Novel Differential Drive Steering System with Energy Saving andNormal Tire Using Spur Gear for an Omni-directional Mobile Robot

Yuki Ueno, Takashi Ohno, Kazuhiko Terashima,Hideo Kitagawa, Kazuhiro Funato and Kiyoaki Kakihara

Abstract— Holonomic omnidirectional mobile robots are use-ful because of their high level of mobility in narrow or crowdedareas, and omnidirectional robots equipped with normal tiresare desired for their ability to surmount difference in level aswell as their vibration suppression and ride comfort. A caster-drive mechanism using normal tires has been developed torealize a holonomic omnidiredctional robot, but some problemshas remain. Here we describe effective systems to control thecaster-drive wheels of an omnidirectional mobile robot. Wepropose a Differential-Drive Steering System (DDSS) using dif-ferential gearing to improve the operation ratio of motors. TheDDSS generates driving and steering torque effectively fromtwo motors. Simulation and experimental results show thatthe proposed system is effective for holonomic omnidirectionalmobile robots.

I. INTRODUCTION

An omnidirectional robot is highly maneuverable in nar-row or crowded areas such as including residences, offices,warehouses and hospitals. This technology can be applied toan autonomous mobile robot in a factory, a wheelchair and soon. Several kinds of omnidirectional mobile robots and theirapplications have been developed [1]-[5]. However, theserobots realized their omnidirectional motion by using specialwheels such as mechanum wheels, ball wheels, omni-disksand omni-wheels. To improve the ride comfort, vibrationsuppression, slippage reduction and ability to surmount dif-ferent in level, omnidirectional robots equipped with normaltires are needed. Arai proposed an omnidirectional vehicleequipped with normal tires [6]. However, it was a non-holonomic vehicle that had to adjust the direction of thewheels before changing the moving direction of the vehicle.Holonomic omnidirectional vehicles, which can move inany direction without changing the direction of the tiresbeforehand, equipped with normal tires include a dual-wheeltype and a caster-drive (active-caster) type [7], [8]. The dual-wheel type has problems as follows. The number of wheelsis limited to two, and it is impossible to get high friction orto adapt to a rough terrain by a synchronous drive of manywheels. Moreover, a passive wheel is needed to stabilize theposture of the vehicle.

Y. Ueno, T. Ohno and K. Terashima are with System and Con-trol Labolatory, Production System Engeneering, Toyohashi Univer-sity of Technology, Hibarigaoka, Tempaku, Toyohashi, Aichi, Japanueno,terasima@syscon.pse.tut.ac.jp

K. Hideo is with the Department of Electrical Control Engineering,Gifu National College of Technology, Kamimakuwa, Motosu, Gifu, Japanhkita@gifu-nct.ac.jp

K. Funato and K. Kakihara are with the KER CO., LTD, Yutakagaoka,Toyokawa, Aichi, Japan kfunato,kkakihara@ker-jp.com

The caster-drive wheel has an offset between the steeringaxis and the center of the wheel. The wheel can move inany direction by controlling the steering axis and the driv-ing wheel independently by using two motors. Holonomicomnidirectional motion of a robot can be realized by usingtwo or more caster-drive wheels. However, the caster-drivewheel also has a problem as follows. When the vehicle is insteady motion, including straight motion and rotation withconstant curvature, only the driving motor works and thesteering motor becomes idle. When the vehicle changes itsmoving direction, high load is applied to the steering motor.Therefore, high power is required both for the driving andsteering motors, which means the vehicle’s mass is increased.

The aim of our research is therefore to develop a holo-nomic omnidirectional mobile robot with a caster-drivewheel minimizing the motor power by using the interferenceof the output of two motors. A new gearing mechanismis proposed to realize the interference. Furthermore, theproposed method is shown to be effective, because it ispossible to surmount difference in level and it absorbs thevibration by using normal tires.

II. PRINCIPLEA. Omnidirectional Motion Using a Caster-Drive Wheel

Fig. 1 shows the caster-drive wheel. Two degrees offreedom can be generated in the steering axis using thismechanism. Longitudinal force is generated by the wheelrotation. Lateral force is generated with the rotation of thesteering axis because of the friction between the floor andwheel. The position and orientation of the wheel can berepresented by the position Ow(Xw, Yw) of the steering axisand the orientation θw as shown in Fig. 1.

