Collocation and inversion for a reentry optimal control problem

Transcription

1 Collocation and inesion fo a eenty optimal contol poblem Tobias NECKEL Chistophe TALBOT Nicolas PETIT 3 - École Polytechnique, 98 Palaiseau Cedex Fance TobiasNeckel@cnesf - Cente National d Études Spatiales, CNES - DDA/SDT/SP Ey, Rond Point de l Espace 93 EVRY Cedex Fance ChistopheTalbot@cnesf 3 - Cente Automatique et Systèmes, École Nationale Supéieue des Mines de Pais 6, bd Saint-Michel 757, Pais Cedex 6, Fance petit@casensmpf Abstact The pupose of this aticle is to poide the eade with an oeiew of an inesion based methodology applied to a shuttle atmospheic eenty poblem The poposed method oiginates in the seach fo computationally efficient tajectoy optimization as an enabling technology fo esatile eal-time tajectoy geneation The technique is based on the nonlinea contol theoy notion of inesion and flatness This point of iew allows to map the system dynamics, objectie, and constaints to a lowe dimensional space The optimization poblem is then soled in the lowe dimensional space Eentually the optimal states and inputs ae ecoeed fom the inese mapping Intoduction The pupose of this aticle is to poide the eade with an oeiew of an inesion based methodology applied to a shuttle atmospheic eenty poblem This poblem has a 6 states, contols nonlinea dynamics with teminal and initial constaints and a teminal cost function Aeodynamics models linea fo lift and quadatic fo dag ae consideed Gaity and ai density ae modelled accoding to the classic non otating spheical eath potential and exponential models The poposed method oiginates in the seach fo computationally efficient tajectoy optimization as an enabling technology fo esatile eal-time tajectoy geneation Tajectoy geneation of unmanned aeial ehicles is an example whee the tools of eal-time tajectoy optimization can be extemely useful In [9, 3, ], this new technique was pesented and used to sole such poblems In [] this methodology was applied to fomation flight of micosatellites unde J gaitational effect Following the same ideas the eal time tajectoy geneation of a plana missile was addessed [] with simila dag and lift models The technique is based on the nonlinea contol theoy notion of inesion [7] and flatness [3, 4] This point of iew allows to map the system dynamics, objectie, and constaints to a lowe dimensional space The optimization poblem is then soled in the lowe dimensional space Eentually the optimal states and inputs ae ecoeed fom the inese mapping The example teated in this epot has inteesting featues Fist it is moe complex in tems of dimensionality and nonlineaities than the peiously cited examples Second the dynamics ae not flat In othe wods it is not possible to fully inet the system dynamics This paticula situation desees a caeful teatment of the paametization of the states aiables Numeical esults ae gien, and a compaison with existing techniques fo this example [] is gien In shot, the poposed appoach appeas tacktable, but could be impoed futhe by paying moe attention to the choice of the nonlinea pogamming sole and the finite dimensional epesentation that ae used Backgound infomation In this section we pesent the geneal famewok of inesion-based collocation methods fo numeical solution to optimal contol poblems Most of this mateial can be found in [3] We addess the simple single-input case which is by fa the most easy and emphasizes the ole of inesion Optimal Contol Poblem Conside the single input nonlinea contol system ẋ = fx + gxu, R t x R n, R t u R whee all ecto fields and functions ae smooth functions It is desied to find a tajectoy of [t, t f ] t x, ut R n+ that minimizes the cost Jx, u =φ f xt f, ut f + φ xt, ut + tf t Lxt, utdt, whee L is a nonlinea function, subject to a ecto of initial, final, and tajectoy constaints lb ψ xt, ut ub, lb f ψ f xt f, ut f ub f, lb t Sx, u ub t,

