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Abstract

Mobile robots are widely used in many applications, such as transportation, military domain, searching, guidance, rescue, hazard detection, and carpet cleaning. Because mobile robots usually carry limited power sources, such as batteries, energy saving is an important concern. In particular, differential wheeled mobile robots that have two independently movable wheels are popular because of relatively low mechanical complexity to achieve three-dimensional motion on a ground. This article presents a new wheeled device with a redundant drive system which consists of three independently movable actuators and two planetary gears to connect the actuators to two wheels. The proposed system is able to continue its motion safely when one of the motors breaks down for some causes and also able to operate over a longer period of time due to energy saving consumption property. Experimental results show that the proposed method is effective in saving energy approximately by 20.45% for linear motion and 13.05% for circular motion compared to a conventional one.

Keywords

Wheeled mobile robot redundant drive system planetary gear energy saving

Introduction

Mobile robots are widely used in many different applications, such as transportation, military domain, searching, guidance, rescue, hazard detection, and carpet cleaning.^1–3 Robots usually carry limited power sources, and it is generally impossible to add new supply of energy while working, so energy conservation is an important concern for mobile robots.^4–8

Nowadays, Japan’s declining birthrate and aging population are having a profound impact on the society, economy, and culture. So, an essential task for Japan society is the improvement of the quality of life for the elderly and disabled people and to support their self-movement.⁹ The need of personal mobility supporting personal movement for the aging society is increasing.

Wheeled mobile robots (WMRs) are popular because of relatively low mechanical complexity and energy consumption. In last few decades, many researches have been conducted for energy saving in mobile robots.^6,10–16 WMRs usually have many components, such as sensors, batteries, motors, controllers, and motor drivers. There are several ways to achieve energy saving such as improving the power efficiency of motor power supply and using energy-efficient motors and energy-minimizing control.^13,17,18

A differential wheeled mobile robot (DWMR) involves two independently driven wheels on a common axis.¹ The robot moves straight forward or backward when the wheels rotate at the same speed, and the robot follows a curved path along the arc of an instantaneous circle, around the instantaneous center of rotation (ICR) when one wheel rotates faster than the other.¹ The robot turns about the midpoint of the two driving wheels when both wheels rotate at the same velocity in opposite directions. When one of the motors breaks down or has damage for some causes, the robot rolls around the wheel which is connected to the damaged motor. Eventually, there will be a risk of the passenger’s life because the passenger cannot continue the motion.

Energy saving is also a considerable topic in mobile robots for increasing the operating time using the same battery capacity. Duleba and Sasiadek¹⁹ discussed a modification of the Newton algorithm applied to nonholonomic motion planning for energy optimization. However, the amount of consumed energy cannot be estimated. Ueno et al.¹⁵ proposed a differential drive steer system for the caster drive wheel of an omni-directional mobile robot. However, the system is expensive because they used two motors for each wheel. As mobility demands limited independent power sources, energy management is the key factor to complete tasks. To overcome this, our research group designs a wheeled system consisting of a redundant drive system to continue its motion safely when one of the motors breaks down for some causes.^20,21 In addition, the redundancy provides the use of each motor at its efficient operating condition. The system uses three motors and two planetary gear systems, which have high transmission efficiency and durability for high power.

Main contribution of this article is as follows: a new design of WMRs is presented in this article to guarantee a secure motion even when one drive system does not work. In addition, its effectiveness for energy saving is analyzed to increase the operating time using the limited battery capacity. Furthermore, simulation and experimental results demonstrate that the proposed method is practically effective. The experimental results in this article show that the method is effective in saving energy approximately by 20.45% for linear motion and 13.05% for circular motion compared to a conventional drive system.

This article is organized as follows. A description of redundant wheeled drive system is included in section “Redundant wheeled drive system.” Section “Controller design” presents the controller design for redundant drive system. Section “Simulation and experimental results” presents the experimental results. Section “Conclusion” provides concluding remarks.

Redundant wheeled drive system

System structure

In general, a DWMR has two driving wheels which are attached to both sides of the device, and each wheel is driven by one motor independently. In order to realize a redundant wheeled drive system, we designed a wheeled system consisting of a redundant drive, as shown in Figures 1 and 2.

Figure 1.

Redundant wheeled drive system.

Figure 2.

Experimental device (backside).

