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Nonlinear regulation and path tracking of a wheeledmobile robot in polar coordinatesTai‐Yu Wang a , Ching‐Chih Tsai b & Jia‐Lun Pang a

a Department of Electrical Engineering , National Chung Hsing University , Taichung,Taiwan 402, R.O.C.b Department of Electrical Engineering , National Chung Hsing University , Taichung,Taiwan 402, R.O.C. Phone: 886–4–22851549 ext. 222 E-mail:Published online: 04 Mar 2011.

To cite this article: Tai‐Yu Wang , Ching‐Chih Tsai & Jia‐Lun Pang (2005) Nonlinear regulation and path trackingof a wheeled mobile robot in polar coordinates, Journal of the Chinese Institute of Engineers, 28:6, 925-933, DOI:10.1080/02533839.2005.9671067

To link to this article: http://dx.doi.org/10.1080/02533839.2005.9671067

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Journal of the Chinese Institute of Engineers, Vol. 28, No. 6, pp. 925-933 (2005) 925

NONLINEAR REGULATION AND PATH TRACKING OF

A WHEELED MOBILE ROBOT IN POLAR COORDINATES†

Tai-Yu Wang, Ching-Chih Tsai*, and Jia-Lun Pang

ABSTRACT

This paper presents regulation and path tracking control methods for wheeledmobile robots (WMRs) associated with kinematic models in two-dimensional polarcoordinates. The proposed two regulation control laws are designed via the Lyapunovindirect and direct stability theorems in order to asymptotically achieve nonlinearregulation. With the Lyapunov-based stabilization law, a novel path tracking controlmethod is proposed to achieve reference path following missions on a global scaleexcept for an arbitrary small region around the origin; in particular, two special pathtracking examples are analyzed with large initial tracking errors. Computer simula-tions and experimental results are described to confirm the efficacy of these proposedcontrol approaches.

Key Words: nonlinear control, path tracking, polar coordinates, posture stabilization,wheeled mobile robots.

†Based on an awarded paper presented at Automation 2005, the8th international conference on automation technology, Taichung,Taiwan, R.O.C. during May 5-6, 2005.

*Corresponding author. (Tel: 886-4-22851549 ext. 222; Email:cctsai@dragon.nchu.edu.tw)

The authors are with the Department of Electrical Engineering,National Chung Hsing University, Taichung, Taiwan 402, R.O.C.

I. INTRODUCTION

The tracking controls of nonholonomic wheeledmobile robots (WMR) have become increasingly im-portant in the recent past. Tracking control problemsfor wheeled mobile robots are often classified intotwo categories: path tracking and trajectory tracking.The trajectory tracking problem is formulated as therobot exactly tracks a desired reference path at a givenvelocity; that is, for trajectory tracking, the referencepath is defined as the function of time. Many sophis-ticated control approaches, such as optimal controlby Soueres et al. (1996), nonlinear control schemesby Kanayama et al. (1990) and Dixon, et al. (2001),smooth time-varying control by Pomet (1992) andSordalen et al. (1992), chained system by Samson(1995), backstepping by Jiang et al. (1997), robustcontrol by Oelen et al. (1994), adaptive control byFukao et al. (2000), sliding-mode control by Blochet al. (1994), fuzzy control by Bentalba et al. (1997)and adaptive neural control by Fierro et al. (1998),

have been proposed to achieve trajectory trackingcontrol for WMRs. Most of the aforementionedapproaches have been devoted to developing trajec-tory tracking control laws based on the kinematiceuations of the mobile robots in Cartesian coordinates,but seldom in polar coordinates. Chwa et al. (2002)addressed the trajectory tracking problem in polarspace; they used a sliding-mode tracking controllerto steer a nonholonomic wheeled mobile robot incor-porating its dynamic effects and external disturbances;the controller was shown effective, having fastresponse, good transient performance and robustnesswith regard to parameter variations.

