Tracking Control of a Mobile Robot using aNeural Dynamics based Approach∗

Guangfeng Yuan, Simon X. Yang† and Gauri S. MittalSchool of Engineering, University of Guelph

Guelph, Ontario, N1G 2W1, Canada

Abstract

In this paper, a novel tacking control approach is pro-posed for real-time navigation of a nonholonomic mo-bile robot. The proposed tracking controller is basedon the error dynamics analysis of the mobile robot anda neural dynamics model derived from Hodgkin andHuxley’s membrane model of a biological system. Thestability of the control system and the convergence oftracking errors to zeros are guaranteed by a Lyapunovstability theory. Unlike many tracking control meth-ods for mobile robot where the generated control ve-locities start with large initial velocities, the proposedneural dynamics based approach is capable of gener-ating smooth, continuous robot control signals withzero initial velocities. In addition, it can deal withthe situation with a very large tracking error. Theeffectiveness and efficiency are demonstrated by com-parison and simulation studies.

1 Introduction

Real-time tracking control of a mobile robot is a veryimportant issue in mobile robotics. Due to slippage,disturbance, noise, vehicle-terrain interaction and sen-sor errors, it is very difficulty to avoid the errors be-tween the desired and actual robot paths. How toeffectively control a mobile robot to precisely track adesired trajectory is still an open question in robotics.In a 2-dimensional (2D) Cartesian workspace, the lo-cation of a mobile robot can be uniquely determinedby three variables, the spatial position (xc, yc) of therobot center C and its orientation θc with respect toC, which is referred as a posture of the mobile robot(see Fig. 1).

There has been many studies on tracking control ofa mobile robot in recent years (e. g., [1]-[8]). The ex-∗This work was supported by Natural Sciences and Engi-

neering Research Council (NSERC) of Canada.†All correspondence should be addressed to S. X. Yang.

Email: syang@uoguelph.ca.

isting tracking control methods for a mobile robot canbe classified into five categories: (1) sliding mode [1];(2) linearization [2]; (3) backstepping [4, 5], (4) neu-ral networks [6]; (5) fuzzy systems [7]; and (6) neuro-fuzzy systems [8]. The control algorithm using slidingmode is complicated and computationally expensive.The generated velocity command with respect to timeis not a smooth curve [1], which may lead to discon-tinuousness in the robot velocities. The linearizationbased methods (e.g., [2]) requires a small initial errorbetween the target and actual robot positions. Thebackstepping based tracking controllers (e.g., [4, 5])are the most commonly used approach. They are verysimple and the system stability is guaranteed by aLyapunov stability theory. In addition, some of thebackstepping based controllers can deal with arbitrar-ily large initial error. However, the generated robotvelocity commands using those conventional controlapproaches start with a very large value, and suffersfrom velocity jumps when sudden tracking errors oc-cur, i.e., the required accelerations and forces/torquesare infinitely large at the velocity jump points, whichis not practically possible. Fierro and Lewis [3] pro-posed a novel controller based on backstepping tech-nique for a mobile robot by generating torque signalsusing computed torque control or a three-layer neu-ral network based control, which can solve the im-practical problem of large initial velocities. But bothcontrol methods are computationally complicated. Inaddition, the computed torque control required theexact robot model that mostly is not available, whilethe neural network require on-line learning in orderto make the robot perform properly. Recently Zhanget al. [4] proposed a controller based on backsteppingand neural network, where the backstepping is used fortracking control, while the neural network is for com-pensating the robot dynamics. However, the mobilerobot also starts with a very large initial velocity, andthe algorithm is computationally expensive. The fuzzyrules based tracking control approaches (e.g., [7, 8])can solve the problem of large initial robot velocities,

but it is very difficulty to formulate the fuzzy rules,which are usually obtained by trial and error. Theexisting neural networks based tracking control algo-rithms (e.g., [3, 6, 8]) for a mobile robot require eitheron-line and/or off-line training procedures before thecontrollers are capable of controlling the robot prop-erly, which also add computational cost.

