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: [email protected]
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.
11
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)
References
Bingtuan Gao., Yong Liu., Ning Xi., Yantao Shen. and Hesheng Wang, Developing an Efficient Calibration System for
Joint Offset of Industrial Robots, Journal of Applied Mathematics(2014).
Bazaz, SA. and Tondu, B, On-line Computing of a Robotic Manipulator Joint Trajectory with Velocity and Acceleration
Constraints, Proceedings of the IEEE International Symposium on Assembly and Task Planning, No.97(1997),
pp.1-6.
Chen, YC., Solving robot trajectory planning problems with uniform cubic B-splines, Optimal Control Applications&
Methods, Vol.12, No.4(1991), pp.247-262.
Constantinescu, D. and Croft, EA, Smooth and time-optimal trajectory planning for industrial manipulators along
specified paths, Journal of Robotic Systems, Vol.17, No.5(2000), pp.233-249.
Denavit J. and Hartenberg R S., A kinematic notation for lower pair mechanisms based on matrices, Journal of Applied
Mechanics, Vol.21, No.5(1995), pp.215-221.
Kang Cunfeng,.Wang Hongwei,. Zhang Pengfei., Li Shujin. and Chen Shujun, Study and Realization of Tool
Coordinate Frame Calibration for Welding Robots, Journal of Beijing University of Technoloy, Vol.42,
No.1(2016), pp.30-34 (in Chinese).
Labanowski, J., Development of under-water welding technique, Welding International, Vol.25, No.12 (2011),
pp.933-947.
Liu Peng, Song Tao, Yun Chao. and Gao Zhihui., Study of kinematics analysis and trajectory planning for welding
robot, Journal of Mechanical & Electrical Engineering, Vol.30, N0.4(2013), pp.390-394(in Chinese).
Liu, HS. Zhang, Y. and Zhu, SQ., Novel inverse kinematic approaches for robot manipulators with piper criterion based
geometry, International Journal of Advanced Robotic Systems, Vol.13, No.5(2015), pp.1242-1250.
Lin,C.-S, Chang P-R. and Luh, J.Y.S, Formulation and optimization of cubic polynomial joint trajectories for industrial
robot, IEEE Transactions on Automatic Control, Vol.28, No.12(1983), pp.1006-1074.
Martin, N. and Bertol, D., Neural control applied to the problem of trajectory tracking of mobile robots with
uncertainties, 10th Brazilian Symposium on Neural Networks (2008), pp.117-122.
Manocha, D. and Canny, J.F., Efficient inverse kinematics for general 6R manipulators, IEEE Transactions on Robotics,
Vol.10, No.5 (1994), pp.648-657.
Peng Yuqing., Li Mu. and Zhang Yuanyuan, Mobile Robot obstacle avoidance based on improved fuzzy algorithm,
Journal of Computer Applications, Vol.35, No.8(2015), pp.2256-2260.
Rowe, M. and Liu, S., Recent developments in underwater wet welding, Science and Technology of Welding and
Joining, Vol.6, No.6 (2001), pp.387-396.
Rai, Jaynendra Kumar. and Tewari, Ravi., Quintic polynomial trajectory of biped robot for human-like walking, 6th
International Symposium on Communications, Control and Signal Processing (ISCCSP) (2014), pp.360-363.
Rocha, CR. Tonetto, CP. and Dias., A comparison between the Denavit–Hartenberg and the screw-based methods used
in kinematic modeling of robot manipulators, Robotics and Computer Integrated Manufacturing, Vol.27,
No.4(2011), pp.723-728.
Sanpeng Deng, Zhongmin Wang, Peng Zhou, and Hongbing Wu.,Research on Path Planning of Mobile Robot Based
On Genetic Algorithm, Advanced Materials Research,Vol.819(2013), pp.379-383.
Schroer, B.J.and Rezapour, A., Calibration of Robots Used in High Precision Operations, Robotics, Vol.4, No.2 (1988),
pp.131-143.
Saramago, SFP. And Junior, VS, Optimal trajectory planning of robot manipulators in the presence of moving obstacles,
Mechanism and Machine Theory, Vol.35, No.8(2000), pp.1079-1094.
Wang, W.,Wang, G. and Yun, C, A calibration method of kinematic parameters for serial industrial robots, Industrial
Robot-An International Journal, Vol41, No.2 (1997), pp.157-165.
Xuguang Wang. and Red, E, Robotic TCF and rigid-body calibration methods, Robotica, Vol.15(1997), pp.633-644.
Yang, SX. and Max Meng., Real-time Collision-free Path Planning of Robot Manipulators using Neural Network
Approaches, Autonomous Robot, Vol.9, No.1(2000), pp.27-39.
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