Bulletin of the JSME

Journal of Advanced Mechanical Design, Systems, and ManufacturingVol.12, No.1, 2018

Paper No.16-00720© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0008]

Lintao XIANG*, Xiaopeng XIE* and Xiaohui LU* *School of Mechanical and Automotive Engineering

South China University of Technology,

No. 381 Wushan Road, Tianhe, Guangzhou, 510640, China

E-mail: luabc@scut.edu.cn

Abstract

The underwater welding robots are replacing humans in several harsh working environments however further

strategies are required to achieve better control of robotic motion in order to extend their utility. This paper

presents a smooth trajectory control strategy to improve the welding quality and efficiency using an underwater

welding robot to perform the arc welding process. First, a mathematical model of the underwater welding robot

is established using the D-H parameter method. Second kinematics equations for the movement of the robot

are deduced. To improve the accuracy of the trajectory, the tool coordinate system is calibrated using the

six-point method. Finally, linear interpolation with parabolic transition is combined with a six-dimensional

space vector to develop a Cartesian space trajectory planning for the robot, which can ensure a smooth welding

process. The results show that by using the above control strategy for underwater welding experiments, a

smooth welding seam is achieved, which improves the weld quality and shortens the time taken to complete

the weld.

Keywords :

1. Introduction

The underwater welding robot is a device that can be moved underwater with a visual and perceptual system,

which can replace humans in underwater operating environments. Under the underwater welding process of robot, the

motion parameters and welding parameters of underwater welding robot need to be adjusted to adapt to the underwater

environment due to the influence of water environment and underwater pressure. At present, underwater welding robots

are used to complete a number of dangerous operations (Rowe and Liu, 2001); however the tasks that can be performed

are limited due to the complexity of the underwater environment. As such, current robotic welding systems are mainly

used for non-destructive testing of welds and crack repairs (Labanowski, 2011). More widespread use will require

greater control over trajectory planning. In this study, the trajectory planning of an existing underwater welding robot is

investigated with the aim of improving its accuracy. Welding robot trajectory planning allows the robot to move from

an initial position to a target location of the process smoothly and at a certain speed and acceleration, within a certain

timeframe. Two methods for trajectory planning exist: joint space planning and Cartesian space planning (Nardenio and

Douglas, 2008). The former method has been used extensively in industrial robotics systems but presents a number of

disadvantages.

Joint space planning based on cubic polynomial interpolation has been applied to determine the position and

velocity of the manipulator. However it does not take into account continuous planning of acceleration (Liu et al.,

2013). The quintic polynomial interpolation method has been applied to the transition angle of the curve in order to

achieve smooth transitions of robot motion. Despite these improvements, problems still exist such as complex

calculations and low efficiency (Rai et al., 2014). Chen et al uses an uniform cubic B-splines method as a planning

function and while this method ensures consistency of the joint angle, angular velocity and acceleration, it is

computationally expensive and the functional equations highly complex (Chen, 1991). Other advanced algorithms such

1

An optimal trajectory control strategy for underwater welding robot

Received: 28 December 2016; Revised: 13 September 2017; Accepted: 25 December 2017

Underwater welding robot, Tool coordinate system calibration, Posture, Trajectory planning

2© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0008]

Xiang, Xie and Lu, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.12, No.1 (2018)

as fuzzy algorithm (Peng et al., 2015), genetic algorithm (Deng et al.,2013) and neural network (Simon and Max, 2000)

have also been used to improve the planning efficiency, however each has its own limitations and drawbacks.

The robot examined in this study is currently used to repair the crack of the spent fuel pool when it is broken due to

the external impact loads or stress in the nuclear power stations. To further optimize the welding quality, a smooth

trajectory control strategy using Cartesian space planning is proposed. A mathematical model of the welding robot is

established and kinematic analysis of the welding robot is performed using the D-H parameter method (Denavit and

Hartenberg, 1995). Formulae are deduced for both the forward and inverse kinematics of the robot and the posture

equation of the robot is simplified. In addition, to improve the accuracy of the welding torch, the tool coordinate

system is calibrated by the six-point method (Schroer, et al., 1988). Finally, the Cartesian space planning method

combined with a six-dimensional space vector is used to improve the trajectory planning accuracy of the underwater

welding robot. Underwater welding experiments are performed to verify the accuracy of the proposed motion control

strategy.

