Real Time Trajectory Generation for Differentially Flat Systems

Copyrighl © 1996 IFAC 13th Triennial World Congress, San Francisco. USA

2b-13 2

REAL TIMJ<; TRAJECTORY GENERATION FOR DIFFERENTIALLY FLAT SYSTEMS· Michiel van Nieuwstadt

Richard M. Murray

Mechanical Engineering, California Institute of Technology

Abstract. This paper considers the real time trajectory generation problem of how to generate, possibly with some delay, a full state space and input trajectory in rea] time from an output trajectory that is given online, whilf! allowing a tradeotf between stability and performance. The paper propo~es two algorithms that solve t.hf! real time trajectory generation problem for differentially flat systems wiUI (possibly non-minimum phase) zero dynamics, and discusses some interesting propcrtics. It explicit.ly addres.<>es the tradeoff between stability and performance. The algorithms are validil.t~d in sjmlllati{)n~ and experiments with a vectorcd thrust ducted fan aircraft. Keywords. Nonlinear control, flight control I trajectory planning

1. INTRODUCTION In this pape-r we describe aJgorithms to generate trajectories in real time for differentially flat nonlinear systems. This is part of a genera] control paradigm for nOlllinear ~ysteIIls depicted in Figure 1.

..

..

I

[""<2\:'

.. '

!I~"<:r<>L,u,

-

.

.,-=h~dUI~d

"'nom,

u"o,,"

[

+

h .. ~,.,.

cDn~_I~r

. ,.,

_

._. .- ... _, .. ,

Fig. 1. Paradigm for nonlinear control. ThiS framework typical1y applies to vehicle control I whence the term "piloC fo-r a de.sired t.rajectory. In this framework, the pilot gives a rate of change of the tracking variables. The tra.jedory generation module generates a nominal state space trajectory a.nd a nominal con-

trol input. The t.rajectory gew~ration can happen amine, in pseudo real time (i.e. at a rate a few orders of magnitude slower than the sampling rate), or in real time, depending on the particular problem. In the real time case, the trajedory is being updated at t.he same rate as new pilot input becomes available, with some delay due to computation time. A sch(·duled linear c.ont.roller is used to correct for errors. In the ca..;;e of non-minimum phase systems the delay is also necessary to keep the internal dynamics bounded. Related work is reported in (Meyer et al. ] 994, Marlin et al. 1991, van Nieuwsladl and Murray 1995), wilh the main difference that trajectory gene.rat.ion is performed essentially offiine. Trajectory generation for nonminimum phase systems is di~cussed in (Chen ]994, Devasia and Paden 1994). This approach results in allticausal trajectories and is thl~rcforc Cl fortiori omine. The main contributions of this paper are two algorithms . for real time t.rajectory genef;Ltioll for differentially flat systems with unstable zero dyaarnics l with an analysis of their convergence properties. We explicitly address the tradeoff between performanc.e and stability for this class of systems, in the form of a wl~ighLcd cost minimization.

'" Research supported in part by NSF Craflt CMS-9502224 and AFOSR Grant. F49620--95-1-0419

2301

2. DIFFERENTIAL ~' LATNESS Oifferential flatness was originally introduced by (Fliess

et at. 1992). in a differential algebraic context. In (van Nieuwstadt et al. 1994) it wa..'" reinte rpreted in an exterior different.ial systerml t:let.t.ing, Ttw irnporLani property of flat systems is that we can find o utputs SUl':h that wc can express all states and input.s in terms of those outputs and t.hr.ir derivatives . The number of flat. outputs we aTe a.l\owed to pick is eq ua l to ilK lIumber of inputs. That is, we havt~

:i: = f(x , u) h(x) z (x) (x , u) = 1b( z, i , ... , z
=

(1)

for some lIlap 1/1. Here y a re the Lra.c..:king outputs and z a.re the fiat output!;. We st.ack the flat. outputs and their derivatives in the flat flag Z. They are not nec.essarily the same. Tracking the outputs of in terest will result in possibly unstabJ e zero dynamic.s. Pnramctrizing the flat o utputs allows us to trade off the magnitude of the zero dynamics with th e tracking error.