2 espectiely Fo conciseness, we will efe to this optimal contol poblem as min Jx, u x,u subject to 3 ẋ = fx + gxu, lb cx, u ub Diffeent appoaches Classical collocation One numeical appoach to sole this optimal contol poblem is the diect collocation method outlined by Hagaes and Pais in [6] The idea behind this appoach is to tansfom the optimal contol poblem into a nonlinea pogamming poblem This is accomplished using a time mesh t = t < t < < t N = t f 4 and appoximating the state x and the contol input u as piecewise polynomials ˆx and û, espectiely Cubic polynomial may be chosen fo the states and a linea polynomial fo the contol on each inteal epesents a good choice Collocation is then used at the midpoint of each inteal to satisfy Equation Let ˆxxt T,, xt N T and ûut,, ut N denote the appoximations to x and u, espectiely, depending on xt T,, xt N T R nn and ut,, ut N R N coesponding to the alue of x and u at the gid points Then one soles the following finite dimension appoximation of the oiginal contol poblem 3 min F y = Jˆxy, ûy y R M subject to ˆx fˆxy, ûy =, lb cˆxy, ûy ub, t = t j + t j+ j =,, N 5 whee y = xt T, ut,, xt N T, ut N, and M = dim y = n + N Inese dynamic optimization In [5] Seywald suggested an impoement to the peious method see also [] page 36 fo an oeiew of this method Following this wok, one fist soles a subset of system dynamics in 3 fo the the contol in tems of combinations of the state and its time deiatie Then one substitutes fo the contol in the emaining system dynamics and constaints Next all the time deiaties ẋ i ae appoximated by the finite diffeence appoximations to get xt i = xt i+ xt i t i+ t i p xt i, xt i = q xt i, xt i } i =,, N The optimal contol poblem is tuned into min F y y R M subject to p xt i, xt i = q xt i, xt i whee y = xt T,, xt N T, and M = dim y = nn As with the Hagaes and Pais method, this paameteization of the optimal contol poblem 3 can be soled using nonlinea pogamming The dimensionality of this discetized poblem is lowe than the dimensionality of the Hagaes and Pais method, whee both the states and the input ae the unknowns This induces substantial impoement in numeical implementation see again [5] fo an implementation of the Goddad poblem 3 Poposed Numeical Appoach In fact, it is usually possible to educe the dimension of the poblem futhe Gien an output, it is geneally possible to paameteize the contol and a pat of the state in tems of this output and its time deiaties In contast to the peious appoach, one must use moe than one deiatie of this output fo this pupose When the whole state and the input can be paameteized with one output, one says that the system is flat [3] When the paameteization is only patial, the dimension of the subspace spanned by the output and its deiaties is gien by the elatie degee of this output Definition [7] A single input single output system { ẋ = fx + gxu y = hx is said to hae elatie degee at point x if L g L k f hx =, in a neighbohood of x, and fo all k < L g L f hx whee L f hx = n i= h x i f i x is the deiatie of h along f Roughly speaking, is the numbe of times one has to diffeentiate y befoe u appeas Result [7] Suppose the system 7 has elatie degee at x Then n Set φ x = hx φ x = L f hx φ x = L f hx If is stictly less than n, it is always possible to find n moe functions φ + x,, φ n x such that the mapping φx = φ x φ n x 6 7

3 has a Jacobian matix which is nonsingula at x and theefoe qualifies as a local coodinates tansfomation in a neighbohood of x The alue at x of these additional functions can be fixed abitaily Moeoe, it is always possible to choose φ + x,, φ n x in such a way that L g φ i x =, fo all + i n and all x aound x The implication of this esult is that thee exists a change of coodinates x z = z, z,, z n such that the systems equations may be witten as ż = z ż = z 3 ż = z ż = bz + azu ż + = q + z ż n = q n z whee az is nonzeo fo all z in a neighbohood of z = φx In these new coodinates, any optimal contol poblem