In practice, it is not necessary for wheeledmobile robots to reach prespecified postures at aspecified instant, but it is of practical significance toaccurately follow the geometric path which is inde-pendent of time. This kind of tracking is referred toas path tracking. Much work has been done on pathtracking using the Cartesian kinematic model, butlittle attention has been paid to the polar model. Pathtracking problems of the WMR in polar coordinateshave been investigated by few researchers. Aicardiet al. (1995) introduced a Lyapunov-based control ap-proach for path following control of wheeled mobilerobots in polar coordinates. Wu et al. (1999) usedbackstepping to design a path tracking controller, butassumed that the linear velocity command always

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926 Journal of the Chinese Institute of Engineers, Vol. 28, No. 6 (2005)

remained constant. Because many paths can be eas-ily expressed in polar coordinates, new path trackingmethods deserve further investigation for findingmore useful and pragmatic strategies.

Aside from tracking problems, regulation orpoint stabilization (PS) problems of WMRs have beensolved by several control schemes. It has been shownthat the regulation of WMR described in Cartesianspace cannot be stabilized by means of a smooth time-invariant state feedback control as pointed out byBrockett (1983). To overcome this difficulty, sev-eral smooth nonlinear regulation methods in Carte-sian coordinates have been well discussed by Dixonet al. (2001). Aicardi et al. (1995) addressed the pointstabilization problem in polar coordinates and ex-tended their proposed method to solve for the pathtracking problem.

Following the work of Aicardi et al. (1995), thispaper proposes two new regulation laws for point sta-bilization of the WMR in polar coordinates. The twopoint stabilization methods are established based onthe Lyapunov stability theorems. With the proposedLyapunov-based regulation rule, a novel path track-ing law is then proposed to follow any desired pathsexpressed in polar coordinates. The developed methodsare expected to be useful in steering WMR to exactlyfollow any reference paths described in polar coordinates.

The remainder of the paper is organized asfollows. Section II briefly describes the kinematicsmodels of the WMR. Section III aims at developingthe two kinematic regulation control laws based onthe polar coordinates. In Section IV, a novel pathtracking controller for WMR has been proposed tohandle the path tracking control problem. In sectionV experimental and simulation results are shown toillustrate the effectiveness and feasibility of the pro-posed control methods. Section VI concludes the pa-per and states directions for future work.

II. KINEMATICS OF WHEELED MOBILEROBOTS

Generalized wheeled-mobile robots undernonholonomic constraints in the Euler-Lagrange for-mulation can be described as follows.

M(q)q + V(q, q)q + G(q) = B(q)τ + AT(q)λλ (1)

where q ∈ Rn is the generalized coordinates, ττ ∈ Rr isthe torque control input vector, λλ is the constraint forcevector, M(q) ∈ Rn × n is the symmetric and positive definiteinertia matrix, V(q, q) ∈ Rn × n is the centripetal andcoriolis matrix, G(q) ∈ Rn is the gravitation vector,B(q) ∈ Rn × r is the input transformation matrix, andA(q) ∈ Rm × n is a matrix related to nonholonomicconstraints. In the following, r = n – m is assumed.

The nonholonomic kinematic constraints of therobots are expressed by

A(q)q = 0 (2)

and the following relation can be obtained.

A(q)S(q) = 0 (3)

where S(q) ∈ Rn × r is composed of linearly indepen-dent vectors in the null space of A(q). From (2) and(3), there exists a r = n – m dimensional vector µµsuch that

q = S(q)µµ (4)

where µµ corresponds to the internal state variable,defining variables q and µµ in (4) as q = [xc yc θc]

T andµµ = [υc ωc]

T. Moreover, (4) represents a nonholonomicconstraint on the rolling motion of the WMR withoutslipping, where υc is a forward linear velocity, θc =ωc is an angular velocity of the mobile robot. Wherethe xc(t), yc(t) and θc(t) represent the position andorientation of the WMR, respectively; thus, the ma-trix S(.) ∈ R3 × 2 is defined in the following

