Robotics and Autonomous Systems 57 (2009) 433–440

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Robotics and Autonomous Systems

journal homepage: www.elsevier.com/locate/robot

Holonomy in mobile robotsJ.A. Batlle, A. Barjau ∗Department of Mechanical Engineering, Polytechnic University of Catalunya, Barcelona, Spain

a r t i c l e i n f o

Article history:Received 5 May 2008Accepted 10 June 2008Available online 24 June 2008

Keywords:Mobile robotsOmnidirectional wheelsHolonomyMobility

a b s t r a c t

The search for a simple and accurate odometry is a main concern when working with mobile robots.This article presents a general analysis of the problem and proposes a particular solution to improve theodometry. The three crucial kinematical aspects of mobile robots (mobility, control, and positioning) arereviewed in detail for vehicles based both in conventional and in omnidirectional wheels. The latter caseis more suitable from a maneuvering point of view as it provides the robot frame with the three DegreesOf Freedom (DOF) of plane motion without singular configurations. Moreover, a suitable design of theomnidirectional wheels leads to a strictly invariant Jacobian matrix and thus to a linear control equationwith constant coefficients. It is shown that such vehicles may have a holonomic behavior when movingunder suitable kinematical restrictions without constraining their trajectory. In that case, the odometryis algebraic (instead of integrative) and thus more accurate. An application case is presented.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Mobile robots – platforms that can move autonomously – areincreasingly used in flexible manufacturing industry and serviceenvironments. Though their basic kinematics is well-known, someaspects are still open to discussion.Three different (though related) kinematical aspects have to

be considered when designing a robot: mobility, control andpositioning [1]. The first one deals with the possible motions thatthe robot may follow to reach a final configuration. Mobile robotsmust be able to reach any position on its plane of motion, withany orientation. Thus their framemust have the three IndependentCoordinates (IC) of the general plane motion of the rigid body.This can be achieved by means of 2 DOF provided the robotframe kinematics is non-holonomic, but maneuvering is required.Maneuvering canbe avoidedwhenever themobile robot has 3DOF.Mobility is enhanced by the use of omnidirectional wheels insteadof conventional wheels.The second aspect deals with the choice of the kinematical

variables – generalized velocities or generalized coordinates – thatwill be directly controlled by the drivers to introduce the requiredrobot motion. If both the control variables and the variables usedto describe the robot frame motion are generalized velocities,the relationship between them is linear and given by means ofa Jacobian matrix. If the Jacobian matrix is invariant – that is,independent from the robot configuration – the control is quitesimple.

∗ Corresponding author.E-mail address: ana.barjau@upc.edu (A. Barjau).

0921-8890/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.robot.2008.06.001

Finally, the third aspect – positioning – considers the localiza-tion system, included in the robot and used to estimate the ac-tual robot pose (position and orientation), necessary to achieve anautonomous operation [2]. Localization is one of the fundamentalproblems in mobile robot navigation.The most widely used positioning methods for accurately

mapped environments are based on Kalman filtering [3,4]. Theyintegrate a prediction phase, based on the odometric data and therobot kinematics, and a correction (or estimation) phase that takesinto account external measurements.The aim of this paper is to establish kinematical conditions

which simplify and make more accurate the odometry and theKalman filtering. After a quick overview of the basic elementsof mobile robot kinematics (Section 2), Sections 3 and 4 aredevoted to mobility and control considerations respectively.Sections 5 and 6 present briefly the odometry and the Kalmanfiltering, and explain how a holonomic behavior can simplifyand improve their accuracy. It will be shown that mobile robotswith omnidirectional wheels may have a holonomic behaviorwhen moving under suitable kinematical restrictions withoutconstraining their trajectory. Finally, Section 7 presents anapplication example.

2. Kinematics description of a mobile robot

From the kinematics point of view, usual mobile robots may bedescribed as a framewith 3 or more wheels. Two different familiesof wheels can be found: conventional and omnidirectional wheels.Among the conventional ones, one can find (Fig. 1):• Fixed wheels, with both centre and axis fixed to the frame. Inorder to guarantee 2 DOF for the robot frame, they have to becoaxial, and may be drive or auxiliary wheels.

