automated tool trajectory planning of industrial robots
TRANSCRIPT
-
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
1/18
ORIGINAL ARTICLE
Automated tool trajectory planning of industrial robots
for painting composite surfaces
Heping Chen &Ning Xi
Received: 27 September 2005 / Accepted: 8 August 2006 / Published online: 2 November 2006# Springer-Verlag London Limited 2006
Abstract Automatic trajectory generation for spray painting
is highly desirable for todays automotive manufacturing.
Generating paint gun trajectories for free-form surfaces to
satisfy paint thickness requirements is still highly challenging
due to the complex geometry of free-form surfaces. In this
paper, a CAD-guided paint gun trajectory generation system
for free-form surfaces has been developed. The system utilizes
the CAD information of a free-form surface and a paint gun
model to automatically generate a paint gun trajectory to
satisfy the paint thickness requirements. Complex surfaces are
divided into patches to satisfy the constraints. A trajectory
integration algorithm is developed to integrate the trajectories
of the patches. The paint thickness deviation from the required
paint thickness is optimized by modifying the paint gun
velocity. A paint thickness verification method is also
developed to verify the generated trajectories. The results of
simulations have shown that the trajectory generation system
achieves satisfactory performance. This trajectory generation
system can also be applied to generate trajectories for many
other CAD-guided robot trajectory planning applications in
surface manufacturing.
Keywords CAD-guided . Free-form surface .
Trajectory generation . Spray painting . Industrial robots
1 Introduction
Spray painting is an important process in the manufacture
of many durable products, such as automobiles, furniture
and appliances. The uniformity of paint thickness on a
product can strongly influence the quality of the product.
Paint gun trajectory planning is crucial for achieving the
uniformity of paint thickness and has been an active
research area for many years. Currently, there are two
trajectory generation methods: the typical teaching method
and automatic trajectory generation method.
The typical teaching method is complex, time-consum-
ing, and paint thickness is dependent on the operators skill.
Automated generation of paint gun trajectories is time-
efficient and minimizes paint waste and process time; it can
also achieve the optimal paint thickness.
Automated trajectory generation has been widely studied
for a free-form surface with one patch. Suh et al. [1]
developed an automatic trajectory planning system (ATPS)
for spray painting robots. Their method is based on
approximating the original free-form surface as a number
of individual small planes. The bicubic B-spline algorithm
was applied to generate the geometric model of the surface.
Their simulation showed over- and under-painted areas on a
painted surface. Asakawa et al. [2] developed a teaching-
less path generation method using the parametric surface
to paint a car bumper. The paint thickness is 13 to 28 mm
while the average thickness is 17.7 mm. In addition, they
did not report how to determine the spray overlappercentage and the gun velocity. The coverage problem is
widely studied [35]. By dividing a composite surface into
several patches, Sheng et al. [3] developed an algorithm to
cover a composite surface. Although some coverage
algorithms can guarantee the coverage of every point on a
surface, the problem of paint thickness is not addressed.
Antonio et al. [6] developed a framework for optimal
trajectory planning to deal with the optimal paint thickness
problem. However, the paint gun path and the paint
deposition rate must be specified. In practice, it is very
Int J Adv Manuf Technol (2008) 35:680696
DOI 10.1007/s00170-006-0746-5
H. Chen (*) :N. XiElectrical and Computer Engineering Department,
Michigan State University,
East Lansing, MI 48823, USA
e-mail: [email protected]
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
2/18
difficult to obtain the paint deposition rate for a free-form
surface. Alternatively, some commercial software, such as
ROBCADTM/Paint [7], can generate paint gun trajectories
and simulate the painting process. However, the gun paths
are obtained in an interactive way between the user and
ROBCADTM/Paint, which is inefficient and error-prone.
Achieving uniform paint thickness for free-form surfaces
is still a challenging research topic. The complex geometryof free-form surfaces is one of the main reasons. To achieve
uniform paint thickness for a free-form surface, the spatial
gun position, orientation and velocity should be planned
based on the local geometry of the free-form surface. To
satisfy some constraints, such as thickness and gun
orientation constraints, a complex surface has to be divided
into several patches and sprayed separately [8]. Trajectory
generation for a free-form surface with multiple patches has
not been studied yet due to the complexity of the
intersection parts of neighbor patches. For a free-form
surface with multiple patches, the paint thickness of the
boundary of neighboring patches has to be considered.Typically, the paint gun velocity is kept constant to
optimize the paint thickness for a free-form surface with
one patch [2, 9, 10]. However, to optimize the paint
thickness for a surface with multiple patches, the paint gun
velocity needs to be varied. In this paper, a new trajectory
generation scheme for free-form surfaces is developed such
that the paint thickness requirements are satisfied. Optimi-
zation processes are developed to optimize the paint
thickness for a surface with multiple patches. By modifying
the paint gun velocity, the paint thickness deviation from
the required paint thickness is optimized. A car door, hood
and fender, which are free-form surfaces with one patch,
were used to test the scheme. Simulations were performed
to verify the generated trajectories. Simulations were also
performed for a surface with two flat patches to verify the
optimized parameters.
2 Task conditions and requirements
A general framework for the automated CAD-guided paintgun trajectory generation system is shown in Fig. 1.
First, a gun trajectory planner develops a paint gun
trajectory for a free-form surface on the basis of: (1) a CAD
model of the free-form surface; (2) a gun model; (3)
thickness requirements. The trajectory is then loaded into
ROBCADTM/Paint for experimental painting and input to a
verification module to verify paint thickness for the free-
form surface. Finally, the control commands are down-
loaded into a specific controller to drive the robot to paint
the free-form surface.
