Multi-Agent Forward Reachable Set

• Multi-Agent Forward Reachable Set (MA-FRS) is a framework that computes all possible future states of interacting agents under uncertainty, ensuring forward invariance and safety.
• MA-FRS employs methods such as convex and semidefinite programming, polytopic and ellipsoidal approximations, and online learning for efficient set propagation.
• Distributed computation and barrier certificate enforcement in MA-FRS enable scalable, provable safety in applications like autonomous driving and swarm robotics.

A Multi-Agent Forward Reachable Set (MA-FRS) is a mathematical and algorithmic construct used to characterize, propagate, and certify the regions of state space that multiple interacting agents—each evolving under their own dynamics and uncertainty—can possibly occupy over a future time horizon. MA-FRS methods are foundational for provable safety in multi-agent motion planning, collision avoidance, and robust coordination under uncertainty. Contemporary approaches employ a spectrum of computational machinery: stochastic modeling, conformal and online learning-based calibration, convex and semidefinite programming, distributed algorithms, Lyapunov analysis in measure spaces, and barrier certificate enforcement. MA-FRS thus unifies set-based prediction, agent interaction, and practical planning constraints into a joint, scalable safety certification pipeline for both engineered and learning-enabled multi-agent systems.

1. Mathematical Definitions and Core Principles

The forward reachable set (FRS) for a single agent with dynamics $\dot{x}(t) = f(x(t), u(t), w(t))$ and initial set $X_0$ under admissible input and disturbance is

$$R(t; X_0, U, W) = \Big\{ x(t) \,\Big|\, x(0)\in X_0,\, u(\cdot)\in U,\, w(\cdot)\in W,\, x(\cdot)\,\,\text{solves the dynamics} \Big\}.$$

In the multi-agent context, for $N$ agents with states $x_i$, dynamics possibly coupled by state-feedback or other information sharing, and disturbances $w_i$, the joint multi-agent FRS at time $t$ is defined by the product (or suitably coupled) set of all agents’ reachable states: $$R_{\mathrm{MA}}(t) = R_1(t) \times R_2(t) \times \dots \times R_N(t).$$ In the presence of coupling, distributed controllers, or population-level interactions, this set generally cannot be disentangled; instead, its structure is preserved through polytopic or ellipsoidal approximations, support function propagation, or functional variation in measure space (Thapliyal et al., 8 Oct 2024, Muthali et al., 2023, Cavagnari et al., 12 Mar 2025).

The MA-FRS encapsulates both dynamic uncertainty, arising from system noise and partial modeling, and intent uncertainty, arising from agent stochasticity or environmental unpredictability (Chakraborty et al., 30 Jul 2025, Mai et al., 15 Nov 2025, Yang et al., 9 Sep 2025). The key safety property is forward invariance: any agent executing an admissible control/disturbance process never leaves its designated reachable set, and no two agents' reachable sets intersect if set separation is enforced at all times.

2. Construction Methodologies: Uncertainty Propagation and Set Representations

The computation of MA-FRS proceeds by propagating a set-valued state description for each agent forward in time. Major set representations and propagation techniques include:

• Ellipsoidal and Gaussian-Mixture Approximations: In learning-based approaches, agent trajectories are modeled by Gaussian mixture models (GMM), with convex optimization used to compute minimal area unions of Mahalanobis ellipses that capture prescribed probability mass. Calibration and correction are effected using split conformal prediction, which stretches covariance matrices to enforce empirical coverage (see Section 3) (Chakraborty et al., 30 Jul 2025).
• Polytopic Approximations and Support Functions: For linear or affine agent dynamics, polytopic outer approximations are propagated using hyperplane support evolution (via Pontryagin’s maximum principle and distributed consensus), providing a distributed recipe even in the presence of communication graph constraints and partial local model knowledge (Thapliyal et al., 8 Oct 2024).
• Neural Network-Controlled Systems and Semidefinite Programming: When agents are governed by neural network controllers, forward reachable sets are over-approximated through convex abstractions using quadratic constraints (QC) for input, ReLU activations, and polytopic output, leading to SDPs parameterizing each agent’s reachable output polytope. Distributed computation leverages architectural modularity (Gates et al., 2023).
• Population-Level/PDE Formulation: For crowds, swarms, or continuum-modeled agents, the MA-FRS is lifted to the Wasserstein probability space $P_2(\mathbb{R}^d)$, with the continuity equation and Hamilton-Jacobi analysis dictating the dissipation of Lyapunov functionals over evolving probability measures (Cavagnari et al., 12 Mar 2025).
• Online Learning and Intent Set Refinement: In high-uncertainty, time-varying environments, agent intent sets are learned in real time via event-triggered minimum-volume ellipsoid inference (Löwner-John) from observed actions. The FRS is adaptively refined via Minkowski sum and uncertainty quantification (Yang et al., 9 Sep 2025).