S(q) =cosθc 0sin θc 0

0 1(5)

which gives the robot’s kinematics model

xc = υccos(θc)yc = υcsin(θc)θc = ωc

(6)

By representing the current posture of the ve-hicle with respect to the goal frame in polar coordi-nates (Fig. 1), we easily obtain the following errordynamic equations

e = – υ cos(θ – φ)

θ = υ sin(θ – φ)e

φ = ω

(7)

where e denotes the error distance between the cen-ter point of the WMR and goal position, α denotesthe sight angle to the goal position, and φ stands forthe error angle from the heading orientation to the x-axis of the goal frame, and θ denotes the angle mea-sured between the sight line and the x-axis of the goalframe. Note that, to avoid the singularity, e can notbe zero and |e| > ε > 0.

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T. Y. Wang et al.: Nonlinear Regulation and Path Tracking of a Wheeled Mobile Robot in Polar Coordinates 927

By setting α = θ – φ, (7) becomes

e = – υ cos(α)

α = – ω + υ sinαe

θ = υ sinαe

(8)

Based on the use of the polar coordinates, thekinematics model (8) is actually valid only for non-zero distances with |e| > ε > 0. Both angles, α and θ,are undefined when e = 0, thus implying that the gen-erally existing one-to-one mapping in (8) is actuallylost in correspondence to such a singular point.

In what follows, we propose two regulation con-trol laws for point-to-point motion such that the po-sition errors and orientation errors converge to zeroin an asymptotical manner.

III. KINEMATICS REGULATION CONTROLS

1. Linearization Regulator Design

The first control law to locally stabilize the ki-nematics model (8) is shown as follows,

υω =

re cos(α)k(α + hθ) + α

α + hθ r sin(α)cos(α)(9)

where k, h, and r are three positive control parameters.Substituting the proposed control law (9) into (8), oneobtains the following closed-loop error system

e = – re cos2(α)

α = – k(α + hθ) + (1 – αα + hθ )r cos(α)sin(α)

θ = r cos(α)sin(α)

(10)

Based on the Lyapunov indirect method (linearizationstabilization), we easily show the local asymptoticalstability of the closed-loop error system. From (10),it follows that the origin, [0 0 0]T, is an equilibriumpoint. Thus, a linearization model around the originis easily obtained from

δeδαδθ

=– r 0 00 – k + r – kh0 r 0

δeδαδθ

(11)

whose characteristic equation is determined by

(λ + r)det λ I – – k + r – khr 0

= (λ + r)(λ 2 + (k – r)λ + khr) (12)

According to the Routh-Hurwitz stability criterion,if the parameters k, h, and r are positive and the in-equality 0 < r < k holds in the proposed control law(9), then the origin is locally asymptotically stable.Although it can be proven that the position error is e→ 0 as t → ∞, the error actually remains uniformlybounded, that is, it converges to a small value ε (|ε| >0). The result is summarized in the followingtheorem.

Theorem 1. For the kinematic model (8) controlledby the linearization regulation law (9), the orienta-tion error approaches zero but the distance error re-mains uniformly bounded.

2. Lyapunov-Based Regulator Design

The second nonlinear controller to globally sta-bilize the kinematical model (8) is proposed asfollows.

υω =

υmax(1 – e– e / ρ)cos(α)

kα + 1 – e– e/ρe υmaxcos(α)sinα

α (α + hθ)

(13)

where υmax is the maximum speed of the robot, andρ, k and h are three positive control parameters. Toshow the asymptotical stability of the closed-loopsystem, we consider the subsequent radially un-bounded Lyapunov function candidate,

V = 12 λλ e2 + 1

2α 2 + 12hθ 2 , λλ , h > 0 (14)

The time derivative of (14) along the trajectory of(8) yields

Fig. 1 Regulation of the mobile robot described in Polar coordi-nates

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928 Journal of the Chinese Institute of Engineers, Vol. 28, No. 6 (2005)

V = λee + αα + hθθ

= λe( – υ cosα) + α( – ω + υ sinαe ) + hθ(υ sinα

e )

= – λυ max e (1 – e– e/ρ)cos2α + α( – ω + υ sinαe

+ υ sinααe hθ)

= – λυ max e (1 – e– e/ρ)cos2α – kα 2 ≤ 0

Using Barbalat’s lemma, we conclude that limt → ∞

e = 0and lim

t → ∞α = 0. What follows shows θ → 0 as t → ∞.

