Multistage Clustering-Based Localization for Remote UAV Swarm: A Coalitional Game Framework

In the GNSS-denied localization scenario, only received signals and corresponding distance measurements are available. Based on that, in this letter, we construct a multistage clustering-based model for UAV swarm. We characterize the proposed model as a coalition formation game (CFG) and provide corresponding preference criteria for algorithm design. A tree-like multistage clustering mechanism is adopted based on a coalitional graph game (CGG). Each cluster (local map) performs localization calculation and then merges through neighboring drones (NDs) stage by stage. The simulation results show that the proposed scheme can achieve better localization accuracy than the comparison algorithms and is more robust for irregular network topologies.


I. INTRODUCTION
U NMANNED aerial vehicles (UAVs, also referenced as drones) has the superiority of large-scale, rapid, and flexible deployment. In the GNSS-denied environment, UAV swarms can effectively support communication devices to perform localization. Likewise, the positions of UAV swarms are also crucial for their mission execution.
This letter implements ultra-wideband (UWB) technology, whereby received signals and corresponding distance measurements are available for localization. For swarming localization, multiple dimensional scaling (MDS) [1] is frequently used, especially when obtaining a complete Euclidean distance matrix (EDM). MDS-MAP adds absolute coordinate mapping. The core of MDS is Singular Value Decomposition (SVD). Due to complex electromagnetic environments, such as remote distance, multipath propagation, and NLOS, distance measurements encounter a significant deviation or even failure, which reduces the accuracy of localization technology. MDS-MAP (P) [1] sufficiently alleviates the calculation error by hopbased clustering. Many MDS-based works employ matrix completion methods such as the shortest path information [1] and SVD-MDS [2], [3]. Chen et al. in [4] proposed an improved SMDS and improves the localization accuracy by relaxing the fully connected requirements, which requires angle information. In [5], we improved MDS-MAP (P) by presenting a CFG-based two-stage clustering scheme.
However, all those clustering schemes cannot adapt well to different topologies of networks and sometimes fall into the dilemma of error accumulation. Based on this, we propose a novel clustering-based robust scheme in remote swarming localization, and the work is described as follows: • This letter constructs a multistage clustering-based model. We introduce anging packet losses (RPLs) using received signals and further characterize the O-EDM rate, which is tightly coupled with localization performance. • We present a tree-like multistage scheme for clusteringbased localization. Then, a coalitional game framework is adopted to explore the characteristics of the model. Then, we design localization algorithms based on the proposed game model. Simulation results show that the proposed methods effectively reduce RPLs and achieve better localization performance in different network topologies and remote scenario compared with comparison algorithms. The main differences of our work with others can be summarized as: 1) Different from hop-based clustering [1]- [4], this letter performs clustering for swarm based on the O-EDM rate. 2) Compared with the previous CFG-based clustering scheme in [5], a tree-like multistage clustering scheme based on CGG is designed to learn the optimal clustering and is more robust for irregular network topologies. This letter adopt asymmetric double-sided two-way ranging (ADS-TWR) in the time-of-flight (TOF) localization. Moreover, the physical layer and MAC layer of the system are designed regarding the IEEE 802.15.4-2011 UWB standard [6].
The rest of the letter is organized as follows. Section II shows the system model & problem formulation for multistage clustering-based localization. Section III analyzed a CFG approach for system model, and built a CFG-based cooperative localization scheme. Simulation results are presented in Section IV. Finally, Section V gives the concluding remarks.