434 J.A. Batlle, A. Barjau / Robotics and Autonomous Systems 57 (2009) 433–440

Fig. 1. Different kinds of conventional wheels: fixed wheels (a), steering wheels(b), castor wheels (c).

Fig. 2. Omnidirectional wheels based on peripheral rollers with free rotationaround their axis.

• Steering wheels, which are pivot or pivot-drive wheels. As theyare not self-oriented, their pivot motion has to be controlled.• Castor wheels, which pivot about a vertical castor axis. They areauxiliarywheels usually, but theymaybedrivewheels and theirpivot motion may be controlled.

As far as the omnidirectional wheels are concerned, there is agreat variety of designs [5–8]. Their centre has 2 DOF (Fig. 2(a)),a driven motion and a free motion. The angle α between them isconstant or almost constant. This articlewill focus onwheels basedon peripheral rollers with free rotation about their axis (Fig. 2(b)).The wheels – conventional or omnidirectional – are assumed to

have a non skidding contact with the ground. If the conventionalwheels and the peripheral rollers of the omnidirectional ones areidealized as rigid bodies with a single point Jk of ground contact,the robot kinematics is fully determined by the equations of thevelocities of thewheel centre (or the pivot axis in the case of castorwheels).Let’s assume that the frame motion is described by means of

the velocity of its point O, vGRF (O), and the angular velocity of therobot frame, Ω robotGRF , both relative to the Ground Reference Frame(GRF, Fig. 3). In omnidirectional wheels and conventional fixed orsteering wheels, the velocity of the wheel centre Ck can be relatedeither to the frame kinematics or to the wheel kinematics. Thus:

vGRF (Ck) = vGRF (O)+ Ω robotGRF ∧ OCk = θk ∧ JkCk + wk, (1)

where wk is arbitrary in omnidirectional wheels and is zero inconventional wheels. For a castor wheel articulated at point Qk ofthe robot frame:

vGRF (Qk) = vGRF (O)+ Ω robotGRF ∧ OQ k

= θk ∧ JkCk + (Ω robotGRF + ϕk) ∧ JkQk, (2)

where ϕ is the pivot velocity relative to robot frame.

Fig. 3. Kinematical description of a robot frame.

Fig. 4. Mobile robot with differential kinematics.

3. Mobility: system DOF and robot frame DOF

Mobility is usually associated with the number of DOF. Inmobile robots, it is worth to distinguish between the DOF ofthe robot frame, DOFframe, and those of the robot as a whole(called ‘‘system’’ from now on), DOFsystem. They are different whensteering wheels are used, as the steering angular velocity hasnot a direct influence on the robot frame motion. The difference(DOFsystem − DOFframe

)is equal to theDOFused to steer thewheels.

Concerning the coordinates, the robot frame has 3 IC, asmentionedbefore. The number of system IC is relevant in the positioningproblem, as it will be discussed in Sections 5 and 6.Let us take the example of amobile robot with differential kine-

matics (that is, with 2 fixed wheels – controlled by independentdrivers – and several auxiliary castor wheels, Fig. 4). In this caseDOFsystem = DOFframe = 2, which can be associated with the longi-tudinal velocity v of the midpoint between the wheel centers andthe yaw velocity ψ . As ICframe = 3, the frame kinematics is non-holonomic. The robot can reach any position on the plane with anyorientation but, having just 2 DOF, maneuvering is unavoidable.Let us consider now a robot with tricycle kinematics (Fig. 5). As

the steering wheel is not self-oriented, its steer motion uses oneof the 2 DOFsystem. The other one describes the rotation velocity ofthe robot frame about its Instantaneous Center of Rotation (ICR),which is determined by the steer angle. In this case DOFframe =1 6= DOFsystem = 2. The velocities v and ψ are not independent, asin the robot with differential kinematics. They are related throughthe steer angle γ relative to the robot frame. Four-wheeled robotswith two fixed wheels and two steering wheels are equivalent tothe tricycle case because the two steering wheels share a singleDOF.If all wheels are steering wheels that allow any position for

the frame ICR (something requiring a minimum of two steering

J.A. Batlle, A. Barjau / Robotics and Autonomous Systems 57 (2009) 433–440 435

Fig. 5. Mobile robot with tricycle kinematics.

