SUGISAKAMASANORI

Journal of Robotics, Networking and Artificial Life
Vol. 10(3); December (2023), pp. 275–281
ISSN (Online): 2352-6386; ISSN (Print): 2405-9021
https://alife-robotics.org/jrnal.html

Research Article

Kinetic Analysis of Omnidirectional Mobile Robot with Symmetry
Roller’s Arrangement
Kenji Kimura1, Kazuo Ishii2

Department of Control Engineering, National Institute of Technology, Matsue College,4-4 Nishi-ikuma-cho, Matsue-shi, Shimane, 690-8518, Japan
Graduate School of Life Science and engineering, Kyushu Institute of Technology, 2-4 Hibikino, Wakamatsu-ku, Kitakyushu-shi 808-0196, Fukuoka, Japan

1
2

ARTICLE INFO
Article History
Received 10 November 2023
Accepted 18 May 2024
Keywords
Mobile robot
Omni-roller
Translational motion
Motion analysis of mobile robot

ABSTRACTIn recent years, the efficient mobility of omnidirectional mobile robots is desired in the logistics
industry. Hence, various types of rollers are installed in mobile robots. These include
omnidirectional rollers, which are characterized with excellent omnidirectional mobility and areconsidered easy to control. Herein, considering that the side surface of the mechanism is circular,
the kinematics of the mechanism assuming an arbitrary arrangement on this circle is derived.
Previous research has evaluated the roller arrangement from the perspective of the speed
efficiency of the drive roller. Meanwhile, this research focuses on a mobile robot equipped withthree omni-rollers. Furthermore, focusing only on the translational speed of the robot, its motion
efficiency is evaluated. The distribution of the robot’s translational speed is studied, the area ofthe region is used as the evaluation function, and its behaviors are analyzed.
© 2022 The Author. Published by Sugisaka Masanori at ALife Robotics Corporation Ltd.
This is an open access article distributed under the CC BY-NC 4.0 license(http://creativecommons.org/licenses/by-nc/4.0/).

[9] have been reported in recent times, focusing on thedriving rollers.
In moving mechanisms that use spheres as wheels (ormechanisms that use spheres as transport objects), theoptimum angle of the sphere rotation axis with respect tomovement direction has been determined, assuming anomnidirectional movement [7]. Furthermore, theoptimum location for the two rollers driving the sphere isdemonstrated to be at the equator [8].
Meanwhile, in the mobile mechanism that uses rollers
as wheels, kinematics is represented using a linear
transformation matrix, which maps the input (roller
velocity) to the output (robot velocity) (translation androtation components) [9]. Additionally, mechanisms witha line-symmetrical roller arrangement, as shown in Fig.
1(b)(a)(c), are analyzed. In [10], translational androtational motions are analyzed by adopting the “volume”
of the image through first-order transformations and“area of positive projection” as criteria for evaluating the
speed efficiency in movement.
In [11], an evaluation function was derived for a threeroller configuration with a line-symmetrical structure andwas used to analyze only translational motion. However,
the analysis for mechanisms with an origin-symmetrical

1. Introduction
Recently, industries, such as logistics, have been
looking forward to the practicality of using
omnidirectional moving mechanisms, and several mobile
robots have been developed with this aim. Holonomic
mobile robots have attracted attention owing to their easeof control. This is because omnidirectional movement has
three degrees of freedom, expressed as the sum of a twodimensional translational component and a onedimensional rotational component, which can becontrolled independently. Thus, many movement
mechanisms (with three omni-rollers) have beendeveloped, and research on its kinematics and control has
been conducted [1].
In this research, the RoboCup meddle-size-league
soccer robot is characterized with an omnidirectional
moving mechanism. The RV-infinity [2], Musashi150 [3],and NuBot [4] use the most basic triangular arrangement(equilateral triangle). Although no theoretical studies
have been conducted on roller arrangements, studies on
sphere transport [5], [6] and moving mechanisms [7], [8],Corresponding author E-mail: k-kimura@matsue-ct.jp, ishii@brain.kyutech.ac.jp

275

roller arrangement, as shown in Fig.1(d)(a)(e), remains
unsolved.
In this study, the efficiency of the mobile robot(equipped with three omni-rollers), as shown in Fig.
1(d)(a)(e), is investigated for only the translational
movement. For this purpose, an evaluation function isderived for the general arrangement. The distribution of
the translational velocity of the robot is also investigated,
and the area of the region is used as an evaluation
function to analyze the robot’s behaviors.
The rest of this study is as follows. Section 2 discusses
the kinematics of mobile robots corresponding totransformation mapping. Section 3 derives a moregeneral sectional area function. Section 4 conducts thesimulation. Finally, Section 5 presents the summary andfuture tasks.