II. MULTISTAGE CLUSTERING-BASED LOCALIZATION
SYSTEM MODEL FOR REMOTE UAV SWARM We consider a remote localization scenario for UAV swarm with GNSS denied, which consists of N drones, denoted by N = {1, 2, . . . N}, and N 1 ground stations (GSs), denoted by N 1 = {N + 1, N + 2, . . . , N + N 1 }. System duration is evenly divided into T discrete slots, denoted by T = {1, 2, . . . T }. GSs are placed to: 1) Communicate with drones and give commands; 2) Act as anchors to determine the coordinate system. Communication links among GSs are available. The objective of our scenario is to achieve optimal localization performance for UAV swarm based on relative ranging information. For ease of representation, we identify UAV and GS This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ collectively as communication user (CU). Denote the position of CU n ∈ N 2 = {N 1 ∪ N } as p n,t = [x n,t , y n,t , z n,t ]. The elevation z n,t can be obtained by the barometer equipped with drones. Define D r = [d n,k ] ∈ R N2×N2 as the EDM of the whole network, where d n,k (t) = p n,t − p k,t is the Euclidean geometric distance between CU n & k, k ∈ N 2 ( · 2 is the L2-norm), N 2 = N + N 1 . CUs receive relative ranging information using ranging devices. ADS-TWR minimizes the measurement error in TOF calculation, which does not require the same response time on every device [7]. Define H • = [r n,k ] ∈ R N2×N2 as the observed EDM (O-EDM). Due to remote localization, there occurs losses of ranging values in H • which we call RPLs.
Denote available channels as C = {1, 2, . . . , C}, where B c is the bandwidth of c ∈ C. To accurately describe the ranging and transmission under UAV-assisted communications, this letter introduces the UAV-to-X (U2X) channel model [8], [9]. Suppose there are no interferences since CUs transmit messages in sequence in ADS-TWR. Thus, the received power of the signal from CU n to k over channel c is: where Pt c n is the transmit power of U n, ε 1 t & ε 2 t are the instantaneous fading coefficient. G is the constant power gains factor introduced by amplifier and antenna. For UAV-to-UAV (U2U) channel model, the path loss is given by the Friis equation. Suppose the path-loss factor is α. d −α n,k (t) · ε 1 t represents the pathloss in U2U links. For UAV-to-network (U2N) channel model, the path loss from n to k over channel c is expressed as PL c n,k (t) = P LoS,n (t) · PL c LoS,n (t) + P NLoS,n (t) · PL c NLoS,n (t), where the LoS & NLoS pathloss PL c LoS,n & PL c NLoS,n are given from [9], [10]. P LoS,n (t) = (1 + a exp (−b (θ n,k (t) − a))) −1 is the probability of LoS connection, θ n,k is the elevation angle from n to k, P NLoS,n (t) = 1 − P LoS,n (t). Hence, the received signal-to-noise ratio (SNR) from n to k is: where N 0 is the noise power per unit bandwidth. a n,c (t) = 1 shows that CU n accesses channel c, while a n,c (t) = 0 dose not. Eq. (2) shows that successful communication between CU n & k only occurs when they access the same channel c. Considering that ADS-TWR requires demodulating signals of both bidirectional U2X links for calculation, we define η loc n,k (p n,t , p k,t , p n ) = min η loc n→k (·), η loc k→n (·) (abbreviated as η loc n,k (t), η loc n,k (t) = η loc k,n (t)). To characterize RPL, we define the indicator function φ n,k (t) as follows: where η min is the minimal demodulation threshold of the signal. φ n,k (t) indicates that whether the ranging is successfully performed. Denote μ n,k as the measurement error, then the observed ranging value is expressed as r n,k = φ n,k (t) · (d n,k + μ n,k ). The RPLs of the received signal in the There exists a negative correlation between RPL and localization performance. Define the O-EDM without missing RPLs as a complete O-EDM, we then present the O-EDM rate of the cluster m, which is: where Φ cmp m is the number of ranging values for cluster m with a complete O-EDM. ψ m (t) (Max. 100%) reflects the relationship between RPLs and localization performance intuitively and has a theoretically optimal value. Our work aims to find an optimal solution for clustering ({c n } n∈N2 ) and multistage mechanism ({st m , st + m , st − m } m∈M ) to achieve maximal O-EDM rate of the whole network, which is: The proposed model can achieve optimal localization performance utilizing ψ m (t). The following section models the problem as coalitional games and presents solutions.
III. PROBLEM SOLUTION & ALGORITHM DESIGN This section solves the clustering scheme and then introduces a tree-like multistage mechanism based on coalitional games (CG). Firstly, we present the clustering algorithm under the proposed CG framework (Algorithm. 1, CG-Tree Loc Algo) and then prove the convergence and stability. Then, we adopt the localization algorithm. "Coalition" refers to "Cluster" in the following description.

A. Clustering-Based UAV Swarm for Remote Localization Constructed as a Coalition Formation Game
The proposed localization model satisfies the characteristic of the cooperative game. In this subsection we construct the proposed UAV swarm clustering model as a coalition formation game (CFG), which is characterized as is the utility function, indicating the total payoff generated by any coalition CO, and is characterized as follows: V (CO, Π) = ψ m (t).
Coalition selection rule (CSR is proven that the merge & split effect can be achieved by a certain number of CS rules or switch rules [11], while avoiding the batch strategy selection of CUs. Log-linear learning is adopted for coalition updates to avoid local optimum caused by the diversity of strategy sets. Denote V * = V (CO, Π * ) + V (CO * , Π * ), V = V (CO, Π) + V (CO * , Π). Set β > 0 as the learning parameter, then we describe the update condition as follows: Therefore, even if the CU's strategic choices have stalled, it may still be on track soon. Remark 1: With the preference relation of coalition order and coalition selection/switch rule, G can be converged to the stable coalition partition under Algorithm. 1.
Theoretical analysis can refer to Section III (D) in [5]. Note that O-EDM rate has performance upper bound (100 %), the stable structure is generally the optimal solution of G. After coalition formation, coalitions need to determine their stages, so as to ensure a lower cumulative localization errors.

B. Tree-Like Multistage Mechanism Design Based on Coalitional Graph Games
Learning from the concepts of coalitional graph games (CGG) [12], we innovatively take coalitions as players in Firstly, determine the stage of GSs as Lv.1, says m * . Then, select the coalition farthest from m * , say m 1 ; Next, select the nearest coalition of m 1 , say m 2 as its subordinate coalition. Finally, we obtain optimal U m1 (·), and ε m1,m2 = 1. We then exclude coalition m 1 and continue above cycles until all coalitions are connected.
Next, we present selection mechanism of ND set B. Denote the O-EDM rate from n to coalition m 1 &m 2 as: We similarly used BR to find the best four CUs (using sort function in Matlab) that maximize the O-EDM rate as NDs.
After adjusting the ND sets, the trend of changing value in V (CO, Π) is the same as in ψ n m1,m2 (t). Remark 2: Under the proposed feasible local strategy in Algorithm. 1, the CGG G 2 is proven to be a local Nash network. Generally, pairwise stability exists, which indicates that G 2 can achieve a stable state. That is, no CU can improve its O-EDM rate through adjusting strategy.

