Monotonic self-stabilization and its application to robust and adaptive pattern formation

We introduce, as an enhancement of self-stabilization, the concept of monotonic self-stabilization for distributed systems that ensures that a certain measure of quality of state monotonically improves when the system in an illegitimate state progresses toward a legitimate state. In the concept, the quality is measured by a real-valued function that can be chosen from a certain class of functions. The concept is applied to a multi-robot pattern formation problem in which a group of autonomous mobile robots move from their respective initial positions to the goal positions like a marching band. We solve the problem by presenting two monotonic self-stabilizing pattern formation algorithms, one of which is for FSYNC model and the other is for SSYNC model. The considered real-valued functions to measure the quality take into account both the distance to the goal location and the accuracy of the formation. We present a formal proof of the algorithms' correctness and monotonic self-stability.