Fig. 2 Correspondence omni-roller speed [𝜈𝜈1 , 𝜈𝜈2 , 𝜈𝜈3 ]𝑻𝑻̇ �𝑻𝑻 : (a). mobile
and robot speed elements �𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 , 𝜙𝜙𝜙𝜙
robot and (b). transformation mapping.
𝑓𝑓𝐴𝐴−1 (𝑊𝑊) ∩ �𝑉𝑉𝑥𝑥 𝑉𝑉𝑦𝑦 − plane�= �(𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 , 0)� |𝜈𝜈1 |, |𝜈𝜈2 |, |𝜈𝜈3 | ≤ 1�(1)

𝑓𝑓𝐴𝐴−1 (𝑊𝑊) = �(𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 , 𝐿𝐿𝜙𝜙̇)� |𝜈𝜈1 |, |𝜈𝜈2 |, |𝜈𝜈3 | ≤ 1�

(2)

Where

= {(𝜈𝜈1 , 𝜈𝜈2 , 𝜈𝜈3 )||𝜈𝜈1 |, |𝜈𝜈2 |, |𝜈𝜈3 | ≤ 1}

(3)

̇ �𝑻𝑻 →
Inverse kinematics ( 𝑓𝑓𝐴𝐴 : �𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 , 𝜙𝜙𝜙𝜙[𝜈𝜈1 , 𝜈𝜈2 , 𝜈𝜈3 ]𝑻𝑻 ) is represented as the following Eq. (4).
𝜈𝜈1− sin 𝜃𝜃1
�𝜈𝜈2 � = �− sin 𝜃𝜃2
𝜈𝜈3− sin 𝜃𝜃3

Fig. 1 Mobile robots with three omni-roller arrangement(b)(a)(c), line-symmetry roller arrangement. (d)(e), andpoint-symmetry roller arrangement.

1 𝑉𝑉𝑥𝑥1� � 𝑉𝑉𝑦𝑦 �
1 𝐿𝐿𝜙𝜙̇(4)

Additionally, forward kinematics ( 𝑓𝑓𝐴𝐴−1 :̇ �𝑻𝑻 ) is represented using Eq. (5).
[𝜈𝜈1 , 𝜈𝜈2 , 𝜈𝜈3 ]𝑻𝑻 → �𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 , 𝜙𝜙𝜙𝜙
𝑉𝑉𝑥𝑥
− sin 𝜃𝜃1
1𝑉𝑉� 𝑦𝑦 � =�− sin 𝜃𝜃2det𝑨𝑨− sin 𝜃𝜃3𝐿𝐿𝜙𝜙̇

2. Transformation mapping and kinematics
As shown in Fig. 2(a), the mobile robot with a common
radius 𝐿𝐿 adapted the i-th rollers (𝑖𝑖 = 1, 2, 3) at contact
point 𝑷𝑷𝒊𝒊 , which is positioned as angle 𝜃𝜃𝑖𝑖 on a global 𝑋𝑋–𝑌𝑌
coordinate system (origin 𝑶𝑶).
Here, the roller peripheral speed [𝜈𝜈1 , 𝜈𝜈2 , 𝜈𝜈3 ]𝑻𝑻 is given,cos 𝜃𝜃1 1
cos 𝜃𝜃2 1�
cos 𝜃𝜃3 1

−𝟏𝟏

𝜈𝜈1

�𝜈𝜈2 �𝜈𝜈3(5)

3. Analysis for sectional shape and area
As a special case, a sectional area is analyzed in thecase of isosceles triangle-like–three-roller arrangement in[11]. Meanwhile, in this study, a more general case isanalyzed regarding sectional shapes.

