Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 4 (March 2018) 275–282___________________________________________________________________________________________________________

Motion Improvement of Four-Wheeled Omnidirectional Mobile Robots for Indoor Terrain
Amornphun Phunopas1
Department of Production Engineering, King Mongkut’s University of Technology North Bangkok, 1518 Pracharat 1
Road Bangsue, Bangkok, 10800, Thailand1

Shinichi Inoue2
Department of Interdisciplinary Informatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka
Fukuoka, 820-0067, Japan2
E-mail: amornphun.p@eng.kmutnb.ac.th, inoue.shinichi.1800@gmail.com
www.kmutnb.ac.th

Abstract
The four-wheeled omnidirectional platform is great to use for an indoor mobile robot. It can increasingly move andchange heading directions. However, the robot is easy to slip when it is moving. One or more wheels are sometimes
not touching the ground. This paper approaches to solving these problems by computational simulation in locomotion.
The mathematical models simulate the robot movement. The robot struggles to go to the target with randomly
simulated slips and untouched ground circumstances. The robot can estimate the positions using the Kalman filter
and readjust itself to the planned path. Consequently, this paper demonstrates the motion improvement and compares
the results of decreasing errors.

Keywords: Omnidirectional wheel, Indoor mobile robot, Kalman filter, Motion improvement

There are several techniques to solve the robot motion
slippery of the omnidirectional type wheel. The design
performance of the four-wheeled Omni wheelchair, witha suspension mechanism that utilizes a hydraulic shock
absorber, was evaluated.3 The results claimed that the
proposed mechanism design had lesser wheel slippage
and vibration than the other existing designs. Moreover,
the omnidirectional mobile can be used with thewheelchair platform, which added the human upper bodymodel to the robot model.4 Normally, the dynamic modeland modeling control methods are implemented in thesystem. The motion control has a dynamic model of three
Omni wheels, which is formulated to stabilize theprogrammed motion and to track the trajectory of a robot.
The performance investigations are based on the direct
Lyapunov method and its extensions. Andreyev et al.
created control laws and simulated it by using thenumerical method.5 Similarly, the problem of trajectory
tracking for a four-wheeled omnidirectional robot is

1. Introduction
It is known that an omnidirectional mobile robot is aholonomic motion. It can travel in multi-directions andchange direction rapidly. This platform is famous for use
in the automated guided vehicles (AGVs), which use is
growing in service robots.1 The robots can support homecare. The robots can flexibly add on many functions forworks or for human assistance.
The omnidirectional mobile robot has many wheelsthat are placed in different directions. It has acomplicated mathematical model to control the
movement of robots.2 There is a constraint when moving
on a flat, indoor terrain. However, it is difficult to movethe robot to the desired position autonomously because
the robot's motion is slippery. Therefore, the robot may
miss its planned path or, in the worst-case scenario, getlost due to unrecognized circumstances.

Copyright © 2018, the Authors. Published by Atlantis Press.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
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proposed to generate fuzzy decision rules16
automatically. Odometry sensor is not suitable for suchdrives because of wheel slippage. Kundu et al. used asingle camera for 360-degree scanning of multiple
markers in the area.17 The markers were based on anaugmented reality localization platform. The system
required a scanning time of 5 seconds, and the robot had
to stop traversing. The average error ranging was from
2.5cm to 5cm in a test area of 5m X 5m. Active Beacon
System is a radio frequency signal transceiver that has
multiple ultrasonic sensors which transmit a package toevery beacon. It is used to determine the absolute positionof the robot.18 Algorithms are used to control the robot
by reducing errors. For real implement in the robot,
Hashemi et al. use a model-based PI-fuzzy control for
four-wheeled
RoboCup
Small
Size
Leagueomnidirectional robot.19 The robot had a path planner andassociated low-level control system to satisfy planning
prerequisites and prevent slippage with velocity andacceleration filtering. The robot can generate a real-time
trajectory by constrained dynamic inversion.20 Anotheridea to improve the trajectory tracking of mobile robots
is to use neural networks. The wheel slip error andexternal disturbance forces were defined as uncertainty.21
The modified backpropagation neural networks have
online weight updating laws for robustness. The
boundary of system stability is proven using the
Lyapunov method. The tracking error can be reduced
according to the theoretical analysis of tracking a straightline and a U-shape trajectory.
The four-wheeled omnidirectional mobile robot has
four points to make contact with the floor. Its geometry
is a rectangular shape with four connecting points, which
are supposed to align all wheels in the same plane. In fact,
the floor is not completely flat, and after running therobot many times, each wheel will become eroded
unequally. The robot becomes unbalanced or shaky eventhough all wheels are designed and aligned in the sameplane. It is possible that some wheels will not touch thefloor when it moves. It is possible that the robot will getstuck and not reach its goal. These problems can besolved by mechanically designing it for rough terrain
conditions.22 For example, the omnidirectional mobile
robot employs a suspension system23 for supporting all
wheels to touch the floor in order to cancel out themechanical noise.
The computational algorithm can reduce error fromnoises and make the robot move on its planned path.
Dead reckoning is the motion estimation of the robot baseon speed estimation, direction, and traveling time.24 The
Kalman filter is great to use in noisy systems such as
localization and navigation.25

