a motion control method for omni-directional mobile robots

Research article

A motion control method for omni-directionalmobile robots based on anisotropy

Chuntao LengSchool of Mechanical Engineering, Shanghai JiaoTong University, Shanghai, China, and

Qixin Cao and Charles LoResearch Institute of Robotics, Shanghai JiaoTong University, Shanghai, China

AbstractPurpose – The purpose of this paper is to propose a suitable motion control method for omni-directional mobile robots (OMRs) based on anisotropy.Design/methodology/approach – A dynamic modeling method for OMRs based on the theory of vehicle dynamics is proposed. By analyzing thedriving torque acting on each axis while the robot moves in different directions, the dynamic anisotropy of OMRs is analyzed. The characteristics ofdynamic anisotropies and kinematic anisotropies are introduced into the fuzzy sliding mode control (FSMC) system to coordinate the driving torque as afactor of influence.Findings – A combination of the anisotropy and FSMC method produces coordinated motion for the multi-axis system of OMRs, especially in the initialprocess of motion. The proposed control system is insensitive to parametric vibrations and external disturbances, and the chattering is apparentlydecreased. Simulations and experiments have proven that an effective motion tracking can be achieved by using the proposed motion control method.Research limitations/implications – In order to obtain a clearer analysis of the anisotropy influence during the acceleration process, only the case oftranslation motion is discussed here. Future work could be done on cases where there are both translation and rotation motions.Practical implications – The proposed motion control method is applied successfully to achieve effective motion control for OMRs, which is suitablefor any kind of OMR.Originality/value – The novel concept of dynamic anisotropy of OMRs is proposed. By introducing the anisotropy as an influential factor into theFSMC system, a new motion control method suitable for OMRs is proposed.

Keywords Robotics, Control technology, Torque

Paper type Research paper

1. IntroductionDifferent from traditional mobile robots, omni-directionalmobile robots (OMRs) can achieve translation along anyarbitrary direction without rotation, which results in theiragile performance (Pin and Killough, 1994; Campion et al.,1996). OMRs have been popularly employed in manyapplications, especially in omni-directional wheelchairs(Wada and Asada, 1999), soccer player robots in RoboCupcompetitions (Samani et al., 2004), and service robots (Qiuand Cao, 2008).

With the special mechanism of the omni-directional wheels,the driving torque required for each wheel is different whilethe robot moves along an arbitrary direction, this is calleddynamic anisotropy in this paper. And the speed of eachwheel is different while the robot moves along an arbitrary

direction, this is called kinematic anisotropy. Especially,owing to the different applications (Tlale, 2006; Chatzakoset al., 2006), the arrangement of the omni-directional wheelsis not generally symmetrical, which makes the anisotropymuch more distinct. Owing to the anisotropy of motioncharacteristics, the former motion control applicable fortraditional mobile robots is not suitable for OMRs (Leng et al.,2008). All of them considered the OMR as an isotropicproblem and therefore the potential of OMRs were not fullyexhibited.

Many motion control methods were proposed for OMRs inthe past few years, but some of them only took into accountthe kinematics but not the dynamics. Daniel et al. (1985)presented the analytical model of the platform kinematics anddescribed the results of preliminary open-loop controlexperiments. Wang et al. (2007) solved the time-optimalcontrol problem as a constrained non-linear programmingproblem according to the kinematic model. Feng et al. (1989)described the design and implementation of a distributedservo-control system for the OMR which can execute

The current issue and full text archive of this journal is available atwww.emeraldinsight.com/0143-991X.htm

Industrial Robot: An International Journal37/1 (2010) 23–35q Emerald Group Publishing Limited [ISSN 0143-991X][DOI 10.1108/01439911011009939]

This paper is supported by the National High Technology Research andDevelopment Program of China (863 Program), 2007AA041602-1.

23

arbitrary trajectories specified by a sequence of three degrees-of-freedom (DOF) acceleration commands.

Most work on the control of OMR is based on the dynamicmodel and feedback method. Watanabe et al. (1998)developed the dynamic model of an OMR, and severalcontrol strategies are discussed based on linear controlmethods while the robot dynamics is non-linear (Watanabe,1998). Purwin and D’Andrea (2006) took limited friction andvehicle dynamics into account, described an algorithm tocalculate near-optimal minimum time trajectories for fourwheeled omni-directional vehicles, which is based on arelaxed optimal control problem. Betourne and Campion(1996) devoted to the dynamic analysis of real redundantmobile robots, an output feedback linear control law isderived. Chen et al. (2002) presented an off-road omni-directional robot, which can run on an uneven road andobstacles, and also showed the adaptive control method forthe OMR.

Although many work proposed the motion control based onthe dynamics of OMR, the anisotropy is not introduced intothe control system. Accordingly, the former motion controlmethod cannot exhibit the potential of OMRs.

As a non-linear system for OMR, there are manyuncertainties, such as slip (Williams et al., 2002), non-linearrotational speed output from the DC motor, battery powerfluctuations, etc. Owing to the non-linear and time-varianceresulting from the above influential factors, it is difficult todeduce the exact control system model for the OMR.Therefore, it is necessary to find a robust control method forOMR. As a robust control method, sliding mode control(SMC) can overcome the influence of parametric vibrationsand external disturbances. In the former research, it has beenadopted in many robot control systems already, e.g. Yang andKim (1999) proposed a robust tracking control of non-holonomic wheeled mobile robots using the sliding mode, andthe proposed scheme is robust to bounded externaldisturbances.

It is well-known that SMC uses discontinuous controlaction to drive state trajectories toward a specific hyper-planein the state space, and to maintain the state trajectories slidingon the specific hyper-plane until the origin of the state space isreached (Hung et al., 1993; Hwang, 2002). This principleprovides a guide to design a fuzzy controller for achievingsystem stability and satisfactory performance. Therefore,fuzzy sliding mode control (FSMC) combining two controlprinciples can provide a robust controller for the non-linearsystems (Hwang and Lin, 1992; Hwang and Chang, 2008),such as OMRs.

