Dynamic Model and Control for a Holonomic Omnidirectional Mobile Robot

Autonomous Robots 11, 173–189, 2001c© 2001 Kluwer Academic Publishers. Manufactured in The Netherlands.

Dynamic Model and Control for a Holonomic Omnidirectional Mobile Robot

TAKAAKI YAMADA, KEIGO WATANABE AND KAZUO KIGUCHIDepartment of Advanced Systems Control Engineering, Graduate School of Science and Engineering,

Saga University, 1-Honjomachi, Saga 840-8502, [email protected]

[email protected]

[email protected]

KIYOTAKA IZUMIDepartment of Mechanical Engineering, Faculty of Science and Engineering, Saga University,

1-Honjomachi, Saga 840-8502, [email protected]

Abstract. In the recent past, mobile robots with high mobility have been developed actively. We have alreadyproposed a holonomic and omnidirectional mobile robot using two active dual-wheel caster assemblies and alsoderived the kinematic models for the assembly and the mobile robot. This paper presents dynamic analysis andcontrol for the mobile robot. The dynamic model has been derived based on the forces acting on the steering axis.Then a model-based resolved acceleration controller is constructed. The validity of the model and the effectivenessof the control system are confirmed by experiments using a prototype robot as well as simulations.

Keywords: omnidirectional mobile robot, active dual-wheel caster assembly, holonomic property, dynamic model,resolved acceleration control

1. Introduction

Mobile robots called automated guided vehicles areused to transport tools, parts and products between vari-ous processes in automated factories. In order to realizecooperative tasks with other mobile robots or manipu-lators in narrow and complicated environments, thereneeds an increasing demand for mobile robots withhigh mobility. Mobile robots have been also introducedinto the service industry as guard robots and sweepingrobots. It seems that such useful applications in variousfields will be increased with an improvement of mobil-ity. On the other hand, with the increase of the averagelife span in advanced nations, the welfare is now fo-cused on the powered wheelchairs with high mobilitywhich has no restriction such as a minimum rotationalradius in conventional ones (Sasaki, 1998; Wada andAsada, 1999).

Note however that conventional mobile robots donot have high mobility and cannot move sideways be-cause of the nonholonomic constraint. Mobile robotswith a two independent driving wheels mechanism ora front-wheel steering and rear-wheel driving mech-anism are representative of them (Campion et al.,1996; Nelson, 1989). In contrast, holonomic and om-nidirectional mobile robots have attracted attention,because they can provide two-degrees of freedom trans-lational motion and one-degree of freedom rotationalmotion simultaneously and independently at an arbi-trary position and posture, without stopping. Most ofthese types of mobile robots already proposed up tonow realize the holonomic property by using special-ized wheel mechanisms such as the universal wheel–(Muir, and Neuman, 1987; Asama et al., 1995), the or-thogonal wheel–(Pin and Killough, 1994) and the ballwheel-mechanism (West and Asada, 1995). In these

174 Yamada et al.

mechanisms, the contact area of the wheel with thefloor is relatively small, so that the ability to bear a loadis low and it may scratch the floor due to the utilizationof a barreled roller or a ball wheel. Furthermore, suchmechanisms are complex in structure and the ability toclimb up a step is also low.

In order to solve these problems, Wada and Mori(1997) developed a holonomic and omnidirectionalmobile robot using two steered driving wheels, inwhich the wheel has the offset distance between theaxle and steering axis like a caster and can use a nor-mal tire such as a rubber tire and a pneumatic tire,instead of a specialized wheel mechanism. An ac-tive dual-wheel caster assembly was proposed by Hanet al. (2000), in which a wheeled mechanism has a pas-sive steering axis with offset arranged on the front ofa conventional two independent driving wheels mech-anism, and a holonomic and omnidirectional mobilerobot was realized by using two or more of these as-semblies. This assembly can generate two-degrees offreedom velocity on the passive steering axis, becauseof the difference between the angular velocities of theleft- and right-wheels. In addition, it can of course usea normal tire and is simple in structure.