can be soled by a patial collocation, ie collocating only z, z +,, z n instead of a full collocation z,, z, z +,, z n, u Ineting the change of coodinates, the state and the input x,, x n, u can be expessed in tems of z,, z, z +,, z n This means that once tanslated into these new coodinates, the oiginal contol poblem 3 will inole successie deiaties of z It is not ealistic to use finite diffeence appoximations as soon as > In this context, it is conenient to epesent z, z +, z n as B-splines B-splines ae chosen as basis functions because of thei ease of enfocing continuity acoss knot points and ease of computing thei deiaties Both equation fom the dynamics and the constaints will be enfoced at the collocation points In geneal, w collocation points ae chosen unifomly oe the time inteal [t o, t f ], though optimal knots placements o Gaussian points may also be consideed and ae numeically impotant The poblem can be stated as the following nonlinea pogamming fom: min F y y R M subject to ż + y q + zy = ż n y q n zy = fo eey w lb cy ub whee y epesents the unknown coefficients of the B- splines These hae to be found using nonlinea pogamming 8 4 Compaisons Ou appoach is a genealization of inese dynamic optimization Let us summaize the pesented appoaches One could wite the optimal contol poblem with: Full collocation soling poblem 5 by collocating x, u = x,, x n, u without any attempt of aiable elimination Afte collocation the dimension of the unknowns space is On + Inese dynamic optimization soling poblem 6 by collocating x = x,, x n Hee the input is eliminated fom the equation using one deiatie of the state Afte collocation the dimension of the unknowns space is On Flatness paametization Maximal inesion, ou appoach, soling poblem 8 in the new coodinates collocating only z, z +,, z n Hee we eliminate as many aiables as possible and eplace them using the fist deiaties of z Afte collocation, the dimension of the unknowns space is On + 3 The uled manifold citeion When facing a new system dynamics, it would be inteesting to know wethe these can be fully ineted o not The single-input case pesented befoe is the exception Unfotunately, up today, thee does not exist any flatness citeion Neetheless the following necessay condition can be a handy tool to check wethe one may completely inet a system This necessay condition fo a system to be flat is gien by the following citeion [4] see also [8] Result [4] Assume the system ẋ = fx, u is flat The pojection on the p-space of the submanifold p = fx, u, whee x is consideed as a paamete, is a uled manifold fo all x Eliminating u fom the dynamics ẋ = fx, u yields a set of equations F x, ẋ = that defines a uled manifold In othe wods fo all x, p R n such that F x, p =, thee exists a diection d R n, d such that λ R, F x, p + λd = 3 The eenty poblem In this section we pesent the eenty poblem We detail the nonlinea dynamics, the constaints and the cost function We show that this system is not flat and explain how to paameteize its tajectoies using a educed numbe of aiables and additional constaints Finally we gie a ewiting of the optimal contol poblem in tems of this educed numbe of unknowns 3

4 3 Dynamics As detailed in Betts [], the motion of the space shuttle ae defined by the following set of equations ḣ = sin γ 9 φ = cos γ sin ψ/ cos θ θ = cos γ cos ψ = Dα m γ = Lα ψ = g sin γ m cos β + cos γ g 3 Lα sin β + cos γ sin ψ sin θ 4 m cos γ whee h denotes the altitude, φ the longitude, θ the latitude, the elocity, γ the flight path, ψ the azimuth The two contol ae α the angle of attack and β the bank angle 3 Contol objectie and constaints Hee ou poblem is to maximize the final alue of the θ aiable in a gien time t f The initial conditions ae pescibed as h = 6 ft φ = deg θ = deg = 56 ft/sec γ = deg ψ = 9 deg In the numeical example teated in this epot the final time t f equals 859 s The study is esticted to the tajectoy satisfying h, 89 deg θ 89 