SUGISAKAMASANORI

Journal of Advances in Artificial Life Robotics
Vol. 3(4); March (2023), pp. 242–249
ON LINE ISSN 2435-8061; ISSN-L 2435-8061
https://alife-robotics.org/jallr.html

Research Article

An Analysis of Robot Speed Efficiency for Mobile Robot Adapted
Three Omni Rollers Using Linear Transformation
Kenji Kimura1, Yuki Shigyo2, Kazuo Ishii3

Department of Control Engineering, National Institute of Technology, Matsue College, 14-4 Nishi-ikuma-cho, Matsue-shi, Shimane, 690-8518, Japan
Digital Solution Div, DX Engagement Dept, Fujitsu, 3-5-20 Minamikamata. Ota City, Tokyo 144-0035, Japan
3
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 I N FO
Article History
Received 10 November,2022
Accepted 05 July 2023
Keywords
Omni-roller
Transformation matrix
Motion analysis of mobile robot

ABSTRACT

RoboCup Middle-sized-League soccer robots are mainly equipped with the three omni-roller
mechanism. These mobile robots are expected to function efficiently in fields, such as logistics.
These systems are easy to control and allow omnidirectional movement. However, theoreticalresearch on these systems’ exercise efficiencies has not been conducted. In this research, we
assume a mechanism that can arbitrarily change the roller arrangement based on a circularmechanism and evaluate the roller arrangement from a speed efficiency perspective. Additionally,
we define an evaluation function using the theory of linear transformation to examine the roller
arrangement.

© 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/).

1. Introduction
RoboCup soccer robot incorporates two importantelements. The first is a ball holding mechanism that
dribbles the ball, and the second is a moving
mechanism using three omni rollers. There arestudies on ball holding mechanisms, such as RVinfinity [1], Musashi150 [2], and NuBot [3]. Studies
on a sphere driven by two rollers were conducted [4]、[5], as research on kinematics. Furthermore,
numerous studies are focused on the kinetic energyof rollers for efficient spherical transportation [6], [7],[8], [9].
Considering omnidirectional movement, mobile
robots have recently been developed in various fields,
including physical distribution. Omnidirectionalmotion with nonholonomic or holonomic properties
can produce three degrees of freedom of motion (i.e.,

two translational degrees of freedom and one
rotational degree of freedom). Among them, theholonomic movement mechanism is easy to control
as the wheels are independently driven and haveexcellent mobility in all directions. Using thisproperty, a mobile robot with an equilateral triangle
roller arrangement was developed [10].
Figure 1 shows three omni layout patterns of mobile
robots. Particularly, many of the layout structures ofType (X), adopted in Figure 1(a), are also found in
RoboCup MSL robots [1], [2], [3]. Moreover, Type(Y) and Type (Z) presented in Figures 1(b) and (c)are conceivable. However, not only the RoboCup
MSL robot but also other mobile robots adopt theregular polygon arrangement, which is the mostCorresponding author’s E-mail: k-kimura@matsue-ct.jp, shigyo.yuki@fujitsu.com, ishii@brain.kyutech.ac.jp242

symmetrical polygon, as shown in Figure 1(a).
Besides, no theoretical research has been conductedon this kinematics
In this research, we consider generalizing theconventional equilateral triangle arrangement [11].
Therefore, we derive general kinematics assuming amechanism can arbitrarily change the position of rollers
using a circular mechanism, derive a transformation
matrix that associates input and output, and adopt thevalue of the determinant as the evaluation function.
Additionally, the roller arrangement is evaluated in termsof speed efficiency. Here, “image volume” and“orthographic projection area” are adopted as evaluation
functions.
The rest of this study is as follows: Chapter 2 discusses
the kinematics of mobile robots and defines an evaluation
function. Chapter 3 conducted the simulation. Finally, we
present the summary and future tasks.
Figure 1 Example of an isosceles triangle in three rollers
arrangement pattern for mobile robots.

2. General mobile robot kinematics for threeomni roller
In this section, we derive kinematics (inverse andforward) in a mechanism that can change the positionof rollers arbitrarily. Assuming motion on ahorizontal plane, the kinematics of the robot withreference [11] are similar. Here, the focus is on therelationship between the rotational speed of thedriven omni-wheel and the speed of the robot.
Table 1 shows variables list of this research. Figure
2 shows parameters for mobile robot adapted three omniroller. Curcic shape robot which has a common radius 𝐿𝐿
adapts three omni-wheels. The k-th rollers (𝑘𝑘 = 1,2,3)
contact position 𝒒𝒒𝒊𝒊 on a periphery of robot, which has a

Figure 2 Parameters for mobile robot mechanism adapted threeomni-roller (a) mechanism of mobile robot (b). focus onneighborhood of single roller.

