1. INTRODUCTION

Spherical wheels can be used in mobile and omnidirectional robots. For example, the ball wheel [1] comprises a sphere with a circular-roller ring arranged on the outer circumference of the sphere. The propulsive force is generated by the rotation of the oblique-rotation axis. Further, there is no binding force in the direction orthogonal to the propulsion force as there is with an omni-wheel [2]. The omni-ball [3] solves the constraint of the ball wheel [1] as it enables omnidirectional driving via the passive rotation of a pair of hemispheres. It offers similar driving performance as the omni-wheel [2] and superior step-climbing ability.

In mobile robots, Active-Caster [4] is composed of an upper sphere and a lower sphere, each of which is in contact with two driving rollers. The upper sphere uses driving rollers to transmit dynamic motion to the lower sphere. This mechanism is used for driving and steering of the caster and enables omnidirectional motion. The balanced-ball robot [5] achieves omnidirectional locomotion by spherical driving using three omni-wheel arranged on the upper hemisphere of an equilateral triangle. The CPU-ball robot [6] has four-omni-wheel arranged on the upper hemisphere of a regular quadrilateral to achieve spherical driving and realize omnidirectional locomotion.

The RoboCup middle-size-league soccer robot utilizes a ball-dribbling mechanism to control the rotation of the ball. Most of the RoboCup teams, such as the Turtles [7], implemented two constraint rollers on the upper half of the ball. Due to makes strong friction force and enhanced ball-holding ability, most designs use slip-roller arrangements as are determined heuristically via experiments in the absence of suitable mathematical models.

In a previous study, we developed a non-slip omnidirectional-locomotion kinematics model that accounts for the sphere kinematics and roller arrangement [8]. However, this model cannot be used for mobile robots as they also undergo slip locomotion. In this study, we modify the previously developed kinematics model and present a novel mathematical model of sphere rotational motion by two constraint rollers that allows for slipping.

In this paper, the outline of the section is as follows: Section 2 consider discusses the existence space of angular velocity vector and the sphere kinematics by two roller. Section 3 conducted simulation. Finally, we present the summary and discuss future tasks.

2. THE SPHERE KINEMATICS BY CONSTRAINT ROLLERS

In this section, we introduce the angular velocity vector of the sphere to algebraically model the sphere rotational motion.

2.1. The Existence of Angular Velocity Vector of the Sphere by Single-constraint Roller

As shown in Figure 1, the center O of a sphere with radius r is fixed as the origin of the coordinate system Σ − xyz. Table 1 shows the variables related to the sphere kinematics. 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_i and O are on the same line. ω denotes the angular velocity vector of the sphere. η_i denotes the unit vector along the rotational axis of constraint roller. ν_i denotes the peripheral speed of the constraint roller. Hence, the velocity vector of the sphere νiS with respect to P_i can be represented by Equation (1):

(1)νiS =  ω×Pi

e_i ∈ span{P_i, n_i} is the unit normal vector along νiR . Using νi=  〈νiR, ei〉 ( νiS=νiR : non-slip condition) and Equation (1), ν_i is represented as follows:

(2)νi=〈νiR,  ei〉=〈ω×Pi,  ei〉=−〈ei×Pi,  ω〉=−r〈ηi,  ω〉

Figure 1

The existence of sphere angular velocity vector in case of single-constraint roller.

Σ−xyz	Three-dimensional coordinate system fixed the sphere
〈a, b〉	Inner product with respect to a and b
‖a‖	Norm of vector a
span{X, Y}	The existence space of ω allow for slip
O	The sphere center
Pi	The Position vector of sphere
ηi	The unit vector along the rotational axis of the constraint-roller
ω	The angular velocity vector of the sphere
ωt	The orthogonal projection of ω with respect to span{P1, P2}
ωs	The orthogonal projection of ω with respect to P1 × P2
νiR	The velocity vector of constraint roller
νiS	The velocity vector of the sphere
ζi	Slip velocity of the sphere with respect to νiR
ei	The unit normal vector along νiR
e	Upper unit normal vector of span{P1, P2}
V	Mobile velocity of sphere on xy-plane
{Xi, e}	Normal orthogonal bases on tangent plane of the sphere at Pi
li(νi)	Set that exists of end point of ω
l′(νi)	The orthogonal projection of li(νi) with respect to span{P1, P2}
νi	Peripheral speed of constraint roller
r	The sphere radius
αi	The roller arrangement angle between ηi and span{P1, P2}
φ	Sphere direction
ρ	Angle of sphere rotational axis

Table 1

The variables related to the sphere kinematics

Thus, ω can be satisfied as Equation (3).

