Adaptive Measurement Selection for Scalable Distributed Graph Optimization in Multi-UAV Relative Positioning

In this section, we introduce the DGO mechanism for co-operative relative positioning of UAV swarms, as shown in Fig. 2. The UAV swarm is modeled as a mathematical graph G = (V, E), where the set of vertices V in the graph represents the collection of UAV position vectors, and the set of edges E in the graph represents the measurement constraints between the UAVs, including distance measurement constraints, angle measurement constraints, and state estimation constraints. By decomposing the overall topology of the swarm into multiple subgraphs, distributed optimization computations can be performed locally on each UAV platform. By sharing the local calculation results through inter-UAV communication, this approach improves the computational efficiency while ensuring the accuracy of the results.

Fig. 2.

The DGO scheme for UAV swarm.

The process of DGO is described as follows. Consider a swarm with N UAVs, whose position set is denoted by P = {p₁, p₂,· · · , p_N}. Let di,jm d_{i,j}^m be the measured distance between UAV i and UAV j (i, j ≤ N), which follows a Gaussian distribution: (1) di,jm=hdpi,pj+N0,Σd2 d_{i,j}^m = {h^d}\left( {{{\bf{p}}_i},{{\bf{p}}_j}} \right) + {\bf{N}}\left( {0,\Sigma _d^2} \right) where h^d is the distance measurement function, and Σd2 \Sigma _d^2 is the variance of the Gaussian error. In real-world systems, inter-UAV relative ranging is typically based on onboard UWB nodes, which enables anchor-free and omnidirectional distance measurements.

Furthermore, let θi,jm \theta _{i,j}^m be the measured relative angle between UAV i and UAV j (i, j ≤ N), and we have (2) θi,jm=hθpi,pj+N0,Σθ2 \theta _{i,j}^m = {h^\theta }\left( {{{\bf{p}}_i},{{\bf{p}}_j}} \right) + {\bf{N}}\left( {0,\Sigma _\theta ^2} \right) where h^θ is the angle measurement function, and Σθ2 \Sigma _\theta ^2 is the variance of the Gaussian error. Relative angle measurements are usually performed by onboard vision systems. However, due to the limited FOV of cameras, a single UAV may have difficulty observing all cooperative targets if the number of UAVs in the swarm is large. Consequently, θ_{i, j} may not exist between each pair of UAV i and j in the swarm.

In general, UAVs are equipped with pose estimation systems for single-unit attitude control. Let pis {\bf{p}}_i^s be the position of UAV i obtained from the state estimation system: (3) pis=hspi+N0,Σs2 {\bf{p}}_i^s = {h^s}\left( {{{\bf{p}}_i}} \right) + {\bf{N}}\left( {{\bf{0}},\Sigma _s^2} \right) where h^s is the state measurement function, and Σs2 \Sigma _s^2 is the variance of the Gaussian error. Although the pose estimation system can directly determine the position of the UAV, the estimation error may increase with time and flight distance. This is because pose estimation systems usually rely on inertial sensors and measurement methods, such as inertial measurement units (IMU), optical flow, and VIO. The process of integrating acceleration and velocity measurements can lead to an accumulation of errors. For swarm relative positioning, inter-UAV relative measurements are required to correct for the drift between the individual UAVs.

Based on the aforementioned measurements, a distributed optimization problem for swarm relative positioning can be formulated. In the distributed scheme, each UAV in the swarm (UAV i) considers its own position p_i as the unknown variable to be optimized. We define the distance cost function for UAV i as Jid J_i^d , which can be expressed as (4) Jid=∑j∈Oiddi,jm−pi−p^j2Σd2 J_i^d = \sum\limits_{j \in {\bf{O}}_i^d} {\left\| {d_{i,j}^m - {{\left\| {{{\bf{p}}_i} - {{{\bf{\hat p}}}_j}} \right\|}_2}} \right\|_{{\Sigma _d}}^2} where Oid {\bf{O}}_i^d is a set of UAV indices. If the distance measurement value between UAV i and UAV j is used in the optimization, then index j is stored in Oid {\bf{O}}_i^d . p^j {{\bf{\hat p}}_j} is the position of UAV j estimated by its local distributed optimization. We assume that UAV i can also determine p^j {{\bf{\hat p}}_j} through inter-UAV communication and use it for its local optimization. The detailed information exchange and iterative process can be found in our previous work [21].

