Omni Wheel Arrangement Evaluation Method Using Velocity Moments

Abstract

Wheeled omnidirectional mobile robots have been developed for industrial and service applications. Conventional research on Omni wheel robots has mainly been directed toward point-symmetric wheel arrangements. However, more flexible asymmetric arrangements may be beneficial to prevent tipping over or to make the robot more compact. Asymmetry can also be the result of a motor/wheel failure in a robot with a redundant configuration; in this case, it may be possible to continue operations, but with an asymmetrical arrangement. For controlling such asymmetric arrangements, it is necessary to consider the moment of propulsive force generated by the wheels. Since it is difficult to measure the propulsive force accurately, in this work we model propulsive forces as being proportional to the ground speed of the wheels. Under this assumption, we estimated the robot’s behavior in an asymmetric wheel configuration by considering the balance of the velocity moment, which is the moment of the wheel’s ground speed. By verifying the robot’s behavior with various wheel configurations, we confirmed experimentally that the sum of the velocity moments affects the straightness of the robot and allows us to improve the design of asymmetric wheel arrangements and control during wheel failures.

1. Introduction

Recently, mobile robots have been developed widely in the industrial, medical, and service fields [1]. Various modes of locomotion for these robots have been proposed, including wheeled robots [2], quadruped robots [3], and biped robots [4]. When operating robots in production, robots must be able to move in all directions (for use in confined work environments) and with precise locomotion. In addition, it is preferable for robots to operate with less energy, as it improves efficiency.

Leg-type robots can move in all directions, run over bumps, and have flexible posture control. However, they are generally less energy efficient than wheel-type robots. On the other hand, although ordinary wheeled robots are efficient in operation, they are limited in the direction they can move, making precise movement difficult in some cases. Therefore, the development of omnidirectional wheels has been promoted [5,6], as it allows for precise and energy-efficient robots. Figure 1 shows examples of wheeled omnidirectional mobile robots that are in use or under development for industrial and service fields; such as loading and towing cage carts in factories, patrolling and monitoring at construction sites as well as customer service and advertising activities in commercial facilities, hotels, and airports.

Typical omnidirectional wheels are the Omni wheel and the mecanum wheel. The two wheels are shown in Figure 2. Both wheels achieve omnidirectional movement by sliding the freewheel rollers. An Omni wheel is a wheel with freewheel rollers on a plane perpendicular to its axis of rotation. Many Omni wheels use two rows of freewheel rollers, as shown in Figure 2a, to contact the ground continuously. A mecanum wheel is a wheel with freewheel rollers arranged diagonally, as shown in Figure 2b, which are arranged ingeniously so that the next roller can be grounded by itself. Compared to the Omni wheel and the mecanum wheel, the mecanum wheel performs better over bumps than the Omni wheel because it allows for a car-like wheel arrangement. On the other hand, many products of Omni wheels have slimmer widths per wheel diameters than mecanum wheels. For example, the Omni wheel in Figure 2a is 29 mm in width and the mecanum wheel in Figure 2b is 50 mm in width. As a result, the mecanum wheel is known to have a more significant contact point shift than the Omni wheel when switching the freewheel roller, and the robot is prone to vibration [7]. Since floors in factories, hospitals, and commercial facilities are often flat, the Omni wheel with less vibration is preferable. Another disadvantage common to both was the small load capacity. In response to this problem, the company to which the author belongs has developed a high-load-bearing Omni wheel. Against this background, this manuscript describes the development of a robot using an Omni wheel.

Omni wheel robots can achieve omnidirectional mobility with only three wheels. On the other hand, it is known that stability can be improved by increasing the number of wheels [8]. Attempts have also been made to increase the number of wheels to increase the load-bearing capacity of robots [9]. Because stability and payload are essential in industrial applications, robots with more wheels are considered in this manuscript.

In agricultural machinery, asymmetric structures prevent tipping over and improve work efficiency [10]. Parts may also be arranged asymmetrically in industrial applications to prevent tipping over and compactness. On the other hand, Omni wheel robots are generally designed with point-symmetrical wheel arrangements. Some wheel arrangements cause poor operability [11]. However, we investigate asymmetric wheel arrangements that do not compromise operability to develop more compact robots with greater payload capacity. We define this wheel arrangement as an “easy-to-operate Omni wheel arrangement” in this manuscript.