In this paper, inspired by the unique features of theneural dynamics in Hodgkin and Huxley’s membranemodel [9] for a biological neural system, and based onthe error dynamics analysis of a mobile robot that issimilar to the integrator backstepping, a novel track-ing controller is proposed for real-time navigation of amobile robot. The control signal consists of a compo-nent from a neural dynamics model, and a componentfrom the error dynamics that is similar to a propor-tional (P) control part. The stability of the controlsystem and the asymptotical convergence of trackingerrors to zero are rigorously proved using a Lyapunovstability analysis. Distinct from the previous neuralnetworks based approaches, no learning procedure isneeded in the proposed control algorithm. The pro-posed controller is capable of generating smooth, con-tinuous control signals with zero initial robot veloc-ities. In addition, it can deal with arbitrarily largetracking errors between the target and current robotpostures. To the best of our knowledge, it is the firsttime that a neural dynamics based model is proposedfor tracking control of a mobile robot, where no learn-ing is needed.

2 Model Algorithm

2.1 Nonholonomic Mobile Robot andits Error Dynamics

In a 2D workspace, a nonholonomic mobile robot canbe described by two coordinate systems: the worldcoordinate system {X, 0, Y } and the local coordinatesystem {D,C,L}, where D is the driving direction(longitudinal direction), L is the lateral direction (lat-itudinal direction), and C is the robot center point.The world coordinate system is fixed to the Cartesianworkspace and the local coordinate system is attachedto the mobile platform. A robot posture in the worldcoordinate system can be uniquely determined by avector Pc = [xc yc θc]T , where (xc, yc) denotes thespatial position of the robot center C, and θc is therobot orientation angle with respect to C (see Fig. 1).

A freely movable mobile robot that is referred asholonomic mobile robot has three degrees of freedom(d.o.f.), xc, yc, and θc. However, because of the kine-

X

Y

D

L

0

C

θc

xc

yc

wc

vc

Figure 1: Model of a nonholonomic mobile robot

matical constraint, the degrees of freedom for a non-holonomic mobile robot reduces to two. On the con-ditions of non-slipping, the kinematic constraint of anonholonomic mobile robot is given as

yc cos θc − xc sin θc = 0. (1)

From the motion control perspective, a mobile robothas 2-d.o.f., vc and wc, where vc is the linear ve-locity and wc is the angular velocity of the mobilerobot. For a nonholonomic mobile robot, the relation-ship between velocity in the world coordinate systemPc = [xc yc θc]T and velocity v = [vc wc]T in the lo-cal coordinate system can be described by a Jacobianmatrix as xc

ycθc

= Pc =

cos θc 0sin θc 0

0 1

[ vcwc

]. (2)

The reference path of a nonholonomic mobile robotprovides the target robot posture Pd = [xd yd θd]

T . inthe world coordinate system. The tracking error in thelocal coordinate system is defined as Ep = [eD eL eθ]T ,where eD, eL and eθ are the errors in the driving di-rection, lateral direction and the orientation, respec-tively. The relationship between the tracking errorsin the world and local coordinate systems can be ob-tained by geometrical projection transformation as

Ep=

eDeLeθ

=

cos θc sin θc 0− sin θc cos θc 0

0 0 1

xd − xcyd − ycθd − θc

. (3)

The error dynamics of the mobile robot can be derivedfrom the time derivative of the above posture errorequation in (3) as eD

eLeθ

=

wceL − vc + vd cos eθ−wceD + vd sin eθ

wd − wc

. (4)

2.2 Tracking Control Problem

The function of a tracking controller in this paper isto implement a mapping between the known infor-mation (e.g., the desired information and the sensoryinformation) and the velocity commands designed toachieve the robot’s task. The controller design prob-lem can be described as: given the desired robot pos-ture Pd(t) = [xd(t) yd(t) θd(t)]T and the desired ve-locities vd(t) and wd(t), i.e., the desired state of themobile robot defined as

Xd(t) = [xd(t) yd(t) θd(t) vd(t) wd(t)]T, (5)

design a control law for the linear velocity vc and an-gular velocity wc, which drive the robot to move, suchthat the actual robot state

Xc(t) = [xc(t) yc(t) θc(t) vc(t) wc(t)]T, (6)

will precisely tracking the desired robot state Xd(t),

limt→∞

Xc(t) = Xd(t), (7)

i.e., the tracking error converges to zero while timeapproaches infinitely.