As mentioned above, so far various holonomic andomnidirectional mobile robots have been developed.However, most of them were focused on developingthe omnidirectional mobile mechanism. There are fewstudies on the derivation of a dynamic model and dy-namic model based control method for each mecha-nism. Tang et al. (1996) constructed the dynamic modelfor the holonomic and omnidirectional mobile robot us-ing three orthogonal wheel-mechanisms, which are oneof the specialized wheel mechanisms, and realized a dy-namic control for the robot. It should be noted that themechanism has the problems described above, need-less to say. Lee et al. (1997) proposed a simple dynamicmodel for the holonomic and omnidirectional mobilerobot which has two offset steered driving wheels pre-sented by Wada and Mori (1997) and discussed witha motion tracking for the robot. Note, however, that inthis case it is difficult to regard it as a precise dynamicmodel because the dynamic properties for the wheelsare not considered.

In this paper, we derive a dynamic model for theholonomic and omnidirectional mobile robot whichconsists of two active dual-wheel caster assembliesand deal with dynamic analysis and control basedon the model. This paper is organized as follows:In Section 2, the kinematic models for the active

dual-wheel caster assembly and the omnidirectionalmobile robot are presented. In Section 3, after deter-mining forces acting on the steering axis, the dynamicmodel for the mobile robot is derived by using theirforces. Section 4 presents a resolved acceleration con-trol system based on the dynamic model and Section 5describes a construction and control system of a pro-totype robot. Simulations and experiments using theprototype robot are performed to demonstrate the va-lidity of the dynamic model and the effectiveness ofthe control method in Section 6.

2. Kinematic Model

We present the kinematic models for the active dual-wheel caster assembly and omnidirectional mobilerobot in this section. Note that in order to apply the re-sults in this section to the dynamic analysis in Section 3,the models are derived with respect to the centers ofgravity (c.g.) of the dual-wheel caster assembly andmobile robot.

2.1. Kinematic Model for the Active Dual-WheelCaster Assembly

This section describes the kinematic model for the ac-tive dual-wheel caster assembly as shown in Fig. 1. Thenomenclatures are as follows:

Figure 1. Model of active dual-wheel caster assembly.

Holonomic Omnidirectional Mobile Robot 175

�w(Ow − XwYw) absolute coordinate system�i (Oi − Xi Yi ) dual-wheel caster coordinate

system with the steering axis atthe origin

xi , yi position of the dual-wheelcaster in �w (coordinates of Oi

in �w)φi posture of the dual-wheel caster

(the angle between Xw- andXi -axes)

ωri , ωli angular velocity of the wheel

di distance between the wheel andthe c.g. of the dual-wheel caster

ri radius of the wheelsi offset distance between the axle

and the steering axis

Note here that the subscripts r and l denote the rightand left wheels respectively, and the subscript i is forthe assembly number.

Letting xi = [xi yi ]T and ωi = [ωri ωli ]T , thedirect kinematic model of the active dual-wheel casterassembly is given by

xi = Aiωi (1)

φi = ri

2di(ωri − ωli ) (2)

where

Ai = ri

2

[cos φi − si

disin φi cos φi + si

disin φi

sin φi + sidi

cos φi sin φi − sidi

cos φi

].

From Eq. (1), the inverse kinematic model can bedescribed as

ωi = A−1i xi (3)

where

A−1i = 1

ri

[cos φi − di

sisin φi sin φi + di

sicos φi

cos φi + disi

sin φi sin φi − disi

cos φi

].

Here, φi can be calculated by

φi = 1

siyi cos φi − 1

sixi sin φi (4)

which represents a nonholonomic constraint. It is foundthat there is no singularity, because A−1

i can be deter-mined in any φi according to Eq. (3).

2.2. Kinematic Model for the OmnidirectionalMobile Robot

A holonomic and omnidirectional mobile robot canbe realized by using two or more active dual-wheelcaster assemblies. In this section, we show the kine-matic model of the omnidirectional mobile robot usingtwo active dual-wheel caster assemblies as presentedby Fig. 2. The nomenclatures are as follows:

�w (Ow − XwYw) absolute coordinate system�o (O − XY ) robot coordinate system with

the c.g. of the platform at theorigin

�i (Oi − Xi Yi ) dual-wheel caster coordinatesystem (i = 1, 2)

x, y position of the robot in �w (co-ordinates of O in �w)

φ posture of the robot (the anglebetween Xw- and X -axes)

Li distance between the steer-ing axis and the c.g. of theplatform

Letting x = [x y φ]T and xa = [xT1 xT

2 ]T , the directkinematic model for the mobile robot is obtained by

x = Bxa (5)

Figure 2. Model of omnidirectional mobile robot.

