Pseudo-rigid formation design with curvature limitations

A virtual structure, called the pseudo-rigid formation, for a multi-agent system has been proposed in [1] to deal with the formation problem. For 2-D problems, the underlying configuration space becomes the product of the two dimensional real vector space and the general linear group (R² × GL(2)). It was shown that the objective function in the optimal design for collision avoidance can be represented in terms of the system variables. The optimal formation can be then obtained, which is allowed to translate, rotate, stretch, and shear. However, the design in [1] does not take the limitations on the motion of the agents, such as an airplane or a car, into consideration. In this paper, the limitation of the curvature of the trajectory of the vehicles is imposed and accommodated in the path planning algorithm. While the designed optimal paths may not be the most efficient, they are the most appropriate such that the agents are capable to follow.