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On Feedback Linearization of Mobile Robots

MS-CIS-92-45GRASP LAB 32

Xiaoping YunYosl~ioYamamot o

University of PennsylvaniaSchool of Engineering and Applied ScienceComputer and Information Science Department

Philadelphia PA 19104-6389

June 1992


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On Feedback Linearization of Mobile Robots

Xiaoping Yun and Yoshio YamamotoGeneral Robotics and Active Sensory Perception GRASP) Laboratory

University of Pennsylvania3401 Walnut Street, Room 301CPhiladelphia, PA 19104-6228

BSTR CTA wheeled mobile robot is subject to both holonomic and nonholonomic con-straints. Representing th e motion and constraint equations in the st ate space,this p ape r studies th e feedback linearization of th e dyna mic system of awheeled mobile robot. Th e main results of the pape r are: 1) It is shownthat the system is not input-state l inearizable. 2) If the coordinates of apoint on the wheel axis are taken as the output equation, the system is notinput-output l inearizable by using a static state feedback; 3 ) but is input-output linearizable by using a dynamic sta te feedback. 4 ) If the coordinatesof a reference point in front of th e mobile robot a re chosen as th e ou tp ut equ a-t ion, th e system is inpu t-ou tpu t l inearizable by using a stat ic sta te feedback.5) The internal motion of the mobile robot when the reference point movesforward is asymptotically stable whereas the internal motion when the refer-

ence point moves backward is un stab le. A nonlinear feedback is derived foreach case where the feedback linearization is possible.

This work is in pa.rt supported by NSF Grants CISE/CDA-90-2253, and CISE/CDA 88-22719, Navy Grant N0014-88-K-0630, NATO Grant CRG 911041, AFOSR Grants 88-0244and 88-0296, Army/DA AL 03-89-C-0031PR1, an d th e University of P ennsylvania R esearchFoundation.


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IntroductionThe feedback linearization of nonlinear systems has been extensively studied in the liter-ature [I, 2 3 4, 51 Broadly speaking, there are two types of linearization: input-sta telinearization and input -output linearization. Necessary and sufficient conditions have beenestablished for each type of linearization [6, 71. For a given nonlinear system, these condi-tions can be checked to determine if the system is linearizable. Two types of feedback arecommonly employed for the purpose of linearization: sta tic sta te feedback and dynamicsta te feedback. Th e dynamic state feedback is more general and includes the s tatic s tat efeedback as a special case. Consequently, the conditions for the dynamic s ta te feedback aremore complicated.

In this paper , we study the feedback linearization of a wheeled mobile robot. Due to thefact that the wheeled rnobile robot is nonholonomically constrained, the wheeled mobilerobot possesses a number of distinguishing properties as far as the feedback linearization isconcerned. In particular, we will first show that the dynamic system of a wheeled mobilerobot is not input-sta te linearizable. We then study the input-output linearization of thesystem for two types of output equations which a re chosen for the trajectory tracking ofthe mobile robot. The first output takes the coordinates of the center point on the wheelaxis, and the other output takes the coordinates of a reference point in front of the mo-bile robot. With the first output equation, we should that the system is not input-outputlinearizable by using a static state feedback but is input-output linearizable by using adynamic st ate feedback. The dynamic feedback achieving the input -output linearizationis constructed following the dynamic extension algorithm [7, 81. With the second type ofoutput equation, the system is input-output linearizable by simply using a stat ic sta te feed-back. Nevertheless, the internal dynamics of the system is not always stable. Specifically,when the reference point is controlled to move backward, the internal motion of the systemis unstable.

Although motion planning of mobile robots have been an active topic in robotics inthe past decade [9 10, 11, 12, 131, the study on the feedback control of mobile robots isvery recent [14, 15, 161. The work which is most closely related to the present study is byd Andrea-Novel e t al. [17] who studied full linearization of wheeled mobile robots. Sincethey used a reduced model, the motions of mobile robots are not completely characterized.In particular, the nonlinear internal dynamics, which are a major topic of this study, areexcluded from the motion equations. Bloch and McClamroch [18] showed that a nonholo-nomic system, including wheeled mobile robot systems, cannot be stabilized to a singleequilibrium point by a sniooth feedback. Walsh e t al [I91 suggested a control law to sta-bilize the nonholonomic system about a trajectory, instead of a point. Other relevant workincludes [20 211 which proved that systems with nonholonomic constraints are small-timelocally controllable.


