Kinetic Modeling and Simulation of Traditional

Chinese Medicine Packing Robot

Sun Wencheng1 ,Tao Yong2* ,Gao Jinpeng3
1. School of Mechanical and Electrical Engineering, ChongQing University of Arts and Sciences, Chongqing, China.
2. School of Mechanical Engineering & Automation, Beihang University, Beijing, China.
3. College of machinery and automation, Wuhan University of Science and Technology, Wuhan, China.
* e-mail: taoy@buaa.edu.cn

Abstract—Due to its advantages, the parallel robot has wide and slave arm and the radius of static and mobile platforms, we
application in the traditional Chinese medicine packing process. analyze the kinetic influence of parallel packing robot.
The robot dynamics is the basis of robot motion control, and the
establishment of accurate kinetic model plays an important role.
The dynamics model of the traditional Chinese medicine packing
robot is established based on the Lagrange Equation. The motion II. INTRODUCTION OF THE STRUCTURE OF PARALLEL
of medicine packing robot with the plane is proposed. The PACKING ROBOT
influence of load and the structural parameters of traditional
Chinese medicine packing robot on the driving joint torque are
This paper studies the parallel packing robot, as shown by
presented, which can provide reference to the design selection of the model in Fig.1. This robot is an enclosed structure
parallel mechanism and the optimized design of the main consisting of the two components of base (static platform) and
parameters of packing robot. moving platform (mobile platform), and these two platforms
are connected by the branched structure consisting of 3 master
Keywords—traditional Chinese medicine; parallel packing arms and slave arms. The master arm is connected to the
robot; kinetic; Parameter Design reducer installed on the static platform through revolute pair,
and it uses the servo motor equipped with the reducer to drive
I. INTRODUCTION the master arm for rotation, in this way to realize the translation
of platform. The salve arm consists of two parallel bars, and in
Due to its advantages, parallel robot has wide application in order to prevent it from falling off, the parallel bar is fixed with
the industrial packing process[1], and the high-precision tension spring. Its workspace is a cylindrical area, which is
control research is the basic research of its application. High- mainly responsible of grabbing the material from one moving
precision control requires focusing solving the kinetic control platform to another moving platform.
of robot, while the primary to conduct kinetic control is to
build a kinetic equation with efficient algorithm[2]. There are
mainly three methods that can be used for the kinetic modeling
of parallel robot: the Newton-Euler Equation, Lagrange
Equation and Kane Equation[3]. Dasgupta and Mruthyunjaya[4]
considered the component weight and the friction at joint, and
adopted the Newton-Euler Equation to build the kinetic
equation of Stewart platform; Liu Shanzeng et al.[5] built the
kinetic model of 3-RRS parallel robot based on the Lagrange
equation, and analyzed the kinetic characteristics of this
parallel robot; based on the Kane Equation, in Literature[6], the
kinetic model of parallel robot was built.
With the in-depth research on parallel robot, more and
more researchers are studying its kinetic control. Most existing
researches built the parallel robot based on the principle of
virtual work, or study the material and flexibility of bar, or
mainly study the high-precision control, and there are very few
researches on the impact of the change of various parameters Fig. 1. Structure of parallel packing robot.
on the kinetics of parallel robot.
The sketch of one single branched chain of robot is as
Based on the Lagrange Equation method, this paper
shown in Fig.2, and the parallelogram mechanism of slave arm
conducts kinetic modeling of parallel packing robot by
is equivalent to a single bar that passes the centers of the upper
introducing the generalized coordinates and Lagrange
and lower sides. Build the coordinate system o-xyz with the
multipliers, which has simplified the kinetic equation. In
geometric center of static platform as the origin of coordinates;
addition, by changing the load weight, the length of master arm
build the coordinate system p-x’y’z’ with the geometric center

