Formation Control in Multi-Agent Systems

• Formation control is defined as algorithms that coordinate autonomous agents to maintain prescribed geometric configurations using graph-theoretic constraints.
• The method utilizes distributed feedback laws—including distance-based, bearing-based, and augmented Laplacian techniques—to ensure robust exponential convergence.
• Advanced strategies integrate obstacle avoidance, node-failure resilience, and dynamic reconfiguration for applications in UAVs, CAVs, and robotic swarms.

Formation control method encompasses the theory and algorithms that govern the coordinated motion of multiple autonomous agents to achieve and maintain prescribed geometric arrangements. Formation control has matured into a central research area in multi-agent systems, with applications stretching from unmanned aerial and ground vehicle cooperatives to marine sensors, robotic swarms, and connected automated vehicles. Rigorous analysis of information constraints, decentralized feedback, graph rigidity, topology, dynamics (single- and double-integrator, nonholonomic), and real-world uncertainties are essential features of the modern formation control landscape.

1. Rigidity-Theoretic Foundations

Formation control is fundamentally linked to the concept of rigidity in graphs. Given a set of agents whose interaction topology is described by a graph $G=(V,E)$, a formation is rigid if, when certain pairwise distances are imposed as constraints, the only allowable continuous deformations are global translations and rotations (plus scalings if allowed) (Belabbas, 2011). The minimally rigid graphs in the plane are Laman graphs, satisfying $|E|=2|V|-3$ and a sparsity condition on subgraphs. For any assignment of edge lengths subject to triangle inequalities, a generic minimally rigid graph yields finitely many congruence classes of agent placements modulo $\mathrm{SE}(2)$.

The configuration space of $n$ agents in the plane modulo translations and rotations is the manifold $E_n \simeq \mathbb{CP}^{n-2} \times (0,\infty)$. The mapping from agent positions to the set of inter-agent distances is a finite covering map, revealing the topological impossibility of defining a globally stabilizing feedback law for generic distance-based formation on directed minimally rigid graphs—at least two isolated equilibria must exist (Belabbas, 2011).

2. Distributed Feedback and Graph Structures

A rich taxonomy of distributed control laws has emerged, strongly dependent on the interaction topology and measurement model. In decentralized frameworks:

• Distance-based control: Each agent enforces prescribed distances to its neighbors using relative position measurements. For undirected, triangulated Laman graphs, gradient flows of potential functions whose minima encode the target shape yield decentralized feedback laws. These protocols can ensure that all stable equilibria correspond exactly to the admissible shapes (modulo $\mathrm{SE}(2)$) (Chen et al., 2014). The control input takes the block form

$$\dot{x}_i = \sum_{j\in N_i} u(\|x_i-x_j\|, y_{ij})\, (x_j - x_i)$$

where $u$ is constructed to vanish exactly at the desired distance $y_{ij}$ and ensures repulsion at short range and attraction at long range.

• Angle-constrained and bearing-based control: When available, inter-agent bearing or angle measurements enable control laws invariant to scaling. The angle-constrained paradigm, particularly under Leader-First-Follower (LFF) directed graphs, admits distributed controllers that guarantee global exponential stability—the system's convergence rate is set by the slowest follower's prescribed angle (Li et al., 2023). The interaction graph can be strictly weaker than a Laman graph, e.g., only requiring minimal persistence.
• Augmented Laplacian/complex Laplacian: For both planar and 3D formations, matrix-weighted Laplacians generalize scalar-weight frameworks. For the augmented Laplacian method, matrix weights are constructed to enforce invariance under translation, uniform scaling, and arbitrary-axis rotation in 3D. The kernel of the augmented Laplacian defines the formation-manifold, and a combination of leader and follower control laws achieves exponential convergence to prescribed, maneuverable shapes with low neighbor requirements (Zhou et al., 9 May 2025).
• Pole-placement and eigenstructure assignment: In leader-follower structures with a rooted, cycle-free directed graph, one can synthesize decentralized linear control gains to assign arbitrary closed-loop poles (matching the agent count), and encode the desired formation as an eigenvector associated with a designated zero mode. This method achieves global exponential convergence to the target configuration once the leader is positioned (Mulla et al., 2013).

3. Extensions: Obstacle Avoidance, Node Failure, and Adaptive Topologies

Formation control in realistic settings must address obstacles, uncertain communications, and failing nodes.

• Artificial potential fields (APF) and stress-response mechanisms: Hybrid control laws superpose consensus/rigidity protocols with artificial potentials. Attractive terms pull the formation toward a target, repulsive terms enforce collision and obstacle avoidance, and stress-response mechanisms inject symmetry-breaking perturbations to escape local minima (Zhao et al., 15 Mar 2025). Such methods are validated on various shapes and guarantee exponential formation rigidity and high collision-avoidance success rates.
• Force-based and behavior-based swarming: Fully decentralized protocols incorporate locally sensed artificial inter-agent and agent-obstacle force models. Explicit decentralized state machines for agent initialization, label assignment, perimeter leader re-election, and stigmergic signaling ensure robust, resilient shape maintenance, obstacle negotiation, and fragmentation-reintegration cycles (S et al., 2023).
• Robust formation under failures and communication constraints: Formation control methods for large-scale arrays (e.g., marine seismic vehicles) couple a second-order displacement-based controller, with leader-pinning, to connectively robust topology design. The Molloy-Reed criterion guides adaptive connectivity for node-failure tolerance: with redundancy, the system remains cohesive up to a percolation threshold determined via Monte Carlo methods (Saleh et al., 2023, Chen, 2022).