176 Yamada et al.

where

B = 1

L

L2 0 L1 0

0 L2 0 L1

−sin φ cos φ sin φ −cos φ

L = L1 + L2.

In addition, setting ω = [ωT1 ωT

2 ]T and combiningEqs. (1) and (5), we have

x = BAω (6)

where

A =[

A1 02×2

02×2 A2

].

On the other hand, the condition that X -directionalvelocity components are the same for each dual-wheelcaster assembly can be written in the form

x1 cos φ + y1 sin φ = x2 cos φ + y2 sin φ. (7)

Using Eqs. (5) and (7), the inverse kinematic model canbe written as

xa = B∗x (8)

where

B∗ =

1 0 −L1 sin φ

0 1 L1 cos φ

1 0 L2 sin φ

0 1 −L2 cos φ

.

Furthermore, Eqs. (3) and (8) give

ω = A−1B∗x (9)

where

A−1 =[

A−11 02×2

02×2 A−12

].

3. Derivation of Dynamic Model

3.1. Forces Acting on the Steering Axis

This section presents forces acting on the steering axisto derive the dynamic model for the omnidirectionalmobile robot (see Fig. 3). The nomenclatures are asfollows:

ixgi ,iygi velocities of the c.g. of the dual-wheel

caster in �i

fxi , fyi forces acting on the steering axis in �wi fxi ,

i f yi forces acting on the steering axis in �i

fri , fli driving forces of the wheelf ′ri , f ′

li sideway forces acting on the wheelsgi distance between the axle and the c.g. of

the dual-wheel castermi mass of the dual-wheel casterIi moment of inertia of the dual-wheel

caster about the gravity axis

By summing Xi -directional forces we arrive at theforce balance equation:

miixgi + i fxi = fri + fli . (10)

Similarly we can write the following equation by equi-librium of Yi -directional forces:

miiygi + i f yi = f ′

ri + f ′li . (11)

Figure 3. Forces on active dual-wheel caster assembly.


By summing moments about the c.g. of the dual-wheelcaster assembly, the moment balance relationship isgiven by

Ii φi + ( f ′ri + f ′

li )sgi + i f yi (si − sgi ) = ( fri − fli ) di .

(12)

Also, the nonholonomic constraint gives

iygi = sgi φi (13)

and the time derivative of this equation is expressed as

iygi = sgi φi . (14)

By the use of Eqs. (11), (12) and (14), we have

(Ii + mi s

2gi

)φi + i f yi si = ( fri − fli ) di . (15)

Therefore, defining fwi = [ fri fli ]T , ixgi = [ixgi φi ]T

and ifi = [i fxii f yi ]T , the following equation is obtained

by combining Eqs. (10) and (15):

fwi = Ciixgi + Di

ifi (16)

where

Ci = 1

2

mi

Ii + mi s2gi

di

mi − Ii + mi s2gi

di

Di = 1

2

[1 si

di

1 − sidi

].

Then, introducing the coordinate transformation ma-trix from �i to �w such as

wTi =[

cos φi − sin φi

sin φi cos φi

]

it follows that

fi = wTiifi (17)

where fi = [ fxi fyi ]T . Taking into account that wTi isan orthogonal matrix, Eqs. (16) and (17) give

fwi = Ciixgi + Di

wTTi fi . (18)

The inverse of Di in Eq. (18) can be determined easily,so that we obtain

fi = wTi D−1i

(fwi − Ci

ixgi)

(19)

where

D−1i =

[1 1disi

− disi

].

3.2. Dynamic Model for the OmnidirectionalMobile Robot

In this section, we derive the dynamic model of the om-nidirectional mobile robot using two active dual-wheelcaster assemblies (see Fig. 4). The nomenclatures areas follows:

uri , uli driving torque of the wheelIwi moment of inertia of the wheelki driving gain factorm mass of the platformI moment of inertia of the platform about the

gravity axis

Letting f = [ f T1 f T

2 ]T be the force vector actingon the steering axis in �w, the dynamic property forthe omnidirectional mobile robot can be described as

Mx = Ef (20)

Figure 4. Forces on omnidirectional mobile robot.

178 Yamada et al.

where

M = diag(m, m, I )

E =

1 0 1 0

0 1 0 1

−L1 sin φ L1 cos φ L2 sin φ −L2 cos φ

.