978-1-4673-8979-2/17/$31.00 ©2017 IEEE 1916

of mobile platform as the origin of coordinates; the static The kinetic energy of robot includes the kinetic energy
platform has a radius of R, the mobile platform has a radius of r, of mobile platform and three branched chains.
the length of master arm is l1, the length of slave arm is l2, and 1) Kinetic force of mobile platform
the included angles between the three master arms and the The weight of mobile platform is m0, so the kinetic
coordinate axis are
D1 , D 2 and D 3 respectively. energy of mobile platform can be expressed as:
p ,p ,p 1
The x y z is set to the position of the moving platform K0 m0v p p 2x  p 2x  p 2x (2)
2
center point P in the o-xyz coordinate system.
v p refers to the speed of mobile platform.
2) Kinetic energy of branched chains
The kinetic energy Kli of mechanical arms in k-group includes
the kinetic energy of master arm K1i and the kinetic energy of
slave arm K2i:
Kli K1i  K2i (3)
The master arm only rotates around the motor shaft, the
1 2
weight is m1, and the moment of inertia is Ji mi li , its
3
kinetic energy is:
1 1
K1i J lZ12i J lTi2 (4)
2 2
The kinetic energy of slave arm includes the translation
kinetic energy and the rotational kinetic energy around the
centroid:
1 1
Fig. 2. Sketch of one single branched chain of robot K 2i m2v22i  J 2Z22i (5)
2 2
In which, m2 includes the weight of slave arm and the
III. KINETIC MODELING OF PARALLEL PACKING ROBOT weight of globe joint. The moment of inertia of slave arm
By using the Lagrange Equation to conduct kinetic 1
modeling of parallel packing robot, the obtained kinetic bypassing the centroidal axis is: J2 m2l22 .
equation is simple and direct, which will be convenient for the 12
analysis of robot. For the system with n degrees of freedom, in (2) Solution of system potential energy
principle, n independent generalized coordinates are adequate The system potential energy includes the potential energy
to build the Lagrange Equation of system. However, the of mobile platform and three branched chains of mechanical
parallel packing system has complicated structural control, arms. The surface of static platform is defined as the zero-
therefore, during the kinetic modeling in this paper, M-n potential energy surface, so the potential energy of mobile
independent generalized coordinates were added, and in the platform can be defined as:
meantime, we also added M-n Lagrange multipliers. These p1 m0 gpz (6˅
added generalized coordinates are related to the original n The potential energy of each branched chains of
independent generalized coordinates [7]. The Lagrange mechanical arms is:
Equation can be written as:
1 1
d § wL · wL r
wF p2i  m1 gl1 sin Ti  m2 g pz  l1 sin Ti (7)
¨ ¸¸  W j  ¦ Oi i j 1, 2,..., M (1) 2 2
dt ¨© wq j ¹ wq j i 1 wq j (3) Kinetic modeling
The Lagrange Function is the difference between system
In which, M refers to number of generalized coordinates;
kinetic energy and potential energy:
Fi q1 , q2 ,..., qM ; t 0 (i=1, 2,...,r) refers to r constraint 3
§ 3
·
r
wFi L K 0  ¦ Kli  ¨ p1  ¦ p2i ¸ (8)
equations. ¦ Oi
i 1 wq j
refers the generalized force of i 1 © i 1 ¹
Substitute Formulas (4-7) into Formula (8), we can
constraining force corresponding to the generalized coordinate obtain the Lagrange Function:
qj O
, and i is the Lagrange multiplier.
(1) System kinetic energy solution of parallel packing robot

1917
L
1
2
1 1
m0 p 2x  p 2y  p 2z  J1 T12  T 22  T32  ª¬T12l12  T 22l12  T32l12  3 p 2x  p 2y  p 2z
2 6
external force are W 4 ,W 5 ,W 6 . Substitute the differential
l1T1 pz cos T1  l1T1 px cos D1 sin T1  l1T1 p y sin D1 sin T1  l1T 2 pz cos T 2  l1T 2 px cos D 2 sin T 2
obtained above to Formula (10), eliminate
Oi , and we can
l1T 2 p y sin i T 2  l1T3 pz cos T3  l1T3 px co
i D 2 sin os D 3 sin
cos i T3  l1T3 p y sin i T3 º¼  m0 gpz
i D 3 sin
obtain the torque of three driving joint:
1 1 1 1 1
 m1 gl1 sin T1  m1 gl1 sin T 2  m1 gl1 sin T3  m2 g pz  l1 sin T1  m2 g pz  l1 sin T 2
2 2 2 2 2
1 ª d § wL · wwL Lº ª d § wL · wL º
 m2 g pz  l1 sin T3 « ¨ ¸ » « ¨ ¸ »
« dt © wT1 ¹ wT1 » « dt © wpx ¹ wpx »
2
(9) ªW 4 º « « § »
»
For the parallel packing robot, its generalized coordinates « » « d § wL · wwL L» T 1 « d wL · wL » (12)
W «W 5 » ¨ ¸ J ¨ ¸
are three joint vectors T1 ,T 2 ,T3 ,they are independent. Add « dt wT wT 2 » « dt ¨ wp y ¸ wp y »
«¬W 6 »¼ « © 2 ¹ » « © ¹ »
three generalized coordinates px , p y , pz , they can be « d § wL · wL » « d § wL · wL »
« dt ¨ wT ¸  wT » « ¨ ¸ »
expressed as the function of T1 ,T 2 ,T3 through the kinetic ¬ © 3¹ 3 ¼ ¬« dt © wpz ¹ wpz ¼»
equation, so they are not independent. The added generalized Into the standard form:
coordinates are: W M q q  C q, q q  G q (13)
q { px , py , pz ,T1 ,T2 ,T3} In which:
Therefore, Lagrange Equation (1) can be written as the 1 1 1 1
M q m1  m2 l12 E  m2l1 J T 1P  m0  m2 JJ T  m2l1PT J 1
following 6 equations: 3 6 6
­ d § wL · wL § x· 1 1
3
wF C ¨ q, q ¸ m0  m2 J T 1 J 1  m2l1PT J 1  m2l1 J T 1P
° ¨ ¸ W 1  ¦ Oi i © ¹ 6 6
dt
° © x¹ w p w p x i 1 wpx
° § ª º
·
° d ¨ wL ¸  wL W  O wFi « »
3