Parametric surfaces, such as Bezier, B-Spline and
NURBS surfaces are popular in CAD modeling [11]. Thesesurfaces generally satisfy certain continuity conditions and
smoothness constraints, and most of them have small areas
and low curvatures. Although parametric representation can
accurately model complex surfaces, its local nature results
in difficulties for trajectory planning such as segmentation
of paths due to surface patch boundaries [12]. To obtain
time-efficient paint gun trajectories and sufficiently utilize
the workspace of the painting robot, it is desirable to
generate paint gun trajectories that are independent of
surface patch boundaries. A triangular approximation to a
free-form surface can overcome the patch boundary
limitations. The error introduced in rendering a free-form
surface into triangles can be decreased by reducing the size
Fig. 1 CAD-guided paint gun
trajectory planning system
Int J Adv Manuf Technol (2008) 35:680696 681
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
3/18
of triangles. Figure2 shows the triangular approximation of
half a car hood used in our algorithm implementation.
After a free-form surface is tesselated into triangles, the
normals of the triangles may not point to one side of the sur-
face. Here a method is used to adjust normals of the triangles
such that all normals point to one side of a surface. Figure 3
shows two neighbor triangles A and B whose normals are
reversed on a surface. In a triangle data structure, the normalof a triangle is determined using the sequence of its three
vertices. If the sequence of two vertices in a triangle is
reversed, the normal of the triangle will be reversed.
Suppose triangle A points to the same side as a reference
direction, then the direction of triangle B needs to be
reversed. The normal of triangle A is generated using the
three vertices v0, v1 and v2. The sequence is:
v0!v1!v2!v0 1The sequence for triangle B has to be reversed in order
to reverse the normal of triangle B. The desired sequence of
triangle B should be:
v1!v0!v3!v1 2The original sequence of triangle B is:
v1!v3!v0!v1 3Because the sequence of triangle B is different from the
desired sequence, the sequence of two vertices among the
three has to be reversed. Then the normal direction of
triangle B is reversed.
Different paint gun models have been used [1, 10, 13,
14] in spray painting. Some models are quite simple [1] and
some quite complex [14]. A typical paint gun model [10,
13] that we adopted is shown in Fig. 4(a), where is the
fan angle,h the paint gun standoff andR the spray radius.r
is the distance of a point Q on a surface to the spray
direction. The position and orientation of the paint gun tip
with respect to a fixed Cartesian reference frame can be
specified by a six-dimensional vector, p2R6;p px;py;pz;p;p;pT. The values px, py and pz represent the guntip position with respect to the fixed reference frame. The
valuesp,py andpqdescribe the roll, pitch and yaw angles
[15]. TheZaxis of the tool frame (gun tip frame) is defined
as the gun direction. Once the gun direction is determined,
the Zaxis of the tool frame is defined. The X axis of the
tool frame is the gun moving direction.
To generate a paint gun trajectory requires the knowledge
of paint deposition rate. The profile of the paint deposition ratecan be roughly approximated by parabolic curves [1, 13]. A
typical profile is shown in Fig. 4(b). The paint deposition
rate depends on many parameters, such as the gun standoff,
the flow rate of paint, the atomizing pressure and solvent
concentration. In our study, we assume that these parameters
are fixed [1, 2, 10]. Therefore, the paint deposition rate is
only related to the distancer, namely, the deposition rate can
be expressed as a function ofr, fr.After a free-form surface S with one patch is painted, the
average thickness of the surface should be qd. Also the
thickness deviation of any point s on the surface should be
less than or equal to the required thickness deviation qd, i.e.,
maxs2S
jqdqsj qd 4
whereqs is the paint thickness of a points on the surface.
Fig. 2 The triangular approxi-
mation of half a car hood
A
B
1v
0v
2v
3v
an
bn
Reference
Fig. 3 The normal of triangle B
needs to be reversed
682 Int J Adv Manuf Technol (2008) 35:680696
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
4/18
Assume that a plane is painted and the paint thickness on
the plane is optimized. The average, maximum and
minimum paint thicknesses are qd, qmax and qmin, respec-tively. Assume that the paint on the plane is projected onto
a free-form surface and the maximum deviation angle of the
free-form surface relative to the normal of the plane is bth.
The maximum deviation angle is the maximum angle
between the normals of the free-form surface and the plane.
Hence the paint thickness qs of each point s on the free-
form surface satisfies the following inequality without
considering the gun standoff variation:
qmincosth qsqmax 5If the thickness of each pointqs on the free-form surface
satisfies the task requirements (4), i.e.,
jqsqdj qd 6then
qmaxqdqd 7
qdqmincosth qd 8If Eq. (7) is always satisfied, a threshold angle bthcan be
calculated using Eq. (8). This means, for any free-form
surface, if the maximum deviation angle bsatisfiesb
bth,
the paint thickness of any point on the free-form surface
can satisfy the task requirements (4).
3 CAD-guided trajectory generation algorithm
The gun trajectory planner is the core of the gun trajectory
generation system for free-surfaces. After reading in a CAD
model of a free-form surface, it is split into patches based on
the threshold angle to deal with the local geometry of the free-
form surface. The gun trajectory for each patch will then be
generated based on the thickness requirements and the gun
model. Finally, the generated trajectories are integrated to form
an overall trajectory pattern for the free-form surface.
3.1 Patch forming algorithm
During spray painting, a free-form surface is only coveredby a spray cone at each time instant. The patch forming
method is based on minimizing the maximum deviation
angle of spray cones. To optimize the paint thickness on a
free-form surface, the maximum deviation angle of every
spray cone has to be minimized. Assume that there are N
triangles inside a spray cone that is projected to a plane.
The plane is chosen such that its normal minimizes the
maximum deviation angle within the spray cone. Such a
plane can be found by a search method.
Figure 5(a) shows a spray cone and triangles inside the
cone. First, two triangles, A and B, whose normals have the
maximum angle, are found. The average of the two normalsis used as an initial normal of the plane. The deviation angle
between the normal of A (or B) and the initial normal of the
plane is computed as an initial maximum deviation angle.
The deviation angles for other triangles are found too. If there
are some deviation angles greater than the initial maximum
deviation angle, the normal of the plane is re-calculated using
the normals of A, B and a new triangle, C, whose deviation
angle is the maximum. As such the initial maximum
deviation angle is updated. The process continues until the
deviation angle is less than the initial maximum deviation
angle. Then the normal of the plane is obtained and the
maximum deviation angle for the spray cone minimized.