deg, 89 deg γ 89 deg 9 deg α 9 deg, 89 deg β 89 deg The final point of the tajectoy is defined by the teminal aea enegy management TAEM inteface which is defined by the following elations ht f = 8 ft, t f = 5 ft/s, γt f = 5 deg 33 Physics constants and paametes We use µ = e7 as gaitational constant, Re = 99 ft as the adius of the Eath, S = 69 ft as the aeodynamic efeence suface, h ef = 38 ft and ρ =378 fo the following physics paametes g = µ/ 5 ρ = ρ exp Re/h ef 6 We use C L = a + a α whee α is in deg, a =-74, a =944 Lift is then gien by L = C LSρ 7 Also we note C D = b + b α + b α, whee b =7854, b = -659e-, b = 648e-3 and use it in The mass of the shuttle was chosen as D = C DSρ 8 m = lbs 34 The system is not flat We use the uled manifold citeion pesented in section 3 to poe that the system is not flat Eliminating the contol fom the eenty dynamics yields an equation F x, ẋ = To get this equation we hae to sole fo the unknowns α and β in tems of the states and its deiaties Fist one may pick equation to get Dα = m mg sinγ Then sole accoding to the physical model 8 to get α = b ± b 4b m +g sin γ b + ρs 9 b On the othe hand it staightfowad to sole fo β using equation 3, equation 4 and the fact that 89 deg β 89 deg This gies cos γ ψ cos γ sin ψ sin θ β = actan γ cos γ g Using these last two elations in the eenty dynamics we get the manifold equation F x, p =, whee p = p, p, p 3, p 4, p 5, p 6 T = ẋ satisfy p = sin γ p = cos γ sin ψ/ cos θ p 3 = cos γ cos ψ 3 and Equation 4 Now let us look fo a non-zeo diection d = d, d, d 3, d 4, d 5, d 6 T R 6 such that at a point x, p such that F x, p =, fo all λ R, F x, p + λd = The fist thee equations,, 3 gie which gie p + λd = sin γ p + λd = cos γ sin ψ/ cos θ p 3 + λd 3 = cos γ cos ψ d =, d =, d 3 = Equation 4 gies afte using the simplification sinactan x = x +x Equation 5 This equation must hold fo all λ R Afte taking the squae of the last expession, the squae oot in the last expession inoling d 4 is the only one that still contains a 4

5 p 6 = ρs 8 a + a m cos γ π sin actan cos γp6 b ± b 4b b + mp 4+g sin γ p 5 cos γ g b cos γ sin ψ sin θ ρs + cos γ sin ψ sin θ 4 p 6 + λd 6 = ρs m cos γ a + a 8 π cos γp6 + λd 6 b ± b 4b mp4+λd4+g sin γ b + cos γp 6 + λd 6 b cos γ sin ψ sin θ ρs cos γ sin ψ sin θ + p5 + λd 5 cos γ g + cos γ sin ψ sin θ 5 squae oot tems in λ It can not be matched to anything else in the expession Thus, necessaily, d 4 = Taking the squae of the last equation gies ise to the following second ode polynomial in λ whee λ d 5 + d 6 + λ p 5 d 5 d 5 cos γ g + cos γ p 6 d 6 a 6 cos γ sin ψ sin θ + p 5 p 5 cos γ g + cos γ g + cos γp 6 p 6 cos γ sin ψ sin θ + cos γ sin ψ sin θ c cos γ c = ρs m cos γ 8 a + a π b ± b + 4b mp4+λd4 g sin γ b + ρs b Fo this polynomial to be identically zeo, necessaily we must hae d 5 =, d 6 = Thus the candidate ecto fo a diection of the uled manifold is d = This shows the manifold is not uled and so the system is not flat 35 Paameteization Should the system hae been flat, we would hae been using only quantities same numbe as inputs fo the paametization of all its aiables As we will see in the following, we need 3 quantities instead We now use z = = h + Re z = θ z 3 = φ whee Re is the adius of the Eath Assuming that aound the tajectoy 9 deg < ψ < 9 deg, we ecoe fom and ż3 ψ = actan cos z ż Since 9 deg < γ < 9 deg, we get fom 9 and and then ż cos ψ γ = actan ż z = actan z ż = cos ψ = ż z ż + ż 3 cos z + ż 6 7 ż + z ż + ż 3 cos z 8 It is conenient in the sequel to sole fo the deiaties, γ, ψ These quantites can be obtained eithe by diect 5

