Development of a quadruped mobile robot and its movement system using geometric-based inverse kinematics

Bulletin of Electrical Engineering and Informatics
Vol. 8, No. 4, December 2019, pp. 1224~1231
ISSN: 2302-9285, DOI: 10.11591/eei.v8i4.1623  1224
Journal homepage: http://beei.org/index.php/EEI
Development of a quadruped mobile robot and its movement
system using geometric-based inverse kinematics
Muhammad Aziz Muslim1
, Mochammad Rusli2
, Achnafian Rafif Zufaryansyah3
, B. S. K. K. Ibrahim4
1,2,3
Electrical Engineering Department, Faculty of Engineering, Brawijaya University, Jl. MT. Haryono 167 Malang
65145, Indonesia
4
School of Mechanical, Aerospace and Automotive, Faculty of Engineering, Environment and Computing Coventry
University, Priory St, Coventry CV1 5FB, United Kingdom
Article Info ABSTRACT
Article history:
Received Apr 30, 2019
Revised Jun 10, 2019
Accepted Jul 4, 2019
As the main testbed platform of Artificial Intelligence, the robot plays an
essential role in creating an environment for industrial revolution 4.0.
According to their bases, the robot can be categorized into a fixed based
robot and a mobile robot. Current robotics research direction is interesting
since people strive to create a mobile robot able to move in the land, water,
and air. This paper presents development of a quadruped mobile robot and its
movement system using geometric-based inverse kinematics. The study is
related to the movement of a four-legged (quadruped) mobile robot with
three Degrees of Freedom (3 DOF) for each leg. Because it has four legs, the
movement of the robot can only be done through coordinating the
movements of each leg. In this study, the trot gait pattern method is proposed
to coordinate the movement of the robot's legs. The end-effector position of
each leg is generated by a simple trajectory generator with half rectified sine
wave pattern. Furthermore, to move each robot's leg, it is proposed to use
geometric-based inverse kinematic. The experimental results showed that the
proposed method succeeded in moving the mobile robot with precision.
Movement errors in the translation direction are 1.83% with the average pose
error of 1.33 degrees, means the mobile robot has good walking stability.
Keywords:
Geometric-based
Half rectified sine wave
Inverse kinematics
Quadruped mobile robot
Trot gait pattern
Copyright © 2019 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Muhammad Aziz Muslim,
Electrical Engineering Department, Faculty of Engineering,
Brawijaya University,
Jl. MT Haryono 167, Malang 65145, Indonesia.
Email: muh_aziz@ub.ac.id
1. INTRODUCTION
As the main testbed platform of Artificial Intelligence, the robot plays an essential role in creating
an environment for industrial revolution 4.0. According to their bases, the robot can be categorized into a
fixed based robot and a mobile robot [1]. Fixed based robot is mainly used industry, in the form of robotic
manipulator [2]. While mobile robot, due to its mobility, has broader application field. Such as in farming
[3, 4], surveillance [5, 6], exploration [7, 8], and military [9, 10]. According to the environment in which they
travel, the mobile robot can be categorized as an unmanned ground vehicle (UGV), unmanned aerial vehicle
(UAV) and autonomous underwater vehicle (AUV). Furthermore, according to the device they used to move,
UGV can be classified into a wheeled mobile robot [11] and legged mobile robot [12].
The first problem to be solved in a mobile robot is locomotion. It is about how the robot should
move and how is the mechanism to move [1, 13]. The use of legs for robot mobility is inspired by the animal
world. A mobile robot with six legs (hexapod) was reported as the first walking legged mobile robot,
followed by the mobile robot with four legs (quadruped) around eight years later [14]. The hexapod robot has