The system has two drive wheels with three motors and two planetary gears, which have high transmission efficiency and durability for high power. As shown in Figure 1, motors 1 and 2 are connected to right and left wheels through each planetary gear. Motor 3 is connected to both right and left planetary gears, and it supplies power for both wheels. Planetary gears consist of a planetary carrier and an outer gear that revolve around a sun gear. In this system, the sun gear connects to motor 3, the outer gear connects to motors 1 and 2, and the planetary carrier connects to each wheel.

We assume that like personal mobility vehicle such as electrical wheelchairs, desired linear and angular velocities of the vehicle are given by operators. We further assume that when one motor breaks down, the operator once stops the system and switches the drive mode of into two-motor control mode. Thus, the singularity or instability matter during transition of the drive mode is not considered in this study and is left for future work.

Mobile robot dynamics

This section describes dynamic equations of the WMR. The robot has two identical rear wheels controlled by three independent motors, and two caster front wheels are used for stability. We assume that the sliding of wheels is negligible. This study considers a WMR in Figure 3. The model includes planetary gear property. The dynamics of the mobile robot is given as follows

M_{w} \overset{\cdot}{ν} = D_{R} + D_{L}

(1)

I \overset{\cdot\cdot}{ϕ} = (D_{R} - D_{L}) L

(2)

where M_w, I, $\overset{\cdot}{ν}$ , and $\overset{\cdot\cdot}{ϕ}$ are the mass, moment of inertia, translational acceleration, and angular acceleration of the robot, respectively, and $D_{R}$ , $D_{L}$ , and L represent the driving force applied to the right and left wheels and the distance from WMR’s center point to the driving wheel, respectively.

Figure 3.

Two-wheeled mobile robot.

The relationship between the linear velocity of the WMR, $ν$ , angular velocity of the WMR, $\overset{\cdot}{ϕ}$ , and angular velocity of two driving wheels is given as follows

ν = \frac{R_{w}}{2} ({\overset{\cdot}{θ}}_{L} + {\overset{\cdot}{θ}}_{R})

(3)

\overset{\cdot}{ϕ} = \frac{R_{w}}{2 L} ({\overset{\cdot}{θ}}_{R} - {\overset{\cdot}{θ}}_{L})

(4)

where ${\overset{\cdot}{θ}}_{R}$ and ${\overset{\cdot}{θ}}_{L}$ represent the angular velocity of the right and left wheels, and $R_{w}$ is the radius of wheels. The relationship between ${\overset{\cdot}{θ}}_{R}$ and ${\overset{\cdot}{θ}}_{L}$ and the velocities of three motors is given as follows

{\overset{\cdot}{θ}}_{R} = \frac{R}{R + 1} {\overset{\cdot}{θ}}_{1} + \frac{1}{R + 1} {\overset{\cdot}{θ}}_{3}

(5)

{\overset{\cdot}{θ}}_{L} = \frac{R}{R + 1} {\overset{\cdot}{θ}}_{2} + \frac{1}{R + 1} {\overset{\cdot}{θ}}_{3}

(6)

where ${\overset{\cdot}{θ}}_{1}$ , ${\overset{\cdot}{θ}}_{2}$ , and ${\overset{\cdot}{θ}}_{3}$ represent angular velocities of motors 1, 2, and 3, respectively. R is the ratio of internal gear radius to the sun gear.

Dynamical equations of the motors are given as follows

\begin{matrix} J_{1} {\overset{\cdot\cdot}{θ}}_{1} + C_{1} {\overset{\cdot}{θ}}_{1} = τ_{1} - R R_{s} F_{L} \\ J_{2} {\overset{\cdot\cdot}{θ}}_{2} + C_{2} {\overset{\cdot}{θ}}_{2} = τ_{2} - R R_{s} F_{R} \\ J_{3} {\overset{\cdot\cdot}{θ}}_{3} + C_{3} {\overset{\cdot}{θ}}_{3} = τ_{3} - R_{s} F_{L} - R_{s} F_{R} \end{matrix}

(7)

where $J_{i}$ and $τ_{i}$ , $i = 1, 2, 3$ , represent the moment of inertia and the torque of motor i in Figures 1 and 2, respectively. $C_{i}$ is the viscous friction coefficient, and $R_{s}$ is the radius of the sun gear.