3. THE REAL TIME TItAJI ~CTOIlY GJ;NJ;IlA'l'ION I'ROBL!>;M

The most straightforward approach to tra.jectory tracking 'is to ·silhtrac1. the plant input from the pilot input and feed this error signal t o the controller. This is the so called "onc degree of freedom" approach, We will show in Section 5 that Lhi.s may lead to !)low re!Jponse times . The re~on i!j that we are trying to track a drifting equilibrium configuration wh ich is an unfeasiblc trajectory. Trajedor~' generation tries to remooy thi~ problem by finding a feasibl e full stal.c and input trajectory along which the system can be stabilillecl Usually conlrol objectives are statNt iL,-~ performance criteria subject to stability. For n'al time traj ectory generation wC'. only haY(' a finite t.ime bi~tory of the desired t rajectory availahlp-, and therefore stability as defined in an infillite Lime horizorl doesn ' t m ak(~ sensp.. Instead we can capture the notion of stability a.s some norm bound on the internal dynamics generatt.x:I when following a desired trajcct.ory. The "performance under stability" requirem cllt. then translates to minimizing a weighted norm between tracking error ana magnitude of the internal dynamics. In agreement. with 'Hoo control theory we takp- tbis norm t.o be th e L2 norm 011 a finite tim e int.ervaL ThiS leads to the followirlg cost t.o be minilrli ~ed at each time instant:

1"

, -~

("(x) - Yd(8)r
+ >.J{(x,u)ds

~

where K is an appropriate penalty on thp- internal dynamics, and 'id defines the time horizon, or the delay with which the trajectory is generated. This formulation allows a tradeoff between performance and stability, as seen in (van Nieuwstadt, and Murray 1995). We can increase stability at the exrense of performance by increasing t,t. P- penalty on the internal dynamics (i .r. . ..x). SiDce we ha\'e to minimi7.e the cost in equation (2) at every time instant, we need to do this subject to fixoo initial conditions, namely, the l::iia.te that we happen to be at. In (Grizzle et al. 1994) it was sbown that under fairly mild condif,ions a necessary condition for asymptotic.: t.racking (and t.herefore for «act tracking) is that the system have stable zero dyn 'lmics. The authors prove this by constructing a signal !.hat cannoL be asymptotically trackf.d by non-minimum phase systems. An esse'Jtial fcature of thi~ signal is that it. has a time derivative with infiniLe support, One way to circumvent. the nonminimum phase zero dynamic:!'I requirement is to restrict a.ttention to asymptotic t.racki ng of signals whose derivatives have finite support. More precisely, we define

where t~ is not given in advallce. We require that our trajectory generation scheme achieve asymptotic trac.:king for all signalti in 8. This ('..om~ down to requiring zero steady state error. We assume that to each value of the output Yd , there is an equilibrium value for thr. states ami inputs, i. e. there ex ist (Xd , Ud) such tha.t Yd == h(Xd), f( Xd, Ud) == O. If this is not t.he case , we cannot maintain the output at the desired constant value. We denote th e mapping that maps each output value Vd to a full state and input space equilibrium by Hq, so that. f(Hq(Yd)) '" 0, and h(Eq(Yd)) = Yd . Based on t.he above djscllssi olJ we propose to study the following problem : Problem I. (Real time trajectory generation). FiDd ao algorithm that calculates in real time from Yd(t) a fe= h(Xd(t), Ud(t)) - Yd(t) = 0 for all Vd E S.

