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A combined hole position error correction
method for automated drilling of large-span
aerospace assembly structures
Junshan Hu
College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China and
Wuhu Machinery Factory, Wuhu, China
Xinyue Sun and Wei Tian
College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China
Shanyong Xuan
Wuhu Machinery Factory, Wuhu, China
Yang Yan and Wang Changrui
College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China, and
Wenhe Liao
School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, China
Abstract
Purpose –Aerospace assembly demands high drilling position accuracy for fastener holes. Hole position error correction is a key issue to
meet the required hole position accuracy. This paper aims to propose a combined hole position error correction method to achieve high
positioning accuracy.
Design/methodology/approach –The bilinear interpolation surface function based on the shape of the aerospace structure is capable of
dealing with position error of non-gravity deformation. A gravity deformation model is developed based on mechanics theory to
efficiently correct deformation error caused by gravity. Moreover, three solution strategies of the average, least-squares and genetic
optimization algorithms are used to solve the coefficients in the gravity deformation model to further improve position accuracy and
efficiency.
Findings –Experimental validation shows that the combined position error correction method proposed in this paper significantly reduces the
position errors of fastener holes from 1.106 to 0.123 mm. The total position error is reduced by 43.49% compared with the traditional mechanics
theory method.
Research limitations/implications –The position error correlation method could reach an accuracy of millimeter or submillimeter scale, which
may not satisfy higher precision.
Practical implications –The proposed position error correction method has been integrated into the automatic drilling machine to ensure the
drilling position accuracy.
Social implications –The proposed position error method could promote the wide application of automatic drilling and riveting machining system
in aerospace industry.
Originality/value –A combined position error correction method and the complete roadmap for error compensation are proposed.
The position accuracy of fastener holes is reduced stably below 0.2mm, which can fulfill the requirements of aero-structural
assembly.
Keywords Automatic drilling, Bilinear interpolation surface, Deformation analysis, Genetic optimization algorithm, Hole position error,
Hole position correction, Gravity-induced deformation
Paper type Research paper
The current issue and full text archive of this journal is available on Emerald
Insight at: https://www.emerald.com/insight/0144-5154.htm
Assembly Automation
© Emerald Publishing Limited [ISSN 0144-5154]
[DOI 10.1108/AA-05-2021-0053]
The research was supported by National Key Research and
Development Program of China (Grant No. 2019YFB1310101),
National Natural Science Foundation of China (Grant No. 52005259)
and Youth Science and Technology Innovation Fund of Nanjing
University of Aeronautics and Astronautics (Grant No. 1005-
XAC2003).
The authors would also like to thank the editors and the anonymous
referees for their insightful comments.
Received 14 January 2021
Revised 15 August 2021
6 November 2021
8 December 2021
Accepted 27 December 2021
1. Introduction
Transport planes, carrier rockets, missile launchers and other
aerospace products are large-span thin-walled structures,
which are constructed by crisscrossed girders and stringers as
skeletons and covered by metal or carbon fiber-reinforced
polymer (CFRP) panels as skins. Commonly, the skeletons and
skins are mainly fastened by riveting joints; thus, the position
accuracy of the fasteners is one of the main factors that affect
the service life and safety of the products (Mei et al.,2015;Bi
et al., 2013;Mei et al.,2018). Usually, the nominal positions of
the fastener holes are directly obtained through the theoretical
digital models of the products prescribed by designers (Robert,
2001). However, the actual fastener positions of the products
are often deviated from the theoretical digital models due to
assembly tolerances, gravity deformation errors, etc. (Bi et al.,
2013;Bi and Liang, 2011). Once there are deviations between
the nominal and actual positions of the fastener holes, the stress
evenness around riveting fasteners is changed and the fatigue
life of the aerospace products is reduced (Li et al., 2021).
Hence, such position deviations or errors need to be eliminated
to ensure the position accuracy of the fastener holes.