¦ 0 ª cos T1 º
° dt ¨ wp y ¸ wp y T 1 « » 1 « »
2 i
wp y
° © ¹ G q J « 0 »  2 m1  m2 gl1 «cos T 2 »
i 1

° d § wL · wL 3
wF (10) « 3 » «¬ cos T3 »¼
° ¨ ¸ W 3  ¦ Oi i « m0 g  m2 g »
° dt © wpz ¹ wpz i 1 wpz ¬ 2 ¼
®
° d § wL · wwL L 3
wFi ª  cos D1 sin T1  cos D 2 sin T 2  cos D 3 sin T3 º
° dt ¨ wT ¸  wT W 4  ¦ Oi wT
° © 1¹ 1 i 1 1 P «« sin D1 sin T1 sin D 2 sin T 2 sin D 3 sin T3 »»
° d § wL · wwL L 3
wF «¬ cos T1 cos T 2 cos T3 »¼
° ¨ ¸ W 5  ¦ Oi i
dt
° © 2¹ wT wT 2 i 1 w T2 xx
° M(q) is the inertia matrix, M q q represents the inertia
° d § wL ·  wL W  O wFi
3

¨
° dt wT ¸ ¦ § x· § x·x
¯ © 3 ¹ wT3 wT3
6 i
i 1 C ¨ qˈ
q¸ C ¨ q, q ¸ q
force, © ¹ is the Coriolis matrix, © ¹ represent the
Solve the partial differential of the generalized coordinate
derivative in Lagrange Function (9), obtain the time derivative, centrifugal force and Coriolis force respectively, and G(q) is
and then obtain the partial differential of generalized the gravity matrix. Formula (13) is the second-order vector
coordinate. differential equation, which is the function of joint torque, and
In accordance with the geometrical relationship, we can it is the basic equation of kinetic control based on Lagrange
Equation.
obtain the following three constraint equations of robot:
2
F1 px  r cos D1  R cos D1  l1 cos D1 cos T1 IV. NUMERICAL SIMULATION
 p y  r sin D1  R sin D1  l1 sin D1 cos T1 2
 pz  l1 sin T1
2
l 2
2 0 During the packing process, the robot needs to generate
2
instantaneous acceleration and speed, which has very high
F2 px  r cos D 2  R cos D 2  l1 cos D 2 cos T 2 requirement for the kinetic performance of robot. The torque
 p y  r sin D 2  R sin D 2  l1 sin D 2 cos T 2 2
 pz  l1 sin T 2
2
 l22 0 of three joints is the basis to generate kinetics, and the change
2
rules of torque during motion should be studied to adjust the
F3 px  r cos D 3  R cos D 3  l1 cos D 3 cos T3 change of motion. Numerical simulation can well present the
 p y  r sin D 3  R sin D 3  l1 sin D 3 cos T3 2
 pz  l1 sin T3
2
 l22 0 change of torque at any moment, so we conduct numerical
analysis of the kinetic model built by us.
(11)
During the design of packing robots, there are seven
Then, obtain the differential of generalized coordinate in
variations that could affect the joint torque: the four sizes
Constraint Equations (11).
shown in Fig.2, the radius R of static platform, the radius r of
When the mobile platform of robot is not under any
mobile platform, the master arm length l1 and the slave arm
external force,
W 1 ,W 2 ,W 3 are all zero, and the generalized length l2; as well as three weights: the mobile platform weight

1918
m0, the master arm weight m1, and the slave arm and globe
joint weight m2. The seven parameters of designed packing
robot are as shown in TABLE I.

