Resolvable Group State Estimation with Maneuver Based on Labeled RFS and Graph Theory

In this paper, multiple resolvable group target tracking was considered in the frame of random finite sets. In particular, a group target model was introduced by combining graph theory with the labeled random finite sets (RFS). This accounted for dependence between group members. Simulations were presented to verify the proposed algorithm.


Introduction
Multi-target tracking is widely used in defense and civilian fields. When multiple targets move in the air, they usually perform tasks in formation. The formation can be seen as group targets on a radar screen. When a shoal of fish swim and be detected by sonar. The shoal of fish usually exhibits the characteristics of group targets. In tracking space debris, it also shows the similar characteristics of group targets. Group target tracking can be seen as a special type of multi-target tracking problem. Most of the traditional target tracking algorithms are based on data association methods. When the number of targets increases, the computation time increases sharply. This has become an obstacle to the development of algorithms [1]. The Random Finite Set (RFS) method provides a new direction for research in multi-target tracking. It can avoid the process of data association and has become a research hotspot in the field of multi-target tracking [2,3]. The RFS method on the basis of the Bayes optimality [4,5] and multi-target estimation error [6] presents the theory of global target tracking in complex observation scenes through target set distribution. RFS method is already one of the important research directions of multi-target tracking [7] today. It can be deployed in a wide range of applications through a series of algorithms, such as the Probability Hypothesis Density (PHD) filter [8][9][10], Cardinalized PHD filter (CPHD) [11,12], multi-Bernoulli filter (MeMBer) [6], Generalized Labeled Multi-Bernoulli (GLMB) filter [13][14][15], and its multi-scan version [16,17]. In addition, Reference [18] represents a new breakthrough by demonstrating that the GLMB filter can track in excess of one million targets simultaneously, over one billion data points. It proposes an algorithm that can track more than one million targets per scan simultaneously. Different from these results, we considered the resolvable group tracking issue. This paper is an extended version of our conference paper (Reference [19]).
The PHD filter belongs to the moment approximation filter algorithm. It takes the first order statistical moment of the posterior probability of the multi-target state set to obtain a feasible approximate form. Further, a Gaussian Mixture PHD (GM-PHD) filter for linear Gaussian was proposed in Reference [10]. Subsequently, many scholars have studied the convergence problem [20,21], track consistency problem [22,23], and state extraction problem [24] of the PHD filters and made a series of

Labeled Random Finite Set (RFS)
Mahler introduced the theory of RFS to target tracking in a series of works [8,[53][54][55], where the multi-target state at time k was represented by a finite-set: The uncertainty in a multi-target state is described by random finite set models that captures birth, spawning, death, and motion. The multi-target state transition equation is given by: where S k|k−1 (x) denotes the surviving targets, B k|k−1 (x) denotes the spawned targets, and Γ k denotes the birth targets. The multi-target measurement is the finite subset: The theory of labeled RFS is given in References [13,14]. A labeled RFS is formed by augmenting a mark to the state of each target. In other words, we attach distinct labels ∈ L = {α i : i ∈ N} to different targets, where N is the set of positive integers. Labeled RFS requires that the labels of any two targets are different, i.e., the function: must equal 1. The densities of an LMB RFS and a labeled Poisson RFS are given in Reference [13]. For the LMB RFS, its density is described as:

Graph Theory
A graph G consists of two sets, the set of vertices V and the set of edges E [56]. At time k, the graph can be described by G k = (V k , E k ) [52], where V k , E k a are non-empty finite set. If the edges have direction, the graph is a directed graph; conversely, it is an undirected graph.
A group structure is similar to a graph structure, so we use the asymmetric adjacency matrix to describe the structure of the resolvable group target. This matrix can describe the collaborative relationship between the members of a resolvable group target, such as the parent-child relationship between mutually dependent targets. In the target adjacency matrix: a(i, j) = 1 means target i is the parent node for target j, and a(i, j) = 0 means target i is target j's child node, or target i has no relationship with target j. For example, the group structures in Figure 1 are described by the following asymmetric adjacency matrices:

Graph Theory Model of Labeled RFS
Let any vertex v i in the graph be a labeled state. For a group target, the set of vertices is finite, and we can define edges from the vertices set as follows: Equation (8) indicates the edges are defined on labeled states. When the edges only depend on the labels, the definition reduces to: Equation (9) shows that the graph only depends on the target labels. Hence, the structure of the group is encapsulated by the graph defined on the target labels.

