Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms

Vietnam Journal of Mechanics, VAST, Vol. 41, No. 2 (2019), pp. 141 – 155
DOI: https://doi.org/10.15625/0866-7136/13073
KINEMATIC AND DYNAMIC ANALYSIS OF A SERIAL
MANIPULATOR WITH LOCAL CLOSED LOOP
MECHANISMS
Chu Anh My∗, Vu Minh Hoan
Le Quy Don University of Technology, Hanoi, Vietnam
∗E-mail: mychuanh@yahoo.com
Received: 05 September 2018 / Published online: 01 April 2019
Abstract. This paper addresses the kinematic and dynamic modelling and analysis for a
robot arm consisting of two hydraulic cylinders driving two revolute joints of the arm. The
two cylinders and relevant links of the robot constitute two local closed kinematic chains
added to the main robot mechanism. Therefore, the number of the generalized coordinates
of the mechanical system is increased, and the mathematical modelling is more complex
that requires a formulation of constraint equations with respect to the local closed chains.
By using the Lagrangian formulation with Lagrangian Multipliers, the dynamic equations
are first derived with respect to all extended generalized coordinates. Then a compact
form of the dynamic equations is yielded by canceling the Multipliers. Since the obtained
dynamic equations are expressed in terms of independent generalized coordinates which
are selected according to active joint variables of the arm, the equations could be best
suitable for control law design and implementation. The simulation of the forward and
inverse kinematics and dynamics of the arm demonstrates the motion behavior of the
robot system.
Keywords: hydraulic robot; robot kinematics; robot dynamics; local closed mechanism.
1. INTRODUCTION
Most of industrial manipulators commonly used in industries are usually actuated
by electric motors, such as the welding robots, the assembly robots, etc. The use of elec-
tric motors actuating active robot joints possesses several advantages: easy to control,
high positioning accuracy, and high flexibility. However, if a manipulator is designed to
operate in a large workspace with high loading capability, the use of electric motors for
the design could lead to a very heavy architecture of the robot. Counterweights could
be added to balance to shaking forces. In that case, hydraulic cylinders driving robot
joints is often used for the design. The presence of hydraulic cylinders increases the stiff-
ness of robot structure so that the robot is capable of handling heavy parts in a larger
operational space. Moreover, the counterweights could be avoided since the cylinders
actuating revolute joints play a role of auxiliary links appended to the main structure.
c© 2019 Vietnam Academy of Science and Technology
142 Chu Anh My, Vu Minh Hoan
Though there will be advantages when using hydraulic cylinders for robot designs, the
presence of cylinders in a robot architecture involves complex procedures for the math-
ematical modelling, analysis and control. The addition of cylinders to the conventional
serial kinematic chain of robot arm architecture could constitute local closed kinematic
chains within the entire robot mechanism. Issues related to the hybrid serial–parallel fea-
ture of the robot structure, the geometry, the mass and the inertia of cylinders must be
taken into account.
In the literature, numerous works have been carried out to investigate several as-
pects related to the dynamic modeling and analysis of serial manipulators and paral-
lel robots [1–21]. The fundamentals of kinematic and dynamic modelling and analysis
of serial manipulators can be found in [1, 2], where Denavit-Hartenberg approach was
mostly used for the kinematic modelling and D’Alembert–Lagrange Formulation for the
dynamic modelling. As for more complex robotic systems, there has been a number of
researches dealing with different issues related to the kinematics and the dynamics as
well. The research presented in [3] studied the kinematic and dynamic modelling for
closed chain manipulators, [4] addressed algorithms for the dynamic analysis of serial
robots having a large number of joints, [5] investigated the inverse kinematics and dy-
namics of the redundant robots, [6] studied the dynamics of mobile serial manipulators.
In parallel with researches concerning with the serial robot dynamics, a massive number
of researches related to the dynamics and control of parallel robots has been addressed
as well such as publications [7–21]. The researches [8,10,13] investigated methods for the
inverse and forward dynamic modelling and analysis of the 3-PRS type parallel manipu-
lators. The Screw theory was used in [9], and the matrix approach was employed in [11]
for the dynamics and control of the parallel robots. A general solution to the problem of
dynamic modelling and analysis of parallel robots was presented in [12]. For the issue of
control law design and development, [14] investigated a model-based technique, [15–17]
addressed the robust control algorithms, whereas the s ... havior of the robot system, the DAEs (10, 11) need
to be transformed in a way that the Multipliers are cancelled.
Rewrite Φs =
∂f
∂s
in the following form
Φs =
[
Φq Φz
]
=
[
∂f
∂q
∂f
∂z
]
. (15)
Let’s consider the following expression
RT =
[
E,−ΦTq
(
Φ−1z
)T]
. (16)
Hence
RTΦTs = 0. (17)
Note that
ΦTs =

