A finite element based model predictive controller (FEMPC) is developed and practically implemented for attenuating in-plane vibration of a two flexible link planar manipulator. This FEMPC structure is based on that used in dynamic matrix control (DMC), with the exception that a finite element (FE) model replaces how the predictions are formulated. A linear FE model is developed for each individual link, which is used with the current measured strain and control actions, to predict the response of each link. These predictions are carried out at each time step to address the geometric non-linearities associated with the orientation of the second link and those associated with friction, backlash and compliance of the geared motors. Furthermore, the use of FE modelling enables the control structure to be formulated based on known properties of the system, eliminating the need for open loop testing. The resulting FEMPC scheme is shown to outperform DMC and is capable of providing substantial attenuation of vibration, reducing the mean amplitude of dominant vibration by 92.5% and 15.6%, for the first and second links, respectively. INTRODUCTIONWith lighter linkages and smaller spatial profiles, flexible link manipulators have advantages over their rigid link counterpart, when the payload being moved is relatively lightweight. These advantages include faster accelerations and more efficient operations since less energy is expended to move these lighter flexible linkages. However, the fundamental drawback of these manipulators is their susceptibility to deflection during motion, which leads to inaccurate positioning of the end effector due to the resulting vibration of the linkages. To address this issue active vibration control has been shown to be a potential solution, where actuators, such as piezoelectric plates, are used to counteract the vibration of these linkages, resulting in a more accurate positioning of the end effector.In part 1 of this work model predictive control (MPC) in the form of dynamic matrix control (DMC) was presented as viable control scheme for attenuating vibration of a planar two flexible link manipulator. While shown to work well with short prediction horizons, the selection of short prediction horizons in MPC tends to result in a less stable control as the scheme becomes more sensitive to noise. However, by increasing the prediction horizon the effectiveness of the controller to attenuate vibration is observed to decline. Since DMC control is ideally intended for first order systems, this behavior is viewed as a result of inadequate representation of the dynamics of the linkages when using the dynamic matrix to predict the response of the links. As such, this part shall incorporate an alternative means of modelling the vibration of the linkages into the DMC scheme, this alternative being finite element (FE).

A model predictive controller (MPC) in the form of dynamic matrix control (DMC) is implemented for attenuating in-plane vibrations of a two flexible link planar manipulator. The rotation of the joints and inertia effect of both the joints and links induce vibration. Piezoelectric actuators, mounted in a bimorph configuration, provide the control actions to reduce vibrations. Implementation of this control scheme is shown to provide appreciable attenuation of vibration over the uncontrolled case, increasing the damping ratio for the first and second link by a factor of 5.99 and 3.40, respectively. DMC control is further shown to reduce the mean amplitude of dominant vibrations from the uncontrolled case by 90.0% and 87.4%, respectively, for the first and second links. Furthermore, for the two link setup, this control is shown to outperform the more conventional ProportionalIntegral-Derivative (PID) control and is sufficiently robust to handle an unknown payload.

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