an efficient method for solving the signorini problem in the simulation of free-form surfaces...
TRANSCRIPT
An efficient method for solving the Signorini problem in the simulation
of free-form surfaces produced by belt grinding
Xiang Zhang*, Bernd Kuhlenkotter, Klaus Kneupner
Department of Assembly and Handling Systems, Leonhard-Euler-Strabe 2, Dortmund University, 44227 Dortmund, Germany
Received 9 August 2004; accepted 8 October 2004
Available online 23 November 2004
Abstract
Industrial robots are recently introduced to the belt grinding of free-form surfaces to obtain high productive efficiency and constant surface
quality. The simulation of belt grinding process can facilitate planning grinding paths and writing robotic programs before manufacturing. In
simulation, it is crucial to get the force distribution in the contact area between the workpiece and the elastic contact wheel because the
uneven distributed local forces are the main reason to the unequal local removals on the grated surface. The traditional way is to simplify this
contact problem as a Signorini contact problem and use the finite element method (FEM) to calculate the force distribution. However, the
FEM model is too computationally expensive to meet the real-time requirement. A new model based on support vector regression (SVR)
technique is developed in this paper to calculate the force distribution instead of the FEM model. The new model approximates the FEM
model with an error smaller than 5%, but executes much faster (1 s vs 15 min by FEM). With this new model, the real-time simulation and
even the on-line robot control of grinding processes can be further conducted.
q 2004 Elsevier Ltd. All rights reserved.
Keywords: Belt grinding; Finite element method; Simulation; Support vector regression
1. Introduction
Belt grinding is a machining process with a geome-
trically indeterminate cutting edge. The grinding belt, which
is the cutting tool, consists of coated abrasives and is
attached around at least two rotating wheels. The workpiece
to be ground is pushed onto one of these wheels which is
called contact wheel. Fig. 1 shows the operation of the
belt grinding process. The materials are cut off under
non-permanent touches between the workpiece and the
abrasives. The belt grinding is a variant of the custom
grinding processes in that the outer layer of the contact
wheel is made of soft materials. The soft contact wheel
makes this process very suitable to produce free-form
surfaces due to its adaptation to the workpiece surface. The
demand for sculptured surfaces originates from both
functional and aesthetic requirements. For example, belt
0890-6955/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijmachtools.2004.10.006
* Corresponding author. Tel.: C49 231 755 5624; fax: C49 231 755
5616.
E-mail address: [email protected] (X. Zhang).
grinding with elastic contact wheel is applied practically to
the manufacturing of turbine blades, the complexity and
quality of whose surface are increasing, and also to the
processing of water taps to meet aesthetic demands.
Industrial robots are introduced as an automatic manipulator
in the grinding process to promote efficiency and quality of
products. However, the robot programming is still a
laborious task because of the complexity of the workpiece
surfaces. Although some off-line programming systems are
available, it is still a job highly dependent on the experience
of the programmer. In order to further automatize the
process, simulation of real-time grinding status becomes
necessary. Through the simulation results, grinding path
planning as well as robot reaction adjusting can be executed
automatically.
This paper presents a simulation framework of the free-
form surface belt grinding process and a new solution to the
Signorini contact problem to accelerate the calculation and to
meet the requirement of the real-time simulation. This paper
is organized as follows. Section 2 will introduce the global
grinding model for simple-shaped surfaces and a new-
developed local grinding model for free-form surfaces.
International Journal of Machine Tools & Manufacture 45 (2005) 641–648
www.elsevier.com/locate/ijmactool
Fig. 1. Belt grinding process with the elastic contact wheel.
X. Zhang et al. / International Journal of Machine Tools & Manufacture 45 (2005) 641–648642
The Signorini contact problem is explained in Section 3.
Section 4 describes the new idea, which is based on the SVR
technology, to solve the Signorini problem and the basic
principle of the SVR. The detailed modelling procedures and
numerical experiments are shown in Sections 5 and 6. The
paper is ended with a conclusion in Section 7.
2. Global grinding model vs local grinding model
The most important step in simulation is to get the
removal from the workpiece surface at discrete time points.