𝑻𝑻while the robot translation speed 𝑽𝑽 = �𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 � andrational speed 𝐿𝐿𝜙𝜙̇ ( 𝜙𝜙̇ : robot angular velocity) can be𝑻𝑻calculated. Thus, �𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 , 𝐿𝐿𝜙𝜙̇� is determined.
As shown in Fig. 2(b), correspondences oḟ � are represented using[𝜈𝜈1 , 𝜈𝜈2 , 𝜈𝜈3 ] and �𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 , 𝜙𝜙𝜙𝜙
̇ �𝑻𝑻 →
linear transformation mapping as 𝑓𝑓𝐴𝐴 :�𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 , 𝜙𝜙𝜙𝜙[𝜈𝜈1 , 𝜈𝜈2 , 𝜈𝜈3 ]𝑻𝑻 . 𝑓𝑓𝐴𝐴−1 (𝑊𝑊) = Image 𝑓𝑓𝐴𝐴−1 ∈ 𝐑𝐑3 is aparallelepiped domain from cubic domain 𝑊𝑊.
Furthermore, the sectional domain of 𝑓𝑓𝐴𝐴−1 (𝑊𝑊) forhorizontal plane �𝑉𝑉𝑥𝑥 𝑉𝑉𝑦𝑦 − plane� is as follow.

cos 𝜃𝜃1cos 𝜃𝜃2cos 𝜃𝜃3

3.1. Setup for domain and division276

Cubic domain 𝑊𝑊 is composed following 8-apexes
𝑷𝑷 = (1, 1, 1), 𝑸𝑸 = (−1, 1, 1), 𝑹𝑹 = (−1, −1, 1), 𝑺𝑺 =(1,−1, 1),𝑷𝑷́ = (1, 1, −1), 𝑸𝑸́ = (−1, 1, −1), 𝑹𝑹́ = (−1,−1, −1), 𝑺𝑺́ = (1, −1, −1).
Focusing on 𝐿𝐿𝜙𝜙̇-component of Eq. (5), we define 𝜃𝜃3 =0°.

Where𝜈𝜈1𝑛𝑛1
𝐹𝐹(𝜈𝜈1 , 𝜈𝜈2 , 𝜈𝜈3 ) = �𝜈𝜈2 � ∙ �𝑛𝑛2 �
𝜈𝜈3
𝑛𝑛3

(6)

𝑛𝑛1 = − sin 𝜃𝜃2(7)

𝑛𝑛3 = −sin(𝜃𝜃1 − 𝜃𝜃2 )(9)𝑛𝑛2 = sin 𝜃𝜃1(8)

In the case of translational motion, 𝜙𝜙̇ = 0 is equivalent
to 𝐹𝐹(𝜈𝜈1 , 𝜈𝜈2 , 𝜈𝜈3 ) = 0, which is a linear equation withrespect to 𝜈𝜈1 , 𝜈𝜈2 , and 𝜈𝜈3 . Therefore, geometrically, it
can be represented as a plane with an origin andnormal vector 𝒏𝒏 = [𝑛𝑛1 , 𝑛𝑛2 , 𝑛𝑛3 ]𝑻𝑻 . Thus, the arrangement
position of roller is completely dependent on 𝒏𝒏.
As shown in Fig. 3, 𝐷𝐷 = {(𝜈𝜈1 , 𝜈𝜈2 )| |𝜈𝜈1 |, |𝜈𝜈2 | ≤ 1} is asectional domain of 𝑊𝑊 by 𝜈𝜈3 = 0 and can be
decomposed to the following four parts: 𝐷𝐷++ , 𝐷𝐷−+ , 𝐷𝐷−− ,and 𝐷𝐷+− .
𝐷𝐷++ = {(𝜈𝜈1 , 𝜈𝜈2 )| 0 < 𝜈𝜈1 ≤ 1, 0 < 𝜈𝜈2 ≤ 1}

(10)𝐷𝐷−− = {(𝜈𝜈1 , 𝜈𝜈2 )|−1 < 𝜈𝜈1 ≤ 0, −1 < 𝜈𝜈2 ≤ 0}

(12)𝐷𝐷−+ = {(𝜈𝜈1 , 𝜈𝜈2 )| −1 < 𝜈𝜈1 ≤ 0, 0 < 𝜈𝜈2 ≤ 1}

(11)

𝐷𝐷+− = {(𝜈𝜈1 , 𝜈𝜈2 )| 0 < 𝜈𝜈1 ≤ 1, −1 < 𝜈𝜈2 ≤ 0}

(13)

Fig. 3 Three-roller arrangement poteen corresponding to−+domain 𝐷𝐷++and 𝐷𝐷++.