From the second equation of (8), we obtain

α = – ω + υ sinαe

= – kα – 1 – e– e/ρe υmax

cosα sinαα (α + hθ)

+ υmax(1 – e– e/ρ)cosα sinαeα α

= – kα – 1 – e– e/ρe υmax

cosα sinαα

× hθ

Assume the facts e → 0 as t → ∞ and lime → 0

vmax

×(1 – exp– e/ρ)

e × cosα sinαα × hθ = lim

e → 0vmax

1ρ × cosα

. sinα × e2 θ = 0, then α is uniformly continuous,

thereby implying that limt → ∞

α → 0 and limt → ∞

α = –kα –

vmax1ρ hθ → 0. Moreover, lim

t → ∞θ = vsinα

e as t → ∞,

and this indicates that θ → θ as t → ∞. With thefact that –kα–vmax

1ρh θ → 0 and α → 0 as t → ∞, we

conclude that θ → 0 as time approaches infinity.Note that although the distance error tends to zero

e → 0 as t → ∞, this error actually remains bounded,namely, it converges to a small value ε (|ε| >0). Thefollowing theorem states the above-mentioned result.

Theorem 2. The kinematic model (8) composed of theLyapunov-based regulation law (9) is globally stabilized;the orientation error approaches zero and the distanceerror tends to a small value around the origin.

IV. PATH TRACKING CONTROL

1. Problem Formulation

The path tracking problem is concerned with thegeometrical convergence of any desired path. As

mentioned before, path tracking is regardless of thetranslational velocity of the robot, namely, it is notrequired to track the desired posture in time.

According to the requirement of approximatelyfollowing an assigned smooth path, we introduce thecurvilinear abscissa, S, along the reference path, asshown in Fig. 2., and then obtain the following errorkinematics of the WMR along the desired path

e = – υ cosα + scosθ

α = – ω + υ sinαe – ssinθ

e – sR(s)

θ = υ sinαe – ssinθ

e – sR(s)

(15)

where R(s) is the current curvature radius of the pathand s is denoted by the target velocity introduced bythe assumed motion of the target. Note that the tar-get speed s is made available for the required pathtracking task. Obviously, the target speed s must beless than or equal to the maximum speed (υmax) ofthe WMR, in order to achieve path tracking.

2. Error Analysis of Path Tracking Controllers

(i) Path Tracking with Variable Speed

By letting the target variable s equal kvcos(α)and |α| < π2 , we propose the time-varying path track-ing controller as

y

x

S

e

yc

xc xr

yr

0

Fig. 2 Path tracking of the mobile robot described in Polar coor-dinates

υω =

υmax(1 – e– e/ρ)cosα + k vcosθ

kα +υmax(1 – e– e/ρ)cosα sinα + k vcosθ sinα – ssinθ – es

R(s)eα (α + hθ)

(16)

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T. Y. Wang et al.: Nonlinear Regulation and Path Tracking of a Wheeled Mobile Robot in Polar Coordinates 929

where the parameter kv is positive. Taking the veloc-ity command υ into the time derivative of the dis-tance error e gives

e = –υmax(1 – e–e/ρ)cos2α – kvcosθcosα

+ kvcosθcosα

= –υmax(1 – e–e/ρ)cos2α (17)

Since |α| < π2 , then e → 0 as t → ∞. For studying theasymptotical stability of the overall closed-loopsystem, the following Lyapunov function is consid-ered

