Positioning reversal error ($B={\bar{e}}_{\uparrow }-{\bar{e}}_{\downarrow }$)

A positioning reversal error compares the mean reached positions at one command when approached from opposite directions. Define $e_{i,j,d}=x_{i,j,d}-x_i^{\ast }$ 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.