Formation Control with Collision Avoidance

• FCCA is a multi-agent control problem where agents maintain prescribed formations while avoiding collisions through dynamic safety constraints.
• It integrates methodologies like MPC, control barrier functions, and potential fields to address geometric, communication, and operational challenges.
• Research highlights trade-offs between performance, computational demands, and decentralization, applicable to UAV swarms, aircraft, and ground robots.

Formation control with collision avoidance (FCCA) denotes the class of multi-agent control and planning problems in which a group of agents must maintain a prescribed formation geometry while avoiding collisions with other agents and obstacles, often simultaneously tracking a desired trajectory, reaching a destination, or enclosing a target. Across the literature, FCCA appears in leader–follower and leaderless swarms, rigid and deformable formations, 2-D and 3-D settings, and platforms ranging from point-mass agents and unicycles to quadrotors, rigid bodies, commercial aircraft, and multi-robot ground systems. The topic is defined by a persistent coupling between geometric objectives and safety constraints: obstacle encounters may require temporary formation deformation or dissolution, after which the group must recover the desired structure without violating dynamic, sensing, or communication limits (Yao et al., 22 Jul 2025).

1. Problem scope and canonical formulations

FCCA is typically posed as a constrained multi-agent coordination problem. Recurrent objectives include maintaining desired relative positions, preventing inter-agent and agent–obstacle collisions, preserving connectivity, tracking a reference path, achieving velocity consensus, or reaching a goal with bounded control effort and smooth motion. In some formulations, the target is a moving trajectory or destination; in others, it is a moving target to be enclosed at a prescribed radius and angular spacing. These objectives are explicitly combined in multi-agent reinforcement learning, model predictive control, consensus, barrier-based control, and distance-rigidity frameworks (Yao et al., 22 Jul 2025, Celik et al., 2023, Zheng et al., 3 Sep 2025).

The dynamical models used in FCCA vary substantially. The literature includes single-integrator kinematics $\dot q_i=u_i$, double-integrator models $\dot p_i=v_i,\ \dot v_i=u_i$, second-order nonlinear Lagrangian dynamics, unicycle kinematics, sampled double-integrator dynamics, and nonlinear quadrotor models with thrust and attitude channels. This heterogeneity matters because collision constraints enter at different relative degree and because the admissible control architecture differs between analytic feedback, quadratic-program safety filters, and receding-horizon optimization (Cheng et al., 2013, Tang et al., 2023, Verginis et al., 2017, Wang et al., 11 May 2025).

Graph structure is equally central. FCCA papers model agent interactions by undirected connected graphs, directed graphs with a directed spanning tree, fixed leader-rooted topologies, chain structures, or time-varying strongly connected digraphs. The graph determines what constitutes a local formation variable, which collisions are monitored, whether connectivity must be maintained, and whether the desired formation is encoded as node states, edge states, or rigid distances (Aditya et al., 2023, Epp et al., 2024, Koulong et al., 2024).

2. Formation representations and network geometry

A defining distinction in FCCA is how the formation itself is represented. A common formulation uses desired relative offsets $c_{ij}$, $d^{ij}$, or $\delta_{ij}^*$ on graph edges, so that formation maintenance reduces to stabilizing pairwise relative positions. Leader–follower schemes often prescribe a chain or spanning-tree structure in which each follower regulates only one predecessor, whereas leaderless schemes generate the desired collective motion through local relative measurements and consensus-like reference models (Tang et al., 2023, Wang et al., 11 May 2025).

Graph-theoretic shape descriptors provide a more structural representation. One approach defines the current formation through a weighted graph and compares its normalized Laplacian $\hat{\mathbf L}$ to a desired normalized Laplacian $\hat{\mathbf L}_{\text{des}}$, with formation error

$$f=\left\|\hat{\mathbf L}-\hat{\mathbf L}_{\text{des}}\right\|_F^2.$$

This replaces direct coordinate matching by a graph-level shape error and is particularly natural in multi-agent reinforcement learning, where neighbor coordinates are transformed into formation features using graph/Laplacian structure (Yao et al., 22 Jul 2025).