Motivated by the principle of the SMC, the control lawoften consists of equivalent control and switch control. Theeffect of equivalent control term is to force system state toslide on the sliding surface, and switch control is to derive thestates toward the sliding surface. To improve theshortcomings such as the chattering phenomenon, a fuzzycontroller is employed. Chen (1999) applied geneticalgorithms to learning membership functions for obtainingan optimal fuzzy control, based on SMC, an expert fuzzycontroller synthesized by a collection of fuzzy linguisticcontrol rules is proposed. In Ha et al. (2001), FSMC isapplied to the control of a hydraulically actuated mini-excavator.

The main objective of this work is to propose a motioncontrol method for the OMR. According to the characteristics

of OMR, we designed a fuzzy controller which is combinedwith SMC. In this control method, an influence factor ks isintroduced to the switch control to coordinate the drivingtorque. As a result, the control system is insensitive toparametric vibrations and external disturbances, and thechattering phenomenon is reduced.

With the special mechanism of omni-directional wheels, thedifferent wheel arrangements, and the interaction forcebetween the redundantly actuated wheels, it results in adynamic anisotropy for OMRs. Based on the theory of vehicledynamics, the acceleration and deceleration characteristic ofeach wheel is distinct. Therefore, when the robot moves alongan arbitrary acceleration and direction, the driving torqueacting on each wheel is different, and the rotational speed ofeach wheel is also different according to the kinematics. As aresult, the slip phenomenon during the acceleration process ismost prominent. Accordingly, it is important to pay moreattention to this issue, but there is little research that dealswith this problem for OMR in the past. To make each wheelachieve a coordinated motion, we translate the torque curveand rotational speed curve into an influential factor, which isalso noted as anisotropy factor kT. And the anisotropy factor isintroduced to the switch control to coordinate the drivingtorque of each wheel output by FSMC. By introducing theanisotropy factor resulting from anisotropy, the robot cantrack the target speed precisely and quickly. Experiments werecarried out and the results were promising. Finally, a newFSMC based on anisotropy is proposed in this paper.

2. Motion characteristics analysis2.1 Kinematics modelingThere are many types of omni-directional wheels (Ferriereet al., 1996). In this paper, we discuss the wheel which isshown in Figure 1. This omni-directional wheel assemblyconsists of a pair of “orthogonal wheels” (Pin and Killough,1994). The wheel is composed of a driving roller and passiverollers, and the passive rollers are symmetrically distributedaround the driving roller. It is obvious from Figure 1 that theaxle (Si) of the driving roller intersects the axle (Ei) of passiverollers, and the angle is 908. The wheel is driven by the motorin a direction orthogonal to the wheel axle, and the passiverollers rotate in a direction parallel to the wheel axle. TheOMR in Figure 2 uses these omni-directional wheels.

To design a robot with good performance, it is necessary tobuild the kinematic model for analyzing the velocitycharacteristics of the OMR (Angeles, 2003; Kalmar-Nagyet al., 2004). According to the velocity relationship of thedriving roller and the passive roller as shown in Figure 1, thevelocity of the driving roller center oi can be determined by:

_oi ¼ 2R _uiU i 2 r _fiZ i ð1Þ

Meanwhile the velocity _oi also can be denoted by the velocityof robot center ð_cÞ and angular velocity v, and the equation is:

_oi ¼ _cþ vjdi where j ;0 211 0

" #ð2Þ

Owing to the passive rollers not being driven by a motor, theangular velocity of the passive roller _fi is irrelevant to ourstudy, and can be eliminated during kinematic analysis. Dot-multiplied by the axle vector Ei on both sides of equations (1)

Motion control method for omni-directional mobile robots

Chuntao Leng, Qixin Cao and Charles Lo

Industrial Robot: An International Journal

Volume 37 · Number 1 · 2010 · 23–35

24

and (2), we can derive equation (3) as a general kinematicequation for the OMR:

2R _ui ¼ ½Eijdi ;Ei �t; i ¼ 1; 2; . . . ; n where t ;v

_c

" #ð3Þ

where:di the vector from the point of

robot center C to the pointof driving roller center oi;

di ¼ d(i ¼ 1, 2, . . . , n), R and r the radiuses of the drivingroller and passive roller,respectively;

_ui the regular velocity of thedriving roller; and

Ui and Zi the vectors defined as inFigure 1.

According to the above analysis, the inverse kinematics equationof a three-wheeled OMR (Figure 2) can be deduced as:

_u1R_u2R_u3R

0BB@

1CCA ¼

0 1 d2

ffiffiffi3

p=2 21=2 dffiffiffi

3p

=2 21=2 d

0BB@

1CCA

_x_yv

0BB@

1CCA ð4Þ

And the forward kinematics equation is shown in equation (5):

_x_yv

0BB@

1CCA ¼

0 2ffiffiffi3

p=3

ffiffiffi3

p=3

2=3 21=3 21=31=3d 1=3d 1=3d

0BB@

1CCA

_u1R_u2R_u3R

0BB@

1CCA ð5Þ

The speed of each wheel is different while the robot moves alongdifferent directions, which is called as kinematic anisotropy. Forexample, the absolute speed value of each wheel is shown inFigure 3 while the linear speed is 1 m/s, and the angular speed is0. The value b of the abscissa denotes the motion direction.

2.2 Dynamics modelingThe sum of the kinematic energy K of the robot is given inequation (6) (Viboonchaicheep et al., 2003):

K ¼ 12

·m · ð_x2 þ _y2Þ þ 12

· Jr ·v2 þ 12

· Jd · _u21 þ _u

22 þ _u

23

� �ð6Þ

where:m the total mass of the robot;Jr the robot moment of inertia; andJd the wheel’s moment of inertia around the center of

revolution.

Figure 1 Omni-directional wheel

Passive wheel

Driving wheel

Si SiSi

Ei

Ei

EiZi

Zi

Bi

Ai

Hi

Ui

Ui

Oi

Oidi

X

Y

C

c.

R

rb

gi

q.i

w

f.i

Figure 2 Omni-directional mobile robot

2

(c)(a) (b)

V2V1

V3

Y

c.

1

3

y.

x.