4. TWO ALGOIU'I'HMS FOR TRAJECTOIlY GENERATION

We wm now propose a solution to the above problem. First, we paramctrize the fl a L outputs Zi, i J ... m by

=

z,(1) = (,(x(t)) := Aij
=

(3)

where the cPj (l), j 1, , . N are basis functions . Tbis reduces Lhe problem from finding a function hl an infinite dimensional space to finding a finite set of parameter::!, At each time t we have availabJe t,o us the desired out-

2302

over the time interval [10 ,1,] := [I -1d ,I]. Steering from an initial point in sta.te space to a desired point in state space j" trivial. Wc ha.ve t.o cakulat.c the values of the Hat outputs and their derivatives fmm the desired points in sta.te space and then solve for the coefficients Aij in th e following system of equations: pllt

at sampling time to , the doHC'!d line is the trajectory generated one pf!riod later .

.. ...

... " ' -. ,',,

I tu

(4)

To streamline notation wc write the following ex~ pre~Rionlo1 fo r the ca.<;e of one flat output. only. The multi-output. c:ase follows by repeatedly applying the singe output. ease, since the algorit.hm decouples in the flat outputs. Let <1>(1) be th" I + 1 by N matrix <1>;111)
=

=

=

(z.(I,) , ... ,zl')(tf)) , z = (i o,if) then the constraint in equation (4) can he writ.t.clI
(5) That, is , wc require the coefficients A to be in the plane defin ed by equation (5). The only condition on the basi, fundions is that ~ is full rank, in order for (5) to hll.ve a solution. We can solve these equations at each sample instant to generaLe a trajectory from the current state to the desired output a certain time 'Id. lat.~r. Wc augment this desired output to a desired fnll state and input by mapping it onto an equilibrillm with the mapping Eq . On this t.rajcr;f.ory wc pick a state ~;orrC8ponding to some tiJI~c T E ~lo l lJ) and use this as the instanlR.ous ri~ir~ state for the linear controller. This leads t.o the first al go', ithm : AlgQ~'ithm 1. Given: t.he delay time 7"d" the current flat flag Zo , tlic desirr.cI Out.put. Yd. At each sampling instant

I. :

i,

(I) Let t f = t" to = lk - T", ~ «l>q(Yd(t.))) . (2) Solve to'- <1>(to)A, z/ = (lf)A for A.

=

(3) Let '" (r)A where r '" [Io,t,]. (4) Solve for (x"",) from ". (5) (Xl,"" is the next. desired stat.e and input to feed.forward ai, tim e lk. This algorithm st.~rs us from l.lte current position to an equilibrium state with t.he desired valUC8 for the outputs. We generate a trajt..'Ctory over ihe time interval {tk -1d ,lkL and pick a time T and corresponding point (Xl , ud on this t rajecl.ory. This will be the desired state to steer to. We repeaL this process at every sampling instant. This is illustrated in Figure 2. The solid line is the pilot input., th e da.-;hcd line is the generated trajectory

+r

.....

~

. ..• "

" ,

z;('I (to) = Ai.i
". "'.

... ,

toI + 2r

I -_.- t/ - tn+ld t/

I

+T

Fig. 2. Algorit.hm for real tim e trajectory generation.

This algorithm does not involve the explicit minimization of a cost. fllDct.ion to trade off stability versus performance. We will show through simulations in Sedion 5 that. t.he parameter 1d. regulates this tradeoff. lncreasiug Td will increase stability at th e expense of performance. We can bypas.~ solving for the coefficients Ai; matr,x A by noting that " = (r)- 1i =: I'(r) i o + G(r)l'J.

In

the

(6)

If we execute this sche.mp. every sample instant we get a. dynamical equation for Zl :=: ik+l for eill:h Zo =: ik, namely:

(7) which has the desired output %/(k) time instant k for its input.

= ((Eq(Yd(tk))) at

Proposition 1. There is aT E [to,tf] ,uch that Algorithm 1 achicvc~ real t.ime asymptotic tracking of all desired out.puts in S.

