= Positioning repeatability
\b[Positioning repeatability] describes the scatter of <positions> reached when a motion system repeatedly returns to the same commanded position under specified conditions. It is an application of <measurement repeatability>, a form of <measurement precision>. A system may return to almost the same wrong position each time, so small scatter does not establish <measurement accuracy>.<International vocabulary of metrology by JCGM>[VIM3 entries2.15,2.20 and2.21: precision and repeatability conditions]{ref}
The approach direction belongs in the test description. <Unidirectional positioning repeatability> keeps it fixed; <bidirectional positioning repeatability> includes returns from both directions. To measure the assembled stage, independently observe its final <position> rather than only the motor's <rotary encoder> count. Physik Instrumente describes a static test that lets the platform stop before reading it, using an <interferometer> for <displacement> and an <autocollimator> for angular error, with two reflecting <mirrors> on the stage.<How to Consistently Verify the Performance of Nanopositioning Motion Control Equipment by Physik Instrumente>[§§3.1–3.2, static readout, direction convention and paired-mirror apparatus]{ref}
For an original numerical example, repeatedly command $x_*=10.000\,\mathrm{mm}$. Suppose an independent position readout with known <measurement calibration> gives the errors $e_j=x_j-x_*$, in <micrometres>,
$$
\begin{aligned}
\text{approach from below}:&\quad -0.1,\ 0.0,\ 0.1,\ 0.0,\\
\text{approach from above}:&\quad 1.9,\ 2.0,\ 2.1,\ 2.0.
\end{aligned}
$$
Each direction has the same <sample standard deviation>,
$$
s=\sqrt{\frac{\sum_{j=1}^{4}(e_j-\bar e)^2}{4-1}}=\sqrt{\frac{0.02}{3}}\,\mathrm{\mu m}\approx0.0816\,\mathrm{\mu m}.
$$
Yet their <arithmetic means> are 0 and $2.0\,\mathrm{\mu m}$. The magnitude of their mean difference, a <positioning reversal error> at this target, is \b[2.0 <micrometres>], far larger than the within-direction scatter. Pooling all eight values gives a <sample standard deviation> of $\sqrt{8.04/7}\approx1.149\,\mathrm{\mu m}$; that statistic mixes the directional shift with the smaller return-to-return variation.
\Image[/positioning-repeatability-direction.png]
{title=Two tightly repeated return positions separated by approach direction}
{description=Original chosen errors at one target. Four trials per direction; small bars show mean plus or minus one sample standard deviation. The figure is not measured product performance.}
The calculation uses a declared <sample standard deviation>, not an unspecified manufacturer's tolerance or a universal repeatability convention. A positioning specification must state its approach directions, target positions, settling/readout procedure, load, environmental conditions, statistic and <measurement uncertainty>. A single target does not establish performance over the travel range. The manufacturer's method explicitly separates directional means from scatter and combines information from multiple targets when forming axis-level specifications.<How to Consistently Verify the Performance of Nanopositioning Motion Control Equipment by Physik Instrumente>[§§4.1–4.4, target-specific means and repeatability aggregation]{ref}
<Backlash> is one possible cause of direction-dependent behavior, while <thermal expansion> and <instrumental drift> can move the apparent target over time. A repeatability experiment must distinguish those effects from random scatter. Measuring small returns to one target also answers a different question from <minimum incremental motion>, which asks whether a small commanded change can actually be executed.
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