By rotating the driving wheel with the angular velocity ωw,velocity xw = rωw is generated in the direction of the Xw

axis. Here, r is the radius of the driving wheel. By rotatingthe steering axis with the angular velocity ωl, velocity yw =−lωl would be generated at the center of the wheel in thedirection of the Yw axis. Here, l is the offset distance betweenthe steering axis and the center of the driving wheel in thedirection of Xw. However, reacting velocity yw = lωl isgenerated at the steering axis in the direction of the Yw axis,because the position of the driving wheel is fixed by thefriction with the ground. Therefore, the velocity (xw, yw) ofthe caster-drive wheel can be controlled by changing ωw andωl.

Fig. 2 shows an example of motion. The initial orientationθw of the wheel is set to be θw = π/2 rad in the frame

2010 IEEE International Conference on Robotics and AutomationAnchorage Convention DistrictMay 3-8, 2010, Anchorage, Alaska, USA

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Revolution center

Grounding point O

Y

X

ωw

ωll

rθw

O (x ,y )w w w

xy ww

Fig. 1. Model of the caster-drive wheel

Fig. 2. Lateral motion to right.

O − XY . The motion, as shown in Fig. 2, can be given bychanging ωw and ωl appropriately. Even though the rotatingwheel itself cannot generate lateral motion to the right, thelateral motion of the robot, which is fixed to the steeringaxis, is realized. No wheel has to control the orientation ofthe robot by itself. The direct kinematic equation is denotedby the state vector xw = [xw , yw]T and the input vectoruw = [ωw, ωl]T as

xw = Bwuw (1)

whereBw =

[r cos θw −l sin θw

r sin θw l cos θw

](2)

The inverse kinamatic equation becomes

uw = B−1w xw (3)

whereB−1

w =[

1r cos θw

1r sin θw

− 1l sin θw

1l cos θw

](4)

Holonomic omnidirectional motion (x, y, θ) of a mobilerobot can be achieved by using two or more caster-drivewheels.

B. Differential-Drive Steering System (DDSS)

We developed a useful method for constructing a caster-drive wheel using a Differential-Drive Steering System(DDSS). The DDSS outputs driving and steering velocitiesfrom two motors using differential gearing. Fig. 3 shows theprinciple of the DDSS. The differential gearing mechanismis realized by using five spur gears, referred to as A, B, C, Cand D. A is engaged with C, and B is engaged with C throughthe counter gear. C and C are fixed to each other. The DDSSis a 2-input/2-output system without fixing any component.

ω

ω

ω

ω

ω

D

B

A

C

C

A

B

C

C'

ZB

ZA

ZC'

ZC

D

Counter gear

Fig. 3. Principle of the proposed DDSS

Turn Unit:D

Bevel Gear

Motor

C'

E

C

B

A

Timing Belt

Fig. 4. Mechanism of the proposed DDSS

A and B are independently driven by two motors. C and Dprovide output torque. Fig. 4 shows the mechanism of theDDSS. Torques of two motors are transmitted to the DDSSby a bevel gear, D, which is fixed to the chassis E, providesthe steering torque, and C, which leads to the driving wheelvia the bevel gear, provides the driving torque. Let ωA, ωB ,ωC , ωC and ωD be of the angular velocity of A, B, C, Cand D in Fig. 3, and ZA, ZB , ZC and ZC be the numberof teeth of A, B, C and C, respectively. When ωD = 0, thesteering angular velocity ωl becomes zero, and we obtain

ωA =ZC

ZA

ZB

ZCωB =

ZC

ZAωC (5)

ωD = 0 (6)

When ωC − ωD = 0, the driving angular velocity ωw

becomes zero because C does not rotate between A and B,and we obtain

−ωA = ωB = ωC(= ωC) = ωD (7)

The direct kinematic equation, which derives driving andsteering output uw = [ωw, ωl]T from motor input uP =[ωA, ωB]T , can be described as

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uw =[

ωC − ωD

ωD

]= BP uP (8)

where

BP =

[ZAZB

ZAZC+ZBZC

ZAZB

ZAZC+ZB ZC

− ZAZC

ZAZC+ZB ZC

ZBZC

ZAZC+ZB ZC

](9)

The inverse kinematic equation becomes

uP = BP−1uw (10)

where

BP−1 =

[ZC

ZA−1

ZC

ZB1

](11)

Next, we derive the motor power ratio of the DDSS. Jointtorques TA, TB, TC and TD of A, B, C and D, respectively,are given by[

TC

TD

]=

[ZB

ZC

ZA

ZC

−ZB

ZC

ZA

ZC

][TA

TB

](12)

where the positive direction of each torque is same as thatof angular velocity in Fig. 3.