roller of radius 𝑅𝑅 distance between rollers and floor. 𝛴𝛴 −𝑋𝑋𝑋𝑋 denotes the global coordinate. system (origin 𝑶𝑶) and𝛴𝛴 − 𝑋𝑋́𝑌𝑌́ denotes robot coordinate system (origin 𝑶𝑶́). The
mobile robot orientation angle 𝜙𝜙 is defined as the angle
between 𝑋𝑋-axis and 𝑋𝑋́-axis. {𝒏𝒏𝟏𝟏 , 𝒏𝒏𝟐𝟐 , 𝒏𝒏𝟑𝟑 } represents theunit vector along the roller velocity direction. Roller
peripheral speed 𝑣𝑣𝑘𝑘 is calculated from given 𝑽𝑽 =(𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 )𝑻𝑻 and 𝜑𝜑 denotes robot direction for 𝛴𝛴 − 𝑋𝑋́𝑌𝑌́ and𝜙𝜙̇ denotes robot angular velocity.
2432.1. Derive of inverse kinematics〈𝒏𝒏𝒌𝒌 , (𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 )𝑻𝑻 〉. Moreover. robot rational component 𝑣𝑣𝑘𝑘𝑅𝑅
is equal to 𝐿𝐿𝜙𝜙̇. Therefore 𝑣𝑣𝑘𝑘 is represented as the sum of
the translation component and rotational component as
Eq. (3).

Table 1 Variables list of this researchTwo-dimensional grovel coordinate system𝛴𝛴 − 𝑋𝑋𝑋𝑋
𝛴𝛴 − 𝑋𝑋́𝑌𝑌́

Two-dimensional robot coordinate system〈𝒂𝒂, 𝒃𝒃〉‖𝒂𝒂‖Norm of vector 𝒂𝒂|𝑎𝑎|

Origin of 𝛴𝛴 − 𝑥𝑥𝑥𝑥

𝑶𝑶
𝒒𝒒𝒊𝒊
𝑶𝑶́

unit vector along the velocity vector of roller
Radius of mechanism body

𝐿𝐿

𝑅𝑅

Radius of omni-roller

𝑣𝑣𝑘𝑘𝑇𝑇

Translation components of 𝑣𝑣𝑘𝑘

Angle between 𝑋𝑋-axis and 𝑋𝑋́-axis

𝑣𝑣𝑘𝑘𝑅𝑅

Rotational components of 𝑣𝑣𝑘𝑘

𝛼𝛼𝑘𝑘
𝜙𝜙̇

Robot angular velocity

Invers image of 𝑊𝑊

𝑓𝑓𝐴𝐴−1 (𝑊𝑊)𝑓𝑓𝑨𝑨−1́(𝑊𝑊)Image volume (Volume of 𝑓𝑓𝐴𝐴−1 (𝑊𝑊))(𝒂𝒂𝟏𝟏 , 𝒂𝒂𝟐𝟐 , 𝒂𝒂𝟐𝟐 )Orthographic projection area (Area of
𝑓𝑓𝑨𝑨−1́(𝑊𝑊))Vectors represent of matrix 𝑨𝑨

cos 𝛼𝛼1 − cos 𝛼𝛼2 𝑣𝑣1
sin 𝛼𝛼1 + sin 𝛼𝛼2 � �𝑣𝑣2 �− sin(𝛼𝛼1 − 𝜃𝜃2 ) 𝑣𝑣3(7)

3. Evaluation function for roller arrangement
In this chapter, we make a connection 𝑨𝑨 (in Eq. (5))
with the rollers contact position in mobile robot andevaluate roller arrangement using noble evaluation
function such as “Image volume,” “orthographic
projection area”.
Figure 3 shows linear transformation mapping for

Robot speed are decomposed translation components 𝑣𝑣𝑖𝑖𝑇𝑇
and rotational components 𝑣𝑣𝑘𝑘𝑅𝑅 . Rollers contact position
𝒒𝒒𝒌𝒌 are adapted angle 𝛼𝛼𝑘𝑘 (0° ≤ 𝛼𝛼𝑘𝑘 < 360°) on the circle
that has a radius of body 𝐿𝐿.