2.3 The Proposed Controller

The biological neural model has a certain propertythat can be used to solve the problem of sudden speedjumps. A typical shunting model derived from theHodgkin and Hulexy’s [9] membrane model is given as

dyidt

= −Aiyi + (Bi − yi)Sei (t)− (Di + yi)Sii(t), (8)

where i is the neuron index, yi is the membrane po-tential, Ai represents the passive decay rate, Bi andDi are the upper and lower bounds of the membranepotential, and Sei and Sii are excitatory and inhibitoryinputs to the ith neuron.

By analyzing the backstepping technique basedtracking controller proposed in [3, 4, 5] the sharp speedjumps are caused by the suddenly changes in trackingerrors. So a biological inspired tracking controller isproposed to solve the problem. Substituting Ai = A,Bi = B, Di = D, yi = vs S

+i = f(eD), S−i = g(eD) in

Eq. (8), a velocity dynamics equation with respect tothe error in the driving direction is obtained as

dvsdt

= −Avs + (B − vs)f(eD)− (D + vs)g(eD), (9)

where vs is the velocity component that will be usedin tracking controller design, A is the passive rate of

the velocity, B and D are the upper boundary andlower boundary of the velocity, respectively, Functionsf(x) is a linear-above-threshold function defined asf(x) = max {x, 0} and the non-linear function g(x) isdefined as g(x) = max {−x, 0}. The proposed trackingcontrol law for the linear and angular velocities aregiven as

vc = vs + vd cos eθ, (10)wc = wd + k2vdeL + k3vd sin eθ, (11)

where the k2 and k3 are positive parameters. At thestart period, we need that the speed of the mobilerobot increases exponentially and reaches the desiredspeed v0. According to the time response propertyof the first order system, the reference velocity canbe defined as vd(t) = v0(1 − e−t/τ ), where v0 is thedesired speed, τ is the time constant.

When the proposed tracking controller is used to-gether with a path planner, the output of the pathplanner is the desired robot posture Pd. The currentposture can be obtained from measurement. The sys-tem architecture of the proposed tracking controlleris shown in Fig. 2. The error vector Ep in the localcoordinate systems is obtained through a a transfor-mation matrix Te from the posture error between thecurrent and desired postures in the world coordinatesystem. The input of the path tracker is the error vec-tor and the desired velocities. The output of the pathtracker is the steering commands in linear velocity vcand angular velocity wc.

Motion

Planner

Path

TrackerTe

[vd wd]T

Σ Mobile

Robot-+

Pc=[ xc

] yc

θc

Pd=[ xd

] yd

θd [ vc ] wc

Ep

Figure 2: System architecture of the proposed trackingcontroller

2.4 Stability Analysis

The tracking control system proposed in this paperis asymptotically stable, and the tracking errors con-verge to zeros. In the shunting equation (9), the exci-tatory input item (B − vs)f(eD) forces the output ofthe shunting model to stay below the upper bound Bwhile the inhibitory input item (D+vs)g(eD) guaran-tees that the output of the shunting model stay above

lower bound −D. Therefore, the output of the shunt-ing model is bounded in the finite interval [−D,B].To rigorously prove the asymptotical stability and theconvergence of the proposed tracking control system,a Lyapunov function candidate is chosen as,

V (t) =12

(e2D + e2

L) +1k2

(1− cos eθ)

+12k

(e2v + e2

w) +1

2Bv2s , (12)

where vs is the velocity component defined in Eq. (9),and ev and ew are the auxiliary velocity errors thatwill be defined later in this section. It is obvious thatV (t) = 0 if and only if eD = 0, eL = 0, eθ = 0, ev =0, ew = 0, and vs = 0.