In this study, we focus on domain 𝐷𝐷++ (grey highlightin Fig. 3). It can be decomposed into two parts by a signof 𝑛𝑛3 . From 0° < 𝜃𝜃1 , 𝜃𝜃2 < 360°, and 𝜃𝜃1 ≠ 𝜃𝜃2 , (𝜃𝜃1 , 𝜃𝜃2 )is satisfied with following inequality.
−360° < 𝜃𝜃1 − 𝜃𝜃2 < 360°, 𝜃𝜃1 ≠ 𝜃𝜃2(14)

𝜃𝜃1 < 𝜃𝜃2 < 180° + 𝜃𝜃1(15)

180° + 𝜃𝜃1 < 𝜃𝜃2 < 360° + 𝜃𝜃1(16)

Fig. 4 Classification of the sectional shape of a cube in the case−: (a). hexagon and (b). rhombus shapes.
of domain 𝐷𝐷++

In the case of 𝑛𝑛3 > 0, it is equivalent to 180° < 𝜃𝜃1 −𝜃𝜃2 < 360° and −180° < 𝜃𝜃1 − 𝜃𝜃2 < 0° ( −360° + 𝜃𝜃1 <𝜃𝜃2 < −180° + 𝜃𝜃1 ). Thus, the following inequality is
obtained.
In the case of 𝑛𝑛3 < 0, it is equivalent to 0° < 𝜃𝜃1 −𝜃𝜃2 < 360° and −360° < 𝜃𝜃1 − 𝜃𝜃2 < −180° ( −180° +𝜃𝜃1 < 𝜃𝜃2 < 𝜃𝜃1 ). Thus, the following inequality is given.

Using Eq. (15) and Eq. (16), 𝐷𝐷++ can be decomposed−+
to domains 𝐷𝐷++and 𝐷𝐷++
.
−
𝐷𝐷++
(17)
= {(𝜃𝜃1 , 𝜃𝜃2 )| 𝜃𝜃1 < 𝜃𝜃2 ≤ 360° , 0° < 𝜃𝜃1 ≤ 180°}

Fig. 5 Classification of sectional in 𝐷𝐷++ (a). The case of−+hexagon domain 𝐷𝐷++(b), hexagon domain 𝐷𝐷++(c), andrhombus domain 𝐷𝐷++

+
𝐷𝐷++
(18)
= {(𝜃𝜃1 , 𝜃𝜃2 )| 180° < 𝜃𝜃2 ≤ 𝜃𝜃1 , 0° < 𝜃𝜃1 ≤ 180°}277

3.2. Analysis of sectional shapes

three triangle areas (Fig. 5(a)(b)). Furthermore, in thecase of rhombus, 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (𝜃𝜃1 , 𝜃𝜃2 ) is derived as the sum of
the same eight triangle areas (Fig. 5(c)).

−
is considered.
The sectional shape condition in 𝐷𝐷++

From Eq. (17), (𝜃𝜃1 , 𝜃𝜃2 ) is satisfied because 0° < 𝜃𝜃1 ≤180° and 180° < 𝜃𝜃2 ≤ 360°.
In cases where 𝑛𝑛3 ≠ 0 and 𝜃𝜃2 − 𝜃𝜃1 < 180° ,𝐹𝐹(1,1,1) and 𝐹𝐹(1,1, −1) are calculated as follows:𝐹𝐹(1,1,1) = sin 𝜃𝜃1 − sin 𝜃𝜃2 − sin(𝜃𝜃1 − 𝜃𝜃2 ) > 0

