# Robust measurement method and uncertainty analysis for position-independent geometric errors of a rotary axis using a double ball-bar

Article

## Abstract

In this study, we propose a robust and simple method using double ball-bar to measure position-independent geometric errors of a rotary axis involving single axis control during the measurement. The standard uncertainty for the proposed method is analyzed to quantify the confidence interval of the measurement result. Two measurement paths are planned to measure the position-independent geometric errors, including two offset errors and two squareness errors of a rotary axis. An error synthesis model using homogenous transform matrices and a ball-bar equation to represent the relation between the positions of two balls and the measured distance between them are used. Set-up errors, which are inevitable during the installation of the balls, are modeled as constants and added to the design position of the balls. Their effects on the measurement result are investigated in detail. Furthermore, a novel fixture consisting of flexure-hinges located at the tool nose is developed to minimize the set-up errors of the ball and to robustly keep the position of the ball during measurements. Finally, the proposed method is validated using simulation and is applied to the rotary axis located on a five-axis machine tool.

## Keywords

Rotary-axis Position-independent geometric errors Measurement method Uncertainty analysis Double ball-bar

## Nomenclature

Coli(τ)

i-th column of 4 × 4 matrix τ

L

offset at measurement

R

nominal length of double ball-bar

ΔRij

deviation to the nominal length at i-th measurement and j-th measured value(i=1, 2; j=1, ..., n)

U

measurement uncertainty

ci

command angle [0, 2π] at for axis C(i=1, ..., n)

k

coverage factor

n

sample number

u

standard uncertainty

(x, y, z)

command for positioning

δij, ɛij

positional error and angular error to the i-direction at axis j(i=x, y, z; j = x, y, z, c)

szx, sxz, syz

squareness errors between three linear axes

oic, sic

offset error and squareness error to the i-direction at axis C(i=x, y)

AMc

4 × 4 rotation matrix of axis C

EMi

4 × 4 matrix consisting of positional error and angular error of axis i(i = x, y, z, c)

SMi

4 × 4 squareness matrix of axis i(i = x, z, c)

TMi

4 × 4 translation matrix of axis i(i = x, y, z)

ΔRi

n i × 1 column vector consist of deviation ΔR ij at i-th measurement (i = 1, 2)

{i}

coordinate system of axis i(i = x, y, z, c)

τij

Transform matrix from j-coordinate system to i-coordinate system

wi(wxi, wyi, wzi)

design position of the ball on work-table at i-th measurement(i = 1, 2)

ti(txi, tyi, tzi)

design position of the ball at tool-nose at i-th measurement(i=1, 2)

Δwiwxi, Δwyi, Δwzi)

set-up error at the ball on work-table at i-th measurement(i=1, 2)

Δtitxi, Δtyi, Δtzi)

set-up error at the ball at tool-nose at i-th measurement(i = 1, 2)

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