Bi-Level Formation Control

Updated 21 November 2025

Bi-Level Formation Control is a hierarchical approach that separates decision-making from execution, enabling systematic multi-agent coordination.
It employs discrete planning for assignment and conflict resolution at the upper layer, while the lower layer uses model predictive and adaptive controls for trajectory execution.
Empirical results in road, aerial, and underwater domains demonstrate improved safety, scalability, and dynamic adaptation in complex environments.

A bi-level formation control framework is a hierarchical architecture for multi-agent coordination in which a high-level (upper layer) module generates logical or geometric formation specifications and resolves discrete conflicts, while a low-level (lower layer) module synthesizes and executes control or trajectory planning to realize these specifications, subject to dynamics and safety constraints. This structure addresses scalability, dynamic reconfiguration, collision avoidance, heterogeneous agent capabilities, environmental uncertainties, and real-time execution for multi-vehicle systems in structured, semi-structured, or uncertain environments. Bi-level frameworks are established across domains including road-based automated vehicles, underwater autonomous platforms, and aerial rotorcraft, employing diverse methodologies for upper and lower layers but sharing the essential stratification of decision and execution.

1. Conceptual Structure of Bi-Level Formation Control

Bi-level frameworks decompose coordinated formation tasks into two interacting subsystems:

Upper Level (Decision/Planning Layer): This module is responsible for:
- Generating reference trajectories, slots, or assignments for agents based on global formation objectives, traffic topology, or formation transitions.
- Resolving discrete combinatorial conflicts such as path collisions or assignment ambiguities, often via integer programming, logic rules, or discrete-event systems.
- Reconfiguring formation geometry efficiently in response to environmental or task-driven regime changes (e.g., bottlenecks, merges, obstacle incursions).
- Exposing explicit or implicit priorities or hierarchies among agents.
Lower Level (Execution/Control Layer): This module focuses on:
- Synthesizing optimal or feasible reference-following trajectories that satisfy agent dynamics, kinematic, and actuation constraints.
- Enforcing intra-formation and obstacle constraints as prescribed by the upper layer.
- Implementing local cooperative or decentralized control laws (e.g., MPC, adaptive learning controllers, multi-affine feedback) that guarantee agent-level safety and formation compliance.
- Providing feedback to the upper layer about execution status or emergent residual conflicts.

The separation aligns discrete combinatorial reasoning and geometric planning with agent-level model-based control and adaptation, conferring scalability and robustness properties and simplifying safety proofs (Qian et al., 2016, Cai et al., 2021, Jandaghi et al., 4 Sep 2024, Karimoddini et al., 2011, Cai et al., 2021).

2. Upper-Level Generation: Assignment, Logic, and Discrete Planning

Upper-level modules employ a range of representations and algorithms tailored to the application domain:

Formation Graphs and Trees: Directed spanning trees rooted at a virtual or physical leader define hierarchical information flow. The shape matrix $S$ encodes desired offsets, and adjacency matrices specify leader-follower relationships, crucial for distributed reference-trajectory calculations (Qian et al., 2016).
Slot-Based Relative Coordinate Systems: In traffic or lane-based scenarios, relative coordinate systems (RCS) are attached to a reference vehicle, transforming vehicle positions into grid slots. Upper-level planners assign vehicles to slots using cost-minimizing bipartite assignment (Hungarian algorithm) and map discrete paths through RCS grid cells. Lane preference and agent-specific slot constraints are encoded via large- $M$ penalties in cost matrices (Cai et al., 2021, Cai et al., 2021).
Conflict Resolution: Collision avoidance is handled either by logic-rule-based constraints (selecting half-space inequalities or region partitions to prevent forbidden proximity) or by explicit conflict-detection and resolution algorithms, such as conflict-based search (CBS). In CBS, upper-level planners iteratively impose additional constraints to resolve node, edge, or triangle conflicts in slot occupancy (Cai et al., 2021).
Dynamic Reconfiguration: Logical or combinatorial transitions between distinct formation geometries (e.g., interlaced to parallel structures, lane-dropping zones) are planned as sequences of 1-step-reachable intermediate shapes or as solution paths in discrete assignment+path planning problems. Isomorphic formation transitions preserve agent priority and ensure safe dynamic reconfigurability (Qian et al., 2016, Cai et al., 2021).
Discrete-Event Supervisory Control: In three-dimensional aerial scenarios, spherical partitioning of formation space induces a finite state abstraction (DES), allowing modular supervisor synthesis for reachability, invariance, and collision-avoidance requirements (Karimoddini et al., 2011).