For an omnidirectional mobile robot with the DDSS,steady motion including straight motion and rotation withconstant curvature is achieved by ωl(= ωD) = 0. Whenωl = 0(TD = 0), the joint torques are given from (12) by

TA =ZC

ZB

ZA

ZC

TB (13)

TC = −2ZB

ZCTA (14)

TD = 0 (15)

The power ratio of two motors is given from (5) and (13)by

PA : PB = TAωA : TBωB

= (ZC

ZB

ZA

ZC

TB)(ZC

ZAωC) : TB

ZC

ZBωc

= 1 : 1 (16)

On the other hand, when ωw = 0(TC = 0), the jointtorques are given from (12) by

TA = −ZC

ZB

ZA

ZC

TB (17)

TC = 0 (18)

TD =2(ZB)ZC

TA (19)

The power ratio is given from (7) and (17) by

PA : PB = TAωA : TBωB

=ZC

ZB

ZA

ZC

TBωB : TBωB

=ZC

ZB:

ZC

ZA(20)

When ZC

ZBis equal to ZC

ZA, the power ratio yields

PA : PB = 1 : 1 (21)

III. OPERATIONRTIO OF MOTORS

To examine the operation ratio of motors, we compare theDDSS to a conventional caster-drive wheel. We define theoperation ratio δ of motors as

δ =(Sum of motor power in motion)(Sum of rated power of motors)

(22)

The ratio (PA0 : PB0) of the rated power of two motors usedin the DDSS is set to be 1:1. The ratio of rated power usedin the conventional method is also set to be 1:1, as denotedin Wada [8].

We calculate the operation ratio δ in the case of drivingmotion (TD = 0). Let P be the sum of motor output powerneeded to achieve the motion. The result of the conventionalmethod is PA0 = PB0 = P and δ = P

PA0+PB0= 0.5. The

result of the DDSS is PA0 = PB0 = P2 and δ = P

PA0+PB0=

1.Next, we calculate δ in the case of steering motion (TC =

0). The result of the conventional method is PA0 = PB0 = Pand δ = 0.5. The result of the DDSS is PA0 = PB0 = P

2and δ = P

PA0+PB0= 1.

The output power of motors can be decreased by using theDDSS as a caster-drive wheel because of its high operationratio of motors. This means that the size of the robot can besmaller by using the DDSS.

IV. CONSTRUCTION OF OMNIDIRECTIONALMOBILE ROBOT

We constructed a prototype model of a vehicle withfour DDSS wheels to check the feasibility of the proposedmechanism, as shown in Fig. 5. Effectiveness of the proposedDDSS was confirmed by this apparatus. Fig. 6 and TableI show a figure and the specifications of a platform ofthe omnidirectional mobile robot with four DDSS wheels,respectively. The proposed omnidirectional robot has thecapability of climbing a slope of 12 deg and exceedinga difference of 70 mm while carrying a load of 100 kg.Offset distance, l, can be adjusted in this unit by trialand error. If the offset distance is larger, the straight run

Fig. 5. Photograph of the four-wheeled vehicle built in the authorslaboratory

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TABLE ISPECIFICATION OF THE VEHICLE

Size (D × W × H) 800 × 700 × 1280 (mm)Weight 80 (kg)

Motor power 150 (W) × 8Max. velocity 6 (km/h)

Max. acceleration 2.0 (m/s2)Max. slope angle 12 (deg)

Max. step difference 70 (mm)Max. loading weight 100(kg)Dementer of wheel 210 (mm)

Offset distance 8 - 56 (mm)

Fig. 6. Platform of the proposed omnidirectional mobile robot

becomes stable and the demand value of angular velocityof the steering becomes smaller. In this case, larger torqueis needed in the steering axis. On the other hand, whenthe offset distance is smaller, the straight run is unstable,but the mobility is improving. In addition, the torque inthe steering axis is smaller, while the angular velocity ofthe steering axis becomes larger. There is a contrastiverelationship between torques, angular velocity and stabilityof the wheel. Therefore, it is necessary to think about theoffset distance from the experiment. From that reasons, theoffset distance can be adjusted between 8 mm and 56 mmin consideration of the motor capacity. In this paper, offsetdistance is tentatively given as 25 mm.A. Kinematic Model of a Four-wheeled Vehicle

A caster-drive wheel generates two degrees of freedom onthe demensional plane. To generate omnidirectional move-ment on the ideal flat floor, a vehicle must have at leasttwo wheels to control three degrees of freedom in total,which includes two translation degrees of freedom and anattitude degree of freedom. However, a real road surfacecontains a nonideal floor. In consideration of vehicle stabilityon the nonideal floor, we adopted a control system for a four-wheeled vehicle.