Here, defining the driving force vector fw =[ f T

w1 f Tw2]T for the wheel and setting xg =

[1xTg1

2xTg2]T , Eq. (19) gives

f = D−1( fw − Cxg) (21)

where

D−1 =[

wT1D−11 02×2

02×2wT2D−1

2

]

C =[

C1 02×2

02×2 C2

].

Also, the relationship between ixgi and xi can bewritten as

ixgi = Fi xi (22)

where

Fi =[

cos φi sin φi

− 1si

sin φi1si

cos φi

].

From Eq. (22), the following equation is satisfied:

xg = Fxa (23)

where

F =[

F1 02×2

02×2 F2

].

Additionally, combining Eqs. (8) and (23) gives

xg = FB∗x. (24)

Hence, differentiating Eq. (24) with respect to time, weget

xg = (FB∗ + FB∗)x + FB∗x. (25)

Introducing the driving torque vector ui = [uri uli ]T

for each dual-wheel caster assembly, defining u =[uT

1 uT2 ]T , and assuming that the friction in the axle

can be ignored, the dynamic property for the drivingsystem (Saito and Tsumura, 1990) is expressed as

Iwω + Rfw = Ku (26)

where

Iw = diag(Iw1, Iw1, Iw2, Iw2)

K = diag(k1, k1, k2, k2)

R = diag(r1, r1, r2, r2).

Then, the time derivative of Eq. (9) can be written as

ω = (A−1B∗ + A−1B∗) x + A−1B∗x. (27)

Thus, from Eqs. (20), (21) and (25)–(27), the directdynamic model for the omnidirectional mobile robot isgiven by

x = G′(R−1Ku − G′′x) (28)

where

G′ = {M + ED−1(R−1IwA−1 + CF)B∗}−1ED−1

G′′ = (R−1Iw A−1 + CF)B∗ + (R−1IwA−1 + CF)B∗.

Now let q = [xT φ1 φ2]T be the state variable for themobile robot and u be the control input.

On the other hand, the condition that the X -directional forces acting on the steering axes are thesame for each dual-wheel caster assembly is repre-sented by

fx1 cos φ + fy1 sin φ = fx2 cos φ + fy2 sin φ. (29)

Using this condition and Eq. (20), we obtain

f = E∗M x (30)

where

E∗ = 1

L

L2 + αcc(φ) αsc(φ) − sin φ

αsc(φ) L2 + αss(φ) cos φ

L1 − αcc(φ) −αsc(φ) sin φ

−αsc(φ) L1 − αss(φ) − cos φ


αss(φ) = 1

2(L1 − L2) sin2 φ

αsc(φ) = 1

2(L1 − L2) sin φ cos φ

αcc(φ) = 1

2(L1 − L2) cos2 φ.

Here, from Eq. (18) we have

fw = Cxg + Df (31)

where

D =[

D1wTT

1 02×2

02×2 D2wTT

2

].

Hence, combining Eqs. (25)–(27), (30) and (31), theinverse dynamic model of the omnidirectional mobilerobot can be written as

u = K−1(H′x + H′′x) (32)

where

H′ = (Iw A−1 + RCF)B∗ + (Iw A−1 + RCF)B∗

H′′ = RDE∗M + (IwA−1 + RCF)B∗.

4. Resolved Acceleration Control

The resolved acceleration control system based on thedynamic model is constructed in this section. Whenintroducing the desired values xd = [xd yd φd ]T ofthe position and posture of the omnidirectional mobilerobot, the error vector e = [ex ey eφ]T is defined asfollows:

e = xd − x. (33)

The revised acceleration vector added a PD-servo tothe desired acceleration xd is expressed as

x∗ = xd + Kv e + Kpe (34)

where Kv > 0 is the derivative gain matrix and Kp > 0is the proportional gain matrix. Substituting x∗ into xin Eq. (32), the control input is obtained by

u = K−1(H′x + H′′x∗). (35)

Figure 5. Resolved acceleration control system.

Now, inserting this into Eq. (28), a closed-loop sys-tem is represented by a linear differential equation ofthe second-order of the error vector e:

e + Kv e + Kpe = 0. (36)

Therefore, when physical parameters for a practicalrobot are given, a desired response can be obtained bysetting the derivative gain matrix Kv and the propor-tional gain matrix Kp appropriately. Block diagram ofthe resolved acceleration control system is shown inFig. 5. Note here that R−1 in Fig. 5 denotes the esti-mating part of the inverse dynamic model for the robotand corresponds to Eq. (35).

