Stability analysis on four agent tetrahedral formations

We consider a four agent tetrahedral formation of mobile agents in 3-dimensional Euclidean space. Each agent is required to maintain prescribed inter-agent distances from its neighbors so that they collectively form a desired formation shape, a task which is now called distance-based formation control. Under a common gradient-based control law, there exists an incorrect equilibrium set in which the agents do not achieve the desired formation shape. By investigating the linearized dynamics of the system, we prove that all incorrect equilibria are unstable, which results in that desired formation shape is almost globally asymptotically stable. Numerical simulation results are also included.