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Geometric Cooperative Control of Formations

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Doctoral thesis

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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. The configuration space of the whole formation is then viewed as a fiber bundle on Jacobi shape space. Applications of shape theory to analyze problems, such as the sequential pursuit, provide insights on body motion and shape changes. We derive the controlled system equations using the Lagrange-DAlembert principle. In the resulting Lagrangian equations, the motion of the center of mass of the formation is decoupled from the rotation and shape dynamics. Furthermore, 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. This class of feedback control laws induces interactions that can be computed in a distributed manner. When communication links and GPS are available to each robot, we use Jacobi vectors as feedback for control. We show that formations are set up asymptotically. The system equations on shape space provide possibilities for achieving formations without communication links between team members, under the assumption that each robot uses its sensors to measure shape and rotation of the formation. 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. Using Lyapunov functions, we prove that controlled dynamics converges to an invariant set where desired shape is achieved. We argue that this freedom of choosing gauge frame helps to improve controller performance in a noisy environment.

Subject Categories:

  • Numerical Mathematics
  • Cybernetics
  • Theoretical Mathematics

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