Each of these propagation methods yields a computationally tractable over-approximation of the true MA-FRS, balancing conservatism and efficiency for safe planning (Mai et al., 15 Nov 2025, Kim et al., 2023).

3. Calibration, Adaptation, and Theoretical Guarantees

A fundamental challenge in MA-FRS is guaranteeing that the constructed sets are both sound (do not admit unsafe plans) and complete (do not over-constrain and exclude safe plans). Recent advances deploy the following mechanisms:

• Conformal Prediction Calibration: By analyzing held-out prediction errors, a global or per-agent “stretch” parameter $\eta$ is determined such that the conformally-calibrated reachable set $C_i(t)$ satisfies high-probability coverage guarantees:

$$P_{s,x}[x \in C_i(t)\,\,\,\forall t \in [0,T]] \geq 1 - \alpha$$

for pre-specified miscoverage rate $\alpha$. The non-conformity score used for calibration is strictly determined by normalized Mahalanobis distances (Chakraborty et al., 30 Jul 2025, Muthali et al., 2023).

• Bayesian Online Adaptation: To preserve soundness in out-of-distribution or predictor failure scenarios, MA-FRS methods use Bayesian filtering to dynamically adapt set inflation via posterior belief on predictor trust. If model confidence decays below a threshold, the system falls back to classical (worst-case) Hamilton-Jacobi FRS (Chakraborty et al., 30 Jul 2025).
• Barrier Function Invariance: In control-theoretic settings, high-order control barrier functions (HOCBFs) using MA-FRS as parameterized boundaries ensure forward invariance and safety under the prescribed agent dynamics, with invariance rigorously justified by Lyapunov function dissipation (Kim et al., 2023, Yang et al., 9 Sep 2025).
• Lyapunov-Based Reachability Certificates: In measure-theoretic formulations, forward reachable sets correspond to sublevel sets of Lyapunov functionals that satisfy dissipation inequalities with respect to a Hamiltonian coupling field, allowing rate-of-convergence or finite-time reachability guarantees (Cavagnari et al., 12 Mar 2025).

The combination of these tools enables formal, probabilistic, and worst-case guarantees for the multi-agent safety problem.

4. Distributed and Decentralized Computation

Scalability to large agent populations is achieved via modular and distributed computational strategies:

• Distributed Support Function and Polytope Propagation: Each agent independently propagates its local FRS via local dynamics, local disturbances, and received neighbors’ polytopic approximations, using distributed consensus and maximization algorithms. Convergence is theoretically guaranteed under both static and time-varying communication topologies, with time and space complexity scaling linearly in the number of faces and neighbor degrees (Thapliyal et al., 8 Oct 2024).
• Parallel Semidefinite Programs for Neural Architectures: When agents use distributed neural network controllers, set over-approximation is reformulated as multiple independent SDPs, each corresponding to an agent and its communication subset, reducing computational complexity from cubic in the total system dimension (centralized) to per-agent subproblems (Gates et al., 2023).
• Game-Theoretic Decomposition: In game-theoretic MA-FRS planning, agents’ objectives incorporate local FRS-based collision penalties, with coupling limited to local neighborhoods. Iterative best-response algorithms (e.g., ND-iBR) allow convergence to approximate Nash equilibria in finite time, with average computational burden growing sublinearly with agent count (Mai et al., 15 Nov 2025).

