Kinematic Analysis of a Two-Wheeled
Self-Balancing Mobile Robot
Animesh Chhotray, Manas K. Pradhan, Krishna K. Pandey
and Dayal R. Parhi
Abstract This paper describes the development of a two-wheeled self-balancing
robot and its kinematic analysis. The system architecture consists of two co-axial
wheeled rectangular structure powered by a pair of DC motors. Two separate motor
drivers are controlled by pulse width modulated voltage signals received from the
Arduino microcontroller board. The attitude determination of the robotic platform
can be accomplished by an IMU sensor, which is a combination of accelerometer
and rate gyro. After mechanical system design, the velocity decomposition of the
two wheels and robot body are analyzed to establish the kinematic model. In this
model, local position of the robot is mapped according to the global coordinates.
Finally, the kinematic constraints are established for fixed standard wheels of the
two-wheeled robot.
Keywords Two-wheeled self-balancing robot
constraints
Kinematic model
Kinematic
1 Introduction
Current considerable research on human–robot interaction in the public domain
with real-time responses results in the development of two-wheeled mobile robots.
These robots have better potential for use in indoor environments as personal
A. Chhotray (&) M.K. Pradhan K.K. Pandey D.R. Parhi
Robotics Laboratory, Department of Mechanical Engineering, National Institute
of Technology, Rourkela, Sundergarh 769008, Odisha, India
e-mail: chhotrayanimesh@gmail.com
M.K. Pradhan
e-mail: manas.pradhan141@gmail.com
K.K. Pandey
e-mail: kknitrkl@yahoo.in
D.R. Parhi
e-mail: dayalparhi@yahoo.com
© Springer India 2016
D.K. Lobiyal et al. (eds.), Proceedings of the International Conference
on Signal, Networks, Computing, and Systems, Lecture Notes
in Electrical Engineering 396, DOI 10.1007/978-81-322-3589-7_9
87
88
A. Chhotray et al.
assistance, tour guidance, surveillance, and as cleaning robots in households. It is
much easier to make a two-wheeled robot with tall structure than the bipedal
structure without compromising the ability to turn on the spot with greater agility.
Rather these robots offer higher levels of mobility and maneuverability than their
four-wheeled counterparts due to the ability to negotiate with tight corners and
corridors.
This leads the research on two-wheeled self-balancing robot to be accelerated
over the last decade in many control and robotic research centers [1–3]. These
robots have two co-axial differential drive wheels mounted on either side of an
intermediate rectangular body. The center of mass lies above the wheel axles, which
actively stabilizes the robot, while traversing over steep hills or slopes.
Two-wheeled robots also have smaller footprints and can spin on the spot about a
single axis in various terrains. These robots are characterized by their capacity to
balance on their two wheels by overcoming the inherent dynamics of the system.
This additional maneuverability allows easy navigation on various terrains. These
capabilities of two-wheeled robots have the potentiality to solve numerous challenges in society and industry.
Many researchers have discussed several kinematic analysis approaches for
various types of robots with a different number of wheels [4, 5]. The kinematic
analysis establishes a relation between robot geometry, system behavior, and
control parameters in an environment. In this analysis, robot velocity is generated
after considering the speed of the wheels, rotation angle, and geometric parameters
from the robot configuration [6–8].
2 Mechanical Systems Design
For the robot to be able to balance successfully, it is essential that the sensors
provide reliable information about the inclination of the robotic platform and its
angular velocity. This is also, of course, to ensuring that the control system, motor
drivers, and motors themselves are properly designed. The design process here is
completed in several phases. At first 3D sketches of the rectangular body structure
with wheels is created through CATIA as shown in Fig. 1. The wheels are placed
parallel to each other and driven by two separate motors. As power source battery
are placed as high as possible above the wheel axis to get better stability, the power
transmission from the motor shaft to the wheel axis is achieved by gear chain
assembly.
In the next phase to enable stabilizing control, the two-wheeled robot is
equipped with appropriate actuators and sensors, which are connected with each
other through microcontroller and motor drivers. An Arduino microcontroller board
is having ATmega328 controller and six analog inputs with fourteen digital input or
output pins from which six can be used as PWM outputs. The microcontroller sends
a suitable signal to PWM generator to generate commands of required force for the
DC motor of the two wheels. The sensor used here is an IMU sensor called
Kinematic Analysis of a Two-Wheeled Self-Balancing Mobile Robot
89
Fig. 1 CAD model of the Lab-built two-wheeled mobile robot
MPU6050 which contains 3-axis accelerometer and a 3-axis gyroscope in a single
chip. It is very accurate and captures X, Y, Z axis values simultaneously. Gyroscope
measures the rate of change of angular orientation, i.e., the angular velocity with
respect to a reference frame. Accelerometers are used to find the rate of change of
linear velocity that the body experiences when a force is applied on it. Also two
separate motor drivers of 6–16 V, 20 A capacity are used to control the direction
and speed of the wheels through motors. These drivers can be interfaced with the
microcontroller and receive digital signals as pulse width modulation to control
motor velocity.
