= Bidirectional positioning repeatability
<Bidirectional positioning repeatability> concerns repeated arrivals at a commanded position using both approach directions. Retain the direction labels: the combined behavior can include both return-to-return scatter and a <positioning reversal error>. The manufacturer's method forms direction-specific means and spreads before reporting an axis-level summary over several targets.<How to Consistently Verify the Performance of Nanopositioning Motion Control Equipment by Physik Instrumente>[Sections4.1–4.4, directional means and repeatability aggregation]{ref}
For an original example, independently read the following errors at one target, in micrometres:
$$
\begin{aligned}
\text{from below}:&\quad0.0,\ 0.2,\ 0.4,\\
\text{from above}:&\quad1.0,\ 1.2,\ 1.4.
\end{aligned}
$$
The means are 0.2 and 1.2, and each direction has <sample standard deviation> $s=0.2\,\mathrm{\mu m}$. If a declared analysis pools all six observations, its sample variance decomposes exactly as
$$
s_{\rm all}^2=\frac{(n-1)(s_\uparrow^2+s_\downarrow^2)+\tfrac n2(\bar e_\uparrow-\bar e_\downarrow)^2}{2n-1},\qquad n=3.
$$
Here the within-direction sum of squares is $0.16\,\mathrm{\mu m^2}$ and the between-direction term is $1.50\,\mathrm{\mu m^2}$. Consequently,
$$
s_{\rm all}=\sqrt{\frac{1.66}{5}}\,\mathrm{\mu m}\simeq0.576\,\mathrm{\mu m}.
$$
The larger pooled spread is mostly explained by approach direction. This is an original variance identity with a stated sample-standard-deviation convention, not a replacement definition for every manufacturer's repeatability tolerance.
A physical study repeatedly traverses the target from both sides and records final settled displacement with independent metrology. Report target positions, approach distances, order, load and environmental conditions. A directional mean difference may be almost invisible if only within-direction scatter is reported; pooling without direction labels hides its cause. Neither statistic alone establishes <measurement accuracy>, and one target does not characterize the full travel range.
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