In complex missions such as cooperative surveillance and target tracking, UAVs must not only detect a target but also continuously determine its position and velocity. However, when sensors can obtain only the target's direction and cannot measure range directly, fusing observations from multiple UAVs to accurately track the target's motion becomes a key challenge in cooperative multi-UAV perception.

To address this challenge, a study involving Yuhan Liu of Beihang University proposed a distributed cooperative estimation method. By fusing bearing observations and historical information from different UAVs, the method enables each UAV to estimate the target's position and velocity cooperatively through local computation and information exchange with neighboring UAVs, without directly measuring target range. The study has been validated through simulations and indoor multi-UAV flight tests.

The related paper has been published in IEEE Transactions on Automation Science and Engineering (TASE). In recognition of this research, Yuhan Liu received the Second Prize of the AMOVLAB Campus Scholarship, including a RMB 5,000 award.

Open-source code and paper information are provided at the end of this article for convenient access. If you find this article helpful, please like, share, and recommend it at the end.

 

01

Research Background

A vision sensor can tell a UAV which line of sight the target lies on, but it cannot directly determine the target's range from a single image. Bearings-only measurements are passive, low-cost, compact, and energy-efficient, making them well suited to small UAVs and to indoor spaces, urban canyons, and complex city environments where satellite navigation is limited. Their limitations are equally clear: a single observation point generally cannot uniquely determine a target's three-dimensional position, let alone estimate its velocity at the same time.

When multiple UAVs observe the same target from different positions, they form multiple lines of sight in space. Repeated observations by the same UAV also provide motion cues over time. Combining spatial and historical information can make it possible to recover the target's position and velocity. However, the bearing model is highly nonlinear, and angular errors grow into position errors of different magnitudes as observation range increases, challenging filter accuracy and stability.

Limitations of Existing Approaches

Existing UAV group-management approaches can broadly be divided into centralized, blockchain-based, and multi-center architectures. Each has value, but when a trusted authority goes offline while group membership continues to change, they typically expose four common challenges:

Range ambiguity: A single line of sight determines only direction and cannot reveal where along that line the target is located.

Strong nonlinearity: The combination of target maneuvers, observer motion, and angular noise can cause conventional filtering methods to become biased or converge slowly.

Dependence on a central node: Centralized methods must continuously aggregate observations from the entire network, which can create communication bottlenecks and introduces a single point of failure.

Range-dependent error: The same angular error produces a larger position deviation at longer range, making it difficult for a fixed noise model to assign observation weights accurately.

 

Why Use a Distributed Architecture?

A centralized Kalman filter can process network-wide information at a single node and is therefore often used as a performance benchmark. As the number of UAVs increases, however, continuously transmitting raw observations consumes more communication resources, while computation and reliability remain concentrated at one node. A partitioned architecture introduces regional centers but may separate cross-region geometric constraints through fixed partitions. In a distributed architecture, each UAV performs prediction and update independently, exchanges only state estimates with its neighbors, and gradually reduces inter-node differences through consensus correction. This makes it better suited to multi-UAV networks with sparse communications and changing scale.

 

02

Core Method

To combine the geometric accuracy of optimal triangulation with the scalability of distributed consensus filtering, the study proposes a Consensus-based Distributed Spatiotemporal Cooperative Triangulation Filter (CD-STCTF). It integrates spatiotemporal cooperative triangulation (STCT), Kalman filtering (KF), and a distributed consensus architecture to deliver accurate, robust, and scalable target-motion estimation from bearing-only observations.

Spatiotemporal Cooperative Triangulation

CD-STCTF first incorporates spatial and historical observations into a unified spatiotemporal cooperative triangulation model. Spatial cooperation aggregates the lines of sight formed at the same instant by multiple UAVs at different positions. Temporal cooperation uses the bearing information accumulated previously by each node and links the states at adjacent time steps through a target-motion model. Multiple lines of sight jointly constrain target position, while sequential observations connect changes in position with velocity.

Rather than simply averaging multiple lines of sight or performing an independent triangulation at every time step, the method uses spatial geometry and temporal continuity simultaneously within a unified estimation process. Even when a node or a time step provides only partial observations, historical and neighborhood information can jointly restore the observability of the target state.

Figure 1 Geometric relationship in cooperative multi-UAV bearings-only estimation

Bearings-Only Pseudolinear Modeling

Each observer node knows its own position and the unit bearing vector pointing toward the target, but the unknown range remains hidden along the line of sight. The algorithm constructs a projection plane orthogonal to the bearing vector and projects the geometric relationship between the target and observer onto this plane. This eliminates the unknown range from the measurement equation and rewrites the highly nonlinear bearings-only relationship as a pseudolinear measurement that can be processed by a Kalman filter.