The motor dynamics of the right and left wheels are given as follows

\begin{matrix} J_{L} {\overset{\cdot\cdot}{θ}}_{L} + C_{L} {\overset{\cdot}{θ}}_{L} + R_{w} D_{L} = R_{s} F_{L} + R R_{s} F_{L} \\ J_{R} {\overset{\cdot\cdot}{θ}}_{R} + C_{R} {\overset{\cdot}{θ}}_{R} + R_{w} D_{R} = R_{s} F_{R} + R R_{s} F_{R} \end{matrix}

(8)

where $J_{R}$ and $J_{L}$ represent the moment of inertia for the right and left wheels. $C_{R}$ and $C_{L}$ are the viscous friction coefficients for the right and left wheels.

From equations (1)–(8), we obtain the equation of motion of the WMR as follows

M \overset{\cdot\cdot}{θ} + C \overset{\cdot}{θ} = G τ

(9)

where M is the 3 × 3 inertia matrix, C is the 3 × 3 viscous friction matrix, and G is the 3 × 3 control input matrix, which are as follows

M = [\begin{matrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{matrix}], τ = [\begin{matrix} τ_{1} \\ τ_{2} \\ τ_{3} \end{matrix}]

(10)

\begin{matrix} \overset{\cdot}{θ} = [\begin{matrix} {\overset{\cdot}{θ}}_{1} \\ {\overset{\cdot}{θ}}_{2} \\ {\overset{\cdot}{θ}}_{3} \end{matrix}], \overset{\cdot\cdot}{θ} = [\begin{matrix} {\overset{\cdot\cdot}{θ}}_{1} \\ {\overset{\cdot\cdot}{θ}}_{2} \\ {\overset{\cdot\cdot}{θ}}_{3} \end{matrix}] \end{matrix}

(11)

\begin{matrix} m_{11} = \frac{J_{1}}{R_{R}} + \frac{R_{R} R_{w}^{2}}{4} {M_{w} + \frac{I}{L^{2}}} + J_{L} R_{R} \\ m_{12} = \frac{R_{R} R_{w}^{2}}{4} {M_{w} - \frac{I}{L^{2}}} \\ m_{13} = \frac{M_{w} R_{1} R_{w}^{2}}{2} + J_{L} R \\ m_{21} = \frac{R_{R} R_{w}^{2}}{4} {M_{w} - \frac{I}{L^{2}}} \\ m_{22} = \frac{J_{2}}{R_{R}} + \frac{R_{R} R_{w}^{2}}{4} {M_{w} + \frac{I}{L^{2}}} + J_{R} R_{R} \\ m_{23} = \frac{M_{w} R_{1} R_{w}^{2}}{2} + J_{R} R_{1} \\ m_{31} = \frac{M_{w} R_{R} R_{w}^{2}}{2} + J_{L} R_{R} \\ m_{32} = \frac{M_{w} R_{R} R_{w}^{2}}{2} + J_{R} R_{R} \\ m_{33} = M_{w} R_{1} R_{w}^{2} + J_{L} R_{1} + J_{R} R_{1} + \frac{J_{3}}{R_{1}} \end{matrix}

(12)

G = [\begin{matrix} 1 / R_{R} & 0 & 0 \\ 0 & 1 / R_{R} & 0 \\ 0 & 0 & 1 / R_{1} \end{matrix}]

(13)

R_{1} = \frac{1}{1 + R}, R_{R} = \frac{R}{1 + R}

(14)

C = [\begin{matrix} \frac{C_{1}}{R_{R}} + C_{L} R_{R} & 0 & C_{L} R_{1} \\ 0 & \frac{C_{2}}{R_{R}} + C_{R} R_{R} & C_{R} R_{1} \\ C_{L} R_{R} & C_{R} R_{R} & \frac{C_{3}}{R_{1}} + C_{L} R_{1} + C_{L} R_{1} \end{matrix}]

(15)

where $M_{w}$ is the mass of a WMR.