Other formulations parameterize the formation as a transformed template. In the virtual rigid body representation, robot positions satisfy

$\bm q_i=\mathbf R\mathbf S\bm c_i+\bm t,$

with rotation, anisotropic scaling, and translation acting on a base configuration $\mathcal B=\{\bm c_1,\dots,\bm c_N\}$. Distributed planning then becomes consensus on the parameter vector $\dot p_i=v_i,\ \dot v_i=u_i$0, with collision avoidance implemented as admissible constraints on the scaling variables (Mikkelsen et al., 2024). A related coordinate-centric approach uses the Jacobi transformation to separate shape variables and centroid motion in complex coordinates, so that formation stabilization and collision avoidance can be designed in transformed space and then mapped back through $\dot p_i=v_i,\ \dot v_i=u_i$1 (Sarkar et al., 2015).

Distance-rigidity formulations make the formation a minimally constrained geometric framework. In target enclosing, the desired circular pattern around a moving target is cast as an isostatic graph $\dot p_i=v_i,\ \dot v_i=u_i$2 with $\dot p_i=v_i,\ \dot v_i=u_i$3 edges on $\dot p_i=v_i,\ \dot v_i=u_i$4 vertices, constructed incrementally by a Henneberg method. The associated rigidity matrix $\dot p_i=v_i,\ \dot v_i=u_i$5 has full row rank, and $\dot p_i=v_i,\ \dot v_i=u_i$6, which is used directly in convergence proofs (Zheng et al., 3 Sep 2025). Pose-based rigid-body FCCA extends this logic from positions to full relative pose, coupling translational and orientation formation errors in the same decentralized potential-function framework (Verginis et al., 2017).

3. Safety mechanisms and collision-avoidance models

Safety in FCCA is rarely a single inequality. Many formulations combine inter-agent separation, obstacle avoidance, workspace or boundary avoidance, wake/downwash constraints, connectivity maintenance, and singularity avoidance. Navigation-function designs encode these requirements multiplicatively through a constraint term $\dot p_i=v_i,\ \dot v_i=u_i$7, so that connectivity barriers $\dot p_i=v_i,\ \dot v_i=u_i$8 and collision barriers $\dot p_i=v_i,\ \dot v_i=u_i$9 jointly repel trajectories from unsafe sets. Under limited and intermittent sensing, this is combined with switching logic: an agent follows the negative gradient of its navigation function only when it can sense all required formation neighbors; otherwise it stops (Cheng et al., 2013).

Barrier-based feedback offers a different mechanism. In constructive barrier feedback for second-order leader–follower formations, the control law

$c_{ij}$0

adds a dissipative divergent-flow term to a nominal tracking controller. Because $c_{ij}$1 is the distance-to-safety variable, the term acts directly on the relative closing speed and prevents the safe set $c_{ij}$2 from being crossed, while avoiding the feasibility issues associated with optimization-based barrier-function controllers (Tang et al., 2023).

Control barrier function filtering generalizes safety to acceleration-controlled networks. In collaborative safety-critical FCCA, a nominal distributed formation controller is wrapped by a high-order CBF safety filter built from

$c_{ij}$3

with recursively defined $c_{ij}$4. Safety is no longer enforced purely locally: neighboring agents exchange capability information and negotiate jointly feasible safe actions when one agent’s local safety requirements cannot be satisfied by its own input alone (Butler et al., 2024).

Potential functions remain prevalent, but their semantics vary. Finite cut-off potentials for nano-quadrotors are designed so that repulsion vanishes beyond a cautionary radius, transitions smoothly in an intermediate zone, and becomes strong in a risky zone; formation tracking is simultaneously gated by $c_{ij}$5, so safety progressively suppresses formation control as collision risk rises (Nguyen et al., 2021). Directionally aware collision penalties further refine this idea by weighting pairwise penalties according to whether another UAV lies in the forward path and on an actual closing trajectory, via $c_{ij}$6 (Jond et al., 29 Jun 2025).

A recurring misconception is that collision avoidance is exhausted by Euclidean separation. In commercial aircraft formation planning, wake turbulence regions $c_{ij}$7 and $c_{ij}$8 are explicit safety constraints in addition to separation from intruders and other formation members (Yang et al., 2024). In multi-UAV NMPC, reciprocal avoidance uses an ellipsoidal collision model $c_{ij}$9 with $d^{ij}$0, where the larger vertical axis models the downwash effect (Wang et al., 11 May 2025). This suggests that FCCA safety sets are often platform-specific and anisotropic rather than isotropic distance balls.

