Guanhua Ding, Jinping Sun, Ying Chen, Juan Yu

Abstract: For modern phased array radar systems, the adaptive control of the target revisiting time is important for efficient radar resource allocation, especially in maneuvering target tracking applications. This paper presents a novel interactive multiple model (IMM) algorithm optimized for tracking maneuvering near space hypersonic gliding vehicles (NSHGV) with a fast adaptive sampling control logic. The algorithm utilizes the model probabilities to dynamically adjust the revisit time corresponding to NSHGV maneuvers, thus achieving a balance between tracking accuracy and resource consumption. Simulation results on typical NSHGV targets show that the proposed algorithm improves tracking accuracy and resource allocation efficiency compared to other conventional multiple model algorithms.

Keywords: near space hypersonic gliding vehicle (NSHGV); target tracking; adaptive sampling;interactive multiple model (IMM)

1 Introduction

Time is an essential resource for modern multifunction radars, and the effective allocation of it improves the system performance in practical applications such as wide area surveillance and multiple target tracking. For phased array radars with electronically steered antennas, the beam direction can rapidly change, thus enabling the radar to perform tracking operations on multiple targets simultaneously [1]. Since the system load capacity and processing power are limited, the revisit time (or sampling interval) for different targets should be appropriately determined to achieve maximum efficiency. Therefore, the adaptive sampling control in target tracking tasks is substantial for obtaining the desired tracking accuracy with less time resource consumption. In general, the existing adaptive sampling algorithms determine the revisit time based on factors such as the target’s maneuver status, distance to radar, environmental perturbations, etc.Since the target’s maneuver is one of the primary factors influencing the tracking accuracy,most of the adaptive sampling methods decrease the revisit time for maneuvering targets while increasing that for non-maneuvering ones. Some adaptive techniques (see, e.g., [2, 3]) determine the sampling intervals by selecting among a set of predefined values or solving constraint equations. However, the computational cost is worth noting for these methods as they require executing the filtering procedure repeatedly to obtain the revisit time. Some other noniterative adaptive methods (see, e.g., [4, 5]) improve the computational efficiency by directly calculating the revisit time, but with the cost of system complexity or performance compromise.

In recent years, the development of near space hypersonic gliding vehicles (NSHGV) has attracted worldwide attention [6]. Unlike common aerodynamic and ballistic targets, the NSHGV travels in a unique gliding trajectory with high speed (usually above Mach 5),enabling it to perform strong maneuvers and potentially become a high-speed strike weapon.The major challenge of NSHGV tracking arises from the target maneuver characteristics. For traditional single-model tracking algorithms, the dynamic model usually cannot precisely describe the motion of NSHGV, resulting in accuracy deterioration and filter divergence. To solve this problem, some researchers modified the dynamic model based on the aerodynamic forces or periodic motion features of the NSHGV [7, 8]. Unfortunately, these models usually require more a prior information of the target, and their performance is sensitive to the model parameters.Another more flexible solution is to track the NSHGV with multiple model (MM) methods,such as the interactive multiple model (IMM) [9]and variable structure multiple model (VSMM)algorithms [10, 11]. These methods illustrate NSHGV maneuvers through the combination of several simple dynamic models, obtaining better adaptability than the single-model approaches.However, these methods have higher computational costs than single-model ones and require a proper model set design. More recently, the datadriven trajectory fitting method [12, 13] is proposed for tracking maneuvering targets with continuous-time trajectory functions, which could be potentially applied in NSHGV target tracking.

Currently, the adaptive sampling methods have rarely been discussed for tracking NSHGV targets in the existing references. This paper proposes a novel sampling adaptive multi-model algorithm optimized for NSHGV target tracking.The proposed algorithm is referred to as improved fast adaptive IMM (IFAIMM), which can effectively respond to the target’s sudden maneuvers by adjusting the revisit time based on model probabilities. Simulation results indicate that IFAIMM achieves higher utilization efficiency in radar time resources, obtaining better overall NSHGV tracking accuracy than other conventional MM algorithms.

2 NSHGV Motion Characteristics

According to the existing study [14], the trajectory of NSHGV can be divided into three basic phases: boost, coast, and reentry. After launching, the NSHGV is propelled in the boost phase until the engine stops, then enters the coast or ballistic phase until it returns to the atmosphere.The motion characteristics and dynamic model of these two phases are similar to that of the conventional ballistic targets, which have been discussed in various researches such as [15]. However, the NSHGV travels through a unique oscillatory gliding trajectory during the atmospheric reentry phase, as illustrated in Fig. 1. Therefore,the discussion in this section is mainly focused on the reentry phase.