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hewheel xis

. _ . . ._. . .he xis of symmetry . >

. . . .

he xis of symmetry ._. . .

Figure 1: Schematic of the mobile robot.

2 Dynamics of Wheeled Mobile Robot2 1 Constraint EquationsIn this section, we derive the motion equations and constraint equations of a wheeled mobilerobot whose schematic top view is shown in Figure 1. We assume that the mobile robotis driven by two independent wheels and supported by four passive wheels at the cornersnot shown in Figure 1). Before proceeding, let us fix some notations see Figure 1).

I :cm :

m :I :

the displacement from each of the driving wheels to the axis of symmetry.the displacement from point Po to the mass center of t he mobile robot,which is assumed to be on the axis of symmetry.the radius of the driving wheels.r / 2 b .the mass of the mobile robot without the driving wheels and the rotors ofthe motors.the mass of each driving wheel plus the rotor of its motor.the moment of inertia of the mobile robot without the driving wheels andthe rotors of the motors about a vertical axis through the intersection ofthe axis of symmetry with the driving wheel axis.the moment of inertia of each driving wheel and the motor rotor about thewheel axis.the moment of inertia of each driving wheel and the motor rotor about awheel diameter.

There are three constraints. The first one is that the mobile robot can not move inlateral direction, i e.

i os x sin 1)


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where (x l, x2) s the coordinates of point Po in the fixed reference coordinated frame XI-X2,and 4 is the heading angle of the mobile robot measured from xl-axis. Th e other twoconstraints are that the two driving wheels roll and do not slip:

l cos + k2 sin + b$ = r01i os + k2 sin = r02

where O and O2 are the angular positions of the two driving wheels, respectively.Let the generalized coordinates of the mobile robot be q = (xl, x2, , 01, 02). Th e three

constraints can be written as follows

wherein 4 cos 4 0 0 0-cos4 -s in -b r 0os -sin 4 b 0 r

We define a 5 x 2 dimensional matrix as follows

The two independent columns,of matrix S(q)are in th e null space of matrix A q ) , hat is,A q)S q)= 0. We define a distribution spanned by the columns of S (q )

S(q>= Is1(9> 2 q ) l =

The involutivity of the distribution A determines the number of holonomic or nonholonomicconstraints [21]. If A is involutive, from the Frobenius theorem [22], all the constraints a reintegrable ( thus holonomic). If the smallest involutive distribution containing A (denotedby A* ) spans the entire 5-dimensional space, all the constraints are nonholonomic. Ifdim(A*)= k hen k constraints a re holonomic and t he others are nonholonomic.

To verify the involutivity of A, we compute the Lie bracket of sl(q) and s2(q).

- cb cos 4 cb cos 4cb sin cb sin

C -C1 00 1 -

-rc sin 4


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which is not in the distribution A spanned by sl(q) and s2(q) . Therefore, at least one ofth e constra ints is nonholonomic. We continue to compute th e Lie bracket of sl(q) and s ~ ( Q )

r -rc2 COS 1

which is linearly independent of sl (q ), s2(q), nd s3(q). However, the distribution spannedby s l y ) , s2(q), 3(q) and s4(q) is involutive. Therefore, we have

It follows tha t, among th e three constraints, two of them are nonholonomic and the thirdone is holonomic. To obtain the holonomic constraint, we subtr act equation (2 ) fromequation 3) .

264 r 8, e l (8)Integrating th e above equation and properly choosing the initial condition of 4, O and 01,we have 4 ~ ( 0 , 1 9)which is clearly a holonomic constraint equation. Thus may be eliminated from thegeneralized coordinates. Th e new generalized coordinates are 4-dimensional, which will bedenoted by y again.