Omnidirectional mobile robots move by making all wheels act in concert. When control of any wheel is lost due to the actuator or mechanical failure, the robot’s operability is compromised. When this occurs, a stuck robot can impede the movement of other robots and humans and incur the cost of moving the robot to a location where it can be repaired. Given certain conditions, a robot that has lost its operability after failure may still be able to move straight ahead in at least one direction by utilizing its remaining wheels. If that is the case, combining this movement with any rotation makes it possible to move the robot to an arbitrary location where it can be repaired. In this paper, we examine the straight-line movement of robots with arrangements that do not satisfy the “easy-to-operate Omni wheel arrangement” condition described above.

The contributions of this work are as follows:

• We discuss the relationship between the velocity moment generated during robot movement and robot turning.
• By considering the balance of the velocity moment, we show that complete omnidirectional movement can be achieved even with an asymmetric wheel arrangement.
• We show that a robot can go straight in at least one direction for any wheel arrangement, and, thus, the robot can be controlled even if an actuator is broken.

The remainder of this paper is organized as follows. Section 2 summarizes recent work on Omni wheel robot kinematics and fault tolerance. Section 3 describes the conditions for an “easy-to-operate Omni wheel arrangement” and a straight-line direction estimation method for Omni wheel robots. Section 4 describes the specifications of the robot that allow experiments with various configurations and methods for evaluating the robot’s behavior. Section 5 shows the effectiveness of the proposed wheel configurations and the straight-line direction estimation method based on experimental results. Section 6 discusses the behavior of the robot when the condition of “easy-to-operate Omni wheel configuration” is not met. Finally, Section 7 summarizes the design and control method of the proposed Omni wheel robot with asymmetric placement.

2. Related Work

Many methods have addressed the relationship between the velocity of a robot’s Omni wheels and its center of mass’s linear and rotational velocity, employing inverse kinematic models. Sofwan et al., Kim et al., and Rijalusalam et al. derived inverse kinematic models of a four-wheeled Omni wheel robot with axles facing the center of the robot [12,13,14]. Almasri et al. proposed an inverse kinematic model for robots with any arbitrary number of wheels in a point-symmetric arrangement with axles oriented toward the robot center [7]. Wang et al. generalized this model to be independent of Omni wheel orientation; however, their validation only addresses models with axles oriented toward the center [15]. This generalized inverse kinematics has also been used by Pang et al., and Amudhan et al. in their work on robots with axles oriented toward the center [16,17]. Our previous work shows that robots can move in all directions, even in a point-symmetric arrangement with axles not oriented toward the robot center [18]. Thus, imposing axle orientation and symmetry is not a requirement; but rather a self-imposed rule for robot design. However, this limitation can become an issue for particular applications; for example, Bulgakov et al. mention the need for asymmetrically shaped machines in the agricultural fields [10].

As a method for determining whether a mecanum wheel robot can move in all directions, Wang et al. proposed using the intersections of bottom-rollers axles of any three mecanum wheels on the robot [11]. If the number of intersections is 2 or 3, the robot can achieve omnidirectional motion. However, not all wheel arrangements that meet the requirements will have adequate operability. This unsuitable arrangement is also similar to Omni wheels. Since this loss of operability is due to the robot’s automatic turning, we focused on the relationship between the speed of each wheel and the robot’s turning. Using this relationship, We achieve omnidirectional movement for good operability even in an asymmetric arrangement.