Fig. 1 Typical trajectory of NSHGV

During the reentry period, the NSHGV has a significantly higher velocity than conventional aerodynamic vehicles and can perform more complex maneuvers than ballistic targets. Since it is challenging to model the complicated maneuvers with kinematic equations, the motion of NSHGV is often analyzed with the aerodynamic method.Assuming that the Earth is a homogeneous sphere, and the NSHGV is only affected by aerodynamic drag and lift forces, the gravity, and Coriolis force caused by the Earth rotation.Then, the motion model of NSHGV is given by(see, e.g., [16, 17]):

whereV,θ,σ,?,λ,rare velocity, flight path angle, heading angle, longitude, latitude and geocentric distance, respectively.ωeis the rotation angular speed of the Earth andgis the gravitational acceleration. In practice, the bank angleνis often used as a control variable for NSHGV maneuvers. The lift accelerationLand drag decelerationDare associated with the air densityρ, reference areaSand vehicle massm, calculated by

where the air densityρis approximated by the 1976 U.S. Standard Atmosphere model [18].Besides, the coefficientsCLandCDare determined by the geometry structure parameters of NSHGV, the angle of attackηand the Mach numberM, expressed as

In conclusion, the high velocity and strong maneuvers of NSHGV require the adaptive sampling algorithm to obtain high tracking accuracy with fast responding speed, which is challenging for the existing adaptive methods introduced in the following section.

3 Review of Adaptive Sampling Methods

In general, the design purpose of adaptive sampling tracking methods is to balance the tracking accuracy and the time resource consumption.The revisit time should be determined according to the following considerations:

1) The revisit time should be short enough,so the prediction error is small enough that the target can be illuminated in the next predicted position by the radar beam.

2) The revisit time should be as long as possible so that the radar will have more available time resources for tracking other targets.

Therefore, the adaptive methods need to calculate the longest possible revisit time while achieving acceptable overall accuracy. Most of the proposed algorithms fulfill this purpose based on three methods (see [2, 3, 5]) which are briefly described as follows.

The first method determines the revisit time with the target position residual (or filter innovation). Since the residual directly reflects the unpredictability of the maneuvering target, the revisit time should be decreased with higher position residual, resulting in the following iterative formulations as

whereT(k) is the revisit time at timek,e(k) is the position residual, andσkis the standard deviation of measurement noise. Since the residual can vary rapidly even in non-maneuvering periods, using a first order filter to smooth the revisit time outputs is also suggested in [2].

The second method calculates the proper revisit time from the predicted error covariance.When the target maneuvers, the predicted error covariancePk+T|kincreases with higher uncertainty in the estimates. The next sampling is scheduled when the predicted error covariance exceeds a given thresholdPth. In practice, the threshold is usually selected according to the measurement noise covariance, and the nondiagonal elements are often ignored. Then, the sampling interval has following constraints

where Tr[·] is the matrix trace operator, parameterλbalances the tracking accuracy and system load. Since there is no explicit solution for the sampling interval, the maximumT(k+1) is usually determined with Newton’s method. Both the first and second methods require iterations to determine the proper sampling interval, which increases the computational burden of the method, especially for multiple model algorithms.

The third method utilizes the model probabilities of IMM algorithm to determinate the revisit time. The method applies two dynamic models with different revisit times and obtains the sampling interval as the weighted sum of these time values, formulated by

whereμi(k), i=1,2 are the model probabilities at timek,Ti, i=1,2 are the revisit time values assigned to different models. This algorithm is known as the fast adaptive IMM (FAIMM)because it does not require iterations to determine the revisit time. However, the fixed sampling intervals assigned to each model limit the adaptive performance of FAIMM. More recently,a novel FAIMM using steady state filters(SSFAIMM) is proposed in [19], which further reduces the computational load. Unfortunately,the steady state filter is unsuitable for the NSHGV tracking task since its optimality depends on steady system conditions, which can be violated in maneuvering target tracking scenarios.

4 Improved Fast Adaptive IMM Algorithm

4.1 IMM Framework for NSHGV Tracking

The proposed IFAIMM algorithm is based on the classical IMM framework, which is a suboptimal Bayesian method widely applied in state estimation. The IMM algorithm parallelly operates multiple filters with different dynamic models to overcome the shortage of single model methods.The transfers between these models are assumed to be governed by a Markov chain, and the final state estimate is calculated by

whereMis the number of models andx?ikrepresents estimations from modeli.

A properly designed model set is essential for the IMM algorithm to obtain high tracking accuracy and stability. Considering that the NSHGV has complex maneuvers, the IFAIMM algorithm uses three current state (CS) models with different parameter settings. The CS model, which has been widely applied in maneuver target tracking,is in essence a modified Singer model with adaptive acceleration mean value [20]. Denote the revisit timeT(k) asTkand set the state vector asx=[xx˙x¨ ]T, then the state transform equation of CS model is defined by

where

αis the reciprocal of the maneuver time constant (maneuver frequency), ˉakis the mean acceleration in the current sampling interval. The process noisewkis a zero-mean Gaussian process with covariance matrixQk, which is formulated by

4.2 Adaptive Sampling Logic

1) During the strong maneuvering periods of NSHGV (i.e., the target acceleration is closer to a white-noise sequence rather than a constant value), CS model 1 with the biggestαwill have the highest model probability.