Dynamic Model of Multiple Resolvable Group
If the target has a single parent node, the resolvable group targets dynamic model [7] is given as follows: where F k,l denotes the state transition matrix, Γ k,i is the state noise factor matrix, H k+1 is the observation matrix, ω k,i is the process noise, and υ k+1,i is observation noise. All these are assumed to be Gaussian. For the state x k,i = [p k,x (i),ṗ k,x (i), p k,y (i),ṗ k,y (i)] ∈ X k , p k,x (i) andṗ k,x (i) are the position and velocity of target i on the x-axis, p k,y (i) andṗ k,y (i) indicate the position and velocity of target i on the y-axis.
For b k (l, i), l is the parent node for target i. It contains the direction and distance information between the parent and child nodes. For a resolvable group with fixed formation and structure, see Figure 2.
x x x ȕ ȕ ș v Figure 2. Angle of resolvable group target structure. The node x 1 is a root and also parent node of nodes x 2 and x 3 . Target x 1 moves with velocity vector v 1 , where the angle is θ against x-axis. Besides, the angle of the child node x 2 and its parent x 1 is β 1 . Similarly, the angle between nodes x 1 and x 3 is β 2 .
That is, the angle β(s, d) between the velocity vector v s of the group targets and the position vector v d between the parent and child nodes remain constant against time. This can be illustrated by Figure 3. Let the formation be three nodes: x 1 , x 2 and x 3 , where x 1 is parent node and x 2 and x 3 are two child nodes. If the formation moves from Point A to B and keeps a fixed shape, two conditions should be met. First, the three nodes are with the same velocites. Second, the angle between two vectors, i.e., velocity vector of parent node x 1 and position vector between parent node and child nodes, should be remain unchanged. For instance, the angle β shown in Figure 3. Thefore, we assumed the angle β(s, d) is constant for a fixed formation. At time k, the motion direction for the parent target is given by the angle: Therefore, for group targets, the displacement vector b k (l, i) can be represented as: where R k (l, i) denotes the designed distance between parent node l and child node i. β k (l, i) is the designed angle between nodes l and i. If the parent node x k,l moves in constant velocity (CV) mode and the group is with a fixed formation, then the distance variable R k (l, i) and angles Thus, the displacement vector will be a constant, as in: Nevertheless, under a maneuvering motion model, the dynamic equation Equation (10) is nonlinear due to the displacment vector between node i and its parent node l dependent on the parent state x k,l . Assume that the group has a fixed formation, so the vector can be written according to Equation (14): where sin(θ k (l, i) and cos(θ k (l, i) are given by: Further, we can transfer Equation (15) to According to Equation (20), if c k (l, i) is a costant coefficient and C a is a constant matrix, then b k (l, i) can be seen as a linear transformation of the state x k,l dependent on some constants. In general, the constants are known in advance. For example, for a group with fixed formation and moving in a CT mode, the variables R(l, i), ṗ 2 k,x (l) +ṗ 2 k,y (l), a β,1 (l, i), and a β,2 (l, i) are all constant.
When the target does not have a parent node, its motion is not affected by other targets, and we call it a head node. The displacement vector b k (l, i) = 0 if the target does not have a parent node. If the target has multiple parent nodes and obeys linear motion, the model is expressed as: According to Reference [52], when all targets have the same transition matrix and some others condition hold, F k,l = F k . This is available from Equation (10): where In Reference [52], a new collaborative noise is proposed: Equation (26) suggests that the new noise is only influenced by the collaborative noise. So for each target i, we can build new model from the target state x k,i , adjacency matrix A d , and collaborative noise ω o k,i , using the following a proposition. Before this, a definition is first introduced.
Definition 1 (Reference [52]). A movement of group is said to be simple if it meets the following conditions: • The movement equations are all linear and same, i.e., F k,l = F k . • The movement mode is CV, CT with known turning rate, or the constant acceleration (CA). • The formation of group targets are fixed and, thus, the displacement vector only exists in the position displacement, i.e., b k (l, The collaboration relation of individual targets are of tree graph. This means each vertex has only one father vertex. Proposition 1 (Reference [52]). Suppose that the dynamic model of group targets is given by Equations (11), (23), (24). If the following conditions hold: (1) The group targets' movement are simple; (2) the displacement vector {b k (l, i)} is Gaussian, i.e., It follows from Equation (27) that the collaborative noise ω o k,i is Gaussian with zero-mean and covariance Q 0 k,i . It follows from Equation (28) that the acquisition of Q 0 k,i depends on the adjacency matrix A d , so we can write the state transition probability in the following form: Based on Proposition 1, when a group moves in a maneuvering model, a further result can be given as follows: Suppose that the dynamic model of group targets is given by Equations (11), (23), (24). If the group targets' movement are simple and target i with a parent l then: It follows from Equations (30), (31), (32) that the collaborative noise ω o k,i is Gaussian with µ 0 k (l, i) and covariance Q 0 k,i .