Xw




Volume 37 · Number 1 · 2010 · 23–35

25

The loss energy is expressed as:

D ¼ 12

·Du · _u21 þ _u

22 þ _u

23

� �ð7Þ

where:Du the coefficient of the wheel’s viscous friction.Substituting equation (5) into (6) and utilizing Lagrangianequation:

ddt

›K› _ui

� �2

›K›ui

¼ ti ð8Þ

yields equation (9):

t ¼ M €uþDu_u ð9Þ

where:_u ¼ ½ _u1 _u2 _u3 �T the rotational speed of each wheel; andt ¼ ½ t1 t2 t3 �T the driving torque acting on each wheel.

M ¼

2Aþ Bþ Jd 2Aþ B 2Aþ B2Aþ B 2Aþ Bþ Jd 2Aþ B2Aþ B 2Aþ B 2Aþ Bþ Jd

2664

3775

where A ¼ 2mR2

9; B ¼ JrR2

9d2

Based on the above equations, the state equation is yielded as:

€X ¼ 2NM21DuN21 _Xþ RNM21tþ h ð10Þ

where:

_X ¼ ½ _x _y v �T; N ¼

0 2ffiffiffi3

p=3

ffiffiffi3

p=3

2=3 21=3 21=31=3d 1=3d 1=3d

0BB@

1CCA; and

h ¼ ½ h1 h2 h3 �T is the disturbance, jhi j , H , H is theupper bound of disturbance.

3. Dynamic anisotropy analysis3.1 Dynamics analysisIn other similar research, the wheels are assumed to be rigidbodies and non-deformable disks, and all interaction betweenthe wheel and the ground is assumed to occur through a pointcontact. In reality, most wheels are made of a deformablematerial (such as rubber), so the interaction is through acontact patch and not a point.

Based on the theory of vehicle dynamics (Yu, 1990), whilethe OMR accelerates, the tangential force caused by thecontact deformation between the driving wheel and theground is the traction of the mobile robot. Rolling resistancecouple resulting from the deformation of the wheelcounteracts the rotation of the driving wheel, and thepassive wheel is also subject to effects of the tangential forceand resistance couple. The unitary force diagram and theforce during the accelerating period of OMR are shown inFigure 4. And the parameters are defined in Table I.

According to the force diagram (Figure 4), the dynamicmodel of driving wheel and passive wheel (Yu, 1990) aredefined by following equations:

mdadi ¼ f di 2 F 0 Jd1di ¼ ti 2 f diR2MPd ð11Þ

mpapi ¼ Qp 2 f pi Jp1pi ¼ f pi r 2MPp ð12Þ

where:md the mass of driving wheel; andmp the mass of passive wheel.

Figure 3 Speed value curve of each wheel

1

0.8

0.6

0.4

0.2

0

–10 20 40 60 80 100 120 140 160 180

Speed of wheel 1

Speed of wheel 2

Speed of wheel 3

b (°)

–0.2

–0.4

–0.6

–0.8

V(m

/s)

Figure 4 Force analysis

Y

(a)

(b) (c)

ad apV

F′

R rPd

MPd MPP

Force acting on driving wheel Force acting on passive wheel

Qp Pp

Nd NpNd Np

Force diagram

c

b

yR ad1

a .

ap1fd2

fd1

fd3

fp2

fp1

fp3

fpfd

xR

c

X

t

ed ep




Volume 37 · Number 1 · 2010 · 23–35

26

3.2 Dynamic anisotropyOwing to the special mechanism of omni-directional wheelsand the existence of dynamic anisotropy, the driving torqueacting on each wheel is distinct, when the robot moves in anarbitrary direction.

With inconsistent torque acting on the wheel, the rotationalspeed of the wheels cannot match each other. Then the robotcannot track the target speed during acceleration, and as aresult, the slip phenomenon will be much more prominentduring the acceleration than during other periods of motion.In order to avoid the uncoordinated motion between the threewheels, it is necessary to analyze the driving torque requiredby each wheel while the robot moves at an arbitrary speed anddirection.

Owing to symmetry of the structure, only the case of0 # b # 1808 is discussed in this work. For simplification, it isassumed that the robot moves in direction b without rotation.To analyze the specifics of the driving torque, the drivingtorque acting on each wheel when the robot moves in differentdirections will be analyzed in the following section. In thefollowing section, we will only analyze the case where thedirection of acceleration b is subjected to b , 308, andthe force diagram is shown in detail in Figure 4(a).

As shown in Figure 4(a), in the local coordinate system{c, xR, yR}, suppose that the robot moves in the direction of bonly with the translational motion, and the acceleration is a.Then according to the Newton’s Second Law, we can obtainthe following equations (13) and (14):

f d1 þ f d3 ¼ f d2 ð13Þ

f d1sinðbÞ þ f d2cosða2 bÞ þ f d3cosðaþ bÞ2 f p1cosðbÞ2 f p2sinða2 bÞ2 f p3sinðaþ bÞ ¼ ma

f d1cosðbÞ þ f d2sinða2 bÞ2 f d3sinðaþ bÞ þ f p1sinðbÞþ f p2cosða2 bÞ2 f p3cosðaþ bÞ ¼ 0

ð14Þ

where a ¼ 308.According to equation (12), the tangential counterforce

acting on the passive wheel by the ground at the point ofcontact can be obtained by:

f pi ¼Jpiapi

r2 þMPp

r; i ¼ 1; 2; 3 ð15Þ

In Figure 4(a), the relation between api and a is shown inequation (16):

ap1 ¼ a · cosðbÞap2 ¼ a · cosð90 2 aþ bÞ ¼ a · sinða2 bÞap3 ¼ a · cosð90 2 a2 bÞ ¼ a · sinðaþ bÞ

ð16Þ

Substituting the equations (13)–(16) into equation (11), wecan obtain the driving torque acting on driving wheel ti. Withthe same deduction, we can obtain ti while b varied in(0, 1808). In order to achieve intuitive cognition about thedriving torque, we describe it in Figure 5. And the parametersneeded for the computation are shown in Table II, where allof the values are determined according to the real robot.