Fig. 6. Robot with only steering wheels.

wheels, while the others can be just auxiliary castor wheels, Fig. 6),2 DOF are used to steer the wheels,

γ1 γ2

T, and 1 DOF ψdescribes the robot frame rotation about the ICR (so a minimum ofone driving wheel is required). Thus DOFframe = 1 6= DOFsystem =3. The mobility of this type of robot is equivalent to that of amobile robot with DOFframe = 3 provided that the ICR evolutionis continuous. Discontinuities of the ICR position introduce timedelays associated with the time needed to reorient the wheels.Such delays do not occur when DOFframe = 3.In a robot with castor wheels only (where some pivot motions

may be controlled and some wheels may be driven), DOFframe =DOFsystem = 3. The higher number of DOFframe in this case (ascompared with the single DOFframe of the robot with steeringwheels) enhances the robot mobility. However other problemsarise in such designs: singular configurations may appear whichimpair the robot control (as will be analyzed in Section 4).

4. Control kinematics

The control of a mobile robot is based on a set of variablesdirectly controlled by the drivers. The number of independentcontrol variables is equal to the number of DOFsystem. They maybe generalized velocities vc or generalized coordinates qc,and are related through the robot kinematics to the generalized

Fig. 7. Control of a mobile robot with differential kinematics.

velocities u used to describe themotion of the robot frame.WhenDOFframe = DOFsystem, the u are directly associated with theDOFframe. When steering wheels are used, DOFframe < DOFsystem,and the control coordinates qc are the independent steer angles.The velocities u are not independent because they are relatedthrough these angles.If DOFframe = DOFsystem, the control variables are independent

velocities vc, and their relationship to u is linear and is givenby the Jacobian matrix [J], which is in general configuration-dependent:vc=[J(qc)]u . (3)

If the Jacobian is invariant (configuration-independent), thecontrol is quite simple. This is the case of robots with differentialkinematics, Fig. 7: the rotation velocities θ1 and θ2 of the driversare related to the 2 DOFframe described by the velocities v and ψthrough a Jacobian that depends on the wheel radius r and thewheels separation 2L:θ1θ2

=1r

[1 L1 −L

]v

ψ

. (4)

In robots with steering wheels, DOFframe = 1 < DOFsystem. Inthis case, the control must include that of the steer angles, and itis no longer linear. It must be pointed out that the control of anangle has a bigger time delay than that associated with its angularvelocity. Actually the directly control variables are the angularaccelerations. A time integration leads to the angular velocities anda further time integration leads to the angular values.If the robot has one or more fixed wheels, the set of steering

wheels – one or more – share 1 of the 2 DOFsystem of the robot(Fig. 8). In this case, if the robot framemotion is described throughvL and ψ , one has to be aware of their dependency, as they arerelated through the steer angles which share a single independentcoordinate. For tricycle kinematics, the robot motion may bedescribed by means of the velocity v′ of the steering wheel centreand the steering angular velocity γ , which are independent and canbe directly used as control variables. However, this option is notequally suitable for both directions of motion. When the steeringwheel is the front-forward one, the robot frame motion is stable,that is, tends to ‘‘follow’’ the path of the steering wheel. In therearward motion of the driving wheel, the robot frame motionis unstable and tends to deviate from the path followed by thesteering wheel. The stabilization of this motion complicates therobot control.

436 J.A. Batlle, A. Barjau / Robotics and Autonomous Systems 57 (2009) 433–440

Fig. 8. Robot with one or more fixed wheels.

Fig. 9. Singular configuration in a robot with driven castor wheels.