solved by obtaining the nonlinear error, which is brought
to a quasi-Linear Parameter Varying (LPV). The
simulation results showed the effectiveness of switching
LPV control, using Linear Matrix Inequalities (LMI)based techniques in the tracking of the desired circular
trajectory.6 The utilized trajectory can be generated for
obstacle avoidance; the robot can travel in smoothvehicle velocities near obstacles. Galicki applied the
Lyapunov stability theory, which is used to derive thecontrol scheme.7 The simulations of an omnidirectional
vehicle had collision-free movement with one obstacleand in cluttered task space. Flippo et al. developed aprediction algorithm in a skid steering turn8 to validate
the use of single wheel testing to predict the full fourwheel vehicle system and to control the side-slip angle
estimation based on lateral dynamics control with
optimal steering angle and traction/brake torquedistribution.9 Both trajectory tracking and stabilization
can be synthesized via the well-known adaptive
backstepping approach for the dynamic models of
nonholonomic mobile robots with dynamic effect anduncertainties.10 The robot that can localize or recognize ashifted position can adjust itself to the correct position.
In fact, the robot has sensors to refer to the absolute
position. An omnidirectional mobile platform has aclosed-loop control scheme to control three degrees-offreedom motions. Cooney et al. adapted the optical miceas a sensor providing dead-reckoning for closed-loop
PID control.11 Huang et al. have built in the electronic
compass and the gyroscope sensors to measure themobile platform azimuth and feedback for platformorientation control.12 They control the robot by knowing
the robot rotation and orientation, relating to an
intelligent fuzzy sliding mode controller (FSMC) for
every wheel. The robot has the suspension system ofOmni wheels to prevent non-ground contact on a
unsmooth surface for better control. Research used afault-tolerant odometry feedback using IMU, stereo
camera, and laser scanner to recognize self-position andtrack the desired trajectory.13 Sanada et al. used a camera
as a velocity sensor to calculate optical flow.14 The
correct position is obtained by the optical flow sensor as
opposed to the dead reckoning method. To solve the slip
problem when a vehicle traverses over soft soil, use adownward-looking camera to capture images that containboth the soil surface and the wheel tire surface.15 The
wheel slip can be estimated using the vision-based
optical flow algorithm without wheel odometry
information. The vision-based perception using anomnidirectional mirror can obtain a panoramic view, andthe omnidirectional mobile robot can then move toward
desired targets and avoid the obstacles. The particle
swarm optimization (PSO)-learning algorithm is276

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This research is about to use the Kalman filter to solve
the slip without the wheel touching the ground. The robot
will use sensors and computation to resolve the problems.
The research is done in simulation by creating thepossible conditions of motion randomly using
MATLAB. The first section is about the kinematics
model; next section is the trajectory to define the state of
motion. After that, the simulation section describes thesteps of simulation and the conditions of motion, and thenthe motion improvement section uses the Extended
Kalman Filter and shows the simulation results. Finally,
conclusion section discusses and concludes the resultsfrom the simulation.

Table 1. Eight directions of the robot’s motion
relate to the four wheels’ rotation direction.
Direction
Forward
Left
Right
Back
+Diagonal Right-Diagonal Right+Diagonal Left
-Diagonal Left

𝑤1
𝑉𝑥
𝑤2
𝑉𝑟 = [𝑉𝑦 ] = 𝐽 [𝑤 ]
3
𝜃̇
𝑤4
𝜋

𝐽=

W2

+Diagonal
right

𝑟
2

(1)
𝜋

1 𝑇

4
3𝜋

2𝑅
1

−𝑠𝑖𝑛 (𝜃 + )𝑐𝑜𝑠 (𝜃 + )−𝑠𝑖𝑛 (𝜃 +)𝑐𝑜𝑠 (𝜃 +)𝑐𝑜𝑠 (𝜃 +−𝑠𝑖𝑛 (𝜃 +

4
3𝜋
4
5𝜋

4
7𝜋

)

4
5𝜋

4
7𝜋

)

2𝑅
1

(2)

2𝑅
1

2.2. Inverse Kinematics
π

To inverse Eq. (1) will obtain Eq. (3), which provides
the robot’s velocities.

