control of mechanical systems with rolling constraints

7/28/2019 Control of Mechanical Systems With Rolling Constraints

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Department of Computer & Information Science

Technical Reports (CIS)

University of Pennsylvania Year 1992

Control of Mechanical Systems with

Rolling Constraints: Application To

Dynamic Control of Mobile RobotsNilanjan Sarkar Xiaoping Yun

R. Vijay Kumar

University of PennsylvaniaUniversity of PennsylvaniaUniversity of Pennsylvania, [email protected]

This paper is posted at ScholarlyCommons.

http://repository.upenn.edu/cis reports/504


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Control of Mechanical Systems with RollingConstraints:Application to Dynamic Control of Mobile Robots

MS-CIS-92-44GRASP LAB 320

Nilanjan SarkarXiaoping YunVijay Kuillar

University of PennsylvaniaSchool of Engineering and Applied Science

Computer and Information Science Department

June 1992


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Control of Mechanical Systems with RollingConstraints: Application to Dynamic Control ofMobile Robots

Nilanjan Sarkar, Xiaoping Yun, and Vijay KumarGeneral Robotics and Active Sensory Perception (G RAS P) Labo ratory

University of Pennsylvania3401 Walnut Street, Room 301C

Philadelphia, PA 19104-6228

ABSTRACTThere are many examples of mechanical systems which require rolling contacts be-tween two or more rigid bodies. Rolling contacts engender nonholonomic constraintsin an otherwise holonomic system. In this paper, we develop a unified approach tothe control of mechanical systems subject t o both holonomic and nonholonomic con-straints. We first present a state space realization of a constrained system and showthat i t is not input-state linearizable. We then discuss the input-output linearizationand zero dynamics of the system. This approach is applied to the dynamic control ofmobile robots. Two types of control algorithms for mobile robots are investigated:(a) trajectory tracking, and (b ) path following. In each case, a smooth nonlinearfeedback is obtained to achieve asymptotical input-output stability, and Lagrangestability of the overall system. Simulation results are presented to demonstrate theeffectiveness of the control algorithms and t o compare the performance of trajectorytracking and path following algorithms.


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1 IntroductionThere are many esamples of mechanical sys tems which require rol l ing contacts between two ormo re rigid bodies. Th ese include wheeled vehicles such as convent ional automobi les , unconvent ionalact ively coordina ted robot ic sys tems such as plan etary rovers [ l l ] , m; inipulators grasping an object[ 25 ] ,an d legged locomotion sys tems [12]. In this pa per , the focus is on svs tems in which th e roll ingcontact i s maintain ed pass ively through external forces such as gravi tat ional forces . T his i s t rue ofalmost all wheeled vehicles.

Rol ling conta cts between two r igid bodies engender nonholonomic con st raints in a n otherwiseholonomic sys tem. T he control of const rained mechanical sys tems in th e robot ics l i terature hasbeen most ly s tud ied in t he con text of force control , and for the special case in which th e contactsbetween a robo t ma nipu lator and i t s environm ent are modeled by holonomic const raints [24, 261.The control of mechanical sys tems wi th nonholonomic const rained has only been s tudied veryrecent ly. Bloch an d McC lamroch [2] f irs t dem onstra ted tha t a nonholonomic sys tem canno t bes tabi l ized to a s ingle equi l ibr ium point by a sm oo th feedback. The y also showed tha t th e sys temis small-t ime locally controllable [3]. Campion e t al. [4] showed that the sys tem is control lableregardless of the s t ru ctu re of nonholonomic const raints .Wheeled mobi le robots are typical examples of mechanical sys tems wi th nonholonomic con-s t raints . Al though navigat ion [ lo , 211 and planning [ I , 14, 22, 91 of mobi le robots have beeninves t igated extens ively over t he pas t decade, th e work on dyn amic control of mobi le rob ots wi thnonholonomic con st raints i s much more recent [6 , 19, 71.

I t i s wel l known that the mot ion of a mechanical sys tem may be descr ibed by a set of, say 11,general ized coordinates and di fferent ial equat ions of mot ion relat ing th e coordinates t o externa lforces and mom ents . If the sys tem is subject to , say m, holonomic constraints, m of the general -ized coordinates may be el iminated f rom the mot ion equat ions , al though the el iminat ion processis of ten cumbersome. Th is resul ts in a reduced order for the mot ion equa t ions [17]. T h e s tatespace representat ion is qui te s imple and th e analys is and des ign of control lers for such a mechan-ical sys tem is well unders tood and do cumen ted. On th e other ha nd , if the sys tem is subje ct tosa y k nonholonomic const raints , the number of general ized coordinates can not be reduced by k.Therefore, before wel l -known s tate space based control methods can be employed, an al ternat iveapproach is necessary to represent the mot ion and const raint equat ions in the s tate space.

In this paper , we present a unif ied approach to the control of mechanical sys tems subject tobo th holonomic an d nonholonomic const raints . We fi rs t character ize th e const rained sys tems inthe s t a t e space an d formula te the contro l problem of such sys tems as a s tan da rd nonl inear systell lcontrol problem. We show tha t the sys tem s are not inpu t-s ta te l inearizable if at leas t one of thecon st raints is nonholonomic. Since the s tat e of th e sys tems can not b e mad e asym ptot ical ly s tableby smo oth feedback, we pursue feedback con trol meth ods which achieve asympto t ical inp ut-o utp uts tabi l i ty.

Applying the control methods to dynamic control of mobi le robots , we discuss two broad cat -egories of ou tp ut equat ions . In th e fi rs t category the ou tp ut vector consis ts of a sub set , say p, ofa set of generalized coordina tes . Th us th e sys tem is des igned to follow a des i red t rajectory, p d ( t )where t is t he t ime. This is called trajectory tracking i n th is paper . In the o ther ca tegory , thehis tory p d ( t ) i s no t a s i m por t an t a s t he pa t h , pd(s ) . Here s i s any convenient parameter (say anarc- length var i able) tha t paramet r i zes the pa th . Th e t r a j ec tory t racking cont ro l s cheme i s shown inFigure 1 (a) for a vehicle subje ct to nonholonomic const raints '. T h e des i red p ath is a s t raight l ine

- p p p p p'T he theo ry and th e de ta i l s of implemen ta t ion a re d i s cus s ed in the fo llowing s ec t ions .

