SUGISAKAMASANORI

Journal of Robotics, Networking and Artificial
Life Vol. 9(1); June (2022), pp. 87–92
ISSN (Online): 2352-6386; ISSN (Print): 2405-9021
https://alife-robotics.org/jrnal.html

Research Article

Efficiency Problem of Spherical Robot in Transfer Kinetic Energy
Kenji Kimura1, Kazuo Ishii2

Department of Control Engineering, Matsue College of Technology, 14-4, Nishi-ikumacho, Matsue,Shimane, 690-8518, Japan
Graduate School of Life Science and engineering, Kyusyu Institute of Technology, 2-4 Hibikino, Wakamatsu-ku, Kitakyushu 808-0196, Fukuoka,
Japan
1
2

ARTIC LE INFOArticle History
Received 10 November 2020
Accepted 20 June 2022

Keywords
Angular velocity vector of the sphere
Angle of sphere rotational axis
Total kinetic energy

ABSTRACTPrevious spherical mobile robots were driven by two rollers with a fixed rotational axis, which
restricts the angular velocity vector of the sphere to two dimensions. Three-dimensional freedom
is expected to improve the rotational diversity of the sphere. This study proposes a sphericalmobile robot with a variable roller-rotational axis that allows the movement of three degrees offreedom. Furthermore, the kinetic energy of transporting the sphere by the rollers is minimized
by an optimization procedure and it is shown that the kinetic energy of sphere transport is efficient.
© 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/).

considered in [3]. In the ball dribbling mechanism [4], therollers are arranged in the upper hemisphere, where theyhold the balls by friction (See Figure 1 (c)). The twodimensionality of the sphere’s angular velocity vector inthese mechanisms was affirmed in kinematic studies of asphere with slipping [5]. In addition, the kinematics ofthe reverse motion was verified in a demonstration
experiment [6].
Among the three-roller cases (𝑁𝑁𝑤𝑤 = 3), OWMPs [7]deployed in highway maintenance move the sphereswithin three constrained rollers (See Figure 1 (d)).The
rollers are arranged on the equator parallel to thehorizontal plane, and the sphere can be rotated in all
directions by generating an angular velocity vector on theplane (note that the existence space of the angularvelocity vector is two-dimensional).
A ball-balanced robot [8] has three unconstrainedrollers (omni-rollers) arranged in a regular triangularconfiguration (See Figure 1 (e)). CPU-Ball Bot [9] hasfour unconstrained rollers fixed in a square configuration(See Figure 1 (f)).

1. Introduction
Many types of robots, such as omnidirectional mobile
robots and sphere-transported robots, are based on
spherical
motions.
Therefore,variousroller
arrangements have been proposed for mobile robot
applications.
Table 1 shows the usage statuses of single sphericalrobots operated by different mechanisms, and thedimensions of the existence spaces of their angularvelocity vectors. Figure 1 shows the numbers of rollers
𝑁𝑁𝑤𝑤 arranged per wheel in each mechanism, and theircontact types.
In the ACROBAT-S [1] mechanism with 𝑁𝑁𝑤𝑤 = 2, thecaster of each sphere is driven by two roller drives (seeFigure 1 (a)). The wheel chair mechanism [2] has threespheres (Figure 1 (b)). The rollers are arranged on theequator, generate an angular velocity vector on thehorizontal plane, and can move in all directions. The
angular velocity vector of the sphere has twodimensional freedom. This situation is theoreticallyCorresponding Author E-mail: k-kimura@matsue-ct.jp, ishii@brain.kyutech.ac.jp87

object conveyance. It is desired to transport with highkinetic energy efficiency.
In this study, we optimize the total kinetic energy of
two rollers. Section 2 calculates the angle of therotational axis of the sphere, and theoretical formula ofminimizes the sum of the kinetic energies of the tworollers. Section 3 presents a simulation of total kinetic
energy, and Section 4 summarizes the results and suggests
ideas for future work.

