Cooperative exploration of level surfaces of three dimensional scalar fields

We develop strategies for a group of mobile sensing agents to cooperatively explore level surfaces of an unknown 3D scalar field. A cooperative Kalman filter is constructed to combine sensor readings from all agents and give estimates of the field value and gradient at the center of the formation formed by the sensing agents. The formation formed by the agents is controlled to track curves on a level surface in the field under steering control laws. We prove that the formation center can move to a desired level surface and can follow a curve with known frame and curvatures. In particular, we present results on tracking lines of curvature on a desired level surface, revealing the 3D geometry of the scalar field. Taubin's algorithm is modified and applied to detect and estimate principal curvatures and principal directions for lines of curvature. We prove the sufficient and necessary conditions that ensure reliable estimates using Taubin's algorithm. We also theoretically justify the minimum number of agents that can be utilized to accomplish the exploration tasks. Simulation results demonstrate that a line of curvature on a desired level surface can be detected and traced successfully.