π/2
q

Right

𝜋
𝜋
−𝑠𝑖𝑛 (𝜃 + )𝑐𝑜𝑠 (𝜃 + )
4
4
3𝜋
3𝜋
𝑤1
−𝑠𝑖𝑛 (𝜃 + ) 𝑐𝑜𝑠 (𝜃 + )1
𝑤
4
4
𝑉𝑤 = [𝑤2 ] =
5𝜋
5𝜋
3
𝑟
−𝑠𝑖𝑛 (𝜃 + ) 𝑐𝑜𝑠 (𝜃 + )
𝑤4
4
4
7𝜋
7𝜋
[− 𝑠𝑖𝑛 (𝜃 + 4 ) 𝑐𝑜𝑠 (𝜃 + 4 )

3π/2

Left

W4
CW
CCW
CW
CCW
CW
CCW
0
0

[− 𝑠𝑖𝑛 (𝜃 + 4 ) 𝑐𝑜𝑠 (𝜃 + 4 ) 2𝑅 ]Where W1 is the angular velocity of wheel 1, W2 is the angularvelocity of wheel 2, W3 is the angular velocity of wheel 3, andW4 is the angular velocity of wheel 4. R is the radius of the robot
from the center to the wheel, r is the radius of the wheel, V x is
the robot’s velocity on the x-axis, Vy is the robot’s velocity on
the y-axis, and 𝜃̇ is the heading angular velocity of the robot.

Forward
W1

W3
CCW
CCW
CW
CW
0
0
CCW
CW

The input of forward kinematics regards the wheels’speed, and the output regards the robot’s velocities as in
Eq. (1) to simulate the robot's motion. Converting thefour wheels’ speed into the robot’s linear velocity
requires the Jacobian matrix as seen in Eq. (2). The
parameters R and r are set to 10 cm and 5 cm respectively
for the simulation.

The omnidirectional mobile robot uses holonomic
motion and has four wheels to move in multiple
directions. It can be controlled through the planned pathin three degrees of freedom. There are two kinematics
models: the first is forward kinematics, and the second isinverse kinematics. The robot platform uses the forward
kinematics to determine the robot’s velocities (𝑉𝑥 , 𝑉𝑦 , 𝜃̇ )and the inverse kinematics to determine the wheel's
angular velocities (𝑤1 , 𝑤2 , 𝑤3 , 𝑤4 ). The robot motion isin the XY coordinate as in Fig. 1. The robot's heading
angle is determined by 𝜃. The robot can move throughthe planned paths of motion from the kinematics models.
As such, there are eight basic directions foromnidirectional movement in Table 1. Each wheel turns
in different directions. CW is clockwise, and CCW is
counterclockwise.

+Diagonal
left

W2
CCW
CW
CCW
CW
CCW
CW
0
0

2.1. Forward Kinematics

2. Kinematics

y

W1
CW
CW
CCW
CCW
0
0
CW
CCW

W4
W3
-Diagonal
right-Diagonal
left
Back

𝑅
𝑉𝑥
[𝑉𝑦 ]
𝑅 𝜃̇

𝑅

𝑅]

(3)

x

The planned path of the robot's velocities is assigned as
input parameters to obtain the speed of four wheels.

Fig. 1. The robot’s configuration and eight directions of
motion.

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There are two effects from changing the conditions of
motion randomly. First, the robot position shifts from theplanned path, and second, the robot cannot reach thegoal's position. There are eight conditions of motion as

2.3. Trajectory model
The robot can move smoothly. The trajectory is for path
planning, which determines speed and acceleration ofeach wheel. The trajectory advantage is to increase andslow down velocity properly. The robot can rotate tochange directions while it is moving. Accordingly, thetrajectory of path planning will be applied to the inverse
kinematics and then control the speed of four wheels. In
reality, the robot has to have odometry sensors for
feedback control to control the wheels’ speed and therobot's direction. For simulation, the velocity of four
wheels can be applied to drive the animated robot. The
trajectory uses a polynomial function of time.
Polynomials are simple to compute and can provide therequired smoothness and boundary conditions using
Peter Corke’s MATLAB library.25 A quintic (fifth-order)
polynomial is used,
𝑆(𝑡) = 𝐴𝑡 5 + 𝐵𝑡 4 + 𝐶𝑡 3 + 𝐷𝑡 2 + 𝐸𝑡 + 𝐹
(4)
4
3
2
̇
(5)
𝑆(𝑡) = 5𝐴𝑡 + 4𝐵𝑡 + 3𝐶𝑡 + 2𝐷𝑡 + 𝐸
𝑆̈(𝑡) = 20𝐴𝑡 3 + 12𝐵𝑡 2 + 6𝐶𝑡 + 2𝐷
(6)

