similar to the true process and is pretty persuasive. The simulation results demonstrate that the self-adaptive noise variance method has the best tracking performance. However, in the highly maneuvering period, all models have the error jumping problem. Controlling the filter jumping error is the key to improve the near-space and high-speed target tracking performance in the future.



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