Leader-Follower Synchronization of Mechanical Systems
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In this thesis a nonlinear output feedback scheme for leader-follower synchronization of rigid robot manipulators and ships is developed. The scheme is also applicable to other systems with Lagrangian dynamics in the same form as for these systems. The objective of the synchronization scheme is that the follower shall follow the motion of the leader when only position measurements are available for the leader and the follower. This is achieved by using nonlinear observers to estimate the velocities of both the leader and the follower. The leader observer is a high gain observer that does not need a physical model of the leader, while the follower observer is based on the Lagrangian dynamics of the follower. Both observers are known from the literature, but are here analysed in a new way by new Lyapunov function candidates and by applying a theorem of uniform ultimate boundedness developed in this thesis. The theorem of uniform ultimate boundedness makes it possible to find separate bound for each part of a state vector, which makes it possible to find a much smaller bound for the position observer errors than for the velocity observer errors. The usage fo this theorem was found to be essential in order to show uniform ultimate boundedness of the synchronization errors. In addition to follower observer is analysed by adding a disturbance term and uncertain model parameters, which was not included in earlier analysis. The follower observer is also modified by saturation of the velocity in the Coriolis term in the observer in order to make it globally asymptotically stable under the condition that the trajectory to be observed is bounded. The synchronization controller is developed based on observer backstepping of the follower observer. The synchronization scheme is analysed by a known small gain theorem that is somewhat modified in this thesis, such that bound are expressed in terms of the norms of the subvectors of the inputs instead of directly in terms of the norms of the inputs. It is shown that by using small gain theorem it easier to see which errors are influenced by which gains than by using a Lyapunov function candidate to analyse the overall system. It also makes it easier to replace the type of observers with other types of observers. The use of the small gain theorem gives a result for the synchronization scheme that is similar to the separation principle for linear observer-controller schemes. The synchronization errors and observer errors are shown to be uniformly ultimately bounded when disturbances and uncertain model parameters are included, with bounds that can be decreased to any value by increasing the observer and controller gains. The allowable value for the norm of the initial errors can be increased to any value by increasing the the observer and controller gains for all errors except for the follower position observer errors. With no disturbances or uncertain parameters the origin of the overall error dynamics is asymptotically stable. The scheme is qualitatively verified by simulations and experimental results. However, the theoretical bounds for given observer and controller gains are much higher than the bounds in the simulations and experiments. This is because in general Lyapunov analysis gives a quite conservative result, because the observers have filtering properties that are not captured by the Lyapunov analysis. A synchronization scheme was also developed for the synchronization of satellite attitudes represented by quaternions, with no uncertain model parameters. Global uniform ultimate boundedness of the errors was shown when disturbances were included, and global asymptotic stability with no disturbances. The scheme was qualitatively verified by simulations.