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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.

Unidirectional positioning repeatability describes the scatter of repeated returns to one commanded position when the final approach direction is held fixed. Specify the complete reset-and-return procedure so that each trial approaches under comparable conditions. PI distinguishes this directional scatter from its summaries that include reversal effects.^[ref]

For an original example, always approach from below. At a 2 mm target, suppose the position errors are $-0.4,-0.2,0.0\mu \mathrm{m}$; at an 8 mm target they are $1.6,1.8,2.0\mu \mathrm{m}$. The means are $-0.2$ and $1.8\mu \mathrm{m}$, respectively, while both sample standard deviations areThe same repeatability statistic accompanies quite different mean errors along the travel. A common shift of every reading at a target changes its mean but not its within-target spread.

For illustration, an explicitly declared specification taking the largest tested directional sample deviation would report $0.2\mu \mathrm{m}$ for these two data sets. A specification using a full width or a multiple of that deviation would have a different numerical value. PI's own published convention states which directional spreads and target aggregation it uses; compare like definitions rather than only their units.^[ref]

Observe final platform motion with an independent position readout, not just the motor's rotary encoder. Allow the chosen settling time and keep load and environment controlled. Consistently approaching from one side does not establish bidirectional positioning repeatability, and a small return scatter does not show that a small incremental command can be executed: that is the separate question of minimum incremental motion.