From Eq.s (4), (9), (10), and(11), the time deriva-tive of the Lyapunov function L becomes

V (t) = eDeD + eV eV +1k2eθ sin eθ

+1k

(evev + ewew) +1Bvsvs

= −vseD −k3

k2vd sin2 eθ +

1k

(evev + ewew)

+1B

[−A− f(eD)− g(eD)] v2s

+1B

[Bf(eD)−Dg(eD)] vs, (13)

If we choose the constants B = D in the shuntingequation, Eq. (13) can be rewritten as

V (t) = −k3

k2vd sin2 eθ +

1k

(evev + ewew)

+1B

[−A− f(eD)− g(eD)] v2s

+ [f(eD)− g(eD)− eD] vs. (14)

From to the definition of f(eD) and g(eD), if eD ≥ 0,then f(eD) = eD and g(eD) = 0. Thus we have,

[f(eD)− g(eD)− eD] vs = [eD − 0− eD] = 0. (15)

Similarly, if eD < 0, then f(eD) = 0 and g(eD) =−eD. Thus we have

[f(eD)− g(eD)− eD] vs = [0− (−eD)− eD] = 0.(16)

Therefore, Eq. (14) can be rewritten as

V (t) = −k3

k2vd sin2 eθ +

1k

(evev + ewew)

+1B

[−A− f(eD)− g(eD)] v2s (17)

To prove the convergence of the robot velocities,the auxiliary velocity error variables, ev and ew in Eq.(12), are define as

ev = vc − vd, (18)ew = wc − wd. (19)

Considering the nonlinear feedback acceleration con-trol input as

av = vd + k(vd − vc), (20)aw = wd + k(wd − wc), (21)

where k is a positive parameter, and av and aw are theactual accelerations of the robot, i.e., namely, av = vc,aw = wc. Note that if the second term on the righthand of Eq.s (20) and (21) is crossed out, the controllaw is usually called perfect velocity tracking, which isnot possible for a real robot. By doing time-derivativeon both sides of Eq.s (18) and (19), and using Eq.s(20) and (21), we have

ev = vc − vd = av − vd = −kev, (22)ew = wc − wd = aw − wd = −kew. (23)

By substituting Eq. (22) and (23) into Eq. (17), wehave

V = −k3

k2vd sin2 eθ − (e2

v + e2w)

+1B

[−A− f(eD)− g(eD)] v2s . (24)

Obviously we have −(e2v + e2

w) ≤ 0. Since param-eters k2 and k3 are positive numbers, and the de-sired linear velocity vd is a positive constant, wehave −(k3/k2)vd sin2 eθ ≤ 0. From the definitionof functions f(eD) and g(eD), we have f(eD) ≥ 0and g(eD) ≥ 0. In addition, the parameters Aand B are nonnegative constants. Thus we can[−A− f(eD)− g(eD)] v2

s/B ≤ 0. Therefore, the timederivative of the Lyapunov function candidate V inEq. (24) along all the system trajectories is not largerthan zero, i.e. V ≤ 0. The proposed tracking controlsystem for a mobile robot is stable.

The velocity error dynamic equations for mobilerobot satisfy the following relationship: vc → vd,wc → wd as t → ∞. Obviously, by assuming thatvd > 0, V (t) ≤ 0 and the entire error Ep, ev, ew, vs arebounded. From Eq.s (4), (18), (19) and V (t) ≤ 0,we can infer that ‖Ep‖, ‖Ep‖, ‖ev‖, ‖ev‖, ‖ew‖,‖ew‖, ‖vs‖ and ‖vs‖ are bounded. Thus we have‖V (t)‖ < ∞. Since V (t) does not increase andconverges to certain constant value. By Barbalat’s

lemma, V (t) → 0 as t → ∞, from which we can de-duce that ev → 0, ew → 0, vs → 0 as t→∞. By usingEq. (9) and the input-output property of the shuntingmodel, we can infer that if output converges to someconstant value (zero), the input is supposed to go toa constant value (zero), namely, eD → 0 as vs → 0.From the first term in Eq. (24), k3

k2vd sin2 eθ = 0, we

have eθ → 0 as t → ∞. From Eq. (4), as eθ → 0,we have wd − wc = 0. From Eq. (11), ew → 0as t → ∞, thus k2vdev = 0. From the assumptionvd > 0, we have eV → 0 as t → ∞. Thus the equi-librium point is Ep = 0, ev = 0, ew = 0. Therefore,the tracking control system is asymptotically stable.From the Jacobean transformation in Eq. (2), it isobvious that xd − xc = 0 and yd − yc = 0 from theresults: eD = 0andeL = 0. Therefore, we have

limt→∞

(Xd(t)−Xc(t)) = 0 (25)

Therefore, the proposed control algorithm is asymp-totically stable and the tracking errors are guaranteedto converge to zeros.