In this context, the simulation of belt grinding processes is
more difficult than such precise operating processes as
milling, because the removal of surface cannot be obtained
directly by CAD data and tool shape. In machining, the
grinding wheel rotates and the belt rubs and strikes the
workpiece surface. Because the shape and distribution of
abrasives on grinding belt is non-uniform and rather
disorderly, the belt grinding process is also considered as
a cutting process with an indeterminate cutting edge. In
addition, the elasticity of the contact wheel causes strong
force variation between the contact wheel and the work-
piece. Thus getting removals is not simply a geometric
computation, but a quite experience-based process.
Fig. 2. Local removal simulati
Several parameters simultaneously effect on the final
global removal of the workpiece surface, for example,
material of the grinding belt, elasticity of the grinding
wheel, temperature and so on. Below is a linear grinding
model by Hammann [1,2]
r Z CAKAkt
Vb
VwlwFA (1)
where r, the material removal rate; CA, the constant decided
by experiments; KA, the combination constant of resistance
factor of the workpiece and grinding factor of the belt; kt,
grinding belt wear factor; Vb, grinding velocity; Vw,
workpiece infeed speed; lw, width of the grinding area;
FA, the acting force between the contact wheel and the part.
The factors in formula (1) are one-valued. A series of
grinding experiments can be executed to get the constant CA
of a particular belt grinding system. Among those factors,
the most important one is the resulting force FA between the
workpiece surface and the elastic contact wheel. Roughly
speaking, the global removal is linearly related to the global
force given that other factors are unvaried.
The global grinding model defined in formula (1) views
the workpiece as a whole and does not consider the statuses
in a local view. Every parameter to be studied in the global
grinding model is quantified by only one value, such as
global force, global removal, etc. The global grinding model
is both applicable to manufacturing of simple-shaped
workpiece surfaces and useful to qualitative study of the
grinding process. But for free-form surface grinding, the
rough linear global relation in Hammann’s model is no
longer adequate specially when a high quality surface is
required. Not only the global removal but also the local
removals from workpiece surface are needed to be known.
To get information in detail, the grinding area at first needs
to be divided into a mesh and then the local situations on
each mesh point need to be defined. For example, the local
forces are represented by the contact forces on all mesh
points in the contact area and the local removals are alike
given by a matrix in which each element indicates
on of free-form surface.
Fig. 3. Signorini contact problem.
X. Zhang et al. / International Journal of Machine Tools & Manufacture 45 (2005) 641–648 643
the removal on one mesh point. In this paper, the local
situation is also named distribution. For instance, the force
distribution means the local forces on the mesh points in the
grinding area.
The local removals or removal distribution of the
workpiece surface result from local situations by contact.
Fig. 2 illustrates the local removals simulation. Firstly, the
force distribution is calculated according to the contact
situation defined by the geometric information of both the
workpiece and the contact wheel. Then, the calculated force
distribution is combined with other grinding parameters to
solve the local removals on the free-form workpiece surface.
Achieving local forces is the most time-consuming step
in the belt grinding simulation flow. It is normally regarded
as a Signorini contact problem and solved by the help of
the FEM.
3. Signorini contact problem
A contact problem between an elastic body and a rigid
body was first carefully studied by Signorini. The left part of
Fig. 3 shows a typical Signorini contact situation. F is the
boundary of the elastic body, FF is the area, where the
external forces p are imposed; FD is the fixed area that
cannot be moved and deformed and FC is the contact area.
