Multiple-axis parallel kinematics piezo-flexural stages, used in variety of scanning probe microscopy and manipulation applications, suffer not only from the hysteresis nonlinearity and the effects of system dynamics, but also from the nonlinear coupling of one axis action on the motion of the other axes. Motivated by these shortfalls, a Lyapunov-based adaptive robust control methodology is proposed for the simultaneous tracking control of multiple-axis parallel kinematics piezo-flexural micro/nano-positioning systems in the presence of parametric uncertainties and unknown or partially known hysteresis and coupling nonlinearities. For this, a double-axis parallel-kinematics piezo-flexural stage with sub-nanometer resolution capacitive position sensors is considered for the system analysis and controller verification. The hysteresis response and the coupling effect of the stage are studied through a number of experiments, followed by a mathematical formulation for the system nonlinear equation of motion. Adopting the variable structure (sliding mode) control method with a parameter adaptation strategy, an adaptive robust controller is then derived with its asymptotic stability guaranteed through the Lyapunov stability criterion. To avoid the chatter phenomenon, which is a common problem in the sliding mode control, the hard switching function of the controller is replaced by a soft switching function. Through theoretical analysis, it is demonstrated that a parallelogram-type zone of attraction can be explicitly formed for the proposed soft variable structure controller, to which the error phase trajectory converges. Therefore, a uniformly ultimately bounded closed-loop system response is achieved. The proposed controller is eventually implemented on the piezo-flexural stage, demonstrating the effective double-axis tracking control of the stage, even in the absence of a representative hysteresis model and despite parametric uncertainties. Results indicate good agreement between experiments and theoretical developments.

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