Modeling and Simulating a Mathematical Tool for Multi-robot Pattern Transformation Blesson Varghese Active Robotics Laboratory, School of Systems Engineering, University of Reading, Reading, Berkshire, UK, RG6 6AY. e-mail: b.varghese@reading.ac.uk Gerard T. McKee School of Systems Engineering, University of Reading, Reading, Berkshire, UK, RG6 6AY. e-mail: g.t.mckee@reading.ac.uk Abstract—The work reported in this paper is motivated by the need to investigate general methods for pattern transformation. A formal definition for pattern transformation is provided and four special cases namely, elementary and geometric transformation based on repositioning all and some agents in the pattern are introduced. The need for a mathematical tool and simulations for visualizing the behavior of a transformation method is highlighted. A mathematical method based on the Moebius transformation is proposed. The transformation method involves discretization of events for planning paths of individual robots in a pattern. Simulations on a particle physics simulator are used to validate the feasibility of the proposed method. Keywords-mathematical transformation; moebius transformation; pattern formation; pattern transformation I. INTRODUCTION Swarm robotics researchers have explored over years methods for establishing and maintaining patterns. The motivation to investigate pattern formation methods originates from models evident in nature. One outcome of pattern formation research in robotics is the formulation of general methods. For example, the leader-follower [1], virtual point [2] and behavioral approaches [3], the most commonly cited are general methods for pattern formation. In nature, there are slow or fast phenomena that result in the change of patterns. For example, the change of grain patterns of rocks is a slow process while the transformation of shape of a swarm of bees is an instantaneous process. Though nature provides illustrations for pattern transformation, there seems to be minimal effort towards investigating general methods for swarm pattern transformation. Pattern transformation in multi-robots is an appropriate response to obstacles for unhindered motion. Transformation of patterns is synonymous with the reconfiguration of patterns. Reconfiguration is achieved by repositioning agents in the multi-robot. A transformation is intended to reconfigure the pattern geometrically which can be achieved by repositioning all or only a subset of the agents. The term reconfiguration is also associated with modular robotic systems. Algorithms to transform the shape of modular robots are considered in [4] -- [7]. However, it is necessary to draw distinction between reconfiguration in modular systems and transformation of multi-robot patterns considered in this paper. Firstly, in modular robot systems physical connectivity between modules exists ensuring modules in close vicinity of adjacent modules. On the other hand in swarm systems, wireless connectivity is established between swarm units, thereby geographically dispersing the swarm units in space. Secondly, reconfiguring in modules is constrained by being able to reposition on the periphery of an attached module. Thirdly, reconfiguration in modules is not strictly geometrical in nature, while transformation of multi-robot reported in this paper is strictly geometry oriented. Formation morphing [8] is another term for pattern transformation. However, formation morphing is used to express the notion of swapping roles / positions within a swarm, which is different to the concept of pattern transformation discussed in this paper. Some researchers have reported pattern transformation in the context of pattern formation studies. In [8] a stable virtual leader pattern transforms to a different pattern by the addition of a morphing force. Illustrations of transformation and mathematical notations for computation of forces in the pattern are presented. The transformation technique claims to facilitate pattern change by allowing participating agents to find their own equilibrium. However, the morphing procedure for transforming pattern is not defined nor appears to be general. An algorithm reported in [9] is capable of transforming patterns in response to a command issued by a human operator. The command is issued to a single robot and causes a chain reaction in the neighboring robots resulting in a global transformation. Pattern transformation from a parabola to a sine curve is illustrated. Though the notion of transforming patterns is presented, the transformation method remains unaddressed. A relative distance versus orientation model for transformation is reported in [10]. The strategy involves varying the orientation value to globally transform a triangle to a line formation. Though a positional transformation is not executed on all participating agents (one position remains unaltered), a global geometrical transformation is achieved. This strategy is specific to the scenario when a