3 Simulation Studies

3.1 Tracking a Straight Path

The desired robot path is a straight line described asy = 5 and x = 0; the robot starts at (0, 3.2, 0). Thedesired cart is supposed to forward speed set out atpoint (0, 5, 0). So the initial error is (0, 1.8, 0). Timevaries from 0 to 10s; the parameters are chosen as:k2 = 5, k3 = 2, A = 5, B = 3, D = 3, vd = 1, wd =0. During the beginning period, the parameters usedin the first order exponent equation for linear speed areas: k = 1, τ = 0.5. The sampling time is 0.01 second.The actual path is denoted with dash dot line and thedesired path is the solid line as shown in Fig. 3A. Thesmooth and reasonable variation of speeds is demon-strated in Fig. 3B. The errors of longitude, lateral andorientation converge to zero as time approaches to in-finity shown in Fig. 3C. Fig. 3D shows the change ofthe orientation of the mobile robot.

3.2 Tracking a Circular Path

Consider a circular path, x2 + y2 = 1, as shown inFig. 4, and assume that the desired linear and angularvelocities are 1 and 1 respectively. The desired vir-tual car proceeds at such velocities and sets out frompoint (1, 0, 0). The actual robot starts at point (0,-.6, 0) . this means that the initial error is (-1, -0.6, 0).During the beginning period, the parameters used in

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

X(m)

Y(m

)

A

0 1 2 3 4 5 6 7 8 9 101

0.5

0

0.5

1

1.5

2

2.5

3

Time(s)

Vel

ocity

(m

/s a

nd r

ad/s

)

BAngular Speed

Forward Speed

0 1 2 3 4 5 6 7 8 9 101.5

1

0.5

0

0.5

1

1.5

2

Time(s)

Err

or (

m a

nd r

ad.)

CLogitudinal Error

Lateral Error

Orientational Error

0 1 2 3 4 5 6 7 8 9 100.2

0

0.2

0.4

0.6

0.8

1

1.2

Time(s)

Ori

enta

tion(

rad)

D

Figure 3: Tracking a straight path. A: dynamic track-ing performance; B: The generated linear and angu-lar velocities; C: The tracking errors in the longitudi-nal and lateral directions, and in orientation; D: Thevarying orientation of the mobile robot.

the first order exponent equation for linear speed are:k = 1, τ = 0.05.The parameters used in the controllerare: A = 5, B = 3, D = 3, K2 = 2, k3 = 4. the sam-pling time is equal to 0.01 second. As illustrated inFig. 4A, the robot chooses a perfect angularity to trackthe circular path. Fig. 4B shows the smooth and rea-sonable velocity curves versus time. The speeds firstincrease and then track the desired speeds. The track-ing errors approach to zero as time goes to infinityshown in Fig. 4C, Fig. 4D shows that the orientationincreases until it equals 2π and then increases from0. The robot advances about more than one and halfcircles shown in Fig. 4D during 10 seconds.

3.3 Comparison to a ConventionalBackstepping Model

In this section, the backstepping technique based con-trol law is compared to the biological neural networkbased tracking controller proposed in this paper. Aswe can see in the following equation (26), the suddenlychanging errors result in the speed jump as shown inthe Fig. 5B . The backstepping technique based track-ing control law used by [3] is given as[

vcwc

]=[

c1eD + vd cos eθwd + c2vdeL + c3vd sin eθ

], (26)

where c1, c2, and c3 are the parameters. The trackingcontrol law defined in Eq. (26) is simulated here usingthe same parameters as the controller proposed in this

1.5 1 0.5 0 0.5 1 1.5 2 2.5 3 3.51

0.5

0

0.5

1

1.5

2

2.5

3

X(m)

Y(m

)

A

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

Time(s)V

eloc

ity(m

/s a

nd r

ad/s

)