V = 12λe2 + 1

2α 2 + 12hθ 2 , λ , h > 0

With (15), (16), and (17), the time derivate of V be-comes

V = λee + αα + hθθ

= – λeυmax(1 – e– e/ρ)cos2α + α [ – ω + υ sinαe

– ssinθe – s

R(s)] + hθ[υ sinαe – ssinθ

e – sR(s)]

= – λeυmax(1 – e– e/ρ)cos2α – kα 2 ≤ 0

which clearly indicates that α → 0, e → 0 as t → ∞.However, with the proposed path tracking control(16), it is not easy to show that the orientation errorθ tends to zero as t → ∞. In the following two cases,we show that the orientation error θ tends to zero as t→ ∞, but the distance error can not conform to zeroas time approaches infinity.

(ii) Line Path Tracking with Constant Speed

To achieve the control goal of line path track-ing at constant speed, we modify the control laws υin (13) as follows

υω =

υmax(1 – e– e/ρ)cos(α) + k v

kα + 1 – e– e/ρe υmaxcos(α)sinα

α (α + hθ)

(18)

where the positive constant kv is added in order tomake the steady-state value of θ and α approach zero.However, the control law will result in a nonzerosteady- state error ess, which will be analyzed in thesequel.

The curvature radius R(s) equals infinity as thepath is a line. The error kinematic equations are nowreduced to the following form

e = – υ cosα + scosθα = – ω + υ sinα

e – ssinθe

θ = υ sinαe – ssinθ

e

(19)

where s = constant = s0 < υmax. The reference linepath is expressed by

θ r = θ0xr(s) = x0 + a1syr(s) = y0 + b 1s

(20)

where a1 = cosθr; b1 = sinθr. The proposed controllaw (18) is used to steer the WMR to follow the de-sired line path, but the steady state error ess is notequal to zero when s > kv > 0. However, it is easy toshow that e = 0, α = 0, θ = 0. The substitution of thecontrol velocity law υ in (18) into the time deriva-tive of the distance error e yields

e = – υmax(1 – e– e∞/ρ)cos2α∞ + (scosθ∞ – k vcosα∞)

= 0 (21)

Because cosα∞ = 1 as α∞ = 0, and θ → 0 as time ap-proaches infinity, i.e., θ∞ = 0, the steady state errorsess can be calculated by substituting α∞ = 0, θ∞ = 0into (21). Thus, one obtains υmax(1 – e–ess/ρ) – (s – kv)= 0 which leads to the steady state error ess = ρln.

υmaxυmax – (s – k v)

≥ 0. Obviously, if s is equal to kv,

then the steady state error ess becomes zero.

(iii) Circular Path Tracking with Constant Speed

For this case, the curvature radius R(s) equal a con-stant r and the error kinematic equations of (15) become

e = – υ cosα + scosθα = – ω + υ sinα

e – ssinθe – s

rθ = υ sinα

e – ssinθe – s

r

(22)

and the reference circular path is described by

θ r(s) = sr + π

2xr(s) = (x0– r) + r cosθ r

yr(s) = y0 + r sinθ r

(23)

where r is the radius of the circle, x0 and y0 are theinitial positions. With an analysis similar to that donein Section IV.2.(ii), it yields α∞ ≠ 0 and θ∞ ≠ 0.Hence, the steady state error ess by using (21) can becalculated by taking the e = 0, α = 0, θ = 0 and α∞ ≠

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930 Journal of the Chinese Institute of Engineers, Vol. 28, No. 6 (2005)

0, and θ∞ ≠ 0 into (21). Thus, we solve for ess and get

e∞ = ρlnυmax ⋅ cosα∞

υmaxcosα∞ – (s ⋅ cosθ∞ – k vcosα∞) ≠ 0 .

Note that if the curvature radius R(s) is equal to a con-stant r, then circular path tracking with constant speedresults in the steady-state position and orientation errors.