2. Kinematics analysis of underwater welding robot

2.1 Establishment of the welding robot model

The underwater welding robot used in this study belongs to the family of six-degree-of-freedom joint robots having

six axes: S, L, U, R, B and T. The structure of the robot is presented in Fig.1.

Fig. 1 Structure diagram of welding robot

The coordinate frame of each joint is established according to the D-H method. The coordinate frame of the robot

at zero position is shown in Fig.2.

2

a1

z0

y1

x0,x1

a2

y2

a3

x2,x3

x4

z3 y4,x5

z5,z6

x6

d4

d6

01

2

3

4

5

6

Fig. 2 D-H coordinate system of welding robot

2© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0008]

Xiang, Xie and Lu, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.12, No.1 (2018)

2.2 Analysis of the forward kinematics

In the D-H parameter method, i

represents the rotating joint angle, ia represents the length of the link, i

represents the rotating angle of the link, and id represents the offset distance of the adjacent link. The structural

parameters of the robot are presented in Table 1.

Joint

code Link i i [°]

ia [mm] id [mm] i [°] Range[°]

S 1 90 150 0 θ1 -170°~170°

L 2 0 570 0 90° + θ2 -90°~150°

U 3 90 130 0 θ3 -170°~190°

R 4 -90 0 640 θ4 -180°~180°

B 5 90 0 0 -90° + θ5 45°~-225°

T 6 0 0 95 θ6 -360°~360°

According to the parameters of the link shown in Table 1, the product of the adjacent link transformation matrix 0

1T ,

1

2T , 2

3T , 3

4T and 5

6T can be calculated as:

0 0 1 2 3 4 5

6 1 2 3 4 5 6

0 0 0 1

x x x x

y y y y

z z z z

T T T T T T T

n o a p

n o a p

n o a p

(1)

The terminal position of the robot ( , ),x y z

p p p can be obtained from Eq. (1) as

6 1 2 3 4 5 1 2 3 4 5 1 4 5 1 2 3 5 1 2 3 5 4 1 2 3 1 2 3

3 1 2 3 1 2 3 2 1 2 1 1

( ) ( )

( )

xp d c c c c s c s s c s s s s c c c c c s c c d c c s c s c

a c c c c s s a c c a c

6 1 2 3 4 5 1 2 3 4 5 1 4 5 1 2 3 5 1 2 3 5 4 1 2 3 1 2 3

3 1 2 3 1 2 3 2 1 2 1 1

( ) ( )

( )

yp d s c c c s s s s c s c s s s c c c s s c c d s c s s s c

a s c c s s s a s c a s

6 2 3 4 5 2 3 4 5 2 3 5 2 3 5 4 2 3 2 3

3 2 3 2 3 2 2

( ) ( )

( )

zp d s c c s c s s s s s c c c c d s s c c

a s c c s a s

The theoretical value of the terminal position of the robot is simplified in the actual welding process; therefore,

after installing the torch to the appropriate position on the manipulator, the sixth axis position can be kept unchanged.

Then welding robot therefore becomes five degrees of freedom, which reduces the complexity of solving the equations.

The simplified terminal position of the robot can be calculated as:

4 1 2 3 1 2 3 3 1 2 3 1 2 3 2 1 2 1 1( ) ( )xp d c c s c s c a c c c c s s a c c a c

4 1 2 3 1 2 3 3 1 2 3 1 2 3 2 1 2 1 1( ) ( )yp d s c s s s c a s c c s s s a s c a s

4 2 3 2 3 3 2 3 2 3 2 2( ) ( )zp d s s c c a s c c s a s

The terminal posture of the robot can be obtained from Eq. (1) as

x 1 23 4 1 4 5 1 23 5 6 1 4 1 23 4 6 n =[( ) ] ( )c c c s s c c s s c s c c c s s