Fig. 2. The simulation procedures.

where S is function of robot motion (𝑥, 𝑦, 𝜃), 𝑆̇ is derivative of
S as robot velocity (𝑉𝑥 , 𝑉𝑦 , 𝜃̇) , 𝑆̈ is derivative of 𝑆̇ as robot
acceleration (𝑎𝑥 , 𝑎𝑦 , 𝜃̈), t is interval time in step sampling, and(A,B,C,D,E,F) are coefficients.

3. Simulation
The simulation is implemented by a kinematics modeland a trajectory model. The planned path is created forthe robot motion, a straight line and a curved line. The
trajectory is ready to use to control the robot's position,
velocity, and acceleration using Eq. (4-6). Then, the robot
converts its body velocity to four wheels’ speed using Eq.
(3). To display the robot's motion, in animation, see Eq.
(1). The Kalman filter is used to improve the robot's
motion and simulate every condition of motion as in Fig.
2.
3.1. Conditions of Motion
The robot may not move as the planned path because offriction and external force. However, the robot normallyhas the feedback control to compensate for the motor’s
speed to maintain the speed input and direction of therobot. Another problem is the robot possibly getting
stuck and being unable to move out from the trap field.
However, the wheels still continue as the assigned
wheels’ speed.

Fig. 3. There are eight possible cases involving the four
omnidirectional wheels. This shows the rotation direction ofeach omnidirectional wheel, clockwise and counterclockwise.
In case 1, not all wheels touch the ground. This is not used in
the simulation.

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shown in Fig. 3. The wheels can change the states of
motion, as in Table 2, the four wheels drive in Case 2, thethree wheels drive in Cases 3 through 6, and the twowheels drive in Cases 7 and 8.

The planned path trajectory is assigned to the robot's
moving path profile. This omnidirectional mobile robot
cannot move along to the goal position when it is inmotion case condition 3-8. The robot needs to adjust its
position to the goal even when the robot goes into a trap
field. Motion improvement aims to help the robot
approach the goal position using the Kalman filter.

Table 2. The omnidirectional wheels touch and do
not touch the floor in the conditions of motion.
Eight cases have been determined.
case
W1
W2
W3
W4
1
0
0
0
0
2
1
1
1
1
3
1
1
1
0
4
1
0
1
1
5
1
1
0
1
6
0
1
1
1
7
1
0
1
0
8
0
1
0
1
Note: 1 is touching and 0 is not touching the ground.

4.1. Kalman Filter
The Kalman filter was invented by Rudolf Kalman. It isa well-known algorithm that is used to estimate theoutput from a system model and sensor measurements.
The Kalman filter was invented for the linear system, butmany problems are non-linear. Thus, it needs to linearize
the system model. Estimating the measurements of a nonlinear system requires a tool known as the extended
Kalman filter (EKF). This paper uses the EKF for the
four-wheeled omnidirectional mobile robot. The
algorithm is iterative to predict, and an update follows
Eq. (7-11). However, much research is needed in order toimplement the EKF within the omnidirectional mobile
robot. Guidelines following the algorithm's steps aredescribed in Ref. 26-27. A difference in this work ascompared to other works is the use of random states of
motion for the state process.
Predict:−𝑥̂𝑡− = 𝑓(𝑥̂𝑡−1, 𝑢̂𝑡− ) + 𝑤𝑡
(7)
−
𝑇
𝑇
(8)
𝑃𝑡 = 𝐹𝑥 𝑃𝑡−1 𝐹𝑥 + 𝐹𝑣 𝑈𝑡 𝐹𝑣 + 𝑄𝑡
Update:𝐾𝑡 = 𝑃𝑡− 𝐻𝑡𝑇 (𝐻𝑡 𝑃𝑡− 𝐻𝑡𝑇 + 𝑅𝑡 )−1(9)𝑥̂𝑡 = 𝑥̂𝑡− + 𝐾𝑡 (𝑧𝑡 − 𝑧̂𝑡 )(10)
𝑃𝑡 = (𝐼 − 𝐾𝑡 𝐻𝑡 )𝑃𝑡−
(11)

In the example, W1 and W3 are stuck, and the robot
cannot keep moving in the desired direction in case 8.
The robot needs to get out by struggling to the left andright, back and forth, as shown in Fig. 4.
In motion case 2, all wheels are touching the floor
while the robot is moving. This is the ideal condition formotion. However, not all wheels can touch the ground all
the time in cases 3 through 8.