Generally, the skeletons and skins of aerospace products are
first fixed on the assembly fixture to establish their relative
position prior to connecting. Then, small reference holes are
made to preassemble them together, followed by drilling and
riveting operations (Zhang et al.,2018). More importantly, the
reference holes are also used as geometric features to compare
with theoretical digital models to correct for the deviations
between them. That is, the positions of the reference holes in
ready-to-assembly components are detected through the vision
system; then, position deviations of the fastener holes are
calculated by the position deviation of reference holes (Zhu
et al.,2014;Tian et al.,2014). Based on the position deviation
of the reference holes and the position relationship between the
reference holes and the fastener holes, an error function is
established, and the position errors of the fastener holes are
compensated to improve the drilling accuracy (Zhao et al.,
2017). Thus, correction results are significantly related to the
selection of reference holes, and it could work better when
products possess substantial stiffness to resist assembly
deformation.
Practically, the aerospace products are thin-walled chambers
with complex curved surface. They are assembled with
manufacturing tolerance and clamping errors, which easily lead
to deviations of hole positions between theoretical models and
real components. From the perspective of the surface shape of
the aerospace components, Zhu et al. (2013) constructed a
bilinear error surface using the deviation vectors of the
reference holes. The nominal positions of the fastener holes
were compensated with linear interpolation, which greatly
improved the drilling position accuracy of the assembly parts
with sharp curvatures. To improve the drilling accuracy of
aircraft cylindrical parts, Bi et al. (2015) proposed an
interpolation Coons surface error correction method, which
introduced normal vectors of the reference holes. To guarantee
the hole margin, Wang et al. (2015) presented a correction
method for hole positions based on hole margin constraints and
Shepard interpolation to ensure the distance between the hole
center and the boundary of trench in skeletons. As for hole
perpendicularity, Zhang et al. (2012) proposed an algorithm
based on the principal that four non-coplanar points could
define a unique sphere tangent in spatial geometry. The normal
vector of the curve surface of a workpiece at the drilling points
was deduced by four displacement sensors installed on the end-
effector in a flexible drilling system. Yan and Cheng (2015)
presented a hole modification method, which could avoid the
reliance on integral precision of the whole structures. The
locations of holes along the line between two inspected
preassembly holes could be calculated via the proposed spatial
coordinate transformation matrix. Shi et al. (2020) exhibited a
correction strategy based on Kriging interpolation through
which the position deviations of the fastener holes could be
predicted. The predicted hole position errors were highly
dependent on the quantity and layout of reference holes. Based
on the combination datum theory, an on-position
measurement method was developed for position error
compensation by Zeng et al. (2021), and the final coaxiality
error of the gimbal was reduced to 0.022 mm. These hole
position error correction methods which compensate position
deviation in view of geometric features are suitable for
automatic drilling and riveting system.
Since aerospace products are large-span structures with low
stiffness, they are easily deformed or distorted under clamping
force, machining force, gravity effects, etc., during assembly
process. Thus, hole position error can be also compensated
from the perspective of deformation. Cheng et al. (2011)
presented a two-stage and eight-state hierarchical model of
multi-state riveting process for thin-walled structures. Bases on
this model as well as the “N-2-1”positioning principle,
positioning error of each stage was analyzed according to
manufacturing error, position accuracy and the mismatch
error. Zhang et al. (2012) established a product deformation
model for automatic carriers, in which the gravity deformation
formula of the drilling point was constructed according to the
spatial relationship between drilling points and the deformation
curve of the beam. The iterative optimization method was used
to compensate the hole position errors of product deformation.
Lu and Islam (2013) developed a simplified method to
compensate thermally induced volumetric error by modeling
the positioning error as functions of ball-screw nut temperature
and travel distance. The average absolute and relative errors
were reduced by 30.44
m
m and 77%, respectively. To improve
local drilling precision and quality, Jie (2013) developed a
simplified analytical model to clarify the interlayer gap
formation mechanism as well as the effects of related factors for
drilling stacked metal materials. It indicated that the interlayer
gap size had a significant effect on interlayer burr size. Liu et al.
(2020) established a mathematical model of interlayer gap with
bidirectional clamping forces, based on which the optimization
of the bidirectional clamping forces was performed to reduce
the degree and non-uniformity of the deflections in stacked
plates. Pogarskaia et al. (2020) proposed a new geodesic
algorithm for the fastener pattern optimization in the A350
fuselage assembly process. The proposed algorithm was
allowed to perform optimization 50 times faster than the local
variations used before due to its non-iterative procedure.