1 23 4 1 4 5 1 23 5 6 1 4 1 23 4 6 n =[( ) ] ( )y s c c c s c s s s c c c c c s s

Table 1 D-H parameters of the welding robot

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2© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0008]

Xiang, Xie and Lu, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.12, No.1 (2018)

z 23 4 5 23 5 6 4 23 6 n =( ) ss c c c s c s s

1 23 5 1 23 4 1 4 5 6 1 4 1 23 4 6( ( ) ) ( )xo c s s c c c s s c s s c c c s c

1 23 5 1 23 4 1 4 5 6 1 4 1 23 4 6( ( ) ) ( )yo s s s s c c c s c s c c s c s c

23 4 5 23 4 6 23 4 6( )zo s c c c s s s s c

1 23 4 1 4 5 1 23 5( )xa c c c s s s c s c

1 23 4 1 4 5 1 23 5( )ya s c c c s s s s c

23 4 5 23 5za s c s c c

where, cos( )iic , sin( )iis , ( 1,2 6)i .

2.3 Analysis of inverse kinematics

There is no universal solution to the inverse kinematic equation of robot (Manocha and Canny, 1994) because it is

extremely complicated and can easily produce multiple solutions (Rocha et al., 2011). Since the last three rotating joint

axes of the robot intersect at a single point, the robot meets the Pieper Criterion for this structure (Liu et al., 2015). As

such, the algebratic manipulation with closed method can be used: the first three joints of the robot determine the

terminal position of the robot and the last three joints determine the terminal posture. The calculation is performed as

follows:

Equation (1) is multiplied by 0 1

1T

on both sides, such that,

0 1 6 1 2 3 4 5 1

1 0 2 3 4 5 6 6T T T T T T T T

(2)

namely

4 5 23 5 23 3 4 4 5 23 5 23 61

4 5 23 5 23 23 4 4 5 23 5 23 611

6

4 5 4 4 5 61

0 0 0 1

x

y

z

c c c s s c s c s c c s p

c c s s c s s c c s c c pT

s c c s s p

(3)

In Eq. (3), 61

0z

p ; 4 2361 3 23 2 2ys d cp a s a ; 61 4 23 3 23 2 2xp d s a c a c ;

Letting

44434241

34333231

24232221

14131211

0

6

10

1

rrrr

rrrr

rrrr

rrrr

TT (4)

Equating the corresponding elements at (3, 4) from Eq. (4):

1 1 0x yp s c p , then

1arctg 2( / )y xp p (5)

Multiplying equation (2) by1 1

2T

on both sides,

4

2© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0008]

Xiang, Xie and Lu, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.12, No.1 (2018)

TTTT 2

6

6

0

10

1

11

2 (6)

Letting the elements at (1, 4) and (2, 4) be equal in Eq.6), the value of 2

and 3

is then calculated using Eq. (7)

2 2 2

1 2 22

2 2 2 2

1 2 2 2 1 2

2 2 2

1 2 23

2 2 2 2

1 2 2 1 2 2

arctg[( ) ][ ( ) ]

arctg[ ( ) ][ ( ) ]

e e a

e a e e e a

e e a

e e a e e a

(7)

In Eq. (7), 1arctg( / )z xp p ; 3 4arctg( / )a d ;

2 2

1 1x ze p p ;2 2

2 3 4e a d .

Letting the corresponding elements in (1, 3) and (2, 3) on both sides of equation (6) be equal

543532

543352112 )(

scsccac

sccscasasacc

z

zyx (8)

Then, the value of4

,5

and 6

can be obtained from Eq. (9)

1 1

4

2 3 1 1 23

2

23 1 1 23

5

23 1 1 23

1 1

6

1 1

arctg( )( )

1 [ ( ) ]arctg 2

( )

arctg( )

x y

x y z

x y z

x y z

x y

x y

s a c a

c c c a s a a s

s c a s a c n

s c a s a c n

s n c n

s o c o

(9)

In the actual situation, an inverse solution closest to the current orientation of the robot is usually chosen.