Controller design

Distribution controller

This study applies distribution and state feedback controllers to the WMR in Figure 2. Figure 4 shows the block diagram of the control system. The reference translational speed $ν_{ref}$ and turning speed ${\overset{\cdot}{ϕ}}_{ref}$ of the vehicle are given by operators. These velocities are typically transformed into desired velocities of left and right wheels using equations (3) and (4). Thus, we do not have to consider the nonholonomic constraints in this study. Personal mobility vehicles such as electrical wheelchairs are typically operated by commercial PD/PID controllers of each motor, and stability is guaranteed by these controllers. Hence, we assume that the stability is guaranteed by commercial PD/PID controllers. The reference rotational speed ${\overset{\cdot}{θ}}_{iref}$ , $i = 1, 2, 3$ , of each motor is obtained by the distribution controller. First, we introduce a fixed ratio d of angular velocities of the wheels to the velocity of motor 3 as follows

R_{1} {\overset{\cdot}{θ}}_{3 ref} = \frac{ν_{ref}}{R_{w}} . \frac{1}{d}

(16)

By substituting equation (3) into equation (16), we obtain the angular velocity ${\overset{\cdot}{θ}}_{3 ref}$ of motor 3 as follows

{\overset{\cdot}{θ}}_{3 ref} = [\begin{matrix} \frac{1}{(2 d R_{1})} & \frac{1}{(2 d R_{1})} \end{matrix}] [\begin{matrix} {\overset{\cdot}{θ}}_{Lref} \\ {\overset{\cdot}{θ}}_{Rref} \end{matrix}]

(17)

By substituting equation (17) into equations (5) and (6), we obtain the angular velocities ${\overset{\cdot}{θ}}_{1 ref}$ and ${\overset{\cdot}{θ}}_{2 ref}$ of motors 1 and 2 as follows

[\begin{matrix} {\overset{\cdot}{θ}}_{1 ref} \\ {\overset{\cdot}{θ}}_{2 ref} \end{matrix}] = \frac{1}{R_{R}} [\begin{matrix} 1 - 1 / (2 d) & - 1 / (2 d) \\ - 1 / (2 d) & 1 - 1 / (2 d) \end{matrix}] [\begin{matrix} {\overset{\cdot}{θ}}_{Lref} \\ {\overset{\cdot}{θ}}_{Rref} \end{matrix}]

(18)

when $d = 1$ , only motor 3 is used for the motion, while only motors 1 and 2 are used for the motion for $d = \infty$ . For any other value of d, three motors are used for the motion.

Figure 4.

Block diagram of the control system.

State feedback controller

In this study, we assume that commercial motor controllers are based on PD control, and we apply the state feedback controller to the proposed WMR as the feedback controller. Feedback gains are obtained by the pole placement method. We consider the following controller

τ = K x_{e}

(19)

where $τ$ is the control inputs. K is the 3 × 6 feedback gain matrix as follows

K = [\begin{matrix} k_{p 11} & k_{p 12} & k_{p 13} & k_{d 11} & k_{d 12} & k_{d 13} \\ k_{p 21} & k_{p 22} & k_{p 23} & k_{d 21} & k_{d 22} & k_{d 23} \\ k_{p 31} & k_{p 32} & k_{p 33} & k_{d 31} & k_{d 32} & k_{d 33} \end{matrix}]

(20)

where $k_{pij}$ and $k_{dij}$ , $i = j = 1, 2, 3$ , are motor angle and velocity feedback gains, respectively, and $x_{e}$ is the difference between the reference and actual values of each state variable

x_{e} = [\begin{matrix} θ_{1 ref} - θ_{1} \\ θ_{2 ref} - θ_{2} \\ θ_{3 ref} - θ_{3} \\ {\overset{\cdot}{θ}}_{1 ref} - {\overset{\cdot}{θ}}_{1} \\ {\overset{\cdot}{θ}}_{2 ref} - {\overset{\cdot}{θ}}_{2} \\ {\overset{\cdot}{θ}}_{3 ref} - {\overset{\cdot}{θ}}_{3} \end{matrix}]

(21)

Simulation and experimental results

In this study, the authors verified the proposed method by means of simulation.²¹ We experimentally applied the proposed method to a WMR, as shown in Figure 2. The control law in equation (19) was implemented using C# language on a laptop PC (OS: WINDOWS 8.1, CPU: 2.3 GHz) with sampling time of 50 ms. In addition, two optical encoders whose resolution is 0.352°/count are attached to motors 1 and 2, and an optical encoder whose resolution is 0.18°/count is attached to motor 3 to measure angular velocities of motors. A pulse counter board with two channels of 24-bit up/down counters and a Digital to Analog converter (DA) board with 16-bit resolution are used for control. Motor drivers are also used in a current control mode, which supplied electric current depending on the voltage commanded from the laptop PC to each motor.