Regarding fault tolerance for omnidirectional mobile robots, most literature has focused on mecanum wheels, with only a few studies on Omni wheels. Alshoman et al. have achieved fault tolerance of a four-wheel mecanum wheel robot by fuzzy control [19]. This study shows that the failed wheel can reach its destination with or without obstacles, assuming it is free to rotate. On the other hand, it has yet to be shown that the control is effective in different wheel arrangements. Sahin et al. show that fault tolerance can be achieved using a weighted pseudo-inverse of the Jacobian matrix derived for kinematics in a symmetric four-wheeled mecanum wheel robot [20]. This study shows that it can cope with actuator output anomalies. Mishra et al. validated fault-tolerance control for one- and two-wheel failures of a four-wheel mecanum wheel robot [21]. This study assumes that the failed wheel is free to rotate, as in Alshoman et al. As in Sahin et al., a weighted pseudo-inverse matrix is used for fault tolerance control. The results show that fault tolerance was influential in the case of a single-wheel failure but could have worked better in the case of a two-wheel failure, depending on the wheel configuration.

Considering that the same is true for Omni wheels, the method mentioned before may not apply to asymmetric arrangements. Furthermore, because of the complex structure of Omni wheels, it is difficult to predict their exact behavior. Therefore, we estimate the direction in which the robot can move in a straight line as a last resort. Finally, the possibility of controlling the robot as if it were a differential wheeled robot by this method is discussed.

3. Omni Wheel Arrangement Evaluation Method Using Velocity Moments

It is common practice that omni wheel robots have symmetric wheel arrangements [5,6,12,13,14,15]. However, it is desirable to arrange the wheels as flexibly as possible, as there may be cases where it is preferable to have a less symmetrical wheel arrangement for specific applications.

Depending on the wheel arrangement, the robot may spontaneously turn without following previously proposed inverse kinematic models.n One reason is that the inverse kinematic models proposed so far are derived from a geometric approach and do not consider the dynamic loads applied to the robot. If the torque exerted on the robot’s center of mass by the propulsive force generated by each wheel is not balanced, the robot turns even under straight-line control. We define “easy-to-operate” as a state where the vehicle can be correctly controlled in a straight line without causing this unexpected behavior.

Since the Omni wheel is a complex structure with multiple freewheel rollers, it is difficult to determine the propulsive force generated by the Omni wheel in a precise manner. Therefore, we focused on the wheel’s ground velocity instead of the propulsive force. All wheels must reach the target velocities simultaneously to control the robot correctly. Therefore, in accelerating the wheels at the same time, the accelerations of the wheels are proportional to the target velocities. Moreover, the wheel’s propulsive force is proportional to the wheel’s acceleration. Thus, we assume that the propulsive forces generated by the Omni wheels are proportional to the wheel’s ground velocity. Using this assumption, we can compute the moments that the velocity exerts on the robot’s center of mass, velocity moments. If the moments are balanced, the robot would follow the straight-line motion correctly, otherwise, it would turn as it moves.

Note that under our assumption, if the wheel’s velocity vector is V. Then, the distance from the velocity vector to the origin is r, and the moment due to the wheel’s velocity is proportional to $Vr$, which is different from the angular velocity $\omega =\frac{V}{r}$.

3.1. Equation of Balance for Velocity Moments

From the schematic diagram of the Omni wheel robot shown in Figure 3, the moment generated by each wheel’s ground speeds V is expressed as $V{l}^{{}^{\prime }}$: where $l^{{}^{\prime }}$ represents the distance to the robot turning center relative to the velocity vector. Moreover, $L^{{}^{\prime }}$ is expressed by using the position $(x,y)$ and angle $\theta$ of the Omni wheel as follows:

$$\begin{array}{cc}\hfill l^{{}^{\prime }}& =xsin\theta -ycos\theta \hfill \\ \hfill & =lsin(\theta -\beta )\hfill \end{array}$$

(1)

where $l=\sqrt{x^2+y^2}$ and $\beta =arctan\left(\frac{y}{x}\right)$ are the distance and direction of the Omni wheel to the robot turning center.