The two nonholonomic constraints arei 1 s i n ~ - i 2 c o s ~ 0il os 4 i in 4 cb( 8 2 )

where cb as defined early. The second nonholonomic constraint equation in the aboveis obtained by adding equations 2 ) and 3) . It is understood that 4 is now a short-handnotation for c(O1 02) rather than an independent variable. We write these two constraintequations in matrix form

A q)Q 0 (13)where is now defined in equation 10) and A(q) is given below


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2 2 Dynamic EquationsWe use the Lagrange formulation to establish equations of motion for the mobile robot.The tota l kinetic energy of the mobile base and the two wheels is

1 1 1I< -m(i: +i: +mCcd(J1 2 ) i 2 cos 1 sin ) + i ~w B: + 8; + 2 ~ ~ 2 ( B 1 2 ) 2 (15)where

Lagrange equations of motion for the nonholonomic mobile robot system are governedby 23

where q; is the generalized coordinate defined in equation (10)) f s the generalized force,a j is from the constraint equation (14), and X1 and X2 are the Lagrange multipliers. Sub-stituting the total kinetic energy (equation (15)) into equation (16), we obtain

m i l m,d($ sin + d2cos ) Xl sin + A 2 cos (17)mi 2+ m, d( $c os $- ~2 si n ) -X1cos++X2sin+ (18)

m,cd(i2 cos .1 sin ) + (Ic2+ 1~)01 c2& T cbX2 (19)-m,cd(i2 cos il in ) ~ ~ B ~(Ic2+1, ~~ 72 cbA2 (20)

where and T~ are the torques acting on the two wheels. These equations can be writtenin the matrix form

M(q)ir'+ V q74.1 E(q)7 AT(q)X (21)where A(q) is defined in equation (14) and

0 -m,cd sin m,cd sin 10 m m,cd cos -m,cd cosM q ) -meed sin 4 m,cd cos 4 Ic2+ I, c21 mccd sin 4 -mccdcos - I C ~ I c 2 + I W 1V q74.

-m,dd2 cos --m,dd2 sin q

00

0 0


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2 3 State Space RealizationIn this subsection, we establish a sta te space realization of the motion equation 21) andconstraint equation 13). Let S q)be a 4 x 2 matrix

cb sin cb sin q0

whose columns are in the null space of A q) matrix in the constraint equation 13), i.e.A q)S q) 0. From the constraint equation 13), the velocity q must be in the null spaceof A q). It follows that q E span{sl q), sz q)),and that there exists a smooth vectorq [ql 772]Tsuch that

S q)rl 23)and

S q)i 24)For the specific choice of S q) matrix in eqation 22), we have q 1,where 0 [jl j21T.

Now multiplying the both sides of equat ion 21) by ST q)nd noticing th at s ( ~ ) A ~ ( ~ )0 and ST q)E q) 2X2the 2 x 2 identity matrix), we obtain

Substituting equation 24) into the above equation, we have

By choosing th e following st at e variable

we may represent the motion equation 26) in the stat e space form

where

It is noted that the dependent variables for each term have been omitted in the aboverepresentation for cla,rity. All the term s are functions of th e st ate variable only. Since qis not part of th e sta ,te variable, it is replaced by S q) q.


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3 Input-State LinearizationIn this section, we study the input-state linearization of the control system 28) usingsmooth nonlinear feedbacks. To simplify the discussion, we first apply the following sta tefeedback

where r is the new input variable. The closed-loop system becomes; f (x) gl( x)p (30)

where

Theorem 1 S y s t e m 30) i s not inpu t s t a t e l i near i zable by a sm oo th s ta t e f eedback .Proof: If the system is input-sta te linearizable, it has to satisfy two conditions : thestrong accessibility condition and the involutivity condition [7, p.1791. We will show thatthe system does not satisfy the illvolutivity condition.

Define a sequence of distributions

Then the involutivity condition requires that the distributions Dl , D2 , D6 be allinvolutive, with 6 being the dimension of the system. Dl ~ ~ a n { ~ l )s involutive since g1is constant. Next we compute

It s easy to verify tha t the distribution spanned by the columns of S (q) is not involutive.(Actually, if the distribution were involutive, the two constraints (11) and (12) wouldbe holonomic.) It follows that the distribution D2 ~ ~ a n { ~ l ,jlgl) is not involutive.Therefore, the system is not input-state linearizable.Corollary 1 S y s t e m 28) i s n ot in put s ta t e l i near izable by a s mo oth s t a t e feedback .Proof: A proof similar to that of Theorem 1 can be carried out. Alternatively, system(30) can be regarded as a special case of system 28) .


