Distributed Time-Varying Optimization for Predefined-Time Affine Formation in MASs

Abstract

Time-varying robotic tasks require a formation to adapt its collective motion while achieving desired affine geometry within a user-specified time. This paper studies a dynamic optimized affine formation problem that couples distributed time-varying optimization with the physical dynamics of multiagent systems. A predefined-time optimization control (PTOC) algorithm is developed to integrate leader-side optimization and stress-matrix-based affine formation within a unified closed loop. Unlike existing optimization-based affine-formation methods with asymptotic convergence, the proposed approach guarantees that the leaders reach the optimizer compatible with the affine constraints and that the entire formation reaches the corresponding affine configuration within a user-specified time. The analysis is established under uniformly strongly convex and smooth local cost functions with bounded time variation, which ensures the well-posedness of the Hessian-based control law. Numerical simulations and experiments demonstrate the predefined-time formation adaptation achieved by the proposed method. Note to Practitioners—Multi-robot tasks generally require a formation to adjust its collective motion as task conditions change while maintaining the desired geometry and meeting a specified completion time. Conventional methods usually compute an updated reference first and then command the robots to track it, without directly guaranteeing when the entire formation reconfiguration will be completed. The proposed PTOC method integrates time-varying task optimization and affine formation control within the same feedback loop. It allows the designer to specify the time within which both the task-driven motion adaptation and formation reconfiguration are completed, which is valuable for time-critical inspection, escort, and cooperative transportation. The method is demonstrated on a six-robot platform and currently requires smooth strongly convex objectives with available gradient and Hessian information.