5. Prototype Robot

5.1. Specifications

In this section, we describe the specifications for theprototype robot of the omnidirectional mobile robotusing two active dual-wheel caster assemblies. Eachwheel is driven by DC motor independently and a drivercircuit for the motor is mounted on the dual-wheelcaster assembly in the prototype robot. The positionand posture of the prototype robot can be estimated bydead reckoning using the equation (6) of the kinematicmodel and information from the encoders arranged onthe wheels and the steering axes. A passive caster isinstalled on the front of the steering axis for each as-sembly as an auxiliary wheel in order to keep the bal-ance of the active dual-wheel caster assembly and theprototype robot. The appearances of the active dual-wheel caster assembly and the omnidirectional mobilerobot are presented by Figs. 6 and 7 respectively. Mea-sured values of the physical parameters of the prototyperobot are shown in Table 1. The driving gain factors, k1

and k2, in Table 1 are used to correct the measurement

180 Yamada et al.

Figure 6. Appearance of active dual-wheel caster assembly.

Figure 7. Appearance of omnidirectional mobile robot.


Table 1. Physical parameters.

d1, d2 0.13 [m]

r1, r2 0.05 [m]

s1, s2 0.075 [m]

sg1, sg2 0.0595 [m]

Iw1, Iw2 0.00407 [kgm2]

m1, m2 6.51 [kg]

I1 0.103 [kgm2]

I2 0.101 [kgm2]

L1 0.39 [m]

L2 0.41 [m]

m 22.0 [kg]

I 3.65 [kgm2]

k1, k2 2.6

error of physical parameters and the modeling error,and they are decided by trial and error. The size ofplatform is given as follows:

Width: 0.6 [m]

Length: 1.1 [m]

Height: 0.34 [m]

Figure 8. Block diagram of prototype robot control system.

5.2. Control System

Control commands from a personal computer are givenin voltages through D/A converter on an interface board(RIF-01). These voltages drive the DC motors via drivercircuits, so that driving torque is occurred. While,pulses from the encoders are counted by UPP (Uni-versal Pulse Processor) on the interface board. Thesecounts are transmitted to the computer. Block diagramof a prototype robot control system is presented byFig. 8. Note that GO ji (i = 1, 2, j = r, l) indicatesa driving restorative constant, whereas GI ji and GIsi

express detecting restorative constants in Fig. 8. Theirdetailed explanations are introduced below.

5.2.1. Driving Restorative Property. A driving res-torative property represents a relation between atheoretical output and an input to the D/A converter sothat a calculated driving torque in a controller agreeswith a practical output. Thus, the driving restorativeproperty is given as an inverse function of a drivingproperty.

Voltage have been applied to the resting wheel, afterthat mean angular velocities were measured at 33 points

182 Yamada et al.

Figure 9. Relationship between voltage and angular velocity for each wheel.

over the range of −8 [V] to 8 [V] so that the angularvelocities of the wheels are almost constant as shown inFig. 9. It is found that an input voltage Vji [V] is nearlyproportional to the angular velocity of the wheel ω j i

[rad/s] in the region of −4 [V] to 4 [V] according toFig. 9. Therefore, this relationship can be described asfollows:

Vji = G ′j i ω j i (37)

where G ′j i [Vs/rad] is a proportional constant. Approxi-

mating a relationship between Vji and ω j i with a linearfunction through the origin, the proportional constantG ′

j i for each wheel is given by

G ′r1 = 0.8254 [Vs/rad]

G ′l1 = 0.8390 [Vs/rad]

G ′r2 = 0.7915 [Vs/rad]

G ′l2 = 0.8653 [Vs/rad].

Substituting Eq. (37) into the relationship between thedriving torque u ji [Nm] and the angular accelerationω j i [rad/s2] of wheel, we have

u ji = Iwi ω j i

= Iwi

G ′j i

V j i (38)

where Iwi [kgm2] is the moment of inertia of the wheel.Since the driving restorative property is the inversefunction of Eq. (38), we get

Vji = G ′j i

Iwi

∫ t

0u ji dt. (39)

This is represented in discrete-time as follows:

Vji = G O ji

n∑k=0

u ji (k) (40)


Figure 10. Simulation results.

where u ji (k) denotes the constant driving torque fromkT to (k + 1)T in discrete-time and the drivingrestorative constant GO ji [V/N m] is written by

GO ji = G ′j i T

Iwi. (41)

Here, T expresses a sampling period. In case where Tis 20 [ms], GO ji were obtained by

GOr1 = 4.06 [V/N m]

GOl1 = 4.12 [V/N m]

GOr2 = 3.89 [V/N m]

GOl2 = 4.25 [V/N m].

