International Journal of Computer Applications (0975 8887)
Volume 113 No. 3, March 2015
Kinematic Model of a Four Mecanum Wheeled Mobile
Robot
Hamid Taheri
Bing Qiao
Nurallah Ghaeminezhad
College of Electronic and
Information
NUAA Nanjing
China
College of Astronautics
NUAA
Nanjing
China
College of Automation
NUAA
Nanjing
China
ABSTRACT
2. KINEMATIC
This paper introduces omnidirectional Mecanum wheels and
discusses the kinematic relations of a platform used four
Mecanum wheels. Forward and Inverse kinematic is been
derived in this paper. Experimental and analytically results are
obtained and 8 different motions without changing the robots
orientation is achieved.
Figure 2 shows the configuration of a robot with four
omnidirectional wheels.
Keywords
Omnidirectional, four Mecanum wheeled robot, mobile robot,
kinematic.
1. INTRODUCTION
Omni-differential locomotion is being using in current mobile
robots in order to obtain the additional maneuverability and
productivity. These features are expanded at the expense of
improved mechanical complication and increased complexity
in control mechanism. Omni-differential systems work by
applying rotating force of each individual wheel in one
direction similar to regular wheels with a different in the fact
that Omni-differential systems are able to slide freely in a
different direction, in other word, they can slide frequently
perpendicular to the torque vector. The main advantage of
using Omni-drive systems is that translational and rotational
motions are decoupled for simple motion although in making
an allowance for the fastest possible motion this is not
essentially the case.
1.1 Mecanum Wheels
Mechanum wheel also called Ilon wheel and Swedish wheel is
a more common omnidirectional wheel designs, invented in
1973 by Bengt Ilon a Swedish engineer [1]. In this design,
similar to the Omni wheel, There are a series of free moving
rollers attached to the hub but with an 45 of angle about the
hub's circumference but still the overall side profile of the
wheel is circular. See Figure 1[2].
Omnidirectional motion can be reached by mounting four
Mecanum wheels on the corners of a four-sided base. Because
of the angled rollers, the mechanical design is much more
difficult, but due to the smoother transfer of contact surfaces a
higher loads can be supported [3].
Fig 2: Wheels Configuration and Posture definition
The configuration parameters and system velocities are
defined as follows:
, , , robots position (x, y) and its orientation angle
(The angle between X and );
X G Y, inertial frame; x,y are the coordinates of the reference
point O in the inertial basis;
, robots base frame; Cartesian coordinate system
associated with the movement of the body center;
, coordinate system of ith wheel in the wheels center
point ;
O, , the inertial basis of the Robot in Robots frame and
= { , } the center of the rotation axis of the wheel ;
, is a vector that indicates the distance between Robots
center and the center of the wheel th;
, , , half of the distance between front wheels and
half of the distance between front wheel and the rear wheels.
, distance between wheels and the base (center of the robot
O);
, denotes the radius of the wheel i
wheels center to the roller center)
(Distance of the
, denotes the radius of the rollers on the wheels.
, the angle between O and XR ;
Fig 1: Mecanum wheel design
International Journal of Computer Applications (0975 8887)
Volume 113 No. 3, March 2015
, the angle between S and XR ;
, the angle between and ;
1
0
1
0
0
.
1
= .
0
1
eq. 6
Where:
[rad/s], wheels angular velocity;
[/], = 0,1,2,3 , is
corresponding to wheel revolutions
the
velocity
vector
, the velocity of the passive roller in the wheel i;
.[ i ]T , Generalized velocity of point in the
frame ;
[ i ]T, Generalized velocity of point in the frame
;
x, y [m/s] - Robot linear velocity;
[rad/s] - Robot angular velocity;
eq. 7
From (eq.3) and (eq.5), the inverse kinematic model can be
obtained:
, = 0,1,2,3.
