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Evaluation of the Roller Arrangements for the Ball-Dribbling Mechanisms Adopted by RoboCup Teams

The middle-size league soccer competition is an important RoboCup event designed to promote advancements in Artificial Intelligence (AI) and robotics. In recent years, soccer robots using a dribbling mechanism, through which the ball is controlled using two driving rollers, have been adopted by teams worldwide. A survey conducted during the 2017 World Cup in Nagoya revealed that the teams determined their roller arrangements heuristically without the use of a formal mathematical process. In this study, we focus on sphere slip speed to develop a mathematical model for sphere rotational motion, allowing for slip. Using this framework, we derived the relationship between the sphere slip and mobile speeds and evaluated the roller arrangements used by the participating teams.


INTRODUCTION
RoboCup is an international project that promotes innovation in Artificial Intelligence (AI), robotics, and related domains. It represents an attempt to develop AI and autonomous robotics research by providing a fundamental problem that can be solved by integrating a wide range of technologies. In particular, the middle-size league soccer competition is a RoboCup event providing a dynamical environment in which many robots cooperate with humans within the same space.
Various developments have been made in the RoboCup middle-size league soccer robot [1,2]. These robots generally have a mechanism similar to that of an omnidirectional movement mechanism using three-omni wheels [3] or that using four-omni wheels [4] and so on, and in recent years, this mechanism is involved in controlling the rotation of a ball. The ball holding mechanism is used.
In the past, several teams have equipped their robots such as arm type [5,6], bar type [7,8] and single-wheel type [9] with ball dribbling mechanisms for controlling the rotation of the ball. To be useful in soccer play, the dribbling performance of such mechanisms must be skillful. However, they cannot grasp a ball and control. Recently, dribbling devices use rollers to grasp the upper half of the ball to enable them to maintain a pull on it while driving in reverse. Friction is generated between the ball and rollers by a spring mounted on a supporting lever. To maintain dribbling, it is essential that the roller and ball rotate with approximately the same speed and that the ball moves in the same direction as the robot. Many robots, such as that deployed by the Turtles team [10], rely on ball handling force and use a roller arrangement in which there is no slip between the roller and ball. To determine their roller arrangement, the Turtles rely on heuristic measures and experimental results. Although slip causes loss of speed in a moving sphere, the resulting friction can improve the ball holding power and rotational stability. Accordingly, slip is an important factor.
The authors conducted an investigation of the ball dribbling mechanisms employed by the teams at the 2017 middle-size league soccer competition in Nagoya, Japan. Table 1 lists our survey results, including team name and roller type, shape, and arrangement angle. The four shapes [□△◇] in the "Symbol" column indicate the roller angles used by the respective teams. Figure 1 shows the rotational axis, ball velocity, and roller velocity in reverse motion of a dual-roller arrangement. The RV-Infinity [11] roller arrangement in Table 1 involves no slip (see Figure 1a) because the roller velocity corresponds to the ball velocity; i.e., the rollers' rotational axes are aligned on a plane that includes the origin of the ball's sphere [12,13]. In another approach, CAMBADA avoids slippage through the use of unconstrained rollers (omni-rollers).
Other teams, namely the Turtles [10], Falcons, Musashi150 [14], NuBot [15], and Water, have adopted roller arrangements that allow for slip to occur (see Figure 1b) through a differential between roller and ball velocity. Based on analysis of sphere rotational motion in which slip is allowed, we developed a sphere kinematics model in which dual-constraint rollers allow for slipping [16].
In this study, we validated the model introduced in Kimura et al. [16] and evaluated the roller arrangements used by the competition teams from the standpoint of sphere speed, efficiency, and sphere slip speed (ball-holding power).  The remainder of this paper is organized as follows. In Section 2, we introduce a sphere kinematics model that allows for slipping. In Section 3, we validated the model introduced in Kimura et al. [16] and theoretical formula of sphere slip velocity vector. In Section 4, we considered the distribution of slip velocity vector and the roller arrangement of a ball dribbling mechanism based on the results of a robotic experiment. Finally, in Section 5, we present a summary and discuss future research.

THE SLIP VELOCITY VECTOR OF THE SPHERE
We quantify the slip between the sphere and roller.
As shown in Figure 2a, the center O of a sphere with radius r is fixed as the origin of the coordinate system ∑ − xyz. The i-th constraint roller is in point contact with the sphere at a position vector P i and is arranged such that the center of mass of the roller P C , P i and O are on the same line. v denotes the angular velocity vector of the sphere. g i denotes the unit vector along the rotational axis of constraint roller. ν i the denotes peripheral speed of the constraint roller. a i (−90° ≤ a i ≤ 90°) denotes roller arrangement angle between g i and span{P 1 , P 2 } which has unit normal vector e. The great circle C G passes thorough P 1 1 2 for all n n (2) where,

VERIFICATION OF THEORETICAL FORMULA
In this chapter, we verified theoretical formulas including sphere kinematic (Section 3.1) and sphere slip velocity vector (Section 3.2) in reverse motion.

