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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.^[ref]

For an original example, independently read the following errors at one target, in micrometres:The means are 0.2 and 1.2, and each direction has sample standard deviation $s=0.2\mu \mathrm{m}$. If a declared analysis pools all six observations, its sample variance decomposes exactly asHere the within-direction sum of squares is $0.16\mu {\mathrm{m}}^2$ and the between-direction term is $1.50\mu {\mathrm{m}}^2$. Consequently,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.

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.^[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.^[ref]

For an original numerical example, repeatedly command $x_{\ast }=10.000\mathrm{mm}$. Suppose an independent position readout with known measurement calibration gives the errors $e_j=x_j-x_{\ast }$, in micrometres,Each direction has the same sample standard deviation,Yet their arithmetic means are 0 and $2.0\mu \mathrm{m}$. The magnitude of their mean difference, a positioning reversal error at this target, is 2.0 micrometres, far larger than the within-direction scatter. Pooling all eight values gives a sample standard deviation of $\sqrt{8.04/7}\approx 1.149\mu \mathrm{m}$; that statistic mixes the directional shift with the smaller return-to-return variation.

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.^[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.

A positioning reversal error compares the mean reached positions at one command when approached from opposite directions. Define $e_{i,j,d}=x_{i,j,d}-x_i^{\ast }$ for target $i$, trial $j$ and approach direction $d$. With $\uparrow$ denoting approach in the increasing-coordinate direction, use the signed target-specific quantityState this sign convention. PI's axis-level quantity called maximum positioning backlash instead takes the maximum absolute directional mean difference over the tested targets.^[ref]

For an original two-target example, measured mean errors at $x^{\ast }=2\mathrm{mm}$ are $0.3\mu \mathrm{m}$ from below and $-0.7\mu \mathrm{m}$ from above, giving $B_1=1.0\mu \mathrm{m}$. At $x^{\ast }=8\mathrm{mm}$, means of $-0.4$ and $0.9\mu \mathrm{m}$ give $B_2=-1.3\mu \mathrm{m}$. ThusA small signed average can conceal substantial reversal effects that change sign along the travel.

Read final load position independently, for example with an interferometer, after an agreed settling time. The observed difference does not identify a unique physical cause: backlash, elastic relaxation or instrumental drift can contribute. For an original drift model $e_d(t)=a_d+vt$, an unbalanced timing schedule givesAt $v=0.01\mu \mathrm{m}{\mathrm{s}}^{-1}$ and mean measurement times 50 s and 150 s, drift alone contributes $-1\mu \mathrm{m}$. Alternate or balance the approach timing when testing that confounding effect. Unidirectional positioning repeatability measures within-direction scatter rather than this difference of means.