1


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Figure 1: (a) T rajector ies and (b ) paths for the geometr ic center Po.an d th e init ial posit ion is not on th e pa th. Th e locus of a reference point o n the vchicle is shown inthe figure for different desired forward velocities. By forward velocity we mean th e com ponen t ofthe velocity of th e reference point pe rpendicular to th e axis of the wheels in a preferred "forward"direction. If a trajectory tracking control algorithm is employed, the path of the reference point isnot a "sm ooth m erge". I t is possible tha t the vehicle may first go in one direction an d then th eopposi te di rect ion depending on the def ini t ion of pd ( t ) . While this may be acceptable and evendesirable for some applications, for road following or path following, the paths shown in Figure 1( b ) i s more appropr iate2. Here the des i red output i s specif ied in terms of the path p d ( s )and t hespeed a long the pa t h , S. Th is is termed dynanaic path following i n th i s paper .

For each case, we develop a nonlinear feedback which realizes the input-output l inearizationand in pu t-ou tpu t decoupling. At the sam e time, we show tha t th e zero dynamics of the sys tem isLagrange s table. A c omp uter s imulat ion of a mobi le robot i s used to s tud y the control of mechanicalsys tems wi th roll ing c ontacts . Both types of control schemes are inves tigated throu gh numericalexper ime nts an d thei r performances are compared. I t is concluded that a dyna mic path-followingscheme is more appropriate for vehicle control applications.

2 Theoretic Formulation2.1 Dynamic Equations of MotionConsider a mechanical sys tem with n generalized coordinates q subject to m bi lateral c onst raintswhich a,re in t he form

C( 9 , Q)= 0 (1)If a c onst raint eq uat io n is in th e form C;(q)= 0, or can be integrated in to this form, i t represents aholonomic constraint. Otherwise i t represents a kinematic (as opposed to a geometr ic) const raint% m o r e e x h a u s t i v e s t l t d v fo l lo ~ v s n Sec t i on 4


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Figure 2: Two Rigid Bodies in Contactand is termed nonholonomic.

We as sume tha t we have k holonomic and m - k nonholonomic independen t c onst raints , all ofwhich can b e wr i t t en in th e form

A(q)4 = 0 ('4where A ( q ) s a n m x iz ful l- rank ma tr ix . Let s l (q) , . - ., sn-,(q) be a set of sm oo th an d linearlyindependent vector f ields in . fV(A), the null space of A(q), i.e.,

Let S ( q ) be t he full rank m atr ix m ade up of these vectors

and l e t A be th e dis t r ibut io n spann ed by these vector fields

It follows that q E A . A may or may not be involutive. For tha t reason, we let A * be th esmal les t involut ive dis t r ibut ion containing A. I t i s c l ear tha t d im(A) 5 d i m ( A * ) . T h e r e a r ethree possible cases (as observed by Campion, e t al. in [4]). Firs t , i f k = m, that i s , al l thecons t ra in t s a re l~olonom ic , hen A is involutive itself. Second, if k = 0 , t h a t is , al l th e const raintsare nonholonomic , then A * spa ns th e entire space. Finally, if 0 < k < m , t h e k constraints arein tegrable an d k components of the general ized coordinates may be el iminated f rom the mot ionequat ions . In th i s case , d im(A8) = n - k.2.2 Contact Between Rigid Bodies2.2.1 Spatial CaseConsider two bodies in con tact a t a point P, as shown in Figure 2. We use S1 and S z t o deno t e t hesurfaces of the two b odies, respectively. Let SIP be an open and connected subset of S1containingt he po i nt P. T h e n t h e p ai r ( f i ,Ul) i s cal led a coordinate sys tem of SIP f there exis ts an opena j l (u )subset U1 of R2 and an inver t ib le map f l : Ul -+ Slp such tha t the par t i a l der iva t ives du land are l inear ly independ ent for a ll u = ( u l , l ) E U1. We choose an or thogonal coordinate


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sys t em so th a t t he m et r i c tensor i s d i agonal. Le t MI be the squa re root of the m etric tensor for Sia t p o i nt P i n t h e coord ina t e sys tem ( f i ,Ul).All t h e no ta t ion for S2can b e defined simi larly. T he con tact point o n S1 o r S 2 ) is specified byt he coord ina t es u1 and v l (o r uz and vz). n ord er t o completely speci fy th e co ntact configuration

we need a fifth variable 6, hich can be the angle between th e t angent t o t h e u l -coord ina t e curvea n d t h a t t o t h e u2-coord ina te curve a t t he contac t po in t , measured about t he ou tward-poin t ingn o rm a l t o S1. T h u s4 = [111 V l Uz V2 $I T (4 )

cons t i t u t es a set of generalized coordinates.Let (v,, v,, v Z ) be the veloci ty of the point p o n S2 relat ive t o th e point p o n 5'1, an d (w,, w,,w,) the angular veloci ty of S2 relat ive to S1. The contact kinemat ic equat ions have been derivedby Montana [15]. For rolling contact, since v, = 0 and v, = 0 , we obtain the rol l ing const raintequa t ion [27]

R+A I l i l - M 2 u 2= 0 ( 5 )where

- i n $ - cos$It can be rewri t ten in the form of Equat ion (2 ) if

IWe choose th e S (q ) ma. tr i x (defined in Equat ion ( 3 ) ) as follows:

where r -4

MTenow compute the Lie brackets

where

Therefo re, the d ist r ibut ion spann ed by t he vector f ields sl (q ) , sz(q), an d ss(q ) is not svolutivesince s4 ( q ) and s5 (q) a re no t in t h e d is t r ibu t ion . Fur ther , t h rough s5(q) span the en t i re5-dimensional configuration space. It follows that the two rol l ing const raints are nonholonomic.Note that for pure rol l ing, that i s , i f the spin mot ion w, = 0 in ad ditio n t o v, an d v, bein g zero , asimi lar approach shows that al l three const raints are nonholonomic.


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2.2.2 Planar CaselTl and U2 are now op en subsets of R and th e contac t conf igurat ion is specif ied by t ~ v o oordinatesq = [u l uzlT .The kinemat ic equat ions of rol l ing contact , Equat ion (5) reduces to

where Mi =2. h e A(q) ma tr ix is clear ly

and the S(q ) mat r ix , which spans the null space of A(q), is S(q ) = [M 2 ~ 4 1 1 ~ .he d i s tr ibutionspan ned by S (q ), a single vector f ield, is tr ivially involutive. Therefore we get th e well-known resultth at th e rol ling const rain t for the planar case is holonomic.2.3 Sta te Space RepresentationWe now consider the mechanical system with constraints given by (21, whose equations of motionare descr ibed by

M(9)i i + V ( q , 9) = E(q).r - A T ( q ) ~ (8 )where M(q) i s the n x n iner t ia matr ix, V(q,q) i s the vector of pos i t ion and veloci ty dependentforces, E ( q ) is the n x r i npu t t r ansformat ion m at r ix3 , T i s the r -dim ensional inpu t vector , A(g) asin Equ at ion (2) is the in x n Jacobian mat r ix , and X is the vector of constrain t forces.

