Laser Detection Manipulator Stability Control Based on Inversion
Control
Yan Ding, Kang Zhao, Zhongwei Ji and Tingrui Liu
*
College of Mechanical & Electronic Engineering, Shandong University of Science & Technology, Qingdao 266590, China
Keywords: Tracking, 3D Modeling, Laser Inspection Manipulator.
Abstract: In this paper, the trajectory tracking control problem of the laser detection manipulator based on inversion
control adjusts the position of the camera under the movement of the 3R manipulator arm. Firstly, the 3D
modeling software was used to build a structural model of the laser detection manipulator and describe
the motion state of the manipulator arm. Secondly, according to the Lagrange dynamic theory, the driving
moment of the three joints of the 3R manipulator arm was derived, the external interfering torque was
introduced, and the dynamic model was established. An inversion controller of the system was then
established based on the 3R manipulator arm dynamic model. Finally, the model was simulated by
MATLAB, and the experimental results show that the control rate can achieve a good trajectory tracking
goal.
1 INTRODUCTION
In recent years, with the continuous progress of science
and technology, people's requirements for machines
are getting higher and higher, the production and
processing accuracy of machine parts is also getting
higher and higher. In the process of parts processing
and parts maintenance, in order to ensure that the
accuracy of parts meets the requirements, it is often
necessary to surface them quality inspection. The
current detection method is mainly based on manual
inspection and sampling inspection, large
interferometers and small detectors. However, many
existing detection instruments have the disadvantages
of large size and immovability, and manual detection
often faces a harsh environment and cumbersome
detection steps, which is seriously affected Parts
processing efficiency and increased labor costs.
Therefore, the research of flexible and high-precision
detection instruments has always been the focus of
attention of experts in this field.
Motion control based on the 3R manipulator arm
will be the key to solving the problem of efficient and
stable detection of laser inspection manipulators. In the
process of movement of multi-joint manipulator arms,
due to the existence of external interference, there will
often be a certain error in the movement of the
manipulator arm, which will cause great trouble to the
detection work requiring higher precision. In recent
years of research at home and abroad, scholars have
proposed some control methods, such as iterative
control methods, sliding mode control methods, fuzzy
control algorithms, etc. (Gao, 2022; Jin, 2018; Tian,
2021) to achieve the goal of anti-interference.
In previous studies, some scholars have divided the
structure into three structures when analyzing the
movement of the manipulator arm: "rigid
linkage-flexible joint", "flexible linkage-rigid joint"
and "flexible linkage-flexible joint" (Wu, 2021). The
"flexible linkage" structure is generally suitable for
long connecting rods and large linkage flexibility of
the mechanical arm. For short and rigid linkages, its
deformation due to its own flexibility during the
movement is negligible. Kanellakopoulos
(Kanellakopoulos, 1991) mentions an inversion
control method that splits a complex system into
simple subsystems that can control the tracking error to
a very small extent, effectively implementing tracking
control on the manipulator arm (Ruan, 2014). In this
paper, a movable integrated laser detection
manipulator is proposed, and the inversion control
method is adopted for its motion control, which
realizes the precise tracking control of the laser
detection manipulator in adjusting the movement of
the camera.
Ji, Z., Ding, Y., Zhao, K. and Liu, T.
Laser Detection Manipulator Stability Control Based on Inversion Control.
DOI: 10.5220/0012150200003562
In Proceedings of the 1st International Conference on Data Processing, Control and Simulation (ICDPCS 2023), pages 109-113
ISBN: 978-989-758-675-0
Copyright
c
 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)
109

(a) Original location (b) Working position
Figure 1: Laser detection manipulator movement process.
Figure 2: Brief diagram of the 3R manipulator arm mechanism.
2 MODELING OF THE
STRUCTURE OF THE LASER
INSPECTION MANIPULATOR
When the laser inspection manipulator performs the
inspection task, it is necessary to adjust the position of
the camera to meet its requirements directly above the
beamsplitter. Figures 1 (a)-(b) depict the process of
simulating the manipulator's movement from its
original position to its working position through UG
modeling.
In the process of laser manipulator moving towards
the workpiece under test, due to inertia and other
external disturbances, the position of each
manipulator arm in the working position often
changes, especially at the three manipulator arms that
control the position of the camera. Therefore, this
paper mainly uses the three manipulator arms as
research objects to analyze the motion and control
problems of laser detection manipulators. A
schematic diagram of the mechanism of the
three-degree self-restraint manipulator arm studied is
shown in Figure 2.
Based on the Lagrange dynamics, the 3R
manipulator arm dynamics equation is established,
and the driving moments of the three moving joints
τ
1
,
τ
2
and
τ
3
the sum, are as follows:
()()cos()()sin()
cos( ) sin( ) ( ) cos
mmmr mmrr mmrr
mrr mrr m m m gr
τθθθθθθθ
θθθ θθθ θ
=++ ++ −++ −
+−+−+++
2 2
1 12311 23122 12 23122 12
2
313 3 1 3 313 3 1 3 1 2 3 1 1
  
