Hierarchical error model toestimate motion error .pdf
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ORIGINAL ARTICLE
Hierarchical error model to estimate motion error of linear
motion bearing table
Gaiyun He
1
& Guangming Sun
1
& Heshuai Zhang
1
& Can Huang
1
& Dawei Zhang
1
Received: 18 February 2017 /Accepted: 5 June 2017 / Published online: 23 June 2017
#
Springer-Verlag London Ltd. 2017
Abstract This study presents a general and systematic ap-
proach for motion error estimation of a linear motion bearing
table based on hierarchical idea. The approach is implemented
in the following four steps: (1) dividing the errors of a linear
motion system into four tiers, namely Datum Tier, Guideway
Tier, Slider Tier, and Table Tier; (2) measuring form errors of
Guideway Tier using the proposed method that combines the
displacement sensors and laser interferometer; (3) developing
a map of the form errors of Guideway Tier and motion errors
of Slider Tier using the Hertz contact theory and a transfer
function method; and (4) formulating a map between the mo-
tion errors of Slider Tier and the error of Table Tier using a
direction cosine matrix. The advantage of this approach is that
it does not require the assumption that the form error phases of
the two guide rails are the same, and thereby provides a more
accurate model of motion error. A typical linear motion bear-
ing table is considered as an example to illustrate the general-
ity and effectiveness of the proposed approach.
Keywords Hierarchical error model
.
Linear motion bearing
table
.
Rail form error
.
Motion error
1 Introduction
W ith the progress of industrial development, numerical control
machine tools, as material carriers of advanced manufacturing
technology, are widely used in machining manufacturing [1–7].
Linear motion bearing tables are importa nt part of a machine tool
and play a crucial role there in owing to their high positioning
accuracy and carrying capacity [8–11]. The motion error of a
linear motion bearing table is mainly affected by guideway form
error , and this has a direct influence on the accuracy of the ma-
chined parts [12]. Therefore, an accurate model is essential to
estimate the error of a linear motion bearing table in the design
and manufacturing process. This, in turn , will allow the adoption
of suitable measures to improve the accuracy via component
tolerance design, manufacturing, and assembly techniques.
Over the past few decades, a considerable amount of inten-
sive research has focused on estimating the error of a linear
motion bearing table. Díaz-Tena et al. developed a methodol-
ogy for the assessment of the geometrical accuracy of a multi-
axis machine based on the D–Hmethod[13]. Majda proposed
a model for rolling guideways with geometric errors and con-
sidered aspects of the practical use of the characteristics of
joint kinematic errors in models for volumetric error in a
medium-sized machine tool [14]. Zha et al. studied motion
straightness of hydrostatic guideways by considering the ratio
of pad center spacing to guide rail profile error wavelengths
[15]. Shamoto et al. proposed a transfer function method to
develop a map of the guideway form errors and table motion
errors and verified its effectiveness through an experiment
[16]. However, the study only focused on the motion error
of a table in a two-dimensional plane [17, 18]. Based on this,
Khim et al. presented a model that extended the motion errors
of the aforementioned study to a three-dimensional space by
using a transfer function method. However, this study did not
consider guide parallelism [19]. Kim et al. proposed an im-
proved transfer function method that accounted for the guide-
way parallelism error in the prediction motion errors of the
table [20]. However, this model assumed that the phase and
amplitude of form errors of different guide rails were the same,
and this is evidently different from the actual conditions. Thus,
* Gaiyun He
hegaiyun@tju.edu.cn
1
Key Laboratory of Mechanism Theory and Equipment Design of
Ministry of Education, Tianjin University, Tianjin 300072, China
Int J Adv Manuf Technol (2017) 93:1915–1927
DOI 10.1007/s00170-017-0635-0
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