Pattern Formation by Robots with Inaccurate Movements

\textsc{Arbitrary Pattern Formation} is a fundamental problem in autonomous mobile robot systems. The problem asks to design a distributed algorithm that moves a team of autonomous, anonymous and identical mobile robots to form any arbitrary pattern $F$ given as input. In this paper, we study the problem for robots whose movements can be inaccurate. Our movement model assumes errors in both direction and extent of the intended movement. Forming the given pattern exactly is not possible in this setting. So we require that the robots must form a configuration which is close to the given pattern $F$. We call this the \textsc{Approximate Arbitrary Pattern Formation} problem. We show that with no agreement in coordinate system, the problem is unsolvable, even by fully synchronous robots, if the initial configuration 1) has rotational symmetry and there is no robot at the center of rotation or 2) has reflectional symmetry and there is no robot on the reflection axis. From all other initial configurations, the problem is solvable by 1) oblivious, silent and semi-synchronous robots and 2) oblivious, asynchronous robots that can communicate using externally visible lights.