=
eq. 8
As 0 , 0 < < 2 , 0,
0 hence, by merging equations 4 and 6 the robots base
velocity (at point O) related to the rotational velocity of the ith
wheel can be obtained from eq. 9.
1 . 1 .
, = 0,1,2,3.
eq. 9
According to eq.3 and eq.4 there is a relationship between
variables in each robots wheels frames and its center. And
with the inverse kinematic, the velocity of the system can be
obtained by implementing ir the linear velocity and i the
rotational speed of wheel ith in eq.10 and the contrary in
eq.11.
Fig 3: Parameters of ith wheel
According to Figure 3(b), we can calculate the velocity of the
wheel i and the tangential velocity of the free roller attached
to the wheel touching the floor:
=
45
, = [4], = 0,1,2,3.
eq. 1
According to Figure 3(b) and considering the equations
(eq.1) , the velocity of the wheel in the frame , can be
derived by:
= + cos .
=
sin
cos
eq. 10
eq. 11
1 . 1 . ,
cos
sin
sin
cos
+ = ( )1 .
sin
cos
1
0
0
1
Considering the fact that = cos and = , and
assuming that the wheels are in a same size, the
transformation matrix is:
= sin .
Where =
= +
eq. 2
eq. 12
The transformation matrix from velocities of the ith wheel to
its center:
sin
.
cos
eq. 3
According to Figure 3(a) and Figure 2, the velocity of the
wheels center translated to the XROYR coordinate system can
be achieved by equation 7.
cos
sin
sin
cos
. eq. 4
Then, the transformation matrix from the ith wheels center to
the robot coordinates system can be obtained from equation
5.
cos
sin
sin
.
cos
[5]
eq. 5
eq. 13
Since there is a relation between independent variables
and in each joint and the systems angular and linear
velocity, assuming that there is no wheel slipping on the
ground, the system inverse kinematic can be obtained by
eq.14.
Since the robots motion is planar, we also have:
International Journal of Computer Applications (0975 8887)
Volume 113 No. 3, March 2015
1
2
1
3 =
4
cos (1 1 )
sin (1 1 )
1 sin (1 1 1 )
1
cos (2 2 )
1
sin (2 2 )
1
2 sin (2 2 2 )
2
cos (3 3 )
2
sin (3 3 )
2
3 sin (3 3 3 )
3
cos (4 4 )
3
sin (4 4 )
3
4 sin (4 4 4 )
By replacing the parameters of Table 1 in matrix (eq. 15) and
considering eq.14 we have come up with:
1 1
1
1
1
1
1 1
eq.14
eq.15 shows the Jacobian matrix for the systems inverse
kinematic:
cos (1 1 )
sin (1 1 )
1 sin (1 1 1 )
1
cos (2 2 )
1
sin (2 2 )
1
2 sin (2 2 2 )
2
cos (3 3 )
2
sin (3 3 )
2
3 sin (3 3 3 )
3
cos (4 4 )
3
sin (4 4 )
3
4 sin (4 4 4 )
= +
1
1
eq. 15
eq. 16
Analyzing the motion of a four Mecanum wheeled robot
brings out the following conclusion: According to the inverse
kinematic, there is a relationship between velocities in each
joint and the robots center velocity, thus, the velocity of the
robots center is reflected by and obtained from an individual
wheels velocity. According to the robot kinematic, inverse
kinematics can be achieved when the rank of the system is
less than the rank of the Jacobian matrix for each wheel of the
robot that reduces the degree of freedom of the robots joints.
Hence in a four Omni-differential design, the kinematic works
with following conditions:
R Jacobian full column rank, i.e. if rank (R) = 3, the
robot performs a better movement.
The rank of the Jacobian matrix column
dissatisfaction, i.e. if the rank (R) <3, the robot can
only move in a singular form and cannot achieve
all-directional movement.
3 =
1sw
2sw
3sw
4sw
4
4
4
4
2
2
2
2
eq. 18
( + )
( + )
.