0 0 0 0
− cos θ3 − sin θ3 0 0
0 0 − cos θ6 − sin θ6
−l22 sin θ2 l22 cos θ2 0 0
l3 sin θ3 + d4 sin θ3 −l3 cos θ3 − d4 cos θ3 0 0
0 0 −l51 sin θ5 l51 cos θ5
0 0 l6 sin θ6 + d7 sin θ6 −l6 cos θ6 − d7 cos θ6

,
(18)
Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms 151
R =

1 0 0
0 1 0
0 0 1
0 − 1
l22 sin (θ2 − θ3) 0
0 − cos (θ2 − θ3)
(d4 + l3) sin (θ2 − θ3) 0
0 0
1
l51 sin (θ6 − θ5)
0 0
cos (θ6 − θ5)
(d7 + l6) sin (θ6 − θ5)

, (19)
s˙ = Rq˙, (20)
s¨ = Rq¨+ R˙q˙. (21)
Substituting (17), (20), (21) into (10) yields
M¯q¨+ C¯q˙+ G¯q = τq, (22)
where M¯ = RTM (s, t)R, C¯ = RT
(
M (s, t) R˙+C (s, s˙, t)R
)
, G¯ = RTg (s, t) , and τq =
RTτ (t).
Eq. (22) is expressed in term of independent generalized coordinates, q = [θ1d4d7]
T.
Notice that all the formulations above are implemented and demonstrated in Maple envi-
ronment. The following section shows the numeric solutions of the forward and inverse
dynamic issues.
3.1. Forward dynamics simulation
The dynamical parameters of the robot system are given in Tab. 4
Table 4. The parameters of the robot
Link
Center of gravity
Mass
Inertia
xC yC zC Ixx Iyy Izz Ixy Iyz Izx
1 0 lC1 0 m1 I1x I1y I1z 0 0 0
2 lC2 0 0 m2 I2x I2y I2z 0 0 0
3 0 0 lC3 m3 I3x I3y I3z 0 0 0
4 0 lC4 0 m4 I4x I4y I4z 0 0 0
5 lC5 0 0 m5 I5x I5y I5z 0 0 0
6 0 0 lC6 m6 I6x I6y I6z 0 0 0
7 0 lC7 0 m7 I7x I7y I7z 0 0 0
l1 = 0.7 m; l10 = 0.4 m; l11 = 0.3 m; l2 = 0.6 m; l20 = 0.2 m; l21 = 0.4 m; l22 = 0.4 m; l23 =
0.2 m; l3 = 0.4 m; l4 = 0.4 m; l5 = 1.2 m; l51 = 0.8 m; l52 = 0.4 m; l6 = 0.4 m; l7 = 0.4 m.
152 Chu Anh My, Vu Minh Hoan
We assume that lC1 = l1/2 = 0.35 m; lC2 = l2/2 = 0.3 m; lC3 = l3/2 = 0.2 m; lC5 = l5/2
= 0.6 m; lC6 = l6/2 = 0.2 m; lC4 = l4/2 = 0.2 m; lC7 = l7/2 = 0.2 m; −5pi6 ≤ θ1 ≤
5pi
6
;
0.1 ≤ d4 < l4; 0.1 ≤ d7 < l7, m1 = 80 kg; m2 = 60 kg; m3 = 20 kg; m4 = 10 kg; m5 = 50 kg;
m6 = 20 kg; m7 = 10 kg.
I1x = m1l21/12; I1z = m1l
2
1/12; I2x = 0; I2y = m2l
2
2/12; I2z = m2l
2
2/12; I3x = m3l
2
3/12;
I3y = m3l23/12; I3z = 0; I4x = m4l
2
4/12; I4y = 0; I4z = m4l
2
4/12; I5x = 0; I5y = m5l
2
5/12; I5z =
m5l25/12; I6x = m6l
2
6/12; I6y = m6l
2
6/12; I6z = 0; I7x = m7l
2
7/12; I7y = 0; I7z = m7l
2
7/12.
The applied torque/forces are given as
τ1 (t) = 1.5× sin (2t) , F4 (t) = 50 (20+ t) , F7 (t) = 30 (20− t) .
Fig. 9 shows the time evolution of θ1 (t), d4 (t) and d7 (t).
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
times [s]
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
teta1
d4
d7
Fig. 9. The time evolution of θ1 (t), d4 (t) and d7 (t)
3.2. Inverse dynamics simulation
To demonstrate the inverse dynamic analysis, two cases of simulation are consid-
ered. The inputs of the simulation are given as
θ1 (t) =
pi × t
18
, d4(t) = 0.1+ 0.005× t, and d7(t) = 0.1+ 0.02× t.
For the first case, the mass of the link 3, 4, 6 and 7 equals to zero. In the second case,
the mass of the link 3, 4, 6 and 7 are given as m3 = 20 kg, m4 = 10 kg, m6 = 20 kg and m7 =
10 kg.
Figs. 10–12 show the results of the inverse dynamic analysis. The “black” curves are
the time evolution of the computed torque/forces corresponding to the first case, while
the “gray” ones are for the second case.
Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms 153
0 5 10 15
times [s]
-0.02
-0.018
-0.016
-0.014
-0.012
-0.01
-0.008
-0.006
M1
1
M1
2
Fig. 10. τ1(t)
Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms 13 
Fig 9. The time evolution of 1 t , 4d t and 7d t 
Inverse dynamics simulation 
To demonstrate the inverse dynamic analysis, two cases of simulation are considered. The inputs of 
the simulation are given as 
 1
18
t
t