𝐹𝐹(1,1, −1) = sin 𝜃𝜃1 − sin 𝜃𝜃2 + sin(𝜃𝜃1 − 𝜃𝜃2 )= sin 𝜃𝜃1 + sin(−𝜃𝜃2 ) + sin(𝜃𝜃1 + (−𝜃𝜃2 ))+( |𝜃𝜃1 − 𝜃𝜃2 | ≠ 180°, (𝜃𝜃1 , 𝜃𝜃2 ) ∈ 𝐷𝐷++) (24)− (𝜃𝜃𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆
1 , 𝜃𝜃2 ) = �𝑓𝑓𝑨𝑨−1 (𝑻𝑻𝑷𝑷𝑷𝑷 ) × 𝑓𝑓𝑨𝑨−1 �𝑻𝑻𝑸𝑸𝑸𝑸 ��+ �𝑓𝑓𝑨𝑨−1 �𝑻𝑻𝑸𝑸𝑸𝑸 �𝑏𝑏 × 𝑓𝑓𝑨𝑨−1 �𝑻𝑻𝑸𝑸𝑸𝑸́ ��(21)= sin 𝜃𝜃1 (1 + cos(−𝜃𝜃2 ))(23)+ sin(−𝜃𝜃2 ) (1 + cos 𝜃𝜃1 ) > 0+ ‖𝑓𝑓𝑨𝑨−1 (𝑻𝑻𝑺𝑺𝑺𝑺́ ) × −𝑓𝑓𝑨𝑨−1 (−𝑻𝑻𝑹𝑹𝑹𝑹 )‖

(20)

(22)

From 1 + cos(−𝜃𝜃2 ) > 0 and 1 + cos 𝜃𝜃1 > 0 ,+�𝑓𝑓𝑨𝑨−1 �𝑻𝑻𝑸𝑸𝑸𝑸 � × 𝑓𝑓𝑨𝑨−1 �𝑻𝑻𝑸𝑸𝑸𝑸́ ��(19)= sin 𝜃𝜃1 + sin(−𝜃𝜃2 ) + sin 𝜃𝜃1 cos(−𝜃𝜃2 )+ cos 𝜃𝜃1 sin(−𝜃𝜃2 )+ (𝜃𝜃𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆
1 , 𝜃𝜃2 ) = �𝑓𝑓𝑨𝑨−1 (𝑻𝑻𝑹𝑹𝑹𝑹 ) × 𝑓𝑓𝑨𝑨−1 �𝑻𝑻𝑸𝑸𝑸𝑸 ��+ ‖𝑓𝑓𝑨𝑨−1 (𝑻𝑻𝑺𝑺𝑺𝑺́ ) × −𝑓𝑓𝑨𝑨−1 (−𝑻𝑻𝑷𝑷𝑷𝑷 )‖−( |𝜃𝜃1 − 𝜃𝜃2 | ≠ 180°, (𝜃𝜃1 , 𝜃𝜃2 ) ∈ 𝐷𝐷++) (25)𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (𝜃𝜃1 , 𝜃𝜃2 ) = 4‖𝑓𝑓𝑨𝑨−1 (𝑻𝑻𝑹𝑹𝑹𝑹 ) × 𝑓𝑓𝑨𝑨−1 (0, 0, 1)‖( |𝜃𝜃1 − 𝜃𝜃2 | = 180°, (𝜃𝜃1 , 𝜃𝜃2 ) ∈ 𝐷𝐷++ ) (26)

Where 𝑻𝑻𝑹𝑹𝑹𝑹 (𝑡𝑡𝑅𝑅𝑅𝑅 , −1, 1), 𝑻𝑻𝑸𝑸𝑸𝑸 (−1,𝑡𝑡𝑄𝑄𝑄𝑄 ,1), 𝑻𝑻𝑺𝑺𝑺𝑺 (1,𝑡𝑡𝑆𝑆𝑆𝑆 ,1),

Thus, 𝑷𝑷(1,1,1) and 𝑷𝑷́(1,1, −1) are up word on ofplane’s normal vector direction. Thus, the sectional shapeof the domain is hexagon (Fig. 4(a)).
In the case where 𝑛𝑛3 = 0, 𝒏𝒏 is parallel to �𝑉𝑉𝑥𝑥 𝑉𝑉𝑦𝑦 −plane�. Thus, the sectional shape of the domain isrhombus (Fig. 4(b)).
𝜋𝜋 ∩ 𝑊𝑊 is satisfied as the following property.
Property1
The sectional shape has a line symmetry with respect tothe origin and can be divided into the following twoclasses:

𝑻𝑻𝑷𝑷𝑷𝑷 (𝑡𝑡𝑃𝑃𝑃𝑃 , 1, 1), and 𝑻𝑻𝑸𝑸𝑸𝑸́ (−1, 1, 𝑡𝑡𝑄𝑄𝑄𝑄́ ).
𝑡𝑡𝑅𝑅𝑅𝑅 =𝑡𝑡𝑄𝑄𝑄𝑄 =− sin 𝜃𝜃2 + sin(𝜃𝜃1 − 𝜃𝜃2 )sin 𝜃𝜃1