( + )
( + )
eq. 19
eq. 20
And
1
1
=
1
4
( +
1
1
1
( + )
1
1
1
1
( + )
( + )
1
2
3
4
eq. 21
Longitudinal Velocity:
= 1 + 2 + 3 + 4 .
eq. 22
Transversal Velocity:
t = 1 + 2 + 3 4 .
eq. 23
Angular velocity:
= 1 + 2 3 + 4 .
4 +
eq. 24
The resultant velocity and its direction in the stationery
coordinate axis (x, y, z) can be achieved by the following
equations (eq. 25, 26):
+ .
eq. 25
eq. 26
4. EXPERIMENTAL RESULTS
Table 1. Robot Parameters
( + )
+ + .
( + )
+ + ,
Typical Mecanum four system shown in Figure 2; the
parameters of this configuration are shown in table 1. In this
configuration wheels sizes are the same.
1
1
+ + + ,
= tan1
( + )
1
1
+ ,
3. FOUR MECANUM
OMNIDIRECTIONAL SOLUTION
( + )
1 =
4 =
Wheels
1
1
1 1
1
1
1
1
1 1
1
2
1
3 =
4
2 =
2.1 The Relation between Motions and the
Translation MATRIX
eq. 17
According to equations (10) and (11) for Forward and Inverse
kinematics there is:
And for the forward kinematic according to the eq.10, we
have:
1
2
3
4
( + )
( + )
,
( + )
( + )
We used four Mecanum wheels in our project. The wheel
topology was the same as figure 2. The direction and the
velocity of the diagonal wheels were set independently. Using
the same speed in each wheel at the same time during the
operation led us to get eight directions for the robots motion
without changing its orientation. By changing the velocities of
the diagonal wheels we achieved a motion between 0 to
360. For example, to accomplish a transversal motion to the
right, the right wheels were rotated against each other
inwardly, while the left wheels were rotated against each other
outwardly (See Figure 4). By using the same technique we
achieved all eight different motions shown in Figure 4.
International Journal of Computer Applications (0975 8887)
Volume 113 No. 3, March 2015
backward
Fig 4: Motions of Omnidirectional platform
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Right
diagonal
backward
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5. CONCLUSION
In our test we assumed that there was no wheel sleep and by
driving the robot to the left we achieved 0.046 rad per second,
which was multiplied by each wheels velocity. The table
(table 2) shows the results of driving the robot in 8 different
directions without changing its orientation. In order to achieve
the same speed for diagonal directions the velocities of two of
the wheels were greater than other cases since the velocity in
the two other wheels was 0.
Table 2. Experimental and Analytical Results
A mobile platform with four omnidirectional wheels was
introduced in this paper. The results were systematically
obtained by using kinematic equations that were similar to
those achieved from the experimental results. The results
show that the platform performs full omnidirectional motions.
This shows that by using Mecanum wheels in the platform the
robot can achieve any direction between to without
changing its orientation.
6. REFERENCES
Direction
Wheel1
Forward
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Backward -5
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Left
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Right
-5
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-4.87
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Left
diagonal
forward
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Left
diagonal
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Wheel2 Wheel3
Right
diagonal
forward
Wheel4
[1] O. Diegel, A. Badve, G. Bright, J. Potgieter, and S. Tlale,
Improved Mecanum Wheel Design for Omni-directional
Robots, no. November, pp. 2729, 2002.
[2] I. Doroftei, V. Grosu, and V. Spinu, Omnidirectional
Mobile Robot - Design and Implementation,
Bioinspiration and Robotics Walking and Climbing
Robots, no. September. I-Tech, 2007.
[3] R. P. A. van Haendel, Design of an omnidirectional
universal mobile platform, National University of
Singapore, 2005.
[4] T. A. Baede, Motion control of an omnidirectional
mobile robot, 2006.
[5] X. Li and A. Zell, Motion control of an omnidirectional
mobile robot, 2006.
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