To satisfy the paint thickness requirements, the maxi-
mum deviation angle of a patch found should be less than
the threshold angle. Neighboring relationship of triangles is
applied to form patches. Figure 5(b) shows a neighbor
triangle A is added to a patch.
A seed triangle is arbitrarily chosen as the first triangle
of a patch. Before adding a new neighbor triangle, a
A B
C
Seed
A
(a) (b)Fig. 5 (a) The normal of a plane generation algorithm. ( b) The patch
forming algorithm
h
Qr
R
)(rf
r
(a) (b)Fig. 4 (a) The paint gun model; (b) The paint deposition rate profile
Int J Adv Manuf Technol (2008) 35:680696 683
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
5/18
maximum deviation angle in a spray cone, centered at the
center of the seed, is calculated. If the maximum deviation
angle is less than the threshold angle, the triangle is added
into the patch. Otherwise, it is discarded. After all triangles
are found, they are taken as seeds and new triangles are
added into the patch. The process is repeated until no more
neighbor triangles can be added into the patch. The result is
the first patch found. If there are remaining triangles, a seedis chosen from the remaining triangles and the patch
forming method is applied again to form a new patch.
The process for patch forming continues until no triangles
are left. As a result, the free-form surface is divided into
one patch or several patches. The patch forming algorithm
can be used to automatically form patches for most of the
automotive parts that have edges.
3.2 Trajectory generation algorithm for a surface with one
patch
A gun trajectory includes gun position, orientation andvelocity. A bounding box method [3] is adopted here to
generate the gun path. For a given patch, a bounding box of
the patch is a box which contains the whole patch exactly.
Figure6(a) shows a bounding box and a patch.
The FRONT direction of the bounding box is the
opposite direction of the area-weighted average normal
direction of the patch, i.e.,
naPN
i1SinikP
Ni1Sinik
9
where,na is the average normal of the surface; ni and Si arethe normal and the area of the ith triangle (i1; :::;N),respectively. The RIGHT direction of the bounding box is
along the longest edge direction. However, the bounding box
method works well for a patch with a regular shape. An
improved bounding box method is proposed here to generate
a tool path for patches with irregular shapes. A patch is cut
using a series of planes which are perpendicular to the RIGHT
direction of the bounding box. A series of intersection lines
will be generated on the patch. Each intersection line will be
divided into segments using the spray width, which is the
distance between two neighboring paths. A series of points
will then be generated on each line. The points will be
connected along the RIGHT direction. A path is then
generated for the patch. Figure6(b) shows a path generated
using the improved bounding box method.
The gun velocity and spray width are determined by
optimizing the painting process on a plane. Figure 7 showspaint accumulation of a point w on a plane while a paint
gun moves. The overlap distance d is the distance of the
overlapped part of two neighboring spray cones, as shown
in Fig. 7. The relationship between the spray width L and
the overlap distance d is:
L2Rd 10
Later on, we will use overlap distance instead of spray
width.
Theorem 1 Given a paint gun profile, the paint thickness ona plane is related to the paint gun velocity and the overlap
distance. Moreover, the paint thickness is inversely propor-
tional to the paint gun velocity.
Proof For each point on the plane, there are at most two
neighboring paths which contribute to the paint thickness of
the point. The thickness qxof a point due to the two pathscan be expressed as:
q x; d; v q1 x; v q1 x; v q2 x; d; v
8
-
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
6/18
where t1 and t2 are the painting time for the two
neighboring paths; r1 and r2 are the distance of the point
to the gun center. t1, t2, r1 and r2 are:
t1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2 x2=p
; t2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2 2Rdx 2=q
r1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit 2 x2q ; r2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit
2 2Rdx 2q 13
Substitute t yv in Eq. (12), we obtain:q1 x; v 2
Rt01
0 f r01
dy 0xRq2 x; d; v 2
Rt02
0 f r02
dy Rdx2Rd14
where
t01ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2 x2p
; t02ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2 2Rdx 2q
r01ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
y2 x2p
; r02ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
y2 2Rdx 2q 15
According to (14), q1x; v and q2x; d; v are inverselyproportional to the paint gun velocity. Submitting Eq. (14)
into Eq. (11), we obtain that the paint thickness on a planeis inversely proportional to the paint gun velocity and is
also related to the overlap distance.
To find the optimal velocityvand the overlap distanced,
the mean squared error of the thickness deviation from the
required thickness qdmust be minimized, i.e.,
mind20;R;v
E1d; v R2Rd
0 qdqx; d; v2dx 16
The maximum paint thickness and minimal paint thickness
have to be optimized too because they will determine the paint
thickness deviation from the average paint thickness.
mind20;R;v
E2d; v qmaxqd2 qdqmin2 17
From Eqs. (16) and (17), we have:
mind20;R;v
Ed; v 12RdE1d; v E2d; v 18
Lemma 1 The minimization of Ed; v is only related to theoverlap distance d.
ProofFrom Theorem 1, qx; d; v is inversely proportionalto the paint gun velocity v. Let:
qx; d; v 1vx; d 19
As such, the maximum and minimum paint thicknesses
can be expressed as:
qmax 1vmaxd; qmin
1
vmind 20
To find minimal Ed; v,@Ed; v
@v 0 21
From Eqs. (18), (19), (20) and (21), we obtain:
v
12Rd R
2Rd0 r
2x; ddxr2maxd r2mindqd 12Rd R2Rd0 rx; ddxrmaxd rmind: 22
The paint gun velocity can be expressed as a function of
the overlap distance d. Therefore, the minimization of
Ed; v is only related to the overlap distance d.A golden section method [16] is adopted to calculate the
overlap distance and paint gun velocity.