Based on the equation relating wheel angular velocity to ground speed: $V=r\omega$, and previous study [18], the ground velocity of the wheels when advancing the robot in the x and y directions is determined by the inverse kinematics of the n-wheeled Omni wheel robot as follow:

$$\left(\begin{array}{c}{V}_1\\ V_2\\ V_3\\ ⋮\\ V_n\end{array}\right)=-\left(\begin{array}{ccccc}cos\left({\theta }_1\right)& & sin\left({\theta }_1\right)& & l_{1}sin({\theta }_1-{\beta }_1)\\ cos\left({\theta }_2\right)& & sin\left({\theta }_2\right)& & l_{2}sin({\theta }_2-{\beta }_2)\\ cos\left({\theta }_3\right)& & sin\left({\theta }_3\right)& & l_{3}sin({\theta }_3-{\beta }_3)\\ ⋮& & ⋮& & ⋮\\ cos\left({\theta }_n\right)& & sin\left({\theta }_n\right)& & l_{n}sin({\theta }_n-{\beta }_n)\end{array}\right)\left(\begin{array}{c}{V}_x\\ V_y\\ \mathsf{\Omega }\end{array}\right)$$

(2)

where $V_x$ and $V_y$ are the velocities of the robot’s center of mass in the x and y directions, and $\mathsf{\Omega }$ is the angular velocity of the robot’s center of mass.

The balance equation for the velocity moments to move straight ahead with the velocity $V=(V_x,V_y)$ is expressed by Equation (3).

$$\begin{array}{cc}\hfill \sum ^n{V}_i{l}_i^{{}^{\prime }}& =\sum ^n-(V_{x}cos\left({\theta }_i\right)+V_{y}sin\left({\theta }_i\right))l_{i}sin({\theta }_i-{\beta }_i)=0\hfill \\ \hfill & =V_x\sum ^n-cos\left({\theta }_i\right)l_{i}sin({\theta }_i-{\beta }_i)+V_y\sum ^n-sin\left({\theta }_i\right)l_{i}sin({\theta }_i-{\beta }_i)\hfill \end{array}$$

(3)

where the first and second terms in the second line of Equation (3) represent the sum of the velocity moments in the x- and y-directions straight ahead, respectively.

The following section will discuss how the above equations are applied to evaluate wheel arrangement and estimate the straight-ahead direction.

3.2. Derivation of “Easy-to-Operate Omni Wheel Arrangement”

Suppose that in an n-wheel Omni wheel robot, the position and orientation of $n-1$ wheels are fixed. When moving straight ahead in the x and y directions, the velocity moment generated by the remaining wheel must satisfy the following equations. $V_x$ and $V_y$ are common values for all wheels so that they can be removed from both sides.

$$cos\left({\theta }_n\right)l_{i}sin({\theta }_n-{\beta }_n)=-\sum ^{n-1}cos\left({\theta }_i\right)l_{i}sin({\theta }_i-{\beta }_i)$$

(4)

$$sin\left({\theta }_n\right)l_{i}sin({\theta }_n-{\beta }_n)=-\sum ^{n-1}sin\left({\theta }_i\right)l_{i}sin({\theta }_i-{\beta }_i)$$

(5)

From the above equations, ${\theta }_n$ and $l_{n}sin({\theta }_n-{\beta }_n)$ are obtained as follows:

$${\theta }_n=arctan\frac{-{\sum }^{n-1}sin\left({\theta }_i\right)l_{i}sin({\theta }_i-{\beta }_i)}{-{\sum }^{n-1}cos\left({\theta }_i\right)l_{i}sin({\theta }_i-{\beta }_i)}$$

(6)

$$l_{n}sin({\theta }_n-{\beta }_n)=\frac{-{\sum }^{n-1}cos\left({\theta }_i\right)l_{i}sin({\theta }_i-{\beta }_i)}{cos{\theta }_n}=\frac{-{\sum }^{n-1}sin\left({\theta }_i\right)l_{i}sin({\theta }_i-{\beta }_i)}{sin{\theta }_n}$$

(7)

By expanding $l_{n}sin({\theta }_n-{\beta }_n)$, we see that the omni wheel position $(x,y)$ is obtained as follow:

$$\begin{array}{cc}\hfill l_{n}sin({\theta }_n-{\beta }_n)& =l_{n}cos{\beta }_{n}sin{\theta }_n-l_{n}sin{\beta }_{n}cos{\theta }_n\hfill \\ \hfill & =xsin{\theta }_n-ycos{\theta }_n\hfill \end{array}$$

(8)

It can be seen that the velocity moment balance can be satisfied by placing the Omni wheel on the straight line that satisfies the Equation (8).