In view of the advantages of the above two kinds of position
compensation methods, the present research proposes a
combined method aiming at the position error correction of
Position error correction method
Junshan Hu et al.
Assembly Automation
fastener holes for large-span aerospace products. The gravity
deformation error and the non-gravity deformation error are
divided according to the features of position errors. The
bilinear interpolation surface model is constructed to correct
the non-gravity deformation error of the product. Meanwhile,
the gravity deformation model based on the elasticity theory is
established to correct the gravity deformation error. The
average, the least square and the genetic optimization
algorithms are used to optimize the gravity deformation model
to furtherly improve the position accuracy of the drilling. The
correctness of the combined methodology is verified by the
simulation and experiment examples.
2. Establishment of position error correction
model
2.1 Principle of position error correction
The large-span air vehicle launching container together with
the self-developed drilling–riveting combined machining tool is
illustrated in Figure 1 to elaborately present the position error
correction method. The machining tool is composed of the
machine tool guideway, the moveable machine tool platform,
the multifunctional end effector, a vision system constructed by
a Gocator 3210 industrial camera and two rotary fixtures. The
moveable machine tool platform carries the multifunctional
end effector and runs along the machine tool guideway,
forming a five-axis machining tool. Two rotary fixtures carry
the air vehicle launch container for drilling and riveting
operation. The launch container is a rectangular box
constructed by aluminum alloy skeletons and covered with thin
skins. Each surface of the launch container is flat with a
geometric dimension of about 7.0 m 1.0 m. Since there are
four flat surfaces in the launch container to be drilled, the pair
of rotary fixtures are required to turn over the container to
make each drilling surface faced with the multifunctional end
effector. Thus, there are not only assembly errors and gravity
deformation errors in the box but also systematic errors such as
distortion deformation caused by asynchronous rotation of two
rotary fixtures. The error composition is complex and needs to
be skillfully compensated.
The error correction process of hole position for the
launch container is elaborately illustrated in Figure 2.First,
the hole position errors are divided into gravity deformation
error and non-gravity deformation error, and the latter
includes rotary tooling positioning error, assembly error,
etc. Then the bilinear error surface model is built to
compensate for the non-gravity deformation error.
Comparatively, the position error distribution of gravity
deformationismorecomplicated.Itisnecessarytoanalyze
it in detail based on the mechanics’theory and establish the
simplified gravity deformation formula. Third, three
algorithms are used to optimize the model to furtherly
improve the position error correction accuracy. Finally, the
hole position error is corrected by the proposed combined
model.
2.2 Construction of non-gravity deformation error
Typical aerospace structures, such as fuselage panels and
launch containers, appear to be smooth globally in the
drilling region and flat locally in the vivacity of the drilling
spot. The reference holes are usually defined at the
four corners of the drilling region, as shown in Figure 3.
Thus, the drilling region can be expressed in terms of a
bi-parametric surface:
Su;v
ðÞ
¼xu;v
ðÞ
;yu;v
ðÞ
;zu;v
ðÞðÞ
(1)
where parameters uand vare the spatial position of the drilling
points relative to the corner control reference holes. If P
00
,P
01
,
P
11
and P
10
are defined as four boundary control points of the
bi-parametric surface, the points in the processing surface can
be rewritten as:
Su;v
ðÞ
¼1u;u
½
P00 P01
P10 P11
1v
v
(2)
According to equation (2), the nominal surface can be
represented as S
0
(u,v). Similarly, the actual surface can be
represented as S
1
(u,v), and a mapping relationship is
established between nominal and actual surfaces:
Figure 1 Air vehicle launch container and the corresponding drilling and riveting combined machine tool
Position error correction method
Junshan Hu et al.