To achieve uniform paint thickness, the gun direction has
to be determined based on the local geometry of a free-form
surface. Figure 8 shows part of a gun path, sample points
and a spray cone. The distance between two sample points
along the gun path can be chosen according to the
computing time and paint thickness variation requirements(Here it is the radius of a spray cone). The over-spray points
are used to deal with the insufficient thickness problem
occurring at the edges of a part.
The gun direction generation method is the same as the
generation of the normal of a plane in Sect. 3.1, except that
the gun direction is the opposite direction of the normal of
the plane. For each gun center with the determined gun
direction, a local maximum deviation angle is found. For a
gun path composed of a series of gun center points, a global
maximum deviation angle is computed, which is the
maximum value of all maximum deviation angles. If the
global maximum deviation angle of a free-form surface isless than the threshold angle bth, the paint thickness must
satisfy Eq. (5) for every point on the free-form surface
assuming a constant gun standoff.
3.3 Trajectory integration algorithm for a surface
with multiple patches
For the trajectory generation of multiple patches, the
overlap distance and paint gun velocity should be kept the
same as those in one patch except the boundary of
Gun moving directionPart
Spray cone
Sample point
Gun path
Over-spray point
Fig. 8 The gun position, sample points and a spray cone
Int J Adv Manuf Technol (2008) 35:680696 685
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
7/18
neighboring patches. Here we discuss a free-form surface
with more than two patches first.
Figure9 shows a surface with three patches. Suppose the
paint thickness between any two patches is optimized. The
only part we need to consider is the thickness around
intersection point P. Since the paint thickness between patch
2 and patch 3 is optimized, we can merge them into one
patch: patch I, on which the paint thickness is optimized.
Since the paint thickness between patch 1 and 3, patch 1 and
2 is optimized, the paint thickness between patch I and patch
1 must be optimized too. Therefore, the paint thickness on
the area around the intersection point P is optimized. This
method can be applied to a surface with more than three
patches that meet at one common point. In general, if the
paint thickness between any two patches is optimized, the
paint thickness on a surface is optimized as well.
The paint thickness optimization of two patches is much
more complicated than that of one patch because the paint
thickness on the neighboring area has to be considered.
According to the criteria that the main part of a paint gun path
is parallel or perpendicular to the intersection line, three
different cases are studied: parallel-parallel case (PA-PA);
parallel-perpendicular case (PA-PE); perpendicular-perpendic-
ular case (PE-PE). Figure10shows the three different cases.
The three different cases will be discussed individually.
First we will discuss the PA-PA case as shown in Fig. 11. In
this case, we need to optimize the distance h between the
two paths. Because the two paths are symmetric, the dis-
tances of the two paths to the intersection line are the same.
Suppose the angle between the two patches are a, then the
paint thickness at the intersection part can be expressed
using Eq. (14):
qx; h q1x q2x; hcos 0xhq1xcosq2x; h h< x2h
23
where qx; h is the paint thickness at the intersection part;q1x is the paint thickness due to the path on patch 1;q2x; h is the paint thickness due to the path on patch 2.Then, the error function can be formulated as:
minh20;R
E
h
1
2h
E1
h
E2
h
24
where:
minh20;R
E1 h R2h
0 qdq x; h 2dx
minh20;R ;
E2 h q0maxqd 2 qdq0min 2 25
where q'max and q'min are the maximum and minimum
material quantities on the neighboring part of the two
patches, respectively. By optimizing Eq. (24), the optimized
distance h can be obtained.
If a path is parallel to the intersection line between two
patches, the paint thickness on the intersection part due tothe parallel path is evenly distributed. Therefore, before we
discuss the optimization processes for PA-PE, the paint
thickness on a patch in a perpendicular case (PE) is
optimized. Figure 12(a) shows a path which is perpendic-
ular to an intersection line.
The paint gun velocity cannot be kept constant in the PE
case to optimize the paint thickness. To optimize the paint
thickness in the PE case, we divide a path into segments as
shown in Fig.12(a). In each segment, the paint gun velocity
is considered to be fixed. In the figure, there are six pieces
of the path, P1, ..., P6. P2, P3 and P6 are divided into i1segments, respectively. The corresponding velocities arev0; . . . ; vi, respectively. P1, P4 and P5 are divided into k
Patch2 Patch3
Patch1
P
Fig. 9 A surface with three patches
Patch1
Patch2
Path
Path
Patch1
Patch2
Path
Path
Patch1
Patch2
Path
Path
(a) (b) (c)
Fig. 10 (a) Case 1: parallel-
parallel case. (b) Case 2: paral-
lel-perpendicular case. (c) Case
3: perpendicular-perpendicular
case
Intersection line
0
xh
Patch 2
Patch 1
h
X
Fig. 11 Parallel-parallel case
686 Int J Adv Manuf Technol (2008) 35:680696
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
8/18
segments, respectively. The corresponding velocities are
vi1; . . . ; vik, respectively. d0 is arbitrarily chosen. Theminimum thickness and maximum thickness occur on Part I
when x 0 and x 2Rd, respectively. Because thepaint thickness is periodic along X axis, we only need to
consider the paint thickness at x2 0; 2Rd. The paintthickness on Part I is calculated using the contribution of
the six pieces of the path:
P1 :qp1
x;j
1
jR2Rd2k j
i
2Rd2k ji1 f 2R