2) On the contrary, CS model 3 with the smallestαwill take advantage in the weak maneuvering periods (when the target has nearconstant acceleration).

Therefore, we propose an improved fast adaptive method that adjusts the target revisit time using model probability outputs of the IMM algorithm, described as follows.

Assuming that the minimum and maximum sampling interval areTminandTmax, the unit time step is ΔT, the model probability at timekisμi(k), i=1,2,3. Then, the adaptive sampling logic is summarized as Algorithm 1.

Algorithm 1 Adaptive sampling logic T(k) μi(k), i=1,2,3Tk[1:n]Input: , , .T(k+1) Tk+1[1:n]Output: , .Tk+1[1:n-1]=Tk[2:n]. (19)μ3(k)＞max(μ1(k),μ2(k))if Tk+1[n]=min(T(k)+ΔT+,T,μ3(max). (20)μ1(k)＞max(μ2(k)k))else if Tk+1[n]=max(T(k)-ΔT-,Tmin). (21)else Tk+1[n]=T(k). (22)endif Tk+1[1:n] T*[1:n]Sort in ascending order to get .T(k+1)Calculate by:T(k+1)=mean(T*[1:l]). (23)

In (20)-(22), the temporary revisit timeTk+1[n]is determined according to the probabilities of three CS models. In order to avoid abrupt changes in sampling intervals, the queueTk[1:n]is used as a moving average filter to smooth the outputs from (20)-(22). At timek, the adaptive logic calculates the next revisit timeT(k+1) by averaging thelsmallest values inTk+1[1:n]. Consequently, the adaptive logic inclines to decrease the revisit time faster than increase it, resulting in a higher response speed to sudden maneuvers of the target. Compared with the conventional FAIMM algorithm, our proposed IFAIMM has more flexible parameter settings which enable advanced adjustments for adaptive characteristics (see section 4.3 for further discussion).

4.3 Implementation Issues

To improve the performance of IFAIMM, the practitioners should pay attention to the following issues.

Firstly, the parameters for the adaptive logic(queue lengthn, increasing time step ΔT+,decreasing time step ΔT-, average numberl) can be adjusted for different adaptive characteristics.For example, the sampling interval will increase slower with a longer queue, and decrease faster with a smaller average number. These adaptive logic parameters can be optimized heuristically based on the application requirements.

Moreover, the target maneuvers should also be considered in determining the maximum revisit timeTmax. If the revisit time is excessively increased during the weak maneuvering periods,the radar may lose the target when it performs abrupt maneuvers. The determination ofTmaxshould base on various factors including the radar beam width, target maximum acceleration,target distance, etc. However, the detailed calculation method is beyond the scope of this paper.

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Finally, when the IFAIMM algorithm is applied in the three-dimensional space, the estimation filters in different axis of the Cartesian coordinates should be decoupled. The overall revisit time is calculated by

whereTx(k), Ty(k) andTz(k) are obtained in coordinate directionsx, yandz.

5 Performance Evaluation

In this section, the performance of IFAIMM,FAIMM, IMM and hybrid-grid VSMM (HGMM,see [11]) is evaluated by Monte Carlo simulations in a simplified NSHGV tracking scenario.The tracking accuracy and time resource utilization efficiency are compared between these four algorithms.

To perform a comprehensive evaluation of the algorithms, we apply two benchmark NSHGV trajectories in the following simulation.The first trajectory shown in Fig. 2 represents the boost and coast phase of a NSHGV target(referred to as target 1). Considering that the NSHGV rarely performs horizontal maneuvers during these periods, the motion of target 1 is modeled in the vertical plane. The generated trajectory is then converted to three-dimensional NE-U coordinates. The motion model of target 1 is formulated by

Fig. 2 Trajectory of NSHGV target 1

whereV,θ,r,Raandmare velocity, flight path angle, geocentric distance, horizontal range and vehicle mass, respectively.FTis the thrust force,Iis the specific impulse, and?is the angle between the thrust force direction and the vehicle’s central axis.LandDcan be calculated with (2).

For target 1, assuming that the total mass at launchingm0is 15 000 kg, the fuel massmfis 11 000 kg, the specific impulseIis 2 600 N·s/kg and the thrust forceFTis 260 kN. With these parameters, the engine works fort=mfI/FT=110 sin total. Target 1 launches at sea level, flies vertically in the first 20 s, and then travels with a constant thrust angle?=4°until the engine stops at 110 s. After that, target 1 enters the coast phase and ends the flight at 300 s.