Assembly Automation
DSu;v
ðÞ
¼S1u;v
ðÞ
S0u;v
ðÞ
¼1u;u
½
DP1DP2
DP4DP3
1v
v
(3)
where DP
1
,DP
2
,DP
3
and DP
4
are error vectors between
nominal and actual corner reference holes. The required error
correction vectors DSbetween nominal and actual corner
reference holes can be obtained once parameters (u
,v
) are
substituted into the equation (3). Based on the above
deduction, the bilinear error surface correction model
established for non-gravity deformation errors is represented as
follows:
DX1u;v
ðÞ
¼DSxu;v
ðÞ
DY1u;v
ðÞ
¼DSyu;v
ðÞ
DZ1u;v
ðÞ
¼DSzu;v
ðÞ
8
>
<
>
:
(4)
2.3 Construction of gravity deformation error
2.3.1 Spatial posture analysis of components
As mentioned earlier, the launch container is a large-span
structure. The gravity could cause beam bending, which is
similar to a beam with simply supported ends. The surface
switch on the rotary fixtures would distort the container in axial
direction, which complicates the gravity deformation. As given
in Figure 4, the local coordinate system OXYZ and the
workpiece coordinate system O
1
X
1
Y
1
Z
1
are defined,
respectively, to compensate for the gravity deformation error of
the launch container. In the local coordinate system, the X-axis
is parallel to machine tool guideway, and the Y-axis is parallel
to direction of gravity. The Z-axis can be determined by the
right-hand rule. In the workpiece coordinate system, the
reference points in the drilling surface are selected and
measured using the vision system firstly. Then, a workpiece
plane is constructed according to these measured points, and
the normal direction of the plane is defined as the Y
1
-direction
[Figure 4(b)]. The X
1
-axis coincides with the central axis of the
workpiece plane. The Z
1
-axis is determined by the right-hand
rule.
The gravity components of the launch container in the
workpiece coordinate system are illustrated in Figure 4(c),
which can be presented as:
Gx¼Gsin
b
Gy¼Gsin
a
cos
b
Gz¼Gcos
a
cos
b
8
>
<
>
:
(5)
where
a
and
b
are the deflection angles from Y-andX-axes,
respectively.
To figure out parameters
a
and
b
, the reference points
shown in Figure 4(b) are used to fit the workpiece plane using
the least-squares function (Arun et al.,1987). Generally, the
workpiece plane is defined by the following equation:
z¼a0x1a1y1a2(6)
where a
0
,a
1
and a
2
are parameter that define the plane. The
distance sum Sbetween the reference points and the workpiece
plane is can be calculated by:
S¼X
n1
i¼0
a0x1a1y1a2z
ðÞ
2(7)
High accuracy of workpiece plane could be obtained if
equation (7) meets the following conditions:
@S
@ak
¼0;k¼0;1;2 (8)
which could be turned into equation group as follows:
X2a0xi1a1yi1a2zi
ðÞ
xi¼0
X2a0xi1a1yi1a2zi
ðÞ
yi¼0
X2a0xi1a1yi1a2zi
ðÞ
zi¼0
8
>
>
>
<
>
>
>
:
(9)
Once the workpiece plane is obtained through above equations,
the workpiece coordinate system and angles of
a
and
b
can be
calculated.
2.3.2 Construction of gravity deformation formula
According to equation (5), the gravity components of the
launch container in the workpiece coordinate system is defined
as a gravity vector (G
x
,G
y
,G
z
). Since the position error in
X
1
-direction is not sensitive to the gravity effect, the
Figure 3 Schematic diagram of hole position deviation
Figure 2 Hole position error correction flowchart
Position error correction method
Junshan Hu et al.
Assembly Automation
deformation of the drilling points along the X
1
-direction can be
ignored. Besides, the deformation of the drilling points along
the Y
1
-direction in the launch container can be compensated
by the axial feed of the end effector, so the deformation in Y
1
-
direction can also be ignored. Thus, the main factor of the
position deviation is induced by gravity in Z
1
-direction.
Considering that the launch container is a hollow rectangular
large-span box with one end hinged and the other end rolled, it
can be considered as a collection of many simply supported
beams as illustrated in Figure 5. The self-gravity of the
container is uniformly distributed in these beams, along which
the distributed force of gravity is q. According to mechanics
theory, the deformation curve of the simplified beam collection
of the container in the Z
1
-direction can be calculated by:
Z1h
ðÞ
¼ qh
24EI l32lh21h3
ðÞ
0hl
ðÞ (10)
where lis the length of the launch container beam; Eis the
elastic modulus of the container material; Iis the moment of
inertia of the cross-section; and his the distance between the
drilling point and the left hinged end.