If all the robot wheels are steering wheels, they share 2 of the3 DOFsystem (Fig. 6). If the robot motion is described by means ofvL, vT , ψ

, it must be taken into account that they share a single

DOF, and that they are related through the steer angleswhich sharetwo independent coordinates. In this case the control becomesnonlinear and includes the control of the steer angles.If all wheels are castor wheels, DOFframe = 3, and consequently

the velocities u =vL vT ψ

T – longitudinal and transversevelocities of a given frame point, and yaw velocity – are relatedto the 3 independent motor velocities (driving or steering)vc, according to Eq. (2), by means of a Jacobian dependingupon the castor angles ϕk relative to the robot frame. Thesingular configurations, mentioned in the previous section, appearwhenever det [J] = 0. In the singular configurations, the directionsof the castor for the driving wheels and the axis of the wheels withcontrolled pivot motion intersect at the ICR (Fig. 9).Mobile robots with omnidirectional wheels may have

DOFframe = 3 and, as the control variables are generalized veloc-ities – those associated with the driven motions – the control islinear. The Jacobian can be either strictly invariant or almost invari-ant. The omnidirectional wheels best suited for mobile robots arethosewith peripheral rollers (Fig. 2(b)). Their free rotationprovidesthe wheel with the free motion in a direction defining a constantangle α with the driven motion.The well-known designs with α = 45 and α = 90 – with

two planes and three or more rollers in each of them, Fig. 10 – [9,10], lead to a Jacobian which is not strictly invariant and dependson the driven angle rotated by the wheels. For α = 45, thefinite width of the rollers is responsible for the displacement of theposition of the roller–ground contact relative to the robot frame.Forα = 90, the ground contact alternates from one roller plane tothe other, and the Jacobian depends on the plane position relativeto the robot frame. Moreover, there is a small overlap of groundcontact between rollers of the same wheel during the transition of

Fig. 10. Omnidirectional wheels with peripheral rollers for α = 45 and α = 90 .

the ground contact from one roller to the other, and this makesskidding unavoidable when modifying the robot’s orientation. Ifthe number of rollers per plane is reduced to one, they becomespherical (Fig. 11(a)).To avoid the drawbacks of using the two planes of rollers,

the wheel may be redesigned as shown in Fig. 11(b), with therollers centers aligned in the directions of the driven motion. Thisdesign implies a same driven motion for both roller centers whilethe robot is modifying its orientation. Under this condition, thedifferences arise in the rollers free rotation. With such wheelsthe Jacobian matrix [J] relating the longitudinal and transversevelocities vL and vT of point O and the yaw angular velocity ψ , isstrictly invariant:θ1θ2θ3

= [J]vLvTψ

. (5)

5. Positioning kinematics

Odometry is a widely used localization method because of itslow cost, high updating rate, and reasonable short path accuracy.Usually, the robot pose is given through the position of a point O,rGRF (O), and the frame orientation,ψ , both relative to the GRF, andit is obtained through a time integration, relative to the GRF, of thecorresponding velocities vGRF (O) and ψ (yaw angular velocity).However, the unbounded growth of the integration errors as

the robot moves is unavoidable and a significant inconvenience.To cope with this, several sensors are used to detect distinctfeatures of a known environment. This information, given by a setof variables

qexp

, is fused with the odometric data rRF (O) and ψ

to obtain the optimum pose estimation.

J.A. Batlle, A. Barjau / Robotics and Autonomous Systems 57 (2009) 433–440 437

Fig. 11. Different designs of omnidirectional wheels with peripheral rollers forα = 90 .

The most commonly used methods for accurately mappedenvironments are based on Kalman filtering. Through this filter,the odometric data – described by means of the state transitionfunction – are fused with the external measurements which aredescribed by means of the observation function.When both functions are linear and the perturbation errors

on both odometric and observation data are Gaussian, the LinearKalman Filter (LKF) [3] leads to the optimum estimation. However,usually both functions are nonlinear, and the Extended KalmanFilter (EKF) [4] has to be used. In the EKF, both equations arelinearized and the perturbation errors are assumed to be Gaussian.Consequently, the ‘‘optimum’’ solution is only an approximatedone, not only because of the linearization but also because theGaussian errors become non Gaussian due to the actual nonlinearbehavior of both odometry and observation.In an integrative odometry, the robot frame configuration is

obtained from the control variables, which are motor velocitieswhen omnidirectional wheels are used. A question appearsconcerning the relationship between the increment of thecoordinates describing the robot frame configuration, ∆qframe,and the angles rotated by the drivers, ∆θmotor. If it is a bilateralone, the odometry is algebraic. This calls for the holonomy of therobot system composed of the frame and the set of independentdrivers. That is, it requires that the linear relationship betweenqframe and

θmotor

can be integrated and thus translates into an

algebraic relationship between ∆qframe and ∆θmotor.