PROOF. We will show that F(r) is stable for appropriate choice of T, and then that the steady state error is zero for Yd E S . Since we constructed the F(r) , G(r) to stf:er us from Zo t.o i J I it. follows that G(I. f) ~ 0 a.nd F(t,) = I . So for T = tf all eigenvaluc. o f F(T) are at the origin. Sine~ the eigenval1Jf~s of F(T) are continuous functions of T, t.here cxist~ H T E {t o.t,] such that. t.he eigenva.lues of F( T) arc in the open unit circle. Now Yd E 8 means that there is a ka such that. i 'j(k) is a constant, say if, for all k > k s . Therefore Zk converges to a constant valm~, say Zoo whidl wi1l be a multiple of zf due to lillearity of (7). So Zoc = "(iJ, where I depends only on F and G. Since there is a traj ec tory from Zoo to zf t.hat. will bring us closer io Zj for appropriat.e choice of T, we have 'Y = 1 for t.hat. value of 1" . Then we have limk-+oo ZII: = Zt· Picking ld = t" - t" _1 in t.he above scheme corresponds to a one step deadbeat. controller. This requir~ large c.onl.rol signals which might satura.t.e the actuators. Clearly, ther~ is a tradt:off between performance and control effort and bandwidth . Nore that the matrices F(r) and G(r) are fixed 0 nc.e T is selected, and can

2303

be compu led ahead of lime. We should rnenlion lhal it is 1I0t hard to find r such th at F(r ) is stable. In fact, it requires cOJl~)dcr ah lr. effort to construct a set of ba.sis fuuctioll6 and a T sudt that F(T) i~ uJlstable. For polynomial basis functions allY r E lto, 1,[ will do. This follows from the fact t hat the degree of a polynomial is an upper bound on the number of it.s zeros. Next we a.ugment. t.his algorithm wit.h a.n additional m in-

imization that a.llows tradeoff betweell stabilit.y and performance as discussed in SectiOIt :L The cost criterion Lakes the form; J

= mln

1'1

to the desired output in one step. Regardles.c: the values fo r All from continuity of z) -=
(9) that if zJ(k) is a. constant for k ;::: k~l say zl, we achieve convergence to a cOllslaD t va]ue for Zk. Similarto the proof of Proposition 1 we l'an show that t his constal1l value ha..;;; to be if.

80

It might seem curious at

(y(A, s) - Yd(S))' (y(A, s) - Yd(S))

t oo

HK(A, . )d. (8) s ub.i~ct

to Zo = <1>(lo)A, Z, c= <1> (If )A . Here Y is lhe tracking ou tpu t , a nd Yd the dl~ired t.racking output. j( is a. fundion that bounds th e internal dynamic:;, We eau perrorll thil5 minimization by finding a pa rticular solution -that satisfies th e initial and fina.l cons t.rAints: Ao Cl>t z• and parametrizing the general solution as A = Au + <}>1. A 1 where 4>.1 i!> a. ha..~i 8 for the nullspace of <1>. Thi!> optimi7',at.ion prohlem i!> i n general nonJin ear and nom:ollv ex. We therefore have 1.0 resort to an iterati ve scheme. Since the optimization ha.c;; 1.0 he perform ed in real t ime, we mi ght not be able complete th e mini m ization proclXi ure and have to prccmpt the procedure. We will ~how 'th a t this will not. result in loss of convergence. This leads to th e followjng algorit,hrn: Algorithm 2. Given: the delay time '[d, th e current Hat flag zo ) t. he desired output Yd. At each sampling instant tk:

=

=

(1) tct I, I. , to Ik - T d , Z f «liq(y(lk))) . (2) Find a part icula r solu t ion li D lo '0 <1>(lo)A, zf . (t,)A. (3) Optimizc A, to minimi.e J in equaLion (8) . (4) LetA ~ Ao+NlA I' (5) 'Let 'I = <1>(T)A where r tC (10, If l . (6) Solve for (Xl, "I) from i,. (7) (X I ,U,) is l he next desired slate and input to feedforward at time tk.