In the OWMP, A ball-balanced and CPU-Ball Bot
mechanisms [7]-[9], the wheel stands are spherical. In
particular, A ball-balanced [8] and CPU-Ball Bot [9]mechanisms can rotate the sphere in three-dimensions
using the passive rotation characteristics of the nonrotating direction of the omni-roller.
Such three-dimensional freedom of the angularvelocity vector of a sphere will improve the motion
diversity of the sphere.
However, unconstrained rollers can slip, degrading theaccuracy of the control. Therefore, the use ofconstraining rollers is desired.
As shown in Figure 1 (c), this problem can be solved
by freeing the rotation axis of the constraining roller,
enabling three-dimensional rotation of the sphere.

2. Total kinetic energy
2.1 Angular velocity vector of the sphere
The center 𝑶𝑶 of a sphere with radius r is fixed as theorigin of the coordinate system 𝛴𝛴 − 𝑥𝑥𝑥𝑥𝑥𝑥 . The 𝑖𝑖 thconstraint roller (i = 1 or 2) is in point contact with thesphere at a position vector 𝑷𝑷𝒊𝒊 and is arranged such thatthe center of mass of the roller 𝑷𝑷𝒊𝒊 and 𝑶𝑶 are on the sameline. 𝝎𝝎 denotes the angular velocity vector of thesphere. 𝜼𝜼𝒊𝒊 denotes the unit vector along the rotational
axis of constraint roller. sphere direction 𝜑𝜑 (0° ≤ 𝜑𝜑 <360°) is the angle from 𝑥𝑥-axis.
Now, given the sphere mobile velocity 𝑽𝑽 (the centervelocity of sphere)

Table 1 Existence space of angular velocity vector forsphere mobile robot
Mechanismstatuses

Rotational
Dimension

ACROBAT-S[1]Caster2Wheel2Wheel chair [2]Ball dribbling mechanism [4]OWMPs [7]Sphereconveyance
Wheel

2

𝑽𝑽 = ‖𝑽𝑽‖[cos 𝜑𝜑

2

A ball-balanced robot [8]Wheel3CPU-Ball Bot [9]Wheel

3sin 𝜑𝜑0]𝑇𝑇(1)𝝎𝝎́ denote orthogonal projection of 𝝎𝝎 with respect to𝑥𝑥𝑥𝑥 −plane. 𝝎𝝎́ is perpendicular to 𝑽𝑽 and represented asEquation (2) (𝑽𝑽 is depend on 𝝎𝝎́ ).
𝝎𝝎́ =‖𝑽𝑽‖[−sin 𝜑𝜑𝑟𝑟0]𝑇𝑇(2)tan 𝜌𝜌]𝑇𝑇(3)cos 𝜑𝜑And. 𝝎𝝎́ is orthogonal projection of 𝝎𝝎 with respect to𝑥𝑥𝑥𝑥 − plane.The angle of sphere rotational axis 𝜌𝜌( −90° ≤ 𝜌𝜌 ≤ 90° ) is the angle between 𝝎𝝎 and 𝝎𝝎́ .
𝑇𝑇Therefore, 𝝎𝝎 = �𝜔𝜔𝑥𝑥 , 𝜔𝜔𝑦𝑦 , 𝜔𝜔𝑧𝑧 � has one-dimensional offreedom with parameter 𝜌𝜌 (𝜔𝜔𝑧𝑧 = ‖𝑽𝑽‖ tan 𝜌𝜌 /𝑟𝑟).
𝝎𝝎 =

‖𝑽𝑽‖
[−sin 𝜑𝜑𝑟𝑟

cos 𝜑𝜑Generally, these depend on the orthogonal projection
vector of 𝑽𝑽 , it has one-dimensional of freedom withrespect to 𝜔𝜔𝑧𝑧 = ‖𝑽𝑽‖ tan 𝜌𝜌 /𝑟𝑟.

Figure 1 Type of roller arrangement for sphere mobilerobot

2.2 Kinetic energy of the two rollers

the angular velocity vector of a sphere will improve themotion diversity of the sphere. However, a mechanism
adapting two constrained rollers is suitable in a spherical
88

Consider two rollers with radius 𝑅𝑅 , moment 𝐼𝐼, androller’s angular velocity 𝜔𝜔𝑖𝑖 at 𝑷𝑷𝒊𝒊 . The summed kinetic
energy of the rollers is given by Eqs.(4).