Remark 1.
It should be noted that the mean µ 0 k (l, i) is the bias of the designed displacement b k (l, i) and estimated displacement F k (x k,i −x k,l ). For a fixed formation, i.e., the matrix C a (l, i) is constant, the coefficient c k (l, i) is commonly time-varying for maneuver movement. However, if parent node velocity ṗ 2 k,x (l) +ṗ 2 k,y (l) can be gotten, then the coefficient c k (l, i) can be estimated. For example, for CT movement, the parent node velocity is a constant. Otherwise, the predicited value can be adopted.
Another point is the relation between covariance and adjacency matrix. From Equations (31), (32), to calculate the means and covariance, the parent vertex l should be first known. This is dependent on adjacency matrix Equation (6). In this paper, the adjacency matrix is defined on the label space and known in prior. That is, in the predicted stage, the adjacency matrix can be gotten and adopted according to the predicted labels. In contrast, if the adacency matrix is unknown, and it needs to be estimated according to the predicted states. In general, the adacency relation is based on the target states, or the motion information. A detailed discussion can be found in Reference [52].

State Prediction
For a resolvable group target x k , its prediction density is: If p(x k,i ) = N (x k,i , µ k,i , P k,i ), then: Therefore, the state covariance is related to the adjacency matrix A d .

State Update
The predicted density p k+1,i is Gaussian, and the corresponding posterior function is: The numerator of Equation (37) is derived by: where

The GLMB Filter
Under the standard multi-target transition and measurement model, the δ-GLMB filter [14,15] is an exact solution to the optimal Bayes multi-target filter. First, let: and let π be a δ-GLMB density: The δ-GLMB prediction density to time k + 1 is given by: where ω (I + ,ξ) + Note that ω B (I + ∩ B) is the weight of the birth labels (I + ∩ B), and ω ξ S (I + ∩ L) is the weight of the survival labels (I + ∩ L). p B (·, l) is the density of the newly born target, and p for each (I,ξ),Θ (M) = ζ (1) , · · · , ζ (M) is the element M of the highest weight Θ.ω (I,ξ,θ (i) ) , ω (I,ξ,θ (i) ) is the weight after truncation.

The UKF GLMB Filter
The constant turn model is a nonlinear motion model with unknown turning rate. The unscented Kalman filter (UKF) is an algorithm for nonlinear filtering proposed in Reference [57]. This paper used the UKF to perform prediction and update for individual track following the CT model in the GLMB filter algorithm. The UKF filtering algorithm is summarized in Table 1.
2. Sigma parameter point prediction: (1) Matrix parameter s (j) k|k−1 ,n ,P j k|k−1 ,n and target existence probability r k|k−1 ,n can be seen in Table 2.
(2) Other parameters S (j) k|k−1 ,n ,C (j)xz k,n and K (j) k,n can be seen in Table 2.

Get target status ox (j)
k,n ; covariance matrix P (j) k,n ; target existence probability r (j) k,n seen in Table 2.

Efficient Implementation of the GLMB Filter
The Gibbs GLMB algorithm [15] combines the prediction and update steps of the GLMB filter, which effectively improves the efficiency of the truncation process. So, we introduced it to solve the problem of resolvable group target tracking under nonlinear condition.
where ω (I,ξ,J,θ) k from some distribution π. It should be noted in the predicted density p (ξ) k|k−1 (·, l) that the collaboration noise ω o k,i (Equation (30)) is adopted, instead of process noise ω k,i , as in Equation (25).

The Algorithm Implementation and Settings
The group target state estimation is complicated due to the limited observations and collaboration between targets. In this paper, we defined the collaboration on the labels of individual targets. Although a target label is just a "temporary identity", the identity contains the collaboration information (group structure) modeled in the adjacency matrix. We tried to use the "temporary identity" to get the group structure and show it is important in estimating target states if it is known in prior.
Therefore, the adjacency matrix was assumed to be known. In contrast, the number of targets and sub-groups were all unknown and needed to be estimated. For simplicity, we did not consider the estimation of sub-groups, which was considered in Reference [52]. Interested readers can refer to it for further information.