=

=

5. SIMULi\TIONS Sirnulations are performed wit. h an approximate flat model of the Caltech ducted fan . The ducted fan is a simplified model of the pitch dynam ics of a vcctored thrust. aircraft. See (Ka.ntncr et al. 1995) for a detailed dc~crjDti()l1 of this apparat.u~. The ductcd fan is mounted o n a stand wif.h .::t counterweight that moves in as the fan moves up. T his results iD inertial masses mx and my in the z and y direction respectively, that C'h ange with the y coo rdin ate. We do not take UJ e variat.ion of these inerti al masses with y into account Lut t.ake t.heir value around hover. The count erweight. also resulLt; in an effecti ve weight m: differellt than t.he lIJasses in x and y di rection. We can apply any force 0 11 the cenl.er of mass hy adjusti ng t.he magnit.ude and t.h e direction of the thrust . A fter shifting the control va riables to compensate for ~ rav i ty, and decomposin g t.h em into a parallel and perpendicular component, the eq uations of motion are: mxx == -m z 9 sin (J + cos OUI - sin (JU2 rrtyY = mzg(cos () - 1) + $;1 n 0«1 + cos BU2

Note tha.t th e opt.imiz;ation over A 1 can be preernpted if computati'o n time rUJls out. Proposition fJ. There is a T f: (t o, t j] S11Ch t hat Algorithm 2 achi eves real t.ime fL
PROOF.. We will show t hat

sight tha t convergence

J. On second thought this is quite advantageous since we cannot guarantee that Lbe optimization of J converges in the allotted computation t ime. Preemption of the rninimj:.o,aiion win not res ult in loss of convergence . The additiolJal opLimization a.llows us t.o get better performance (in the sense of a. lower cost criterion .1) if the c:omputaf.ion time aJlowH it.

=

=

fir~t

of Algorithm 2 does not dep end on the cost criterion

i ) cOI1\'c rgaJ to a constant value for const.a.nt. zf : and then that this constant value (.'
JO =

TU,

(10)

where (x , y) are the coordinate!:; ccnter of the center of mass , () is the angle with thl ~ vertical I Ul is the force perpendicu lar to the fan shroud , «2 is the forr:e parallel to the fan shroud l r is the distance between t he Cf.~ nter of mass and the point where th e force is a pplied , 9 is tbe gra.vitational constant, m z ) my is the inertial mass of the fan in the (x , y) direction resp~tivelYI nlzg is the weight of the fan , and J is the mom ent. of inertia. The tracking outputs are the (x, 11) coordin;J.t~ of th e cClller of mass. Analogous to (Martin et al. 1994) t.h e fl a t outputs are

2304

xf =x -

J

."

- - SIn[]

rn,'

Yt

J = y+ -- 00.0. nlyr

(11)

~

Note that. these outputs are not fixed in body coordinates. The variable () can be cxpres8ed in terms of t.he flat output.s as LanO

-m.x. X }

= myth + rn"g'

0." _ 0 . ... ..

0 .0

:!: •

(12)

From 0 and th e fla.t uutputs we can find the other states and the inputs. We simulate online pilot input as a file from which successive samples are read every sam p le instant. The pilot command is input to a trajectory generation mouule, whose o utpul. serves as a no mina l trajecLory. \Ve wrap a simple static LQn. cont.roller around a noolioea r model of the fa ll . This cont roller wa.c; dC'..signoo to stabilize hover. S~ (1
T he nonlincar mood used for tiilIlulatjon takes into account t he . aerody namic. drag , ine rt ial effect.ij from the rotat.ing propellor, chaugiIlg inertias with a.ltitude , and viscous fridion . This model i ~ more elaborate than the flat approximation u~ e d to generate t.h r. nominal trajcctorie~,

First we show how th e duc.ted fan behaves without feedforward. The pilot input is llsed 1.0 gCllerate an error signa l to l. he coot.roller. At each time instant we stabilize around t he equilibrium point generated by setting x alld y equal to th e pilot inpu t, and all other states equal to n. T his is th(~ conventional "onc degree of freedom"controller. Figure 3 shows that th e t rajec tory fol lowed by the fan lags far behind the desi red trajectory.