Bulletin of Electr Eng and Inf ISSN: 2302-9285 
Development of a quadruped mobile robot and its movement system using… (Muhammad Aziz Muslim)
1225
advantageous of good static stability [15], while a quadruped robot has better speed and mobility compared
to the hexapod robot [16].
Since the robot has many legs, movement coordination through gait pattern is a must for smooth
movement. Gait pattern is a repetitive pattern of each legs movement (translation or rotation) by the
movement of the robot body so that the robot can move from one position to another position [14].
Simulation and experiment to get a gait pattern by Genetic Algorithm were given by [17]. Generation of gait
pattern via central pattern generator (CPG) was proposed by [18, 19], and an adaptive approach of CPG was
introduced by [20]. Integration of trot gait to CPG was presented by [21]; the combination of trot and walk
gait was proposed by [22]. Analysis on the stability of generated trajectory by gait pattern was substantially
discussed in [23, 24].
Gait pattern is a guideline to generate a trajectory for the leg tip of the mobile robot [25]. In this
paper, we propose to use half rectified sine wave to generate trajectory guided by a simple trot gait. Since this
trajectory must be followed by the tip of the robot’s legs (end effector), it is necessary to employ an inverse
kinematic method to provide joint angle given end effector trajectory. As inverse kinematic method, we
propose to decompose the inverse kinematics geometrically into a more straightforward calculation.
2. RESEARCH METHOD
There are two purposes for this research. First is to build a legged mobile robot capable of exploring
an indoor environment and the second is to design a control system for the movement. A mobile robot
comprises of sensors, processor and locomotion system. Sensors provide information surrounding
environment. This information proceeds by the processor and action are decided accordingly. For a mobile
robot, the decision is about robot movement, provided by the locomotion system. Mobile robot realization
consists of building stage and testing stage. The robot building steps consist of mechanical part building and
electronic part building. Testing steps consist of mechanical functionality test and electrical functionality test
(including sensor and actuator linearity test). The robot building will be reported in this section, while the
testing step will be reported in section 3.
Mobile robot movement is designed by path planning and response of sensory information.
According to the mobile robot movements method, there are two main categories, wheeled mobile robot and
legged mobile robot. The legged mobile robot is classified according to the number of legs for movement and
degrees of freedom for each leg. This research is focused on a four-legged mobile robot with three degrees of
freedom (3 DOF) for each leg. The detail on robot movement design will be described in this section.
2.1. Robot mechanical design
Figure 1 depicts the mechanical design of the mobile robot base. Robot center represented by the
white circled area in robot base. Servo motor positions are arranged uniformly so that the robot able to turn
90º or 180º without changing the position of the body, just by changing the movement direction. The
horizontal distance among servos is 9.405 cm, and the distance among servos will be 13 cm if measured
crossing the center. This servos arrangement is decided in order to give enough space for a 220 mAh
Lithium-Polymer (Lipo) battery and to give free movement for each leg. Using this construction also enable a
designer to add a spacer for strengthening robot structure.
Leg construction is shown in Figure 2. This leg is a 3 degree of freedom (3DOF) body comprises of
3 servo motors, namely coxa servo, femur servo, and tibia servo. These part names are originally the names
an arthropod leg parts. From Figure 2, the length of the femur (length_femur) is 4 cm while the length of the
tibia (length_tibia) is 5, 6 cm. Coxa servo rotational axis is intersecting with the rotational axis of femur
servo, setting zero offsets for coxa servo. While rotational servo of tibia and femur are parallel to each other
(as shown in Figure 2(a)), these mechanical setting will be essential parts for the derivation of
geometric-based inverse kinematic. Angle measurement is defined in Figure 2(b). Femur and tibia angle are
measures according to an axis connecting the rotational axis of femur and tibia. This definition will further
simplify the calculation of inverse kinematics.

