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We study the problem of distance-based formation shape control for autonomousagents with double-integrator dynamics. Our considerations are focused onexponential stability properties. For second-order formation systems under thestandard gradient-based control law, we prove local exponential stability withrespect to the total energy by applying Chetaev's trick to the Lyapunovcandidate function. We also propose a novel formation control law, which doesnot require measurements of relative positions but instead measurements ofdistances. The distance-only control law is based on an approximation ofsymmetric products of vector fields by sinusoidal perturbations. A suitableaveraging analysis reveals that the averaged system coincides with themulti-agent system under the standard gradient-based control law. This allowsus to prove practical exponential stability for the system under thedistance-only control law.

Exponential and practical exponential stability of second-order formation control systems