(3)〈ηi,ω〉=−νir

Further, from the property of the constraint roller (i.e., that slip does not occur in side direction of the roller), ω must be on span{η_i, P_i}. Thus, Eq. (3) indicates that ω is constructed as the sum of (ν_i/r)η_i and P_i (see Figure 1). Thus, ω cannot be uniquely determined using a single roller. However, the end point set of ω can be represented as a line set as follows:

(4)li(νi)={ω|(−νir)ηi+kPi,  k∈ℝ}

2.2. The Existence of Angular Velocity Vector of the Sphere by Two-constraint Rollers

Let the roller arrangement the angle α_i (−90° ≤ α_i ≤ 90°) between η_i and span{P₁, P₂}. Using the normal orthogonal base {X_i, e} on tangent plane of the sphere at P_i, η_i can be represented as Equation (5).

(5)ηi=Xicos αi+esinαi

where

(6)Xi=r−1e×Pi,e=P1×P2‖P1×P2‖

In this section, we consider location between l₁(ν₁) and l₂(ν₂) which depend on parameter ν₁, ν₂ (when α₁, α₂ are fixed) and determined the rotational axis of the sphere.

Using Equation (4) (as i = 1, 2), as shown in Figure 2a, if a pair of ν₁, ν₂ exists such that l₁(ν₁) and l₂(ν₂) have points in common, the end point of ω can be uniquely determined. Using Equation (4) (as i = 1, 2), ω must be on span{P₁, η₁}∩span{P₂, η₂}. On the other hand, as shown in Figure 2b, if a pair of ν₁, ν₂ exists such that l₁(ν₁) and l₂(ν₂) have no points in common, slip can occur.

Because the sphere rotational axis is defined with respect to the arbitrary parameters ν₁ and ν₂, Q_o ∈ ℝ³ can be determined such that the sum of the squared distances between Q ∈ ℝ³ and l_i(ν_i) (i = 1, 2) is minimized. Then, Q_o corresponds to the mid-point between l₁(ν₁) and l₂(ν₂) [see Appendix (A)]. Using X_i, P_i, orthogonal projection of l_i(ν_i) with respect to span{P₁, P₂} is represented as Equation (7).

(7)l′1(νi)={ω|(−νicosαir)Xi+kPi,k∈ℝ}

Figure 2

The location of l₁(ν₁) and l₂(ν₂). (a) A pair of ν₁, ν₂ exists such that l₁(ν₁) and l₂(ν₂) have points in common. (b) A pair of ν₁, ν₂ exists such that l₁(ν₁) and l₂(ν₂) have no points in common.

The end point of ω_t (the orthogonal projection of ω with respect to span{P₁, P₂}) is represented as common point of l′1(ν1) and l′2(ν2) [see Appendix (B)].

(8)ωt=1‖P1×P2‖[(ν2cosα2)P1−(ν1cosα1)P2]

A heights l_i(ν_i) from span{P₁, P₂} (i = 1, 2), is represented as follows:

(9)−ν1sinα1r,−ν2sinα2r

Using Equation (9), ω_s (the orthogonal projection of ω with respect to P₁ × P₂) is represented as mean of these.

(10)ωs=−ν1sinα1+ν2sinα22rP1×P2‖P1×P2‖

Thus, ω can be represented with respect to ν₁ and ν₂ ∈ ℝ.

(11)[ωx,ωy,ωz]T=ωt+ωs=1‖P1×P2‖[(ν2cosα2)P1−(ν1cosα1)P2]   −ν1sinα1+ν2sinα22rP1×P2‖P1×P2‖

where

(12)Pi=r[cosθ1,icosθ2,i,sinθ1,icosθ2,i,sinθ2,i]T

Specifically, when α_i = 0°, the second term of the right-hand side of Equation (11) vanishes. Thus, for all ν₁, ν₂, ω ∈ span{P₁, P₂}. In other words, the sphere has omnidirectional locomotion without slip in the expanded form of kinematics model [8].

