GEOMETRIC COOPERATIVE CONTROL OF FORMATIONS
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Robots in a team are modeled as particles which obey simple, second order dynamics. The whole team can be viewed as a deformable body with changing shape and orientation. Jacobi shape theory is applied to model such a formation. We derive the controlled system equations using the Lagrange-D'Alembert principle. Control forces on each robot are combined and reorganized as controls for the center, for rotation and for shape changes. From a shape-theoretic point of view, general feedback control laws are designed to achieve desired formations. The system equations on shape space provide possibilities for achieving formations without communication links between team members equipped with sufficient sensing ability. We allow each robot freedom to establish a coordinate system in which shape dynamics of the whole formation is computed. Without knowing such coordinate systems of other robots, each robot is able to perform cooperative control. This is made possible by a class of gauge covariant control laws. We argue that freedom of choosing gauge frame helps to improve controller performance. When all robots are required to have common constant speed, the control forces have to be of gyroscopic nature. Previous works of Justh and Krishnaprasad has inspired us to study the obstacle avoidance and navigation problem from a point of view of formation shape control. We achieve gyroscopic control laws to achieve boundary following behavior when the particle encounters an obstacle. The "steady state" trajectory of the particle forms a Bertrand pair with the boundary curve of the obstacle. This steady state behavior correspond to a relative equilibrium for a non-autonomous system on special Euclidean groups. Our control law achieves asymptotic convergence of the non-autonomous system dynamics. The boundary following behavior is a building block for robot navigation in a cluttered environment. Based on the configuration of the obstacles and the target, we may construct virtual boundary curves by analyzing sensory data. Such virtual boundary curves lead the robot to the target without collision. We have also studied the problem of establishing a formation of satellites with periodic shape changes near an elliptic earth orbit. We propose a control law that would set up a given formation near a given orbit. This law also allows a satellite formation to achieve orbit transfer.