d
2 y
x dyP2 :qp2 x;j 1j
Rj1i1d0j
i1d0f
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Rd
2 x 2 d0y 2q
dy
P3 :qp3 x;j 1jRj1
i1d0j
i1d0f
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Rd
2 x 2 d0y 2q
dy
P4 :qp4 x;j 1jR2Rd
2k ji
2Rd2k
ji1 f 2Rd
2 yx
dyP5 :qp5 x;j 1j
R2Rd2k
ji 2Rd
2k ji1 f
3 2Rd 2
yx
dy
P6 :qp6 x;j 1jRj1i1d0j
i1d0f
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2Rd 2 x 2 d0y 2r !dy
26
As such, the thickness on Part I is:
qI x Xi
j0 qp2 x qp3 x;j qp6 x;j
Xik
ji1 qp1 x;j qp4 x qp5 x;j 27
When optimizing the paint thickness on Part I, we
consider the thickness on Part II in Fig. 12(c) too, which
can be expressed using P0, P3, P4 and P5 as follows:
P0 :q0p0x 1
Z R0
fRyxdy
P3 :q0p3x;j 1
j
Zj1i1d0
j
i1d0fjyxjdy
P4 :q0p4x;j 1
j
Z 2Rd2k
ji
2Rd2k
ji1f
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid0x2 y2
q dy
P5 :q0p5x;j 1
j
Z 2Rd2kji
2Rd2k
ji1frdy
where : r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid0x2 32Rd2
y 2s28
0
0v
iv
1+iv kiv +x
0d
dR 2
P1
P2P3
P4 P5
P6
v
X
P0
Patch
Intersection line
X0 x
I P4
P3
0
xIIP3
XP4
(a)
(b) (c)Fig. 12 Paint gun velocity optimization for a perpendicular path
0v
iv
1+iv kiv +h
Patch 2
Patch 1
Intersection line h
I
II
III
Fig. 14 Perpendicular-perpendicular case
Pmax
0v
iv1
+
i
vki
v+
x
z
1h
Patch 2
Patch 1
Intersection line2
h I II
Pmin
Fig. 13 Parallel-perpendicular case
Int J Adv Manuf Technol (2008) 35:680696 687
http://-/?-http://-/?- -
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
9/18
Hence the thickness of Part II is:
qIIx q 0p0x Xij0q0p3x;jXik
ji1 q0
p4x;j q0p5x;j 29
According to Eqs. (16), (17), (18), we have:
minvj;j20;ik
Evj 12RdEIvj
1
d0EIIvj EIIIvj
30where EI, EII and EIII are defined as:
EIvj Z 2Rd
0
qdqIx2dx
EIIvj Z d0
0
qdqIIx2dx
EIIIvj q0maxqd2 qdq0min231
where q'max and q'min are the maximum and minimum
thicknesses on Part I and II.
The problem described by Eq. (30) is a multi-variable
unconstraint optimization problem. The steepest-descent
algorithm [17] is adopted here to optimize Eq. (30). The
optimized velocities are obtained such that the paint
thickness is optimized. After that, the maximum thickness
point Pmax and the minimum thickness point Pmin can befound for Part I.
Figure 13 shows the PA-PE case. Suppose the angle
between the two patches area, the paint thicknesses on Part
I and II can be calculated:
qI x; h1; h2
q1 x q2 x; h1; h2 cos0xh1
q1 x cosq2 x; h1; h2 h1< xh1h2
8>>>>>:
qI x; h1; h2
q01 x q02 x; h1; h2 cos
0xh1q01 x cosq02 x; h1; h2
h1< xh1h2
8>>>>>:
32
where qIx; h1; h2 and qIIx; h1; h2 are the paint thicknesson Part I and II, respectively; q1xand q2x; h1; h2are the
paint thicknesses on Part I due to the paths on patch 1 and
patch 2, respectively; q'1x and q'2x; h1; h2 are the paintthicknesses on Part II due to the paths on patch 1 and patch
2, respectively. A method similar to the perpendicular case
(Eqs. (26) and (27)) is used to calculate the paint thickness.
Then the error function can be developed:
minh1;h2
E h1; h2 1h1h2 EI h1; h2 EII h1; h2 EIII h1; h2
33
10 20 30 40 50 6030
35
40
45
50
55
60
65
70
Distance (mm)
Thickn
ess(um)
Fig. 15 The optimized paint thickness on a surface with one patch
5 10 15 20 25 30 35 40 45 50 55
30
35
40
45
50
55
60
65
70
Distance (mm)
T
hickness(um)
0 10 20 30 40 50 60 70 80 90
16
18
20
22
24
26
28
30
32
Angle
D
istanceh(mm)
(a) (b)Fig. 16 Optimization results for PA-PA case. (a) Paint thickness when30o. (b) The relationship betweenh and
688 Int J Adv Manuf Technol (2008) 35:680696
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
10/18
whereEIh1; h2,EIIh1; h2and EIIIh1; h2are defined as:
EIh1; h2 Z h1h2
0
qdqIx; h1; h22dx
EIIh1; h2 Z h1h2
0
qdqIIx; h1; h22dx
EIIIh1; h2 q0maxqd2 qdq0min2
34
where q'max and q'min are the maximum and minimum
thicknesses on Part I and II, respectively. By optimizing Eq.
(33), we obtain h1 and h2.
For the PE-PE case, according to the relative position of
the path in patch 1 and that in patch 2, the optimized paint
thickness is different. To get the optimized paint thickness on
a surface with two patches, the paint thickness due to one
path should compensate the paint thickness due to the other.
Figure14shows the PE-PE case. After calculating the paint
thickness on Part I, II and III using the similar method in
the perpendicular case, the similar error function in the PA-
PE case can be formed. In this case, the distance h and the
velocities have to be optimized together to achieve the
optimized paint thickness. The steepest-descent algorithm
[17] is adopted here to optimize the paint thickness.
0 5 10 15 20 25 30 35 40 4530
35
40
45
50
55
60
65
70
Distance (mm)
Thickn
ess(um)
II
I
(a) (b)
0 10 20 30 40 50 60 70 80 900
5
10
15
20
25
30
35
40
45
50
Angle
Distanc
eh(mm)
Fig. 17 Optimization results for parallel-perpendiculr case. (a) Paint thickness when 30o. (b) The relationship between h1, h2 and
0 5 10 15 20 25 30 35
30
35
40
45
50
55
60
65
70
Distance (mm)
T
hickness(um)
0 10 20 30 40 50 60 70 80 90
0
2
4
6
8
10
12
14
16
18
20
Angle
Distanceh(mm)
(a) (b)Fig. 18 Optimization results for PA-PE case. (a) Paint thickness when30o. (b) The relationship between h and
Int J Adv Manuf Technol (2008) 35:680696 689
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
11/18
4 Implementation and experimental verification
4.1 Paint thickness optimization
Suppose the required average thickness qd is 50m and
the thickness deviation percentage is 35%. In this case, the
paint thickness deviation qd5035%17:5m. Thespray radius R50 mm. The paint deposition rate is
fr 110
R2 r2m=s 35
After optimization, the gun velocity and the overlap
distance were found to be:
v323:3 mm=s; d39:2 mm 36The maximum and minimum thicknesses are:
qmax52:02m; qmin48:05m 37The optimized paint thickness is shown in Fig.15.