5.2.2. Detecting Restorative Property. Rotationalangles of the axles and steering axes in the prototype

robot are converted into pulse signals by the arrangedencoders respectively. These pulses are counted byUPP on the interface board and these counts are usedin the controller. In this case, the angular velocity ω j i

for each wheel is described as

ω j i = GI ji Pji . (42)

Also, a relative angular velocity φ′i [rad/s] and a relative

angle φ′i [rad] of the steering axis to the platform are

given by

φ′i = GIsi Psi (43)

φ′i = GIsi Psi T (44)

where Pji and Psi are the numbers of the pulses in theaxles and steering axes for the sampling period T re-spectively, and the detecting restorative constants GI ji

184 Yamada et al.

and GIsi [1/s] are represented as

GI ji = 2π

QwT(45)

GIsi = 2π

Qs T. (46)

Here, Qw and Qs are the numbers of the occurredpulses while the axles and steering axes make onerotation respectively. In case where T is 20 [ms], GI ji

and GIsi were obtained as follows:

GI ji = 0.223 [1/s]

GIsi = 0.126 [1/s].

5.3. Calculation of Angle and its Rateof Steering Axis

As previously described, the prototype robot can detectthe relative angular velocity φ′

i and the relative angle φ′i

of the steering axis to the platform using informationfrom the encoder. However, φi and φi are necessary toperform the calculation in the controller and estimatethe position and posture of the robot by dead reckoning.How to calculate the angular velocity φi and angle φi

in the prototype robot is described in this section.The angular velocity φi of the steering axis can be

calculated by

φi = φ + φ′i . (47)

When defining the initial posture of the omnidirectionalmobile robot as φ(0) and the initial posture of the dual-wheel caster as φi (0), the difference between them, φi ,is given by

φi = φi (0) − φ(0). (48)

The rotational angle φi of the steering axis, namely theposture of the dual-wheel caster can be calculated asfollows:

φi = φ + φ′i + φi . (49)

Note that, in Eqs. (47) and (49), φ can be determined by

substituting ω given by the detecting restorative prop-erty into Eq. (6) of the kinematic model and φ can beobtained by the integration.

6. Simulation and Experiment UsingPrototype Robot

6.1. Simulation

Before experiments were carried out, a variety of simu-lations have been performed to illustrate the validity ofthe inverse dynamic model for the omnidirectional mo-bile robot. We show a representative simulation in thissection. Now consider the straight line as the desiredtrajectory. Table 2 presents the desired values, and the

Figure 11. Errors of position and posture in simulation.


Figure 12. Experimental results for prototype robot.

conditions of simulation are represented by

Simulation time: 30 [s]Sampling period: 20 [ms]Initial state of robot: q = [0 0 π/2 π/2 π/2]T ,

q = [0 0 0 0 0]T .

Table 2. Desired values in simulation.

xd 0.008t [m/s] for 0 ≤ t < 10 [s]0.08 [m/s] for 10 ≤ t ≤ 30 [s]

yd 0.008t [m/s] for 0 ≤ t < 10 [s]0.08 [m/s] for 10 ≤ t ≤ 30 [s]

φd −π/60 [rad/s] for 0 ≤ t ≤ 30 [s]

The measured values of the physical parameters shownin Table 1 were used, except that the driving gain fac-tors, k1 and k2, were set to 1.0. The gain matrices weredetermined as

Kv = diag(2.0, 2.0, 2.0)

Kp = diag(1.0, 1.0, 1.0).

Figure 13. Errors of position and posture in experiment.

186 Yamada et al.

Figure 14. Behaviors of dual-wheel caster assemblies and omnidirectional mobile robot in the neighborhood of (a) 0 [s], (b) 15 [s], (c) and (d)30 [s], (e) 45 [s], and (f) 60 [s].