Fig. 7 shows a schematic model of the four-wheeledvehicle. The kinematic model is described follows:

xv

yv

θv

=

14 0 − 1

4 (xva sin θv + yva cos θv)0 1

414 (xva cos θv − yva sin θv)

14 0 − 1

4 (xvb sin θv + yvb cos θv)0 1

414 (xvb cos θv − yvb sin θv)

14 0 − 1

4 (xvc sin θv + yvc cos θv)0 1

414 (xvc cos θv − yvc sin θv)

14 0 − 1

4 (xvd sin θv + yvd cos θv)0 1

414 (xvd cos θv − yvd sin θv)

T

xa

ya

xb

yb

xc

yc

xd

yd

(23)

OX

YA(x ,y )A(x ,y )vava vava

D(x ,y )D(x ,y )vdvd vdvd

B(x ,y )B(x ,y )vbvb vbvb

C(x ,y )C(x ,y )vcvc vcvc

θ aa(x ,y )bb bb

(x ,y )aa aa

(x ,y )cc cc

(x ,y )dd dd

θ vv

X

Y

vv

vv

ωvv O (x ,y )vv vvvv

Fig. 7. Model of the vehicle

rx&

ry&1−

wB

wrω

lr

wB

s

1 w

x&

y&s1

s

1y

x

1−

PB

PBl

w

A

Bω

ω

ω

ω

ω

θ

Fig. 8. Block diagram of the single wheel

Then, the inverse kinematic model is described as follows:

xa

ya

xb

yb

xc

yc

xd

yd

=

1 0 −yva cos θv − xva sin θv

0 1 xva cos θv − yva sin θv

1 0 −yvb cos θv − xvb sin θv

0 1 xvb cos θv − yvb sin θv

1 0 −yvc cos θv − xvc sin θv

0 1 xvc cos θv − yvc sin θv

1 0 −yvd cos θv − xvd sin θv

0 1 xvd cos θv − yvd sin θv

xv

yv

θv

(24)Therefore, when xv , yv and θv(= ωv) are given, the

velocity(xa, ya, xb, yb, xc, yc, xd, yd) of four wheels iscalculated by using (24), where xv , yv and ωv indicatethe velocity in the X-direction, Y-direction and the angularvelocity of the vehicle, and xva, yva, xvb, yvb, xvc, yvc, xvd

and yvd indicate the coordinate of each wheels on the localcoordinate system, respectively.

V. SIMULATION AND EXPERIMENT

A. Simulation of the Single Wheel

To check the kinematic model of the DDSS, we simulatedthe motion of the wheel. Fig. 8 shows a block diagram ofthe control system. If ordered velocity xr and yr were given,the angular velocities of gears ωA and ωB are calculated

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0 1 2 3 4 5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Xposition (m)

Yposi

tion (

m)

0 2 4 6

0

0.5

1

1.5

time (s)

x (

m/s

)

0 2 4 6-1

0

1

time (s)

y (

m/s

)

. .w w

Wheel

0 2 4 6-10

0

10

ωA

(ra

d/s

)

0 2 4 6-10

0

10

ωB

(ra

d/s

)

0 2 4 6

0

10

20

ωw

(ra

d/s

)

0 2 4 6-10

-5

0

ωl (

rad/s

)

Fig. 9. Simulation results of the single wheel

by using (11). θw is the steerage angle of the wheel whichis the integral calculus value of steerage angular velocityωl. However, in fact, the wheel angle is detected from theabsolute encoder attached to the steering axis.

Translational motion toward the +X direction with amaximum velocity of 6 km/h (=1.67 m/s) and a maximumacceleration of 0.5 m/s2 was examined. The initial value ofθw was set as 1.57 rad (=90 deg) in this case.

Fig. 9 shows the simulation result, where, xw, ˙yw, ωA,ωB , ωw and ωl indicate the velocity in the X-direction, Y-direction, the angular velocity of gear A and the gear B,the angular velocity of the wheel and the steering axis,respectively. As shown from the simulation result, the wheelcan move to reference directions instantly and independentlyof the initial angle of the wheel. In addition, graphs of theangular velocity of two motors, ωA and ωB , show that theDDSS can be generated driving and steering torques by usingtwo motors simaltaneously as indecated by ωl and ωw.

B. Simulation and Experiment with the Four-wheeled Vehicle

To verify the mobility of the vehicle, we examined its X-axis direction translation movement. Initially the wheels arepointed in different directions, and the value of θv was setto be 1.57 rad in this case.