We allow for k of the m co nstraints in Equ at ion (2) to be holonomic. Since the const rained veloc-i ty is always in th e 11~11pace of A ( q ) , it is possible to define n- m velocities v(t) = [vl v2.9 .v,-,,Isuch tha t

Q = S( q ) v ( t ) (9)The se veloci ties need not b e integrable but t hey can be regarded as being t ime d er ivatives of n - mquasi-coordinates p1 , p2, . . ., p,-, [IT]. For example, we can choose the qu as i -coordinates so th atv = LL= S + q , w h er e S+ is a generalized inverse of S.

Different iat ing Equat ion (9) , subst i tut in g th e expression for q into (8) , an d premult iplying byS T , we have

s T ( M s i , ( t ) + M S V ( ~ ) V ) = S ~ E T (10)Note that s ince S E i V ( A ) . s ~ A ~ / \anishes in this equ at ion.

Using th e s ta te space var iable s = [qT vTIT, we have

where f 2 = ( S T h l ~ ) - l ( - ~ T ~ f ~ ~ /~ v ) . A s sum ing t ha t t he num ber of ac t ua t o r i npu t s is g r ea t ert ha n o r equa l t o t he num ber o f t he degr ees of f reedom of the mechanical sys tem ( r > n - m ) , a nd( s ~ A , I S ) - ' S ~ E has rank n - m , we may apply the following nonlinear feedback"3 E ( q ) s an iden t i ty ma tr ix in m ost cases . However, if the genera l ized coordin ates are chosen to be some variableso the r tha n t he join t va r iab le s, o r if t he re a re pa ss ive jo in t s w i thou t ac tua to rs , i t is no t an iden t i ty m a t r ix .4W hi le i t i s convenien t to us e th e gene ra li zed inve rse t o re so lve th e redundancy , i t i s p roduc t ive t o us e s u rp lus

i n p u t s t o c o nt ro l t h e i n t e r a d o n f o rc es a n d m o m e n t s [28, 131. In th i s pape r , fo r th e mos t pa r t , we wi ll no t be conce rnedwit11 redundant sys tems.


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Th e s t a t e equat ion s impli fi es to the form

where

2.4 Control PropertiesTh e following two proper t ies of the sy s tem (13) have bee11 established in 111 for th e special case inwhich all constraints are nonholonomic.Theorem 1 The nonholonom ic s ystem (13) is controllable.Theorem 2 The equilibrium point x = 0 of the nonholon omic sys tem (1 3) can be made Lagmngestable, but can not be made asymptotically stable by a smooth state feedback.

In the rest of this section, we discuss the more general case in which Equation ( 2 ) consists ofboth holonomic a n d no~tholonomic ons t raints .Theorem 3 The system in Equation (13) is not input-state linearizable b y a state feedback i f oneor more constraints are nonholonomic.Proof: T he sy s tem h as to sat is fy two condi t ions in order to be inpu t-s tate l inear izable: th e s t rongaccessibil i ty condition and the involutivity condition [16, p. 1791. It is shown below th at th einvolutivity condition is not satisfied.

Define a sequence of distr ibutions

The n th e involu t iv i ty condit ion r equir es th a t the d i s t r ibut ions Dl , D z, . . . ,D2n-nz ar e a11 involutiv e.Note tha t the dimension of the s tate var iable is 2 n - m. D l = s p a n { g ) is involutive since g iscons tant . Next we compute

Since the dis t r ibut ion A spanned by t he columns of S(q) i s not involutive, the dis t r ibut ion D2 =s pa n {g , L f g } s no t involutive. Therefore, the sys tem is not inp ut-s t ate l inearizable.Al though a sys tem with nonholonomic const raints is not inp ut-s tate linear izable, i t may b einp ut-o utp ut l inearizable if a proper set of out pu t equat ions are chosen. Consider the pos i t ion

control of the sys tem, i.e., the output equat ions are funct ions of pos i t ion s tate var iable q only.Since the num ber of th e degrees of f reedom of the sys tem is ins tantaneou sly n - m, we may havea t m ost 12 - m i ndepende nt pos i tion ou tpu t s equat ions .


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T h e necessary an d sufficient condit ion for inp ut-o utp ut l inearizat ion is tha t th e decoupl ing m at r ishas ful l rank [16]. W i t h t he ou t pu t equa ti on ( 14 ) , t he decoupling ma t r i s @(x) or the sys tem is the( n - m ) x (n - m ) m at r i x

@(Q) = Jh(q)S(q) (15)where Jh = is t h e (n - m ) x n Jacobian mat r ix . @(z) s nonsingular i f the rows of J h a rei ndependen t o f t h e row s of A(q).

To character ize the zero dynamics and achieve input-output l inear izat ion, we int roduce a newstate space var iable z defined as follows

where is an m - d i m en s i on4 f unc ti on s uch t h a t [J : J;] has full rank. I t i s easy to ver ify th atT(x) s indeed a diffeomorphism [27] a n d t h u s a valid s t a t e space t r ans formation . Th e s ys temunder th e new s t a t e var iable z is characterized by

Utilizing the following state feedback

we achieve input-output l inear izat ion as well as input-output decoupl ing by not ing the observablepar t of the sys tem il = 22 i2 = v Y = z1Th e zero dynamics of t h e sys tem is (obtained by subst i tut ing zl = 0 and z2 = 0 ) [23]

which is clear ly Lagrang e s tab le but not a.symptotical ly s table.

3 Dynamics and Control of a Mobile Platform3.1 Constraiilt EquationsIn orde r to i l lus t rate th e methodology, we consider a mobi le rob ot s imilar to the LAB MATE^mobile plat form. I t ha s two dr iving wheels on an ax is which passes throug h th e vehicle geometr iccenter as shown in Figure 3. The y are powered by D.C. motors . T h e plat form h a s four passivewheels (ca s tors) on each corner .

T he following nota t ion wil l be used in t he pa per (see Figure 4).5LABMATE is a trdeanlark of Transition Research Corporation.