The relative angle cost function for UAV i is defined as Jiθ J_i^\theta , which is given by (5) Jiθ=∑j∈Oiθθi,jm−∠pi,p^jΣθ2 J_i^\theta = \sum\limits_{j \in {\bf{O}}_i^\theta } {\left\| {\theta _{i,j}^m - \angle \left( {{{\bf{p}}_i},{{{\bf{\hat p}}}_j}} \right)} \right\|_{{\Sigma _\theta }}^2} where ∠(·,·) represents the relative angle operator. The set Oiθ {\bf{O}}_i^\theta represents the index set of cooperative targets that can be observed by UAV i.

Similarly, we define the cost function for state estimation as follows: (6) Jis=pi−pisΣs2 J_i^s = \left\| {{{\bf{p}}_i} - {\bf{p}}_i^s} \right\|_{{\Sigma _s}}^2

Based on (4)–(6), we formulate the distributed optimization problem as follows: (7) p^i=argminpikdJid+kθJiθ+ksJis {{\bf{\hat p}}_i} = \mathop {\arg \min }\limits_{{{\bf{p}}_i}} \left( {{k_d}J_i^d + {k_\theta }J_i^\theta + {k_s}J_i^s} \right) where p^i {{\bf{\hat p}}_i} is the estimated position of UAV i. k_d, k_θ and k_s are user-defined weighting coefficients. In previous research, the method for optimal weight selection was analyzed based on the CRLB and Bayes’ theorem. The results show that theoretically optimal performance can be achieved when the weights are inversely proportional to the variances. The weighting coefficients can be dynamically adjusted based on the variances.

After optimization, each UAV shares its estimated local position with other UAVs via inter-UAV communication so that all UAVs can acquire the relative positions of cooperative targets at each time step.

For small-scale swarms, each UAV can perform a DGO with the relative measurement results of all other cooperative targets. In our previous work, we conducted experiments on DGO in a 4-UAV swarm and demonstrated the effectiveness of the proposed method [21]. However, when the swarm size is large, it is necessary to select a limited number of suitable measurement conditions for computation due to the limited sensing capabilities and computational power of a single UAV. More specifically, the swarm must be able to adaptively adjust Oid {\bf{O}}_i^d and Oiθ {\bf{O}}_i^\theta in (4) and (5) based on the swarm topology and sensing accuracy to balance inter-location precision and computational efficiency. The proposed measurement constraint selection method is presented in the following section.

First, we give the definitions of the FIM and the CRLB. Suppose y ∈ ℝ^q is continuous random variable that depends on x ∈ ℝⁿ. The FIM is given by (8) FIMx=𝔼∂lnpy|x∂xT∂lnp(y|x)∂x {\bf{FIM}}\left( {\bf{x}} \right) = {\mathbb{E}}\left[ {{{\left( {{{\partial {\rm{\;ln\;}}p\left( {{\bf{y}}|{\bf{x}}} \right)} \over {\partial {\bf{x}}}}} \right)}^T}\left( {{{\partial {\rm{\;ln\;}}p({\bf{y}}|{\bf{x}})} \over {\partial {\bf{x}}}}} \right)} \right] where 𝔼[·] is the expectation operator and p (·) is the probability density function. We define x^y {\bf{\hat x}}\left( {\bf{y}} \right) as an unbiased estimator of x; therefore, we have (9) 𝔼ex=𝔼x^y−x=0 {\mathbb{E}}\left[ {{\bf{e}}\left( {\bf{x}} \right)} \right] = {\mathbb{E}}\left[ {{\bf{\hat x}}\left( {\bf{y}} \right) - {\bf{x}}} \right] = {\bf{0}}