4. Control and planning paradigms

FCCA admits a wide range of control architectures. Some derive explicit decentralized feedback laws; others solve optimization problems online or offline; still others separate nominal formation generation from safety filtering or tracking. The dominant paradigms differ mainly in where coupling is resolved: directly in analytic feedback, in distributed consensus variables, or in constrained optimization.

Paradigm	Representative formulation	Representative papers
Navigation / potential functions	Goal term plus barrier term $d^{ij}$1	(Cheng et al., 2013, Verginis et al., 2017, Nguyen et al., 2021)
Barrier / safety filtering	Constructive barrier feedback or CBF-QP-like filter	(Tang et al., 2023, Butler et al., 2024)
MPC / NMPC	Finite-horizon tracking with collision constraints	(Celik et al., 2023, Wang et al., 11 May 2025)
Global optimization	MILP or flatness-based finite-horizon OCP	(Yang et al., 2024, Jond et al., 29 Jun 2025)
Game / consensus / hybrid	LQDTG, OtA-consensus, hybrid supervisors	(Aditya et al., 2023, Epp et al., 2024, Karimoddini1 et al., 2014)

Model predictive control treats FCCA as a constrained finite-horizon optimization problem. Generic MPC formulations use

$d^{ij}$2

with penalties on tracking error and control effort over prediction and control horizons, while obstacle and inter-agent avoidance enter as future-state constraints (Celik et al., 2023). Distributed NMPC refines this by combining a reference formation layer with local nonlinear tracking and explicit obstacle, inter-UAV, workspace, velocity, and input constraints; in leaderless UAV swarms, the reference layer itself is generated from local relative measurements under a directed spanning tree (Wang et al., 11 May 2025).

At the planning end of the spectrum, MILP and flatness-based methods emphasize structured, horizon-wide solutions. Commercial aircraft formation avoidance is posed as a mixed-integer linear program over discrete-time dynamics, with a total objective containing maneuver, avoidance, drag, and smoothness costs and with big-$d^{ij}$3 disjunctive separation constraints against intruders (Yang et al., 2024). Flatness-based finite-horizon planning instead exploits linear dynamics in flat coordinates, solves the formation OCP analytically by Pontryagin’s principle, and then adds a collision-constrained LQR-type tracking layer (Jond et al., 29 Jun 2025).

Distributed formulations avoid centralized coupling in different ways. The LQDTG approach moves from node agents to fictitious edge agents, so relative dynamics on graph edges admit decoupled state-dependent Riccati equations whose outputs are then mapped back to node controls by a distributed gradient-descent procedure (Aditya et al., 2023). Over-the-air consensus uses wireless superposition to update formation variables $d^{ij}$4 from simultaneous transmissions, turning interference into a communication primitive for centroid agreement in collision-aware formation control (Epp et al., 2024). Hybrid supervisory control takes a formal discrete-event view, abstracting continuous UAV motion into polar partitions and synthesizing modular supervisors for reaching formation, keeping formation, and collision avoidance (Karimoddini1 et al., 2014).

5. Learning-based and adaptive FCCA

Learning-based FCCA has largely concentrated on how to encode multi-objective priorities without destabilizing training. In multi-agent reinforcement learning, FCCA can be formulated as an MDP $d^{ij}$5 under centralized training and decentralized execution, with PPO-based actor–critic updates and observations containing goal state, motion state, nearby obstacles, and neighboring formation information. A salient contribution is to treat the LLM not as an online controller but as a reward designer that progressively prioritizes reaching the destination, then obstacle avoidance, then formation maintenance, and finally stable motion and time efficiency (Yao et al., 22 Jul 2025).

That framework also argues that raw reward values are a poor feedback signal for reward revision. Instead, reward refinement is driven by structured task metrics: Success Rate, Hazard Incidents, Formation Error, Total Time, and Average Acceleration. The paper explicitly warns that raw rewards can be misleading because an LLM might merely increase coefficients or reshape the reward numerically without improving actual task performance (Yao et al., 22 Jul 2025). A plausible implication is that FCCA learning systems require an external notion of task semantics, not just scalar return maximization.