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/ACTUATEDWHEELS

x-y:X-Y:Po:

I,:I, :a :d :

FOURPASSIVEk

Figure 3: An exam ple of a wheeled platform (top view)the wor ld coord ina te sys tem;the coord ina te sys tem f ixed t o th e ca r t as shown in Figure 4;th e geometr ic cente r wi th coord ina tes (x,,o ) which is the intersectionof the a.xis of sy m me try with th e driving wheel axis;th e center of inass of the platform with coo rdinates (x,,,);a vir tua l reference point a .t ta .ched to th e platform with coordinates ( x i , l);the dis tance between ei ther driving wheel and the axis of symmetry;the radius of each driving wheel;the mass of the platform without the driving wheels and the rotors of the DC motors ;th e mass of each driving wheel plus th e rotor of i ts moto r;th e mome nt of inert i a of th e platform without t he driving wheels and therotors of th e motors ab ou t a vert ical axis through PC;th e mom ent of inert ia of each wheel and the motor ro to r abou t the whee l ax is ;the m oment of ine r t ia of each whee l and the motor ro to r abo u t a wheel diameter;th e length of th e platform in the direct ion perpendicular t o the driving wheel axis ;the d is tance f rom Po o PC long the posi t ive X -axis .

If we ignore the passive wheels, the configuration of the platform can be clescribed by fivegeneral ized coordina tes . T hese are the three variables th at describe the posi t ion and orientat ion ofthe platform an d two variables tha t specify the angu lar posit ions for the driving wheels . Therefore ,le t

Q = ( x C , Y C , + , ~ T , ~ I )where (x,,,)s the coordinates of the center of mass PC n th e wor ld coord ina te sys tem , and # isthe head ing ang le of th e p la t fo rm as shown in Figure 4. 8, and 8, are th e angular posi t ions of theright and left driving wheels respectively.

Assum ing th e driving wheels roll (and do not s l ip) there are three constraints . Firs t , th e velocityof the point Po f th e platform m ust b e in the direct ion of the axis of symm etry, the X-axis:


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Figure 4: Notat ion for the geom etry of th e mobi le plat formFurth er, if th e driving wheels do not s l ip,

iCos 4 + gc sin 4 + b& = ~ 4 ,i c c o s 4 + j rc s in 4 - b$ = re,T he thre e cons t ra in t s can be w r i t ten in t h e form:

where - in + 0 4 d0 0 1o s d - in 4 - 6 T 0- c o s 4 - s i n + b 0 T

T h u s t h e m e c h a n i c al sy s te m has two degrees of freedom.It i s s t raight forward to verify t ha t th e following m atrixI (b cos 6 - d sin #) c(b cos C + d sin 4 ) -c ( b s i n + + d c o s # ) c ( b s i n 4 - d c o s d )S(q)= [ ~ l ( c l ) ,2 ( 4 ) l = C -C1 00 1 -


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sa t isf ies A(q)S(q) = 0 , where th e cons tan t c = %. C o m p u t i n g t h e Li e b ra ck et of s ~ ( q )nd s2 (q )we ob ta in r - r c s i n # 1

which is not in the d is tr ibut ion A spanned by s l (q ) and s2 (q ) . The re fo re , a t l ea s t one o f theconstra in ts is nonholonomic. We continue by computing the Lie bracket of s l (q) and ss(q)

which is l inear ly inde pend ent of s l (q ) , s2( q) , and ss(q) . However , we can verify th at t he d is tr ibut ions pa nn ed by s l (q ) , s 2 (q ) , s3(q) a nd ~ ~ ( 9 )s involutive. Therefore, we have

s ' l (q) = [ ~ l ( c l ) , 3 ( 9 ) 1=

I t fo llows th at , amo ng tlie tllrce constraints, two of tlicrn are nonholonornic and the third one isl~o lonomic .To ob ta in the l io lonomic cons t rain t , we sub t rac t Equa t ion (24 ) f rom Equation (23) .

-TC 2 C O S ~- r c 2 sin 4

000-

In teg ra t ing th e above equa t ion we have

where c l is a con stant of in tegra t ion. Th is is c lear ly a holonomic constra in t equat ion. Note that 4,0, an d O r can b e def ined in such a way th at c l may be taken t o be zero.3.2 Dynamic EquationsWe now derive th e dynam ic equat ion for the mobile p la tform. Tlle Lagrange equat ions of motionof the p la tform with the Lagrange mult ip l iers X I , X2, an d X3 are given by

m i c + 2m,d($ sin 4 + $2 cos 4 ) - X1 si n 4 - A2+ X3) cos 4 = 0 (29)mij, - 2m,d($ cos q5 - d2 in 4) + X1 cos 4 - A2 + X3) sin 4 = 0 (30)

2mW d(Zc in 4 - j, c o s 4 ) + 14 - dX1 + b(X3 - X2) = 0 (31)IW8,+ X2r = TT (32)1 ~ 8 ,+ X ~ T rl ( 3 3 )

where


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a n d r, a n d rl are the torques act ing on the wheel axis generated by the r ight and left motorsrespectively. Th ese five equat ion s of mo tion can easily be writ ten in t i le form oi' i..il11;iti0118). T h emat r i ces h l (q) , I, . (q , i), and E (q ) a re given by:

-21(q) =

If we choose 8, a n d B I to b e th e two quasi -cordinates,

- nz 0 2 m w d s i n 4 O 00 m -2m,dcos@ 0 02 m w d in @ -2mwd cos 4 I 0 00 0 0 I w 00 0 0 0 It" -

J , 7 ( q , ) =

a n d we can veri fy that Equat ion 9 is sat isfied. Then the state variable is the following vector:

Using this s ta te variable, the dy na ~n ics f the mobi le plat form can be represented in the sta te spaceform, Equat ion (11).

- 2rnWd$' cos 4 -

-

2mwd$2 sin @000

3.3 Output Equations

0 0 -0 01 00 1

While the state equat ions of a dynamic system are uniquely determined by i t s dynamic character-i s t ics , the o utp ut variables ar e chosen in such a way th a t t h e t asks to be per formed by the dynamicsystem can be conveniently specified and the controller design can be easily accomplished. Forexam ple, if a six degree-of-freedom ro bot m anipu lator is to perform pick-an d-place or trajec toryt racking task s, th e six-dimensional joint posi tion vector or t he six-dimensional Ca rtesian posi tionand o rientat ion vector i s normal ly chosen as the o utp ut vector. In this sect ion, we present a nu mbe rof possible choices for out pu t variables of the cont rol system for th e mobi le plat form and discusseach case.