4. Fuzzy sliding mode controlOwing to the uniqueness of an omni-directional wheel, thewheel moves in any direction with three DOF on a two-dimensional plane. And the interaction force between theredundantly actuated wheels results in slip phenomenoneasily, which affects the motion control. As a non-linearcomponent for DC motor, it is difficult to achieve multi-axiscoordinated motion control. Also, because the power for DCmotors is supplied by a portable battery, the non-linearcharacteristic of system is more apparent when the powerdecreases with respect to time during operation.

Owing to the above influential factors, the OMR exhibitsnon-linear and time-variant characteristics. It is difficult tobuild the exact mathematical model for the OMR motioncontroller. To find a more robust control for OMRs, firstwe propose a SMC method to overcome the uncertainty in

Figure 5 Driving torque acting on each wheel

2

1.8

1.6

1.4

1.2Torq

ue (

Nm

)

1

0.8

0 20 40 60 80 100 120

b (°)

140 160 180

Driving torque acting on wheel 1Driving torque acting on wheel 2Driving torque acting on wheel 3

Table I Parameter definition

Pd, Pp Gravitational loads acting on driving wheel and passive wheelNd, Np Normal counter-force acting on the driving wheel and passive

wheel by the groundfdi, fpi Tangential counter-force acting on the driving wheel and

passive wheel by the ground at the point of contacti Sequence number of the wheelF0, Qp Component of forces acting on the driving wheel and passive

wheel by the driving axis and passive axisMPd , MPp Rolling resistance moments acting on the driving wheel and

passive wheel1di, 1pi Angular accelerations of the driving wheel and passive wheeladi, api Components of acceleration of the driving wheel center and

passive wheel centerJd, Jp Moments of inertia of the driving wheel and passive wheelti Driving torque of the motor

Table II Parameters for driving torque (SI units)

P (N) R (mm) r (mm) a (m/s)2 Jd (kg m2) MPp (N m) MPd (N m)

196 0.2 0.04 1 0.011 0.392 0.147




Volume 37 · Number 1 · 2010 · 23–35

27

this paper. But following this is the chattering phenomenon,which proves to be a considerable factor of interest forpractical applications. In order to solve the problem, fuzzycontrol is introduced into the sliding mode controller here.

4.1 Sliding mode controlThe robot is a typical non-linear system, and there aremany unpredictable external disturbances. Owing to thecharacteristics of fast response and insensitive to parametricvibrations and disturbances, the SMC method is mainlyapplied to robot control in the recent years (Yu and Xu,2000).

As in the commonly used SMC, the control law consists ofequivalent control and switch control as:

t ¼ teq þ tsw ð17Þ

The former is to force the system state to slide on the slidingsurface, and the latter is to derive the states toward the slidingsurface.

We define the sliding surface is s ¼ s1 s2 s3h i

¼ lTE,where l is the sliding surface coefficient vector. E ¼ ½e_e �T isthe error state vector, e is the tracking error. According to theeffect of equivalent control, in sliding mode the equivalentcontrol is described when the state trajectory is near s ¼ 0,and to yield an equivalent control law to keep state remainingon the sliding surface, it can be obtained by letting _s equalzero.

For an OMR system, _s ¼ lT _Eþ €Xd 2 €X, where Xd is thetarget trajectory. Substituting the state equation (10) into ityields the equivalent control:

teq ¼ ðMN21ðlT _Eþ €XdÞ þDuN21 _XÞR

ð18Þ

The purpose of switch control tsw is to guarantee the statesmove toward the sliding surface. Here, we define the switchcontrol as:

tsw ¼ MN21j sgnðsÞR

ð19Þ

where j ¼ ½ j1 j2 j3 �T; ji . 0, sgn( · ) is the Sign function.Substituting the above equations into _s ¼ lT _Eþ €Xd 2 €X,

we can obtain:

si_si ¼ si · ð2ji · sgnðsiÞÞ2 si · hi ¼ 2ji jsi j2 sihi ð20Þ

To guarantee the control system is stabilized, a Lyapunovfunction candidate is given as:

V ¼ 12s2i ð21Þ

Then differentiate V with respect to time. Based on thestability of SMC condition, we have:

_V ¼ si_si , 0 ð22Þ

According to the above analysis, we can achieve a stable SMCsystem when ji $ RH.

4.2 Fuzzy sliding mode controllerTo reduce susceptibility to parametric vibrations and externaldisturbances, the control output is operating or chattering at ahigh frequency. The chattering may result in a bad influence

on motors and other equipments of the servo system. And thetracking error may be chattering near zero point, accordinglyto increase the steady-state error. As a result of chattering,parasitic oscillation will occur, finally the control systemperformance will be influenced (Ha et al., 2001). To reducethe chattering phenomenon in the sliding mode, fuzzy tuningschemes are employed in this paper. A coordinatingparameter ks is introduced into the traditional SMC toreduce the chattering phenomenon. We define the control lawas:

t ¼ teq þ ks · tsw ð23Þ

For the stability of control system, as analyzed in Section 4.1,ksji $ RH must be guaranteed.

When the rotational speed of any wheel is close to the targetspeed, it is better to reduce the value ks to reduce the speedwhile crossing the sliding mode surface. And it is better tokeep a big value ks to ensure high velocity, when the rotationalspeed has not reached the target speed. Therefore, taking thetracking error of speed for each wheel ewi as input, and takingthe coordinating parameter ks as output, we design the fuzzysliding mode controller. The robot system consists of threewheels, so there are three independent fuzzy controllers. Inthe following section, we will describe the design of the fuzzycontroller.

Defining the universe of discourse for input variable as ewifor each wheel fuzzy controller in [2a1,a1], and the universeof discourse for output variable ks in [0,b4], where ai and bi arenoted in Figures 6 and 7, i ¼ 1, 2, 3, 4. The fuzzy subset oflinguistic value for input and output variables is set as {PB,PM, PS, ZO, NB, NM, NS}.