(a)

(b)

Figure. 2 Existence of the sphere angular velocity vector with respect to sphere movile velocity.(a) 𝝎𝝎𝟏𝟏 is determined(𝝎𝝎𝟏𝟏 is lay on intersection 𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬{𝑷𝑷𝟏𝟏 , 𝑷𝑷𝟐𝟐 } and line 𝒍𝒍 . (b) 𝝎𝝎𝟐𝟐 is determined as minimizes the summed kinetic energiesof the two rollers.
𝐸𝐸 = 𝐼𝐼(𝜔𝜔12 + 𝜔𝜔22 )=𝐼𝐼(‖𝝎𝝎 × 𝑷𝑷1 ‖2 + ‖𝝎𝝎 × 𝑷𝑷2 ‖2 )𝑅𝑅2

Substituting Eqs.(6) into Eqs.(4)， we obtain 𝐸𝐸 = 𝐸𝐸1
as Eqs.(7).

(4)

𝐸𝐸1 =

Using the angular velocity vector of a sphere, we now
derive the kinetic energy of Type-I (with both rotationalaxes fixed on the same plane) and Type-II (withvariable rotational axes) configurations(indicated inFigure 2 (a)(b)).

(𝑷𝑷𝟏𝟏 × 𝑷𝑷𝟐𝟐 )𝑥𝑥 sin 𝜑𝜑 − (𝑷𝑷𝟏𝟏 × 𝑷𝑷𝟐𝟐 )𝑦𝑦 cos 𝜑𝜑�(𝑷𝑷𝟏𝟏 × 𝑷𝑷𝟐𝟐 )𝑧𝑧

And.

𝒆𝒆𝟑𝟑 = [0, 0, 1]𝑇𝑇 , 𝝎𝝎́ = �𝜔𝜔𝑥𝑥 , 𝜔𝜔𝑦𝑦 , 0�

(8)

𝑇𝑇(9)

𝝎𝝎 × 𝑷𝑷𝑖𝑖 = (𝝎𝝎́ + 𝜔𝜔𝑧𝑧 𝒆𝒆𝟑𝟑 ) × 𝑷𝑷𝑖𝑖(10)‖𝝎𝝎 × 𝑷𝑷𝑖𝑖 ‖2(11)= 〈𝝎𝝎́ × 𝑷𝑷𝑖𝑖 + 𝜔𝜔𝑧𝑧 (𝒆𝒆𝟑𝟑 × 𝑷𝑷𝑖𝑖 ) , 𝝎𝝎́ × 𝑷𝑷𝑖𝑖 + 𝜔𝜔𝑧𝑧 (𝒆𝒆𝟑𝟑 × 𝑷𝑷𝑖𝑖 )〉(6)(𝑷𝑷𝟏𝟏 × 𝑷𝑷𝟐𝟐 )𝑥𝑥 sin 𝜑𝜑 − (𝑷𝑷𝟏𝟏 × 𝑷𝑷𝟐𝟐 )𝑦𝑦 cos 𝜑𝜑‖𝑽𝑽‖�−sin 𝜑𝜑 , cos 𝜑𝜑 ,�(𝑷𝑷𝟏𝟏 × 𝑷𝑷𝟐𝟐 )𝑧𝑧𝑟𝑟where𝝎𝝎 = 𝝎𝝎́ + 𝜔𝜔𝑧𝑧 𝒆𝒆𝟑𝟑

= 𝝎𝝎́ × 𝑷𝑷𝑖𝑖 + 𝜔𝜔𝑧𝑧 (𝒆𝒆𝟑𝟑 × 𝑷𝑷𝑖𝑖 )(5)

By substituting Eqs. (5) into Eqs. (3)， we obtain 𝝎𝝎𝟏𝟏
as Eqs. (6).
𝝎𝝎𝟏𝟏 =(7)(B) Case of Type-Ⅱ
As shown in Figure 2 (b), we determine 𝜌𝜌 = 𝜌𝜌1 , 𝝎𝝎 =𝝎𝝎𝟐𝟐 that minimizes the summed kinetic energies of thetwo rollers, and calculate the minimum energy (𝐸𝐸 = 𝐸𝐸2 ).
To this end, we first express 𝝎𝝎 as the sum of 𝝎𝝎́ and𝜔𝜔𝑧𝑧 𝒆𝒆𝟑𝟑 .