2.4 Calibration of welding robot tool coordinate system

The calibration of the welding robot tool coordinate frame is used to obtain the relationship between tool

coordinate frame {E} and the final link coordinate frame {T}. The machining error of the weld torch itself and the

deviation relative to the end of the manipulator all have an effect on the accuracy of the trajectory planning and must be

taken into account (Wang, 1997; Gao et al.,2014; Wang et al., 1997).

The tool coordinate frame is calibrated with six-point calibration method in this paper (Kang et al., 2016). A robot

tool coordinate frame algorithm is studied, which uses the least squares method to calculate the tool center point (TCP)

position and tool coordinate frame (TCF) orientation.

2.4.1 The position calibration of TCP

The relationship between the base coordinate system {B}, the manipulator end coordinate system {E} and the

welding gun end coordinate system {T} is as follows:

TTT B

T

E

T

B

E (10)

The matrix in Eq. (10) was further expanded to.

101010

T

B

i

B

TT

EE

TEi

B

i

B

E pRpRpR (11)

In Eq.(11), B

E iR is the rotation matrix for four different orientation of the manipulator end (i=1,2,3,4); Ei

B p

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2© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0008]

Xiang, Xie and Lu, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.12, No.1 (2018)

represents the four position vector of robot in the terminal coordinate system, E

TR is rotation matrix of the welding

gun, T

E p is position vector of the welding gun, B

T iR is rotation matrix of four points in the end of the welding gun,

and T

B p is position vector of four points in the tool coordinate system.

In Eq. (11), the relationship of the corresponding elements in (1,2) is as follows:

Tz

B

Ty

B

Tx

B

Ez

B

Ey

B

Ex

B

Tz

E

Ty

E

Tx

E

zzz

yyy

xxx

p

p

p

p

p

p

p

p

p

aon

aon

aon

(12)

Four different points are shown in Fig. 3.

Fig. 3 Calibration points of the robot

The data of the terminal position and posture of the robot is obtained empirically and is shown in Table 2.

Point 1 [°] 2 [°] 3 [°] 4 [°] 5 [°] 6 [°]

Point1 3.5222 25.0937 -8.6107 -12.038 0 0

Point2 -3.5026 25.0856 -14.3244 11.0624 0 0

Point3 0.3496 13.9951 3.6773 -1.1378 -24.000 0

Point4 0.3651 29.3825 -17.5817 -1.1378 8.2001 0

The mathematical relationship between the four calibration points is shown in Eq. (13).

2 1

2 11 2

2 3 2 1

3 4

4 3

B B

Ex Ex

B BEB BEy EyTxE E

B B E B B

E E Ty Ez Ez

B B EE E Tz

B

Ez Ez

p p

p ppR R

R R p p p

R R p

p p

(13)

The least squares method is then used to solve the above matrix equation, and the result is calculated as:

252.2587

158.8522

117.4985

E

Tx

E

Ty

E

Tz

p

p

p

(14)

Table 2 Data for robot at joint angle (point1 ~point 4).

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2© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0008]

Xiang, Xie and Lu, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.12, No.1 (2018)

2.4.2 The posture calibration of TCP

The orientation matrix of TCP is calibrated in the X and Z direction as shown in Fig. 4. Under the same pose, the

robot is first allowed to move 100 mm from calibration point 4 in the +X direction to calibration point 5. Then the robot

is moved 100 mm from calibration point 4 in the +Z direction to calibration point 6.

Fig. 4 Calibration points of the robot in X and Z direction

The data of terminal coordinate system is shown in Table 3.