The design principle for the fuzzy controller rule isdescribed as follows. As a fundamental requirement, thetarget reach and rapidity must be guaranteed. Accordingly,when the rotational speed is far away from target speed, theamplitude of switch control term should be big, whereas itshould be reduced when the rotate speed is close to targetspeed. According to the conventional fuzzy condition andfuzzy relation, we consider the following fuzzy control rules:

Figure 6 The membership function of input

NB NM NS ZO PS PM PB1

0.8

0.6

0.4

0.2

0

–a1 a1–a2 a2–a3 a3–a4 a40ewi

Deg

ree

of m

embe

rshi

p




Volume 37 · Number 1 · 2010 · 23–35

28

Ri : If ewi isAi ; then ks isBi

where:Ri denotes the ith rule;Ai the ith fuzzy set; andBi the ith output.The fuzzy control rules are shown in Table III.

Without loss of generality, the membership functions offuzzy sets are all considered as triangle typed and shown inFigures 6 and 7 for input and output, respectively. Obviously,the definition of adjoining membership functions are overlapand symmetrical. And we employ the centroid method as ade-fuzzification method to transform the fuzzy output intoexact output in this paper.

4.3 Simulation resultsIn order to validate the feasibility of application to OMR systemfor FSMC, we conducted simulation comparisons for theeffectiveness between fuzzy SMC and traditional SMC. Thesimulation is realized by Matlab/Simulink. The simulation objectis the three-wheeled OMR shown in Figure 2. To be as close tothe practical effect as possible, the parameters used in thesimulation are determined according to the real robot. The mainpart of the initial setup for the simulation is shown in Table IV,other parameters are shown in Table II.

The control block diagram is shown in Figure 8. To emulatethe real case, we introduce the Gaussian Function disturbanceinto the system state equation, which is defined as:

hiðtÞ ¼ A · exp 2ðt 2 cÞ2

2b2

!ð24Þ

where A ¼ 8, b ¼ 0.5, c ¼ 3. And the upper bound ofdisturbance is defined as H ¼ maxðjhðtÞjÞ þ h, where h ¼ 1.

In the simulation, we compare the chattering phenomenaresulting from the fuzzy SMC and traditional SMC, bytracking a circular trajectory, also the anti-disturbance abilityand the trajectory tracking effect is presented. The circulartrajectory is defined as:

xd ¼ cos tyd ¼ sin tfd ¼ t

8>><>>: ð25Þ

As shown in Figure 9, the traditional SMC also can eliminatethe disturbance influence, and it can achieve good trackingresults. But the control output appears with significantchattering phenomenon, which will result in bad influence forthe real robot system, while there is almost no chatteringphenomenon for the SMC combined with fuzzy control.According to the comparison, it has proven the advantage forthe FSMC weakening the chattering phenomenon.

5. FSMC based on anisotropy5.1 Control methodThe interaction force between wheel and ground is not takeninto account for the dynamics model used in the controlsystem above. It leads to difficulty of exhibiting the realresponse of each wheel in the real robot motion. Therefore,there are some limits in the practical robot control with thecontrol method above.

Owing to the special mechanism of omni-directional wheelsand the different wheel arrangement, the dynamic anisotropy isdistinct for OMRs. The force acting on each wheel is differentwhile the robot moving in different direction, i.e. the drivingtorque acting on each wheel is different. Influenced by thefrictional force, which results from the contact between the wheeland the ground, it also leads to the dynamic anisotropy. With theguarantee that the fuzzy SMC is stable, by introducing theinfluence factor of dynamic anisotropy into the control system,the driving torque acting on each wheel is coordinated while therobot moves in different direction. As a result, it can achieve a

Figure 7 The membership function of output

ZO PS PM PB1

0.8

0.6

0.4

0.2

0

b4b3b2b10

ks

Deg

ree

of m

embe

rshi

p

Table III Rule for FSMC

ewi PB PM PS ZO NB NM NS

k PB PM PS ZO PB PM PS

Table IV Setup for simulation (SI units)

Jr (kg m2) Jp (kg m2) jdij (m) Du a1 a2 a3 a4 b1 b2 b3 b4 ji

0.3562 0.5482 £ 1024 0.22 0.3 10 4 1 0.5 0.2 0.6 1 1.5 13




Volume 37 · Number 1 · 2010 · 23–35

29

good speed and trajectory tracking with rapidity and accuracy,also the slip phenomenon can be reduced to some extent.

In order to ensure that the robot moves along the targetdirection throughout the motion process, whether during theacceleration process or during the process of uniform motion,the rotational speed of wheels should be coordinated witheach other, or else the robot will not move in the specifiedtarget direction. While the robot moves only with thetranslation, there will be some form of rotational motionoccurring without coordinated motion control, especiallywhen it has not reached steady state motion. Therefore, thetarget rotational speed and the force condition of each wheelmust be taken into consideration during the motion control.

According to the analysis above about the driving torqueacting on different wheels while the robot moves in differentdirections, the following conclusion can be deduced. Whenthe driving torque needed for any wheel is big, thecorresponding kT should be big, vice versa. And for afaster target speed, the higher the acceleration should be,i.e. the corresponding driving torque should be higher orvice versa. Accordingly, the speed and driving torquecharacteristics of each wheel are processed as the influentialfactor together.

For the three-wheeled OMR discussed in this paper,according to the analysis on the kinematics in the abovesection, we can obtain the absolute speed value of each wheel,while it moves in the speed of 1 m/s in some arbitrarydirection. Owing to the symmetry, only the case of 0-1808 ispresented here.

First of all, we obtain the absolute value of speed (Figure 3),and convert both the value of speed and driving torque(Figure 5) into [0, 1], which are denoted by V(b) and T(b),respectively. And then, the influential factor kT, which is alsoknown as the anisotropy factor, is defined as:

kT ¼ nvV ðbÞ þ nTT ðbÞ ð26Þ

where:nv and nT the influential weights of speed and driving

torque, respectively.

For example, when both of them are defined as 50 percent,then the influential factor curve is shown in Figure 10.

Taking into consideration, the anisotropy characteristicsand the contact interaction force, we introduce the influentialfactor kT into the control system. The control law is defined as:

t ¼ teq þ ðks þ kT Þtsw ð27Þ

To ensure that the system introduced by anisotropy factor isstable, according to the analysis in Section 4.1, we should alsoensure that ðks þ kT Þji $ RH.