Therefore, Equations (6) and (8) can be used to determine the nth Omni wheel and its position and orientation. As can be seen from the fact that $V_x$ and $V_y$ were removed in determining Equations (4) and (5), the wheel arrangement obtained satisfies Equation (3) at all velocity $V=(V_x,V_y)$.

3.3. Estimation of Straight-Line Direction

Even if unbalanced, we assume any Omni wheel robot of any wheel arrangement can go straight in at least a particular direction $X^{{}^{\prime }}$, where the angle from the X axis to the $X^{{}^{\prime }}$ axis is $\varphi$. Since the ground speed of each wheel when moving straight ahead in the $X^{{}^{\prime }}$ axis is obtained by the wheel’s orientation from the $X^{{}^{\prime }}$ axis, the equation of balance for the velocity moment is expressed as follow:

$$\sum ^n-V_{x}cos({\theta }_i-\varphi )l_{i}sin({\theta }_i-{\beta }_i)=0$$

(9)

To Solve the above equation for $\varphi$, $\varphi$ is obtained as follow:

$$\varphi =arctan\left(-\frac{{\sum }^{n}cos\left({\theta }_i\right)l_{i}sin({\theta }_i-{\beta }_i)}{{\sum }^{n}sin\left({\theta }_i\right)l_{i}sin({\theta }_i-{\beta }_i)}\right)$$

(10)

Thus, the robot’s straight-line direction $\varphi$ can be estimated by the ratio of the sum of the velocity moments in the x- and y-directions. Since Equation (10) is valid for any number of wheels, it can be applied to any wheel arrangement. In other words, even a robot that cannot move normally due to motor failure may be controlled to move straight ahead.

The minimum number of wheels for this straight-line estimation is two since Equation (9) only needs to be satisfied. Moreover, since this method does not consider the friction caused by the failed wheel, the failed wheel must not interfere with the robot’s movement, as in the case of a wheel falling off. Therefore, this method does not apply to all robot failures, but it may help to increase the number of measures that can be taken in the event of a failure.

4. Verification

A robot was built to confirm that the proposed wheel arrangement with balanced velocity moments does not impair operability, and its performance was verified. The same experiment was also conducted when the balance was not satisfied to verify whether the direction of straight-line motion could be estimated.

In the remainder of this section, we describe the robot’s hardware configuration and the methodology used for the experiment.

4.1. Hardware Configuration

The fabricated robot is shown in Figure 4. As shown in Figure 5, multiple screw holes are drilled at equal intervals in the board that serves as the foundation, and by combining them with holes drilled in the board that fixes the wheels, various wheel arrangements can be realized.

The hardware configuration of the robot is shown in Figure 6. Stepping motors are used to drive the robot because it is easy to control its speed. An Arduino Due is used to generate the pulse-width-modulated (PWM) signals required to control the motor drivers. In addition, a Raspberry Pi 4 was used as the control PC. While it is possible to directly generate PWM signals using the Raspberry Pi 4’s general-purpose input/output pins, eliminating the need for the Arduino Due, we decided to include the Arduino for its more stable PWM signal generation. To control several motors, the Raspberry Pi 4 requires the use of PWM signals generated by software, which are prone to errors due to delays caused by communication and other factors. To avoid this possible source of errors, we developed a speed control library that is not affected by these delays, using hardware-generated PWM signals from the Arduino Due. This library can specify signals up to 500 kHz, which is more precise than the 10 kHz wiringpi (http://wiringpi.com/reference/software-pwm-library/, accessed 19 January 2023), the Raspberry Pi’s library. The library for speed control is available on GitHub (https://github.com/hijimasa/stepper_pulse_generator_for_arduino_due, accessed on 16 January 2023). The power to the motor drivers and electronic boards is supplied externally to withstand long hours of experimentation. The Raspberry Pi 4 and Arduino are connected via USB. An AR marker is placed in the center of the robot. The AR marker is used to acquire the robot’s position during the motion verification experiment. The AR marker was generated by a Robot Operating System package (http://wiki.ros.org/ar_track_alvar, accessed on 28 November 2022). ROS was also used to control the robot and acquire the marker location during the experiment.