Distributed deep reinforcement learning has also been used for FCCA with local sensing only. A DDPG-based controller combines formation-tracking cost with a modified stream-function-inspired obstacle avoidance term built from lidar detections and locally constructed virtual cylinders. This design retains smooth obstacle-avoiding trajectories without requiring global obstacle information, and the observation vector explicitly includes preprocessed obstacle-cylinder features rather than forcing the policy network to infer them end-to-end (Qiu et al., 2021).

Adaptive control provides a third learning-like route. In distributed adaptive consensus for heterogeneous nonlinear agents, neural networks approximate unknown agent dynamics, leader dynamics, and disturbances through $d^{ij}$6, $d^{ij}$7, and $d^{ij}$8, while potential-based repulsion terms $d^{ij}$9, $\delta_{ij}^*$0, and $\delta_{ij}^*$1 encode collision and obstacle avoidance. The closed-loop analysis is cast in terms of a composite Lyapunov function over synchronization errors, NN weight errors, and graph-weighted performance variables, yielding CUUB formation tracking under the stated assumptions (Koulong et al., 2024).

6. Applications, validation, and persistent challenges

FCCA validation spans simulation-only studies, hardware demonstrations, and domain-specific planning case studies. LLM-guided MARL has been validated in Gazebo and in a real-world Mecanum-wheel robot setup using OptiTrack and NVIDIA Jetson AGX Orin; the reported iteration sequence improved from 60% success and formation error 74.6 at iteration 0 to 100% success and formation error 27.2 at iteration 3, and the comparison table reported 95% success, 11.5 s mission time, and formation error 26.7 for the LLM-guided policy versus 93%, 14.5 s, and 37.4 for the human-designed reward and 79%, 19.2 s, and 35.2 for ORCA-F (Yao et al., 22 Jul 2025). Crazyflie experiments have likewise shown time-varying formation acquisition and obstacle avoidance using finite cut-off potentials (Nguyen et al., 2021), and seven-UAV hardware experiments with Vicon have demonstrated leaderless distributed NMPC in cluttered environments (Wang et al., 11 May 2025).

Application domains materially shape FCCA problem statements. In commercial aviation, the main illustrated scenario is a side intruder approaching at $\delta_{ij}^*$2, and the model shows that two-aircraft formations can use lateral-only avoidance, whereas three-aircraft and larger formations may require vertical maneuvers and position swapping during recovery (Yang et al., 2024). Urban multi-UAV FCCA has been studied with a semi-centralized probabilistic Lloyd/CVT formation planner combined with distributed pigeon-inspired obstacle avoidance, including 3-D maneuvers through buildings, static obstacles, and dynamic obstacles (Ahmadvand et al., 2024). Rigid formation teleoperation under uncertainty has been validated by four MRS UAVs moving through cylindrical obstacles while enforcing a probabilistic upper bound on inter-robot collision (Mikkelsen et al., 2024).

The field’s persistent challenges are equally explicit. MPC is repeatedly noted to face real-time implementation difficulty, especially as vehicle count and constraint count increase (Celik et al., 2023). MILP planning is effective but computationally expensive, with reported solution times of about 30 minutes for the 2-aircraft case, several hours for the 3-aircraft case, and 1–2 days for 5-aircraft or more (Yang et al., 2024). Reward-shaping approaches can fluctuate between iterations and may lose track of early requirements as task complexity increases (Yao et al., 22 Jul 2025). Potential-based and hybrid consensus methods may converge to local minima or non-target steady states in symmetric geometries (Epp et al., 2024). Some constructive barrier methods admit unstable undesired equilibria on the safety boundary even though the desired equilibrium is asymptotically stable (Tang et al., 2023). Under intermittent sensing, convergence may be only uniformly ultimately bounded unless the switching sequence provides sufficiently many full-sensing intervals (Cheng et al., 2013).

These limitations indicate that FCCA is not a single solved synthesis problem but a family of tradeoffs among geometry, safety, computation, and decentralization. A consistent theme across recent work is architectural separation: reference-generation layers, safety-filter layers, and low-level tracking layers are increasingly decoupled so that formation objectives and collision constraints can be handled by different mechanisms with distinct guarantees (Wang et al., 11 May 2025, Butler et al., 2024, Jond et al., 29 Jun 2025).