[Paper Review] Arbitrary pattern formation by asynchronous opaque robots on infinite grid

Manash Kumar Kundu, Pritam Goswami+2 more|arXiv (Cornell University)|May 6, 2022

Optimization and Search Problems4 citations

TL;DR

This paper presents a deterministic algorithm for Arbitrary Pattern Formation (APF) using asynchronous, opaque, luminous robots on an infinite grid with 8 persistent colors and one-axis agreement. The robots operate in Look-Compute-Move cycles, avoid collisions via sequential movement, and achieve target pattern formation in O(kD) moves by electing a leader and forming a compact line before moving to target coordinates.

ABSTRACT

Arbitrary pattern formation ($\mathcal{APF}$) by mobile robots is studied by many in literature under different conditions and environment. Recently it has been studied on an infinite grid network but with full visibility. In opaque robot model, circle formation on infinite grid has also been studied. In this paper, we are solving $\mathcal{APF}$ on infinite grid with asynchronous opaque robots with lights. The robots do not share any global co-ordinate system. The main challenge in this problem is to elect a leader to agree upon a global co-ordinate where the vision of the robots are obstructed by other robots. Since the robots are on a grid, their movements are also restricted to avoid collisions. In this paper, the aforementioned hardness are overcome to produce an algorithm that solves the problem.

Motivation & Objective

• To solve Arbitrary Pattern Formation (APF) for asynchronous, opaque, luminous robots on an infinite grid with obstructed visibility.
• To overcome the challenge of leader election without global coordinates, using only local visibility and one-axis agreement.
• To prevent collisions during movement by enforcing sequential robot movement and structured target positioning.
• To design a scalable algorithm that works for all solvable initial configurations under an adversarial asynchronous scheduler.
• To extend prior work on APF in obstructed visibility models to the grid environment, which is more constrained than the plane.

Proposed method

• Uses a two-phase approach: first forming a compact line (LH1) using a leader-elected configuration.
• Employs a leader robot with a 'leader' light to establish a reference coordinate system based on one-axis agreement.
• Implements sequential movement: only one robot moves at a time, ensuring no collisions during transitions.
• Assigns unique target coordinates to each robot based on their position relative to the leader and the target pattern.
• Uses a TargetMove() procedure that guides robots vertically and horizontally to their target positions, updating lights only upon arrival.
• Relies on persistent lights (8 colors) for coordination and state tracking, enabling robots to determine their role and target in the pattern.

Experimental results

Research questions

• RQ1 Can Arbitrary Pattern Formation be achieved with opaque, asynchronous robots on an infinite grid under obstructed visibility?
• RQ2 How can a leader be elected and a global coordinate system established without a shared reference frame?
• RQ3 What mechanisms ensure collision-free movement when robots are restricted to grid edges and cannot see beyond blocked lines of sight?
• RQ4 How can the algorithm guarantee formation of any arbitrary pattern regardless of initial configuration?
• RQ5 What is the time complexity of the algorithm in terms of robot moves, and can it be bounded?

Key findings

• The algorithm solves APF for all solvable initial configurations on an infinite grid with opaque, asynchronous, luminous robots using only one-axis agreement and 8 colors.
• The algorithm ensures collision-free movement by enforcing sequential robot movement and structured path planning.
• The worst-case time complexity is O(kD) moves, where k is the number of robots and D = max{m, n, M, N, k} (m, n and M, N are dimensions of the smallest enclosing rectangles of initial and target configurations).
• A compact line is formed on a reference line (LH1) to enable leader-based coordination and target assignment.
• All robots except the leader and the last robot reach their targets via a two-step movement: vertical alignment followed by horizontal adjustment.
• The leader robot moves last to the final vacant target position, completing the pattern formation.