= Positioning reversal error
{title2=$B=\bar e_{\uparrow}-\bar e_{\downarrow}$}
A <positioning reversal error> compares the mean reached positions at one command when approached from opposite directions. Define $e_{i,j,d}=x_{i,j,d}-x_i^*$ for target $i$, trial $j$ and approach direction $d$. With $\uparrow$ denoting approach in the increasing-coordinate direction, use the signed target-specific quantity
$$
B_i=\bar e_{i,\uparrow}-\bar e_{i,\downarrow}.
$$
State this sign convention. PI's axis-level quantity called maximum positioning backlash instead takes the maximum absolute directional mean difference over the tested targets.<How to Consistently Verify the Performance of Nanopositioning Motion Control Equipment by Physik Instrumente>[Section4.3, maximum absolute difference of directional means]{ref}
For an original two-target example, measured mean errors at $x^*=2\,\mathrm{mm}$ are $0.3\,\mathrm{\mu m}$ from below and $-0.7\,\mathrm{\mu m}$ from above, giving $B_1=1.0\,\mathrm{\mu m}$. At $x^*=8\,\mathrm{mm}$, means of $-0.4$ and $0.9\,\mathrm{\mu m}$ give $B_2=-1.3\,\mathrm{\mu m}$. Thus
$$
\max_i|B_i|=1.3\,\mathrm{\mu m},\qquad\frac{B_1+B_2}{2}=-0.15\,\mathrm{\mu m}.
$$
A 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 gives
$$
B_{\rm observed}=(a_\uparrow-a_\downarrow)+v(\bar t_\uparrow-\bar t_\downarrow).
$$
At $v=0.01\,\mathrm{\mu m\,s^{-1}}$ and mean measurement times 50 s and 150 s, drift alone contributes $-1\,\mathrm{\mu m}$. Alternate or balance the approach timing when testing that confounding effect. <Unidirectional positioning repeatability> measures within-direction scatter rather than this difference of means.
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