g is the interval between the elastic body and the rigid body,
i.e. relative position of two objects. If the rigid body is
translated towards the soft body, the soft body will be
deformed according to the geometry of the rigid body (see
right part of Fig. 3). The deformation of the soft body is
indicated by the strain u. Assume that the both bodies do not
penetrate into each other, then the following initial
boundary conditions
vu
vnðu KgÞ Z 0 on FC
u Z 0 on FD
8<: (2)
together with force balance equation
Ku Cp Z 0 on F (3)
hold, where the n is the normal direction vector and K is the
stiffness factor of the elastic body. Besides Eqs. (2) and (3),
the energy minimization principle must be resorted to solve
the elastic deformation u. The elastic body deforms in a way
that tends to minimize its strain potential energy when in
contact with the rigid body, requiring that the conditions (2)
and (3) are satisfied at the same time. The total energy of the
elastic body can be written as
J Z1
2aðu; uÞK f ðuÞ (4)
in which the first item is the potential deformation energy
and the second item f(u) is the energy derived by external
forces. The traditional FEM provides a numerical solution
to the Signorini contact problem. It discretizes the elastic
body into some small elements whose strain are vi. Then the
energy description (4) is rewritten as the discrete element
form
Je ZX
i
1
2vT
i Avi KpT ðviÞ
� �(5)
where A indicates the elasticity field. The numerical solution
is actually to minimize the overall energy (5) subject to
the contact boundary conditions (2) and force balance
condition (3).
Solving this optimization problem is very computation-
ally expensive since a compact mesh is required to ensure the
calculating precision. Blum [3,4] and Suttmeier [5] worked
out a FEM model that considers this contact problem as a
Signorini contact problem. Although having adopted an
optimized mesh discretization strategy [3], it still requires
about 15 min for the calculation of one contact situation. This
is far away from the demand for the real-time simulation of
belt grinding processes, not to mention adjusting the robot’s
X. Zhang et al. / International Journal of Machine Tools & Manufacture 45 (2005) 641–648644
reaction in time. The calculating of the force distribution
becomes a bottleneck in the real-time simulation flow.
4. A new method for solving the Signorini
contact problem
Two branches are under research nowadays in order to
accelerate the calculation. The first branch is to optimize the
mesh division; the other is to improve the convergent rate
and stability of optimization algorithms. Both cannot avert
the iteration steps each time when a new contact situation is
given. To overcome this, a learning machine is introduced to
approximate the well-established FEM model in this paper.
Even though an optimization process is also necessary in the
training phase, it can finish the calculation of one contact
situation in a very short time because the time-consuming
transaction is put into the training phrase, not in the run time
any more.
The support vector machine (SVM) [6], an efficient
regression method, is adopted as the learning machine in
this paper. The SVM was firstly invented to deal with the
classification problems and applied successfully in an OCR
(optical character recognition) project by AT & T
Laboratory. After that, it was also extended to solve
regression and prediction problems. So far, the SVM is
successfully applied in both classification applications
(SVC) [7–10] and regression problems (SVR) [11,12].
The results of these projects show that SVM is quite
competitive compared to other existing methods, for
instance, Neural Network.
4.1. Setting of regression problem
Consider approximating the following data set:
ðx1; y1Þ; ðx2; y2Þ;.; ðxn; ynÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}n
(6)
xn 3Rl and yn 3R, where l is the dimension of the input
xi and the output yi is a scalar The regression function takes
the form
f ðx;aÞ a2L
where L is the parameter space. The goal of the regression is
to find a function f(x,a0) that minimizes the risk function
Rðf Þ ZXn
iZ1
Cðyi; f ðxiÞÞCcjjf jj2R (7)
where C is the cost function or the loss function. The first
item in Eq. (7) indicates the total penalty imposed on output
errors, jjf jj2R is the regularization item of the result or the
smoothness of the regression function f. c is the tradeoff
factor between two items. The introduction of regularization
item into the risk function aims to enhance the
model’s resistance to the noisy input and to improve the
generalization of the regression function. Different
regression methods differ mainly in two aspects. One is
how to formulate the risk function R(f); the other is how to
find an effective numerical algorithm to get a0.
4.2. Support vector regression
Nonlinear SVR function takes the form
f ðxÞ Z w$FðxÞCb with F : Rl 1F; w2F (8)
where w is a vector, b is a bias scalar and F is the feature
space that normally has a higher dimension than the input
dimension l. F defines how the input x is transformed from
the input space to the feature space. The main idea of
nonlinearity is to map the vectors from the input space to the
feature space F and then apply the linear SVR in this feature
space. When the input space equals the feature space the
SVR is degraded to the linear case.