matrix 𝐴𝐴 and correspondence two space ie. 𝑓𝑓𝑨𝑨 :̇ �→(𝑣𝑣1 , 𝑣𝑣2 , 𝑣𝑣3 ). Now. We define 𝑊𝑊 as cubic
�𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 , 𝜙𝜙𝜙𝜙(1)

domain that has a square, 2 [𝑚𝑚/𝑠𝑠] on each side as

From unit vector 𝒏𝒏𝒌𝒌 is perpendicular to 𝒒𝒒𝒌𝒌 and 𝒏𝒏𝒌𝒌 isrepresented as Eq. (2).
𝒏𝒏𝒌𝒌 = (−sin 𝛼𝛼𝑘𝑘 , cos 𝛼𝛼𝑘𝑘 )𝑻𝑻− cos 𝛼𝛼1 + cos 𝛼𝛼3− sin 𝛼𝛼1 + sin 𝛼𝛼3− sin(𝛼𝛼3 − 𝛼𝛼1 )(See Appendix(A))−1projection vector of 𝒂𝒂𝒌𝒌 with respect tohorizontal plane
Volume of 𝑓𝑓𝐴𝐴−1 (𝑊𝑊)Area of 𝑓𝑓𝑨𝑨−1́(𝑊𝑊)𝒒𝒒𝒊𝒊 = 𝐿𝐿(cos 𝛼𝛼𝑘𝑘 , sin 𝛼𝛼𝑘𝑘 )𝑻𝑻

(5)

(6)

where
det𝑨𝑨 = −sin(𝛼𝛼1 − 𝛼𝛼2 ) − sin(𝛼𝛼2 − 𝛼𝛼3 )− sin(𝛼𝛼3 − 𝛼𝛼1 )Orthographic projection domain of 𝑓𝑓𝐴𝐴−1 (𝑊𝑊)𝐷𝐷𝑉𝑉𝑉𝑉𝑉𝑉

𝐷𝐷𝐴𝐴

cos 𝛼𝛼2 − cos 𝛼𝛼3
� sin 𝛼𝛼2 − sin 𝛼𝛼3−sin(𝛼𝛼2 − 𝛼𝛼3 )Cubic domain that has a square, 2 [𝑚𝑚/𝑠𝑠]

Linear transformation mapping of matrix 𝑨𝑨

𝑓𝑓𝐴𝐴

𝐷𝐷𝑉𝑉

𝑉𝑉𝑥𝑥
� 𝑉𝑉𝑦𝑦 � = (det𝑨𝑨)−𝟏𝟏
𝐿𝐿𝜙𝜙̇

Position angle of 𝒒𝒒𝒌𝒌

𝑊𝑊

𝑉𝑉𝑥𝑥
1
𝑉𝑉
1� � 𝑦𝑦 �
1 𝐿𝐿𝜙𝜙̇Here., we put (3,3) matrix right side of Eq. (5) to 𝑨𝑨. Eq.
(5) is solved to (𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 , 𝐿𝐿𝜙𝜙̇)𝑇𝑇 as follow.

Roller peripheral speed

𝑣𝑣𝑘𝑘

cos 𝛼𝛼1cos 𝛼𝛼2cos 𝛼𝛼32.2. Derive of forward kinematics

𝜙𝜙

𝒂𝒂𝒌𝒌́

𝑣𝑣1
− sin 𝛼𝛼1𝑣𝑣
� 2 � = �− sin 𝛼𝛼2𝑣𝑣3− sin 𝛼𝛼3Rollers contact position
Origin of 𝛴𝛴 − 𝑋𝑋́𝑌𝑌́

𝒏𝒏𝒊𝒊

(4)

Substituting Eq. (2) with Eq. (4) in each 𝑣𝑣𝑘𝑘 (𝑘𝑘 = 1,2,3) ,(𝑣𝑣1 , 𝑣𝑣2 , 𝑣𝑣3 )𝑻𝑻 is represented using matrix.

𝑧𝑧-component of vector 𝒂𝒂

Mobile velocity of sphere on 𝑥𝑥𝑥𝑥-plane

𝑽𝑽

(3)

= 〈𝒏𝒏𝒌𝒌 , (𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 )𝑻𝑻 〉 + 𝐿𝐿𝜙𝜙̇

Absolut value of real member 𝑎𝑎(𝒂𝒂)𝑧𝑧𝐷𝐷𝐴𝐴𝐴𝐴𝐴𝐴𝑣𝑣𝑘𝑘 = 𝑣𝑣𝑘𝑘𝑇𝑇 + 𝑣𝑣𝑘𝑘𝑅𝑅

Inner product with respect to vector 𝒂𝒂 and 𝒃𝒃

following.