V. SIMULATIONS AND EXPERIMENTALRESULTS

1. Simulation Results and Discussion

The following subsection is devoted to examin-ing the feasibility and effectiveness of the proposedtwo regulators. With the proposed regulators, sev-eral computer simulations were performed to exploreall the possible behaviors of the error variables (e, αand θ), where the parameter υmax was 20, ρ was 20, hwas given by 10, and k was selected as 10. To exam-ine the performance of the proposed linearizationregulator, Fig. 3 shows that the position errors andheading errors both quickly converge to zero. Fig. 4depicts the simulation results of the Lyapunov-basedregulator. Through the simulation results, the pro-posed two regulators have been shown effective insteering the robot to reach its destinations.

Note that the simulation results without exter-nal disturbance seem pretty good, but in practice theremight exist some steady state errors, which may showup during the work shown in the next experimentalsubsection.

2. Experimental Results and Discussion

The experimental system was equipped with aceiling-mounted fixed camera whose output was

connected to a host computer, and a WMR with twodifferent colorful, round marks. These two markswere used to periodically provide the position and ori-entation information of the WMR for tracking the ref-erence trajectory. The dead-reckoning capability ofthe WMR was provided by two encoders mounted inthe driving wheels that providing the dead-reckoninginformation when the vision posture information wasnot available. Fig. 5 shows the system block diagramof the vision-based mobile robot. Fig. 6 depicts theexperimental results of the linearization regulatorfrom the initial posture x0 = 73 [Pixels], y0 = 4 [Pixels]and θ0 = 110 [degrees] to the final posture x0 = 54[Pixels], yr = 22 [Pixels], θr = 160 [degrees]. Fig. 7displays the experimental results of the Lyapunov-based regulation controller from the initial posturex0 = 104 [Pixels], y0 = -8 [Pixels] and θ0 = 120[degrees] to the final posture xr = 78 [Pixels], yr = 30[Pixels], θr = 180 [degrees]. In comparison to theperformance of the two proposed regulators, they arecapable of reaching the goal efficiently and quickly.

The following experiments were conducted toexamine the performance of the proposed path tracking

30

25

20

15

10

5

0

-50 5 10 15

Time

20 25

Initi

al v

alue Alpha

Theta

Error distance

(73,4,1100) --> (54,22,160)

Linearization controller

Fig. 3 Simulation result of the linearization regulator

16

14

12

10

8

6

4

2

0

-20 21 3 5 7 94 6

Time

8 10

AlphaThetaError distance

e0 = 14.8Alpha0 = 5Theta0 = 3

Regulation with Lyapunov-based controller

Fig. 4 Simulation result of the Lyapunov-based controller

Main computer withtransmitter

Reference workingfield path

WMR with wireless receiver

Fixed CCD camera

Fig. 5 Vision-based WMR control experimental system

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controller (16) for a mobile robot. Fig. 8 shows theexperimental results of the line path tracking from theinitial posture x0 = 108 [Pixels], y0 = 27 [Pixels] and θ0

= 250 [degrees] to the starting point of the line with xr

= 77 [Pixels], yr = -9 [Pixels], θr = 160 [degrees]. Fig.

30

25

20

15

10

5

0

-550 5545 6560

X coordinate frame (Pixels)

Y c

oord

inat

e fr

ame

(Pix

els)

70 75 80

Start point

(54,22,160)

Goal point

Regulation with linearization controller

(73,4,110)

Fig. 6 Experimental results of the proposed linearization regulator

40

35

30

25

20

15

10

5

0

-5

-10

-1575 8070 9585 90

X coordinate frame (Pixels)

Y c

oord

inat

e fr

ame

(Pix

els)

100 105 110

Start point

(78,30,180)

Goal point

Regulation with Lyapunov-basedcontroller

(104,-8,120)

Fig. 7 Experimental results of the proposed Lyapunov based regu-lator

35

30

25

20

15

10

5

0

-5

-10

-15

-20200 8040 60

X coordinate frame (Pixels)