5.2 Experiment resultsIn order to verify the effectiveness of the FSMC based onanisotropy, we carried out some experiments on the realrobot. The experiment results using improved FSMC methodare compared with the results using traditional FSMCmethod. Line trajectory and circular trajectory trackingmotions are discussed in the experiments.

In the experiments, the omni-directional middle-size soccerrobot (Figure 2) is used, and the omni-directional wheels areall side rollers (diameter 80 mm). The weight of the robot isabout 20 kg, and the dimensions are 45 £ 45 £ 80 mm.

The experiment is carried out on the green-carpeted fieldfor RoboCup middle-sized robots (12 £ 8 m, with reference tothe RoboCup rules before 2007).

We carried out some experiments with different influentialweights, then chose the experimental result with besteffectiveness of control and determined the correspondinginfluential weights as the final values. Here, the values arenv ¼ 0.4 and nT ¼ 0.6.

When the robot is moving during the acceleration process, dueto the existing anisotropy of each wheel, it is difficult to achievecoordinated motion. To exhibit the motion error resulting fromthe non-synchronous motion during the acceleration process, weset the rotational speed at 0, i.e. there is only translation.

For the line trajectory tracking motion, we let the robotmove at 1 m/s along a 458 direction, and speed of each wheelis shown in Figure 11(a) and (b). Though the rotational speedis set at 0, according to Figure 11(c) and (d), we can see thathigh angular speed occurs during the acceleration process.It results in the motion direction deviating from the target

Figure 8 Control system block diagram

Trajectorygenerator

du/dt+–

SMC

Fuzzy control

FSMC

Anisotropy

Improved FSMC

Omni-Robotplant

ks, kT

(xr, yr, fr)

(xd, yd, fd)

ewi

e

e·

S

tsw

teq




Volume 37 · Number 1 · 2010 · 23–35

30

Figure 9 Simulation result comparisons

1

0.8

0.6

0.4

0.2

–0.2

Y (

m)

–0.4

–0.6

–0.8

–1–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 10

X (m)

–0.5

0.5

1

Err

or (

m)

0

0 0.5 1 1.5 2 2.5 3 3.5T (s)

(a) Trajectory result using SMC

Trajectory error along X axis

–0.5

0.5

1

Err

or (

m)

0

0 0.5 1 1.5 2 2.5 3 3.5T (s)

Trajectory error along X axis

–0.01

0.020.03

Err

or (

m)

0.010

0 0.5 1 1.5 2 2.5 3 3.5T (s)

Trajectory error along Y axis

–0.05

–0.02

–0.04

0

0.02

Err

or (

m)

0 0.5 1 1.5 2 2.5 3 3.5T (s)

Trajectory error along Y axis

–0.01

0.020.03

Err

or (

m)

0.010

0 0.5 1 1.5 2 2.5 3 3.5T (s)

Orientation error

–0.02

0.02

0

0.04

0.06

Err

or (

m)

0 0.5 1 1.5 2 2.5 3 3.5T (s)

Orientation error

0

1

0.8

0.6

0.4

0.2

–0.2

Y (

m)

–0.4

–0.6

–0.8

–1–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 10

X (m)

Theoretical trajectory

Simulation trajectory

(b) Trajectory result using FSMC

(c) Trajectory error using SMC (d) Trajectory error using FSMC

0

0.5

1.5

1

2

0 1 2 3 4 5 6T (s)

–1.8

–0.9

0.9

0

1.8

0 1 2 3 4 5 6T (s)

(e) Driving torque curve using SMC (f) Driving torque curve using FSMC

Driving torque acting on wheel 1Driving torque acting on wheel 2

Driving torque acting on wheel 3

0

Torq

ue (

Nm

)

Torq

ue (

m)




Volume 37 · Number 1 · 2010 · 23–35

31

direction when the robot speeds up to the target speed asshown in Figure 11(e). In the case of no sensors installed ontothe robot such as a vision sensor, the robot will move in thewrong direction. In the experiments, we only use encoder datafor control feedback. The trajectories of the experimentresults are calculated from the encoder data of each wheel.From Figure 11(g) to (h), we can find out that after speedingup to the target speed, the robot can move along a line, andthe rotational speed error is constrained to a limited range.

For the circular trajectory tracking motion, we let the robotmove at 1.1 m/s, and the radius of the circular trajectory is2 m. During the motion the rotational speed of each wheel(Figure 12(a) and (b)) is changing with respect to time. FromFigure 12(c) to (d), it is clear that angular speed errors occur,especially during the acceleration process. According totracking results in Figure 12(e)-(h), the trajectory error isdistinct especially with the FSMC method not based on theanisotropy.

Comparing the experimental results using traditionalFSMC and FSMC based on anisotropy, we can concludethat both the line trajectory tracking and circular trajectorytracking, the tracking error of the latter method is smallerthan the former one. For example, in line trajectory trackingmotion with traditional FSMC method, the rotational speederror is significant during the acceleration process, while it issmaller with the improved FSMC, comparing with (c) and (d)of Figure 11. Owing to the speed variation in all motionprocesses for circular trajectory tracking motion, the largerotational speed error exists throughout for traditionalFSMC, while it is not the case for the improved FSMC,comparing with (c) and (d) of Figure 12.

From the line trajectory and circular trajectory trackingmotion experiment results, when the rotational speed of thewheel is changing or accelerating, the wheels cannot achievecoordinated motion, and a large angular speed error occurs.For example, in the line trajectory tracking motion, the targetspeed of wheel 2 is higher than the other two, and the drivingtorque required is the largest. Accordingly, wheel 2accelerates slowly with traditional motion control, and itcannot achieve synchronous motion with the other twowheels. After introducing the anisotropy factor, the angular

speed error resulted from the non-synchronous motion isdistinctly reduced.

In the line trajectory tracking motion experiments, althoughthe ground is uneven, the robot control system is insensitiveto parametric vibrations and external disturbances. When therobot sped up to the target speed and moves at a constantspeed, the robot can overcome the influence of non-linearcharacteristics, and the angular speed error is constrained to alimited range. According to the experiment results above, wecan conclude that by introducing the anisotropy factor intothe FSMC method, it can achieve invariant control and acoordinated motion quickly.