4.2. Experimental Evaluation

To evaluate the operability of the wheel arrangements, we task the robot to move in a given direction. The materials and conditions used in the experiments are listed in

Table 1. The materials and conditions used in the experiments.

Material / Condition	Detail
Omni Wheel	(4 inches) 100 mm Double Aluminum Omni Wheel Bearing Rollers ${}^1$
Wheel Material	Rubber
Flooring	Lonseal’s Lonleam Plain ${}^2$
Standard Applied to Flooring	JIS A 5705
Flooring Material	vinyl resin
Flooring Size	2 mm (thickness) × 1.82 m × 3 m
Flooring laying method	direct placement (no adhesion)
Camera	Logitech’s C930e ${}^3$
Camera Mounting Height	2.17 m
Room Temperature	$24{}^{\circ }\mathrm{C}$

¹https://www.nexusrobot.com/product/4-inch100mm-double-aluminum-omni-wheel-bearing-rollers-14054.html, accessed on 28 November 2022. ² Lonseal’s Lonleam Plain, https://www.lonseal.co.jp/products/floor/pl/, accessed on 28 November 2022. ³ https://www.logicool.co.jp/ja-jp/products/webcams/c930e-business-webcam.960-001199.html, accessed on 28 November 2022.

. As mentioned before, trajectories are measured using AR markers attached to the robot. Specifically, the robot’s position is measured by tracking the AR marker using a camera 2.17 m above the ground. A Logitech C930 camera was selected for this purpose (https://www.logicool.co.jp/ja-jp/products/webcams/c930e-business-webcam.960-001199.html, accessed on 28 November 2022.). The measurements were taken every 200 ms.

To check the robot’s behavior not only in the front-back and left-right directions but also in the diagonal direction, the robot’s movement direction was set to 8 directions, and the path length was set to 1 m. The path length was selected to fit the camera’s field of view. The front of the robot is assumed to be the x-axis positive direction, and the robot’s initial position is set to the origin.

The target speed of the robot is set to 0.1 m/s, 0.2 m/s, and 0.3 m/s and determined based on wheel odometry; no other sensors are employed (the camera is only used to get ground truth locations). For each path and each speed, five measurements were taken. To avoid issues with uneven flooring, the experimental area is covered with a flooring vinyl (Lonseal’s Lonleam Plain, https://www.lonseal.co.jp/products/floor/pl/, accessed on 28 November 2022) that allows for a more homogeneous surface. Once the robot has traveled the predetermined path length, the robot stops.

To verify the “easy-to-operate Omni wheel arrangement”, the wheel arrangements of 4- and 6-wheeled Omni wheel robots were obtained based on the positions and orientations of the 3- and 5-wheels obtained by random numbers. The wheel arrangements generated by shifting the positions and orientations of the wheels from the above wheel arrangements were also used to validate the straight-line direction estimation. The wheel arrangements used are shown in

Table 2. Wheel arrangements of 4-wheeled Omni wheel robots.

Configuration	Wheel Number	x [m]	y [m]	$\mathit{\theta }$ [deg]	$\mathit{\varphi }$ [deg]
E-to-O	1	−0.191	−0.191	−120	-
	2	0.071	−0.191	−60
	3	0.191	0.131	30
	4	−0.281	0.221	120
Not E-to-O 1	1	−0.191	−0.049	−120	1.67
	2	0.071	−0.191	−60
	3	0.292	0.180	0
	4	−0.109	0.251	60
Not E-to-O 2	1	−0.191	−0.191	−120	−59.3
	2	0.071	−0.191	−60
	3	−0.191	0.109	−120
	4	0.191	0.131	30
Not E-to-O 3	1	−0.191	−0.191	−120	−79.0
	2	0.071	−0.191	−60
	3	−0.191	0.109	−120
	4	0.191	0.049	−30

and

Table 3. Wheel arrangements of 6-wheeled Omni wheel robots.

