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INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 6, ISSUE 09, SEPTEMBER 2017 ISSN 2277-8616
285
IJSTR©2017
www.ijstr.org
Inverse Kinematic Analysis Of A Quadruped
Robot
Muhammed Arif Sen, Veli Bakircioglu, Mete Kalyoncu
Abstract: This paper presents an inverse kinematics program of a quadruped robot. The kinematics analysis is main problem in the manipulators and
robots. Dynamic and kinematic structures of quadruped robots are very complex compared to industrial and wheeled robots. In this study, inverse
kinematics solutions for a quadruped robot with 3 degrees of freedom on each leg are presented. Denavit-Hartenberg (D-H) method are used for the
forward kinematic. The inverse kinematic equations obtained by the geometrical and mathematical methods are coded in MATLAB. And thus, a program
is obtained that calculate the legs joint angles corresponding to desired various orientations of robot and endpoints of legs. Also, the program provides
the body orientations of robot in graphical form. The angular positions of joints obtained corresponding to desired different orientations of robot and
endpoints of legs are given in this study.
Index Terms: Quadruped robot, D-H parameters, forward kinematic, inverse kinematic, MATLAB, kinematic program, simulation.
———————————————————
1 Introduction
Quadruped robots are an important place in robotic and their
popularity are increasing. Quadruped robots are more
complex in structure, more difficult to control than wheeled and
crawler robots. They have a complex structure, which leads to
a higher instructional in terms of robotics and control theory.
Less energy consumption, good stability and locomotion on
uneven and rough terrain are main advantages of quadruped
robots. Low speed, difficult to build and control, need on-
board power are limitations of quadruped robots. The number
of study about quadruped robots has increased in recent years
due to better performance in challenging terrain conditions.
The example of current studies about quadruped robots;
BigDog [1] is developed by Boston Dynamics, HyQ2Max [2] is
developed by Semini et al., Jinpong [3] is developed by Cho et
al. In quadruped robots, a good kinematic model is necessary
to stability analysis and trajectory planning of system. There
are two types of kinematic analysis: forward and inverse
kinematics analysis. In the forward kinematic analysis, the joint
variables are given to find the location of the body of the robot.
In the inverse kinematic analysis, the location of the body is
given to find the joint variables necessary to bring the body to
the desired location. Detailed studies on the kinematic
analysis of quadruped robots are available in the literature.
Potts and Da Cruz [4], presented both the forward and inverse
kinematics model of a complex quadruped robot named
Kamambare with singularity analysis.
Oak and Narwane [5], showed the kinematic modelling of a
quadruped robot leg with four bar chain mechanism. Zhang
et.al [6], propose a structure design of quadruped robot using
a mammalian animal with the kinematics analysis, also
optimized a leg of robot and simulated in ADAMs. Anand [7],
provided insights into the kinematic analysis of a quadruped
robot capable of both walking and functioning as a machining
tool. Ganjare et.al [8], designed a quadruped robot with two
joints of the leg, it enable to perform two basic motions: lifting
and stepping at medium speed on flat terrain. Chenet.al [9],
introduced a hydraulic quadruped robot and built kinematic
model of the robot in part of their study. In this paper, a
MATLAB program that calculates forward and inverse
kinematics of a quadruped robot corresponding to desired
different orientations of robot and endpoints of legs and
demonstrate the 3-D robot form in graphical is developed.
Kinematic equations are obtained by Denavit-Hartenberg
method [10] and analytical solutions [11].
2 KINEMATIC ANALYSIS
The quadruped robot is a robotic system that consists of a
rigid body and four legs with three degrees of freedom (each
leg has the same structure). The links of legs are connected to
each other by rotary joints. The physical model of the
quadruped robot is shown Fig. 1. The parameters of robot are
given in Table 1.
____________________________
Muhammed Arif Sen, Department of Mechanical
Engineering, Selçuk University, Konya, Turkey,
marifsen@selcuk.edu.tr
Veli Bakırcıoğlu, Technical Science Vocational
Education and Training School, Aksaray University,
Aksaray, Turkey, vbakircioglu@aksaray.edu.tr
Mete Kalyoncu, Department of Mechanical
Engineering, Selçuk University, Konya, Turkey,
mkalyoncu@selcuk.edu.tr
Fig. 1. The physical model of the quadruped robot
INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 6, ISSUE 09, SEPTEMBER 2017 ISSN 2277-8616
286
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As shown in Figure 1, depending on the legs coordinates, the
robot body can have different configurations. For this reason,
the kinematic equation between the rotational movements (φ,
ψ, ω) around the centre of body’s coordinate system (xm, ym,
zm) and the coordinate system of each endpoint of leg (x4, y4,
z4) is investigated. Initially, to determine the position and
orientation of the robot centre of body in the workspace, the
transformation matrix is obtained in Eq.5 using the rotation
matrices (Eq. 1, Eq. 2, Eq. 3, Eq. 4).