From the thickness deviation percentage and Eq. (8), the
threshold angle is calculated:
bth45:6o 38After performing the optimization process for the PA-PA
case, the optimized paint thickness fora30o is shown inFig.16(a). The relationship between the optimized distance
h and the angle a is shown in Fig. 16(b).
In the velocity optimization process for the perpendicular
case, i5, d02R and k6 are chosen. The optimizedpaint gun velocities (mm=s) are:
0272:2; 1333:1; 2459:2; 3336:4;4226:7; 5355:3; 6547:2; 7690:8:
39For the PA-PE case, the optimized paint thickness for
a30o is shown in Fig. 17(a). The relationship betweenthe optimized distance h1, h2 and the angle a is shown in
Fig.17(b).
For the PE-PE case, the optimized paint gun velocities
(mm=s) when a30o are:0252:0; 1308:4; 2425:2; 3311:5;4209:9; 5329:0; 6506:7; 7639:6:
40The optimized paint thickness fora30o is shown in
Fig.18(a). The relationship between the optimized distance
h and the angle ais shown in Fig. 18(b).
The maximum and minimum paint thicknesses for the
three cases when a30o are shown in Table1.
4.2 Implementation and trajectory verification
The algorithm was implemented in C++ on a PC with
PentiumTM III 860 MHZ processor. Three parts, part of a
car hood, fender and door, shown in Figs. 2,19(a) and (b),
respectively, were used to test the algorithm. The triangular
approximation was exported from GID (http://gid.cimne.
upc.es/) with an error tolerance of 2 mm. The car hood,
fender, and door have 3320, 9355 and 4853 triangles,
respectively. Using the patch forming method, the car hood,
fender and door form only one patch, respectively.
Table 1 The simulation results
Case Minimum thickness (mm) Maximum thickness (mm)
PA-PA 48.8 51.5
PA-PE 40.7 59.4
PE-PE 44.6 55.8
2 2.5 3 3.5
4 4.50.6
0.8
1
1.2
1.40.5
0.6
0.7
0.8
0.9
1
Length (m)
Width (m)
Height(m)
2.5
3
3.5
4
0.8
1
1.2
1.4
1.6
1.80.45
0.5
0.550.6
0.65
0.7
0.75
0.8
0.85
0.9
Length (m)Width (m)
Height(m)
(a) (b)Fig. 19 The triangular approximation of (a) a car fender; (b) a car door
690 Int J Adv Manuf Technol (2008) 35:680696
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://gid.cimne.upc.es/http://gid.cimne.upc.es/http://gid.cimne.upc.es/http://gid.cimne.upc.es/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
12/18
Once the overlap distance is determined, the gun paths
were generated using the improved bounding box method.
The generated gun paths are shown in Fig. 20(a)(c) for the
car hood, fender and door, respectively.
The gun direction was found for each sample point and
the global maximum deviation angles were calculated for
each part, respectively. The number of triangles and global
maximum deviation angles for each part are shown in
Table2. The global maximum deviation anglesbin Table2
are less than the threshold angle bthfor the three parts. This
0.60.8
11.2
1.41.6
1.8
0
0.2
0.4
0.6
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Length (m)Width (m)
He
ight(m)
2 2.5 3 3.5 4
4.50.6
0.8
1
1.2
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Length (m)Width (m)
Height(m)
2.5
3
3.5
4
0.8
1
1.2
1.4
1.6
1.80.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
Length (m)Width (m)
Height(m)
(a) (b)
(c)Fig. 20 The generated path for (a) a car hood; (b) a car fender; (c) a car door
Table 2 The calculated parameters
Part Triangles b
hood 3320 13:1o
fender 9355 29:7o
door 4853 44:6o
hih
Plane
Projected
point
Free-formsurface
an
in
i
sFig. 21 The simulation surface and its projected plane:h is the given
gun standoff; h i is the real gun standoff; na is the gun direction; n i is
the normal of a triangle
Int J Adv Manuf Technol (2008) 35:680696 691
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
13/18
means the generated gun trajectories can satisfy the
thickness requirements.
Simulations were performed to verify the generated
trajectories. Figure 21 shows a model used to simulate the
paint thickness on a free-form surface. A plane is generated
using the gun direction and gun standoff. The paint
thickness on the plane is calculated. Then the paint
thickness on the plane is projected to a free-form surface.
The projected paint thickness qs of a point s on the free-
form surface can be calculated as:
qs
q
h
hi
2
cosi
41
where q is the paint thickness of the projected point on the
plane; hi is the actual gun standoff and bi is the deviation
angle of the point.
After the simulation, the paint thickness of randomly
chosen points on the car hood, fender and door were
computed and shown in Fig. 22(a)(c), respectively. The
average, maximum, and minimum paint thicknesses were
calculated and shown in Table 3. The simulation results
demonstrate that the average, maximum and minimum
50 100 150 200 250 300 350 400 450 50040
42
44
46
48
50
52
Points
Thickn
ess(um)
54
56
58
60
50 100 150 200 250 300 350 400 450 50030
35
40
45
50
55
60
65
70
Points
Thick
ness(um)
50 100 150 200 250 300 350 400 450 50030
35
40
45
50
55
60
Points
Thickness(um)
(a) (b)
(c)Fig. 22 The simulation result of paint thickness for (a) a car hood; (b) a car fender; (c) a car door
Table 3 The simulation results
Part Average
thicknessq (mm)
Maximum
thicknessqmax (mm)
Minimum
thicknessqmin (mm)
Hood 50.0 53.9 46.0
Fender 49.1 55.0 42.6
Door 49.4 54.9 35.6
692 Int J Adv Manuf Technol (2008) 35:680696
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
14/18
thicknesses for the car hood, fender and door all satisfy the
thickness requirements. The trajectories generated using our
new algorithm can achieve required paint thickness.