Figure 10 depicts the behaviors of the dual-wheelcaster assemblies and omnidirectional mobile robot,the position and posture of the robot, and the pos-tures of two assemblies. Figure 11 shows the errorsof the position (ex , ey) and posture (eφ) of the robot.From these figures, it is found that although there ex-isted an error in the posture for the first few seconds,the desired and simulation results agreed well on thewhole.

6.2. Experiment Using Prototype Robot

In order to demonstrate the effectiveness of the abovecontrol system and the validity of the dynamic model, avariety of experiments using the prototype robot wereconducted. A representative experiment is describedonly in this section. Consider the desired trajectorycombining circular arcs. The desired values are pre-sented by Table 3 and the experimental conditions areshown as follows:

Experimental time: 60 [s]Sampling period: 20 [ms]Initial state of robot: q = [0 0 0 0 0]T ,

q = [0 0 0 0 0]T

The measured values of the physical parameters shownin Table 1 were used. The gain matrices were decidedas

Kv = diag(2.6, 2.6, 2.6)

Kp = diag(1.69, 1.69, 1.69).

Figure 12 shows not only the position and posture ofthe robot but also the postures of two dual-wheel casterassemblies. Figure13 shows the errors of the position

Table 3. Desired values in experiment.

xd (π/60) cos(π t/30) [m/s] for 0 ≤ t < 15 [s]−(π/60) sin(π t/30) [m/s] for 15 ≤ t < 30 [s](π/60) sin(π t/30) [m/s] for 30 ≤ t < 45 [s](π/60) cos(π t/30) [m/s] for 45 ≤ t ≤ 60 [s]

yd (π/60) sin(π t/30) [m/s] for 0 ≤ t < 15 [s]−(π/60) cos(π t/30) [m/s] for 15 ≤ t < 30 [s](π/60) cos(π t/30) [m/s] for 30 ≤ t < 45 [s](π/60) sin(π t/30) [m/s] for 45 ≤ t ≤ 60 [s]

φd 0.0 [rad/s] for 0 ≤ t < 15 [s]π/60 [rad/s] for 15 ≤ t < 30 [s]0.0 [rad/s] for 30 ≤ t < 45 [s]−π/60 [rad/s] for 45 ≤ t ≤ 60 [s]

and posture of the robot. From these results, we found alittle position error in the neighborhood of 0 [s], 15 [s],30 [s] and 45 [s] due to transient state, however, theprototype robot pursued the desired trajectory. Also,it is found that the desired and experimental values ofthe posture of the robot agreed well, though there ap-peared a little error during transition periods and someoscillations while keeping φ = 0 [rad] for the first15 [s] and φ = π/4 [rad] from 30 [s] to 45 [s]. In addi-tion, Figure 14 depicts the behaviors of the dual-wheelcaster assemblies and omnidirectional mobile robot inthe neighborhood of some interesting points of the ex-perimental trajectory. From these figures, we could seeinteresting motions of two assemblies, in which suchmotions are a feature of the omnidirectional mobilerobot with active dual-wheel caster assemblies.

7. Conclusions

In this paper, we have derived the dynamic model forthe holonomic and omnidirectional mobile robot withtwo active dual-wheel caster assemblies, in order torealize a dynamic control for the mobile robot. Theprocedure for deriving the dynamic model was as fol-lows: The forces acting on the steering axis were firstdetermined and the dynamic model for the omnidirec-tional mobile robot was derived by using their forces.Then, the resolved acceleration control system basedon the dynamic model was constructed, and the simu-lations and experiments using the prototype robot wereperformed. Their results showed the validity of thedynamic model and the effectiveness of the controlsystem.

In the future works, it will be necessary to investi-gate the design of the controller which can realize thedynamic control adaptively even if the physical param-eters for the robot are changed.

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Takaaki Yamada was born in Tokuyama, Japan, on March 23, 1976.He graduated from Tokuyama College of Technology, Japan in 1996.He completed its Advanced Course and received the Bachelor ofEngineering degree in electrical engineering from NationalInstitution for Academic Degrees, Japan in 1998. He received theMaster of Engineering degree in advanced systems control engineer-ing from Saga University, Japan in 2000. He is currently a Ph.D. can-didate in the Department of Advanced Systems Control Engineering,Graduate School of Science and Engineering, Saga University, Japan.

