FORMULA: FORmation MPC with neUral barrier Learning for safety Assurance

Qintong Xie; Weishu Zhan; Peter Chin

cs.ROApr 6, 2026

FORMULA: FORmation MPC with neUral barrier Learning for safety Assurance

Authors: Qintong Xie, Weishu Zhan, Peter Chin

Organizations: Thayer School of Engineering, Dartmouth College, NH 03784. · Department of Computer Science, The University of Manchester, Manchester M13 9PL, United Kingdom.

Abstract

Multi-robot systems (MRS) are essential for large-scale applications such as disaster response, material transport, and warehouse logistics, yet ensuring robust, safety-aware formation control in cluttered and dynamic environments remains a major challenge. Existing model predictive control (MPC) approaches suffer from limitations in scalability and provable safety, while control barrier functions (CBFs), though principled for safety enforcement, are difficult to handcraft for large-scale nonlinear systems. This paper presents FORMULA, a safe distributed, learning-enhanced predictive control framework that integrates MPC with Control Lyapunov Functions (CLFs) for stability and neural network-based CBFs for decentralized safety, eliminating manual safety constraint design. This scheme maintains formation integrity during obstacle avoidance, resolves deadlocks in dense configurations, and reduces online computational load. Simulation results demonstrate that FORMULA enables scalable, safety-aware, formation-preserving navigation for multi-robot teams in complex environments.

Figures & tables

Fig. 1: Cooperative formation and safety-aware navigation for multiple mobile robots.

Fig. 2: Overview of the proposed FORMULA framework. Left: each robot runs a distributed MPC-CLF optimizer for formation stability and a NN–CBF module for safety, with an event-triggered deadlock-resolution mechanism. Right: one of the multi-robot simulation environments.

Fig. 3: Formation control in obstacle-dense environments. (a) Planar trajectories of the leader (red) and followers (magenta, blue, green, cyan) among randomly placed obstacles (gray circles). (b) Formation tracking error $\|\boldsymbol{e}_{\mathcal{N}_{i}}\|$ over time for all followers, showing a temporary increase during obstacle negotiation followed by convergence as the formation re-stabilizes.

Followers	Method	Safety rate $\uparrow$	Avg. form. err. [m] $\downarrow$	Avg. min. dist. [m] $\uparrow$
2	APF	0.297	1.090	0.059
	MPC-CBF	0.755	1.129	0.417
	CLF+CBF-QP	0.717	0.787	0.409
	Proposed	0.863	0.705	0.490
4	APF	0.220	0.925	0.078
	MPC-CBF	0.456	1.045	0.353

TABLE I: Safety rate, average formation error, and average minimum distance for different controllers and team sizes. Each entry is averaged over one representative rollout.

Fig. 4: Deadlock resolution in FORMULA . Colored trajectories show four robots safely crossing at an intersection near the origin. The event-triggered mechanism perturbs the nominal states $\hat{\boldsymbol{x}}_{i}$ when $E_{i}<0$ , breaking potential deadlocks while maintaining safety and formation coherence.

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This paper proposes a fully decentralized model predictive control (MPC) framework with control barrier function (CBF) constraints for safety-critical trajectory planning in multi-robot legged systems. The incorporation of CBF constraints introduces explicit inter-agent coupling, which prevents direct decomposition of the resulting optimal control problems. To address this challenge, we reformulate the centralized safety-critical MPC problem using a structured distributed optimization framework based on the alternating direction method of multipliers (ADMM). By introducing a novel node-edge splitting formulation with consensus constraints, the proposed approach decomposes the global problem into independent node-local and edge-local quadratic programs that can be solved in parallel using only neighbor-to-neighbor communication. This enables fully decentralized trajectory optimization with symmetric computational load across agents while preserving safety and dynamic feasibility. The proposed framework is integrated into a hierarchical locomotion control architecture for quadrupedal robots, combining high-level distributed trajectory planning, mid-level nonlinear MPC enforcing single rigid body dynamics, and low-level whole-body control enforcing full-order robot dynamics. The effectiveness of the proposed approach is demonstrated through hardware experiments on two Unitree Go2 quadrupedal robots and numerical simulations involving up to four robots navigating uncertain environments with rough terrain and external disturbances. The results show that the proposed distributed formulation achieves performance comparable to centralized MPC while reducing the average per-cycle planning time by up to 51% in the four-agent case, enabling efficient real-time decentralized implementation.

Abstract

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