Since the launch container is composed of skins and
supporting structures such as stringers and skeletons, it is hard
to determine the parameters in the equation (10) directly.
Thus, the equation (10) is directly reflected by the following
equation:
Z1h
ðÞ
¼ah l32lh21h3
ðÞ
0hl
ðÞ (11)
where ais the deformation coefficient. In this way, the position
error model of gravity deformation can be denoted by:
DXh
ðÞ
¼Z1h
ðÞ
sin
b
DYh
ðÞ
¼Z1h
ðÞ
sin
a
cos
b
DZh
ðÞ
¼Z1h
ðÞ
cos
a
cos
b
8
>
<
>
:
(12)
2.3.3 Solution for deformation coefficient
The accurate establishment of gravity deformation equation (11)
is significant to the compensation accuracy of the required error
correction vectors in equation (3). The drilling workpiece surface
of the launch container is illustrated in Figure 6,wherethe
fastener holes and the reference holes are arranged trimly. The
position errors DPof reference holes consist of the gravity
deformation error DGand the non-gravity deformation errors
DS.
To solve the gravity deformation coefficient a, the position
errors need to be separated since the correction principles of the
two kinds of position errors are different. It can be seen from
Figure 6 that the reference holes (black points as given in
Figure 4 Schematic diagram of the local and workpiece coordinate systems
Figure 5 Simplification of the force analysis model of the launch
container
Position error correction method
Junshan Hu et al.
Assembly Automation
Figure 6) at four corners of the workpiece surface are fixed in
the conformal frames. The gravity deformation error in these
local regions can be ignored. Therefore, a non-gravity bilinear
interpolation error surface model DS(u,v) can be constructed
to calculate the non-gravity position errors of the reference
holes (red points as given in Figure 6). The theoretical errors
DS(a) are calculated from the equation (11). The deformation
coefficient can be determined by defining DG=DG(a), the
solution process as shown in Figure 7.
However, there will be multiple deformation coefficient
solutions since a large number of reference holes are arranged
on the workpiece surface of the launch container. Thus, the
average, least square and genetic optimization algorithms are
used comparably to figure out the optimal deformation
coefficient.
2.3.3.1 Average algorithms. In average algorithms, the actual
gravity deformation errors DG
i
(i= 1,2,...,n) and the
theoretical gravity deformation errors DG
i
(a)(i= 1,2,...,n)of
the reference holes are equal:
DG1¼DG1a
ðÞ
DG2¼DG2a
ðÞ
...
DGn¼DGna
ðÞ
8
>
>
>
>
<
>
>
>
>
:
(13)
By applying the average method, the optimal deformation
coefficient ais denoted as follows:
a¼a¼1
nX
n
i¼1
ai(14)
2.3.3.2 Least squares algorithms. It is assumed that DG
i
(i=1,
2, ...,n)andDG
i
(a)(i=1,2,...,n) are equal; the
equation (13) can be rewritten as:
kDGk¼kDGa
ðÞ
k¼jah l32lh21h3
ðÞ
j0hl
ðÞ
(15)
Let y=kDGkand x=jh(l
3
2lh
2
1h
3
)j; the equation (15) can
be furtherly rewritten as:
y¼ax (16)
The coefficient acan be obtained by linear fitting by the least-
square method (Peng et al., 2016). It is supposed that:
S¼X
n
i¼1
yiy
ðÞ
2¼X
n
i¼1
yiaxi
ðÞ
2(17)
The derivative is taken on both sides of equation (17).By
setting the derivative to be zero, the deformation coefficient a
can be obtained:
a¼nXyixiXyiXxi
nXxi2XxiÞ2
(18)
2.3.3.3 Genetic optimization algorithms. The genetic
optimization algorithm for solving the deformation coefficient
is illustrated in Figure 8 (Cheng and Yang, 2012). Obviously,
the deformation coefficient ais the design variables of the
genetic optimization model. The optimal solution of ais
randomly distributed around theoretical item q/24EI, so the
range of ais set from 0 to q/12EI. According to the deformation
coefficient solution strategy, the smaller the difference between
DG
i
(i=1,2,...,n) and DG
i
(a)(i=1,2,...,n), the more
accurate the deformation coefficient a. Therefore, the objective
function is defined as:
Figure 6 Drilling workpiece surface of the launch container
Figure 7 Flowchart of deformation coefficient solution
Position error correction method
Junshan Hu et al.
