Formation Control in Agents' Local Coordinate Frames for Arbitrary Initial Attitudes

In this article, we address formation tracking control in agents' body coordinate frames while imposing no assumptions, such as positive definiteness on agents' initial attitude matrix. Difficulty in achieving formation control is exacerbated when agents are unable to access their absolute attitudes relative to a common frame of reference, and initial attitude matrices are required to be restricted to special sets. Addressing this issue, we present a distributed fixed-time attitude synchronization law, which utilizes only relative attitude measurements to align the agents' coordinate frames almost globally in finite time and locally in fixed time. The proposed law ensures the global asymptotic convergence of formation tracking error for displacement-based formation control in a leader–follower topology. We also establish global uniformly asymptotic stability for bearing-only formation tracking. A Lyapunov-based attitude estimation algorithm is proposed to estimate the attitudes up to a common reference frame in finite time without using Gram–Schmidt orthogonalization. For nonholonomic agents, formation stabilization in agents' local coordinate frame is studied using these attitude estimates and bearing-only measurements. Finally, to illustrate the applicability of our results, simulations are presented.