The kinematic equation the centre of body’s coordinate system
(xm, ym, zm) and The Main Coordinate System of each leg (x0,
y0, z0) is given by the transformation matrices given in Eq.6,
Eq.7, Eq.8, Eq.9. The positions and orientations of each leg
can be calculated according to the position and orientation of
the robot's body.
Forward kinematics of robot, deals with the relationship
between the positions, velocities and accelerations of the
robot links. Inverse kinematics is the process of finding the
values of the joint variables according to the positions and
orientations data of the endpoint of robot. In other words, in
order to move the robot endpoint to the desired position, it is
necessary to determine the rotational values of the joints with
inverse kinematic analysis. The forward and inverse
kinematics of analysis one leg of a quadruped robot are
described in detail. The legs are in different orientations with
each other but in the same structure, so it is sufficient to
investigate the forward and inverse kinematics analysis of a
single leg. Fig. 2 shows the coordinate systems and the
angular positions of the right front leg joints. The Denavit-
Hartenberg parameters for forward kinematics of the leg are
given in Table 2.
TABLE 1. THE PARAMETERS OF ROBOT
Physical
Dimensions
The Length of Robot
L=1 [m]
The Width of Robot
W=0.4 [m]
The Length of Side Swing Joint
L1=0.1 [m]
The Length of Hip Joint
L2=0.4 [m]
The Length of Knee Joint
L3=0.4 [m]
Coordinate Systems
The Coordinate System of centre of Body
[xm, ym, zm]
The Main Coordinate System of Each Leg
[x0, y0, z0]
The Coordinate System of Side Swing Joint
[x1, y1, z1]
The Coordinate System of Hip Joint
[x2, y2, z2]
The Coordinate System of Knee Joint
[x3, y3, z3]
The Coordinate System of Endpoint of Leg
[x4, y4, z4]
Variables
The Yaw Angle of Robot
ϕ
The Pitch Angle of Robot
ψ
The Roll Angle of Robot
ω
The Angle of Side Swing Joint
θ1
The Angle of Hip Joint
θ2
The Angle of Knee Joint
θ3
INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 6, ISSUE 09, SEPTEMBER 2017 ISSN 2277-8616
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Each transformation matrices are given in Eq.
10, Eq. 11, Eq. 12, Eq. 13 and the forward kinematic matrix
is given in Eq. 14. The elements of the forward kinematic
matrix of a leg are given in Table 3.
After obtained the transformation matrices and forward
kinematic matrices required for the inverse kinematic solution
of the Quadruped Robot, the inverse kinematic analysis is
performed using analytical methods. Equations expressing the
angular position of joints (θ1, θ2, θ3) are obtained and given in
Eq.15, Eq.16 and Eq.17. There are nonlinear equations in the
solution of inverse kinematics problems. For every
mathematical expression computed, there may not be a
physical solution. Also, there may be more than one solution for
the legs endpoint to go to the desired position. For this reason,
the legs of the robot (1 and 3) and the leg of the robot (2 and 4)
have been realized in the same kinematic structure but in
different configurations.
TABLE 2. THE PARAMETERS OF DENAVIT-HARTENBERG
Link
α i-1
a i-1
di
θi
0-1
0
L1
0
θ1
1-2
-π/2
0
0
-π/2
2-3
0
L2
0
θ2
3-4
0
L3
0
θ3
TABLE 3. THE ELEMENTS OF THE FORWARD KINEMATIC
MATRIX
Fig. 2. The Coordinate Systems of Leg Joints
INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 6, ISSUE 09, SEPTEMBER 2017 ISSN 2277-8616
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IJSTR©2017
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(Legs for 1 and 3)
(Legs for 2 and 4)
3 CONCLUSION
In this study, forward and inverse kinematic analysis of a
quadruped robot is investigated. Denavit-Hartenberg method
is used for forward kinematics, analytical solutions is used for
inverse kinematics. The obtained kinematic equations are
transferred to MATLAB environment and a program is
developed to obtain the angular position of joints of each legs
corresponding to the different robot orientations and different
coordinate systems of endpoint of legs. In order to test the
program, 3 different inverse kinematic analyses are performed
and the results obtained from the program are presented in
graphical form. As a result, the program which developed in
this study performs efficiently the kinetic analysis of a
quadruped robot. Furthermore, the robot form can be display
by using the program easily. In addition, this study will
contribute to the work such as dynamic analysis, walking
analysis on the four-legged robot. In addition, this study will
contribute to the other studies about dynamic analysis, walking
analysis on quadruped robots.