These distributed frameworks exploit the weak coupling structure and leverage recent advances in parallelization and coordination theory.

5. Safety-Embedded Planning and Barrier Enforcement

The computed MA-FRS are integrated directly into the trajectory planning loop through a variety of constraint enforcement schemes:

• Occupancy Set Aggregation: For motion planning, the union over all agents' MA-FRS at each time $t$ defines the time-varying dynamic occupancy region. Ego trajectories are declared safe only if they remain strictly separated from $S_{\text{all}}(t)=\bigcup_i S_i(t)$ for all $t$ (Chakraborty et al., 30 Jul 2025, Muthali et al., 2023).
• Barrier Certificate Constraints: In both model-based and learning-based settings, the MA-FRS are enforced through control or optimization constraints that penalize (via exponential or hard constraints) any intersection between the ego agent’s forecasted state and another agent’s FRS tube. Discrete barrier updates ensure forward invariance conditionally on the barrier’s strict positivity (Kim et al., 2023, Yang et al., 9 Sep 2025).
• Bi-Convex and ADMM-Based Contingency Optimization: In contingency planning, barrier constraints with respect to MA-FRS are embedded in trajectory parameterization and solved using consensus ADMM, supporting both real-time solve rates and recursive feasibility (Yang et al., 9 Sep 2025).

Such integration allows planners to synthesize policies that are both efficient and rigorously certified safe even in the presence of non-modeled behaviors and online learning dynamics.

6. Applications and Empirical Validation

MA-FRS methodologies have been demonstrated across a variety of safety-critical and high-dimensional applications:

Application Domain	MA-FRS Formulation Highlights	References
Urban/mixed traffic autonomous driving	GMM+conformal MA-FRS, Bayesian OOD filter, nuScenes/Waymo datasets	(Chakraborty et al., 30 Jul 2025, Muthali et al., 2023)
Multi-robot and swarm coordination	HOCBF-parameterized ball over-approx, QP enforcement, real-time layered planners	(Kim et al., 2023)
Quadrotor swarms, 4WD vehicles	Ellipsoidal linearized FRS, barrier cost, potential games, decentralized best-response	(Mai et al., 15 Nov 2025)
Distributed controls with neural nets	Local-SDP decomposition, QC abstractions, vehicle platoons, power grid AGC	(Gates et al., 2023)
Crowd/continuum models	Wasserstein space FRS, Lyapunov functionals, viscosity HJB, convergence rates	(Cavagnari et al., 12 Mar 2025)
Event-based intent and contingency planning	Online ellipsoidal FRS refinement, FRS-barrier ADMM, real-world microvehicle tests	(Yang et al., 9 Sep 2025)

Empirical results report millisecond-to-subsecond solve times (even for online learning or Bayesian components), near-nominal probabilistic coverage of realized agent behavior, significant improvement in safety with reduced conservatism, and successful deployment in both simulated and hardware environments (Chakraborty et al., 30 Jul 2025, Mai et al., 15 Nov 2025, Yang et al., 9 Sep 2025).

7. Limitations and Research Directions

While MA-FRS provides the foundation for certifiable safety, several open problems and limitations persist:

• The “curse of dimensionality” constrains Hamilton-Jacobi or full compositional set-based approaches, motivating ongoing research into efficient set representations, abstractions, and sampling (Muthali et al., 2023).
• Learning-based FRS estimation requires continual calibration against distribution shift and nonstationarity, motivating hybrid Bayesian and fallback mechanisms (Chakraborty et al., 30 Jul 2025).
• The trade-off between conservatism and operational efficiency is an ongoing topic; event-based online intent learning and adaptive constraint modulation are current leading directions (Yang et al., 9 Sep 2025).
• In dense and highly coupled scenarios, scalability and decentralized convergence require further advances in parallel solvers, graph sparsification, and robust consensus (Thapliyal et al., 8 Oct 2024, Mai et al., 15 Nov 2025).

MA-FRS remains a rapidly evolving field at the interface of control theory, safety-critical learning, distributed algorithms, and formal verification. Continued empirical and theoretical advances are expected to further enhance the fidelity, efficiency, and autonomy of multi-agent systems across scales and domains.