A Formal Modular Synthesis Approach for the Coordination of 3-D Robotic Construction with Multi-robots

Abstract: In this paper, we deal with the problem of coordinating multiple robots to build 3-D structures. This problem consists of a set of mobile robots that interact with each other in order to autonomously build a predefined 3-D structure. Our approach is based on Supervisory Control Theory, and it allows us to synthesize from models that represent a single robot and the target structure a correct-by-construction reactive controller, called supervisor. When this supervisor is replicated for the other robots, then the target structure can be completed by all robots

• The paper introduces a novel DES framework for synthesizing decentralized, correct-by-construction controllers for multi-robot 3-D construction.
• It employs modular automata for structure and robot models with iterative state-space pruning to enforce nonblocking, task-observer, and reciprocal properties.
• The approach leverages event relabeling for scalable supervisor replication, eliminating centralized planning and ensuring deadlock-free operation.

Formal Modular Synthesis for 3-D Multi-Robot Construction Coordination

Problem Formulation and Theoretical Framework

The paper "A Formal Modular Synthesis Approach for the Coordination of 3-D Robotic Construction with Multi-robots" (2512.16555) presents a discrete-event systems (DES) theoretical framework for synthesizing decentralized, reactive controllers—supervisors—for automated multi-robot coordination in 3-D construction scenarios. The approach emphasizes correct-by-construction guarantees and formally scalable design via modular replication of controllers based on Supervisory Control Theory (SCT).

The target problem is formalized as follows: a team of mobile robots constructs an arbitrary 3-D structure, defined as a grid-based function mapping cell coordinates to brick heights, under specified mechanical and movement constraints. Notably, the framework is agnostic to specific structure shapes, except for minimal physical feasibility, contrasting with "closed path"-restricted prior art. Each robot is equipped with a local buffer, restricted movement, brick placement conditioned on height and physical access, and must operate under limited observability and communication, as formalized in explicit problem assumptions (A1–A4). Notably, collision avoidance is delegated to a lower-level subsystem, decoupling high-level supervisory synthesis from real-time physical contention.

Modular Automata Construction

The methodology is structured around two core modeling components:

• Structure Automaton ($\mathcal{T}$): Represents the structure as a nonblocking finite-state automaton, where each state corresponds to a feasible configuration, and transitions are labeled by possible brick placements indexed by cell coordinates. State reachability and permissibility conditions are codified through formal guards evaluating local height configuration constraints. The structure automaton is independent of both robot count and robot internal state.
• Robot Automaton ($\mathcal{R}[i]$): Encodes each robot’s locally feasible behaviors as an automaton constructed modularly from extended finite automata templates reflecting navigation (guarded motion within the grid), loading, brick placement conditions, and step-climbing constraints based on relative brick heights. All events are differentiable as either local (navigation, load/unload) or task-synchronous (brick placement at a cell).

The product $\mathcal{T}[i] \| \mathcal{R}[i]$ forms the local plant model for supervisor synthesis.

Supervisor Synthesis and Properties

Supervisor synthesis is cast as computing a maximal controllable, nonblocking subautomaton fulfilling three essential properties:

1. Co-accessibility: Every state allows feasible progress to a marked (task-completed, robot-exited) state via some event sequence, ensuring deadlock prevention and solution completeness.
2. Task-observer: The robot, at any intermediate structure state, must be able to either execute or grant permission for brick placements required for progress, enforcing comprehensive task monitoring and non-interference.
3. Total reciprocity: Permissions and behavioral constraints are mutually symmetric across all robots with respect to brick placement events, assuring correct behavior under supervisor replication.

The supervisor automaton $\mathcal{S}[i]$ is synthesized through iterative state-space pruning to enforce these properties on $\mathcal{T}[i] \| \mathcal{R}[i]$. For scalability, after isolation and synthesis of $\mathcal{S}[i]$ for a canonical robot model, supervisors for the remaining robots are generated via event-index relabeling.

A formal theorem is provided: If all robot supervisors satisfy nonblocking, task-observer, and reciprocal properties, then the composed system is guaranteed to be nonblocking, i.e., the target structure is always fully constructed and all robots return to the base.

Numerical Analysis and Example Scenarios

The construction time complexity for synthesizing a supervisor is $\mathcal{O}(N^2 + N^2M)$, with $N$ states and $M$ transitions in the joint automaton. This complexity is dominated by the structure size (grid dimensions and target heights). The framework does not target extremely large structures due to inherent state space explosion; however, once synthesized, the controller is statically replicable for arbitrary robot numbers, permitting scale-out in deployment.

A detailed example demonstrates that the approach disables robot actions in potentially blocking configurations (e.g., placing bricks that could physically entrap a robot), and only enables actions that preserve collective task feasibility. Blocking states are pruned automatically via supervisor synthesis, even in the face of highly permissive policies, without requiring explicit deadlock analysis by the designer.

Practical and Theoretical Implications

From a practical perspective, the approach yields decentralized, reactive controllers with correct-by-construction guarantees for a broad class of robot collectives and target structures. Coordination is ensured solely via local supervisor logic and limited communication (event broadcasting for state updates). Scalability is achieved through modular synthesis and controller replication, obviating the need for centralized mission planning or communication bottlenecks.

Theoretically, this work advances multi-robot construction by formally separating robot behavior modeling, global task modeling, and controller synthesis, while guaranteeing deadlock-freedom and structural liveness. The permissive, decentralized architecture aligns with bio-inspired collective systems but formalizes their coordination via DES and SCT rather than behavioral heuristics or template-based compilers—addressing notable limitations in prior swarm robotics and collective construction approaches.

Future Directions

Potential research directions include: integration with lower-level collision avoidance and uncertainty models, abstraction for large-scale structure synthesis to combat state explosion (e.g., symmetry exploitation, compositional abstraction), real-time adaptation to environment changes, and extension to on-the-fly structure re-targeting (dynamic task specifications). The formal modular methodology may also be generalized to other multi-robot cooperative tasks beyond construction.

Conclusion

This paper provides a rigorous, modular approach for supervisor synthesis in multi-robot 3-D construction. Through discrete-event systems modeling and SCT-based synthesis, it achieves decentralized, correct-by-construction coordination with provable liveness and task completion guarantees, high flexibility, and practical scalability conditioned on the tractability of the automata for the target structure (2512.16555).