Fig. 10 shows the simulation result of the four-wheeledvehicle. The rectangle, triangle and four smaller rectanglesin the graph indicates the vehicle body, front of the vehicle

1 2 3

4 5 6

7 8 9

0 0.5-0.5

0

0.5

X position (m)

Y p

osi

tion (

m)

t = 0.00 s

t = 0.80 st = 0.70 st = 0.60 s

t = 0.50 st = 0.40 st = 0.30 s

t = 0.20 st = 0.10 s

Fig. 10. Simulation result of the vehicle

Motor Driver

Motor Driver

Motor Encoder

Motor Encoder

ω

ω

DDSS 1Driving

Steering

ωw

ω l

θ1

1-1

1-2

Absolute

Encoder

DDSS1 Controller

DDSS2 Controller

DDSS3 Controller

DDSS4 Controller

PLC Unit

θ 2

θ 3

θ 4

Fig. 11. Control system of the vehicle

and wheels, respectively. From the results, the vehicle canmove to goal direction instantly, despite the difference in theways the wheels point, by solving inverse kinematics of (24)and (3).

Fig. 11 is a block diagram of the control system ofthe vehicle. A programmable logic controller (PLC) unit isused for controlling the vehicle. The PLC unit has variousexclusive units such as the CPU unit, logic input and outputunit and analog input and output unit.

Each wheel angle is detected from an absolute encoderattached to the steering axis by using the input unit. Tocontrol DC motors, an exclusive motor driver is used. Therotation speed of DC motors is controlled by the motordriver, and reference for the revolution speed is given byanalog voltage from the analog output unit. The CPU unitis used for resolving the kinematic equation from the angleof each wheel and reference velocities and obtaining eachmotor’s angular velocity. The calculated angular velocity istransformed to analog voltage and transmitted to the motordriver, and the vehicle is controlled. When experiments are

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0 1 2 3

0

1

2

3

X position (m)

Y p

osi

tio

n (

m)

-0.5

0

0.5

xv (

m/s

)

-0.5

0

0.5

yv (

m/s

)

-10

0

10

ω (

rad/s

)

Time (s) Time (s)0 20 40 600 20 40 60

-10

0

10

-10

0

10

-10

0

10

ωA

-2 (

rad/s

)ω

(ra

d/s

)

ω (

rad/s

)C

-1A

-1

C-2

A B C D A B C D

. .

A

B

D

C

: Moving direction: Reference: Experiment

Fig. 12. Experimental result of the vehicle

performed, the x-y position of the vehicle can be computedfrom the encoder signal.

Fig. 12 shows the experimental result of the four-wheeledvehicle. In addition, xv , yv, ωA−1, ωA−2, ωC−1, ωC−2

indicates the velocity in the X-direction, Y-direction, theangular velocity of motors in wheel-A (motor:A-1 and A-2) and the angular velocity of motors in wheel-C (motor:C-1 and C-2), respectively. We examined autonomous movingby using a velocity pattern that was generated beforehand.The pattern of motion was at the first translation movementforward, the second translation movement rightward andfinally the translation movement to 45 degrees in the leftrear side.

From the result, we see that there is a little error trajectorywith respect to the reference. It is considered to be the resultof the large amount of friction between the tire and thefloor because the error trajectory is generated when the movedirection changes. We should be able to avoid this problemin future by adding a feedback control system. On the otherhand, when the control system of a vehicle consists of amanual control system such as a joystick or power assistsystem, the vehicle does not require strict trajectory control,because the operator is doing fine adjustment of the vehiclethemselves. The error of the angle of the vehicle in thetranslation movement affects its operability, so we need toimprove stability of the straight-run in the future.

VI. CONCLUSIONS AND FUTURE WORKSA. Conclusions

In this paper, a novel Differential-Drive Steering System(DDSS) is proposed for the caster-drive wheel of a holo-nomic omnidirectional mobile robot, and omnidirectionalmobile robot based on the proposed mechanism is newlybuilt. The DDSS can provide a high operation ratio of motorscompared to a conventional caster-drive wheel. Numericalanalysis and experiments by building a prototype novelvehicle showed the effectiveness of the DDSS.

B. Future Works

Future works include the following:• Posture control on rough terrain.• Evaluation of vibration test on the ground.• Application to an omnidirectional wheelchair.

VII. ACKNOWLEDGMENTS

We would like to express my gratitude to Prof. Miyoshiand Dr. Yoshiyuki Noda for their advice to develop thisresearch.

This work was supported by the 21st Century COEProgram ”Intelligent Human Sensing”, and also G-COE”Frontier of Intelligent Sensing” , and furthermore by AichiScience & Technology Foundation Research Aid Project of2008 and 2009 year. We would like to address the sincerethanks.

REFERENCES

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