The SVR uses a new cost function, the 3-insensitive
cost function, to evaluate the discrepancy between
the observation y and the prediction f(x), which has the form
C3ðy; f ðxÞÞ Z jy K f ðxÞj3
Z0 if jy K f ðxÞj%3
jy K f ðxÞjK3 otherwise
((9)
where 3 is defined by user. The 3-insensitive cost function
has its physical meaning that no cost will be imposed if the
difference is smaller than a given value 3. When the value
of 3 is decreased to zero the 3-insensitive cost function
turns out to be the traditional Laplacian cost function. The
specific risk function of the SVR can be written as the
form
Rsvrðf Þ ZXn
iZ1
ðjy K f ðxÞj3ÞCcjjwjj2 (10)
Therefore, the learning process of the SVR is to find the
values w, b that minimize the Rsvr(f) in formula (10).
Thanks to the 3-insensitive cost function, it is not necessary
to get the w explicitly because w can implicitly expressed
by some of the input vectors xi that are called support
vectors [6,13].
The optimization problem (10) can be transformed to
another mathematical form—Dual Formulation that makes it
not necessary to assign the mapping rule F explicitly but just
needs to select a general kernel function instead. From users’
point of view, the kernel function, the 3 value and the tradeoff
factor c should be selected before training the model.
5. Contact modelling using SVR
The input of the FEM model is the initial contact
situation g in formula (2) and Fig. 3; The output is the local
Fig. 4. Basic assumption.
Fig. 5. Part point selection.
X. Zhang et al. / International Journal of Machine Tools & Manufacture 45 (2005) 641–648 645
force distribution. In this work, an uniform discrete form is
used to describe the local initial contact situation, force
distribution and the final local removals. The grinding area
is firstly divided into a mesh with even intervals. The values,
such as g, forces or removals, on mesh points reproduce the
real local conditions approximately. The mesh interval is a
control factor of the precision. The smaller the interval, the
higher is the precision. In our project, the contact area is
limited to a 50!50 mm2 square area and the mesh is evenly
spaced with 1 mm interval. Both local contacts and local
forces are represented by a 50!50 matrix. The FEM model
plays a role that transforms the local contacts matrix g to a
local removal matrix F in a nonlinear way. The SVR is
employed to find such a nonlinear transform by learning the
calculation results of the ‘slow’ FEM model.
Unfortunately, it is not plausible to take the whole local
contact situation, i.e. the 50!50 matrix, as the input of the
SVR. Good results or generalization with such a high input
space dimension cannot be expected because of the curse of
the dimensionality [14,15]. Therefore, we use a data
presentation strategy that is based on a local assumption.
The force on one mesh point is affected only by contact
situation of its surrounding mesh points inside a finite
size area.
The force at one contact point is a function of initial
contact conditions of its surrounding points inside an area
named function area. This is illustrated in the Fig. 4. The left
part is the local contacts in which different colors indicate
different values and right part is the local forces shown in
the same manner. The area enclosed by a black square is the
function area of the mesh point P, whose force is to be
calculated. Obviously, if the function area is large enough or
is the whole contact area, the assumption is undoubtedly
correct.
In this way, the input dimension of the SVR model equals
to the square of the function area size. A big enough
function area would be a guarantee of correctness of the
basic assumption above. However, a big function area will
also cause a high dimension of the input. Through training
experiments, 11 mm is a reasonable size of the function
area. So the input dimension of the SVR model is 121 and
the output is a force value. Each contact mesh point
generates one input/output pair to train the SVR model.
In addition, part point selection (PPS) is used to reduce
the data redundancy in the input vector and further decrease
the input dimension. The PPS, as its name implies, takes
only part of mesh points in the function area instead of all
points as the input of the SVR model with preference.
Table 1
Experiments result of one model
Training pairs 3 c g nSV Mean Simu.