According to the Cramér-Rao theorem, the following inequality holds: (10) Cov x^y=𝔼exexT≥ CRLBx {\rm{Cov}}\;\left[ {{\bf{\hat x}}\left( {\bf{y}} \right)} \right] = {\mathbb{E}}\left[ {{\bf{e}}\left( {\bf{x}} \right){\bf{e}}{{\left( {\bf{x}} \right)}^T}} \right] \ge \;{\bf{CRLB}}\left( {\bf{x}} \right) where Covx^y {\rm{Cov}}\left[ {{\bf{\hat x}}\left( {\bf{y}} \right)} \right] denotes the covariance of x^y {\bf{\hat x}}\left( {\bf{y}} \right) , and CRLB (x) is the Cramér-Rao Bound matrix, which can be given by (11) CRLBx=FIM(x)−1 {\bf{CRLB}}\left( {\bf{x}} \right) = {\bf{FIM}}{({\bf{x}})^{ - 1}}

Let x be the set of state vectors related to swarm relative positioning, while y denotes the set of measurement values. According to the Gaussian noise model, we have (12) y=hx+v,v∼N0,R {\bf{y}} = h\left( {\bf{x}} \right) + {\bf{v}},{\bf{v}} \sim N\left( {{\bf{0}},{\bf{R}}} \right)

Then, the FIM can be given by (13) FIMx=HxTR−1Hx {\bf{FIM}}\left( {\bf{x}} \right) = {\bf{H}}{\left( {\bf{x}} \right)^T}{{\bf{R}}^{ - 1}}{\bf{H}}\left( {\bf{x}} \right) where H (x) = ∂h (x) / ∂x. Let x_i = [x_i, y_i]^T be the state vector of UAV i. Considering the measurement between UAV i and UAV j, let yi,j=di,jm,θi,jmT {{\bf{y}}_{i,j}} = {\left[ {d_{i,j}^m,\theta _{i,j}^m} \right]^T} be the set of relative distance and relative angle measurements between UAV i and UAV j. The measurement function h^ij (·) can be given by (14) hijxi=(xi−xj)2+yi−yj2atan xi−xjyi−yj {h^{ij}}\left( {{{\bf{x}}_i}} \right) = \left[ {\matrix{ {\sqrt {{{({x_i} - {x_j})}^2} + {{\left( {{y_i} - {y_j}} \right)}^2}} } \cr {{\rm{atan\;}}\left( {{{{x_i} - {x_j}} \over {{y_i} - {y_j}}}} \right)} \cr } } \right] where atan(·) is the arctangent operator. We define ri,j=ri,jx, ri,jy=xi−xj,yi−yj {{\bf{r}}_{i,j}} = \left( {r_{i,j}^x,\;r_{i,j}^y} \right) = \left( {{x_i} - {x_j},{y_i} - {y_j}} \right) as the relative state vector between UAV i and UAV j.