Let Pl be the reference point on the mobile platform. Lie choose Pl t o be a virtual point on theasis of symmetry displaced through a distance L from the center of mass PC s shown i n Figure 4( L can be posi t ive, negat ive, or zero). I t s coordinates are denoted by (x l ,y r) :X ( = xc f L c o s 4gi = yc + L sin 4

Since the sys t em ha s two inpu ts , we may choose any two ou tpu t variables. We co~ lside rhe followingfour t ypes of ou tpu t equa t ions :Type 1: Y = h (9 ) = [xl y1ITType 11: y = h(q)= [xl ITType 111: y = h (q ) = [y r 4ITType IV: !J = / L I Z ) = [ h l ( q ) h ? ( v ) I T


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T h e T y p e I ou tpu t equ a t ion resu l ts in a t ra jec to ry t rack ing con tro l sys tem which has been s tud iedin [6, 191. Th e cor responding decoupling mat r ix fo r th is ou tpu t i s

whereQ l l = c(b cos 4 - d + L) s in 4 )Q = c(b cos qi + (d + L ) si n 6 )@ 21 = c(b sin q5 + (d + L ) c os 4 )

= c(b sin 4 - (1 + L) cos 4 )d+L)S in ce t h e d e t e r min a n t of the decoupling ma t r ix i s de t (@(q))= -%, i t is s ingular i f and only ifL = -d, th at is , if point PI coincides with point Po. Therefore , t ra jec tory tracking of the point Po is

no t possible as poin ted ou t in [19]. This is c learly due t o the presence of nonholonomic cons traints .Choosing L n o t equal t o -d, we may decouple and l inearize the system as follows. T h e derivativeof the decoupl ing mat r ix is (no t ing tha t is a function of 4 only)

Since ST^ = 1 2 x 2 r the nonlinear feedback (Equations (12) a n d (20) ) in this case simplifies to

a n du = @-'(g)(v - &( q ) v )

T he l inear ized and decoupled subsys tems a re

T y p e 11a n d T y p e I11 system s are s imilar . For a Type I1 outp u t equa t ior l , the decoupling mat r i s

where and Q12 are defined by E quations (37) and (38) . I t s de te rmin an t i s de t (@ rr(q ) ) =-2c2b cos 4. all is nonsingular if cos q5 # 0. Similarly , the decoupling ma tr ix for T yp e I11 ou tpu t isnonsingular if sin 4 # 0. Thu s , i t i s possib le to decouple and l inear ize th e sys tem wi th T yp e I1 andT y p e 111ou tp ut s in a large region of the s ta te space. However, i t is not convenient t o specify controltasks wi th these two types o f ou tpu ts . We discuss Typ e IV o u tp u t e q u a t ion s a n d t h e d y n a mic pathfollowing prob lem in the n es t sec tion be low.


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3.4 Dynamic Path FollowingIf we analyze automobile manoeuvring, the two most important requirements ; i r(> I follow theroad (o r p a th ) by s tay ing as c lose to th e pa th as poss ib le and to main ta in the c ic \~ red o rwardvelocity . W ith this in min d, we would l ike to choose an out pu t equation with two variables: theshor tes t d is tance of a re fe rence po in t on th e mobile p lat fo rm f rom the u es ~r ed c~ t i li i , he forwardvelocity . By doing so. we formu late a dynam ic pa th fol lowing problem [20] instea ; ( '1 t ra jectorytracking problem. In a t ra jec to ry t rack ing p rob lem, th e desi red t im e h is to ry o f the ou tpu t va r iab lesis specif ied. T herefore , in th is case, the task is no t only t o reach a poin t bu t alsc. t o reach i t a t aspecif ied t im e insta nt . In a p at h fol lowing problem, however, the geom etry of th e par 11 is specified.In th is case , i t i s more im por ta n t to fo llow the p a th c lose ly tha n t o reach po in ts on the pa th a tspecif ied t ime instan ts . By specifying th e desired forward velocity , we ( indirect ly) ensure th at thevehicle reaches desired points on the path.

In th is sect ion , me achieve pa th following by appro priately choosing hl a n d h2. hl is tlefined asth e s h o r t e s t d i s ta n c e f r om th e p o in t P, on the mobi le p la tfo rm to th e desi red pa th . Th e fo rmula t ionis qu i te genera l s ince Pl can b e any where on t he vehicle , a l though we prefer to c lloose Pi on th eX-ax is . However , no te tha t for an arb i t ra ry pa t h , the re i s no c losed fo rm express ion l o r the shor tes td is tance f rom PI t o t h e p a th . W e de fine h 2 to be t he com ponen t o f the ve loc ity of PI along the-Y-axis. We call this the forward velocitv.

We f irs t consider two basic paths: a s t ra igh t l ine pa th and a c i rcu la r pa th . .-I closed formexpress ion fo r th e d is tance f rom a po in t to the p a th can be easi ly ob ta ined in either case.

We firs t consider a c ircular path. Let Pf be the center of the c ircular path whose coordinatesa re deno ted by ( x f , y f ) in th e world coord ina te sys tem. Le t R be t he rad ius o f the c i rcu la r pa th .We choose h l as follows:

Note th a t t he shor tes t d is tance from po in t PI to th e c i rcu la r pa th i s th e abso lu te va lue o f h l ( q ) .H e re ( x f , yj) a n d R a r e c o n s t a n t s a n d x i a n d yl are re la ted to t he s ta te va r iab les , x, , gc, n d 4, byE q u a t io n s (34) a n d (35). T he forward velocity of the platform is given by

h2(v)= x, cos 4 + y, sin 4 = I-(vl+ v2)2 ( 4 7 )I t i s c lea r tha t we have a Type IV o u t p u t . T h e d e c o u ~ l i n gmatr ix fo r th is ou tpu t equa t ion i scomputed as fol lows.

whereyc - yf + L sin 4L cos +(y c - yl) - L sin $(sc - sf)


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Therefore , the decoupl ing m at r ix i sJh , ( q ) S ( q )

= [ Jh, ]a n d th e d e t e r min a n t o f i9 is

I t fol lows that the decoupling matr ix is s ingular i f ( 1 ) L = -d (po in t Pl coincides with point Po ) ,o r (2 ) ( y c- y f ) cos 4 = ( x c - z) sin 4 ( the heading direct ion of the platform or the X-axis isnormal t o th e c i rcu la r pa th ) . Whi le th e f ir s t cond i tion is due to th e nonholonomic cons t ra in t o fth e p l a t f o r m, a n d t h e s e c on d c o n d it io n is d u e t o t h e f a c t t h a t t h e d i r e ct io n a lo ng th e p a th i s nr'explicitly specified in th e fun ction h l ( q ) . T h u s wh e n th e X- a x i s i s n o rma l t o t h e p a th , s p e cn y in gth e forward velocity does not uniquely specify th e pat h direct ion. We also note t ha t in order toavoid this t yp e of sing ular ity it is beneficial to have L > -d if the forward velocity is positive.Motion in the reverse direct ion can be accomplished with L < - d and a negative forward velocity.