Simulations
For this simulation experiment, the radar sensor is adopted to track group targets. The UKF-GLMB and UKF-Gibbs GLMB filters are used for comparison. Two experiments were compared in this section. In experiment 1, two filters were used to track group targets with cooperative. And in experiment 2, they were used to track group targets with cooperative to non-cooperative. In both simulations, the state vector is x k = [p k,x ,ṗ k,x , p k,y ,ṗ k,y ]. The dynamic function and the radar observations are given by: where: And the initial state of the two parent targets are:

Scenario 1
For this simulation, we used Gibbs GLMB filter and GLMB filter for comparison. In the simulation, the group targets are shown in Figure 4, including two sub-groups. The distance between any parent and its child vertices was 100 m. Each sub-group target contained four targets, i.e., {(x 1 , 1 ), · · · , (x 4 , 4 )} as sub-group 1 and {(x 5 , 5 ), · · · , (x 8 , 8 )} as sub-group 2. The two sub-groups are independent of each other. Let the adjacency matrices for the two sub-groups be known and given by: The monitoring range of the experiment was [−π/2, π/2; 0 m, 3000 m]. The experiment lasted 100 s, sub-group 1 was born at the time of k = 0 s and disappeared at the time of k = 70 s, and sub-group 2 was born at k = 20 s and disappeared at k = 100 s. The covariance of the observed noise R = diag[0.0012100]. The covariance of process noise Q = diag[0.040.040.04]. The real trajectory of the target is shown in Figure 5. The curve represents the trajectory, the circle represents the starting point, and the triangle represents the end point. In this experiment, GLMB and Gibbs GLMB were used to estimate them, respectively. The state estimation obtained by UKF-GLMB filtering algorithm is shown in Figure 6 and by UKF-Gibbs GLMB filtering is shown in Figure 7, the OSPA distance is shown in Figure 8 and by UKF-Gibbs GLMB filtering is shown in Figure 9. The number of targets is estimated in Figure 10 and by UKF-Gibbs GLMB filtering is shown in Figure 11. It can be seen from Figures 6 and 7 that both filters could accurately estimate the motion state of each target at each moment. It can be seen from the OSPA loc and OSPA Dist in Figures 8 and 9 and Table 3 that the Gibbs GLMB filter was better in the state estimation effect of the target. It can be seen from the OSPA Loc and OSPA Dist in Figures 8 and 9 that the Gibbs GLMB filter was better in the state estimation effect of the target. For the estimation of the number of targets, it can be seen from Figures 10 and 11 that both could effectively track the number of targets, but Gibbs GLMB performed better, and it can be seen from the OSPA Cad in Figures 8 and 9 that the delay time of SCA-GLMB in the tracking process was less than the GLMB filter. In addition, we recorded the running time of the two filtering algorithms in ten experiments, as shown in Table 3. As can be seen from Table 1, the time required for the GLMB filtering algorithm to run once was 264.5118 s, while that for Gibbs GLMB was 43.4962 s. So, we can see that the average efficiency of the Gibbs GLMB filter was more than six times faster than the GLMB filter.   True Estimated Figure 11. The estimated number of targets by Gibbs GLMB filter.

Scenario 2
In this subsection, the group targets, including two sub-groups, are shown in Figure 12. Similarly, each sub-group target contained four sub-targets and the adjacency matrix was the same as Scenario 1. The two sub-groups were mutually independent. Sub-group 1 was born at time k = 1 s and survived in  Figure 12, where the curve shows the trajectory, the circle represents the starting point, and the triangle is the end points. The state estimation obtained by UKF-GLMB filtering is shown in Figure 13 and the UKF Gibbs GLMB filtering is shown in Figure 14. The OSPA distance is shown in Figure 15 and by UKF Gibbs GLMB filtering is shown in Figure 16. The number of targets is shown in Figure 17 and the UKF Gibbs GLMB filtering is shown in Figure 18. From Figures 13 and 14, we know that both filters can accurately estimate the motion state of each target. The Gibbs GLMB filter was better in the state estimation as shown by Figures 15 and 16. For the number of targets, it can be seen from Figures 17 and 18 that both can effectively estimate the number of targets and Gibbs GLMB performed better in speed. This can be seen from the OSPA metric plotted in Figures 15 and 16. As in Scenario 1, we recorded the running time of the two algorithms. In ten experiments, as shown in Table 4, the time cost for GLMB filtering was 284.1470 s, while for Gibbs GLMB it wa 23.7307 s. In tracking performance, the two algorithms had a close performance.

Conclusions
In this paper, we concentrated on multiple resolvable group targets tracking. Different from the previous work, we considered the collaboration relation of the group. We incorporated graph theory into labeled RFS to model collaborative dependence between targets in the same group. Based on original GLMB filter and Gibbs GLMB filter, we considered the resolvable group target estimation, where the new collaboration noise under maneuver was modeled. Simulations on nonlinear example validatde the soundness of the proposed algorithm.