Specifically for situations in which the information about the relative distance or relative angle between the UAVs is not available, we have (15) hijxi=ri,j2atan xi−xjyi−yj T, j∈Di∩Θiri,j2 0 T, j∈Di−Θi 0 atan xi−xjyi−yj T, j∈Θi−Di {h^{ij}}\left( {{{\bf{x}}_i}} \right) = \left\{ {\matrix{ {{{\left[ {\matrix{ {{{\left\| {{{\bf{r}}_{i,j}}} \right\|}_2}} & {} & {{\rm{atan}}\;\left( {{{{x_i} - {x_j}} \over {{y_i} - {y_j}}}} \right)} \cr } \;} \right]}^T},\;\;\;\;j \in {{\bf{D}}_i} \cap {\Theta _i}} \hfill \cr {{{\left[ {\matrix{ {{{\left\| {{{\bf{r}}_{i,j}}} \right\|}_2}} & {} & {\;\;\;\;\;\;\;\;\;0} \cr } \;\;\;\;\;\;\;\;\;\;\;\;} \right]}^T},\;\;\;\;j \in {{\bf{D}}_i} - {\Theta _i}} \hfill \cr {{{\left[ {\;\;\;\matrix{ 0 & {} & {\;\;\;{\rm{\;atan}}\;\left( {{{{x_i} - {x_j}} \over {{y_i} - {y_j}}}} \right)} \cr } \;} \right]}^T},\;\;\;\;j \in {\Theta _i} - {{\bf{D}}_i}} \hfill \cr } } \right. where ri,j2=xi−xj2+yi−yj2 {\left\| {{{\bf{r}}_{i,j}}} \right\|_2} = \sqrt {{{\left( {{x_i} - {x_j}} \right)}^2} + {{\left( {{y_i} - {y_j}} \right)}^2}} is the 2-norm of r_i,j, D_i is the set of indices of the UAVs capable of performing relative distance measurements with UAV i, and Θ_i is the set of indices of the UAVs capable of performing relative angle measurements with UAV i. The measurement matrix H^ij (x_i) can be expressed as (16) Hijxi=ri,jxri,j2ri,jyri,j2ri,jyri,j22−ri,jxri,j22,j∈Di∩Θiri,jxri,j2ri,jyri,j200,j∈Di−Θi00ri,jyri,j22−ri,jxri,j22,j∈Θi∩Di {{\bf{H}}^{ij}}\left( {{{\bf{x}}_j}} \right) = \left\{ {\matrix{ {\left[ {\matrix{ {{{r_{i,j}^x} \over {{{\left\| {{{\bf{r}}_{i,j}}} \right\|}_2}}}} & {{{r_{i,j}^y} \over {{{\left\| {{{\bf{r}}_{i,j}}} \right\|}_2}}}} \cr {{{r_{i,j}^y} \over {\left\| {{{\bf{r}}_{i,j}}} \right\|_2^2}}} & {{{ - r_{i,j}^x} \over {\left\| {{{\bf{r}}_{i,j}}} \right\|_2^2}}} \cr } } \right],} \hfill & {j \in {{\bf{D}}_i} \cap {\Theta _i}} \hfill \cr {\left[ {\matrix{ {{{r_{i,j}^x} \over {{{\left\| {{{\bf{r}}_{i,j}}} \right\|}_2}}}} & {{{r_{i,j}^y} \over {{{\left\| {{{\bf{r}}_{i,j}}} \right\|}_2}}}} \cr 0 & 0 \cr } } \right],} \hfill & {j \in {{\bf{D}}_i} - {\Theta _i}} \hfill \cr {\left[ {\matrix{ 0 & 0 \cr {{{r_{i,j}^y} \over {\left\| {{{\bf{r}}_{i,j}}} \right\|_2^2}}} & {{{ - r_{i,j}^x} \over {\left\| {{{\bf{r}}_{i,j}}} \right\|_2^2}}} \cr } } \right],} \hfill & {j \in {\Theta _i} \cap {{\bf{D}}_i}} \hfill \cr } } \right.

We assume that each UAV uses the data from N_o cooperative UAVs for DGO computations. Let O_i be the set of indices of all UAVs that participate in the local computation of UAV i, along with its own index. Then we define the measurement quality evaluation function for all cooperative targets as (17) Fi=∑j∈Oi−lnFIMijxi {F_i} = \sum\limits_{j \in {{\bf{O}}_i}} { - {\rm{\;ln\;}}\left( {\left| {{\bf{FI}}{{\bf{M}}^{ij}}\left( {{{\bf{x}}_i}} \right)} \right|} \right)} where |FIM^ij (x_i)| denotes the determinant of the FIM.