𝑊𝑊 = {(𝑣𝑣1 , 𝑣𝑣2 , 𝑣𝑣3 )||𝑣𝑣1 |, |𝑣𝑣2 |, |𝑣𝑣3 | ≤ 1}

(2)

Thus. Translation component 𝑣𝑣𝑘𝑘𝑇𝑇 is length of orthogonal
projection vector of 𝑽𝑽 with respect to 𝒏𝒏𝒌𝒌 .Thus. 𝑣𝑣𝑘𝑘𝑇𝑇 =(a) Three-dimensional image domain

244

(8)

From inverse linear transformation mappinġ � , image of cubic
𝑓𝑓𝐴𝐴−1 : (𝑣𝑣1 , 𝑣𝑣2 , 𝑣𝑣3 ) → �𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 , 𝜙𝜙𝜙𝜙
domain 𝑊𝑊 is equal to parallelepiped domain in three-dimensional coordinate space. Using inverse matrix oflinear transformation 𝑨𝑨−1 = (𝒂𝒂𝟏𝟏 , 𝒂𝒂𝟐𝟐 , 𝒂𝒂𝟑𝟑 ) , Inverse
image 𝑓𝑓𝑨𝑨−1 (𝑊𝑊) is represented as follows.
(9)

𝑓𝑓𝐴𝐴−1 (𝑊𝑊) =��𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 , 𝐿𝐿𝜙𝜙̇ �� 𝑣𝑣1 𝒂𝒂𝟏𝟏 + 𝑣𝑣2 𝒂𝒂𝟐𝟐 + 𝑣𝑣3 𝒂𝒂𝟑𝟑 , |𝑣𝑣1 |, |𝑣𝑣2 |, |𝑣𝑣3 | ≤ 1�wherecos 𝛼𝛼2 − cos 𝛼𝛼3𝒂𝒂𝟏𝟏 = (det𝑨𝑨) � sin 𝛼𝛼2 − sin 𝛼𝛼3 �,− sin(𝛼𝛼2 − 𝛼𝛼3 )− cos 𝛼𝛼1 + cos 𝛼𝛼3𝒂𝒂𝟐𝟐 = (det𝑨𝑨)−𝟏𝟏 � − sin 𝛼𝛼1 + sin 𝛼𝛼3 �−sin(𝛼𝛼3 − 𝛼𝛼1 )Figure 3 Linear transformation mapping correspondence
between three rollers speed {𝑣𝑣1 , 𝑣𝑣2 , 𝑣𝑣3 } and robot (mobile
speed (𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 )), and rotational speed 𝜙𝜙̇.

−𝟏𝟏

cos 𝛼𝛼1 − cos 𝛼𝛼2𝒂𝒂𝟑𝟑 = (det𝑨𝑨)−𝟏𝟏 � sin 𝛼𝛼1 + sin 𝛼𝛼2 �−sin(𝛼𝛼1 − 𝛼𝛼2 )the projection area of domain 𝑓𝑓𝑨𝑨−1 (𝑊𝑊) for horizontal
plane as Eq. (13).

(10)

𝑓𝑓𝑨𝑨−1́ (𝑊𝑊) =Besides, Using matrix property. det(𝑨𝑨−1 ) is equal to−𝟏𝟏(det𝑨𝑨) . linear transformation property is that
determinant is equal to the volume ratio from two
space. In this research, we define the volume of
𝑓𝑓𝑨𝑨−1 (𝑊𝑊) as 𝐷𝐷𝑉𝑉 (𝛼𝛼1 , 𝛼𝛼2 , 𝛼𝛼3 ). Moreover, using det(𝑨𝑨−1 ),image volume is calculated as follows (SeeAppendix(B)).
𝐷𝐷𝑉𝑉 (𝜃𝜃1 , 𝜃𝜃2 , 𝜃𝜃3 ) = 8 /det𝑨𝑨 [𝑚𝑚/𝑠𝑠]3

��𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 �� 𝑣𝑣1 𝒂𝒂𝟏𝟏́ + 𝑣𝑣2 𝒂𝒂𝟐𝟐́ + 𝑣𝑣3 𝒂𝒂𝟑𝟑́ , |𝑣𝑣1 |, |𝑣𝑣2 |, |𝑣𝑣3 | ≤ 1�wherecos 𝛼𝛼2 − cos 𝛼𝛼3𝒂𝒂𝟏𝟏́ = (det𝑨𝑨)−𝟏𝟏 � sin 𝛼𝛼2 − sin 𝛼𝛼3 �0− cos 𝛼𝛼1 + cos 𝛼𝛼3𝒂𝒂𝟐𝟐́ = (det𝑨𝑨)−𝟏𝟏 � − sin 𝛼𝛼1 + sin 𝛼𝛼3 �
0(11)cos 𝛼𝛼1 − cos 𝛼𝛼2𝒂𝒂𝟑𝟑́ = (det𝑨𝑨)−𝟏𝟏 � sin 𝛼𝛼1 + sin 𝛼𝛼2 �0Particularly, we analyze a case that contact position
triangle {𝒒𝒒𝟏𝟏 , 𝒒𝒒𝟐𝟐 , 𝒒𝒒𝟑𝟑 } represents the isosceles triangle
as 𝒒𝒒𝟏𝟏 𝒒𝒒𝟑𝟑 = 𝒒𝒒𝟐𝟐 𝒒𝒒𝟑𝟑 . As shown in Eq. (11), det 𝑨𝑨 is threeparametric function with respect to 𝛼𝛼1 , 𝛼𝛼2 and 𝛼𝛼3 .
Substituting Eq. (11) with 𝛼𝛼1 = 𝜃𝜃, 𝛼𝛼2 = 180° − 𝜃𝜃, and
𝛼𝛼3 = 270°, we obtain Eq. (12)8𝐷𝐷𝑉𝑉 (𝜃𝜃) =2 cos 𝜃𝜃 + sin 2𝜃𝜃