Configuration	Wheel Number	x [m]	y [m]	$\mathit{\theta }$ [deg]	$\mathit{\varphi }$ [deg]
E-to-O	1	−0.208	−0.058	−135	-
	2	0.101	−0.221	−30
	3	0.221	−0.079	30
	4	0.071	0.191	60
	5	−0.071	0.311	120
	6	−0.182	−0.256	−112.7
Not E-to-O	1	−0.150	0.082	90	28.6
	2	0.101	−0.281	−30
	3	0.281	0.221	30
	4	−0.251	0.311	120
	5	0.019	0.199	−150
	6	0.221	−0.281	−30

.

Table 4. Sum of velocity moments per unit command speed [m]—4-wheeled Omni wheel robots.

Configuration	Direction [deg]
	0	45	90	135	180	$-135$	$-90$	$-45$
E-to-O	−0.0029	0.0008	0.0040	0.0049	0.0029	−0.0008	−0.0040	−0.0049
Not E-to-O 1	−0.0130	0.3071	0.4474	0.3255	0.0130	−0.3071	−0.4474	−0.3255
Not E-to-O 2	0.2855	0.3217	0.1694	−0.0821	−0.2855	−0.3217	−0.1694	0.0821
Not E-to-O 3	0.4714	0.3980	0.0915	−0.2686	−0.4714	−0.3980	−0.0915	0.2686

and

Table 5. Sum of velocity moments per unit command speed [m]—6-wheeled Omni wheel robots.

Configuration	Direction [deg]
	0	45	90	135	180	$-135$	$-90$	$-45$
E-to-O	0.0002	0.0001	−0.0001	−0.0002	−0.0002	−0.0001	0.0001	0.0002
Not E-to-O	−0.3789	0.2230	0.6943	0.7589	0.3789	−0.2230	−0.6943	−0.7589

show the sum of speed moments per unit command speed in the arrangements used. The sum in the 45 [deg] and 135 [deg] directions was obtained using Equation (9). As shown in Figure 5, the Omni wheel was designed to be fixed to some degree freely, but because it is not wholly free, E-to-O does not allow the sum of the velocity moments to be completely zero. Therefore, E-to-O was designed to have the lowest sum of velocity moment to the extent feasible.

5. Results

The traversed paths for each arrangement are shown in Figure 7 and Figure 8. The experimental results confirm that the robot goes straight when the “easy-to-operate Omni wheel arrangement” is used, and the robot turns when it is not. Even when the “easy-to-operate Omni wheel arrangement” is not used, the estimated straight-line direction is still straight ahead.

To evaluate the angular velocities at each condition, we obtained the mean and standard deviation by extracting the measurement results for half of the travel time from around the center where the velocities are expected to be stable. The mean and standard deviation of the obtained angular velocities are shown in Figure 9 and Figure 10.

Figure 9 and Figure 10 also show that the angular velocity is smaller than other arrangements and the trajectory is less distorted in the case of the “easy-to-operate Omni wheel arrangement”. Furthermore, it can be seen that the angular velocity increases proportionally with the command velocity. Comparing the sum of velocity moments and angular velocity from Table 4 and Table 5, it can be seen that the larger the sum of measured moments, the larger the angular velocity.

The relationship between the sum of the velocity moments generated by the command velocity and the angular velocity is shown in Figure 11. The figure shows that the two values are proportional and that the proportionality constant depends on the wheel arrangement. The proportionality constant can be the sensitivity to the sum of the velocity moments.

6. Discussion

In the previous section, we showed that it is practical to consider the balance of velocity moments in the design of asymmetric Omni wheel robots. On the other hand, there are cases in which it is difficult to completely satisfy the balance of velocity moments, as in the present experiment. In such cases, the sensitivity of the sum of the velocity moments is vital because it is possible to obtain the permissible sum of the velocity moments from the permissible angular velocities. This section discusses this sensitivity.

The experimental results confirm that the sensitivity varies with the wheel arrangement and the number of wheels. As values to evaluate wheel placement, the

Table 6. Average and root-mean-square of the distance l of the wheel from the origin and the distance $l^{{}^{\prime }}$ between the wheel’s resulting velocity vector and the origin in each arrangement.