The spray gun trajectories generated were also exported
to ROBCADTM/Paint to simulate the painting process. The
robot model used is an ABBTM irb6K30-75. The work-cell
setup is shown in Fig. 23(a) for painting a car hood. A part
of gun path is shown in Fig. 23(b). A part after painting isshown in Fig.23(c). The simulation results showed that the
generated trajectories can be applied to paint free-form
surfaces.
A surface with two flat patches are generated and
rendered into triangles. The trajectory for each patch is
generated and the optimized parameters are applied to
calculate the paint thickness using the simulation model.
The part rendered into triangles is shown in Fig. 24.
The paths of the part are generated for the PA-PA, PA-PE
and PE-PE cases. Figure 25 shows the path and paint
thickness when a30o for PA-PA case. Figure 26 forPA-PE case. Figure 27for PE-PE case.
The maximum and minimum paint thicknesses for thethree cases when a30o are shown in Table4.The results shown in Tables1and4 are quite close. This
means the optimized parameters can optimize the paint
thickness at different cases. From the optimization results
and the verification results, we can see that the optimized
paint thickness for the PA-PA case and the PE-PE case is
quite uniform. The paint thickness deviation from the
required thickness is about 5m. However, the paint
thickness deviation for the PA-PE case is quite large, which
is about 10m. Therefore, in trajectory generation, the
PA-PE case should be avoided.
4.3 Paint thickness optimization
The lower and upper bounds are used to describe the paint
thickness deviation of free-form surfaces. They are defined as:
qmaxqmaxqdqminqdqmin 42
where qmax and qmin are the maximum and minimum paint
thicknesses on a free-form surface, respectively. According
1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4-1
-0.8 -0.6
-0.4 -0.2
0 0.2
0.4 0.6
0.8 1
-0.8
-0.6
-0.4
-0.2
0
Width (m)
Length (m)
Height(m)
Fig. 24 The part with two flat patches when30o
Fig. 23 (a) The ROBCAD
simulation system; (b) A part of
a gun path; (c) A painted part
(part of a car hood)
Int J Adv Manuf Technol (2008) 35:680696 693
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
15/18
-1000
-800
-600
-400
-200
0
200
-1000-800-600-400-200
0200400600
8001000
-500
-400
-300
-200
-100
0
Length (m)
Width (m)
Height(m)
0 100 200 300 400 500 600 700 80040
42
44
46
48
50
52
54
56
58
60
Points
Thickness(um)
(a) (b)Fig. 26 Verification results for PA-PE case. (a) The Path. (b) The paint thickness
-1000
-800
-600
-400
-200
0
200
400
-1000-800-600-400-2000
2004006008001000
-500
-400
-300
-200
-100
0
Length (m)
Width (m)
Height(m)
50 100 150 200 250 300 350 400 450 50030
35
40
45
50
55
60
65
70
Points
Thickness(um)
(a) (b)Fig. 25 Verification results for PA-PA case. (a) The Path. (b) The paint thickness
-1000
-800
-600
-400
-200
0
200
400
-1000
-500
0
500
1000
-600
-400
-200
0
Length (m)
Width (m)
Height(m)
50 100 150 200 250 300 350 400 450 50030
35
40
45
50
55
60
65
70
Points
Thic
kness(um)
(a) (b)Fig. 27 Verification results for PE-PE case. (a) The Path. (b) The paint thickness
694 Int J Adv Manuf Technol (2008) 35:680696
-
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
16/18
to Eq. (5), the upper bound of the paint thickness is
controlled by the optimized maximum paint thickness on a
plane. However, the lower bound of the paint thickness is
controlled by both the optimized minimum paint thickness
on a plane and the maximum deviation angle of a free-form
surface. If a free-form surface has a bigger maximum
deviation angle, the lower bound will be much larger than
the upper bound. The paint thickness simulations show that
the lower bounds of the car fender and door are much larger
than the upper bounds. A method is developed to optimize
the paint thickness deviation by modifying the gun velocity.
For a sample point shown in Fig. 8, suppose themaximum deviation angle is bmax, the minimum deviation
angle is bmin. Then, the velocity is modified to be:
v'vcosbmaxacosbmin1a 43
where v'is the modified velocity, a is a constant to control
the paint waste.
Simulations are performed only for the car fender and
door since the upper and lower bound of the car hood are
quite close. a is chosen to be 2. Figure 28(a) and (b) show
the simulation results for the car fender and door,
respectively. Table 5 shows the maximum, minimum andaverage thicknesses of the modified method. The results
show that the lower bounds of the modified method are
decreased for the car fender and door. The lower bound of
the car door is 10m, instead of 15m for the original
method. That of the car fender is 4:9m, instead of 7:6m.The thickness deviation is decreased from 30% to 20% for
the car door, and 15% to 10% for the car fender,
respectively. Therefore, the proposed method can optimize
the paint thickness deviation.
4.4 Comparison with other methods
Simulations with fixed gun direction (Case 2) were also
performed to show the advantages of the gun directiongeneration method proposed in this paper (Case 1). The
fixed gun direction was determined using the search
method in Section3.1for the whole car hood. So was the
fixed gun direction of the car fender and door. The results
are shown in Table6.
The global maximum deviation angle for each part in
Case 1 is much smaller than that in Case 2. The deviations
of the average thickness to the required thickness are 0, 0.9
and 0.6mmfor the car hood, fender and door respectively in
Case 1. However they are 1.1, 1.8 and 2.4 mm in Case 2.