Y c

oord

inat

e fr

ame

(Pix

els)

100 120

(108,27,250)

Actualpath

Initial path

Referencepath

Line path tracking

Reference pathstart point(77,-9,160)

Fig. 8 Experimental result of the proposed path tracking control-ler with line path

15

10

5

0

-5

30 35 40 45

X coordinate frame (Pixels)

Y c

oord

inat

e fr

ame

(Pix

els)

50 55

(51,0,45)Ref. start point

Reference path

Actual path

Circular path tracking Initial point

(52,-9,90)

Fig. 10 Experimental line path tracking with variable speed

Fig. 9 Experimental result of the proposed path tracking control-ler for circular path

35

30

25

20

15

10

5

0

-5

-10

-15-10 100 20 40 60 8030 50

X coordinate frame (Pixels)

Y c

oord

inat

e fr

ame

(Pix

els)

70 90

(60,18,15)Ref. start

point

Actual path

Reference path

Line path tracking withvariable speed

Initial point(84,-7,170)

9 demonstrates the experimental results of the circularpath tracking from the initial posture x0 = 52 [Pixels],y0 = -9 [Pixels] and θ0 = 90 [degrees] to the startingpoint of the circle with xr = 51 [Pixels], yr = 0 [Pixels],θr = 45 [degrees]. Fig. 10 presents the experimentalresults of the line path tracking with variable speed fromthe initial posture x0 = 84 [Pixels], y0 = -7 [Pixels] andθ0 = 170 [degrees] to the starting point of the line withxr = 60 [Pixels], yr = 18 [Pixels], θr = 15 [degrees].These results indicate that the proposed tracking con-trol (16) has been shown capable of exactly followingcircular and straight-line paths.

VI. CONCLUSIONS

This paper has proposed methodologies for pointstabilization and path tracking of a nonholonomic

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932 Journal of the Chinese Institute of Engineers, Vol. 28, No. 6 (2005)

wheeled mobile robot in polar coordinates. Two newkinematics point-to-point control laws in polar coor-dinates have been proposed in order to achieveasymptotical point stabilization, and they are appli-cable to derive the path tracking control law for awheeled mobile robot. The proposed path trackingcontroller has been used to steer the mobile robots toasymptotically follow the desired paths. For line andcircular paths, the nonzero steady-state distance track-ing errors can be calculated if the robot produces con-stant-speed motions. The experimental vision-basedmobile robot, together with the proposed control laws,has been successfully used to steer the WMR as itfollows designed reference paths. An interesting topicfor future work might be to develop the dynamic pathtracking control laws in polar coordinates.

NOMENCLATURE

A(q) ∈ Rm × n matrix related to nonholonomic con-straints

B(q) ∈ Rn × r input transformation matrixe distance error variableG(q) ∈ Rn gravitation vectorh, k, r positive control parameterskv positive constantM(q) ∈ Rn × n symmetric and positive definite in-

ertia matrixq ∈ Rn generalized posture vectorq ∈ Rn generalized velocity vectorR(s) current curvature radius of the paths curvilinear abscissaS(q) ∈ Rn × r linearly independent vectors in the

null space of A(q)V Lyapunov functionV(q, q) ∈ Rn × n centripetal and coriolis matrixx, y, xr, yr Cartesian positionsxc(t), yc(t) the position of the WMRα sight angle error variableφ heading orientation error variableγ constant deviation angle from the

Cartesian coordinates x-axisλλ constraint force vectorµµ internal state variableν, νr linear velocitiesθ orientation w.r.t. the goal frameθc(t) orientation of the WMRττ torque control input vectorυmax maximum speed of the robotω, ωr angular velocities

ACKNOWLEDGMENTS

This study was supported by the National Sci-ence Council of Taiwan, the Republic of China un-der Grant NSC 93-2213-E-005-037.

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Manuscript Received: Jun. 17, 2005and Accepted: Jul. 05, 2005

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