Assembly Automation
fa
ðÞ
¼minX
n
i¼1
kDGia
ðÞ
DGk
s:t:a¼aja20;q=12EI
½ðÞ
8
>
>
<
>
>
:
(19)
In view of the fact that the calculation result may be stable in a
locally optimal solution in a short time, a certain amount of
calculation algebra N
i
is chosen as the termination condition.
Finally, the optimal deformation coefficient a
0
is determined.
3. Validation for position error correction method
3.1 Position accuracy evaluation by simulation
approach
During drilling process, the positions of the fastener holes
change with the deformation of the drilling workpiece surface in
the launch container, and their absolute positions could not be
measured directly in assembly site. Since the deformation of the
workpiece could be tracked effectively in simulation
environment, the deformation of the launch container in
drilling process is simulated in ABAQUS software platform to
acquire the position error of fastener holes.
3.1.1 Simulation setup
The three-dimensional (3 D) finite element model of the
launch container with a total geometric dimension of
7.0 m 1.0 m 1.0 m is presented in Figure 9. The container
together with two conformal frames is simplified as a thin-
walled large-span rectangular box and modeled by 3 D shell
elements S4R. The edge length and thickness of the shell
element is set to be about 0.1 m and 0. 05 m, respectively, by
homogenization of skeletons and skins. The left and right ends
of the launch container are hinged as illustrated in Figure 5.
The gravity is loaded on all elements, and a slight rotational
disturbance is exerted to the right end surface to simulate the
discordant rotation of two rotary fixtures in Figure 1.The
material properties of aluminum alloy are listed in Table 1.
After the simulation, the deformation of the simplified
launch container is presented in Figure 9(b),fromwhichthe
deformed position data of the drilling points are abstracted
with a row space of 0.4 m and a column space of 0.2 m to
validate the position error correction model as constructed
in Figure 2.
3.1.2 Position accuracy of error correction
The accuracy evaluation of the non-gravity deformation error
correction is presented in Figure 10. When no correction is
applied to the position of drilling points, both the gravity
deformation and container distortion cause the deviation of
drilling position. The curves of uncorrected errors exhibit the
same increasing trend as the drilling points move father away
from the starting point at the left end of the launch container.
The maximum position error even reaches 40.0 mm. This is
because the left end is hinged and there is no X-axis
rotational degree of freedom, and the distortion of the
container accumulates along the number increasing
direction and becomes much bigger at the right end. Also,
the position error of rows r1 and r4 is higher than rows r2
and r3. The reason is that rows r1 and r4 are on the edge of
the processing surface where the drilling hole position is
more sensitive to the torsion of the workpiece. As the
position errors corrected with the non-gravity deformation
model, actually the distortion deviations of the drilling
points are compensated. As a result, the error curves
approximately conform to the law of gravity deformation of
the simplified launch container given in Figure 5,andthe
maximum position error is reduced to 5.0 mm, revealing
that the non-gravity deformation errors are effectively
corrected.
Figure 8 Flowchart of genetic optimization algorithm for solving deformation coefficient
Figure 9 Boundary conditions and deformation of the simplified launch container
Position error correction method
Junshan Hu et al.