Where 𝑥̂ is the estimated state. F is the state transition matrix(translation between states). 𝑢̂ is the control variables. 𝑤 is themodelled random white noise. 𝑃 is the state variance matrix(error of estimation). 𝑈 is the control input error covariance.
𝑄 is the process variance matrix (error due to process). 𝑧 is themeasurement variables. 𝐻 is the measurement matrix (mapping
measurements onto state). 𝐾 is the Kalman gain. 𝑅 is themeasurement variance matrix (error from measurements).
Subscripts are as follows: t current time period and t-1 previous
time period. Superscript (-) is intermediate steps.

Fig. 4. The omnidirectional mobile robot attempts to move
under the conditions of motion case 8. The trap field, or the
freewheel, is random in the simulation. This means that theomnidirectional wheel is not touching the floor.

The EKF framework has various sensors for the step
update that is referred to as a "sensor fusion.27" Thevalues from simulation compared to the original sensors
are the heading angle from a compass, the yaw rate from

4. Motion Improvement

279

Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 4 (March 2018) 275–282___________________________________________________________________________________________________________a gyroscope, the wheel’s speed from an encoder, thefreewheel check from the wheel's rotation and the robot's
movement using an encoder and an accelerometer, andtarget bearing from a camera.

heading angle stopped at π ⁄4 rad. Moving under theconditions of motion randomly, the robot shifted from theplanned path and had not reached the goal position when
the time ended. The robot’s actual position is shown in
Fig. 5.
Accordingly, each wheel of the robot had the wheel’s
speed, as shown in Fig. 6. The wheel’s speed dropped tozero in the trap field in case of free running, or the wheel
did not touch the floor.

5. Simulation Results
The planned path was a linear path from the start position(x = 20, y = 20) to the goal position (x = 150, y = 150). It
started moving to the +diagonal right, as in Table 1, andthen attempted to rotate itself because the heading angle
was changed. The robot’s heading angle started from 0rad, as in the configuration in Fig. 1. Finally, the robot’s

Fig. 7. The robot’s actual positions x and y shifted from theplanned path. The estimated position x and y from the EKF had
a better motion.
Fig. 5. The robot randomly moved under the eight conditionsof motion, following the planned path from the start position to
the goal position.

The total simulation time was 0.5 seconds, with every
0.01 second per step. After applying the EKF to the robot,
the estimated positions x and y were smooth and could
reach very close to the goal position as shown in Fig.7.

Fig. 6. The four wheels’ speed trajectory is related to theplanned path. W1 and W2 rotated counterclockwise. W3 and W4
rotated clockwise.

Fig. 8. The robot’s heading angle simulated the actual position.
The estimated and measured heading angle were the motion
improvement from the EKF.

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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 4 (March 2018) 275–282___________________________________________________________________________________________________________The robot maintained the position on the planned
path. However, the robot’s heading angle frommeasurement and estimation changed a lot as shown in
Fig. 8.
The robot in x and y position could be compared in

The robot randomly changed its states of motion in fourwheel, three-wheel, and two-wheel drive. The main error
came from the robot motion model, which changed thenumber of the driving wheel. Generally, the EKF reduces
the effects of slip noise and sensor noise in the deadreckoning problem. However, the simulation resultshows that the EKF produces better robot motion along
the planned path. The robot struggles to traverse its
planned path with its assigned trajectory.
The behavior of the robot during the real test may
considerably differ from the simulation. Noises fromsignals are reduced, but the dynamic motion of the robot
may not work like the control signals. For real tests infuture work, the omnidirectional mobile robot has to have
speed control sensors to monitor its wheel speed, and it
will be important to check the freewheel and the floor
touchless condition when some of the wheels rotate, but
the robot does not move, or it moves in the wrong
direction.

Fig. 9. Comparing the motions of the robot from path planning,measurement, estimation, and actual position.

Acknowledgements

Fig. 9. The measured position was noisy and affected by
the error of the actual position.
For the motion improvement, each wheel’s speed was
generated from the estimated positions. The estimatedwheel’s speed was very noisy compared to the actualThis work was funded by Science and Technology
Research Institute, King Mongkut’s University ofTechnology North Bangkok.
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