 
(1)
o
1
θ
2
θ
3
θ
3
m
2
m
1
m
1
r
2
r
3
r
x
y
ICDPCS 2023 - The International Conference on Data Processing, Control and Simulation
110

()()cos()()sin()
cos( ) sin( ) ( ) cos
m mr m mrr m mrr
mrr mrr m m gr
τθ θθθ θθθ
θθθ θθθ θ
=+ ++ −−+ −
+−+−++
2 2
2232223121 12 23121 12
2
323 3 2 3 323 3 2 3 2 3 2 2
  
 
(2)
cos( ) sin( ) cos( )
sin( ) cos
mr mrr mrr mrr
mrr mgr
τθ θθθ θθθ θθθ
θθθ θ
= + −− −+ −
−−+
22
3 33 3 313 1 1 3 313 1 1 3 323 2 2 3
2
323 1 2 3 3 3 3
   

(3)
Considering the interference moment, apply the kinetic equation Eq. (1)-Eq. (3) Simplified to:
NNNe
MI DI K ττ +++=
 
(4)
where
N
M
is the inertia matrix,
N
D
is the
centrifugal force and the Gossonian force vector,
N
K
is the gravitational vector,
e
τ
is the
interference moment vector,
τ
is the joint driving
torque vector,
I
is the vector of the joint angle
variable.
3 CONTROLLER DESIGN
3.1 Equations of State Space
Kinetic equation Eq. (4) Consider
[
]
1123
T
I θθθ =
and
2123
T
I θθθ

=

 
,
then the equation of its state-space form is:
12
21
II
I
UI R P Cu

=


=− − − +




(5)
where
1111
,,,
N
NNNNeN
UMDRMKPM CM τ
−−−−
====
,
u τ =
.
3.2 Invert the Controller Design
By introducing the desired trajectory, designing an
inversion controller, and controlling the change of the
driving torque, the purpose of tracking control is
achieved.
Set the error vector of the system
E
to:
[]
11
1123 2 2
33
d
T
d
d
i
EAI ee e i
i
θ
θ
θ
−


=−= = −


−

(6)
where
[
]
123
T
ddd
Ai i i=
is the desired state
trajectory.
The error vector is derived and combined with Eq.
(5) to obtain:
12
E
AI AI=−=−


(7)
Build Lyapunov function:
1
1
2
=
T
VEE
(8)
Derive it and bind it to Eq. (7) to obtain:
()
12
TT
VEEEAI== −


(9)
According to the Lyapunov stability principle,
1
V
is positive and
1
V

is negative.
21
A
IkE−=−

(10)
where
1
0>k
.
combine it with Eq. (9) to obtain:
11
0=− <

T
VkEE
(11)
Set the error vector of the system as:
2
F
HI=−
(12)
Make the second desired trajectory is
2
H
I=
,
combine it with Eq. (10) to obtain:
1
H
AkE=+

(13)
Build Lyapunov function:
21
1
2
=+
T
VV FF
(14)
Derive it and bind it to Eq. (11) to obtain:
Laser Detection Manipulator Stability Control Based on Inversion Control
111

𝑉


= −𝑘

𝐸

𝐸 + 𝐹

𝐸 + 𝐹

 (15)
According to Lyapunov's stability theorem, we can
know
2

V
is negative.
𝐸 + 𝐹

= −𝑘

𝐹 (16)
where
2
0>k
and consider Eq. (15) to obtain:
𝑉


= −𝑘

𝐸

𝐸−𝑘

𝐹

𝐹 <0 (17)
The controller system meets the stability
conditions of Lyapunov, when the desired trajectory
is
A
set to the origin, and the asymptotic
F
is stable
at the origin, it can be guaranteed that
2
I
and
1
I
approache tracking trajectory gradually.
Derive Eq. (12) and bind it to Eq. (5) and Eq. (16)
to obtain:
𝐹