 , 4 ( ) 0.1 0.005d t t , and 7 ( ) 0.1 0.02d t t . 
For the first case, the mass of the link 3, 4, 6 and 7 equals to zero. In the second case, the mass of the 
link 3, 4, 6 and 7 are given as 3m =20 kg, 4m =10 kg, 6m =20 kg and 7
m =10 kg.
Figs 10, 11 and 12 show the results of the inverse dynamic analysis. The “red” curves are the time
evolution of the computed torque/forces corresponding to the first case, while the “green” ones are for 
the second case.
Fig 10. 1( )t Fig 11. 4( )F t 
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
times [s]
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
teta1
d4
d7
0 5 10 15
times [s]
- .
- .
- .
- .
- .
- .
- .
-0.006
M1
1
M1
2
0 5 10 15
times [s]
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
F
4
 [
N
]
F4
1
F4
2
Fig. 11. F4(t)
My Anh Chu, Hoan Minh Vu 14
Fig 12. 4( )F t 
It can be seen that there is a considerable change of the computed torque/forces when considering the 
mass of the hydraulic cylinders in the dynamic model of the robot system. 
4. CONCLUSION
The kinematic and dynamic equations for a particular type of robot have been formulated. It has shown 
that when the mass and inertia of a hydraulic cylinder driving a revolution joint of a robot are considered, 
such the cylinder and relevant links of the robot constitute a local closed mechanism appended to the 
main robot architecture. By taking into account this particular feature of the hydraulic robot, the 
kinematic and dynamic modelling and analysis of the robot are more accurate. This could help to design 
more efficient and more effective control laws for the arm. 
Acknowledgment: This research is funded by Vietnam National Foundation for Science and 
Technology Development (NAFOSTED) under grant number 107.04-2017.09. 
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system.
4. CONCLUSION
The kinematic and dynamic equations for a particular type of robot have been for-
mulated. It has shown that when the mass and inertia of a hydraulic cylinder driving
a revolution joint of a robot are considered, such the cylinder and relevant links of the
robot constitute a local closed mechanism appended to the main robot architecture. By
taking i to account this particular featur of the hyd aulic robot, the kinematic and dy-
namic modelling and analysis of the robot are more accurate. This could help to design
more efficient and more effective control laws for the arm.
154 Chu Anh My, Vu Minh Hoan
ACKNOWLEDGMENT
This research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 107.04-2017.09.
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