Collision-Free Transport Plans

• Collision-free transport planning is a framework that defines trajectories for multiple agents to avoid collisions with each other and with obstacles.
• It integrates combinatorial, decentralized, and optimization-based methods to ensure deadlock avoidance and efficient navigation in dynamic, real-world settings.
• Advanced techniques like MPC, convex relaxations, and formal certification provide robust, real-time guarantees of collision avoidance and system scalability.

A collision-free transport plan specifies how a collection of agents, robots, or vehicles traverse a shared environment such that their movements never result in collisions—either with each other or with obstacles. In technical terms, collision avoidance must be achieved over trajectories occupying shared spatial and temporal resources while ensuring that all agents reach their assigned goals. Collision-free transport planning integrates combinatorial, optimization-based, and decentralized algorithms under strong correctness guarantees, often for multi-robot, multi-agent, or cyber-physical systems in real-world, dynamic environments.

1. Collision-Free Transport: Concepts and Problem Structure

Collision-free transport planning entails specifying trajectories for multiple agents such that for every pair of agents and for all time, their swept volumes are disjoint and do not intersect any static or dynamic obstacles. The formal abstraction involves:

• Configuration space: Often denoted as $F(\mathbb{R}^d, k)$ for k agents in $\mathbb{R}^d$, with obstacles further removing forbidden regions, e.g., $F(\mathbb{R}^d - Q_r, k)$.
• Resource graph models: Discretize space into nodes/resources, with agents requesting and moving between these nodes (as in automated guided vehicles (AGVs) on factory floors (Marino et al., 2011)).
• Temporal extension: Trajectories represented as space-time functions $\pi_i(t)$, occupancy regions $R_i(\pi_i)$, or, for density control, as mappings over tempo-spatial probability distributions.

A collision-free plan is then operationally characterized by the requirement that for all (i, j), $R_i(\pi_i) \cap R_j(\pi_j) = \emptyset$ and $R_i(\pi_i) \cap \mathcal{O} = \emptyset$, where $\mathcal{O}$ denotes obstacle regions.

2. Distributed and Decentralized Coordination Paradigms

Decentralized approaches eliminate the need for a centralized controller, allowing agents to coordinate using only local information and communication protocols:

• Resource arbitration and finite state machines: In distributed AGV coordination (Marino et al., 2011), each agent operates an event-driven finite state machine with states for requesting, waiting, moving, and replanning. Collision avoidance is achieved by mutually exclusive access grants to micro (node-level) and macro (region-level) resources, resolved via local “sign-board” communication containing agent status, priorities, and positions.
• Priority-based schemes and deadlock avoidance: Access to resources is scheduled by static or dynamic priority values; tie-breakers rely on agent IDs, timers, and local information. Mathematical sampling constraints ensure mutual exclusion and preclude deadlocks and livelock by strictly bounding the number of agents per resource to the spatial capacity.
• Continuous communication and negotiation: Vehicles in leaderless traffic scenarios (Wartnaby et al., 2019) share “desired” (unconstrained) and “planned” (collision-aware) trajectories with each other, iteratively influencing each agent’s chosen path via weak, soft penalization in a decentralized model predictive control (MPC) framework. Cooperative, collision-free behaviors emerge from distributed broadcasting and mutual trajectory avoidance adjustments.
• Token-based best response: The COBRA framework (Čáp et al., 2015) passes a shared token containing space-time trajectory plans, so each agent’s best-response plan avoids occupancy conflicts, with completeness and deadlock-freeness established when endpoints are configured according to valid infrastructure conditions.

3. Formal Guarantees and Theoretical Foundations

Guaranteeing collision-freeness and completeness of transport plans requires formal methods encompassing topological, combinatorial, and optimization-theoretic analysis:

• Topological complexity and motion planners: Multitasking planners (Zapata et al., 2019, Zapata et al., 2019) partition the (multi-agent, multi-stage) configuration space into the minimal number of regions—determined precisely by higher topological complexity constants, $TC_n(F(\mathbb{R}^d, k))$—and construct continuous local motion planners (sections) that connect prescribed waypoints via paths with explicit, collision-avoiding deformations.
• Explicit mathematical constraints: Path updates, resource competitions, velocity profiles, and sampling intervals are always tightly bound by formal mutual exclusion, mutual access, and occupancy constraints, often written in terms such as

$$\|\mathbf{x}_{i}(t) - \mathbf{x}_{j}(t)\| \geq d_{\text{safe}}, \quad \forall t,~\forall i \neq j$$

and, in discrete resource models, access transitions dependent on verified conditions from local communication.

• Deadlock/livelock avoidance: For distributed protocols (Marino et al., 2011), upper bounds on agent count per resource, together with priority and timer policies, yield guaranteed absence of deadlock (stall) and livelock (perpetual motion without progress).
• Convex relaxations and certification: Convex-hull and ellipsoidal/polytope covers (safe flight corridors) (Wu et al., 13 Jun 2024) and sums-of-squares hyperplane certificates (Amice et al., 2023) offer tractable surrogates and algorithmic means to verify or enforce collision-free evolution, even for high-dimensional or nonlinear configuration spaces.