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4 Input Output Linearization nd DecouplingAlthough the dynamic system of a wheeled mobile robot is not input-sta te linearizable asshown in the previous section, it may be input-output linearizable. In this section, westudy the input-output linearization of two types of outputs . First, the coordinates ofthe center point Po are chosen as the output equation. It will be shown tha t t he input-output linearization is not possible by using stat ic sta te feedback, but is possible by usinga dynamic stat e feedback. Second, the coordinates of a reference point P, in front of themobile robot is chosen as the output equation. In this case, the input-output linearizationcan be achieved by using a stat ic sta te feedback. Nevertheless, the internal dynamics whenthe mobile robot moves backwards is unstable.

4 1 Controlling the Center Point oSince the mobile robot has two inputs, we may choose an output equation with two inde-pendent components. A natural choice for the outpu t equation is the coordinates of thecenter point Po, . e .

Together with this output equation, we will consider the st at e equation 30), assuming thatthe nonlinear feedback 29) is applied to cancel the dynamic nonlinearity. To verify if thesystem is input-output linearizable, we compute the time derivatives of y .

where cb cos cb cosS1 x) cb sin cb sinSince jl is not a function of the input p we differentiate once more.

where the second term on the right-hand side is evaluated to be

in~ 1 i x ) r c2b v: 7:) cos ]Now that ij is a function of the input p the decoupling matrix of the system is Sl x). SinceSl x) s singular, the system is not input-output linearizable and the output can not bedecoupled by using any stat ic s ta te feedback [6, 14, 151.


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4 2 Dynamic Feedback ControlAs shown above, the mobile robot under th e ou tpu t equation 31) is not inp ut-ou tputlinearizable with a ny s tat ic feedback of t he form

Nevertheless th e inpu t-outp ut linearization may be achieved by using a dyna mic feedbackof t h e fo rm [7, 24, 2 5, 26, 81

We follow th e dyn amic extension algorithm [7, pp.258-2691 to derive fE ., , g t . , . , a . , ,and P . , if they exist at all . We divide the algorithm in three steps.Step 1: Since th e rank of th e decoupling matri x S l x ) in equation 32) is one, we f irstapply a s tatic feedback to linea,r ize an d decouple one outpu t from th e others. For themobile robot , there are two outpu ts y [yl y 2 ] T We choose to linearize y and decouplei t f rom y2 Sub stituti ng the following static feedback into equation 32)

the closed-loop input-output map is then

It is clear that iil ul , tha t is , the f ir st ou t put y l is linearized and controlled only by ulThu s ul can b e designed to a,chieve the performance requirements for y l On the o therhand , y2 is still nonlinear. Fu rth er, it is also driven by ul .

Step 2: We subst i tu te the s ta t ic feedback 36) in to equat ion 30) to obtain the new s t ateequat ion


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Figure 2: Dynamic feedback controller of a mobile robot.We now differentiate the second output with respect to the new state equation

f 2 ( x )+ 9 2 ( x ) ~ ,oping that u2 will appear in the derivative of y2. In the following differ-entiation, is treated as a (time-varying) parameter.

v uba3(x,s)+ (X,~)VC a?x)+ x)u k

jl2 = cb(71+ 772) sin2 1

= c I + tan u l(3' = 3 2 2 sin q u12 b( 72 ) ( 1 ) 1 2 ) ~+I 7 2 ) - 4cos b

+ cos 7:) tan + G+ tan ti1+ 2c2b(v1+ v 2 ) u2cos d

a (@+ b l x ) ~ x) ~ x ) T

It is seen that u2 appears in the third-order derivative of y2. We note that IJ?) as thefollowing structure YP = Q l x )+Q ~ ( x ) u I +3Gl + Q4 2 (40)where Q ; ( x )can be easily identified.