(13)

(14)

Thus. 𝐷𝐷𝐴𝐴 (𝛼𝛼1 , 𝛼𝛼2 , 𝛼𝛼3 ) is function depended on roller
contact position (𝛼𝛼1 , 𝛼𝛼2 , 𝛼𝛼3 ) and it has two case by signof (𝒂𝒂𝟐𝟐́ × 𝒂𝒂𝟑𝟑́ )𝑧𝑧 (𝒂𝒂𝟏𝟏́ × 𝒂𝒂𝟑𝟑́ )𝑧𝑧 .

(12)

𝐷𝐷𝐴𝐴 =

And. Theoretically, 𝐷𝐷𝑉𝑉 (𝜃𝜃1 , 𝜃𝜃2 , 𝜃𝜃3 ) takes minimal value
when {𝒒𝒒𝟏𝟏 , 𝒒𝒒𝟐𝟐 , 𝒒𝒒𝟑𝟑 } is equilateral triangle (SeeAppendix(C)).

⎧
⎪

⎨
⎪
⎩

4‖𝒂𝒂𝟏𝟏́ × 𝒂𝒂𝟐𝟐́ ‖ + 4�(𝒂𝒂́ 𝟏𝟏 − 𝒂𝒂𝟐𝟐́ ) × 𝒂𝒂𝟑𝟑́ �[ (𝒂𝒂𝟐𝟐́ × 𝒂𝒂𝟑𝟑́ )𝑧𝑧 (𝒂𝒂𝟏𝟏́ × 𝒂𝒂𝟑𝟑́ )𝑧𝑧 < 0]4‖𝒂𝒂𝟏𝟏́ × 𝒂𝒂𝟐𝟐́ ‖ + 4�(𝒂𝒂́ 𝟏𝟏 + 𝒂𝒂𝟐𝟐́ ) × 𝒂𝒂𝟑𝟑́ �[ (𝒂𝒂𝟐𝟐́ × 𝒂𝒂𝟑𝟑́ )𝑧𝑧 (𝒂𝒂𝟏𝟏́ × 𝒂𝒂𝟑𝟑́ )𝑧𝑧 > 0]

(See Appendix(D))(b) Two-dimensional
orthographic
projection
domainNext, 𝒂𝒂𝒌𝒌́ is the projection vector of 𝒂𝒂𝒌𝒌 with respect tothe �𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 �-plane. Following 𝑓𝑓𝑨𝑨−1́ (𝑊𝑊) can be defined
245

(15)

(a) Case of Type(X)(b) Case of Type(Y)Figure 4 Three-dimensional shape of inverse image of 𝑊𝑊(c) Case of Type(Z)(a) Case of Type(X)(b) Case of Type(Y)Figure 5 Two-dimensional shape of 𝑓𝑓𝑨𝑨−1́(𝑊𝑊)(c) Case of Type(Z)Figures 4 and 5 and show the shape 𝑓𝑓𝐴𝐴−1 (𝑊𝑊) of
Eq.(9) and 𝑓𝑓𝐴𝐴−1́(𝑊𝑊) of Eq.(13), respectively. forType(X), Type(Y), and Type(Z). Table 2 shows value
𝐷𝐷𝑉𝑉 (𝜃𝜃) of Eq.(12) and 𝐷𝐷𝐴𝐴 (𝜃𝜃) of Eq. (15) in Type(X)–(Z).
And Figure 6 shows behavior of 𝐷𝐷𝑉𝑉 (𝜃𝜃) and 𝐷𝐷𝐴𝐴 (𝜃𝜃) withrespect to parameter 𝜃𝜃.
As shown in Figure 4, Shape of 𝑓𝑓𝐴𝐴−1 (𝑊𝑊) is aparallelepiped, which is getting stretched with respect todirections 𝑉𝑉𝑥𝑥 -axis and 𝐿𝐿𝜙𝜙̇-axis.
As shown in Figure 5, Shape of 𝑓𝑓𝐴𝐴−1́(𝑊𝑊) is hexagons,
which is getting stretched with respect to 𝑉𝑉𝑥𝑥 -axis
direction.
As shown in Table 2, Type (X) has the smallest value
𝐷𝐷𝑉𝑉 (𝜃𝜃). Additionally, since Type(Z) is the largest valueof 𝐷𝐷𝑉𝑉 (𝜃𝜃), the moving volume in translational motion iseffective. And similar results were obtained for theevaluation 𝐷𝐷𝐴𝐴 (𝜃𝜃).
As shown in Figure 6, 𝐷𝐷𝑉𝑉 (𝜃𝜃) and 𝐷𝐷𝑉𝑉 (𝜃𝜃) take minimalvalues when 𝜃𝜃 = 30°.
In this way, we geometrically obtain the
distribution and property of 𝑓𝑓𝐴𝐴−1 (𝑊𝑊) and 𝑓𝑓𝐴𝐴−1́(𝑊𝑊).
Furthermore, we get property of image volume
𝐷𝐷𝑉𝑉 (𝜃𝜃) and orthographic projection area 𝐷𝐷𝐴𝐴 (𝜃𝜃) and weclear that effective and ineffective roller arrangement.