4. Hierarchical and Optimization-based Methods

Many collision-free transport plans exploit a layered architecture:

• Workspace convexification and hierarchical decoupling: The multi-manipulator payload system (Tallamraju et al., 2019) builds a high-level deformable virtual bounding box (via MPC with potential fields) to carve a collision-free convex workspace, solves a nonlinear program for payload rolling/orientation, then employs decentralized MPC for individual agent paths within a regionally partitioned workspace, with analytic inverse kinematics closing the loop.
• Safe flight corridors from overlapping convex sets: Algorithms dynamically adjust overlapping polytopes/ellipsoids along nominal paths, iteratively enlarging feasible regions and updating waypoints with heuristic penalties (path length, control effort), as in the joint cover optimization strategy (Wu et al., 13 Jun 2024). Final trajectories are optimized as polynomials constrained to remain within the convex corridors.
• Space-time graphs of convex sets (ST-GCS): Holistic formulations that embed both space and time as dimensions in the planning graph allow explicit encoding of dynamic obstacle motion, velocity, and causality constraints (Osburn et al., 13 Aug 2025). Piecewise BÉzier curves with all control points within the appropriate convex set ensure trajectories globally respect both spatial and temporal safety requirements.

5. Scalability, Robustness, and Real-world Implementation

Collision-free planning must scale to large agent populations and diverse operational settings:

• Complexity bounds: Many decentralized and time-extended roadmap approaches (e.g., COBRA (Čáp et al., 2015)) exhibit quadratic time complexity in the number of robots (O(n²)), as opposed to the exponential blowup common in fully coupled centralized planners, enabling practical deployment for large fleets.
• Robustness to communication failures: Local communication protocols (sign-boards, reference-free broadcasted trajectories) naturally tolerate packet loss and partial observations, as only neighboring agents affect contention.
• Domain applicability: Methods have been demonstrated in settings such as industrial AGV fleets, vehicle traffic with connected and automated vehicles, mobile manipulator formations, aerial swarms, and congested warehouse or factory layouts, with empirical evaluation confirming the ability to avoid deadlocks, minimize congestion, and preserve transport efficiency across diverse topologies and environments.
• Cost, interpretability, and modularity: The decoupling of structural planning (e.g., assignment, workspace inflation) from trajectory optimization (e.g., polynomial or clothoid-based curves (Oh et al., 2023)) allows the system to remain cost-efficient (low computational and hardware demand), interpretable (succinct parameter encoding of plans for V2X sharing), and modular (composable with higher-level task planners).

6. Extensions and Future Directions

Recent research highlights several dimensions along which collision-free transport planning continues to advance:

• Density control and mean-field interactions: OT-based frameworks now explicitly embed congestion aversion/collision penalties (e.g., convex congestion functionals (Kazeykina et al., 14 Mar 2025)), often via entropic regularized or mean field control schemes, with scalable solution algorithms like Sinkhorn-Frank-Wolfe yielding optimal, collision-averse transport plans in high-density agent regimes.
• Flow-matching and path-dependent constraints: New paradigms optimize entire transport paths (not just endpoints), e.g., by learning neural velocity fields that co-evolve with control policies and incorporate trajectory-level collision costs (Duan et al., 8 Oct 2025). This approach bridges mean-field games, Schrödinger bridge formulations, and classic OT, permitting multiagent coordination that jointly achieves endpoint matching and strict pathwise safety.
• Certification and verification: Advances in sums-of-squares optimization allow for exact, real-time certification of non-collision for complex polynomial motion plans (even for highly articulated systems), ruling out false positives rigorously (Amice et al., 2023).
• Dynamic, unstructured, and non-cooperative environments: Strategies now accommodate non-cooperative, coordination-free settings through topological diversity and stochastic routing (assigning paths in distinct homology classes (Wang et al., 2022)), dynamic replanning, and robust potential-field local controllers, thereby sustaining high performance even amid dense, unpredictable traffic and partial information.

7. Challenges and Comparative Insights

Challenges persist in tuning cost function weights, ensuring stability under symmetric agent situations (which may induce plan oscillations in decentralized systems (Wartnaby et al., 2019)), and scaling exact convex cover or graph-of-sets decompositions to extreme environment complexity. Choice of methodology is dictated by application—fine-grained centralized optimization may yield globally optimal results for small-to-medium problem instances, while decentralized, communication-limited, or combinatorial-topological approaches are paramount in large, variable, or privacy-constrained deployments.

In summary, collision-free transport planning synthesizes algorithmic, combinatorial, and continuous optimization tools to produce robust, scalable, and mathematically verified plans for multi-agent navigation. The field continues to evolve with the integration of optimal transport, density control, advanced certification, and decentralized reasoning, establishing the foundations for safe, efficient, and adaptable transport in complex, real-world environments.