The choice of upper-level module is tightly coupled to agent dynamics, communication topology, and the nature of formation transitions required by the application scenario.

3. Lower-Level Trajectory Planning and Control

The lower level implements agent-specific controllers and planners subject to the outputs of the upper layer:

Model Predictive Control (MPC): Car-like vehicle platforms employ an MPC controller that tracks reference trajectories generated from the upper layer. The vehicle is modeled (e.g., kinematic bicycle in the Frenet frame), with the cost comprising quadratic penalties for trajectory tracking error and control effort, and constraints encompassing state bounds, dynamic feasibility, lateral acceleration limits, intra-formation logic rules, and obstacle boundaries. All constraints are formulated as affine or convex quadratic functions, enabling real-time execution via standard solvers (Qian et al., 2016).
Discrete-Continuous Trajectory Synthesis: In lane-based multi-agent formations, real-world continuous trajectories are generated to interpolate RCS slot assignments. Between discrete time steps, trajectory segments are connected by curvature-continuous Bézier curves or by direct collocation, and local optimal-control problems are solved to minimize actuation effort and constrain drift. Stability is addressed with cascaded feedback (e.g., Stanley controller for lateral tracking; velocity feedback for longitudinal) or trajectory-centric MPC (Cai et al., 2021, Cai et al., 2021).
Adaptive and Learning-Based Control: For agents subject to uncertain or nonlinear dynamics (e.g., underwater vehicles), the lower layer integrates decentralized deterministic-learning controllers. Formation error coordinates are tracked with control laws augmented by radial basis function (RBF) neural networks, supporting online adaptation and memory retention. Backstepping designs ensure convergence, with Lyapunov-based proofs securing boundedness despite total model uncertainty (Jandaghi et al., 4 Sep 2024).
Multi-Affine Feedback: For three-dimensional rotorcraft, low-level path planners utilize multi-affine control laws within spherical state partitions. By precomputing control vectors at partition vertices, the framework ensures local invariance or guarantees exit through specified facets in the partitioned state space. This supports a rigorous bisimulation to the associated discrete supervisor, establishing correctness by construction (Karimoddini et al., 2011).
Execution Protocol and Communication: Each agent (vehicle, AUV, UAV) computes its own control action in a distributed manner (real-time MPC cycle, local learning loop, continuous feedback), often using only local information and intermittently received upper-level directives. Upon solving, planned trajectories are broadcast to immediate followers, supporting tight information flow without full centralization (Qian et al., 2016, Jandaghi et al., 4 Sep 2024).

4. Collision Avoidance and Safety Mechanisms

A central challenge in bi-level frameworks is scalable, reliable, and real-time collision avoidance:

Logic-Rule-Based Separation: Logic rules assign a priority list ensuring information flows hierarchically (child has lower priority than parent), and for each pair $(i,j)$ , only a single affine separation constraint partitions the state space to prevent overlap. Obstacle avoidance further augments the set of constraints with paraboloidal or polygonal representations (Qian et al., 2016).
Slot Conflict Detection: Grid-based upper layers preclude vertex, edge, and triangle conflicts by enforcing node exclusivity and planning conflict-free assignments. The CBS, for example, is complete and optimal under mild assumptions, but computational cost scales exponentially with the formation size and number of conflicts, motivating clustering or suboptimal variants for large formations (Cai et al., 2021).
Supervisor-Generated Avoidance: In 3-D aerial systems, the discrete-event supervisor rapidly issues avoidance commands (e.g., azimuthal shift) upon detection of potential collision alarms, with supporting multi-affine control laws guaranteeing exit from hazardous regions (Karimoddini et al., 2011).
Learning-Based Robustness: In uncertain underwater domains, decoupling estimation from control in the bi-level structure is shown to accelerate convergence and avoid coupling instabilities, yielding resilience in dynamic or unmodeled flow variations. Pre-learned neural network weights can be redeployed after restarts, skipping periods of poor safety typical in single-layer adaptives (Jandaghi et al., 4 Sep 2024).
Rigorous Feasibility: By constructing the lower-level constraints to directly encode half-space or region restrictions and mapping formation transitions to 1-step-reachable graphs, the frameworks ensure feasibility and safety are preserved throughout dynamic reconfiguration (Qian et al., 2016).