Step : Noting equation 40 ) , y2 will be linearized if we apply th e following feedback

T-+

with v being the reference input. However, this feedback depends on u l , which can beeliminated by introducing an integrator on th e first input channel. Formally, we utilize the

X


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following dynamic feedback

where is one-dimensional and

After applying the above dynamic feedback we finally obtain two linearized and decoupledsubsystems:

It is noted tha t the first subsystem is now of th ird order due to the introduction of theintegrator on its input channel. This concludes the dynamic extension algorithm. Theresulting extended system hence is decouplable with static state feedback.The overall dynamic feedback control of the mobile robot is depicted in Figure 2. Thefirst feedback 29) is to cancel the dynamic nonlinearity in order to simplify th e subsequentdiscussion. Th e second feedback 36) is to linearize yl and also decouple it from y2 . Thethird feedback represented by equations 42) and 43) is to linearize y2.

Finally we comment on the invertibility of the system [27 28 291. Since the differentialoutput rank p* of this particular system is computed by [ ]

which is equal to the number of outputs the system is right-invertible [27]. This guaranteesthe success of the above dynamic extension algorithm since a right-invertible system canalways be locally decoupled via a dynamic state feedback [27]. Furthermore since thedifferent output rank is equal to the number of inputs the system is also left-invertible128, 29 301.4 3 Look Ahead ControlIn Section 4.1, we showed that the center point o of the mobile robot cannot be controlledby using a static feedba.cl


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an alternative control method. The method is motivated from vehicle maneuvering. Whenope rating a vehicle, a driver looks at a point or an a rea in front of the vehicle. We define areference point P which is L distance (called look-ahead distance) from Po (see Figure 1).We take th e coordinates of P in the fixed coordinate frame as the o utput equation, i . e .

X l Lcosdy h(x) x2 L sinTo verify if the system is input-output linearizable with this output equation, we computethe derivatives of y .

d h . dhl - x = -da dx f l x ) gl(x)p)cb cos 4 cL sin cb cos q cL sin 4

i c c o osin L cos 4 ] :j: ]Since y is not a function of th e input p we differentkite it once more.The input shows up in the second order derivative of y. Clearly, the decoupling matrix inthis case is @ (x ).Since the deternlinant of @ (x) s (-2c2bL), it is nonsingular as long as thelook-ahead distance L is not zero. It follows that t he system can be input -outp ut linearizedand decoupled [6]. The nonlinear feedback for achieving the i nput-o utput linearization anddecoupling is

P ( x ) u (x)v) (47)Applying this nonlinear feedback, we obtain

Therefore, the mobile robot can be controlled so that the reference point P . tracks a desiredtrajectory. Th e motion of the mobile robot itself, particularly the motion of the center pointP o is determined by the internal dynamics of the system which is the topic of the nextsection. We note that the look-ahead control method degenerates to the control of thecenter point if L 0.4 4 Internal ynamicsTh e previous section addresses the input-outpu t properties of the mobile robot with t helook-ahead control output equation 46) . In this section, we proceed to study the behaviorof th e internal dynamics including the zero dynamics of th e system. For a general discussionof internal dynamics and zero dynamics, see Chapter of [31]

We first construct a diffeomorphism by which the overall system can be represented inthe norm form of nonlinear systems [31]. Since the relative degree of each output is two,


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we may construct four components of t he needed diffeomorphism from the two ou tputs andits Lie derivative, i .e. hl(x) , Lfh l(x ), hz(x) and Lfh 2(x ). Since the state variable xis six dimensional, we need two more components. We choose th e two components to be01 and 02. Thus the proposed diffeomorphic transformation would be

To verify that T x)s indeed a diffeomorphism, we compute its Jacobian.

It is easy t o check tha t has full rank1. Thus T( x) is a valid sta te space transformation.xTh e inverse transformation TV1 z)s given by

We partition the sta te variable z into two blocks

After applying the feedback 47) , the system of the mobile robot is represented in thefollowing normal form.

The terms denoted by do not affect the computation of the r ank .