Point 1 [°] 2 [°] 3 [°] 4 [°] 5 [°] 6 [°]

Point4 0.3651 29.3825 -17.5817 -1.1378 8.2001 0

Point5 0.3844 21.5129 -6.8109 -1.1271 -14.4999 0

Point6 -1.7308 18.1141 -2.4108 -1.1271 -14.4991 0

The direction vector of the tool coordinate system in +X is as follows:

5 4

5 4

5 4

B B

Ex Ex

B B

Ey Ey

B B

Ez Ez

X

p p

p p

p p

(15)

The direction vector of the tool coordinate system in +Z is as follows:

46

46

46

Ez

B

Ez

B

Ey

B

Ey

B

Ex

B

Ex

B

pp

pp

pp

Z (16)

Y-axis vector can be obtained by the right-hand rule: Y Z X . The results of the vectors x, y, and z are as

follows:

0.4498

0.0030

0.8931

X

,

0.8703

0.2259

0.4376

Y

,0.4911

0.3313

0.8057

Z

(17)

Theoretical value of the tool coordinate frame is calculated as follows:

1

0.4498 0.8703 0.4911 1249.79770.0030 0.2259 0.3313 220.25230.8931 0.4376 0.8057 543.6175

0 0 0 1

E B BT E TT T T

(18)

Table 3 Data of robot joint angle (point 4~point 6).

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2© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0008]

Xiang, Xie and Lu, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.12, No.1 (2018)

The actual value of the tool coordinate system is obtained by measuring the size of the welding gun.

1

0.4498 0.8703 0.4911 1249.80760.0030 0.2259 0.3313 220.20210.8931 0.4376 0.8057 543.7025

0 0 0 1

ETT

(19)

Therefore, a better precision can be achieved using this calibration method.

3. Trajectory planning analysis of underwater welding

Trajectory planning methods include the joint trajectory and Cartesian trajectory methods. With joint trajectory

planning, it is necessary to specify the position of the robot at the start and end points. First, the path point is converted

to the joint angle vector value by the inverse kinematics formula. Then, the corresponding track function for each joint

is fitted, which ensures each joint move from the starting point, once through all path points and finally reaches the

target point (Constantinescu and Croft,2000;Saramago and Junior,1998; Bazaz and Tondu,1997).

In this study, the path between the adjacent path points is a straight line in this paper. Although the joint space

method guarantees the calculated path will pass through all points, the spatial path is not a straight line, and therefore,

does not meet the actual welding needs. The Cartesian space method, on the other hand, uses the pose-time function to

describe the path trajectory thereby ensuring the trajectory between each path point as a straight line.

Each path point is determined by the expected posture of the tool coordinate system relative to the table coordinate

system. These path points are described by the transfer matrix of E relative to T without having to compute the inverse

solution.

3.1 Linear interpolation algorithm based on spatial parabolic transition

3.1.1 The representation of robot end orientation

Taking into account that the motion trajectory of the robot is influenced by the underwater environment, the linear

interpolation method with parabola transition is proposed to achieve a stable and accurate trajectory

In the process of linear interpolation with parabola fitting, in the linear part of each segment, three components of

the spatial position are linearly changed, and the end of the welding torch moves in a straight line through space. If the

rotation matrix is used to represent the pose of each path point, then the pose components can not be linearly

interpolated.

Taking into account the above situation, an effective pose representation method can be presented: equivalent axes

coordinate system representation is used to define the torch end posture combined with Cartesian space position vector,

which forms a column vector '' of 6×1.

Suppose there is an intermediate point, its transformation matrix relative to the table coordinate system isS

AT . The

torch end position and posture are defined by AORG

S p and S

AR in the coordinate system {A}.

The rotation matrix S

AT can be expressed by equivalent axes coordinate: ( , )S

A SAR K , thus the column vector

is expressed as: , ,( )S

AORG X Y Zp

S

A

S

AORGS

AK

p

（20）

In Eq. (15),S

AK is the unit direction vector；

SA is the rotation angle value；

S S

A A SAK K .

Each path point is represented by Eq. (20), the component value of S

A can change smoothly from the starting

position to the target position over time. In each linear section, the three components of the position are linearly varied,

so it can be linearly interpolated. The velocity and angular speed at the end of the torch will vary smoothly at each path

point.