5. Dynamic Reconfiguration and Adaptability

Bi-level formation control architectures explicitly address the challenge of changing formation geometries, environmental constraints, and agent heterogeneity:

Isomorphic Formation Switching: Adherence to fixed agent-priority orders allows any isomorphic formation to be reached via finite sequences of feasible intermediate shapes, guaranteeing stability and collision-free transitions (Qian et al., 2016).
Slot Assignment and Sequence Interpolation: Smooth switches are obtained by interpolating between old and new grid assignments, using continuous sequences of grid slots and ensuring continuity in spatial occupation. Deadlock and non-deadlock conflicts during transition are algorithmically detected and corrected (Cai et al., 2021, Cai et al., 2021).
Adaptive and Environment-Independent Control: By embedding local learning via RBF neural networks or modular distributed observers, the lower layer remains robust to variations in dynamics, topology, and partial observability, while the upper layer quickly propagates new formation objectives in response to macro-environmental changes (Jandaghi et al., 4 Sep 2024).
Hybrid Discrete-Continuous Design: In systems where switching occurs between discrete events (e.g., reach-formation, collision-alarm), an interface layer translates symbolic supervisor actions into continuous control objectives, with closed-loop modularity supporting mission reliability and rapid adaptation (Karimoddini et al., 2011).

6. Empirical Outcomes and Performance Analysis

Simulation and experimental investigations provide quantitative demonstrations of the effectiveness and scalability of bi-level frameworks:

Domain	Key Performance Metrics	Typical Outcomes
Road Vehicle	Travel Time, Fuel Economy	Up to 47% reduction in mean trip time, 20–30% reduction in fuel use under congestion
UAV/Helicop.	Convergence, Accuracy, Safety	Formation convergence at ~1m accuracy, reliable avoidance in HIL simulation
AUV	Error Norms, Adaptation Speed	Steady-state tracking error $O(10^{-2}$ m), rapid adaptation to modeled/unmodeled changes

In high-volume traffic, bi-level formation controllers maintain near-free-flow travel even as baseline methods suffer congestion and queue formation; for more than 1200 vehicles per hour per lane, classical methods exhibit 50% increases in travel time, while bi-level methods remain near optimal (Cai et al., 2021, Cai et al., 2021). In hardware-in-the-loop environments for aerial vehicles, correctness properties proven at the supervisory level carried through to physical closed-loop responses (Karimoddini et al., 2011). For underwater missions, local RBF learning and estimator-controller decoupling yielded error convergence and resilience across a range of environmental uncertainties (Jandaghi et al., 4 Sep 2024).

7. Extensions, Limitations, and Variants

Bi-level frameworks are extendable to mixed agent types, broader environmental scenarios, and enhanced objectives:

Scalability: For large formations, the upper layer's combinatorial complexity necessitates clustering, approximate CBS variants, or real-time parallelization to maintain tractability.
Extensible Constraints: Additional specifications, such as lane-type exclusions, dynamic lane closures, on-ramp merges, or intersection management, are encoded at the upper level through assignment and slot logic (Cai et al., 2021).
Adaptive Reuse: Persistent storage of local dynamic knowledge enables rapid redeployment after restarts or agent replacement, particularly in learning-centric frameworks (Jandaghi et al., 4 Sep 2024).
Hybrid and Modular Design: The use of discrete-event abstractions enables modular supervisor design and correct-by-construction synthesis, but requires careful coordination with the continuous execution layer to avoid conservative over-abstraction or latency-induced failures (Karimoddini et al., 2011).
Limitations: Real-time performance is maintained for modest group sizes (e.g., $N\leq6$ for CBS), with loss of optimality or rising computational delays at large $N$ (Cai et al., 2021). Conflict-avoidance guarantees are contingent on perfect detection, communication robustness, and timely execution of transition phases. Some methods require prior knowledge or design of priority orders and may not trivially generalize to fully arbitrary graph topologies without loss of distributedness (Qian et al., 2016).

A plausible implication is that continued advances in distributed optimization, convexification of logic rules, and integrated learning-control architectures will further extend the applicability and resilience of the bi-level formation control paradigm across mixed vehicular and robotic networks.