6. ConclusionDifferent from traditional mobile robot, it is difficult to achievea coordinated motion for the multi-axis system of OMRs duringthe acceleration process with the existence of anisotropy, so asnot to be able to achieve exact translational motion along somedirections. Taking into consideration, the friction and otherinteractive forces between the wheel and ground, we analyzedthe relation between driving torques acting on each wheel, andproposed a novel concept of dynamic anisotropy for OMRs inthis paper. We introduced an anisotropy factor into FSMCmethod based on the anisotropy. Then a new FSMC method isproposed in this paper. By coordinating the driving torque withthe anisotropy factor, each wheel can speed up to the targetspeed, and the resulting motion trajectory is greatly improved.With the proposed control method, the robot control system isinsensitive to parametric vibrations and external disturbances,and the chattering phenomenon is reduced. Simulations andexperiments have proven that each wheel can achieve acoordinated motion, and an effective motion tracking can beachieved by using the proposed motion control method.

In order to obtain a clearer analysis of the anisotropyinfluence during the acceleration process, only the translationmotion case is discussed in this paper. The case with bothtranslation and rotation motion will be dealt with in futureresearch.

Figure 10 Anisotropy factor

1

0.8

0.9

0.7

0.5

0.3

0.1

0.6

0.4

0.2

0 20 40 60 80 100

Wheel 1

Wheel 2

Wheel 3

120 140 160 180

b (°)

kT




Volume 37 · Number 1 · 2010 · 23–35

32

Figure 11 Experimental data of line tracking motion

1

0.5

–0.5

–1

0 1 2 3T (s)

Experimental speed of wheel 1Theoretical speed of wheel 1

(a) Speed curves of wheels using FSMC (b) Speed curves of wheels using FSMCbased on Anisotropy



4 5

0

1

0.5

–0.5

–1

0 1 2 3T (s)

4 5

0

1

0.8

0.6

0.4

0.2

0

–0.2

1

0.8

0.6

0.4

0.2

0

–0.2

0 1 2 3T (s)

Experimental speed along X axisTheoretical speed along X axis

(c) Speed curves of robot using FSMC (d) Speed curves of robot using FSMCbased on Anisotropy

Experimental speed along Y axisTheoretical speed along Y axis

Experimental translation speedTheoretical translation speed

4 5 0 1 2 3T (s)

4 5

4.5

4

3.5

3

2.5

2

0.5

0

1.5

1

4

3.5

3

2.5

2

0.5

0

1.5

1

0 10.5 1.5 2.5 3.52 3X (m)

Theoretical trajectory

(e) Trajectory curve using FSMC (f) Trajectory curve using FSMCbased on Anisotropy

Experimental trajectory

4 0 10.5 1.5 2.5 3.52 3X (m)

4

0.2

0.1

0.15

Ang

le (

rad)

0

–0.05

0.05

0.015

0.01

–0.015

–0.01

–0.005

0.005

0

0 1 2 3T (s)

(g) Orientation error using FSMC (h) Orientation error using FSMCbased on Anisotropy

54 0 1 2 3 54T (s)

Ang

le (

rad)

V (

m/s

)

V (

m/s

)

V (

m/s

)/A

ngel

(ra

d)

V (

m/s

)/A

ngel

(ra

d)

Y (

m)

Y (

m)




Volume 37 · Number 1 · 2010 · 23–35

33

Figure 12 Experimental data of circular trajectory tracking motion

1.5

0.5V

(m

/s)

–0.5

–1.50 3 6 9

T (s)

(b) Speed curves of wheels using FSMCbased on Anisotropy

Experimental speed of wheel 1

Theoretical speed of wheel 1





12 15

–1

1

0

1.5

0.5

V (

m/s

)

–0.5

–1.50 2 4 6 8

T (s)10 12

–1

1

0

1.5

0.5

V (

m/s

)/A

ngle

(ra

d)

–0.5

–1.50 3 6 9

T (s)

(c) Speed curves of Robot using FSMC (d) Speed curves of Robot using FSMCbased on Anisotropy

Experimental speed along X axis

Theoretical speed along X axis

Experimental speed along Y axis

Theoretical speed along Y axis

Experimental rotation speed

Theoretical rotation speed

12 15

–1

1

0

1.5

0.5V

(m

/s)/

Ang

le (

rad)

–0.5

–1.50 2 4 6 8

T (s)10 12

–1

1

0

0.1

0

Ang

le (

rad)

–0.6

–0.80 3 6 9

T (s)

(g) Orientation error using FSMC (h) Orientation error using FSMCbased on Anisotropy

12 15

–0.7

–0.1

–0.2

–0.3

–0.4

–0.5

0.020

Ang

le (

rad)

–0.160 2 4 6 8

T (s)10 12

–0.02–0.03–0.04–0.06–0.08–0.1

–0.12–0.14

4

–2

–2 –1.5 –1 –0.5 0.5 1.510X (s)

(e) Trajectory curve using FSMC (f) Trajectory curve using FSMCbased on Anisotropy

Theoretical trajectory Experimental trajectory

2 –2 –1.5 –1 –0.5 0.5 1.510 2

–4

2

0

4

–2

–4

2

0

X (s)

(a) Speed curves of wheels using FSMC

Y (

m)

Y (

m)




Volume 37 · Number 1 · 2010 · 23–35

34

ReferencesAngeles, J. (2003), Fundamentals of Robotic MechanicalSystems: Theory, Methods, and Algorithms, 2nd ed.,Springer, New York, NY.

Betourne, A. and Campion, G. (1996), “Dynamic modelingand control design of a class of omnidirectional mobilerobots”, IEEE International Conference on Robotics andAutomation, Minneapolis, MN, pp. 2810-15.

Campion, G., Bastin, G. and D’Andrea Novel, B. (1996),“Structural properties and classification of kinematic anddynamic models of wheeled mobile robots”, IEEETransactionson Robotics and Automation, Vol. 12 No. 1, pp. 47-62.

Chatzakos, P., Markopoulos, Y.P., Hrissagis, K. and Khalid, A.(2006), “On the development of a modular external-pipecrawling omni-directional mobile robot”, Industrial Robot:An International Journal, Vol. 33 No. 4, pp. 291-7.

