Edge potential residuals in low-complexity aggregation behaviors

In multiagent robotics, formation control problems usually address the maintenance of inter-agent distances equal to a set of predefined values. This problem requires either agents to be distinguishable or some combinatorial optimization to be solved in order to find the most convenient target-to-agent assignment. However, in many applications or mission stages, group connectivity and collision free trajectories are in fact the only desired requirements, while the robots themselves are allowed to freely generate their spatial distribution. To this end, the simplest possible case corresponds to all agents trying to maintain the same relative scalar distance. However, since it is not possible to have more than d+1 equally separated points in a d-dimensional space, we investigate solutions where interagent distances are as close as possible to desired one. We introduce a quadratic potential proportional to the error from the desired distance and equilibrium with non-null potentials are investigated. Stability of the behavior resulting from our protocol is described using elements from rigidity theory and final results are presented in form of numerical simulations.