A single bearing observation provides only two independent constraints, so one node cannot independently recover the complete state comprising three-dimensional position and three-dimensional velocity. CD-STCTF therefore incorporates local pseudolinear measurements, the target-motion model, historical states, and neighbor estimates into the filtering process. Compared with the conventional EKF approach of repeatedly linearizing around the current estimate, this orthogonal projection preserves the original geometric constraints and reduces sensitivity to the initial range estimate.

Information-Matrix-Based Consensus Filtering

The research team first uses maximum a posteriori estimation (MAP) to incorporate observations from different UAVs and different time steps into a unified model. By exploiting the model's structure, it then converts the batch computation that would otherwise require centralized processing into a recursive Kalman filter that runs at each time step.

During operation, each UAV mainly processes its own observations and fuses the state estimates of directly connected neighbors only during consensus correction, eliminating the need to upload raw data from the entire network to a central node. Each update consists of four steps:

Prediction: Use the target-motion model and the previous estimate to obtain a prior prediction of the current state and its uncertainty.

Local update: Correct the target's position and velocity using the node's pseudolinear bearing or bearing-rate measurement.

Noise adaptation: Dynamically reconstruct the measurement-noise covariance from the predicted range so that observation weights vary with the geometric conditions.

Neighborhood consensus: Fuse the state estimates of one-hop neighbors and correct inter-node differences so that results across the network gradually converge.

Range-Adaptive Noise Modeling

Angular error in a bearings-only measurement does not correspond to a fixed position error. For the same line-of-sight deviation, a more distant target produces a larger transverse position error in space. If a fixed noise covariance is always used, the filter may become overconfident in long-range observations and assign distorted fusion weights.

Because the true target range is unknown during tracking, CD-STCTF computes an approximate range from the current predicted target position and the observer-node position, then dynamically updates the measurement-noise covariance at every time step. Short-range observations with better geometry receive more appropriate weights, while long-range observations are down-weighted. For CD-STCTF-R, the noise model also accounts for bearing-rate error and uncertainty caused by relative velocity.

Bearing-Rate Enhancement

The basic CD-STCTF uses only the target direction. The enhanced CD-STCTF-R additionally introduces bearing rate, namely the rate at which the line-of-sight direction changes over time. Through a range-decoupled pseudolinear relationship, the paper converts bearing rate into a measurement constraint on relative velocity between the target and observer, adding two independent measurement degrees of freedom for the velocity state.

This is not a matter of estimating position first and then taking a simple difference. Instead, measurements directly related to velocity are incorporated into the same filtering framework. Both simulation and flight-test results show that the two versions achieve similar position accuracy, while CD-STCTF-R produces markedly more stable velocity estimates. Its average computational cost is approximately 31% to 34% higher than that of the basic version, but its maximum update time remains below 1.5 ms.

Technical Highlights

Unified geometry and filtering: Embeds a high-performance spatiotemporal triangulation model into a recursive Kalman filter, retaining geometric constraints while continuously estimating the target state.

Complementary spatial and temporal information: Uses both the spatial lines of sight formed by multiple UAVs at the same instant and the historical bearing information accumulated over successive time steps to mitigate the range ambiguity of a single observation.

Distributed consensus: Each node processes only local measurements and exchanges state estimates with one-hop neighbors. It requires neither a global nor a regional fusion center and does not transmit raw images.

Dynamic range-based weighting: Updates the measurement-noise covariance from the real-time predicted range, avoiding the assumption that long-range bearing observations are as reliable as short-range observations.

Enhanced velocity observation: CD-STCTF-R introduces bearing rate to directly constrain relative target velocity, focusing on the velocity component that is difficult to obtain under bearings-only conditions.

 

03

Simulation Validation

Two Scenarios

The study places 10 observers in a 100 m x 100 m x 100 m three-dimensional space. Each simulation lasts 100 s with a sampling interval of 0.1 s. Each node communicates by consensus only with its three nearest neighbors.

Scenario 1: Static observers track a target ascending along a helical trajectory.

Scenario 2: The observers also move while tracking a target following a three-dimensional figure-eight maneuver, testing the algorithm's stability under complex relative motion.

Figure 2 Comparison of 3D trajectory evaluation results in Scenario 1

Figure 3 Comparison of 3D trajectory evaluation results in Scenario 2

Error Comparison

The research team compares CD-STCTF and CD-STCTF-R with seven methods: CKF, CIKF, CMKF, HCMCIKF, GTSAM, STT, and STTR. The centralized CKF achieves the lowest mean errors in the table. Among the distributed methods, CD-STCTF and CD-STCTF-R have an advantage, and the enhanced version incorporating bearing rate yields the clearest improvement in velocity estimation, with overall performance approaching the centralized benchmark.