Chen, J.Y. (1999), “Expert SMC-based fuzzy control withgenetic algorithms”, Journal of The Franklin Institute,Vol. 336 No. 4, pp. 589-610.

Chen, P., Mitsutake, S., Isoda, T. and Shi, T. (2002), “Omni-directional robot and adaptive control method for off-roadrunning”, IEEE Transactions on Robotics and Automation,Vol. 18 No. 2, pp. 251-6.

Daniel, D., Krogh, B. and Friedman, M. (1985), “Kinematicsand open-loop control of an ilonator-based mobile platform”,Proceedings of the 1985 IEEE International Conference onRoboticsand Automation (ICRA’85), pp. 346-51.

Feng, D., Friedman, M.B. and Krogh, B.H. (1989), “Theservo-control system for an omnidirectional mobile robot”,IEEE International Conference on Robotics and Automation,Vol. 3, pp. 1566-71.

Ferriere, L., Raucent, B. and Campion, G. (1996), “Designof omnimobile robot wheels”, Proceedings of the IEEEInternational Conference on Robotics and Automation,Minneapolis, MN, USA, pp. 3664-70.

Ha, Q.P., Nguyen, Q.H., Rye, D.C. and Durrant-Whyte, H.F.(2001), “Fuzzy sliding-mode controllers with applications”,IEEE Transactions on Industrial Electronics, Vol. 48 No. 1,pp. 38-46.

Hung, Y., Gao, W. and Hung, J.C. (1993), “Variablestructure control: a survey”, IEEE Transactions onIndustrial Electronics, Vol. 40 No. 1, pp. 2-22.

Hwang, C.L. (2002), “Robust discrete variable structurecontrol with finitetime approach to switching surface”,Automatica, Vol. 38 No. 1, pp. 167-75.

Hwang, C.L. and Chang, N.W. (2008), “Fuzzy decentralizedsliding-mode control of a car-like mobile robot indistributed sensor-network spaces”, IEEE Transactions onFuzzy Systems, Vol. 16 No. 1, pp. 97-109.

Hwang, G.C. and Lin, S.C. (1992), “A stability approach tofuzzy control design for nonlinear systems”, Fuzzy Sets andSystems, Vol. 48 No. 3, pp. 279-87.

Kalmar-Nagy, T., D’Andrea, R. and Ganguly, P. (2004),“Near-optimal dynamic trajectory generation and control ofan omnidirectional vehicle”, Robotics and AutonomousSystems, Vol. 46 No. 1, pp. 47-64.

Leng, C.T., Cao, Q.X. and Huang, Y.W. (2008), “A motionplanning method for omni-directional mobile robot based

on the anisotropic characteristics”, International Journal ofAdvanced Robotic Systems, Vol. 5 No. 4, pp. 327-40.

Pin, F.G. and Killough, S.M. (1994), “A new family of omni-directional and holonomic wheeled platforms for mobilerobots”, IEEE Transactions on Robotics and Automation,Vol. 10 No. 4, pp. 480-9.

Purwin, O. and D’Andrea, R. (2006), “Trajectory generationand control for four wheeled omnidirectional vehicles”,Robotics and Autonomous Systems, Vol. 54 No. 1, pp. 13-22.

Qiu, C.W. and Cao, Q.X. (2008), “Modeling and analysis ofthe dynamics of an omni-directional mobile manipulatorssystem”, Journal of Intelligent and Robotic Systems: Theoryand Applications, Vol. 52 No. 1, pp. 101-20.

Samani, A.H., Abdollahi, A., Ostadi, H. and Rad, S.Z.(2004), “Design and development of a comprehensiveomni-directional soccer player robot”, International Journalof Advanced Robotic Systems, Vol. 1 No. 3, pp. 191-200.

Tlale, N.S. (2006), “On distributed mechatronics controllerfor omni-directional autonomous guided vehicles”,Industrial Robot: An International Journal, Vol. 33 No. 4,pp. 278-84.

Viboonchaicheep, P., Shimada, A. and Kosaka, Y. (2003),“Position rectification control for Mecanum wheeled omni-directional vehicles”, The 29th Annual Conference of theIEEE Industrial Electronics Society, Vol. 1, pp. 854-9.

Wada, M. and Asada, H.H. (1999), “Design and control of avariable footprint mechanism for holonomic omnidirectionalvehicles and its application to wheelchairs”, IEEETransactions on Robotics and Automation, Vol. 15 No. 6,pp. 978-89.

Wang, S.M., Lai, L.C., Wu, C.J. and Shiue, Y.L. (2007),“Kinematic control of omni-directional robots for time-optimal movement between two configurations”, Journal ofIntelligent and Robotic Systems, Vol. 49 No. 4, pp. 397-410.

Watanabe, K. (1998), “Control of an omnidirectional mobilerobot”, Proceedings of the 2nd International Conference onKnowledge-based Intelligent Electronic Systems, pp. 51-60.

Watanabe, K., Shiraishi, Y. and Tzaffestas, S.G. (1998),“Feedback control of an omnidirectional autonomousplatform for mobile serve robots”, Journal of Intelligentand Robotic Systems, Vol. 22 No. 3, pp. 315-30.

Williams, R.L., Carer, B.E., Gallina, P. and Rosati, G.(2002), “Dynamic model with slip for wheeledomnidirectional robots”, IEEE Transactions on Robotics andAutomation, Vol. 18 No. 3, pp. 285-93.

Yang, J.M. and Kim, J.H. (1999), “Sliding mode control fortrajectory tracking of nonholonomic wheeled mobilerobots”, IEEE Transactions on Robotics and Automation,Vol. 15 No. 3, pp. 578-87.

Yu, X.H. and Xu, J.X. (2000), Advances in Variable StructureSystems, World Scientific, Singapore.

Yu, Z.S. (1990), Automoblie Theory, China Machine, Beijing.

Corresponding authorChuntao Leng can be contacted at: [email protected]




Volume 37 · Number 1 · 2010 · 23–35

35

To purchase reprints of this article please e-mail: [email protected] visit our web site for further details: www.emeraldinsight.com/reprints

a motion control method for omni-directional mobile robots

Documents