4. Simulation
This section presents the simulation findings (behavior
and shape), including the evaluation values as image
volume and orthographic projection area .
Simulations were performed at the three various roller
arrangement Type(X)-(Z) that 𝒒𝒒𝟏𝟏 , 𝒒𝒒𝟐𝟐, and 𝒒𝒒𝟑𝟑 arearranged a symmetry triangle shaped (See Figure1).
They are set up 𝛼𝛼𝑘𝑘 (𝑘𝑘 = 1,2,3) as follows:
Type(X): {𝛼𝛼1 , 𝛼𝛼2 , 𝛼𝛼3 } = {30°, 150°, 270°}
Type(Y): {𝛼𝛼1 , 𝛼𝛼2 , 𝛼𝛼3 } = {0°, 180°, 270°}
Type(Z): {𝛼𝛼1 , 𝛼𝛼2 , 𝛼𝛼3 } = {210°, 270°, 300°}
and 𝐿𝐿 = 1[m].
Table 2 Comparison of three roller arrangement patten by theevaluation values 𝐷𝐷𝑉𝑉 (𝜃𝜃) and 𝐷𝐷𝐴𝐴 (𝜃𝜃)Type (X)Type (Y)Type (Z)𝐷𝐷𝑉𝑉 (𝜃𝜃) [𝑚𝑚/𝑠𝑠]3

𝐷𝐷𝐴𝐴 (𝜃𝜃) [𝑚𝑚/𝑠𝑠]2

9.24

13.85

3.08
4.00

4.61
6.00

246

Figure 7 Calculation of volume of parallelepiped domain(a)Substitute the cofactor of 𝑎𝑎𝑖𝑖,𝑗𝑗 with 𝑎𝑎�
𝚤𝚤,𝚥𝚥 , We obtain thefollowing using the cofactor.
𝑨𝑨−1 =
where

𝑎𝑎�
1,1
1
�𝑎𝑎�
2,1
det𝑨𝑨
𝑎𝑎�
3,1

𝑎𝑎�
1,2
𝑎𝑎�
2,2
𝑎𝑎�
3,2

𝑎𝑎�
1,3
𝑎𝑎�
2,3 �
𝑎𝑎�
3,3

(A,1)

𝑎𝑎�
1,1 = 𝐿𝐿 cos 𝛼𝛼2 − 𝐿𝐿 cos 𝛼𝛼3𝑎𝑎�
2,2 = −𝐿𝐿 sin 𝛼𝛼1 + 𝐿𝐿 sin 𝛼𝛼1𝑎𝑎�
1,2 = −𝐿𝐿 cos 𝛼𝛼1 + 𝐿𝐿 cos 𝛼𝛼3𝑎𝑎�
2,1 = 𝐿𝐿 sin 𝛼𝛼2 − 𝐿𝐿 sin 𝛼𝛼3

(b)

(A,2)

𝑎𝑎�
1,3 = 𝐿𝐿 cos 𝛼𝛼1 − 𝐿𝐿 cos 𝜃𝜃1𝑎𝑎�
2,3 = 𝐿𝐿 sin 𝛼𝛼1 − 𝐿𝐿 sin 𝛼𝛼2Figure 6 Behavior of evaluation functions with respect toparameter 𝜃𝜃 (a). Case of 𝐷𝐷𝑉𝑉 (𝜃𝜃) [𝑚𝑚/𝑠𝑠]3 (b). Case of 𝐷𝐷𝐴𝐴(𝜃𝜃)[𝑚𝑚/𝑠𝑠]2