In Scenario 1, CD-STCTF-R has a mean position error of 1.46 m and a mean velocity error of 0.29 m/s. Compared with HCMCIKF, it reduces the mean velocity error from 2.78 m/s to 0.29 m/s, an improvement of approximately 90%. Compared with STTR, it reduces the error from 0.40 m/s to 0.29 m/s, an improvement of 27.5%. In Scenario 2, the mean position and velocity errors of CD-STCTF-R are 1.04 m and 0.29 m/s, respectively. The basic version has a mean velocity error of 1.39 m/s, again showing that bearing rate mainly improves velocity estimation.

Figure 4 Position and velocity errors over time in Scenario 1

Figure 5 Position and velocity errors over time in Scenario 2

Scale and Disturbance Tests

When network size increases from 5 to 15 observers, the errors of both algorithms remain stable overall, indicating that the local-neighbor-only design has scaling potential. In Scenario 2, the observer angular velocity is also increased from 0.1 rad/s to 0.5 rad/s. The new algorithms maintain low errors even as node motion accelerates. Across multiple randomized trials, CD-STCTF achieves a median position error of approximately 1.2 m, while CD-STCTF-R achieves a median velocity error of approximately 0.3 m/s.

Figure 6 Mean errors for different network sizes

Figure 7 Mean errors at different observer velocities

Figure 8 Error distributions across multiple trials

Parameters and Real-Time Performance

A larger consensus gain is not always better. The sensitivity experiment tests values from 0.001 to 8.0 and shows that a gain of 1.0 delivers the best overall performance. When the gain rises to 4.0 and 8.0, both position and velocity errors increase markedly. This agrees with the stability conditions given in the paper and indicates that consensus parameters must be selected with both theoretical bounds and the actual network in mind.

Figure 9 Time-series errors under different consensus gains

Figure 10 Mean errors under different consensus gains

In terms of runtime, CD-STCTF has an average update time of 0.146 to 0.155 ms across the two scenarios, while CD-STCTF-R takes 0.196 to 0.203 ms. The maximum update time of both algorithms remains below 1.5 ms. Even with bearing-rate processing, both algorithms still meet the real-time requirements of high-frequency online estimation.

 

04

Flight-Test Validation

Flight Platform

To further validate the proposed CD-STCTF/CD-STCTF-R algorithms beyond simulation, we conducted real-world experiments on an indoor multi-UAV platform. The test platform included a motion capture system (MCS) and UAVs customized from the AMOVLAB Z410 research UAV. In this experiment, the observer UAVs formed a distributed sensing network, while the target UAV flew within the capture area. Each observer UAV was equipped with a flight-control computer and a data-transmission radio, and the required local information was exchanged over the wireless network at 40 Hz. The MCS provided synchronized reference data for algorithm execution and performance evaluation. The flight-test deployment was also supported by the open-source Prometheus autonomous UAV project.

Figure 11 Real-world flight-test platform

Figure 12 Real-world flight-test system architecture

Test Results

During the test, multiple observer UAVs flew in small-radius circles to maintain the distributed observation network, while the target UAV followed a maneuvering trajectory. The ground station generated noisy bearing measurements from the real-time positions recorded by the motion capture system and sent them to the observer nodes. Each node ran CD-STCTF or CD-STCTF-R locally and completed its estimate using only its own data and messages from neighboring nodes.

Figure 13 Flight snapshots at four time points during the real-world test

These snapshots show the overall flight sequence and the spatial relationship between the observer UAVs and the target UAV. The trajectory comparison in Figure 14 shows that the CD-STCTF/CD-STCTF-R estimates closely follow the MCS ground-truth trajectory throughout the experiment, indicating that the proposed methods can maintain reliable motion estimates under real sensing noise, communication delays, and model mismatch. Figure 15 further shows that both position and velocity errors remain bounded after a brief transient phase. The hardware experiments also demonstrate that although CD-STCTF and CD-STCTF-R achieve comparable position-estimation accuracy, CD-STCTF-R performs better in velocity estimation, consistent with the preceding algorithm analysis. These results agree with the simulation study and support the practical feasibility of the proposed distributed bearings-only target-motion estimation method.

Figure 14 Comparison of algorithm estimates and motion-capture ground truth

Figure 15 Position and velocity errors in the real-world test

Validated through simulations and real-world flight experiments, CD-STCTF outperforms existing state-of-the-art methods in accuracy, convergence, and noise robustness, providing a scalable and practical solution for autonomous multi-agent surveillance. Future research will extend this framework to large-scale heterogeneous networks, multimodal sensor fusion, and delay-tolerant mechanisms for asynchronous wireless environments.