The second trajectory shown in Fig. 3 represents the reentry phase of a NSHGV target(referred to as target 2). The trajectory is generated with the motion model in section 2, and then converted to the N-E-U coordinates. As shown in Tab. 1, the parameters for target 2 are determined according to the experiment data of CAV-H hypersonic vehicle [22].

Fig. 3 Trajectory of NSHGV target 2

Tab. 1 Parameters of NSHGV target 2

During the reentry flight, the angle of attackηis set to 10°and the bank angleνis set to 20°.The initial state of target 2 is set as

And the reentry flight is maintained for 500 s. The simulated trajectory shows that target 2 follows the typical oscillatory gliding characteristic of NSHGV.

For the IFAIMM, FAIMM, IMM and HGMM in comparison, the state vector of the target is defined as

wherevkis zero-mean Gaussian white noise with covariance matrixR, (xr,yr,zr) is the radar position. Since the measurement model is nonlinear,all four MM algorithms use cubature Kalman filter [23] as sub-filters.

To control the variables, the three IMM algorithms in comparison have the same model set, sub-filters, and initial states. The only difference is in the sampling control method. The common parameters for these IMM algorithms are listed in Tab. 2.

For FAIMM, the minimum revisit timeTminis assigned to the CS model 1, and the maximum timeTmaxis assigned to the CS model 3.Since the model set includes three models instead of two in the original FAIMM algorithm, the revisit time is determined by

which is a normalized form of (8). The additional parameters for IFAIMM and FAIMM are adjusted to obtain comparable total sample times(around 300 samples for trajectory 1 and 360 samples for trajectory 2). See Tab. 3 and Tab. 4 for the parameter values.

Tab. 2 Common parameters

Tab. 3 IFAIMM parameters

Tab. 4 FAIMM parameters

In HGMM, the maneuver frequencyαof CS model is chosen to generate the continuous model space. The fixed model set (coarse grid)MC={αC1, αC2, αC3}includes 3 CS models, and the moving model setMF={αF1, αF2}is formed with fixed fine grid distance (see [11], section III.A for more details). In the simulation, the coarse grid is defined as

and the coarse grid distance isd=0.045 s-1. Then the fine grid can be expressed as

where the model estimates?kis obtained by

Fig. 4 RMSE for tracking target 1

Fig. 5 RMSE for tracking target 2

Fig. 6 Revisit time of IFAIMM and FAIMM for target 1

Fig. 7 Revisit time of IFAIMM and FAIMM for target 2

Tab. 5 Simulation results for tracking target 1

Tab. 6 Simulation results for tracking target 2

First, we compare the tracking performance of IFAIMM, IMM, and HGMM. When the NSHGV performs strong maneuvers (see, e.g.,10 s-25 s in Fig 4, 170 s-220 s in Fig. 5), the IFAIMM provides higher tracking accuracy and lower peak RMSE compared to IMM and HGMM with a similar average revisit time. However, the average tracking error of IFAIMM is slightly higher than the other 2 fixed-sampling algorithms since the adaptive logic allocates fewer samples for weak maneuvers. Note that the HGMM (including 5 CS models) obtains the lowest average RMSE with the cost of a significantly longer execution time. Simulation results listed in Tab. 5 and Tab. 6 also suggest that the IFAIMM provides better overall tracking performance with limited time and computational resources.

Furthermore, the IFAIMM is compared with the FAIMM algorithm. It can be found from the results that the IFAIMM obtains lower peak RMSE than FAIMM, although they have similar time resource consumption (average revisit time). Fig. 6 and Fig. 7 further show that the adaptive logic of IFAIMM is more sensitive to target maneuvers, resulting in a wider adaptive range for revisit time. It is worth noting in Fig. 7 that the IFAIMM adaptive logic increases the revisit time slower than decreases it, which benefits the tracking accuracy and improves algorithm stability. The improved fast adaptive logic is computationally efficient as the execution times for two adaptive IMM algorithms are almost the same. As a conclusion, the IFAIMM provides better overall tracking performance and flexibility compared to FAIMM.

6 Conclusion

In this paper, we present IFAIMM as a novel adaptive sampling algorithm optimized for tracking NSHGV targets. The algorithm applies an effective sampling control logic to the IMM framework, which utilizes the probability of different CS models to dynamically adjust the target revisit time. Simulation results indicate that the IFAIMM algorithm obtains superior overall performance than the fixed sampling IMM and the conventional FAIMM algorithm. Consequently, the proposed algorithm achieves higher utilization efficiency for radar time resources,which benefits its practical applications for tracking NSGHV and other maneuvering targets.