This paper presents a micro XY stage that employs electrostatic comb-drive actuators and achieves a bi-directional displacement range greater than 225 μm per motion axis. The proposed XY stage design comprises four rigid stages (ground, motion stage, and two intermediate stages) interconnected via flexure modules. The motion stage, which has two translational degrees of freedom, is connected to two independent single degree of freedom intermediate stages via respective parallelogram (P) transmission flexures. The intermediate stages are connected to the ground via respective clamped paired double parallelogram (C-DP-DP) guidance flexures. The C-DP-DP flexure, unlike conventional flexures such as the paired double parallelogram flexure (DP-DP), provides high bearing direction stiffness (K b) while maintaining low motion direction stiffness (K m) over a large range of motion direction displacement. This helps delay the onset of sideways instability in the comb-drive actuators that are integrated with the intermediate stages, thereby offering a significantly greater actuation stroke compared to existing designs. The presented work includes closed-form stiffness analysis of the proposed micro-stage, finite elements simulation, and experimental measurements of its static and dynamic behavior.

This paper presents an investigation of zeros in the SISO dynamics of an undamped three-DoF LTI flexible system. Of particular interest are non-minimum phase zeros, which severely impact closed-loop performance. This study uses modal decomposition and zero loci to reveal all types of zeros — marginal minimum phase (MMP), real minimum phase (RMP), real non-minimum phase (RNMP), complex minimum phase (CMP) and complex non-minimum phase (CNMP) — that can exist in the system under various parametric conditions. It is shown that if CNMP zeros occur in the dynamics of an undamped LTI flexible system, they will always occur in a quartet of CMP-CNMP zeros. Consequently, the simplest undamped LTI flexible system that can exhibit CNMP zeros in its dynamics is a three-DoF system. Motivated by practical examples of flexible systems that exhibit CNMP zeros, the undamped three-DoF system considered in this paper comprises of one rigid-body mode and two flexible modes. For this system, the following conclusions are mathematically established: (1) This system exhibits all possible types of zeros. (2) The precise conditions on modal frequencies and modal residues associated with every possible zero provide a mathematical formulation of the necessary and sufficient conditions for the existence of each type of zero. (3) Alternating signs of modal residues is a necessary condition for the presence of CNMP zeros in the dynamics of this system. Conversely, avoiding alternating signs of modal residues is a sufficient condition to guarantee the absence of CNMP zeros in this system.

This paper presents a model to explain complex nonminimum phase (CNMP) zeros seen in the noncollocated frequency response of a large-displacement XY flexure mechanism, which employs multiple double parallelogram flexure modules (DPFMs) as building-blocks. Geometric nonlinearities associated with large displacement along with the kinematic under-constraint in the DPFM lead to a coupling between the X and Y direction displacements. Via a lumped-parameter model that captures the most relevant geometric nonlinearity, it is shown that specific combinations of the operating point (i.e., flexure displacement) and mass asymmetry (due to manufacturing tolerances) give rise to CNMP zeros. This model demonstrates the merit of an intentionally asymmetric design over an intuitively symmetric design in avoiding CNMP zeros. Furthermore, a study of how the eigenvalues and eigenvectors of the flexure mechanism vary with the operating point and mass asymmetry indicates the presence of curve veering when the system transitions from minimum phase to CNMP. Based on this, the hypothesis of an inherent correlation between CNMP zeros and curve veering is proposed.

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This paper presents a micro XY stage that employs electrostatic comb-drive actuators and achieves a bi-directional displacement range greater than 225 μm per motion axis. The proposed XY stage design comprises four rigid stages (ground, motion stage, and two intermediate stages) interconnected via flexure modules. The motion stage, which has two translational degrees of freedom, is connected to two independent single degree of freedom intermediate stages via respective parallelogram (P) transmission flexures. The intermediate stages are connected to the ground via respective Clamped paired Double Parallelogram (C-DP-DP) guidance flexures. The C-DP-DP flexure, unlike conventional flexures such as the paired Double Parallelogram flexure (DP-DP), provides high bearing direction stiffness (Kb) while maintaining low motion direction stiffness (Km) over a large range of motion direction displacement. This helps delay the onset of sideways instability in the comb-drive actuators that are integrated with the intermediate stages, thereby offering significantly greater actuation stroke compared to existing designs. The presented work includes closed-form stiffness analysis of the proposed micro XY stage, finite elements simulation, and experimental measurements of its static and dynamic behavior.