# Nanopositioning Stages - Coupled Motion in Nanopositioners - Supplier Data by Mad City Labs

## Topics Covered

Background

Nanopositioning Stages - Coupled Motion in Nanopositioners – Supplier Data by Mad City Labs

Understanding Coupled Motion and Its Effect on Nanopositioning

Calculating Positioning Errors

Examples of Yaw Pitch and Roll in a Mad City Labs Nanopositioning Stage

Pure Motion Amplifiers

## Background

All Mad City Labs Nanopositioning stages are designed and manufactured with the goal of minimizing or eliminating coupled motion. By minimizing coupled motion, pure linear motion is achieved. Pure linear motion nanopositioning stages give accurate results: the shape and size of objects you measure in microscopy applications are correct, the stage behaves identically running in the forward and reverse directions, and the position along each axis is independent of the position along the other axes.

## Understanding Coupled Motion and Its Effect on Nanopositioning

To understand coupled motion and the effects it has on nanopositioning, it is appropriate to concentrate on the angular coupled motions of roll, pitch, and yaw. In the figure below, the angles of roll, pitch and yaw are defined for a standard Mad City Labs single axis nanopositioning stage.

Figure 1. Schematic showing the relative motions of yaw, pitch and roll.

## Calculating Positioning Errors

The positioning errors due to coupled motion are easily calculated. By way of example we define the X-axis as the translation direction and determine the positioning errors due to Yaw in the XY plane. The positioning errors are dependent on how far the point of interest is located from the stage center, D, and on its angular position, θ, see figure below. The positioning errors are ex=D(Yaw)cos θ and ey=D(Yaw)sin θ. Similar relationships can be derived for roll and pitch.

Figure 2. Relationship between Yaw and distance from stage center.

## Examples of Yaw Pitch and Roll in a Mad City Labs Nanopositioning Stage

Every Mad City Labs Nanopositioning stage is measured for roll, pitch and yaw during final production testing. Examples of such measurements are given in the figures below.  Using these examples we can calculate ex and ey. The maximum Yaw is 3 μrad, if we assume D = 10mm and θ = 45o, then ex=ey= 21 nm. This is correctly interpreted as a relative positioning error over the entire measurement range of 20 μm. The relative error is therefore equal to 0.1% for this example.

Figure 3. Actual measurements of Yaw, Pitch and Roll in a nanopositioning stage.

## Pure Motion Amplifiers

To achieve long range motion in a moderate footprint the piezoactuator motion is often amplified. Amplifying the motion of a piezoactuator can result in parasitic coupled motion. Among these parasitic motions are the introduction of rotations about the X, Y or Z-axis, and the direct coupling of an X axis translation to a Y axis translations. Quite often these parasitic motions are non reproducible making them difficult or expensive to correct for, making it impossible to zoom in on an object or making it impossible to achieve absolute imaging. For example, a number of nanopositioners on the market use parallelograms or other types of amplifiers, which introduce rotational errors. Such a device is shown schematically below, where the coupled motions are evident.

Figure 4. Schematic of parallelogram amplifiers which can introduce rotational errors.

Parallelogram amplifiers and similar devices are not used in Mad City Labs nanopositioners. In Mad City Labs nanopositioners “Pure Motion Amplifiers” are used to achieve the mechanical amplification. Pure Motion Amplifiers are designed by a strict set of rules, which minimize coupled motions between the axis, typically we achieve 10 microradians or less of roll pitch and yaw in our 100 micron scanning stages. This translates to a measuring error of 1 nm in 100 microns, or 0.001% error. Additionally there is no direct coupling between the X and the Y-axis. Therefore the image of a 10 micron x 10 micron square is a 10 micron x 10 micron square using a Mad City Labs nanopositioning system.