Fig. 12. Mobile robot described bymeans of Cartesian coordinates and a yaw anglex y ψ.

As will be proved in Section 6, this holonomic behavioris possible in mobile robots with omnidirectional wheels andinvariant Jacobian matrix. In such a case, positioning is simplerand more accurate because the usual integrative odometry issubstituted by a linear algebraic odometry which leads to a linearstate transition function. This allows the use of a simpler andmoreaccurate Kalman Filter.

6. Holonomic behaviour of a mobile robot

The concept of holonomy in mobile robots has been exploredby several authors [7,11]. Holonomy is related to the generalintegrability of the kinematical constraint equations:

qframe = [K ]θdrivers .

All robots with conventional wheels are nonholonomic, whilerobots with omnidirectional wheels may have a holonomicbehavior. As an example, let’s consider a mobile robot whose poseis described by means of the Cartesian coordinates of its pointO, rGRF (O) =

x y

T, and the yaw angle ψ (Fig. 12). If itscontrol is defined by means of Eq. (5) with an invariant Jacobianmatrix [J], the constraint equations relative to x, y, ψ and theirtime derivatives are given by xyψ

=[cosψ − sinψ 0sinψ cosψ 00 0 1

][J]−1

θ1θ2θ3

. (6)

Though 6 coordinates (x, y,ψ , θ1, θ2, θ3) are implied in Eq. (6), only 5can be independent, as the invariance of the Jacobian matrix leadsto a linear relationship with constant coefficients between ψ andθ1, θ2, θ3 that translates into a linear relationship between∆ψ and∆θ1,∆θ2,∆θ3.Holonomy requires an algebraic relationship between the

increments ∆x, ∆y, ∆ψ and the increments ∆θ1, ∆θ2, ∆θ3 ofthe wheels rotation angle. Obviously such a relationship exists ifψ is kept constant, and this is one of the two cases that will befound in the subsequent general search for restrictions leading toholonomy. Whenever this condition is fulfilled, the relationship islinear and leads to an extremely simple odometric positioning andto a simpler and more accurate EKF.In some robots, odometry is carried out from the angles rotated

by measuring wheels instead of those rotated by the drivingwheels. Being light and suitably articulated to the robot frame,the measuring wheels may provide more accurate data than thedrivingwheels because free from the intense forces involved in thedynamics of the robot motion.

438 J.A. Batlle, A. Barjau / Robotics and Autonomous Systems 57 (2009) 433–440

Fig. 13. Mobile robot described by means of general orthogonal curvilinearcoordinates.

6.1. General analysis

The holonomic behavior obtained from Eq. (6) by keepingψ constant can be the starting point for the general search ofsuitable restrictions. Those restrictions may be easier to discoverwhen using coordinates other than the Cartesian ones (though ofcourse they could be found in any coordinate system). Thus, thegeneralization comes from the formulation of Eq. (6) for any setof orthogonal curvilinear coordinates

q1 q2

T to describe theposition of point O and the corresponding generalized velocitiesq1 q2

T to describe its motion (Fig. 13). The unit vectors e1 ande2 are oriented in the GRF through the angle β , and through theangle δ in the Robot Reference Frame (RRF ):

ΩeiGRF = β, Ω

eiRRF = δ. (7)

The longitudinal and transversal velocities of pointO are relatedtoq1 q2

T through:vLvT

=

[cos δ sin δ− sin δ cos δ

]h1(q)q1h2(q)q2

, (8)

where h1(q), h2(q) are related to the elements of themetric tensor:

[G] =[h21(q) 00 h22(q)

], (9)

and are in general q-dependent.The equation for the kinematical constraints, assuming invari-

ant Jacobian matrix, becomes:h1(q)q1h2(q)q2ψ

=[cos δ − sin δ 0sin δ cos δ 00 0 1

][J]−1

θ1θ2θ3

≡ [T (δ)]

θ1θ2θ3

. (10)