Err (%)
5160 0.04 1 0.4 540 8.56
5160 (PPS) 0.04 1 2 606 8.84
6880 0.04 2 0.6 887 7.71
6880 (PPS) 0.04 1 0.6 675 6.51
10,264 0.04 1 0.6 933 8.23
X. Zhang et al. / International Journal of Machine Tools & Manufacture 45 (2005) 641–648646
It lowers the input dimension without losing much
information because the workpiece surface is assumed to
be continuous in every direction and is varied smoothly, not
abruptly. As Fig. 5 shows, only 41 points (solid points on
four black lines) are selected out from all 121 mesh points in
the 11!11 mm2 function area. It will be shown in Section 6
that using PPS does not degrade the performance of the
regression model.
10,264 (PPS) 0.04 1 2 880 8.22
6. Numerical experiments
180 contact situations are defined to generate sufficient
knowledge database to train the SVR and another 64 for
testing. Then FEM model is used to compute the force
distributions accordingly. There are over 100,000 individual
contact mesh points in contact situations for training.
Linear Kernel, Radial Basis Function Kernel (RBF
Kernel), Polynomial Kernel of degree 2 and 3 have been
tried to establish the SVR model. Except that the linear SVR
is apparently incapable of the task, the other three kernels
perform similarly in this application. However, the RBF is
much faster than the polynomial kernels in training
processes. The sigmoid kernel is not included in evaluation
for two reasons. One is that sigmoid kernel does not always
fulfil the Mercer Condition [6], which is a necessary
condition to be a qualified kernel. The other is that the
sigmoid kernel would perform similarly to RBF kernel
when the q is a very small value [16]. Therefore, only the
RBF kernel is considered in the later experiments of
modelling, which takes the form
KRBFðx; yÞ Z eKgjjxKyjj2 (11)
As mentioned above, over 100 thousand training points
are available that are too large to be involved into training at
one time. So varied batch of contact points are selected
randomly to train the SVR model. Below are the proposed
steps to construct the SVR model.
(1)
Fig.
right
Select training data from all available data.
(2)
Determine the 3 in the 3-insensitive cost function of theSVR and tradeoff factor c in the risk function.
6. One example of simulation results (the left side is the forces calculated by th
side is the error between them).
(3)
e FE
Determine the parameter g in the RBF Kernel.
(4)
Solve the quadratic optimization problem and getsupport vectors.
(5)
Change the 3 and parameters and go back to (2) until thetesting error reaches the minimum.
(6)
Select more training data and repeat the above steps.The training data are generated with and without PPS.
Having been trained, the model is managed to simulate all
the test contact situations. The relative simulation error of
each contact situation is calculated by the formulaPi jFi KF 0
i j
n!max Fi
(12)
where the Fi is the force of one contact point calculated
by FEM model while Fi0 is the result by the SVR model.
Table 1 lists the results of the SVR training and simulation
errors. The mean simulation error in Table 1 is an average
relative simulation error of all testing situations. It is
obvious that PPS does not lower the performance of the
model although it cuts off 2/3 features in the input vector.
Taking more training pairs does not decrease the error any
more when the quantity of training points are already more
than 7000. The best average simulation error of 64 testing
contacts is 6.51%, see Table 1.
Fig. 6 is the simulation result of one contact situation.
The errors occur mainly on the periphery and corner of the
contact area. It can be inferred from figure that the errors are
highly related to the position of the contact point in the
contact area and shape of the contact area itself. The SVR
model may ‘over-learn’ from inner contact points and
‘under-learn’ others because most contact points are inner
M model; the middle is the simulation result of the SVR model and the
Fig. 7. Contact points classification scheme.
Table 3
Experiments result of one model for each kind of points
Point
type
3 c g nSV Relative test-
ing error (%)
1 0.0055 20 0.5 421 3.0
2 0.02 8 0.8 657 4.0
3 0.011 14 1.0 1023 3.0
4 0.02 10 0.8 1081 2.0
5 0.02 8 0.8 659 4.0
6 0.01 20 1.2 657 1.9
7 0.02 10 0.8 1112 2.0
8 0.02 3 1.4 1233 2.8
9 0.011 14 1.0 1027 3.0
10 0.02 10 0.8 1092 1.9
11 0.02 2 1.6 680 2.4
12 0.02 3 0.8 1303 1.8
13 0.02 10 0.8 1119 1.9
14 0.02 3 1.4 1277 2.8
15 0.025 3 0.8 1273 1.8
16 0.03 1 1 1159 2.3
X. Zhang et al. / International Journal of Machine Tools & Manufacture 45 (2005) 641–648 647
points. The idea of improvement is to classify the contact
points into some categories and train one SVR model for
each category.
