Discrete-Time Formation Control With Attitude Alignment: A Geometric Approach Under Ergodic Products

Assuming agents possess sensing capabilities and dynamics relative to their body coordinate frames, this article addresses a displacement-based strategy for discrete-time formation control through attitude synchronization. Unlike the Eulerian approximation, the first challenge lies in developing discrete-time models that capture both position and attitude motions, thereby complementing existing continuous-time systems with a discretized version. Then, two leader-following control protocols are designed, incorporating a jointly connected topology to relax the connectivity requirements. The analysis of these systems often relies on the Lyapunov function, the existence of which may be hard to guarantee. Consequently, the geometric properties of several polytopes are explored under the dynamic topologies and time-varying parameters. By extending the ergodicity results to leader-following scenarios, these properties are characterized as positively invariant sets. This invariance, which is independent of the existence of the Lyapunov function or the solvability of its associated criteria, provides a beneficial tool for analyzing time-variant systems. Utilizing this invariance and some mathematical techniques, the desired formation shape can be achieved in cases where the attitudes of the followers synchronize with the leader. Finally, a numerical simulation validates the results.