𝑡𝑡𝑆𝑆𝑆𝑆 =|𝜃𝜃1 − 𝜃𝜃2 | ≠ 180° ⟹ Hexagon

𝑡𝑡𝑃𝑃𝑃𝑃 =|𝜃𝜃1 − 𝜃𝜃2 | = 180° ⟹ Rhombus

3.3. Derivation of sectional area functionis
𝜋𝜋 ∩ 𝑊𝑊 = Hexagon 𝑻𝑻𝑹𝑹𝑹𝑹 𝑻𝑻𝑺𝑺𝑺𝑺́ 𝑻𝑻𝑷𝑷́𝑺𝑺́ 𝑻𝑻𝑷𝑷́𝑸𝑸́ 𝑻𝑻𝑸𝑸𝑸𝑸́ 𝑻𝑻𝑸𝑸𝑸𝑸
decomposed to the following six parts: △ 𝑶𝑶 𝑻𝑻𝑹𝑹𝑹𝑹 𝑻𝑻𝑸𝑸𝑸𝑸 , △𝑶𝑶 𝑻𝑻𝑸𝑸𝑸𝑸 𝑻𝑻𝑸𝑸𝑸𝑸́ , △ 𝑶𝑶 𝑻𝑻𝑸𝑸𝑸𝑸́ 𝑻𝑻𝑹𝑹𝑹𝑹 , △ 𝑶𝑶 𝑻𝑻𝑷𝑷́𝑺𝑺́ 𝑻𝑻𝑷𝑷́𝑸𝑸́ , △ 𝑶𝑶 𝑻𝑻𝑷𝑷́𝑺𝑺́ 𝑻𝑻𝑺𝑺𝑺𝑺́ ,and △ 𝑶𝑶 𝑻𝑻𝑸𝑸𝑸𝑸́ 𝑻𝑻𝑷𝑷́𝑸𝑸́ . Furthermore, the transformation
mapping of 𝑓𝑓𝑨𝑨−1 shows that 𝑓𝑓𝑨𝑨−1 (𝑶𝑶) = 𝑶𝑶 and𝑓𝑓𝑨𝑨−1 (𝜋𝜋 ∩ 𝑊𝑊) is represented as sum of △𝑶𝑶 𝑓𝑓𝑨𝑨−1 (𝑻𝑻𝑹𝑹𝑹𝑹 )𝑓𝑓𝑨𝑨−1 �𝑻𝑻𝑸𝑸𝑸𝑸 �, △ 𝑶𝑶 𝑓𝑓𝑨𝑨−1 �𝑻𝑻𝑸𝑸𝑸𝑸 �𝑓𝑓𝑨𝑨−1 �𝑻𝑻𝑸𝑸𝑸𝑸́ �, △𝑶𝑶 𝑓𝑓𝑨𝑨−1 �𝑻𝑻𝑸𝑸𝑸𝑸́ �𝑓𝑓𝑨𝑨−1 (𝑻𝑻𝑹𝑹𝑹𝑹 ) , △ 𝑶𝑶 𝑓𝑓𝑨𝑨−1 (𝑻𝑻𝑷𝑷́𝑺𝑺́ )𝑓𝑓𝑨𝑨−1 �𝑻𝑻𝑷𝑷́𝑸𝑸́ � , △𝑶𝑶 𝑓𝑓𝑨𝑨−1 (𝑻𝑻𝑷𝑷́𝑺𝑺́ )𝑓𝑓𝑨𝑨−1 (𝑻𝑻𝑺𝑺𝑺𝑺́ ), and △ 𝑶𝑶 𝑓𝑓𝑨𝑨−1 �𝑻𝑻𝑸𝑸𝑸𝑸́ �𝑓𝑓𝑨𝑨−1 �𝑻𝑻𝑷𝑷́𝑸𝑸́ �.
Area of 𝑓𝑓𝑨𝑨−1 (𝜋𝜋 ∩ 𝑊𝑊) is defined as 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (𝜃𝜃1 , 𝜃𝜃2 )((𝜃𝜃1 ,𝜃𝜃2 ) ∈ 𝐷𝐷++ ). It is decomposed into the following
+
+ (𝜃𝜃
−
𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆
1 , 𝜃𝜃2 ) ( (𝜃𝜃1 , 𝜃𝜃2 ) ∈ 𝐷𝐷++ ) and 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (𝜃𝜃1 , 𝜃𝜃2 )−((𝜃𝜃1 ,𝜃𝜃2 ) ∈ 𝐷𝐷++ ), which are function depended on roller
contact position (𝜃𝜃1 , 𝜃𝜃2 , 0°).
The opposite sides are parallel and have the samelength. Thus, hexagon 𝑓𝑓𝑨𝑨−1 (𝜋𝜋 ∩ 𝑊𝑊) has a symmetry
with respect to origin 𝑶𝑶. Thus, in the case of hexagon,+ (𝜃𝜃−𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆
1 , 𝜃𝜃2 ) and 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (𝜃𝜃1 , 𝜃𝜃2 ) are derived as the sum of− sin 𝜃𝜃1 − sin(𝜃𝜃1 − 𝜃𝜃2 )sin 𝜃𝜃2sin 𝜃𝜃2 + sin(𝜃𝜃1 − 𝜃𝜃2 )sin 𝜃𝜃1sin 𝜃𝜃1 − sin(𝜃𝜃1 − 𝜃𝜃2 )sin 𝜃𝜃2

