Error analysis to minimize cross-axis couplings in 6-DOF motion systems with a single moving part

In multi-axis motion control, cross-axis couplings cause error force and position disturbances in an axis when a desired motion is generated along another axis. Different from the parasitic errors that result from the imperfections of the mechanical bearings and reference surfaces, cross-axis perturbations are caused by errors that occur both statically (geometrical errors) and dynamically (in the transient responses) and are more prevalent in air-bearing and magnetic-levitation (maglev) stages. The parasitic errors are heavily dependent on the sizes of the stage's mechanical components, while the cross-axis perturbations depend significantly on the mover's speed and acceleration. For stages using permanent magnets (PMs) and Lorentz coils, the causes of off-axis forces include 1) errors in the coil turns' straightness, perpendicularity, and parallelism of the motor axes, and 2) errors in the local magnetizations and PMs' fringing effects. The purpose of this paper is to analyze the topologies of 6-degree-of-freedom (6-DOF) single-moving-part stages to minimize cross-axis couplings. The outcome is a stage configuration with reduced couplings and cross-axis perturbations. This is supported by experimental results performed on a newly developed 6-DOF maglev laser-interferometer stage. Its achieved root-mean-square (rms) positioning noise and minimum step size in XY are 3 nm and 10 nm, respectively. Its achieved resolution in out-of-plane rotations is 0.1 mu rad. In addition to the analysis supported by these results, this paper introduces a new measure to represent cross-axis perturbations and to compare the effects of couplings in multi-axis positioning. This measure is entitled the cross-coupling quantity (CCQ) and calculated from the displacement of the stage in the axis of interest, the peak time of the response, and the peak-to-peak (p-p) error in the perturbed axis.