3. SIMULATION

This section presents the simulation results, including the trajectory of the end point of the angular velocity vector, the angle of the sphere rotational axis, the peripheral speed of the constraint roller, and X_i-component of the slip velocity in the given mobile speed of the sphere: ||V||=1 m/s. The conditions are as follows: r = 1 (m), θ_1,1 = 215°, θ_1,2 = 325°, θ_2,2 = 60°, and α₂ = −α₁. Simulations were conducted at the four different angles, α₁ [represented as k; curve color]. Further, ω_k, ρ_k, ν_i,k, and S_i,k (k = 0, 1, 2, 3) were indicated such as α₁ = 0° [k = 0; red curve], α₁ = 10° [k = 1; blue curve], α₁ = 20° [k = 2; green curve], and α₁ = 30° [k = 3; pink curve]. They are calculated from Equations (18), (19), (22), and (28), respectively. As shown in Figure 3a, ellipsoid trajectories ω_k (k = 0, 1, 2, 3) are getting scarp in turn and have a common line parallel to the x-axis.

Figure 3

Comparison in case of α₁ = 0°, 10°, 20°, 30° in simulation. (a) Trajectory of end point of angular velocity vector of the sphere. (b) Angle of the sphere rotational axis. (c) Rollers peripheral speed. (d) X_i-component of slip velocity of the sphere.

As shown in Figure 3b, ρ_k (k = 0, 1, 2, 3) satisfy the inequality |ρ₀| < |ρ₁| < |ρ₂| < |ρ₃| for all φ. Specifically, when φ = 90° or 270°, ρ_k = 0°. Thus, the sphere undergoes pure rotation (forward and backward movement).

As shown in Figure 3c, ν_i,k (k = 0, 1, 2, 3) are satisfied |ν_i,0| < |ν_i,1| < |ν_i,2| < |ν_i,3| (i = 1, 2) For all φ. Specifically, where φ = 0° and 180° (right and left side forward movement), ν_1,k = −ν_2,k (opposite sign). Where φ = 90° and 270°, ν_1,k = ν_2,k and |ν_i,0| = 0.91, |ν_i,1| = 0.93, |ν_i,₂| = 0.96, |ν_i,3| = 1.03 (m/s).

As shown in Figure 3d, S_i,k (k = 0, 1, 2, 3) satisfy the inequality |S_i,0| < |S_i,1| < |S_i,2| < |S_i,3| (i = 1, 2) for all φ. From Equation (26), where 0° < φ < 180°, ζ_1,k and ζ_2,k are face-to-face. In contrast, where 180° < φ < 360°, ζ_1,k and ζ_2,k are back-to-back. Specifically, when φ = 0° and 180°, the sphere slip speed is ||ζ_1,k||=||ζ_2,k||=0 m/s. Specifically, when φ = 90° and 270°, |S_1,k| and |S_2,k| are at their maxima (|S_1,0|=|S_2,0|=0, |S_1,1|=|S_2,1|=0.16, |S_1,2|=|S_2,2|=0.32, and |S_1,3|=|S_2,3|=0.52 m/s).

Further, when α₁ = 0°, |S_1,0|=|S_2,0|=0 for all φ. Thus, in this case, the sphere has omnidirectional locomotion without slip. Moreover, the proposed model includes the previously developed model [8].

4. CONCLUSION

Herein, we consider the existence of an angular velocity vector for the sphere and propose a sphere kinematics model that allows for slipping. In addition, we demonstrate the trajectory of the end point of the angular velocity vectors of the roller speed and slip speed of the sphere in simulations. This model includes the previously developed model [8] and is expected to be applicable to a wide range of mobile robots in a variety of situations.

In future studies, this model should be verified experimentally. Further, it could be applied to simulate the ball-dribbling mechanism.

CONFLICTS OF INTEREST

There is no conflicts of interest.

Authors Introduction

Mr. Kenji Kimura

He received the M.E. (mathematics) from Kyusyu University in 2002. Then he was a mathematical teacher and involved in career guidance in high school up to 2014. He currently, is instructor of International Baccalaureate Diploma Program (Mathematics: applications and interpretation, analysis and approaches) in Fukuoka Daiichi High School, and student in the doctoral program of the Kyushu Institute of Technology. His current research interest spherical mobile robot kinematics, control for object manipulation, and human body motion Bernstein’s degrees-of freedom problem.

Mr. Kouki Ogata

He received high school diploma in Fukuoka Daiichi high school until 2016. He is currently under Graduate School student at Saga University. His current research interest sphere mobile robot kinematics, control, analytical dynamics.

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.