TABLE I. THE PARAMETERS OF THE ROBOT

par num
R/mm 350
r/mm 100
l1/mm 200
l2/mm 600
m0/kg 0.8
m1/kg 1.34
m2/kg 0.27 Fig. 4. Curve of joint torques changing with time under load of 2kg

This paper simulated the motion of packing robot with When the load is 2kg, operate the same trajectory, and we
the plane and provided a simple trajectory plane, and through can obtain the curve of driving joint torques changing with
simulation with simulation software, the curve of the impact time, as shown in Fig.4. Through comparison, we find that the
of the packing robot parameters on the three driving joints was torque is higher with load than during dry running, which is
drawn. The effects of various parameters on the joint torque consistent with the actual situation.
were compared and analyzed. In order to simplify the When the load is 2kg, change the length of the master
computation, we made the mobile platform make circular arm and slave arm of packing robot, conduct analysis, and we
movement with a radius of 0.08m on the plane, and with the can obtain Figs.5 and 6.
master arm at level condition as the criterion, we provide the
following equation of motion[8]:
­ px 0.08cos 2S t
°
® p y 0.08sin 2S t (14)
° p 0.51
¯ z
During dry running, the curve of the torques of three
driving joints changing with time is as shown in Fig.3. In
accordance with Fig.3, we can see that the torques of three
driving joints all present regular change, which is related to
the input motion equation of mobile platform and consistent
with the circular motion with a cycle of 2π made by mobile
platform on the plane. The torques of three driving joints Fig. 5. Curve of uniaxial joint torque changing with time under different
master arm length
present a phase difference of 120 degree, which is consistent
with the 120 degree symmetrical distribution of three driving
joints.

Fig. 6. Curve of uniaxial joint torque changing with time under different
slave arm length
Fig. 3. Curve of joint torques changing with time during dry running
In accordance with Figs.5 and 6, we can see that the
torque presents positive correlation with the length of master
arm, while negative correlation with the length of slave arm.
After changing the radiuses of static and mobile
platforms, we obtain Figs.7 and 8. The torque presents

1919
negative correlation with the radiuses of static and mobile V. CONCLUSIONS
platforms. In this paper, based on the Lagrange method, we
established the kinetic equation of parallel packing box, used
simulation software to simulate the kinetic model of parallel
packing box, proved the accuracy and effectiveness of
Lagrange method, and analyzed the influence of load and the
structural parameters of robot on the driving joint torque,
which can provide reference to the design selection of parallel
mechanism and the optimized design of the main parameters
of packing robot.

ACKNOWLEDGMENT
This work is supported by Beijing Science and Technology
Fig. 7. Curve of uniaxial joint torque changing with time under different Ministry project: Research and Development of Composite
radius of static platform Navigation Automatic Guided Vehicle (AGV) and Automatic
Packing Robot, No: D151100001315002.

REFERENCES
[1] Yang Binjiu,Cai Guangqi.Dynamics simulation of 3-TPT parallel
robot.Journal of HarBin Institute of Technology,vol.11,2009,pp:103-105.
[2] Huang Zhen,Kong Lingfu,Fang Yuefa.Theory and control of parallel
robot mechanism.Bei Jing:China Machine Press,1997.
[3] Yang Yiyong.Mechanical system dynamics control.Bei Jing:Tsinghua
University Press,June,2009.
[4] Dasgupta,B.T.Mruthyunjaya,A Newton-Euler formulation for the
inverse dynamics of the Stewart platform manipulator.Mechanism and
Machine Theory,vol.8.1998.pp:1135-1152.
[5] Liu Shanzeng,Yu Yueqing,Si Guoning etc.Kinematic and Dynamic
Analysis of a Three degree of freedom Parallel Manipulator.Journal of
Fig. 8. Curve of uniaxial joint torque changing with time under different
Mechanical Engineering,vol.8.2009.pp:11-17.
radius of mobile platform
[6] Zhang Guo-wei, Song Wei-gang.A Kane Formulation for the Inverse
Dynamic of Stewart Platform Manipulator.Journal of System
Through analysis of curve, we can see that during design of Simulation,vol.7.2004.pp:1386-1391.
packing robot, if the overall weight of mobile platform and [7] Qiao Wengang,Ding Jin,Zhao Jingsi.Dynamics Modeling and
load are fixed, if the weight of mobile platform can be reduced, Simulation Analysis of 3-DOF Parallel Robot.Tool
it means that the weight of load can be increased. Under the Engineering,vol.50.2016.pp:49-52.
situation of satisfying various conditions such as the workspace [8] Tian Tao.Design and Implementation of a High-Speed Pick and Place
and flexibility, in order to make the motor work under Parallel Robot.Dalian University of Technology.June.2013.
optimum condition, a structure with smaller master arm length
l1 should be designed.

1920