Based on the aforementioned FIM computation process, we can identify the optimal constraints for relative positioning from numerous measurement constraints. Let N_i be the set of indices of all UAVs that can perform relative measurements and communicate with UAV i. For a user-defined N_o, we can derive O_i from N_i: (18) Oi=argminOi Fi,Oi⊆Ni {{\bf{O}}_i} = \mathop {\arg \min }\limits_{{{\bf{O}}_i}} {\rm{\;}}{F_i},{{\bf{O}}_i} \subseteq {{\bf{N}}_i}

Then, we can derive Oid {\bf{O}}_i^d and Oiθ {\bf{O}}_i^\theta from O_i, which are used for the DGO calculations in (6) and lead to the final inter-UAV relative positioning results.

The optimization problem in (18) is a combinatorial optimization task to select a subset of N_o neighbors from the set N_i. Applying a brute-force approach would require evaluating all possible combinations, especially for large-scale swarms. We use a greedy algorithm to solve this problem efficiently. The objective function F_i is an additive sum of individual quality metrics for each neighbor, such that the global minimum can be found by individually minimizing each term. Consequently, a greedy strategy that selects the neighbors with the smallest individual metrics is guaranteed to be optimal. The algorithm proceeds as follows: 1)

For each candidate neighbor j ∈ N_i, compute its individual contribution to the objective function using (18).

The overall complexity is the sum of these steps, where the dominant term is the sorting operation. Therefore, the total computational complexity of our measurement selection algorithm is O (|N_i|log(|N_i|)). This polynomial-time complexity is highly efficient and ensures that the selection process remains computationally feasible for real-time computation, even as the swarm size increases.

A key challenge in swarm positioning is that certain topological configurations can affect the system’s observability. For example, if multiple UAVs are in a collinear or near-collinear arrangement, their relative measurements can become linearly dependent. This dependence may cause the measurement Jacobian matrix H (x) to be rank-deficient. Since the FIM is calculated according to (13), a rank-deficient Jacobian leads directly to a singular or near-singular FIM. In such cases, the system becomes unobservable and the CRLB for the positioning error would approach infinity.

However, the FIM-based measurement selection method proposed here contains its own strategy to mitigate this problem. Our objective function in (18) evaluates the quality of each measurement based on the determinant of its corresponding FIM. If a measurement from a candidate neighbor j produces or enhances a degenerate topology, its associated FIM^ij will be near-singular, so that its determinant approaches zero. Consequently, the quality metric for this neighbor, − ln (|FIM^ij (x_i)|), approaches positive infinity.

During the optimization described in (18), the proposed algorithm tries to select the N_o neighbors with the smallest quality metrics. Therefore, the algorithm automatically penalizes and discards measurements that affect the system observability. This ensures that the subgraph selected for the DGO maintains a strong observability, thus improving the robustness and accuracy of the final positioning solution.

Furthermore, the DGO approach cannot guarantee accuracy if the observability is significantly degraded due to changes in the formation topology. In such circumstances, the solution robustness can be improved by dynamically increasing the weighting of the robot’s self-pose estimate (which is less affected by the formation changes) within the optimization problem.

The computational process of the proposed method is illustrated in Algorithm 1.

To evaluate the effectiveness of the proposed method for large-scale UAV swarm relative positioning problems, we conducted numerical experiments, using the proposed FIM-based method to select the optimal measurement constraints for DGO. As shown in Fig. 3, four groups of swarms with different sizes were used, consisting of 12, 24, 36, and 100 UAVs, respectively. In the first three groups of swarms, the UAVs were arranged in a rectangular formation; in the swarm of 100 UAVs, their positions were randomly distributed within a 200 m × 200 m area. Assume the relative measurement errors between the UAVs are: Σ_d = 0.1m and Σ_θ = 3°, together with a state estimation error of Σ_s = 1m. The simulation model was developed using MATLAB R2022b and ran on a computer equipped with an Intel i5-12500H 2.50 GHz processor and 16 GB RAM. The simulation parameters and error characteristics in this section are consistent with the findings from our previous research [21] in real-world scenarios.

Fig. 3.

UAV swarms used in the simulations. (a) 12-UAV swarm. (b) 24-UAV swarm. (c) 36-UAV swarm. (d) 100-UAV swarm.