The last row in Eq. (10) can be directly integrated, and conse-quently only 5 system coordinates are independent.The holonomic behavior will be obtained if the other two rows

can be integrated. This condition for the right hand side of Eq. (10)is trivially fulfilled if δ is kept constant (δ = δ0). This translates intoa kinematical restriction: the robot angular velocity Ω robotGRF has tobe equal to ΩeiGRF , ψ = β . The Ω

eiGRF can be expressed through the

Christoffel symbols Γ ijk as:

α =h2(q)h1 (q)

(Γ 211(q)q1 + Γ

212(q)q2

)=h1(q)h2(q)

(Γ 121(q)q1 + Γ122(q)q2). (11)

Fig. 14. Holonomic robot frame motion: pure translation (a), motion with ∆ψ =∆ϕ (b).

Expressing the Γ ijk(q) through the metric tensor components andtheir derivatives yields the constraint equation:

ψ = −1h2(q)

∂h1(q)∂q2

q1 +1h1(q)

∂h2(q)∂q1

q2. (12)

Though the DOFsystem reduce from 3 to 2, the number of ICframeremains unchanged unless the left hand side of Eq. (10) can alsobe integrated. Among the more usual coordinate systems, two forwhich this is feasible are the Cartesian and the polar ones,

x y

Tand

r θ

T respectively.6.2. Cartesian coordinates

For these coordinates, the metric tensor is the identity and theorientation angles ψ and δ are identical (except for their sign).Keeping δ = δ0 constant implies a pure translation of the robotframe (Fig. 14(a)). The first and second equations in Eq. (10) can beintegrated analytically and yield:

∆x =3∑i=1

T1i (δ0)∆θi, ∆y =3∑i=1

T2i (δ0)∆θi. (13)

The trajectory of pointO in the robot frame has not been restricted,as∆x and∆y can take independent values. Thus, though being justa translation, the unrestricted geometry of the trajectory makesthis particular motion applicable to real situations.

6.3. Polar coordinates

The elements of the metric tensor are h21 = 1 andh22 = ρ

2, andconsequently ψ = ϕ and thus∆ψ = ∆ϕ. The increment of the ρcoordinate is obtained through the integration of the first equationof Eq. (10):

∆ρ =

3∑i=1

T1i (δ0)∆θi. (14)

The second equation cannot be analytically integrated, but having(∆ρ,∆ϕ) allows its numerical calculation:∫ t2

ttρϕdt =

∮ρdϕ =

∮ρdψ =

3∑i=1

T2i (δ0)∆θi. (15)

However no immediately useful information can be extracted fromit, and can be ignored.The robot frame rotates and the trajectory of point O in the

robot frame has not been restricted (Fig. 14(b)). In this case, therobot frame keeps a constant orientation relative to the polar radialdirection. Again, the unrestricted geometry of the trajectorymakesthis particular motion applicable to real situations.

J.A. Batlle, A. Barjau / Robotics and Autonomous Systems 57 (2009) 433–440 439

∆ρ∆ϕ

=

11+ sin ξ

sin δ0 −12

(1+ sin ξcos ξ

cos δ0 + sin δ0

)12

(1+ sin ξcos ξ

cos δ0 − sin δ0

)−sin ξL

−12L

−12L

∆θ1∆θ2∆θ3

.

Box I.

Fig. 15. The ‘‘SPHERIK-3x3’’ vehicle: general view (a), driven and free wheelsmotion (b).

7. Application case

The previous analysis will be applied to the mobile robot‘‘SPHERIK-3x3’’ (Fig. 15), based on spherical omnidirectionalwheels and financed by the ‘‘Centre de Referència de Tèc-niques Avançades de Producció (CERTAP)’’ of the ‘‘Generalitat deCatalunya’’ (Barcelona, Spain). This vehicle is based on omnidirec-tional wheels with spherical rollers as those shown in Fig. 11(b). Itsframe has the 3 DOF of the planemotion, and its Jacobian is strictlyinvariant:θ1θ2θ3

=[ 0 −1 −L− cos ξ sin ξ −Lcos ξ sin ξ −L

]vLvTψ

. (16)