= 𝐻

−𝐼


= 𝐻

−−𝑈𝐼


−𝑅−𝑃+ 𝐶𝑢 = −𝐸 −
𝑘

𝐹 (18)
Then the inversion control rate is:
𝑢 = 𝐶

𝐸 + 𝐻

+ 𝑘

𝐹 + 𝑈𝐼


+ 𝑅 + 𝑃 (19)
4 SIMULATION ANALYSIS
Adding the inversion control rate obtained above, this
article uses the 3R manipulator arm shown in Figure 2
as the control object for MATLAB simulation
analysis (Xue, 2007; Liu, 2016).
The structural parameters of the 3R manipulator
arm are shown in Table 1.
Suppose the desired trajectory of the three joints is
𝐴 =

𝑠𝑖𝑛( 𝑡) 𝑠𝑖𝑛( 𝑡) 𝑠𝑖𝑛( 𝑡)


, the interference
torque is 𝜏

=

1.2 𝑠𝑖𝑛( 𝑡)1.2𝑠𝑖𝑛( 𝑡)1.2𝑠𝑖𝑛( 𝑡)


, and the
controller controls the parameters are 𝑘

= 𝑘

=
1~100. The simulation results are shown in Figure 3.
Table 1: 3R manipulator arm structure parameters.
Joint quality length Rotation angle
Joint 1
1.0kg
0.5m
1
θ
Joint 2
1.0kg
0.5m
2
θ
Joint 3
3.0kg
0.5m
3
θ
(a) Angular displacement trajectory tracking (b) Angular velocity trajectory tracking
Angle of Link1Angle of Link2Angle of Link3
Angle speed of Link1Angle speed of Link2Angle speed of Link3
ICDPCS 2023 - The International Conference on Data Processing, Control and Simulation
112

(c) Angular displacement tracking error (d) Angular velocity tracking error
Figure 3: Simulation diagram of trajectory tracking of 3R manipulator arm based on inversion control.
From the simulation analysis diagram, it can be
seen that the inversion controller can effectively
control the manipulator arm to track the desired
trajectory, the angle tracking and angular velocity
tracking effect of joint 1, joint 2 and joint 3 is
obviously very good, the angle tracking error can
always be controlled within 0.024𝑚, and the angular
speed tracking error can be controlled within
0.015𝑚 . Inside, the controller can effectively
overcome external interference, so that the laser
inspection manipulator can smoothly and accurately
carry out inspection operations.
5 CONCLUSION
In this paper, the upper three arms of the laser
detection manipulator are used as the research object,
and the dynamic equation of the 3R manipulator arm
is established based on the Lagrange dynamic theory,
and the inversion control method is adopted, the
inversion controller of the motion system is designed,
and its stability is verified. The motion control of the
3R manipulator arm is simulated by MATLAB
simulation software, and the simulation results show
that the controller designed in this paper can
effectively achieve the goal of trajectory tracking and
stabilize the error within the effective range. This
testifies the feasibility of the control law of the design,
and provides a rich theoretical basis for the future
research on improving the detection accuracy and fast
and smooth motion of laser detection manipulators.
ACKNOWLEDGEMENT
he authors gratefully acknowledge the support of the
National Natural Science Foundation of China (no.
51675315).
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Ruan Y, Wang W 2014 Journal of Minjiang University 35
67-73.
Wu K 2021 Taiyuan University of Technology.
Xue D 2007 MATLAB Solutions to Mathematical Problems
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Liu J 2016 Robot Control System Design and MATLAB
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Tsinghua University Press) chapter 6 pp 152–162.
0 5 10 15 20
time(s)
-0.5
0
0.5
Angle error of Link1
Results of angular displacement error
0 5 10 15 20
time(s)
-0.5
0
0.5
Angle error of Link2
0 5 10 15 20
time(s)
-0.5
0
0.5
Angle error of Link3
0 5 10 15 20
time(s)
-0.5
0
0.5
Angle speed error of Link1
Results of angular speed error
0 5 10 15 20
time(s)
-0.5
0
0.5
Angle speed error of Link2
0 5 10 15 20
time(s)
-0.5
0
0.5
Angle speed error of Link3
Laser Detection Manipulator Stability Control Based on Inversion Control
113