In the proposed method, for a given number N_o, we select N_o UAVs from the swarm to participate in the computation process of UAV i by minimizing F_i. Then the relative positions can be solved with DGO. For comparison, we used five other methods to compute the relative positions: 1) state estimation: each UAV computes its position using only its own state estimation system; 2) centralized graph optimization (CGO): a central node collects all measurement data (distance, angle) from each UAV in the swarm and then performs a global graph optimization to compute the relative positions; 3) distributed filtering (DF): each UAV exchanges its state estimate and covariance information with its neighboring nodes to perform prediction, and uses the relative distance and angle measurements to update the steps based on a typical distributed Kalman Filter; 4) DGO-F_i-max: each UAV selects N_o neighbors with the largest F_i values for DGO; 5) DGO-F_i-random: each UAV randomly selects N_o neighboring UAVs for the DGO computations.

In each simulation group, the UAVs follow a circular trajectory at a constant speed of 1 m/s, traveling a total distance of 100 m. For example, the flight trajectories of the aforementioned 12-UAV swarm are shown in Fig. 4. We varied N_o and analyzed the relative positioning error of the aforementioned methods. For each set of parameters, we use the average error value in the flight process to access the final error size. Define the average relative positioning error of the swarm for each test as follows: (19) MAE=1NΣi=1Np^i−p^cen−pi−pcen2 MAE = {1 \over N}\sum\limits_{i = 1}^N {{{\left\| {\left( {{{{\bf{\hat p}}}_i} - {{{\bf{\hat p}}}_{cen}}} \right) - \left( {{{\bf{p}}_i} - {{\bf{p}}_{cen}}} \right)} \right\|}_2}} where p_cen is the position of the geometric center of the swarm, and p^cen {{\bf{\hat p}}_{cen}} is the estimated value of p_cen.

Fig. 4.

The flight trajectory of the swarm.

The results of the numerical experiments are shown in Fig. 5. It can be observed that the magnitude of the state estimation error remains largely consistent for experiments with different numbers of UAVs and different N_o values. Without correcting the inter-UAV relative positions by relative measurements, the state estimation method cannot be directly applied to swarm localization. The errors of the CGO and DF methods are smaller compared to the state estimation methods, but they have inherent drawbacks: the DF method converges slowly when the number of nodes is large, and their estimation errors are sensitive to initial values; CGO uses relative measurements between all UAVs in the swarm, but when the number of nodes is large, a high-dimensional optimization problem of dimension 2*N (N is the number of UAVs) must be solved, which is computationally complex, difficult to converge to the optimal solution, and requires extensive inter-UAV information exchange.

Fig. 5.

Relative positioning error variations with respect to N_o across four methods. (a) Error curves for the 12-UAV swarm. (b) Error curves for the 24-UAV swarm. (c) Error curves for the 36-UAV swarm. (d) Error curves for the 100-UAV swarm.

The remaining three DGO-based methods all show a significant corrective effect on the positioning errors, resulting in an lower overall error than that of the state estimation error. Moreover, the errors of the three methods decrease with increasing N_o. Among the methods mentioned above, the method that maximizes F_i has the largest total error. This is due to the negative correlation between F_i and the theoretical error, which indicates that this method includes the least suitable measurement constraints for localization in the calculation process. The total error of the method in which participants are randomly selected for positioning calculations is smaller than that of the F_i-max method. However, for specific values of N_o, e.g. when N_o = 3, the error becomes unstable. The observed error fluctuations are due to the stochastic nature of the selection process and its interaction with the swarm’s topology. As discussed in Section 2, certain geometric configurations of UAVs can degrade the system’s observability. This leads to a rank-deficient Jacobian matrix and consequently a near-singular FIM. By its very nature, the random selection method occasionally and unintentionally selects such ill-conditioned subsets of neighbors.

Among the four methods, the proposed F_i-min method achieves the minimum error and keeps the error constant within 0.5 m across different configurations of swarm size and N_o. This is because the minimization of F_i ensures that the sensor data included in the calculations makes the greatest contribution to improving accuracy. Therefore, this method is highly suitable for distributed calculation scenarios.