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2© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0008]

Xiang, Xie and Lu, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.12, No.1 (2018)

3.1.2 Derivation of trajectory functions

A set of intermediate points of a joint of a robot is represented by . Let l, m, n be three adjacent path points, the

fitting time interval of path point n is mt .The linear segment time interval is

1lmt between point l and m, the total time

interval is lmt between point l and m,

lm is straight line segment speed, andl is fitting segment acceleration at

point l. Fig .5 shows a straight path.

Fig.5 straight path fitting with parabola transition

Known conditions can be obtained from by Eq. (20) such that.

1

SGN

(

( )

(

) /

) /

/ 2 / 2

lm m l lm

m mn lm m

m mn lm m

lm lm m l

t

t

t t t t

(21)

For l=1, m=2, letting the two speed expressions for the linear part of the velocity be equal:

1 1 2 1 121 1( ) / ( / 2)t t t

The path point parameters are obtained as follows:

1 2 1 1

12 2 1 12 1

2

1 12 12 2 1 1

121 12 1 2

SGN( )

(

2( )

) / ( / 2)

/

/ 2

t t

t t t

t t t t

(22)

In trajectory planning, it is only necessary to provide the intermediate point position and duration of each path

segment. The acceleration is set to a default value. The calculation is then performed using the equation (22).

When time t is within the linear segment, the straight-line segment (track 1) is as follows:

0

l lm

lm

t

(23)

The next region of the linear segment is the parabolic region (track 2):

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2© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0008]

Xiang, Xie and Lu, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.12, No.1 (2018)

( / 2 ) ( / 2 ) / 2

( / 2 )

m lm l lm m l lm

lm m l lm

m

t t t t t

t t t

(24)

The position and posture of the torch end at each path point in the Cartesian coordinate system is obtained by Eq.

(23) and (24) and finally the angle of the first three joints is obtained by Eq. (1) and (4).

4. Underwater welding experiment

Programs for solving the robot kinematics trajectory planning program are developed by C++. After planning each

path point position, velocity, and acceleration, local dry welding is used to simulate the underwater experiment in a

water tank of 2.0 m x1.5m x1.8m. The experimental setup is shown in Fig.6. The underwater welding parameters were

got by several experiments, which included an extended wire length of 10 mm, welding current 75-330A, arc welding

voltage 23-33V, welding speed 3mm/s, wire feed speed 3.56m / min, and pressure in the drainage hood of 0.3MPa.

1. The welding robot；2. Welding torch bracket；3.protective gas；4. Welding torch；5. Drainage hood；6. Experimental water tank

Fig.6 Welding experimental platform

4.1 Underwater straight-line welding experiment

Welding object is a 150mm welding seam. First, 11 path points is obtained through artificial teaching, each joint

angle of the robot in the first path point is obtained according to Eq. (6) and (8).

0 3.4035,11.8173, 4.2815, 0.0508, 0.0134, 0[ ]q

Each joint angle in the last point of the welding seam is as follows：

10

11.6548,8.9879, 7.5455, 0.0508, 0.0073, 0[ ]q

The orientation vector of the first point of the welding seam is given by:

0

1008.51, 59.98, 462.12, 0.05, 0.01, 0[ ]

The orientation vector of the last point of the welding seam is given by:

10

[998.73, 206.01, 487.50, 0.05, 0, 0]

1

2

3

4

6

5

10

2© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0008]

Xiang, Xie and Lu, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.12, No.1 (2018)

The orientation vector was obtained by Eq. (1)

8.27

(0.280, 0.053, 0.958)

( 7.484, 150.236, 23.997, 2.318, 0.442, 7.922)

SA

S

AK

（25）

In the welding process, the torch end is perpendicular to the working plane, so posture of the robot remained

unchanged throughout the process. The three components of the position vector AORG

S p varied linearly with time.

Then pose vector of each path point is obtained by Eq. (25) (n=1、2...10).

SAA

S

A

S

n

n

n

KK

znzz

ynyy

xnxx

0

0

0

(26)

The speed of each path segment, parabolic area time and straight-line area time is obtained from Eq. (21). At each

path point, the corresponding joint angle of the welding robot is calculated by Eq. (5), (7) and (9). Values are presented

in Table 4.

