Local asymptotic convergence of a cycle-free persistent formation of double-integrators in three-dimensional space

We present local asymptotic convergence analysis for a cycle-free persistent formation of double-integrator modeled agents moving in three-dimensional space. Due to the absence of an available common sense of orientation, the agents sense the relative-displacements of their neighbors only with respect to their own local reference frames whose orientations are not aligned, and control the norms of the relative-displacements to stabilize their formation to the desired formation. Under a gradient-based control law for the agents, we prove local asymptotic convergence of the cycle-free persistent formation to the desired formation based on cascade system stability theory. This result is an extension of the existing results on two-dimensional formations of single-integrators.