𝑡𝑡𝑄𝑄𝑄𝑄́ =4.

sin 𝜃𝜃1 + sin 𝜃𝜃2sin(𝜃𝜃1 − 𝜃𝜃2 )

(27)

(28)

(29)

(30)

(31)

Simulation

This section presents the simulation findings,
including the horizontal cross-sectional area 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 , asevaluation values.
Section 4.1 shows the case of a roller arrangementthat has a line symmetry with respect to the 𝑥𝑥-axis.
Section 4.2 shows the case of a roller arrangementthat has a point symmetry with respect to the origin.
Simulations were performed on three various roller
arrangement patterns. Types (A)-(E) are arranged in the
shape of a triangle, corresponding to Fig. 1(a)-(e),respectively.

278

4.1. Case of line-symmetry arrangement with respect tothe 𝒙𝒙-axis
Fig. 6 (b)(a)(c) shows the shape in the case of a triangle
that has a line symmetry with respect to the 𝑥𝑥 -axis.

+ (𝜃𝜃,2𝜋𝜋 − 𝜃𝜃) (substituting Eq. (24) to 𝜃𝜃1 = 𝜃𝜃 and
𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆− (𝜃𝜃,
2𝜋𝜋 − 𝜃𝜃)𝜃𝜃2 = 2𝜋𝜋 − 𝜃𝜃 for 90° ≤ 𝜃𝜃 < 180°) and 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆(substituting Eq. (25) to 𝜃𝜃1 = 𝜃𝜃 and 𝜃𝜃2 = 2𝜋𝜋 − 𝜃𝜃 for
0° ≤ 𝜃𝜃 < 90°) are analyzed. They are set up as 𝜃𝜃𝑖𝑖 (𝑖𝑖 =
1,2,3) as follows:
Type(B):(𝜃𝜃1 , 𝜃𝜃2 , 𝜃𝜃3 ) = (60°, 300°, 0°)
Type(A):(𝜃𝜃1 , 𝜃𝜃2 , 𝜃𝜃3 ) = (90°, 270°, 0°)
Type(C): (𝜃𝜃1 , 𝜃𝜃2 , 𝜃𝜃3 ) = (120°, 240°, 0°) and 𝐿𝐿 =1[m].
Fig. 6(d)(a)(e) show an outline of the velocity
distributions for Type(A), Type(B), and Type(C). Theirshapes are hexagon, square, and hexagon, respectively.
Fig. 7 shows the shape in the case of a triangle that hasa point symmetry with respect to origin 𝑶𝑶 . 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (𝜃𝜃, 2𝜋𝜋 −𝜃𝜃) (substituting Eq. (26) to 𝜃𝜃1 = 𝜃𝜃 and 𝜃𝜃2 = 2𝜋𝜋 − 𝜃𝜃 for90° ≤ 𝜃𝜃 < 180°), as a result, 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (90°) = 4.05[𝑚𝑚/𝑠𝑠]2
and it takes minimal value 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (60°) and 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (120°) =3.85 [𝑚𝑚/𝑠𝑠]2 . Moreover, 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (0°) = ∞ and
𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (180°) = ∞.
The result shows that Type(B) and Type(C) are therollers arrangement with poorest efficiency for onlytranslational motion.