Fig. 7 illustrates the scheme to classify the contact points.
Point P is the contact point to be classified and the area with
gray background is the contact area. A virtual box is drawn
around the point P that has the same size here (can be changed)
as the function area of the point P. Points P1, P2, P3, P4 are
middle points of the four sides of the virtual box. The value of
point Pi takes 1 if the point is inside the contact area or 0
otherwise. Then the value P1P2P3P4C1 is the type No. of the
point P in Tables 2 and 3. For the case in Fig. 7, P1P2P3P4Z0011Z3, so the type No. of point P is P1P2P3P4C1Z4. The
right part of Fig. 7 shows all possibilities of the contact point P.
Thus all contact points can be classified into 24Z16 groups.
Table 2 lists the number of different type of available contact
points for training and testing. The type 16, namely inner
points, has the biggest batch.
There are a few advantages of training one SVR model
for each type of points. First of all, the SVR can
approximate the FEM model with higher precision. Second,
we can get relatively smaller model by classification that
implies a faster calculation. Third, more training pairs can
Table 2
Points statistics
Type Training set Testing set
1 584 56
2 2246 700
3 1764 428
4 5534 2861
5 2246 700
6 1852 172
7 5534 2861
8 9824 4946
9 1764 428
10 5534 2861
11 2596 316
12 11,128 5424
13 5534 2861
14 9824 4946
15 11,128 5424
16 24,976 13,740
Total 102,068 48,724
be involved in the training process to ensure the final
generalization. However, the smoothness of calculation
results by the SVR model will be poor among the adjoining
areas of different type of points. To overcome it, an
overlapping training strategy is adopted. Those points in the
adjoining area of different types are involved in training the
model for all those types. For example, if point P is of type
16, but inside the adjoining area between the type 1 and type
16, then the point P will used to train both models for type 1
and 16. Thus the practical number of training pairs for each
type is more than those listed in Table 2. The virtual box size
is 11 mm with overlapping area 2 mm. Table 3 lists the
training results of 16 kinds of contact points. The final
average simulation error of 64 testing contacts is 4.1%
compared to 6.5% before.
The time required for one contact calculation is about 1 s
on a PC (CPU AMD Athlon 2600, 512 MB memory). The 3 in
Table 3 can be bigger in some degree to reduce the number of
support vectors and hence to accelerate the calculation as
long as the average simulation error is below 5%.
7. Conclusion
This paper demonstrates a local grinding model to
simulate the robot-controlled belt grinding processes,
especially for grinding free-form surfaces. A new force
calculating model is also put forward as an alternative to the
conventional FEM model. Instead of handling the problem in
pure physical way, the new model learns the nonlinear
relation between local contact situation and force distri-
bution. The training experiments show that the approxi-
mation error can be controlled smaller than 5% using the
SVR technology while the time needed for calculation is
reduced from several minutes to about 1 s. This makes the
real-time simulation of the robot-controlled belt grinding
processes possible. As a result, it can facilitate the planning
X. Zhang et al. / International Journal of Machine Tools & Manufacture 45 (2005) 641–648648
of the grinding paths beforehand and can even generate
the robot instructions of movement automatically. Finally,
the on-line force control is possibly incorporated into the
simulation system to obtain a high surface quality in the
future work.
Acknowledgements
This research has been supported by the Deutsche
Forschungsgemeinschaft(DFG) as a part of the research
group 366 (Simulation Aided Offline Process Design
and Optimization in Manufacturing Sculptured Surfaces).
http://www-isf.maschinenbau.uni-dortmund.de/fgfff/.
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