A linear control approach to distributed multi-agent formations in d-dimensional space

The paper presents a linear approach for formation control of multiple autonomous agents in d-dimensional space. The generalized notion of graph Laplacian, associated to a graph with possibly negative weights on the edges, is introduced aiming to solve the formation problem of controlling a network of point agents to form a given pattern (including rotation, translation, and scaling). By assuming the number of agents is larger than d + 1, we derive that a linear distributed control law exists for this purpose if and only if certain algebraic conditions hold (or equivalently, the graph is globally rigid). Next, it is shown that the generalized graph Laplacian used to stabilize the formations can be obtained by solving a convex optimization problem. Further results are also provided to reveal the conditions under which the attained formations are congruent to or just translations of the desired one.