His research interests include intelligent robots, machine learn-ing, and application of soft computing to the robot control. He isa member of IEEE (R&A, SMC, EMB, IE, and CS Societies), theRobotics Society of Japan, the Japan Society of Mechanical Engi-neers, the Society of Instrument and Control Engineers, the Instituteof Electrical Engineers of Japan, and Society of BiomechanismsJapan.

Keigo Watanabe received B.E. and M.E. degrees in MechanicalEngineering from the University of Tokushima in 1976 and 1978,respectively, and a D.E. degree in Aeronautical Engineering fromKyushu University in 1984. From 1980 to March 1985, he was aresearch associate in Kyushu University.

From April 1985 to March 1990, he was an Associate Professor inthe College of Engineering, Shizuoka University. From April 1990 toMarch 1993 he was an Associate Professor, and from April 1993 toMarch 1998 he was a full Professor in the Department of MechanicalEngineering at Saga University. From April 1998, he is now with theDepartment of Advanced Systems Control Engineering, GraduateSchool of Science and Engineering, Saga University.

His research interests are in stochastic adaptive estimation andcontrol, robust control, neural network control, fuzzy control, geneticalgorithms and their applications to the machine intelligence androbotic control. He has published more than 270 technical papers intransactions, journals and international conference proceedings, andis author or editor of 14 books, including Adaptive Estimation andControl (Prentice Hall), Stochastic Large-Scale Engineering Systems(Marcel Dekker) and Intelligent Control Based on Flexible NeuralNetworks (Kluwer). He is an active reviewer of many journals ortransactions, an editor-in-chief of Machine Intelligence and RoboticControl, and editorial board members of the Journal of Intelligent andRobotic Systems and the Journal of Knowledge-Based IntelligentEngineering Systems.

He is a member of the Society of Instrument and Control Engi-neers, the Japan Society of Mechanical Engineers, the Japan Societyfor Precision Engineering, the Institute of Systems, Control and In-formation Engineers, the Japan Society for Aeronautical and SpaceSciences, the Robotics Society of Japan, Japan Society for FuzzyTheory and Systems, and IEEE.

Kazuo Kiguchi received the Bachelor of Engineering degree in me-chanical engineering from Niigata University, Japan in 1986, theMaster of Applied Science degree in mechanical engineering fromthe University of Ottawa, Canada in 1993, and the Doctor of Engi-neering degree from Nagoya University, Japan in 1997.

He was a Research Engineer with Mazda Motor Co. be-tween 1986–1989, and with MHI Aerospace Systems Co. between


1989–1991. He worked for the Dept. of Industrial and Systems Engi-neering, Niigata College of Technology, Japan between 1994–1999.He is currently an associate professor in the Dept. of Advanced Sys-tems Control Engineering, Graduate School of Science and Engineer-ing, Saga University, Japan. He received the J.F. Engelberger BestPaper Award at WAC2000. His research interests include biorobotics,intelligent robots, machine learning, application of soft computingfor robot control, and application of robotics in medicine. He is amember of the Robotics Society of Japan, IEEE (R&A, SMC, EMB,IE, and CS Societies), the Japan Society of Mechanical Engineers,the Society of Instrument and Control Engineers, the Japan Societyof Computer Aided Surgery, International Neural Network Society,Japan Neuroscience Society, the Virtual Reality Society of Japan,and the Japanese Society for Clinical Biomechanics and RelatedResearch.

Kiyotaka Izumi received a B.E. degree in Electrical Engineeringfrom the Nagasaki Institute of Applied Science in 1991, a M.E. degree

in Electrical Engineering from the Saga University in 1993, and aD.E. degree in Faculty of Engineering Systems and Technology fromthe Saga University in 1996.

From April in 1996 to March in 2001, he was a Research Associatein the Department of Mechanical Engineering at Saga University.From April in 2001, he is now with the Department of AdvancedSystems Control Engineering, Graduate School of Science and En-gineering, Saga University.

His research interests are in robust control, fuzzy control, behavior-based control, genetic algorithms, evolutionary strategy and theirapplications to the robot control.

He is a member of IEEE, the Society of Instrument and ControlEngineers, the Japan Society of Mechanical Engineers, the RoboticsSociety of Japan, Japan Society for Fuzzy Theory and Systems, theInstitute of Electronics, Information and Communication Engineersand the Japan Society for Precision Engineering.

Documents

Dynamic Model and Control for a Holonomic Omnidirectional Mobile Robot