The minimum thickness is 46, 42.6 and 35.6 mmfor the car
hood, ender and door, respectively in Case 1, while 40.3,35.5 and 24.3 mm for the car hood, ender and door,
respectively in Case 2. Using the paint thickness optimiza-
Table 4 The simulation results
Case Maximum thickness (mm) Minimum thickness (mm)
PA-PA 47.6 51.6
PA-PE 41,6 59.5
PE-PE 47.5 54.5
50 100 150 200 250 300 350 400 450 50030
35
40
45
50
55
60
65
70
Points
Th
ickness(um)
50 100 150 200 250 300 350 400 450 50030
35
40
45
50
55
60
65
70
Points
Th
ickness(um)
(a) (b)Fig. 28 The simulation result of paint thickness using the thickness optimization method: (a) for a car fender; (b) for a car door
Table 5 The simulation results
Part Average
thicknessq (mm)
Maximum
thicknessqmax (mm)
Minimum
thicknessqmin (mm)
Fender 50.8 56.0 45.1
Door 51.5 59.4 40.1
Int J Adv Manuf Technol (2008) 35:680696 695
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
17/18
tion method, we achieved much better results. The results
show that the gun directions generated on the basis of the
local geometry of free-form surfaces achieved better paint
thickness.
The thickness deviation percentage presented by Asakawa
et al. [2] is more than 58%. In our results, the thickness
deviation percentage is less than 20% using the paint
thickness optimization method. For the car hood, much
better result (8%) is achieved. Although the results presented
by Suh et al. [1] are 20%, the over- and under-painted areaswere not included. Also the paint distribution rate is a
constant instead of a parabolic curve.
5 Conclusion
A new CAD-based paint gun trajectory generation system
for free-form surfaces has been developed. The system
utilizes the CAD information of a free-form surface and a
paint gun model to generate a paint gun trajectory to satisfy
the paint thickness requirements. Simulation results showed
that the paint thickness requirements are satisfied. Thetrajectory integration algorithm can be used to integrate
trajectories for a free-form surface with multiple patches.
Therefore, this method can be used as an off-line trajectory
generation system of free-form surfaces for spray painting
robots. This trajectory generation method can be applied
into many other CAD-guided robot trajectory planning
applications in surface manufacturing, such as spray
coating and spray forming. Our future work will concen-
trate on some practical issues, such as robot constraints and
spray gun constraints because these will also affect paint
distribution.
Acknowledgements The authors would like to acknowledge Na-
tional Science Foundation for its partial support to this work under
Grant IIS-9796300, IIS-9796287 and Ford Motor Company Univer-
sity Research Program. The authors would like to thank Tecnomatix
Inc. for providing us with the software ROBCAD.
References
1. Suh SH, Woo IK, Noh SK (1991) Development of an automatictrajectory planning system(atps) for spray painting robots. In:
IEEE International Conference on Robotics and Automation,
Sacramento, California, pp 19481955, April
2. Asakawa N, Takeuchi Y (1997) Teachingless spray-painting of
sculptured surface by an industrial robot. In: IEEE International
Conference on Robotics and Automation, Albuquerque, New
Mexico, pp 18751879, April
3. Sheng W, Xi N, Song M, Chen Y, MacNeille P (2000) Automated
cad-guided robot path planning for spray painting of compound
surfaces. In: IEEE/RSJ International Conference On Intelligent
Robots And Systems, Takamutsa, Japan, pp 19181923, October
4. Choset H (2000) Coverage of known spaces: the boustrophedon
cellupdar decomposition. Auton Robots 9:247253
5. Penin LF, Balaguer C, Pastor JM, Rodriguez FJ, Barrientos A,
Aracil R (1998) Robotized spraying of prefabricated panels. IEEERobot Autom Mag 5(3):1829
6. Antonio JK, Ramabhadran R, Ling TL (1997) A framework for
optimal trajectory planning for automated spray coating. Int J
Robot Autom 12(4):124134
7. Tecnomatix (1999) ROBCAD/Paint training. Tecnomatix, Michigan,
USA
8. Chen H, Sheng W, Xi N, Song M, Chen Y (2002) Automated
robot trajectory planning for spray painting of free-form surfaces
in automotive manufacturing. IEEE Int Conf Robot Autom 1:450
455
9. Suh S, Woo I, Noh S (1991) Automatic trajectory planning
system (atps) for spray painting robots. J Manuf Syst 10(5):396
406
10. Antonio JK (1994) Optimal trajectory planning for spray coating.
In: IEEE International Conference on Robotics and Automation,San Diego, California, pp 25702577, May
11. Chang TC, Wysk RA, Wang HP (1998) Computer-aided manu-
facturing, 2nd edn. Prentice Hall, New Jersey
12. Lai JY, Wang DJ (1994) A strategy for finish cutting path
generation of compound surface. Comput Ind 25:189209
13. Persoons W, Van Brussel H (1993) Cad-based robotic coating
with highly curved surfaces. In: International Symposium on
Intelligent Robotics (ISIR93), vol. 14, pp 611618
14. Hertling P, Hog L, Larsen R, Perram JW, Petersen HG (1996)
Task curve planning for painting robots. i. process modeling and
calibration. IEEE Trans Robot Autom 12(2):324330, April
15. Fu KS, Gonzalez RC, Lee CSG (1987) Robotics: control, sensing,
vision and intelligence. McGraw-Hill, New York, NY, USA
16. Rao SS (1983) Optimization: theory and application. Wiley, New
York, USA17. Gill PE, Murray W, Wright MH (1981) Practical optimization.
Academic Press, New York, NY, USA
Table 6 The simulation results for fixed gun directions
Part Average
thicknessq' (mm)
Maximum
thicknessq'max (mm)
Minimum
thicknessq'min (mm)
b'
Hood 48.9 54.3 40.3 32:8o
Fender 48.2 54.6 35.5 42:0o
Door 47.6 54.6 24.3 64:2o
696 Int J Adv Manuf Technol (2008) 35:680696
http://-/?-http://-/?-http://-/?-http://-/?- -
8/22/2019 Automated Tool Trajectory Planning of Industrial Robots
18/18