 ISSN: 2302-9285
Bulletin of Electr Eng and Inf, Vol. 8, No. 4, December 2019 : 1224 – 1231
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(a) (b)
Figure 1. Mechanical design of the mobile robot base, (a) Base frame only, (b) Base frame with leg position
Figure 2. Mechanical configuration of robot leg, (a) Rotational direction, (b) Angle measurement of femur
and tibia
2.2. Robot electronics design
Figure 3 shows the electronics of the robot. As depicted in Figure 3(a), an STM32F407VGT6 is
employed the main microcontroller for the system. The microcontroller is connected to the servos (12 units
of Dynamixel AX 12A) via 74LS241 Tri-state buffer to ensure the servos received half-duplex signal needs
for their movement. The microcontroller receives outside information from HC SR04 ultrasonic rangefinder
sensor (provides distance to the object in front of the sensors) and CMPS11 attitude sensor (provides pitch,
roll, and yaw orientation of the robot). The robot can be controlled via a push button or PC serial instruction
via Bluetooth. Figure 3(b) shown the electronics part arranged in a single board and Figure 3(c) shows the
assembled electronic parts.

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(a) (b)
(c)
Figure 3. Electronic parts of the robot, (a) Diagram of the electronic system, (b) The electronic board, (c)
Assembled electronic parts
2.3. Design of mobile robot movement
The legged mobile robot developed in this research has four legs with 3 degrees of freedom each.
Hence it has 12 degrees of freedom (12 DOF). There are two problems to be solved for robot movement.
First, how to coordinate legs for forwarding, backward, left and right movement. Second, given tibia tip
position on how to determine coxa, femur and tibia angle. The answers for the first question lead to the
generation of tibia tip trajectory, and the answer for the second question leads to the inverse
kinematic formulation.
Constrain of robot movement is in its workspace. The workspace could be analyzed by moving the
joints step by step for all possible angle combination. Workspace determination is particularly crucial for
trajectory generation so that the algorithm only generate a reachable position of end effector (tibia tip
position). The robot in this research reach a position in X-Y space by changing the angle of coxa joint and
reach a position in X-Z space by changing femur and tibia angle. Figure 4 shows the workspace of these
two spaces.

 ISSN: 2302-9285
1228
(a) (b)
Figure 4. Leg workspace in X-Y space and X-Z space
Legged-robot movement is performed according to a gait pattern. Gait pattern is a movement pattern
of each leg which is coordinated with the body movement of the robot both translation and rotation which is
done repetitively so that the robot's body can move from one place to another [14]. The gait pattern
implemented in this study is trot gait. Trot gait is a movement pattern using two legs in the diagonal plane
with each other to swing the leg (swing phase), while the remaining two legs are responsible for supporting
the quadruped robot (support phase). Figure 5 illustrates the trot gait pattern used in this study. The distance
traveled by each leg is identical between one leg and the other leg. In one movement cycle, each leg will
experience one swing phase and one support phase with timings according to what can be seen in Figure 5.
For smooth movement, a trajectory generator is added to trot gait pattern. Trajectory generator provides a
smooth transition of end-effector trajectory from starting point to end point. We proposed to use half rectified
sine wave for this purpose. Figure 6 illustrates the trajectory generated by half rectified sine wave combined
with trot gait pattern.
The coordinates generated by the trajectory generator should be converted to angle movement
commands for each joint. These angle movement commands are provided by inverse kinematics calculation.
For simplicity, the inverse kinematics of 3 DOF robot leg decomposed into two sub-problems, inverse
kinematics for coxa joint (1 DOF) and inverse kinematic for femur-tibia joint (2 DOF). Figure 7 depicts the
kinematics of coxa joint. Without loss of generality, it is assumed that each leg is parallel with the center of
the robot, hence  = 90. Hence, the angle of the coxa joint ( ) is;
(1)
Figure 5. Trot gait pattern

