Distributed Nominal Configuration Design for Linear Formations

Linear formation is a recently introduced concept in formation control that characterizes the formation shape through a linear mapping from a high-dimensional nominal configuration to agents' workspace. However, the abstract nature of high-dimensional spaces renders the design of such nominal configurations challenging. This paper transforms the distributed design of nominal configurations into the problem of solving nonlinear equations in distributed manners. It is shown that the gradient-based method may fall into undesired equilibria when local solutions are near the origin. To address this issue, an exponential coordinate mapping is introduced to restrict the solution search to the positive orthant of the state space, thereby avoiding undesired equilibria. In addition, auxiliary variables are employed to shift the desired equilibrium, ensuring the solvability of the nonlinear equation and enhancing the convergence behavior of the system. This equilibrium shift is also shown to be compatible with the original nominal configuration design problem. Finally, numerical simulations are provided to verify the effectiveness of the proposed method.