.. _ 0 .0

Fig. 3. One degree of frr:cdoJll controller. in real time trajectory generation. Figure 5 shows these for a delay Lime of Td 100 sampling periods. It. is clear that. the larger delay results in b etter stahil it.y, i.e. lower magnitude of () a nd the nominal forces, hut poorer performance, since t he d elay is bigger.

=

Figures 4 and 5 both show a. large error in (J right after th e first and second peak of the nominal () trace. Wc suspect this is caused by the inertia changing with altitude, whi ch is not t.aken into accollnt. in the flat model, but is present in t.he simulal iOll model. Note t,hil.t. the errors occur simultaneously wilh a substantial error ill altitude y. We tested Algorithm 2 in si mul ation . It behaves as expected, in the sense that it pl:nalizes the cost. The major problem is that. the optimization is a factor 10 too slow fo r rcali~tic operator sa mpling rates. Im provement. of this op timization is a subj!:d of current research.

,,, . ,.,.,,,, ~:

o. 0

--,

.. 0..

, (\ _,

r

V,,~.,

_---I

L ~., . '-0

In this piot and subsequent plots, the pilot input is deooted by (xp, yp ), the ge nerated desiroo t rajecto ry , which is identical to t he pilot input. in thiij case, is denotea (xd, yd), whereas t he vari iLbl(' lI a rn e without suffi x de.notes the rcal (experimental or simulated) t ime trace of Cl. quantity. Tht! force parallel to t he fan sh roud is deHoted "para", the force perpendic.ular to th e fan shroud is called "fperp" . Next we ·.show for Algorithm 1 plots of t he p ilot input, th e generat.ed trajecto ry: the simulated trajectory for the (x , y) position of the fan , as well a.s the generated and simulated trajecto ry for 0 a nd the nominal forces. The pilot input. is ;\. 1.0 meter step in the r d irection at c.onstant altitude. Figu re 4 shows these for a delay tim e of 1'rJ == 60 sampling periods of c/~ 0.0 1 seconds . We see thrLt. the fan follows the pilot input. much hetter than iu the one deKH.!C of freedom dcsigll . C lea rly: there i::; a n advantage

Fig. 4. Algorithm I , Td

!

0: _

= 60

(1;

= 0.6 s .

:7 ;:-1~ :~F"~~:~:~-':~~~=

'.r

· ..·..0 --

.

,_(,,]

~ ---

..

''"<->~! C. - ',. ' ! _0.° _ ._,v,. '/ -/7\00

-'''0 -

,"

.'

..

..

Fig. ,). Algorithm 1, Td

" '0

z

- - - -..

01 ..... '.1

- - --- -...

:l---'""""'-""""'=:J

~ l-C:-~-··:=": ~:_:J -0--

i

;,

----....

= J 00 X 1~ = 1.0 s.

6. bXPERIM ENTAL DATA

=

Algorit.hm 1 was irnplementp.d on t he experiment.al apparat.u~ , T he pilot input com('s from a joystick with two

2305

degrees of freedom. We run the trajectory generation algorithm at 100 Hz, and the controller at. 200 Hz. The delay time '1~ = 1.0 seconds, corresponding to 100 samples for the trajectory generation algorithms.