Verification of Kinematics
As shown in Table 2, ||V||, j and r are values calculated from v 1 , v 2 = −0.91 (m/s) by Equations (11) and (14) refer to Kimura et al. [16].  (11) and (14) refer to Kimura et al. [16]. ||V|| e , j e and r e are experimental values measured from  In the evaluation case of a = 0°, n 1 e and n 2 e are close to n 1 m and n 2 m , respectively (see Figure 4). As shown in Table 3          In the evaluation case of a = 10°, n 1 e and n 2 e are close to n 1 m and n 2 m , respectively (see Figure 5). As shown in Table 4 In the evaluation case of a = 20°, n 1 e and n 2 e are close to n 1 m and n 2 m , respectively (see Figure 6). As shown in Table 5, n n In the evaluation case of a = 30°, the limitations of the motor drivers and intense dynamical friction that caused heat between the roller surfaces and the sphere obliged us to cease the second and subsequent experiments. However, ||V|| e , j e and r e are partially close to ||V|| m , j m and r m , respectively (see Figure 7). And, n n  Table 6).

Inverse kinematics
Thus, inverse kinematics is validated.  Tables 7-10 show the absolute mean error calculated in interval of 7-8 (s). We consider comparison between ||V|| m , j m , r m and ||V|| e , j e , r e , respectively in detail, see below.  In the evaluation case of a = 0°, ||V|| e , j e and r e are close to ||V|| m , j m and r m , respectively (see Figure 8). As shown in Table 7, |||V|| m − ||V|| e | is <0.06 (m/s), |j m − j e | is <3.9 (°), and |r m − r e | is <6.3 (°).
In the evaluation case of a = 30°, the data have intense behavior due to limitations of the motor drivers. However, ||V|| e , j e and r e are partially close to ||V|| m , j m and r m , respectively (see Figure 11). And, |||V|| m − ||V|| e | is 0.04 (m/s), |j m − j e | is 9.7 (°), and |r m − r e | is 5.1 (°) (see Table 10).
Thus, forward kinematics is validated.

Consideration of Slip Velocity Vector
As shown in Table 11 are experimental values calculated from ||V|| e , j e , r e , n 1 e and n 2 e by Equations (1) and (2).
Using Equation (2)       In the evaluation case of a = 0°, S e 1 and S e 2 are close to S m 1 and S m 2 , respectively (see Figure 12). As shown in Table 12 In the evaluation case of a = 10°, S e 1 and S e 2 are close to S m 1 and S m 2 , respectively (see Figure 13). As shown in Table 13 In the evaluation case of a = 20°, S e 1 and S e 2 are close to S m 1 and S m 2 , respectively (see Figure 14). As shown in Table 14 In the evaluation case of a = 30°, the data have intense behavior due to limitations of the motor drivers. However, S e 1 and S e 2 are partially close to S m 1 and S m 2 , respectively (see Figure 15). And, S S − is 0.08 (m/s) (see Table 15).
Thus, Equations (1)- (3) is validated by experiment. Figure 16 shows the relationship between the sphere slip speed and sphere mobile speed. The horizontal and vertical axes show S i e (i = 1, 2) and ||V|| e , respectively. Under this kinematics model, it assumed that the roller and sphere contact at a single point. In reality, of course, they would contact along a surface, causing, y i to generate a frictional force F i in opposition to y i . Referring to consideration, in which y 1 and y 2 are aligned back-to-back along a great circle, the frictional forces F 1 and F 2 generated by y 1 and y 2 , respectively, would be in and are opposite directions and face-to-face (from Section 3 refer to Kimura et al. [16]).

CONCLUSION
In this study, we verified derive the sphere kinematics that allows for slipping and considered sphere slip velocity vector and evaluated the roller arrangement used by the world teams. As a result, Tech United Turtles and Falcons have adopted optimum roller arrangement in teams of mobile speed efficiency.
In future studies, by applying the ball-dribbling mechanism, this model should be verified experimentally.

CONFLICTS OF INTEREST
The authors declare they have no conflicts of interest.

Dr. Kazuo Ishii
He is a Professor in the Kyushu Institute of Technology, where he has been since 1996. He received his PhD degree in Engineering from University of Tokyo, Tokyo, Japan, in 1996. His research interests span both Ship Marine Engineering and Intelligent Mechanics. He holds five patents derived from his research. His lab got "Robo Cup 2011 Middle Size League Technical Challenge 1st Place" in 2011. He is a member of the Institute of Electrical and Electronics Engineers, the Japan Society of Mechanical Engineers, Robotics Society of Japan, the Society of Instrument and Control Engineers and so on.