The corresponding equation for the kinematical constraints is:h1(q)q1h2(q)q2ψ

= 11+ sin ξ

[cos δ − sin δ 0sin δ cos δ 00 0 1

]

×

0 −

1+ sin ξ2 cos ξ

1+ sin ξ2 cos ξ

−112

12

−sin ξL

−12L

−12L

θ1θ2θ3

. (17)

For the case of Cartesian coordinates and under the kinematicalrestriction ψ = ψ0, the previous equation after integration is:

∆x∆y

=

− sinψ0 − tan ξ cosψ0 −cosψ0cos ξ

sin ξ sinψ0 − cosψ01+ sin ξ

− tan ξ cosψ0 −sinψ0cos ξ

×

∆θ1∆θ2

. (18)

For the case of polar coordinates and imposing δ = δ0 (and so∆ψ = ∆ϕ), the algebraic odometry equation is given in Box I.

8. Conclusions

The three basic aspects of the kinematics of mobile robot basedon conventional and omnidirectionalwheels (mobility, control andpositioning) have been revisited.The kinematical analysis of the different mobile robot designs

has lead to the enunciation of the suitable conditions to be fulfilledin order to obtain a simpler and more accurate odometry.As already discussed in articles by different authors, designs

leading to a robot frame with the three DOF of the plane motionenhance mobility as they eliminate the need of maneuvering. Thisis the case of robotswith only castorwheels and that of robotswithomnidirectional wheels.The control of the frame motion is given by the Jacobian

matrix, relating the frame DOF with the variables directly drivenby the actuators.Whenever thismatrix is configuration dependent,singular configurations may appear. Robots with castor wheelsshow this drawback. Robots with omnidirectional wheels with asuitable design of the omnidirectional wheels, however, have astrictly invariant Jacobian matrix. It is the case of wheels based onspherical rollers with centers aligned in the direction of the drivenmotion.Concerning the robot positioning, the most used methods for

accurately mapped environments are based on Kalman filters,which fuse the odometric data with the external measurements.Odometry is based on the kinematical robotmodel, and it is usuallyintegrative. It has been shown that the odometry may be algebraicand linear formobile robotswith omnidirectional wheels providedthe robot behavior is holonomic.To attain such behavior, it is necessary to reduce the frame

mobility to 2 DOF with suitable kinematical restrictions. Startingfrom a general description of the frame motion through generalorthogonal curvilinear coordinates and their derivatives, a generalcondition to be fulfilled by restrictions leading to holonomy hasbeen obtained. It has been shown that this condition translatesinto a simple constraint when Cartesian or polar coordinates areused. In the former case, it consists in keeping constant the frameorientation in the Ground Reference Frame; in the latter, it resultsin a constant orientation of the radial direction in the FrameReference Frame. None of these constraints restrict the trajectoriesof the frame, and this makes these particular motions useful forpractical application.A consequence of this linear algebraic odometry is that it allows

the use of a Kalman Filter with a linear transition function, whichresults in more accurate predictions in the positioning process.

440 J.A. Batlle, A. Barjau / Robotics and Autonomous Systems 57 (2009) 433–440

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Dr. J.A. Batlle was born in Barcelona in 1943. Hewas awarded his M.S. in Mechanical Engineering in1968 and his Ph.D. in 1975. Upon graduation he joinedthe Polytechnical University of Catalunya where he isProfessor in Mechanical Engineering and head of theresearch line on Mechanics and Acoustics. He is a memberof the Royal Academy of Sciences and Arts of Barcelona(1991), and also a member of the American Society ofMechanical Engineering (1996) and the Acoustical Societyof America (1990). His research work is mainly related tothe mechanics of mobile robots, to percussive dynamics

and to the acoustics of waveguides and woodwinds.

Dr. A. Barjau obtained her diploma in Theoretical Physicsin 1980, and her Ph.D. in 1987. Since then, she hasbeen professor at the Dep. of Mechanical Engineeringof the Polytechnical University of Catalunya (UPC). Since2004, she is coordinator at UPC of a European master ofMechanical Engineering. Her research interests includetheoretical mechanics, mechanics of robots and acousticsof axisymmetrical waveguides, with special application tomusical instruments.