(A) Case of Type- I
As shown in Figure 2 (a), if both rotational axes
lie on the same plane, omnidirectional locomotion ispossible (see [4]). Therefore, the end point of 𝝎𝝎𝟏𝟏 canbe determined as the intersection of 𝑙𝑙 (the lineperpendicular to the horizontal plane passing throughend point 𝝎𝝎́ ) and 𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬{𝑷𝑷𝟏𝟏 , 𝑷𝑷𝟐𝟐 } . The angle of thesphere’s rotational axis 𝜌𝜌1 is then obtained as follows:𝜌𝜌1 = tan−1 �𝐼𝐼(‖𝝎𝝎𝟏𝟏 × 𝑷𝑷1 ‖2 + ‖𝝎𝝎𝟏𝟏 × 𝑷𝑷2 ‖2 )𝑅𝑅2= 𝜔𝜔𝑧𝑧2 ‖𝒆𝒆𝟑𝟑 × 𝑷𝑷𝑖𝑖 ‖2 + 2𝜔𝜔𝑧𝑧 〈𝝎𝝎́ × 𝑷𝑷𝑖𝑖 , 𝒆𝒆𝟑𝟑 × 𝑷𝑷𝑖𝑖 〉 + ‖𝝎𝝎́ × 𝑷𝑷𝑖𝑖 ‖2Using, Eqs.(11), it is represented as quadratic function
with respect to 𝜔𝜔𝑧𝑧 as Eqs.(12)𝑇𝑇89‖𝝎𝝎 × 𝑷𝑷𝑖𝑖 ‖2 + ‖𝝎𝝎 × 𝑷𝑷𝑖𝑖 ‖2 =(12)As shown in Figure 3 (b), 𝐸𝐸1 is minimized at spheredirection
angles𝜑𝜑 =90°and270°,and maximized at 𝜑𝜑 = 0° and 180°. Meanwhile, 𝐸𝐸2 isminimized at 𝜑𝜑 = 0° and 180°, and maximized at 𝜑𝜑 =90° and 270°. And. 𝐸𝐸1 and 𝐸𝐸2 are same value at 𝜑𝜑 =90° and 270°.

(‖𝒆𝒆𝟑𝟑 × 𝑷𝑷1 ‖2 + ‖𝒆𝒆𝟑𝟑 × 𝑷𝑷2 ‖2 )𝜔𝜔𝑧𝑧2+ 2(〈𝝎𝝎́ × 𝑷𝑷1 , 𝒆𝒆𝟑𝟑 × 𝑷𝑷1 〉 + 〈𝝎𝝎́ × 𝑷𝑷2 , 𝒆𝒆𝟑𝟑 ×𝑷𝑷2 〉)𝜔𝜔𝑧𝑧
+ ‖𝝎𝝎́ × 𝑷𝑷1 ‖2 + ‖𝝎𝝎́ × 𝑷𝑷2 ‖2Hence, focusing the coefficients of Eqs.(12), 𝐸𝐸 takes
minimal value 𝐸𝐸2 as Eqs.(13).
𝐸𝐸2 = (‖𝝎𝝎́ × 𝑷𝑷1 ‖2 +‖𝝎𝝎́ × 𝑷𝑷2 ‖2 )−where𝐼𝐼