B

Angular Speed

Forward Speed

0 1 2 3 4 5 6 7 8 9 100.2

0

0.2

0.4

0.6

0.8

1

1.2

Time(s)

Err

or (

m)

C

Longitudinal Error

Lateral Error

Orientational Error

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

Time(s)

Ori

enta

tion

(rad

)

D

Figure 4: Tracking a circular path. A: The dynamictracking performance; B: The generated linear and an-gular velocities; C: The tracking errors in the longitu-dinal and lateral directions, and in orientation; D: Thevarying orientation of the mobile robot.

paper (c1 = 10, c2 = 2, c3 = 4). The Fig. 5 shows thatthe results of the tracking controller defined by Eq.(26). Fig. (5B) demonstrates that the speed changessuddenly rather than gradually from zero when timeequals zero. So this does not hold in practice. In com-parison to Fig. (4B), the proposed path tracking con-troller can obtain a reasonable results. What’ more,the proposed controller in this paper tracks the cir-cle path quicker than the controller proposed in [3]bycomparison of Fig. (4A) and Fig. (5A).

4 Conclusions

In this paper, a novel neural dynamics based trackingcontroller is proposed, which is capable of generatingsmooth and continuous velocity commands with zeroinitial value. The tracking control system is asymp-totically stable, and the tracking errors are guaranteedto converge to zeros. The proposed controller resolvesthe problem of sharp speed jumps at beginning andwhen sudden tracking errors occur.

References

[1] J. M. Yang and J. H. Kim: Sliding mode control fortrajectory tracking of nonholonomic wheeled mobilerobots. IEEE Trans. on Robotics and Automation, 15(3): 578-587, 1999.

1.5 1 0.5 0 0.5 1 1.5 2 2.5 3 3.51

0.5

0

0.5

1

1.5

2

2.5

3

X(m)

Y(m

)

A

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Time(s)

Vel

ocity

(m

/s a

nd r

ad/s

)

B

Forward Speed

Angular Speed

0 1 2 3 4 5 6 7 8 9 100.2

0

0.2

0.4

0.6

0.8

1

1.2

Time(s)

Err

or(m

)

C

Longitudinal Error

Lateral Error

Orientational Error0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

Time(s)

Ori

enta

tion(

rad)

D

Figure 5: Tracking a straight path with a conventionalbackstepping controller. A: The dynamic tracking per-formance; B: The generated linear and angular veloc-ities; C: The tracking errors in the longitudinal andlateral directions, and in orientation; D: The varyingorientation of the mobile robot.

[2] D. H. Kim and J. H. Oh: Tracking control of a two-wheeled mobile robot using input-output linearization.Control Engineering Practice, 7 (3): 369-373, 1999.

[3] R. Fierro and F. V. Lewis: Control of a nonholonomicmobile robot: backstepping kinematics into dynamics.J. of Robotic Systems, 14 (3): 149-163, 1997.

[4] Q. Zhang, J. Shippen and B. Jones: Robust backstep-ping and neural network control of a low-quality non-holonomic mobile robot. Intl. J. of Machine Tools andManufacture, 39 (7): 1117-1134, 1999.

[5] M. Ahmadi, V. Polotski and R. Hurteau: Path Track-ing Control of Tracked Vehicles. Proc. of IEEE Intl.Conf. on Robotics and Automation, San Francisco,USA, pp. 2938-2943, 2000.

[6] V. Boquete, R. Garcia, R. Barea and M. Mazo: Neuralcontrol of the movements of a wheelchair. J. of Intel-ligent and Robotic Systems, 25 (3): 213-26, 1999.

[7] A. Ollero, A. G. Cerezo and J. V. Martinez: Fuzzy su-pervisory path tracking of mobile robots. Control En-gineering Practice, 2 (2): 313-19, 1994.

[8] K. C. Ng and M. M. Trivedi: A neural-fuzzy controllerfor real-time mobile robot navigation. Proc. of theSPIE - The Intl. Soc. for Optical Engineering, 2761:172-183, 1996.

[9] A. V. Hodgkin and A. F. Huxley: A quantitative de-scription of membrane current and its application toconduction and excitation in nerve. J. of PhysiologyLondon, 117: 500-554, 1952.