Intrinsic Geometry-Aware Distributed Continuous-Time Optimization on SO(3) for Multi-agent Systems

This project aims to develop a robust, scalable framework for coordinating complex multi-agent networks operating in constrained environments. The project will bring transformative change by providing a unified mathematical approach that enables these systems to achieve optimal performance in attitude synchronization, cooperative positioning, and high-speed multi-agent alignment. This will be achieved by leveraging geometric principles to simplify how agents process information and make collective decisions without centralized control. While distributed optimization is well-studied, existing methods are largely restricted to flat, Euclidean spaces, failing to account for the unique curved configuration space of rotations, known as a manifold (specifically, the Special Orthogonal Group). Because traditional tools for flat spaces cannot manage this geometry, current methods often suffer from instability and performance lag in dynamic scenarios. The intellectual merit of the project lies in the creation of control laws that respect this curved structure to navigate geometric nonlinearities with guaranteed stability, providing the high-precision coordination required for complex, time-critical operations. The broader impacts of the project include curriculum enrichment and development in distributed optimization, the mentorship of students in science and engineering, and community outreach through scholarly sessions on advanced system coordination. Technically, the objective is to establish a comprehensive intrinsic nonsmooth framework for distributed continuous-time optimization on the Special Orthogonal Group (SO(3)) by developing unified optimization laws that operate directly on the manifold, ensuring representation-independent feasibility and convergence. This research is organized into three integrated thrusts: first, developing the mathematical foundations by performing analytical stability and convergence analysis to prove the forward invariance of safe geodesic regions and establish structural conditions for exact exponential convergence to team-optimal orientations; second, enhancing the framework to account for time-varying cost functions for high-fidelity online tracking, developing fully distributed self-adaptive laws to eliminate global gain tuning, and devising robust strategies to mitigate topological obstructions during dynamic maneuvers; and third, performing experimental validation on networked systems to validate the practical robustness and real-time feasibility of the proposed approach. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.