3 Kinematic Model of Two-Wheeled Mobile Robot
Kinematics is the most basic study to understand the mechanical behavior of the
robot. While designing appropriate mobile robots for performing desired tasks and
to create suitable control software for any instance of robot hardware, prior
knowledge of proper kinematic model is highly essential. Also, Robotic direction
cannot be measured instantaneously; rather it is integrated over time. Also, slippages of wheels add some inaccuracies to the motion. Therefore, to get precise
robot position is always an extremely challenging task. Since each wheel contributes to robot motion and both are coupled together on robot chassis, their
individual constraints combine to figure the overall constraints affecting final robot
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A. Chhotray et al.
motion. Hence, to formulate the kinematic model the actual model is simplified
with the following assumptions.
1. The Robot must be considered as a rigid body irrespective of joints and degrees
of freedom due to wheel and steering assembly.
2. The plane of operation should be treated as horizontal.
3. Robot motion is pure rolling without slipping, skidding, or sliding between
wheel and floor.
4. There should not be any friction for rotation around contact points.
5. Steering axes must be orthogonal to the surface.
6. During motion, the wheel and the horizontal plane is get in touch with a single
point.
The understanding of the robot motion starts with analyzing the individual
contribution of each wheel toward the whole robot motion. Also the effect of
constraints like lateral skidding cannot be overlooked. As the wheels are tied
together according to the robot chassis geometry, their constraints are also combined to affect the overall motion. Therefore, while describing a robot’s navigation
as a function of its geometry and wheel behavior, we have to introduce notations for
its motion in a global reference frame and local reference frame as in Fig. 2. In the
kinematic analysis of a two-wheeled differential robot the position of the robot is
described by a local reference frame as {XL, YL} in a global reference frame {XG,
YG}. If the angle of orientation of the robot is h then its complete location in the
global reference frame is given by
2 3
x
nG ¼ 4 y 5
h
ð1Þ
YG
vl
ωl
Vrobot
rl
vr
θ
yl
ωr
D
D
2
rr
D
ωrobot
2
θ
xl
Fig. 2 Two-wheeled differential robot kinematics
XG
Kinematic Analysis of a Two-Wheeled Self-Balancing Mobile Robot
Mapping between two frames can be established
matrix as
2
cosðhÞ sinðhÞ
RðhÞ ¼ 4 sinðhÞ cosðhÞ
0
0
91
by considering the rotational
3
0
05
1
ð2Þ
Therefore, the local position of the robot can be mapped with respect to global
coordinates as
3
2 3 2
cosðhÞ_x þ sinðhÞ_y
x_
n_ l ¼ RðhÞn_ G ¼ RðhÞ 4 y_ 5 ¼ 4 sinðhÞ_x þ cosðhÞ_y 5
ð3Þ
h_
h_
Let us consider a two-wheeled robot with two independently driven wheels
mounted on the same axis as in Fig. 1. The movement of robot is described by both
translation of center and orientation of centroid. The above two action can be
accomplished by controlling the wheel speeds of left and right wheels. Robot linear
velocity Vrobot(t) and angular velocity xrobot(t) are functions of the linear and angular
velocities of its right wheel, ʋr(t), xr(t) and left wheel, ʋl(t), xl(t), respectively. The
distance between the two wheels is taken as D and the right and left wheel radius are
taken as rr, rl respectively. The rotation angle of the wheel about its horizontal axis is
denoted by ur and ul accordingly.
The robot velocities can be expressed as
Vrobot ¼
vr þ vl
vr v l
; xrobot ¼
2
D
The kinematic equation can be written in
2
3 2
cosðhÞ
x_ ðtÞ
4 y_ ðtÞ 5 ¼ 4 sinðhÞ
_
0
hðtÞ
ð4Þ
global coordinate or initial frame as
3
0 V
0 5 robot
ð5Þ
xrobot
1
And in according to local coordinate it is
2
3 2
3
x_ l ðtÞ
r=2 r=2 xl ðtÞ
4 y_ l ðtÞ 5 ¼ 4 0
5
0
xr ðtÞ
h_ l ðtÞ
r=D r=D
ð6Þ
4 Wheel Kinematic Constraints
The primary step of a kinematic model is to find out the number of constraints that
are affecting the motion of each wheel. By assuming that the wheels always remain
vertical with pure rolling condition and a single point contact the analysis become
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A. Chhotray et al.
Fig. 3 Fixed standard wheel kinematic constraints
quite simplified. Here in local coordinate, velocity in Y axis is always zero; since the
possible motion for wheel is to move back and forth along wheel plane and rotate
around its contact point with the ground plane as in Fig. 2.
Hence from Fig. 3 the kinematic constraints for a two-wheeled robot having
fixed standard wheels can be derived as.
1. Rolling Constraints
½sinða þ bÞ cosða þ bÞðlÞcosbRðhÞn_ l r u_ ¼ 0
ð7Þ
2. Sliding Constraints
½cosða þ bÞ sinða þ bÞ lsinbRðhÞn_ l ¼ 0
ð8Þ
5 Conclusions
The two-wheeled mobile robot can be a better alternative than multi-wheeled and
humanoid robots to work in indoor environments like narrow corridors and tight
corners. The kinematic analysis establishes the relation between position and orientation of the robot in local reference frame to that of global reference frame.
A relation for both linear and angular velocity of the robot has been derived taking
Kinematic Analysis of a Two-Wheeled Self-Balancing Mobile Robot
93
the left and right wheel velocities and robot dimensions. Finally, the kinematic
constraints like rolling and sliding constraints are established for fixed standard
wheels of the two-wheeled robot.
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