Furthermore, we observed the impact of changes in N_o on the localization error. Within the distributed computing framework, increasing N_o improves accuracy but at the cost of higher computational overhead. Therefore, it is important to determine an appropriate value of N_o in order to achieve a balance between accuracy and computational cost. In the four experimental groups shown in Fig. 5, the error of the proposed method decreases significantly with the increase of N_o when N_o < 10. Beyond this threshold, the error converges towards the theoretical minimum and stabilizes.

Fig. 6 illustrates the variation in computation time with N_o. Since the dimensions of the variables to be optimized in the proposed DGO computation do not increase with the number of UAVs, the trend in the computation time shows a relatively smooth variation. Consequently, it is justified to set N_o below 10, as this achieves accuracy close to the theoretical minimum while the computational overhead remains low. In particular, the computation time per step remains at around 10 ms—less than half the time required for N_o = 100. Meanwhile, the computation time per iteration in CGO increases exponentially with the number of drones, which is significantly higher than the proposed method. When the number of drone agents reaches 100, the single-step computation time on a PC is 3.5 s, indicating that it is difficult to meet the real-time requirements of large-scale swarms in engineering practice.

Fig. 6.

The variation of computation time per step with N_o.

We also investigated the adaptability of the proposed method to different error conditions. For the 100-UAV swarm with configurations of N_o = 4, 6, 8, and 10, the parameter pairs (Σ_d, Σ_θ) were assigned values of (0.1 m, 3°), (0.2 m, 6°), (0.3 m, 9°), and (0.4 m, 12°), respectively. Comparative localization errors of the four methods under these heterogeneous error conditions are shown quantitatively in Fig. 7. The results show that the localization errors of the three relative positioning methods progressively increase with increasing sensing errors. However, the proposed method consistently shows significantly lower localization errors under different error conditions, confirming its adaptability to heterogeneous error scenarios. In practical applications, the proposed method is expected to demonstrate robust adaptability for UAVs with different sensing hardware.

Fig. 7.

Relative positioning error for four methods in the 100-UAV swarm under different sensing conditions. (a) N_o = 4. (b) N_o = 6. (c) N_o = 8. (d) N_o = 10.

To evaluate the robustness of the proposed method under different UAV swarm formation topologies, we conducted a simulation experiment for dynamic formation reconfiguration. As shown in Fig. 8, a 24-UAV swarm was used to successively transition from a rectangle formation to a circular formation and then to a V-formation. The entire reconfiguration process lasted 100 seconds. During the transformation, the relative positioning error of the swarm was calculated using the six aforementioned methods.

Fig. 8.

Dynamic formation reconfiguration. (a) Transformation from rectangle formation to circular formation. (b) Transformation from circular formation to V-formation.

As shown in Fig. 9, the error trajectories during the dynamic topology reconfiguration show a remarkable increase in magnitude compared to the static topology cases. Moreover, the errors exhibit a certain degree of random walk behavior due to the frequent communication disruptions and formation adjustments in dynamic environments.

Fig. 9.

Error curves in topology reconfiguration simulation.

As the simulation results show, the proposed method adaptively selects the optimal constraints in dynamic topology changes, which significantly reduces the average localization error. Table 1 summarizes the statistical metrics (mean absolute error (MAE), standard deviation (STD), and maximum error (MAX)) of the six methods, which further confirm the robustness of the proposed approach in dynamic environments.

We recognize the importance of overcoming real-world communication challenges. In our DGO-FIM framework, the UAVs use the last received positions of their neighbors for local optimization. Communication delays and packet loss in real-world scenarios lead to outdated position data and thus to errors that affect the swarm positioning accuracy during maneuvers. A possible solution to compensate for time-delay errors is to integrate the inertial state estimates (prediction) of the UAVs with the DGO results (measurements) using time-delay Kalman filtering [21]. This multi-method integration increases system stability.