We conducted 2 experimentti. Th(~ first. one was the one degree of freedom controller: the pilot signal was used to generate an error signal in t.he output. a.round which the fan was stabilized. The results are depicted in Figure 6. The generated desired trajectory, is identical to the pilot input in this case. In t.he second one, we used the pilot. signal to generate a t.rajectory, a..<; descrihed in this paper. The result.s are depicted in Figure 7. The desired trajectory (xci, yd) is generated' by the trajectory generation module and is no longer equal to t.he pilot. input. (xp, yp). Since t.he pilot input. is given real time the experiments are not rcpeat.ahtt~. We can draw sornc qualitat.ive conclusions though. The real time tra.jectory generation algorithm gives a more aggressive response, at t.he expense of more oscillations in the pitch angle 8. This could likely be remedied by implementing Algorithm 2. Even so, the mean squared pitch error for the real time trajectory gcncratiOI~ is less than for the onc degree of freedom controller. The real time implementation performs consid· erabi y worse than the simulations. We submit. t.hat this is due to floise ifl the pilot input and plant uncertainty. The pilot input from the joystick has a dead zone of about 5 ticks on a total range of 2~iO. The flat model used to generate traject.ories is or course only an approximation of the real dynamics. Also, it. can be seen from the plots that the required actuator bandwidth is high. Limit.ing t.he nominal actuat.or bandwidth can be included as part. of the cost crit.erion by weighting basil:! functiolls with high frequency content different. than basis functions wit.h low frequency content.

Fig. 6. Experimental data: one degree of freedom cont.roller.

7. CONCLUSIONS In fhis paper, wc proposed a formulation for the real time trajectory generation prohlem. We described two algorithms for real time trajectory g{'neration for different,ially flat systems wit.h Ilnsta.ble "ero dynamics, and

Fig. 7. Experiment,al data: real t.ime t.rajectory generat.ion. proved stability and convergence properties. The first algorithm generated a trajectory that steers from the current position to a desired final position given by the pilot. input. Wc can tradc off stabiJity versus performance by varying the delay time. The Sf~cond algorithm steers to a desircd final position while minimizing a cost. criterion 1 that. typically limits the magnitude of the zero dynamICS.

8. REFER F,NCES Chcn, D. (1994). An iterative ,;olution to stable inversion of nonminimum pha.sc systems. In: Pmc. American Control Conference. pp. 2960-2964. Dcvasia, S. and B. Paden (19:)4). Exact. output tracking for nonlinear time-varying systems. In: Proc. IEEE Control and Dec;s;o" Co"fer~nce. pp. 2346-235.5. Fliess, M., J. Levine, Ph. Marl-in and P. Rouchon (1992). Sur lcs systernes non lineaires differenticllement plats. In: C.R. Acad. ,),,,. Pans, t. 315, .'!ene I. pp. 619-624. Gri""le, .LW., M.D. Di Belll~etto and F. LamnahhiLagarriguc (1994). Ncces:'!ary conditions for asympt.ot.ic t.racking in nonlinear systems. lEER Tmnsaclions on Automatic Control 39(9), 1782-1795. Kantner, M., 13. Hodenheimcr, P. Rendot.ti and R.M. Murray (1995). An experiment.al comparison of cont.rollers for a vectorcd thrust, duct.ed fan engine. In: Proc. AmeT1ccm Control Conference. p. 1956. Martin, Ph., S. Devasia and H. Paden (1994). A different look at output. tracking: control of a VTOL aircraft. In: Proc. IEEE Control and Decision Conference. pp. 2376-2381. Meyer, G., L.R. Hunt and R. Su (1994). Nonlinear syst.em guidance in thc presence of transmission zero dynamics. Technical Report Draft 5. NASA. van '1icuwstadl, M. and H.M. Murray (1995). Approximate trajectory generat.ion for different.ially Oat syst.ems with zero dynamics. In: Proc. lEER Control and Decision Conference. pp. 4224-4230. van Nieuwstadt., M., M. Rathinam and R.M. Murray ((994). Differential /latnes. and absolute equivalence. In: Pmc. lEEk' Control and JJer.1.sion Confernlcc. pp. 326 333.

2306