Arrangement	l [m]		${\mathit{l}}^{{}^{\prime }}$ [m]
	Average	RMS	Average	RMS
4-wheeled Not E-to-O 1	0.254	0.261	0.177	0.197
4-wheeled Not E-to-O 2	0.231	0.233	0.173	0.201
4-wheeled Not E-to-O 3	0.223	0.225	0.151	0.179
6-wheeled Not E-to-O	0.297	0.309	0.212	0.278

shows the average and root mean square of the distance l of the wheel from the origin and the distance $l^{{}^{\prime }}$ between the wheel’s resulting velocity vector and the origin in each arrangement. The root-mean-square was introduced to accommodate changes in value due to positive and negative distances. Based on the experimental results with the 4-wheeled robots, the value with the largest Not E-to-O 2 and the smallest Not E-to-O 3 or the value with the largest Not E-to-O 2 and the smallest Not E-to-O 3 is suitable to represent the relationship with sensitivity. Therefore, the root-mean-square of $l^{{}^{\prime }}$ may be the most appropriate value. However, further verification is necessary because $l^{{}^{\prime }}$ is not simply proportional to sensitivity. In addition, the root-mean-square of $l^{{}^{\prime }}$ between 4- and 6-wheeled robots is more significant for 6-wheeled, while the sensitivity is lesser for 6-wheeled, suggesting that the sensitivity becomes lesser as the number of wheels increases.

The experimental results confirm that the wheel arrangement may rotate without following inverse kinematics. The generalized kinematics used in this study was obtained geometrically on the condition that the Omni wheel does not slip. It is considered that the robot rotates due to the slippage of the Omni wheel if the propulsive forces are not balanced. If this hypothesis is correct, the sensitivity to the sum of the velocity moments must be obtained by considering the slippage due to friction on the Omni wheel. Therefore, a mechanical approach that takes friction into account rather than a geometric approach is considered necessary to verify the details of the sensitivity.

7. Conclusions

In production, there is a need for wheeled omnidirectional mobile robots that can perform precise movements in a confined work environment with high efficiency. In addition, mobile robots may need to be asymmetric depending on the application. This manuscript presents a design methodology for omnidirectional mobile robots with asymmetric wheel arrangements. In particular, Omni wheels, which have less vibration than mecanum wheels, are used for wheeled omnidirectional mobile robots. It is known that the operability of Omni wheel robots is reduced by the wheel arrangement. We hypothesize that an imbalance in propulsive forces causes this reduced operability. In addition, we assumed that the propulsive force generated by the wheels depends on the ground speed of the wheels. We then proposed a wheel placement method that takes into account the moment balance of the wheel ground speed on the robot’s turning center.

Omni wheel robots lose operability when control of some of their wheels is lost. We hypothesized that even with this loss of operability, the robot could move straight ahead in at least one direction. In this manuscript, we propose a method for estimating the direction in which the robot will move straight ahead using the balance of velocity moments.

To evaluate the effectiveness of the proposed method, we compare the movements of 4- and 6-wheel Omni wheel robots in 8 directions in arrangements that satisfy and do not satisfy the velocity moment balance.

It was confirmed that when the equilibrium of the velocity moments is satisfied, the 4- and 6-wheel robots go straight according to inverse kinematics. When the equilibrium is not satisfied, the robots spontaneously turn. These experimental results show that the equilibrium of the velocity moments can be used to obtain the “easy-to-operate Omni wheel arrangement” that follows inverse kinematics. It was confirmed that the angular velocity is proportional to the commanded velocity when the equilibrium of the velocity moments is not satisfied. It was also suggested that this angular velocity is proportional to the sum of the velocity moments. Since the velocity moment is proportional to the robot’s velocity, the relationship between the sum of the velocity moments and the angular velocity of the robot was obtained. The results showed that the sum of the velocity moments and the robot’s angular velocity are proportional for the 4- and 6-wheel robots. The proportionality constant of the angular velocity to the sum of the velocity moments is the sensitivity to the sum. Experimental results showed that this sensitivity is affected by the wheel arrangement and the number of wheels on the robot. In particular, the results verified for the wheel arrangement suggest that this sensitivity is related to the root mean square of the distance between the velocity vector generated by the wheels and the robot’s center of mass.