1229
Figure 6. End-effector trajectory generated from the combination of trot gait and half rectified sine wave
Figure 8 shows the inverse kinematics for the femur-tibia joint (2 DOF). The femur length
(length_femur) and tibia length (length_tibia) are constants. These constants will further simplify the angles
calculation in (2) and (3). The inverse kinematic calculation in (1)-(3) in for right side legs, the solution for
the left side legs can be generated accordingly by flipping the angle direction.
(2)
(3)
Figure 7. Kinematics problem of coxa joint Figure 8. The kinematic problem of femur-tibia joint
2.4. Experimental design
In this study, the performance of the mobile robot movement using the proposed scheme will be
tested. Without loss of generality, only forward movement (in the Y direction) will be considered. The
movement of the mobile robot is considered reasonable if the robot's final position is as desired and the
attitude of the robot remains in a state of equilibrium. Robot state of equilibrium is measured using a tilt

 ISSN: 2302-9285
1230
sensor (CMPS11 sensor) which is returned pitch, roll, and yaw orientation of the robot. Figure 9 shows the
experiment scenario for movement performance.
Figure 9. Experiment scenario for movement performance
3. RESULTS AND ANALYSIS
In this experiment, the mobile robot is moving in a lateral direction toward the wall at a certain
distance. Table 1 shows the mobile robot performance of forwarding movement. Label “FM” in the first
column means forward movement traveled by mobile robot toward the wall. The “Error..” in column 2 until
11 is the subtraction of lateral position (in Y-axis) to the set point (herein equals 0). While the last two
columns are an average error for each distance toward the wall. Taking the average of the last column we got
the average error rate is 1,83%, which is considerably small.
Next experiment is a concern with walking stability. It is hoped that the robot moves while
maintaining its pose. A tilt sensor is used for this purpose. The sensor measured yaw, pitch and roll
difference to the reference. Table 2 summarizes the experimental results. As in Table 1, Label “FM” in the
first column means forward movement traveled by mobile robot toward the wall. From the table, it is
understood that pose error also considerably small, just around 1.33 degrees. It could be concluded that
walking stability is good.
Table 1. Performance of forward movement
FM
(cm)
Error in each experiment (cm) Avg
(cm)
Avg
(%)
1 2 3 4 5 6 7 8 9 10
5 0.05 -0.12 -0.36 -0.16 0.05 -0.45 -0.21 0.12-0.39 0.09 -0.14 2.75
10 -0.14 0.28 0.43 0.60 0.10 -0.02 0.29 0.70 -0.29 -0.43 0.15 1.53
15 0.22 -0.96 0.64 -0.05 0.07 0.60 -0.02 -0.31 -0.52 -0.72 -0.11 0.69
20 -0.39 0.33 -0.67 -0.86 -0.55 -0.95 -0.89 -0.22 0.67 0.21 -0.33 1.67
25 -0.93 -0.43 -0.10 -0.63 -0.12 -0.41 0.01 -0.52 -0.96 -0.16 -0.42 1.69
30 -0.67 -0.49 -0.72 -1.53 -0.22 -1.60 -0.93 -1.05 -0.43 -0.26 -0.79 2.63
Average 1.83
Table 2. Pose stability experiment results
FM (cm) Yaw Pitch Roll Average Error (deg)
5 2.8 -0.548 0 0.750667
10 5.2 -0.352 0.167 1.671556
15 3.9 -0.452 0 1.149333
20 0.2 -0.048 0 0.050667
25 5.6 -0.816 0.667 1.816889
30 7.8 -1.152 1 2.549333
Average 1.331407
4. CONCLUSION
This paper presents development of a quadruped mobile robot and its movement system using
geometric-based inverse kinematics. The trot gait pattern method is proposed to coordinate the movement of
the robot's legs, while the end-effector position of each leg is generated by a simple trajectory generator with
half rectified sine wave pattern. Furthermore, to move each robot's leg, it is proposed to use geometric-based
inverse kinematic. The experimental results showed that the proposed method succeeded in moving the
mobile robot with precision. Movement errors in the translation direction are 1.83% with the average pose
error of 1.33 degrees, means the mobile robot has good walking stability

1231
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