Optimal formation problems of robots by the Lagrangian multiplier method

Optimal formation of robots is one of the most important and interesting topics in the fields of robotics, multi-agents, intelligent control and evolutionary computation, etc. The objective of this paper is mainly to discuss the formation problems by solving the Lagrangian method when the number of robots is only equal to 3 and the number of constraints is larger than 1. The considered objective function is selected to be the sum square distance from points in the initial shape to points in the desired shape. More importantly, two theorems and the analytic solution on three typical cases are provided to analyze and discuss the typical formation problems with three robots. Furthermore, one theorem is to highlight that the center of the initial shape is equal to that of the desired shape, and the corresponding theorem can improve the efficiency and reduce the optimization dimension of formation problem.