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where [bsin -cLcos4 -cbcos$-cLsin4z')z ] 2c2bL -cb sin cL cos cb cos4 cL sin

It is understood th at in the expression of w(zl , z2) is a short-hand notation for c z 5z6). Together, the linear state equation (51) and the linear output equation (53) are anequivalent representation of the input-output map (equations (48) and (49)) . Equation 52)represents th e unobservable internal dynamics of the mobile robot under the look-aheadcontrol.

The zero dynamics of a control system is defined as the dynamics of t he system whenthe outputs are identically zero 2 . e . y 0, jl 0, y 0, . If the outpu ts a re identicallyzero, it implies th at z1 0, and the zero dynamics is

Thus , z2 remains constant while the outputs are identically zero. Th e zero dynamicsis stable but not asymptotically stable. In other words, if the reference point P remainsstill , so does the mobile robot (or more specifically, the wheels do not move).

We now look a t the internal dynamics while the reference point is in motion. Morespecifically, we are interested in the internal motion of the mobile robot when it movesstraight forward or backward. Let the mobile robot be initially headed in the positive X1direction. We assume that th e reference point is controlled to move in the negative XIdirection. Th e velocity of the reference point is then

where ~ t ) 0. Substituting this into the internal dynamics 5 2 ) ,we obtaincb sin 4 cL cos

-cb sin q cL cosA solution of this internal dyna,mics is

where cl is a constant . Th at is, the two wheels rota te at exactly the same angular velocityand the mobile platform moves straight in the negative XI direction.


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We now study the stability of the internal motion described by equations (55) and (56).We first change the stat e variable so tha t t he stability of the internal motion in z2 can beformulated as the stability of equilibrium points in 5.

i 2 6 g*We may express the internal dynamics in terms of C T

This system has an equilibrium subspace characterized by

We may not draw any conclusion based on the linear approximation of the internal dynamicswhich has an eigenvalue a t the origin. We will utilize the Liapunov method to establishthe stability condition. Consider the following candidate for a Liapunov function

In a neighborhood of Ec,V(C) 0 f ( E EC, nd V(C) > 0 if EC Thus V(C) is positivedefinite with respect to EC, and may serve as a Liapunov function for testing the stabilityof EC.We compute the derivative of V 5 )with respect to the time

Since e(t) > 0, v ) is also positive definite with respect to Ec. Therefore the equilibriumsubspace EC s not stable.

On the other hand, if the reference point is controlled to move in the positive Xdirection, the velocity of the reference point is

where ~ ( t ) 0 Using the same Liapunov function, we can similarly show that

along the forward internal motion. Therefore, the forward internal motion is stable. Intu-itively, if the mobile platform is pushed at the reference point, the internal motion is notstable. If it is pulled or dragged at the reference point, the internal motion is stable.


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onclusionWe presented a num ber of interesting results on the feedback linearization of th e dynam icsystem of a wheeled mobile robo t. Th e first result reveals th at the s ystem is not in put-st at e linearizable. T h e proof of this result is based on th e fact a wheeled m obile robo t isnonholonomically constrained. T he other results ar e on the input-o utput l inearizat ion anddecoupling of the system. Two types of outputs have been addressed. In the first type ofou tp ut , th e center point of th e mobile robot on t he wheel axis is intended t o be controlled.It has been known that the point on the wheel axis cannot be controlled using a stat icfeedback [14, 151. We show that the center point can be controlled to track a t rajectoryby using a dynam ic nonlinear feedback. Th e dynamic feedback for achieving the in put-output linearization and decoupling has been developed through a three-step algorithm.T he second ou tp ut takes the coordinates of a reference point in front of th e mobile robo t.The input-ouput linearization of the system under this output is possible by simply usinga sta tic nonlinear feedback. T h e last part of th e paper investigates the behavior of theinternal dynamics of the system with the second type of ou tpu t . We showed th at theinternal m otion of th e system is asymptotically sta ble when the reference point is controlledto move forward, but is unstab le when it is controlled to move backward. The se results,togethe r w ith t he results on con trollability and feedback stabilization [18, 20, 14, 15, 161provide a theoretica l found ation for feedback control of w heeled mobile ro bots.

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