And

where

4. Conclusions
In this research, we derived kinematics for a mobile
robot that adapted three omni rollers that can arbitrarilychange the roller arrangement position. Additionally, we
adopted the determinant of the transformation matrix thatassociates the input and output as an evaluation function
to examine the roller arrangement.
The following results were obtained: the evaluation
function was minimized in an equilateral triangle
arrangement (Type(X)). Furthermore, Type(Z) is themost efficient considering the translational motion.
Therefore, motion evaluation of a translational
mechanism is the only future challenge in this field.
As futures works, we’ll conducted experiment using
prototype adapted three omni roller.

det𝑨𝑨 = 𝑎𝑎�
3,1 + 𝑎𝑎�
3,2 + 𝑎𝑎�3,3𝑎𝑎�
3,1 = − sin 𝛼𝛼2 cos 𝛼𝛼3 + cos 𝛼𝛼2 sin 𝛼𝛼3= − sin(𝛼𝛼2 − 𝛼𝛼3 )𝑎𝑎�
3,2 = sin 𝛼𝛼1 cos 𝛼𝛼3 − cos 𝛼𝛼1 sin 𝛼𝛼3= sin(𝛼𝛼1 − 𝛼𝛼3 )

(A,3)

(A,4)

𝑎𝑎�
3,3 = − sin 𝛼𝛼1 cos 𝛼𝛼2 + cos 𝛼𝛼1 sin 𝛼𝛼2= sin(𝛼𝛼2 − 𝛼𝛼1 )(B) Calculation of image volume

As shown in Figure 7, The parallelepiped domain isspanned by {𝒂𝒂𝟏𝟏 , 𝒂𝒂𝟐𝟐 , 𝒂𝒂𝟑𝟑 } . Upward plane spanned by{𝒂𝒂𝟏𝟏 , 𝒂𝒂𝟐𝟐 } have area ‖𝒂𝒂𝟏𝟏 × 𝒂𝒂𝟐𝟐 ‖ and unit normal vector
𝒂𝒂𝟏𝟏 × 𝒂𝒂𝟐𝟐 /‖𝒂𝒂𝟏𝟏 × 𝒂𝒂𝟐𝟐 ‖.

Length of orthographic projection vector ������⃑𝐎𝐎𝐎𝐎 isrepresented as follows.
𝟏𝟏〈𝒂𝒂𝟏𝟏 × 𝒂𝒂𝟐𝟐 , 𝒂𝒂𝟑𝟑〉
OH = ‖𝒂𝒂(B,2)‖

Appendix(A) Calculation of inverse matrix

𝟏𝟏 × 𝒂𝒂𝟐𝟐

247

Therefore. volume is calculated as follows.
𝐎𝐎𝐎𝐎
𝑽𝑽 = ‖𝒂𝒂𝟏𝟏 × 𝒂𝒂𝟐𝟐 ‖ × �����

Moreover, using determinant property,
𝑽𝑽 = 〈𝒂𝒂𝟏𝟏 × 𝒂𝒂𝟐𝟐 , 𝒂𝒂𝟑𝟑 〉 = det[𝒂𝒂𝟏𝟏 , 𝒂𝒂𝟐𝟐 , 𝒂𝒂𝟑𝟑 ]𝑹𝑹+ 𝑷𝑷+ 𝑷𝑷𝑷𝑷 and △𝑸𝑸+ 𝑷𝑷+ 𝑹𝑹+ . Hence, 𝐷𝐷𝐴𝐴 is represented as
Eq. (12).
Case 2: 𝒂𝒂𝟑𝟑́ is lay from 𝒂𝒂𝟏𝟏́ and −𝒂𝒂𝟐𝟐́ . or 𝒂𝒂𝟑𝟑́ is lay from−𝒂𝒂́ 𝟏𝟏 and 𝒂𝒂𝟐𝟐́ .
In this case, we prove this as Case 1, and. 𝐷𝐷𝐴𝐴 isrepresented as Eq. (12).

(B,3)
(B,3)

(C) Minimalize of function in Type(X)Now, we conduct a change of variables (𝛼𝛼1 , 𝛼𝛼2 , 𝛼𝛼3 ) → (A,B, C) as follows:
𝛼𝛼2 − 𝛼𝛼1 = 𝐴𝐴, 𝛼𝛼3 − 𝛼𝛼2 = 𝐵𝐵 ，𝐶𝐶 = 360° − 𝐴𝐴 −(C,4)
𝐵𝐵
sin(𝜃𝜃1 − 𝜃𝜃3 ). We can simplify using 𝜃𝜃3 − 𝜃𝜃1 = 𝐴𝐴 +𝐵𝐵 = 360° − 𝐶𝐶 as follows:sin(𝛼𝛼1 − 𝛼𝛼3 ) = sin(𝐶𝐶 − 360°) = sin 𝐶𝐶