We now consider a stra ig ht l ine pa th. Let the path be described by A x + B?J+ C = 0.

Once aga in , th e shor tes t d is tance f rom po in t Pl t o t h e p a th i s t h e a b s o lu te v alu e of h l . T h e s ec on dc o mp o n e n t of t h e o u tp u t e q u a t io n , hZ, s the sam e as fo r the c i rcu la r pa th . Th e decoupl ing mat r ixc9 h a s t h e s a m e f o rm e s c ep t t h a t Jhl is now replaced by

Th e de te rmin an t of i sdet @ = L , (I3 cos 4 - A sin 4)dm

Once aga in , the decoupl ing mat r ix i s s ingu la r i f L = - d or the X-ax is i s pe rpend icu la r to thes t ra igh t l ine pa th .More generally, if f ( x , y ) = 0 i s an a rb i t ra ry pa th , so lving the shor tes t d is tance f rom po in t f ito t he pa th invo lves so lv ing th e ex t remiza t ion p rob lem

A closed form solution is impossible in a general case. However, an approximate expression forlz l mav be used instea ,d:


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I Linear FeedbackFigure 5: Schematic of th e control a lgori thms

Al though hl in this case is not the tru e dis tance t o the pa th, i t is a meas ure of th e closeness tothe given pa th . In th e two basic p aths discussed above, i t is noted th at th e dis tanc e representat iondiffers from t he p ath descri t ip t ion only by a const ant . Finally , we note tha t D ubins [8] and la terReeds an d S hep p (181 showed th at given any ini t ia l and f inal posi t ion and orientat ion of a car ,there exis ts a family of paths co n~ pos ed f only s tra ig ht l ine and circular arc segments betweenany two po in ts . In f ac t, Du bi ns p ro ve d t ha t t h is fam ily c o nt ai ns t h e ~ - ~ e o d e s i c ~etween the twopoin ts . We use th is resu l t to a rgue tha t any pa t h can be su i tab ly b roken down in to s t ra igh t lineand circular arc segments . Therefore if we are able to control the mobile platform on such basicpa ths and on p iecewise con t inuous pa ths composed o f these two ,we can effectively move from anypos it ion a nd o r ien ta t ion t o any o ther pos it ion and o r ien ta t ion .

For e i the r of th e two bas ic pa th s o r a rb i t ra ry pa t h , by app ly ing the non l inear feedback , Equa t ion(20) , we ob ta in a l inearized and decoupled system in the form

A linear feedback can be designed to make each subsystem stable and to meet the performancespecifications (see ['LG, 283, for example).3.5 Design of the Control AlgorithmsWe presented two types of control a lgori thms for mobile robots: (a ) t ra jec to ry t rack ing ; (b) p a thfollowing. W hile th ey differ in t he selection of ou tp ut eq uations, th e basic scheme is the s a m e a sshown in Figure 5. In the f igure, v d is the reference (desired) values for the outputs , hl a n d hz.Th e non l inear feedback (Eq ua t ion (1 2) ) cance ls th e non linear i ty in t he dynamics so tha t the s t a teequa t ion i s s implied in to the fo rm of Equa t ion (13) . Th is i s represented by the d o t t ed b lock in F igure5 . No te t h a t t h e n o n l ine a r it y in the k inematics remains in th e s impli fied s ta te equa t ion . A second

'An R-geodesic is the minimal length p ath between two points having an average curvatu re everywhere less thano r equal to R - ' . where R I < n fixet i po


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nonl inear feedback (Equ at ion (20)) l inearizes and decouples th e inpu t-outp ut ma p. Th e overallsyste m is thu s decoupled into two linear subsystems. For t rajectory t racking , bo th subsystems a reof second order. In t he case of pa th fol lowing, the distance control subsystem is of second order,and the veloci ty control subsystem is of fi rst order. To stabi l ize these subsystems and to achieveth e desired perform ance, an o uter l inear feedback loop is designed to place th e poles of th e system .

4 Evaluation of Control Schemes4.1 The Dynamic SimulationWe developed a computer simulat ion in order to veri fy the val idi ty of the dynamic model andth e effectiveness of th e control algori thm discussed in the previous sect ions for a mob i le plat formthat i s k inemat i ca l ly s imi l a r t o t he LABMATE. The d imensions and the iner t i a l parameters a rerepresentat ive of t he LAB MA TE pla t form. A ccording to the no ta t ion in t roduced before :

b = 0.75m;d = 0.30m;a = 2.00m;m, = 30.00kg;m, = 1.00kg;I, = 15.625kg-m2;I, = 0.0051;~-nx2;I,,, = 0.0025kg-m2.T he v i r tua l re ference poin t Plwas chosen to be coincident w ith PC. he gains for the l inear outer

loop were designed in such a way that we got an overdamped system for the decoupled posi t ioncontrol subsystem . This i s approp riate if we want t o follow a wal l or a p ath wi th a m edian or adivider. In such a case the p l a tform should not overshoot i t s des ired pa th . On the o th er hand , wecan choose a cri t ical ly-dam ped system i f we want t o fol low a curve on the middle of t he road .4.2 Trajectory tracking versus dynamic path followingConsider a s t ra igh t li ne pa th , y = x, as shown in Figures 6 and 1. The reference point is defineds o t h a t L = 0.0 m eters. T he ini t ial posi t ion i s such tha t

and the ini t ial veloci ty i s zero. The desired forward velocity is 1.414 m/sec . For t he t ra jec toryt racking a lgor i thm ,h l = x l , h 2= YldV 1 = t , 2); = t