Assembly Automation
The accuracy evaluation of gravity deformation error
correction is presented in Figure 11. Obviously, the position
errors are greatly reduced compared with the uncorrected
position errors, revealing that the proposed position error
correction method is effective in compensating large-span
aerospace assembly structures. The position accuracy of three
used algorithms is listed in Table 2; the average position errors
of corrected reference points are reduced to about 0.568 mm,
0.515 mm and 0.475 mm for the cases disposed with the
average, the least-squares and the genetic optimization
algorithms, respectively. This regulation indicates that the
gravity deformation error correction method based on the
genetic optimization algorithm achieves the best results in
alleviating position error to a certain extent, which is attributed
to its wide search range and the multiple search points. This
ability benefits the genetic optimization algorithm to discover
global optimum and avoids trapping in a locally optimal
solution. Therefore, the position errorcorrection method based
on the genetic optimization algorithm is more suitable for
solving the deformation coefficient.
3.2 Position accuracy evaluation by drilling experiment
3.2.1 Experimental setup
The experimental validation is carried out on the self-
developed drilling–riveting combined machining tool as
illustrated in Figure 1, and the real machining tool equipment is
presented in Figure 12. During the drilling process, the actual
positions of the reference holes in the launch container are
firstly got by the vision measurement system. Then, the
positions of the drilling holes are corrected by the proposed
method. The measurement error of the vision system used in
the present research is within 0.035 mm, which is an order of
magnitude lower than the required hole position accuracy.
In the vision measurement process, the template matching
method is used to process the two-dimensional image
information of the reference holes. As shown in Figure 13,
T(m,n) is the template of the hole to be measured, and
S(row,col) is the point cloud of the workpiece surface. The
template T(m,n) overlays on the searched point cloud
S(row,col) and translates throughout the cloud. The area
where the template covers the searched point cloud is called
the subgraph S
ij
(m,n), where (i,j) is the position number of
the subgraph center in the searched point cloud. The
position with the highest similarity between S
ij
(m,n)andthe
template T(m,n) is considered as the position of the hole.
The error method is used to evaluate the similarity between
T(m,n)andS
ij
(m,n)(Zou et al., 2019):
Table 1 Material properties of aluminum alloy used for the launch container
Material parameters E
ms
b
r
Value 68.9 Gpa 0.33 228 Mpa 2730kg/m
3
Figure 10 Accuracy of non-gravity deformation error correction in workpiece surface of launch container
Position error correction method
Junshan Hu et al.
Assembly Automation
Ei;j
ðÞ
¼X
rowm
2
i¼m
2X
coln
2
j¼n
2
jSij m;n
ðÞ
Tm;n
ðÞ
j
m
2<i<row m
2;n
2<j<col n
2
(20)
where the minimum value of E(i,j) is the position of the
matched target hole. It is obvious that the smaller the template,
the faster the matching speed. The matching rate Kof template
matching method is calculated by the following formula
(Jeyasenthil and Choi, 2019):
K¼1Ei;jðÞ
mn
100% (21)
where the higher the matching rate K, the higher the credibility
of the matching process. The matching result of the hole to be
measured is presented in Figure 13(b), where the matching rate
Kis 98.1057%.
3.2.2 Verification of drilling position accuracy
To validate the correction algorithm, a real launch container
workpiece including aluminum alloy and CFRP panels is used
for drilling test, and the drilling region is the part of the launch
container surface. The drilling surface is divided into four
areas, namely A, B, C and D zones, as shown in Figure 14.
The holes in zone A is drilled with the proposed method to
correct the two types of errors together. The holes in zone B
is drilled without the position correction algorithm. The
holes in the zone C drilled with non-gravity deformation
error correction method, and the holes in the zone D is
drilled with gravity deformation error correction method.
The reference hole is defined every 0.5 m interval, and the
nominal drilling hole space is 0.05 m. Since the absolute
position of the actual drilling holes in the workpieces cannot
be measured directly, the position accuracy of the drilling
can be evaluated by relative position of holes, namely,
column spacing and hole row spacing. The position error is
defined by:
Figure 11 Comparison of accuracy of the three used algorithms
Table 2 Statistical results of three used algorithms for gravity deformation
model
Average position error/mm r1 r2 r3 r4
Before correction 20.738 13.800 13.271 19.528
Average algorithm 0.577 0.517 0.596 0.581
Least square algorithm 0.538 0.443 0.543 0.535
Genetic optimization algorithm 0.498 0.425 0.499 0.479
Position error correction method
Junshan Hu et al.