Figure 8 (a) Layout condition of {𝒂𝒂𝟏𝟏́ , 𝒂𝒂𝟐𝟐́ , 𝒂𝒂𝟑𝟑́ } focus on direction
𝒂𝒂𝟑𝟑́ . (b). Line trajectory of direction 𝒂𝒂𝟑𝟑́ of existing hexagonal
domains(C,5)

Thus, Eq. (A,3) is denoted by a simple equation.
det𝑨𝑨 = 𝐿𝐿(sin 𝐴𝐴 + sin 𝐵𝐵 + sin 𝐶𝐶)

References(C,6)(𝐴𝐴 + 𝐵𝐵 + 𝐶𝐶 = 360°)
sin 𝑡𝑡 is an upward convex function with respect to 𝑡𝑡
and it takes a maximal value when 𝑡𝑡 = (𝐴𝐴 + 𝐵𝐵 + 𝐶𝐶)/3= 120°．Thus, 1/ det𝑨𝑨 has maximal value. we findout that 𝐷𝐷𝑉𝑉 (𝜃𝜃) was minimized in the equilateral triangle
arrangement (i.e., Type(X)).
(D) Calculation of 𝐷𝐷𝐴𝐴

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𝑻𝑻

Using column vector 𝒂𝒂𝒌𝒌́ , �𝑉𝑉𝑥𝑥 , 𝑉𝑉𝑦𝑦 � is denoted by a linear
combination of 𝒂𝒂𝟏𝟏́ , 𝒂𝒂𝟐𝟐́ and 𝒂𝒂𝟑𝟑́ as follow.
𝑼𝑼 denote district set as 𝜈𝜈3 = 0 from 𝑓𝑓𝑨𝑨−1́ (𝑊𝑊) as

follows.
𝑼𝑼 = �𝜈𝜈1 𝑨𝑨́𝟏𝟏 + 𝜈𝜈2 𝑨𝑨́𝟐𝟐 �|𝜈𝜈1 |, |𝜈𝜈2 | ≤ 1�(D,1)

We consider that shape when 𝑈𝑈 move on a line trajectoryof direction 𝒂𝒂𝟑𝟑́ (as parameter 𝑣𝑣3 ) as follows. Figure 8(a)shows two cases proving the following Case 1 and Case
2 of a layout condition of {𝒂𝒂𝟏𝟏́ , 𝒂𝒂𝟐𝟐́ , 𝒂𝒂𝟑𝟑́ } with a focus on
the direction 𝒂𝒂𝟑𝟑́ .
Case 1: 𝒂𝒂𝟑𝟑́ is lay from 𝒂𝒂𝟏𝟏́ and 𝒂𝒂𝟐𝟐́ , or 𝒂𝒂𝟑𝟑́ is lay from−𝒂𝒂𝟏𝟏́ and −𝒂𝒂𝟐𝟐́ .
This condition is derived as follows.
{(𝒂𝒂𝟐𝟐́ × 𝒂𝒂𝟑𝟑́ )𝑧𝑧 < 0 and (𝒂𝒂𝟏𝟏́ × 𝒂𝒂𝟑𝟑́ )𝑧𝑧 > 0 }or { (𝒂𝒂𝟐𝟐́ × 𝒂𝒂𝟑𝟑́ )𝑧𝑧 > 0 and (𝒂𝒂𝟏𝟏́ × 𝒂𝒂𝟑𝟑́ )𝑧𝑧 < 0 }⟺ (𝒂𝒂𝟐𝟐́ × 𝒂𝒂𝟑𝟑́ )𝑧𝑧 (𝒂𝒂𝟏𝟏́ × 𝒂𝒂𝟑𝟑́ )𝑧𝑧 < 0]

(D,2)
(D,3)

Figure 8(b) shows when 𝑣𝑣3 is vanished or fixed, 𝑼𝑼 will
be a parallelogram. Thus, the trajectory trac of 𝑼𝑼 in
existing hexagonal domains on opposite sides are parallel.
Thus, Half of the domain is the sum of the area of □248

assembly, Journal of the Robotics Society of Japan. Vol.
17, No. 1,pp. 51-60, 1999.

Authors Introduction
Dr. Kenji Kimura

He is a Lecturer in the Department ofControlEngineering,
National
Institute of Technology, Matsue
College, where he has been since2022. He received his ME(mathematics)
fromKyushuUniversity in 2002 and received his
Ph.D(Engineering) from Kyushu Institute of in 2020.
His research interests are spherical mobile robots.
Mr. Yuki Shigyo

He is an engineer working for FujitsuLtd. He received the BE from Tokyo
University of Science in 2016 and theME (mathematics) from Nagoya
University in 2018. His researchinterests are spherical mobile robot
kinematics and machine learning.

Dr. Kazuo Ishii

He is a Professor at the Kyushu
Institute of Technology, where he has
been since 1996. He received his
Ph.D. degree in engineering from the
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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