𝑅𝑅 2(13)(〈𝝎𝝎́ × 𝑷𝑷1 , 𝒆𝒆𝟑𝟑 × 𝑷𝑷1 〉 + 〈𝝎𝝎́ × 𝑷𝑷2 , 𝒆𝒆𝟑𝟑 × 𝑷𝑷2 〉)‖𝒆𝒆𝟑𝟑 × 𝑷𝑷1 ‖2 + ‖𝒆𝒆𝟑𝟑 × 𝑷𝑷2 ‖2𝜔𝜔𝑧𝑧 = −〈𝝎𝝎́ × 𝑷𝑷1 , 𝒆𝒆𝟑𝟑 × 𝑷𝑷1 〉 + 〈𝝎𝝎́ × 𝑷𝑷2 , 𝒆𝒆𝟑𝟑 × 𝑷𝑷2 〉(14)‖𝒆𝒆𝟑𝟑 × 𝑷𝑷1 ‖2 + ‖𝒆𝒆𝟑𝟑 × 𝑷𝑷2 ‖2𝜌𝜌2 = tan−1 �−(a)〈𝝎𝝎́ × 𝑷𝑷1 , 𝒆𝒆𝟑𝟑 × 𝑷𝑷1 〉 + 〈𝝎𝝎́ × 𝑷𝑷2 , 𝒆𝒆𝟑𝟑 × 𝑷𝑷2 〉�‖𝒆𝒆𝟑𝟑 × 𝑷𝑷1 ‖2 + ‖𝒆𝒆𝟑𝟑 × 𝑷𝑷2 ‖2(15)Substituting Eqs.(15) into Eqs.(3), 𝝎𝝎𝟐𝟐 as Eqs.(16).
𝝎𝝎𝟐𝟐 =(16)𝑇𝑇〈𝝎𝝎́ × 𝑷𝑷1 , 𝒆𝒆𝟑𝟑 × 𝑷𝑷1 〉 + 〈𝝎𝝎́ × 𝑷𝑷2 , 𝒆𝒆𝟑𝟑 × 𝑷𝑷2 〉‖𝑽𝑽‖
�−sin 𝜑𝜑 , cos 𝜑𝜑 , −�‖𝒆𝒆𝟑𝟑 × 𝑷𝑷1 ‖2 + ‖𝒆𝒆𝟑𝟑 × 𝑷𝑷2 ‖2𝑟𝑟(b)3. Simulation

Figure 3 Simulation result. (a) Trajectory of angular velocity
vector of sphere. (b) Totally kinetic energy

This section presents the simulation results 𝝎𝝎1 (SeeEqs.(6)), 𝝎𝝎2 (See Eqs.(16)), 𝐸𝐸1 (See Eqs.(7)) and𝐸𝐸2 (See Eqs.(13)) with parameter 𝜑𝜑 (0° ≤ 𝜑𝜑 < 360°) inthe given sphere mobile speed: ‖𝑽𝑽‖ = 1 [m/s]. The
conditions are as follows: 𝐼𝐼 = 1, 𝑅𝑅 = 00.1[m], ‖𝑽𝑽‖ =1 [m/s] , 𝑟𝑟 = 0.1[m], 𝜃𝜃1,1 = 215°, 𝜃𝜃1,2 = 60°, 𝜃𝜃2,1 =325°, 𝜃𝜃2,2 = 60 °.

3.2 Example of sphere transfer
This subsection simulates the sphere transportation
process. As shown in Figure 4, the robot's transfer routestarts at 𝑶𝑶 and reaches 𝑷𝑷 and 𝑸𝑸. Each path 𝑶𝑶𝑶𝑶 ( 𝜑𝜑 =90°), 𝑷𝑷𝑷𝑷 (𝜑𝜑 = 45°) and 𝑸𝑸𝑸𝑸 (𝜑𝜑 = 0°) is completed aftersome time 𝑡𝑡 (0 ≤ 𝑡𝑡 ≤ 5).
The direction angle and mobile speed of the sphericalwheel are defined as functions of 𝑡𝑡 (see Figure 5).