TABLE 3. THE EXAMPLES RESULTS OBTAINED FROM THE PROGRAM
[x4, y4, z4]=[0, -0.65, 0] [xm, ym, zm]= [0, 0, 0]
θ1
θ2
θ3
φ= 0°
ψ=-15°
ω=0°
Leg Number
1
7.5883°
28.7493°
-29.7695°
2
11.5735°
-33.0804°
100.5692°
3
11.5735°
33.0804°
-100.569°
4
7.5883°
-28.7493°
29.7695°
[x4, y4, z4]=[-0.05, -0.55, 0] [xm, ym, zm]= [0, 0, 0]
θ1
θ2
θ3
φ= -45°
ψ= 0°
ω=-10°
Leg Number
1
-9.7298°
49.8269°
-53.8359°
2
47.7890°
-30.1490°
67.8506°
3
-31.9917°
59.6929°
-69.4310°
4
31.8200°
-36.8724°
82.0530°
[x4, y4, z4]=[-0.15, -0.7, 0.05] [xm, ym, zm]= [0.1, 0.2, -0.3]
θ1
θ2
θ3
φ=-10°
ψ=-10°
ω=15°
Leg Number
1
-51.7965°
30.1317°
-35.2716°
2
-48.0254°
-34.7341°
99.7991°
3
34.9428°
62.5980°
-105.322°
4
43.2869°
-25.3487°
59.5477°
INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 6, ISSUE 09, SEPTEMBER 2017 ISSN 2277-8616
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IJSTR©2017
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ACKNOWLEDGMENT
This study is supported by the Coordinator ships of Selçuk
University’s Scientific Research Projects. (Project No:
16401148)
REFERENCES
[1] Marc Raibert, et al., “Bigdog, the rough-terrain quadruped
robot,” Proceedings of the 17th World Congress The
International Federation of Automatic Control, pp. 10822-
10825, 2008, Seoul, Korea. (Conference proceedings)
[2] Claudio Semini, et al., “Design of the Hydraulically-
Actuated Torque-Controlled Quadruped Robot HyQ2Max,”
IEEE/ASME Transactions on Mechatronics, vol. 22, no. 2,
pp. 635-646, 2017.
[3] Jungsan Cho, et al., “Simple Walking Strategies for
Hydraulically Driven Quadruped Robot over Uneven
Terrain,” Journal of Electrical Engineering & Technology,
vol. 11, no. 5, pp. 1921-718, 2016.
[4] A.S. Potts and J. Jaime da Cruz. “A Kinematical and
Dynamical Analysis of a Quadruped Robot,” Mobile
Robots-Current Trends. InTech, 2011.
[5] S.S. Oak, V.S. Narwane, “Kinematic Modeling Of
Quadruped Robot With Four Bar Chain Leg Mechanism,”
International Journal of Management and Applied Science
(IJMAS), vol. 2, ıssue 1, pp. 24-29, 2014.
[6] Z. Jinrong, C. Wang and J. Zhang, “The Kinematics
Analysis and Configuration Optimize of Quadruped
Robot,” Open Automation and Control Systems Journal,
vol. 6, pp. 1685-1690, 2014.
[7] Raghu Mandayam Anand, “Kinematic analysis of a
quadruped robot capable of working as a machining tool,”
PhD Thesis, The University of Texas at Arlington, 2010.
[8] S.A Ganjare, V.S. Narwane, and U. Deole. “Kinematic
Modeling of Quadruped Robot,” International Journal on
Theoretical and Applied Research in Mechanical
Engineering, vol. 2, ıssue 4, pp. 21-26, 2013.
[9] Chen Xianbao, et al. “Kinematic analysis and motion
planning of a quadruped robot with partially faulty
actuators,” Mechanism and Machine Theory, vol. 94, pp.
64-79, 2015.
[10] J. Denavit and R.S. Hartenberg, “A kinematic notation for
Lower-pair mechanisms based on matrices,” ASME Jappl.
Mechan, pp. 215-221, June 1955.
[11] D.L. Pieper, B. Roth, “The Kinematics of Manipulators
Under Comp. Cont,” Int. Cong. for The Theory of Machine
and Mechanism, vol. 2, pp. 159-168, Zakopane, Poland,
1968.
