Analysis of undirected formation shape control with directional mismatch

A typical formation shape control problem involves point agents sensing relative positions, i.e. directions and distances, of their neighbors and then moving so that these relative positions achieve some prespecified values. Such a procedure requiring as it does sensing of directions implicitly presupposes that all agents have a shared understanding of the direction of north. On the other hand, there may be biases in sensors, variations in the earths magnetic field interfering with compass-based sensing, or drift in inertial sensors, with the result that directions are inconsistently measured or measured with error. This paper considers the consequences of this error, considering first the two agent case in a two-dimensional plane. We show that the agents converge to a fixed, but distorted formation exponentially fast. In contrast to the matched case, the formation is not asymptotically stationary, but rather instead translates with a certain constant velocity. The distance error and the angular error between the actual final formation and the desired formation are explicitly given, as well as the steady state velocity of the formation. Based on the results, estimation algorithm is given to obtain the mismatch angle, which allows a compensation algorithm to be proposed such that the desired formation is achieved with zero steady-state velocity for the formation as a whole. The case of the three-dimensional ambient space is then considered and similar phenomena are observed. Simulations are also provided to validate the theoretical results.