For the path fol lowing algori thm,


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Figure 6: (a ) T ra jec to ry and (b) path of a reference point on a wheeled vehicle.In both cases, as shown in Figure 6 the reference point is able to reach the path and s tay

o n t h e p a t h . Note th a t th e ga ins for the pos it ion variables a re same fo r bo t h cases . T he pa thfollowing algori th m seems to exhibit a gradua l merge while the tra je ctory tracking reacts morequickly and, depending on the gains, i t even forces the reference point in the wrong direct ion.Depend ing on the po in t o f in te res t , the t ra jec to ry t rack ing a lgor i thm a lso resu l ts in cusps in t hetrajectory. For examp le, consider the locus of the geometr ic center , Po, in b oth cases for differentdesired forward velocities as shown in Figures 1 . W ith a T y p e I V o u tp u t e q u a t io n t h e a c tu a lpa th fo l lowed i s smooth bu t the Type I outpu t equa t ion o f ten p roduces a, discontinuity in theslope of th e tra jectory . Th is can also be seen from t he velocity his tory shown in Figure 7. In thisf igure we have shown th e forward velocity corresponding t o case C of Figure 1. W i t h t h e T y p eIV ou tp u t equa t ion the fo rward ve loci ty exh ib i t s a smooth exponentia l response that is typical ofa f irs t order system a s expected (Figure 7 (b ) ) . Th e t ra jec to ry t rack ing a lgor ithm may resu lt indiscontinuities in velocities. As shown in Figure 7 (a ) , th e center of th e vehicle is accelerated andthen decelerated to a s to p twice before monotonically increasing to t he desired velocity . If th eobjective is to follow a desired path , as is the case in ; tu tonomous navigation r2O]. i t is c lear thatthe path fol lowing algori thm is more appropriate . It is possible that t ra jectory tracking may bedesirable in applicat ions in which t im e is a cr it ica l pa ramete r. I t appears t ha t fo r a wide range ofapplicat ions in robotics th a t pa th fol lowing is th e more appr opria te s tra tegy. For th is reason, fromth is po in t on , we concen t ra te on Ty pe IV ou t pu t equa t ions . T he resu l ts of numer ica l e sper im entswith path fol lowing algori thms are presented in the remainder of th is sect ion.4.3 Performance of dynamic path following algoritl~nls4.3.1 Effect of init ia l condit ionsThe in i t ia l condit ion th at most affects th e tra jec tory is the ini t ia l velocity . Hence the tw o rl lostimp or tan t pa rame te rs a re the in i t ia l head ing ang le (d) and the magni tude o f t h e initia l forwa.rd


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Figure 7: Forw ard velocity of the geom etric center Po i n (a ) t ra j ec tory t rack ing ; (b) pa th following.velocity. Figure 8 a ) shows how the mobile platform follows a circular path when i t starts witha forward velocity of 5m/s but with different init ial heading angles (4). Here the init ial referencepoint posit ion is

( ~ 1 , 1 ) (30.0,15.0)an d

We note that the algorithm is singular if the vehicle is oriented along the shortest l ine joiningthe reference point a nd the pa th. In this case an orientat ion wi th q5 = -26.6 deg or 153.4deg leadsto problems. For heading angles other than these, the response is sati sfactory as seen from theFigure 8 (a ) .

Figure 8 (b) depic ts th e system response for a desired circular pa th for different init ial forwardvelocit ies but wi th a cons tant heading angle ( in this case i t was 0 degrees). As expe cted, if the ini tialheading is away from the desired pa th (as in this case), the system exh ibi t s bet ter performancewhen th e ini t ial speed is less.

4.3.2 Effect of i n o d e l i n g u n c e r t a i n t i e sModel ing errors can resul t du e to th e di ff iculty in measuring or est im at ing the geometric, kinemat icor inert ial param eters or from the lack of a complete knowledge of the com ponents of t he sys t em.We simulated model ing errors in the inert ial parameters namely m, and m, and invest igated theperformanc e on a st ra ight l ine pat h as well as on a ci rcular path. For as much as a 100 percentmodel ing error in m, and m,,, the con trol scheme follows the pa th q uite well . Altho ugh a theoreticalrobustness analysis was not performed, i t appears from extensive simulat ions that the scheme isqui t e robus t . F igures 9 ( a ) a n d ( b ) s l ~ ow she response of the system for a st raight l ine a n d circular


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-10.m urn am urn wm urn lam um m.rn u.rn a m 15m( 4 (b )

Figure 8: P a th followed with : (a ) dif ferent ini t ial heading angles ( in degrees) for an ini t ial forwardvelocity of 5 me tersl sec ; (b ) different init ial forward velocit ies in (mete rs/se c) for an init ial headingangle of 0 degrees .pa th respect ively. I t takes a longer t ime in com par ison t o the case when the re is no model l ing error(as seen f rom the pa th C of Figure 1 (b) ) , t o converge to the des ir ed pa th for a s t raight l ine pathwhereas i t fol lows the ci rcular path but only wi th a small constant offset .4.3.3 Piecewise continuous pathsIt is shown in References [8] an d 1181 tha t th e shor tes t path s for wheeled mobi le car ts are composedof circular arcs an d strai gh t lines. T h e result is a piecewise continuous pa th with discon tinuities inth e curvature an d higher derivatives. An exam ple7 of such a pa th is shown in Figure 10 where Aand C are ci rcular arcs and B i s a s t raight l ine. T h e performance of the co ntrol sys tem is shown inFigure 11 . T h e magnified view of the first transit ion point in Figu re 12 shows th a t the per formanceis acceptable - iscontinuities in curvature are negotiated witho ut an y difficulty. l l'e n ote th atthe second t rans i t io n involves a smal ler change in curvatures s ince arc C possesses a smaller radiusof curvature. As a resul t there is almost no de viat ion of the actu al pa th f rom the des i red one.4 .4 A recalibration scheme for mobile robotsIn wheeled vehicles, there is no direct way of obtaining posit ion feedback. The pos i t ion ( andor ientat ion) of the vehicle m ay b e es t imated f rom the pos i t ions or veloci ties of the wheels . Th eremay be smal l e r ror s in the es t imates e i ther due to s lippage and scuff ing or d ue t o er ror s in th e wheelsensors. Since smal l er rors in th e ang ular veloci ties integrated over a large t im e interval result inlarge pos i tion errors , this presents a ser ious problem in control .

'Note that the discussion in References [8] and [IS] s limited to circular arcs of constant curvature. Here, weconsider different curvatures in order to investigate the effect of changing curvature.


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Figure 9: Desired and actual p a th of t h e mo b il e ~ l a t f o r m h en th e r e is a modeling error in m, andm , by 100% for the case of a : (a) stra ight l ine ; ( b ) circle.

X

Figure 10: Desired compos ite pa th .


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Figure 11: Performance of the plat form in composi te pa th fol lowing.

Figure 12: Magnified view of th e first transit io n point in Fig ure 11.


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Figure 13: Sensory noise in the measurement of the angular velocity of : (a) t l lc right wheel ; ( b )the left wheel.

6Jm -a. m - -ssm - -

-w.m -a m - u m - -(rm - (r m -

We consider a si tuation in which the sensed wheel angular velocit ies are different from theactual angular velocit ies. We generated noise using Gaussian distribution and simulated the sensedangu lar velocities a s follows:

u mmmu mlorn

where (8,)n0'3e a n d ( o ~ ) " ~ ' " ~re random s ignals . Th e mean an d s t anda rd devia tion of (6,)"0"~a re 5.0 rad ian / sec and 3.0 radian/sec respect ively and that of (8r)n0tse r e 0.5 radian/sec and 3.0radian/sec respect ively.