3.1 Trajectory of the end point of the angler velocity
vector and Totally kinetic energy

This section presents the simulated trajectory of theend points of the angler velocity vectors 𝝎𝝎1 and 𝝎𝝎2 , andthe kinetic energies 𝐸𝐸1 and 𝐸𝐸2 .
As shown in Figure 3 (a), the ellipsoid trajectory of𝝎𝝎2 lies nearer to the 𝑥𝑥𝑥𝑥-plane than that of 𝝎𝝎1, andboth trajectories cross a common line parallel to the𝑦𝑦-axis.

90° 0 ≤ 𝑡𝑡 ≤ 5
𝜑𝜑(𝑡𝑡) = � 45° 5 ≤ 𝑡𝑡 ≤ 10
0 ° 10 ≤ 𝑡𝑡 ≤ 15

‖𝑽𝑽‖ = �90𝑡𝑡 (0 ≤ 𝑡𝑡 ≤ 1 , 5 ≤ 𝑡𝑡 ≤ 6 , 10 ≤ 𝑡𝑡 ≤ 11)
1 (1 ≤ 𝑡𝑡 ≤ 4 , 6 ≤ 𝑡𝑡 ≤ 9 , 11 ≤ 𝑡𝑡 ≤ 14)
5 − 𝑡𝑡 (4 ≤ 𝑡𝑡 ≤ 5 , 9 ≤ 𝑡𝑡 ≤ 10, 14 ≤ 𝑡𝑡 ≤ 15)

(17)

(18)

Substituting Eqs. (17) and (18) into Eqs. (6) and (16)respectively, we obtain 𝝎𝝎𝟏𝟏 (𝑡𝑡) and 𝝎𝝎𝟐𝟐 (𝑡𝑡),respectively. Moreover, substituting 𝝎𝝎𝟏𝟏 (𝑡𝑡) and 𝝎𝝎𝟐𝟐 (𝑡𝑡)into Eqs. (7) and (13) respectively, we obtain
𝐸𝐸1 (𝑡𝑡) and 𝐸𝐸2 (𝑡𝑡), respectively.
The simulation result is shown in Figure 6. In theinterval [0, 5], the robot is traveling straight, so we have
𝐸𝐸1 (𝑡𝑡) = 𝐸𝐸2 (𝑡𝑡). In the interval [5, 15], 𝐸𝐸2 (𝑡𝑡) is smaller
than 𝐸𝐸1 (𝑡𝑡).

(a)Table 2 The totally kinetic energy efficiency in spheretransport
Migration pathway[𝑎𝑎, 𝑏𝑏][𝑎𝑎,𝑏𝑏]𝑬𝑬𝑖𝑖

𝜑𝜑Type-ⅠType-Ⅱ𝛾𝛾 [𝑎𝑎,𝑏𝑏]

𝛾𝛾 [0,15]

𝑶𝑶𝑶𝑶[0, 5]
90°

1.66

𝑷𝑷𝑷𝑷

[5, 10]
45°

2.55

𝑸𝑸𝑸𝑸

[10, 15]
0°

3.39

1.66

1.49

1.340%

41 %

61 %

34 % (Total energy reduction ratio)(b)
Figure 5 The definition (a). Sphere direction (b). Spheremobile speed

And. we define integration of 𝐸𝐸1 (𝑡𝑡), 𝐸𝐸2 (𝑡𝑡) with respect
to time interval [𝑎𝑎, 𝑏𝑏].
[𝑎𝑎,𝑏𝑏]𝑬𝑬𝑖𝑖

𝑏𝑏= � 𝐸𝐸𝑖𝑖 (𝑡𝑡) 𝑑𝑑𝑑𝑑

(19)

𝑎𝑎

Energy-reduction ratio in [𝑎𝑎, 𝑏𝑏] is defined as follow.
[𝑎𝑎,𝑏𝑏]𝛾𝛾 [𝑎𝑎,𝑏𝑏] = 100(1 − 𝑬𝑬2[𝑎𝑎,𝑏𝑏]/𝑬𝑬1) [%](20)

As shown in Table 2, it is represented by 𝛾𝛾 [0,5] =0 [%], 𝛾𝛾 [5,10] = 41 [%] , 𝛾𝛾 [10,15] = 61 [%]andis[0,15]𝛾𝛾= 34 [%]. Therefore, the kinetic energy of spheretransport was efficient.