− (𝜃𝜃,
Fig. 7 Behaviors of 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆
2𝜋𝜋 − 𝜃𝜃) ( 0° ≤ 𝜃𝜃 < 180°)+ (𝜃𝜃,and 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 2𝜋𝜋 − 𝜃𝜃)(90° ≤ 𝜃𝜃 < 180°).

Fig. 8 Behaviors of 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (𝜃𝜃, 2𝜋𝜋 − 𝜃𝜃)(0° ≤ 𝜃𝜃 < 180°)

4.2. Case of point-symmetry arrangement with respect
to origin
Fig. 6(d)(a)(e) shows the shape in the case of a triangle
that has a point symmetry with respect to origin 𝑶𝑶.
𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (𝜃𝜃, 2𝜋𝜋 − 𝜃𝜃) (substituting Eq. (26) to 𝜃𝜃1 = 𝜃𝜃 and𝜃𝜃2 = 2𝜋𝜋 + 𝜃𝜃 for 0° ≤ 𝜃𝜃 < 180°) is analyzed and is set
up as 𝜃𝜃𝑖𝑖 (𝑖𝑖 = 1,2,3) as follows:
Type(D): (𝜃𝜃1 , 𝜃𝜃2 , 𝜃𝜃3 ) = (60°, 240°, 0°)
Type(A): (𝜃𝜃1 , 𝜃𝜃2 , 𝜃𝜃3 ) = (90°, 270°, 0°)
Type(E): (𝜃𝜃1 , 𝜃𝜃2 , 𝜃𝜃3 ) = (120°, 300°, 0°)Fig. 6 shows an outline of velocity distributions forType(D), Type(A), and Type(E), represented as hexagon,square, and hexagon, respectively.
As shown in Fig. 8, it is symmetry function andtakes minimal value 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (90°) = 4.05[𝑚𝑚/𝑠𝑠]2 . In
addition, 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (0°) and 𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆 (180°) = ∞.
The result shows that Type(A) is the smallest andefficient when only translational motion is assumed.

279

(d) Type(D) (𝜃𝜃1 , 𝜃𝜃2 ) = (60°, 240°)(c) Type(C) (𝜃𝜃1 , 𝜃𝜃2 ) = (60°, 300°)(a) Type(A) (𝜃𝜃1 , 𝜃𝜃2 ) = (90°, 270°)(b) Type(B) (𝜃𝜃1 , 𝜃𝜃2 ) = (120°, 240°)(e) Type(E) (𝜃𝜃1 , 𝜃𝜃2 ) = (120°, 300°)

Fig. 6 Simulation result for the roller arrangement in the case of symmetry (a)(b)(c), line symmetry. (a)(d)(e), andpoint symmetry

References

5. Conclusion

1.

In this study, a function was successfully derived toevaluate the efficiency of only the translational
movement in a mobile mechanism equipped with threeomni-rollers. Furthermore, the distribution of thetranslational velocity of the robot was analyzed throughsimulation, and the roller arrangement with the poorest
efficiency was clarified.
The future works will focus on the efficiency of
movement when the roller direction is changed arbitrarily
in mobile robots.

2.
3.

4.

Acknowledgment
This work was supported by JSPS Grant-in-Aid for
Scientific Research 23 K13279.

5.

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Authors Introduction
Dr. Kenji Kimura

He is a Lecturer at the Department ofControlEngineering,
National
Institute of Technology, Matsue
College, and a Visiting Associate
Professor at Chuo University. He
received his ME (mathematics) from
Kyushu University in 2002 and his
PhD (engineering) from the Kyushu Institute ofTechnology in 2020. His research interests are sphericalmobile robots.
Dr. Kazuo Ishii

He is a Professor in the Kyushu
Institute of Technology, where he has
been since 1996. He received his
Ph.D. degree in engineering from
University of Tokyo, in 1996. His
research interests span both ship
marine engineering and Intelligent
Mechanics. He holds five patents
derived from his research.

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