This i s shown in Figures 13 ( a ) a n d ( b ) for a uni form st raight l ine mot ion. Th e performance ofthe control system is shown in Figure 14 for a ci rcular path. I t i s evident th at the vehicle 's pat hdiverges from th e desired pat h. IIo~vever rom the "sensed path " shown in th e figure i t is clear tha tthe vehicle " thinks" th at i t is close to the desi red path. Th e only remedy to this problem is toprovide some form of end-point feedback.

In F igures 15 and 16 we consider end-p oint feedback a t rates t ha t ar e much lower tha n servo-level sampling ra tes. If end -point feedback is available once every second, th e path (shown in Figure15) exhibi ts a significant improvement (compared t o Figure 14) . Note th at while the sensed posi t ionexhibi ts a discont inui ty, the actua l posi tion and veloci ty are not discontinuous. T he performanceimproves with increased end-point feedback frequency as shown in Figure 16 for a 2 Hz . c:~mpl ingrate. In practic e, vision sy stem s are capable of providing fram e-rates th at are well abovcl 10 Hz.PI.

- - L L-- d ---

35mm.mu mmm

-- 8.d- ---- .ad -- -r


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I I I I I I I xam zoo 4.m 60 0 8.00 I m

Figure 14: Effect of sensory noise on t he performance of th e mobile platform .

F igure 15: Effect of recalibrat ion on t he actu al pa th followed by the p la t fo rm when the recal ib ra t ionfrequency is 1 Hz.


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3.m17-00

aoo

Figure 16: Effect of recal ibrat ion on t he actu al pat h fol lowed by the ~ la t f o rr n hen th e recalibrat ionfrequency is 2 Hz.5 ConclusionWe have presented a general method of cont rol l ing mechanical systems wi th holonomic as wellas nonholonomic const raints . We discussed the inp ut-o utp ut l inearizat ion an d th e zero dynamicsof such system s. For wheeled mobi le plat forms, we derived a nonl inear feedback tha t guaran teesinp ut-o up ut stab i l i ty and Lagrange stabi l i ty for the overal l system. We invest igated two types ofcontrol algori thms: t rajectory t racking and path fol lowing. The dyn am ic pa th following problemwas presented here for the f i rst t ime. Computer simulat ion resul ts were presented to i l lust rateand c om pare th e performan ce of each algorithm. Based on these i t was concluded th at a dynamicpath-following schenle is more a pp rop riate for vehicle control applications. Finally, the effects ofmodel l ing errors and sensor noise ar e invest igated through numerical experiments.

6 AcknowledgementTh is work was in part supp orted by: Airforce gran t AFO SR F49620-85-I


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References[I ] J . Barraqu and an d Jean-Claude Latombe. Nonholonomic mult ibody mobile i .~Lots: ontrolla-bility a nd m otion plann ing in the presence of obstacles. In Proceedings of 1991 In te rna t iona l

Conference on Robotics an d Automation, pages 2328-2335, Sacra men to, C.4: -2pri l 1991.['2] Anthony Bloch and N. H. McC lamroch. Con trol of mechanical system s witlc ~ l a s s i c a l o n-

holonom ic constr aint s. In Proceedings of 28th IEEE Conference on Decision and Control,pages 201-205, Ta mp a, Florida, December 1989.

[3] Anthony Bloch, N. H. McClamroch, and M. Reyhanoglu. Controllabili ty and stabili ty prop-erties of a nonholonomic control system. In Proceedings of 29th IEEE Conference on Decisionand Control, pages 1312-1314, I-lonolulu, Hawaii, Dec ember 1990.

[4] G . Campion , B. d'Andrea-Novel, and G. Bastin . Controllabili ty an d sta te feedback stabilizationof non holonomic mechanical systems. In C. Canudas d e Wit , edi tor , Lecture Notes in Controla n d Inform ation Science, pages 106-24, Springer-Verlag, 1991.

[5] James J. Clark and Nicola J. Ferrier. Control of visual atte ntio n of mobile robots . In IE E EInte rnat ion al Conference on Robotics an d Autom ation, pages 826-831, May 1989.[6] B. d'Andrea-Novel, G. Ba.stin, and G. Campion . Modelling and control of non holonomic

wheeled mobile robots. In Proceedings of 1991 International Conference on Robotics a n dAuto matio n, pages 1130-1135, Sacra men to, CA , April 1991.

[7] C. Can udas d e Wit and R. Roskam. Pa th fol lowing of a 2-DO F wheeled mobile robot underpa th and in pu t torque constra ints . In Proceedings of 1991 Inte rna tion al Conference on Roboticsan d Auto matio n, pages 1142-1147, Sa cram ento, CA, April 1991.

[8] L. E. Dubins. On curves of minimal leng th with a constra int on average curvature , and withprescribed ini tia l and terminal posi tions an d tangents . American Jou rnal of M athematics ,79:497-516, 1957.

[9] T. Fraichard. Sm ooth t ra jectory planning for a car in a structu red world. In Proceedings of1991 Inte rna tion al Conference on Robotics and Automation, pages 318-323, Sacram ento, CA ,April 1991.

[ lo] Y. Koren a nd J . Borenste in . Potent ia l field meth ods and their inherent l imita t ions for mobi lerobot navigation. In Proceedings of 1991 International Conference on Ilobotics and Automa-tion, pages 1398-1404, Sacra men to, CA, April 1991.

[ l l ] V. I


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[14] J. P. Laum ond. Findin g collision-free smo oth trajectories for a non-holonomic mobile robot. Inlo th International Joint Conference on Artificial Intelligence. pages 1120-1 123. Milano, Italy ,1987.

[15] David J . Mo ntana. Th e kinematics of contact and grasp. The International Jo urnal o f RoboticsResearch, 7(3):17-32, Ju ne 1988.

(161 H. Nijmeijer and A. J . van der Schaft. Nonlinear Dynamic Control Systems. Springer-Verla.g,New York, 1990.[I71 L. A. Pars . A Treatise on Analytical Dynamics. Wiley, New York, 1968.[18] J. A. Reeds an d L. A. Shepp. O pt ima l paths for a car tha t goes both forwards and backwards .

Pacific Journal of Mathematics, 145(2):367-393, 1990.(191 C. Sam son an d I


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[28] Xiaoping Yun and Vi jay R. I


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control of mechanical systems with rolling constraints

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