Figure 5 shows an S-shaped velocity profile used as a reference translation velocity of the WMR. The velocity $v (t)$ and the acceleration $a (t)$ of each period are described by the following equations

v (t) = {\begin{matrix} \frac{v_{\max}}{2} (1 - \cos (\frac{π}{T_{a}} t)) & t \in [0, T_{a}) \\ v_{\max} & t \in [T_{a}, T_{a} + T_{c}) \\ \frac{v_{\max}}{2} (1 + \cos (\frac{π}{T_{d}} (t - T + T_{a}))) & t \in [T_{a} + T_{c}, T] \end{matrix}

(22)

a (t) = {\begin{matrix} \frac{v_{\max}}{2} (\frac{π}{T_{a}} \sin (\frac{π}{T_{a}} t)) & t \in [0, T_{a}) \\ 0.0 & t \in [T_{a}, T_{a} + T_{c}) \\ \frac{v_{\max}}{2} (\frac{π}{T_{d}} \sin (\frac{π}{T_{d}} (t - T + T_{a}))) & t \in [T_{a} + T_{c}, T] \end{matrix}

(23)

where $T_{a}$ is the acceleration period, $T_{c}$ is the constant velocity period, $T_{d}$ is the deceleration period, T is the total motion time, and $ν_{\max}$ is the constant velocity. This study considers only smooth trajectories. When non-smooth trajectory is required, the robot must stop once to avoid infinite acceleration. In the straight line motion, the rotational velocity is set as $\overset{\cdot}{ϕ} = 0.0 rad / s$ . The maximum velocity is set as $ν_{refmax} = 1 m / s$ , and $T_{a} = 5 s$ , $T_{c} = 10 s$ , and $T_{d} = 5 s$ . This experiment applies various values of d. The dynamic parameters are shown in Table 1. Sample-decoupled PD controller is applied for each motor i where the proportional and derivative gains are 10 and 1 N m s/rad, respectively. Figures 6 –8 show the simulation results of the control input of each motor. Figures 9 –11 show the simulation results of the translational speed and linear velocity of each motor. In Figure 9, when the value of $d = \infty$ , only motors 1 and 2 operate as expected. In Figure 10, when the value of $d = 1$ , only motor 3 operates as expected. For any other value of d, all of motors work, as shown in Figure 11. The circular motion was obtained by setting the rotational velocity $\overset{\cdot}{ϕ} = 0.5 rad / s$ , as shown in Figure 12, and the path has a velocity profile, as shown in Figure 13. Figures 14 and 15 show the tracking performance from off-trajectory initial conditions, which show the effectiveness of feedback controller.

Table 1.

Dynamics parameters values.

Parameter	Value	Unit
$M_{w}$	46.0	kg
I	0.625	kg m²
$R_{w}$	0.165	m
L	0.275	m
$J_{1}, J_{2}$	0.0149	kg m²
$J_{3}$	0.00181	kg m²
$J_{L}, J_{R}$	0.034	kg m²
$C_{1}, C_{2}$	0.02681	N m s/rad
$C_{3}$	0.01021	N m s/rad
$C_{L}$	0.007134	N m s/rad
$C_{R}$	0.00584	N m s/rad
R	3	–

Figure 5.

S-shaped velocity profile.

Figure 6.

Control input of all motors $(d = \infty)$ , the same profile is obtained for $τ_{1}$ and $τ_{2}$ .

Figure 7.

Control input of all motors (d = 1), the same profile is obtained for $τ_{1}$ and $τ_{2}$ .

Figure 8.

Control input of all motors (d = 3.5), the same profile is obtained for $τ_{1}$ and $τ_{2}$ .

Figure 9.

Simulation result $(d = \infty)$ .

Figure 10.

Simulation result (d = 1).

Figure 11.

Simulation result (d = 3.5).

Figure 12.

Simulation result for circular motion (d = 3.5).

Figure 13.

Simulation result for linear and angular velocities for circular motion, as depicted in Figure 12.

Figure 14.

Simulation result for tracking performance with off-trajectory initial conditions (d = 3.5).

Figure 15.

Simulation result for tracking performance with off-trajectory initial conditions $(d = \infty)$ .

