Arbitrary Pattern formation ( $\mathcal{A}\mathcal{P}\mathcal{F}$) by a swarm of mobile robots is a widely studied problem in the literature. Many works regarding $\mathcal{A}\mathcal{P}\mathcal{F}$ have been proposed on plane and infinite grid by point robots. But in practical application, it is impossible to design point robots. In Bose et al. [Arbitrary pattern formation on infinite grid by asynchronous oblivious robots. Theor Comput Sci. 2020;815:213–227], the robots are assumed opaque fat robots but the environment is plane. To the best of our knowledge, no work till now ever considered the $\mathcal{A}\mathcal{P}\mathcal{F}$ problem assuming opaque fat robots on infinite grid where movements are restricted. In this paper, we have provided a collisionless distributed algorithm and solved $\mathcal{A}\mathcal{P}\mathcal{F}$ using 9 colours.

The ARBITRARY PATTERN FORMATION (APF) is widely studied in distributed computing for swarm robots. This paper deals with the APF problem in an infinite grid under an asynchronous scheduler. In [Bose K, Adhikary R, Kundu MK, et al. Arbitrary pattern formation on infinite grid by asynchronous oblivious robots. Theor Comput Sci. 2020;815:213–227], the authors proposed an algorithm for APF problem in $\mathcal{O}\mathcal{B}\mathcal{L}\mathcal{O}\mathcal{T}$ model under an asynchronous scheduler, but the proposed algorithm was neither time optimal nor move optimal. This work provides two algorithms that solve APF problem in an asynchronous scheduler. The first algorithm is move optimal considering $\mathcal{O}\mathcal{B}\mathcal{L}\mathcal{O}\mathcal{T}$ model and the second algorithm is move and time optimal considering the $\mathcal{L}\mathcal{U}\mathcal{M}\mathcal{I}$ model, where each robot has one light having three distinct colours.