Point 1 [°] 2 [°]

3 [°] 4 [°]

5 [°] 6 [°]

Point0 3.4035 11.8173 4.2815 -0.0508 -0.0134 0

Point1 4.5058 11.2383 4.9592 -0.0508 -0.0135 0

Point2 5.3620 10.7546 5.5217 -0.0508 -0.0136 0

Point3 6.0159 10.5562 5.7514 -0.0508 -0.0136 0

Point4 6.8226 10.0965 6.2810 -0.0508 -0.0136 0

Point5 7.5864 10.2461 6.1088 -0.0508 -0.0136 0

Point6 8.4037 9.8211 6.5966 -0.0508 -0.0136 0

Point7 9.2287 9.4146 7.0609 -0.0508 -0.0132 0

Point8 9.9935 9.6139 6.8338 -0.0508 -0.0073 0

Point9 10.8273 9.2430 7.2563 -0.0508 -0.0073 0

Point10 11.6548 8.9879 7.5455 -0.0508 -0.0073 0

The movement of the welding torch is controlled by a set of joint angles and the results are shown in Fig.7.

Fig.7 diagram of underwater straight-line welding

Table 4 Inverse solution of the path point.

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2© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0008]

Xiang, Xie and Lu, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.12, No.1 (2018)

Fig.7 shows a smooth welding seam and suggests linear path planning using vector method results in high

precision and good welding quality.

4.2 Experiment of underwater polyline welding with parabolic transition

Experiments are performed using a cubic polynomial interpolation (Lin, 1983) and linear interpolation method

with parabolic transitions proposed in this paper. The corresponding each joint angle at seven path points is shown in

Table 5.

Point 1 [°] 2 [°]

3 [°] 4 [°]

5 [°] 6 [°]

Point0 3.5765 8.1979 8.4709 -0.5484 -0.0002 0

Point1 4.6003 7.3056 9.4687 -0.5484 -0.0002 0

Point2 5.2843 6.8931 9.9260 -0.5484 -0.0002 0

Point3 6.9614 6.6622 10.1807 -0.5484 -0.0002 0

Point4 8.5474 7.4963 9.2564 -0.5484 -0.0002 0

Point5 10.2176 7.4105 9.3519 -0.5484 0 0

Point6 12.4211 6.0364 10.8672 -0.5484 0 0

The movement of the end of the welding torch is controlled by a set of joint angles presented in Table 5 and the

results are shown in Fig.8.

Fig.8 Curved welding seam

It can be seen from Fig.8 that the initial and final position of Curve1, which are obtained by the cubic polynomial

interpolation, have some defects such as uneven welding, uneven edges and welding time was 39.3s. Curve2 shows a

uniform and smooth welding seam with a shorter welding time of 20s. Therefore, the proposed trajectory control

strategy resulted in a welding curve that is both smoother and neater.

5. Conclusions

In this study, the forward and inverse kinematics equations of the robot are analyzed and simplified by the D-H

parameter method. On the basis of kinematic analysis, a robot tool coordinate frame calibration algorithm- six-point

method is studied to improve the accuracy of the trajectory, which uses least squares to fit tool center point position and

coordinate transformation to calculate tool coordinate frame orientation. Then an effective six-dimensional vector is

proposed to simplify the orientation of the robot, which provides accurate data of path points for the trajectory planning.

A linear interpolation with parabolic transition is deduced, the trajectory planning of the robot in Cartesian space is

achieved by this method. Finally, a set of contrast experiments is performed using a cubic polynomial interpolation (Lin,

1983) and linear interpolation method with parabolic transitions proposed in this paper. The experiments are

implemented by simulating the actual underwater welding process, the result shows Curve2 is smoother and neater than

Curve1, at the same time, the time spent on Curve 2 is 19.3 seconds less than Curve1's. Therefore, it is shown the

proposed motion control strategy is capable of improving the welding quality and shortening the welding time.

Table 5 Inverse solution of the path point.

Curve 1

Curve 2

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2© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0008]

Xiang, Xie and Lu, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.12, No.1 (2018)

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