4. Maneuvering, Target Tracking, and Adaptive Formations

Modern formation control supports not only static shape acquisition but also dynamic maneuvers and reconfiguration.

• Trajectory planning with virtual structures and cooperative filtering: For formations such as close-flying fixed-wing UAVs, the virtual structure paradigm encodes the entire shape by a fixed frame with each agent assigned a position. Cooperative consensus filters ensure smooth, rigid motion of the entire formation, while robust control rejects inter-vehicle aerodynamic coupling (Zhang et al., 2019).
• Moving target pursuit and encirclement: Formation control coupled to estimator modules (recursive least squares estimators, distributed Kalman filters) enables agents to maintain enclosing formations about moving targets. Pattern generators based on Kuramoto-style coupled oscillators guarantee even distribution along a moving circle, while consensus control synchronizes the positions using only relative displacement and target information (Liu et al., 2023, Liu et al., 18 Oct 2024). The resulting frameworks are robust to GPS denial and intermittent occlusions.
• Shape switching and bi-level planning: For connected automated vehicles in mixed traffic or multi-lane scenarios, bi-level frameworks separate logical (relative coordinate grid, slot assignment and conflict-free path planning) and physical (continuous trajectory tracking) layers. Conflict-based search (CBS) is deployed for discrete collision resolution, while trajectory optimization ensures dynamically feasible implementation. This paradigm supports online formation switching, lane-selection with vehicle preference, and congestion alleviation (Cai et al., 2021, Cai et al., 2021).

5. Theoretical Limitations and Global Behavior

Formation control is subject to topological and information constraints:

• For minimally rigid, directed graphs, global stabilization to a unique configuration is topologically impossible; in the undirected rigid case, only almost-global convergence can be achieved due to the nontrivial covering structure of the configuration space (as established by the LS-category argument) (Belabbas, 2011).
• Stability guarantees are generally local (distance-based protocols), almost-global (rotation-symmetry methods (Martinez et al., 1 Oct 2025), angle-based methods on LFF graphs (Li et al., 2023)), or global when bearing or angle control is possible with sufficient information redundancy.
• Decentralized protocols exploiting only minimal neighbor information and local reference frames can achieve exponential convergence, but complete global convergence is only possible for special classes of graphs and measurement models (Chen et al., 2014, Li et al., 2023).

6. Numerical and Experimental Validation

Formation control methodologies are validated across simulation and hardware platforms:

Method	Representative Metric/Result	Reference
Triangulated Laman	All stable equilibria are target configs, exponential convergence	(Chen et al., 2014)
Angle-constrained LFF	Global exponential convergence rate dictated by slowest angle	(Li et al., 2023)
Augmented Laplacian	Arbitrary 3D rotation/translation/scale, block-matrix exponential convergence	(Zhou et al., 9 May 2025)
APF + SRM	100% obstacle avoidance success, <700 steps to $\epsilon$-formation	(Zhao et al., 15 Mar 2025)
Marine vehicle array	Formation stability under node failures, threshold fraction $f_c$ quantified	(Saleh et al., 2023)
Missile swarm (NCES)	Nash-equilibrium policy, $<0.1$ final error, robust to node failures	(Chen, 2022)
CAV bi-level	$>30\%$ travel-time reduction, provably conflict-free, optimal slot paths	(Cai et al., 2021, Cai et al., 2021)
UAVs (relative localiz)	$<0.2$ m formation centroid error, experiments under occlusion, GPS-denied	(Liu et al., 18 Oct 2024)
Symmetry-based	$n-1$ edge minimal connectivity, exponential convergence to symmetric shape	(Martinez et al., 1 Oct 2025)

Such results show the effectiveness, scalability, and adaptability of rigorous formation control laws across a broad spectrum of agent numbers, dynamics, sensing modalities, and uncertainty regimes.

7. Outlook and Open Problems

Critical directions and challenges remain:

• Universality: Characterizing all information graphs and measurement models for which global convergence (or almost-global excluding measure-zero sets) is possible remains unresolved.
• High-dimensional and nonlinear extensions: Generalization to 3D with arbitrary group action invariances, dynamic shape adaptation, and complex agent dynamics (nonholonomic, underactuated) is ongoing.
• Robustness: Seamless handling of communication failures, intermittent sensing, latency, and adversarial disturbances remains a principal design goal. Adaptive and learning-based architectures (e.g., co-evolutionary strategies (Chen, 2022)) are promising.
• Complete theoretical characterization: Many existing guarantees are either local or almost-global; topological obstruction theorems indicate the foundational importance of symmetry, graph theory, and system invariance in shaping what can be achieved with decentralized protocols (Belabbas, 2011, Chen et al., 2014).

Formation control thus comprises a spectrum of graph- and geometry-theoretic frameworks, distributed feedback synthesis, stability and robustness analysis, and practical design for dynamic, complex, and uncertain environments—supported by a mature and growing body of provable results and field-deployed validation.