C. Localization Algorithm Design
Based on Algorithm. 1, we design a multistage clustering-based UAV localization algorithm (CG-Tree Loc Next, we give the complexity analysis of the algorithm. For Algorithm. 2, since the sort function requires traversing all coalition, the computation complexity for each time slot j is O M 2 ; Clustering reduces the time complexity of MDS-MAP, which is O N ave · N 2 , whereN is the average number of nodes in each clustering. Besides, denote O (Rec) as the computation complexity for recording data each time slot. Thus, the total computational complexity of the proposed CG-Tree Loc Algo is calculated as: e th is the maximum error allowed for successful localization, then LocSR(%)(m) = N succ = {n ∈ N | p n −p n < e th }.
We set e th = 5 m in the simulation. Our simulation introduces MDS-MAP (P) [1] and CFG-based Localization [5] (abbreviated as CFG Loc Algo) as comparison algorithms. Besides, we add multistage clustering mechanism to MDS-MAP (P) (abbreviated as MDS-MAP (P) + tree). Finally, global localization is introduced. All algorithms adopt the matrix completion method from [5]. While ensuring the invariants of the comparison algorithms are maintained, we rerun all the algorithms 100 times to simulate different network topologies and take the average results.

A. Performance Comparisons Considering the Number of Coalitions
In order to compare the impact of the number of coalitions M to the algorithms, we deploy a 340 m × 340 m × 45 m mission area, where L lg /L wd = 1 to avoid the influence of network topology. Then, we set CUs' transmit power Pt = −14.3 dBm). The results are shown in Figure. 2. When M = 3, the number of coalitions is insufficient, resulting in a poor clustering effect for all algorithms. As M increases, the RPLs of all clustering algorithms show a downward trend and gradually slow down (3 ≤ M ≤ 6), which verifies that clustering can effectively alleviate RPLs. When M ≥ 7, RPLs approach the performance lower bound. RPLs of comparison algorithms even become larger due to excessive strategy selections. Compared with MDS-MAP (P) + Tree, the proposed algorithm reduces RPLS by 12.3% on average. Set M = 5, Figure. 3 shows a diagram for CG-Tree Loc Algo. All CUs are divided into five clusters. CO 1 is set to be the first stage (Lv. 1). Map fusion is performed through NDs & CO 2 (CO 3 ) in Lv. 2; Similarly, CO 2 (CO 3 ) performs map fusion through NDs & CO 4 (CO 5 ) in Lv. 3, which is in accord with the proposed multistage clustering-based model.

B. Performance Comparisons Considering Transmit Power
We set border ratio L lg /L wd = 4, where the border is 620 m × 155 m × 45 m mission area. Figure. 4 presents the performance curve. As Pt increases, RMSE of all algorithms decrease, while LocSR increase. Table. I shows that the proposed algorithm achieves the lowest RMSE (2.71m) and the highest LocSR (96.7%) in average. Furthermore, the proposed algorithm achieves better localization performance than MDS-MAP(P) + Tree, demonstrating the CFG-based clustering scheme's advantages in alleviating RPLs.

C. Performance Comparisons Considering Border Ratio
This part investigates the algorithm performances under different network topologies. The parameters are as follows:    Figure. 5, as L lg /L wd gradually deviated from 1:1, the RMSE of all comparison algorithms has increased, and the LocSR has also decreased. As the border ratio L lg /L wd deviates farther from 1:1, the worse the localization performance is. We analyze that the border ratio directly reflects and affects the network topology, and comparison algorithms have poor performance when facing irregular network topology. It is worth mentioning that CG-Tree Loc Algo can consistently maintain the robust localization performance. Statistical results (Table. I) also shows that the CG-Tree Loc Algo can complete the cooperative localization under different network topologies and CUs deployment in the given scenario. Thus, the simulation results confirm the effectiveness and robustness of the proposed multistage clustering-based localization.
V. CONCLUSION This letter constructed a multistage clustering-based model for remote UAV swarm. ADS-TWR was adopted for ranging. Each cluster (local map) performs localization calculation and then merges through NDs stage by stage. We characterized the proposed model as a coalitional game and designed a tree-like multistage clustering scheme. Simulation results evaluated the effectiveness of the proposed coalitional game (CG) framework. The proposed CG-Tree Loc Algo effectively alleviates RPLs, achieves lower RMSE and higher LocSR than the comparison algorithms, and was robust for irregular network topologies. Our work has proved advantageous for remote localization and robust for dynamically changing network topologies. The proposed CG framework offers novel thoughts for other TOF-based localization approaches.