On 3-D Formation Control With Mismatched Coordinates

A typical formation shape control problem involves point agents sensing relative positions, that is, orientations 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 orientations, implicitly presupposes that all agents have a shared understanding of the common orientations. On the other hand, there may be biases in sensors, variations in the earth's magnetic field interfering with compass-based sensing, or drift in inertial sensors, with the result that orientations are inconsistently measured or measured with error. In this paper, we investigate the formation control problem with mismatched coordinates in the 3-D space, considering the consequences of this error. First, the situation of a two-agent formation is considered. 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 depending on the mismatches. The formation distortion between the actual one and the desired one is obtained, as well as the steady-state velocity of the formation for small mismatch orientations. The case of agents with double integrator dynamics is then considered and similar phenomena are observed. Based on the results, an estimation algorithm is given to obtain the mismatch rotation matrix, 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
$n$
-agent formations is finally considered, first with a star graph and then with a general graph. Simulations are provided to validate the theoretical results.