Assembly Automation
DPp¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s0s
ðÞ
21r0r
ðÞ
2
q(22)
where sand rare theoretical value of hole column spacing and
hole row spacing, respectively. s’and r’are their actual
measured value.
The position accuracy of the drilled holes in aluminum alloy
and CFRP workpieces is presented in Figures 15 and 16,
respectively. It is easily observed that position errors of
fastener holes for two kinds of material workpieces could reach
61.3 and 62.4 mm, respectively, when no correction method
is applied to the drilling process. In comparison, the final hole
position errors are reduced to 60.7 mm when only gravity
deformation is corrected, and they are within 60.36 mm when
only non-gravity deformation is corrected. It seems that the
influence of gravity deformation on position error is not as
significant as that of non-gravity. This is because the gravity
deformation is limited locally within 0.5 m between two
reference holes. By applying the position error correction
method, hole position errors are reduced to 60.2 and 60.3mm
for two kinds of materials, which is acceptable for the large-
span aerospace structures. The rest position errors that could
not be compensated by the proposed method are calibration
errors, measurement errors of reference holes, positioning
errors of the machine tool and environmental factors.
The above experimental results show that the position
precision obtained through one single error correction method
is much lower than the proposed combined method, which can
be explained by correction principles of the two methods
illustrated in Figure 17. The bilinear interpolation method
focuses on the correction of approximately linear errors, while
the gravity deformation model mainly corrects the non-linear
errors caused by gravity. Consequently, the error correction
result is poor when the error and its compensation method are
mismatched. It is also worth noting that four non-gravity
deformation correction curves exhibit the slight “U”shape.
The phenomenon can be attributed to the fact that the curve error
Figure 12 Experimental platform of the drilling–riveting combined machining tool
Figure 13 Image matching process of hole position
Figure 14 Drilling surface and hole number of the launch container
Position error correction method
Junshan Hu et al.
Assembly Automation
is corrected using a straight line as illustrated in Figure 17(a).
Meanwhile, the correction results using the gravity deformation
model are irregular. As presented in Figure 17(b), the error
correction is insufficient when the deformation coefficient is small,
under which circumstance the over-correction would occur. By
comparing the four curves, it is possible to conclude that dividing
the errors into two categories and correcting them with different
principle algorithms is completely effective.
In addition, both the traditional mechanics theory method
and the proposed method are used to compensate the same set
of hole position data, the correcting precision is shown in
Figure 18. The position error of drilling holes with the
traditional mechanics theory method could reach 0.606 mm,
while the position error compensated by the proposed
combined error correcting method in in the present research
achieves an accuracy of 0.125 mm. In the assembly of the
aerospace structures, the required position accuracy of the
fastener holes is 0.5mm. Thus, the accuracy of the proposed
error correction method can fulfill the requirement in aircraft
manufacturing.
4. Conclusions
In the present research, a combined method of position error
correction for automatic drilling in aerospace manufacturing is
proposed and validated on the large-span air vehicle launching
container. The category and incentive of position errors in
large-span aerospace structures during assembly is elaborately
analyzed, and the errors are divided into non-gravity and
gravity deformation errors. The non-gravity error is corrected
by the proposed bilinear interpolation surface model based on
the vision measurement data of reference holes set at intervals
Figure 15 Drilling position errors of the aluminum alloy workpiece in the launch container
Figure 16 Drilling position errors of the CFRP workpiece in the launch container
Position error correction method
Junshan Hu et al.
Assembly Automation
in the workpiece surface. The gravity deformation formula is
established using the theory of elasticity, based on which the
gravity deformation error model is further constructed and
optimized by genetic optimization strategy to solve the problem
of low accuracy of gravity deformation formula coefficients.
By implementing the proposed error correction model in the
drilling process of launch container, the position errors of
fastener holes can be limited within 0.2 mm. Compared with
the traditional mechanics theory method, the position error is
reduced by 43.49%. Thus, the accuracy and effectiveness of
the proposed error correction method is experimentally
verified.
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Corresponding author
Wei Tian can be contacted at: tw_nj@nuaa.edu.cn
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Position error correction method
Junshan Hu et al.
Assembly Automation
