Common use of Pattern Formation Clause in Contracts

Pattern Formation. [6] The robots have in input the same pattern, called the target pattern F, described as a set of positions in the plane given in lexicographic order (each robot sees the same pattern according to the direction and orientation of its local coordinate system). They are required to form the pattern: at the end of the computation, the positions of the robots coincide, in everybody’s local view, with the positions of F, where F may be translated, rotated, and scaled in each local coordinate system. Definition 3 (Uniform Flocking) Let r1 . . . rn be a set of robots and let F be the flocking pattern. The set of robots satisfy the flocking specification if the following properties hold: • head/leader emergence eventually robots agree on an unique head (leader), r1; • pattern emergence eventually robots, r2, . . . , rn, form the pattern F; • velocity agreement after any modification of the r0 position, robots in the pattern rotate and translate F in order to converge to the same relative position and orientation between r0 and F as it was before the modification. • no collision any robot motion is collision free. Note the common flavour between the Xxxxxxxx rules and the above properties. No collision property corresponds to the separation rule. Velocity agreement corresponds to the alignment rule and finally leader and pattern emergence are similar to the cohesion property. In the following we combine three different tasks to solve the uniform flocking in systems where robots are asynchronous, do not share the same coordinate systems, are oblivious and uniform. First, we design a novel strategy for equipping a set of robots with a common co- ordinate systems. To this end we propose a probabilistic strategy that creates two singularity points. Then, we combine this module with existing probabilistic election strategies ([1, 2]) in order to create the third singularity point. The motion of this third point will eventually designate the head of the flock (robot r0) and the orientation of the common coordinate sys- tem. Then, the emergent common coordinate system is further used by all the robots but r0 to arrange themselves in a flocking pattern, F, that will further follow the head r0. During the pattern motion, both the head and the common coordinate system are preserved.