The feedback gains used in experimental are shown in Table 2, which are obtained by the pole placement method (poles are located at −2, −2, and −3 (1/s)). Figures 16 –18 show the experimental results of the control input of each motor. Figures 19 –21 show the experimental results of the translational speed and linear velocity of each motor. In Figure 21, when the value of $d = 100$ , only motors 1 and 2 operate as expected. In Figure 19, when the value of $d = 1$ , only motor 3 operates as expected. But, for any other value of d, all of motors will work, as shown in Figure 20. The circular motion was obtained, as shown in Figure 22, and the path has a velocity profile, as shown in Figure 23.

Table 2.

Experimental condition (controller gains).

$k_{pij}$	Value (N m s/rad)	$k_{dij}$	Value (N m s/rad)
$k_{p 11}$	0.14001	$k_{d 11}$	0.10684
$k_{p 12}$	0.00397	$k_{d 12}$	0.00159
$k_{p 13}$	0.01869	$k_{d 13}$	0.02144
$k_{p 21}$	0.00397	$k_{d 21}$	0.00159
$k_{p 22}$	0.14014	$k_{d 22}$	0.10762
$k_{p 23}$	0.01858	$k_{d 23}$	0.02164
$k_{p 31}$	0.01869	$k_{d 31}$	0.02144
$k_{p 32}$	0.01858	$k_{d 32}$	0.02164
$k_{p 33}$	0.03607	$k_{d 33}$	0.01796

Figure 16.

Control input of all motors (d = 1), the same profile is obtained for $τ_{1}$ and $τ_{2}$ .

Figure 17.

Control input of all motors (d = 2).

Figure 18.

Control input of all motors (d = 100).

Figure 19.

Experimental results (d = 1).

Figure 20.

Experimental results (d = 2).

Figure 21.

Experimental results (d = 100).

Figure 22.

Experimental result for circular motion (d = 2).

Figure 23.

Experimental results for linear and angular velocities for circular motion, as depicted in Figure 22.

Energy consumption

The WMR usually carries limited energy sources such as batteries. The battery capacity determines the operating time. Here, we consider the effectiveness of the proposed WMR for energy saving. Energy required for the motors is obtained by the following equation

E = \sum_{i = 1}^{3} \int_{0}^{t} | τ_{i} {\overset{\cdot}{θ}}_{i} | dt

(24)

where E is the total energy consumption. We verified different robot speeds in simulation, as shown in Figures 24 –27. For all cases, $d = 3.5$ provides the minimum energy consumption. Hence, this value of d is appropriate for the designed system. Because the optimum value of d depends on actuator specifications, at that value of d, the energy was reduced by about 13.99% and 14% compared to the case of $d = \infty$ for the linear motion, as shown in Figures 24 and 25. Also, for the circular motion, it was reduced by 10.68% and 13.99%, as shown in Figures 26 and 27. The repeatability was verified by conducting multiply times experiments with the same condition. Figures 28 and 29 show the total energy consumed in all motors for various value of d for the linear and circular motions. The average energy consumption, shown in Figures 30 and 31, in which $d = 2$ , provides the minimum energy consumption. At that value, the energy was reduced by about 20.45% and 13.05% compared to the case of $d = 100$ (typical drive condition by two motors) for the linear and circular motions, respectively.

Figure 24.

Total energy consumption for linear motion (robot speed = 0.668 m/s).

Figure 25.

Total energy consumption for linear motion (robot speed = 1 m/s).

Figure 26.

Total energy consumption for circular motion (robot speed = 0.668 m/s).

Figure 27.

Total energy consumption for circular motion (robot speed = 1 m/s).

Figure 28.

Total energy consumption for linear motion.

Figure 29.

Total energy consumption for circular motion.

Figure 30.

Average energy consumption for linear motion.

Figure 31.

Average energy consumption for circular motion.

Conclusion

This article presents a new wheeled device with a redundant drive system to continue its motion safely when one of the motors breaks down for some causes. The proposed method also aims energy saving for a wheeled drive system. The distribution and state feedback controllers are applied for the wheeled drive system. The distribution controller creates the reference angular velocity of motors. Experiment was conducted to verify the effectiveness of the proposed method, and a distribution ratio $d = 2$ provides 20.45% and 13.05% reduction in the consumed energy compared to the case that only two motors operate for the linear and circular motions, respectively. Theoretically, we do not consider any robustness of the control system to nonparametric uncertainties such as sliding of wheels because the objective of this article is to show the possibility of the proposed new drive system for energy saving and fail safety. We are currently conducting nonlinear robust controller design.

Footnotes

Academic Editor: Hamid Reza Shaker

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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