Figure 6 Simulation result of totally kinetic energy

4. Conclusion

Figure 4

We proposed a non-offset mechanism with threedimensional freedom under the assumption of variablerotational axes of the rollers. The kinetic energyconsumption was compared with that of the previous
fixed type in a simulation of sphere transportation, which
confirmed a large energy reduction in the proposedmechanism.
In future work, we plan to evaluate the kinetic energyof the roller arrangement at an arbitrary contact point on
the upper hemisphere as an evaluation function.

The robot's transfer route91

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[9]

Authors Introduction

M.Wada, K.Kato, "Kinematic modeling and simulationof active-caster robotic drive with a ball transmission(ACROBAT-S)".
2016
IEEE/RSJInternational
Conference on Intelligent Robots and Systems. Daejeon,
2016-12-9/25, IEEE Robotics and Automation Society,
S.Ishida ， H.Miyamoto,“Holonomic Omnidirectional
Vehicle with Ball Wheel Drive Mechanism, 2012The
JapanSocietyofMechanicalEngineers.Vol.78,No.790,
pp.2162-2170, 2012.
S. Chikushi, M. Kuwada, et al., Development of NextGeneration Soccer Robot”Musashi150”for RoboCup
MSL,30th Fussy System Symposium, pp. 624-627,2014
K. Kimura, K. Ishii, Y. Takemura, M. Yamamoto,
Mathematical Modeling and motion analysis of the wheel
based ball retaining mechanism, SCIS & ISIS, pp.41064111, 2016.
K. Kimura, K. Ogata, K. Ishii, Novel Mathematical
Modeling and Motion Analysis of a Sphere Considering
Slipping, Journal of Robotics, Networking and Artificial
Life, Vol.6, issue 1, pp. 27- 32, 2019.
K. Kimura, S. Chikushi, K. Ishii, Evaluation of the Roller
Arrangements for the Ball-Dribbling Mechanisms
adopted by RoboCup Teams, Journal of Robotics,
Nrtworking and Artificial Life, Vol.6, issue 3, pp. 183190, 2019.
Lee,Y.C,Danny,Lee,D.V.,Chung,J.,andVelinky,S.A.,"C
ontrol of a redundant, reconfigurable ball wheel drive
mechanism for an omunidirectional platform",Robotica,Cambridge University Press,Vol.25, pp.385-395, 2007.
M. Kumagai, T. Ochiai: “Development of a robot
balanced on a ball – Application of passive motion
to transport”,Proc. ICRA IEEE(2009), pp. 41064111, 2009.
Endo, Tatsuro. et al. "An omnidirectional vehicle on abasketball". 12th International Conference on Advanced
Robotics. Seattle, 2005-07-18/20, IEEE Robotics andAutomation Society, pp. 573-578, 2005.

Dr. Kenji Kimura

He is a Lecturer in National
Institute of Technology, Matsue
College, where he has been since2022. He received the ME(mathematics) from Kyusyu
University in 2002 and received
his Ph.D. degree in engineeringfrom Kyushu Institute of in 2020.
Then, He was mathematical teacher of International
Baccalaureate Diploma Program (Mathematics) andengineering course chief in Fukuoka Daiichi HighSchool, His research interests spherical mobile robot
kinematics, control for object manipulation.
Dr. Kazuo Ishii
He is a Professor in the Kyushu
Institute of Technology, where hehas been since 1996. He received
his Ph.D. degree in engineeringfrom University of Tokyo, Japan, in1996. His research interests span
both ship marine engineering andIntelligent Mechanics. He holds fivepatents derived from his research. His lab got "Robo Cup
2011 Middle Size League Technical Challenge 1stPlace" in 2011. He is a member of the Institute ofElectrical and Electronics Engineers, the Japan Societyof Mechanical Engineers, Robotics Society of Japan, theSociety of Instrument and Control Engineers and so on.

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