Automated Tool Trajectory Planning of Industrial Robots

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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]
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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

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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

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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



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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


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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

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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

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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

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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


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

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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


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

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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

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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



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


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



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)

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-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



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-


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


Part Average

thicknessq (mm)

Maximum

thicknessqmax (mm)

Minimum

thicknessqmin (mm)

Fender 50.8 56.0 45.1

Door 51.5 59.4 40.1



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

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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

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USA

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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

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Automated Tool Trajectory Planning of Industrial Robots