6 diffeentiation of 6 7 and 8 as ψ + tan dtan ψ ψ = dt = d ż3 cos z dt ż = z 3 cos z ż 3 sin z ż3 z ż ż cos z which gies ψ = + ż 3 ż cos z z3 cos z ż 3 sin z ż3 z ż ż cos z and = z sin γ + cos γ cos ψ z z + ż ż + cos γ sin ψ 9 z 3 z cos z + ż 3 ż cos z ż ż 3 z sin z 3 γ = z cos γ sin γ cos ψ z z + ż ż sin γ sin ψ z 3 z cos z + ż 3 ż cos z ż ż 3 z sin z 3 The lift is computed fom equations 3 and 4 as L =m ψ /z cos γ sin ψ tan z cos γ + γ /z g cos γ/ / sign γ /z g cos γ/ which we note afte substitution with equations 6, 7, 8, 3 and 9 L = f L z, ż, z, z, ż, z, z 3, ż 3, z 3 3 The bank angle can be ecomputed fom the peious expession and equation 3 β = accos γ /z g cos γ//m/l which we note afte substitution with equations 7, 8 and 3 β = f β z, ż, z, z, ż, z 3, ż 3 33 Using the linea model fo lift see appendix, we can sole fo the angle of attack α = L/ρ/ /S a /a which we note afte substitution with equations 8 and 3 and the ai density model fo ρz gien by equation 6 α = f α z, ż, z, z, ż, z, z 3, ż 3, z 3 34 The dag is then ecomputed fom the law D = ρs C D 35 Paameteization constaints The eenty dynamics hae the same nonlinea stuctue as the following simple nonlinea system with 3 states and inputs ẋ = Du ẋ = Lu cos u ẋ 3 = Lu sin u In geneal this system is not flat eg if D and L coespond to dag and lift models In othe wods, not any time function t x t, x t, x 3 t is a tajectoy of the system But the tajectoies of the system, ie time functions t x t, x t, x 3 t, u t, u t solution to the dynamics, indeed satisfy ẋ3 tan u = 35 and L = ẋ ẋ + ẋ 3 signẋ cos u These ae only necessay conditions Sufficient exta conditions ae that ẋ = DL ẋ + ẋ 3 signẋ cos u In ode to sole equation 35, one has to pick the ight detemination of the angle In geneal it can not be assumed that u ] π/, π/[ it is the case in ou example though Let us call u this solution defined up to π A suitable alue has to be such that ẋ = Lu cos u ẋ 3 = Lu sin u To summaize, the tajectoies of the system ae of the fom t x t, x t, x 3 t, L ẋ + ẋ 3 signẋ cos u, u whee x, x, x 3, u ae any abitay function that satisfy ẋ = DL ẋ + ẋ 3 signẋ cos u ẋ = ẋ + ẋ 3 signẋ cos u cos u ẋ 3 = ẋ + ẋ 3 signẋ cos u sin u ẋ3 tan u = ẋ Similaly, in ou case the following constaints must hold Fist the dag and the lift must coespond In othe wods, the dag that is computed fom the lift must be such that m + g sin γ + D = 6

7 Also the sign that appeas in the lift expession has to be taken into account Two additional constaints hae to be satisfied to tansfom the peious necessay condition in a sufficient condition It is assumed that α ] π/, π/[ So u is uniquely defined by the actan function As a summay, the tajectoies hae to satisfy ψ /z cos γ sin ψ tan z cos γ = L cos β/m// cos γ γ /z g cos γ/ = L sin β/m/ 35 Paameteization of the tajectoies The peious elations deied at section 35 ae necessay conditions In othe wods if the time functions t ht, φt, θt, V t, γt, ψt, αt, βt ae solutions of the eenty dynamics then they ae of the fom h = z R e φ = z 3 θ = z = ż + z ż + ż 3 cos z ż γ = actan ż z + ż3 cos z ż3 ψ = actan cos z ż α = f α z, ż, z, z, ż, z, z 3, ż 3, z 3 β = f β z, ż, z, z, ż, z 3, ż 3 Conesely any time function t ht, φt, θt, V t, γt, ψt, αt, βt computed fom the same elations ae not solutions to the eenty dynamics Sufficient exta conditions ae that these functions must satisfy the exta conditions m + g sin γ + C DρS z ż cos z 3 ψ /z cos γ sin ψ tan z cos γ = L cos β/m// cos γ γ /z g cos γ/ = L sin β/m/ + ż = These thee elations can be ewitten, afte substitution with the necessay conditions 6, 7, 8, 3, 3, 33 F z, ż, z, z, ż, z, z 3, ż 3, z 3 = 36 F z, ż, z, z, ż, z, z 3, ż 3, z 3 = 37 F 3 z, ż, z, z, ż, z, z 3, ż 3, z 3 = Rewiting of the optimal contol poblem The poblem is only to find the best time functions [, t f ] t z t, z t, z 3 t so as to maximize z t f unde the following constaints Initial constaints h = z Re 39 φ = z 3 4 θ = z 4 = ż + z ż + ż 3 cos z 4 ż γ = actan z ż + ż 3 cos z 43 ż3 ψ = actan ż cos z 44 Tajectoy constaints must hold fo all t [, t f ] F z, ż, z, z, ż, z, z 3, ż 3, z 3 = 45 F z, ż, z, z, ż, z, z 3, ż 3, z 3 = 46 F 3 z, ż, z, z, ż, z, z 3, ż 3, z 3 = 47 z Re, 89 z 89 ż + z ż + ż 3 cos z, 89 actan ż z ż + ż 3 cos z 89, 9 f α z, ż, z, z, ż, z, z 3, ż 3, z 3 9, 89 f β z, ż, z, z, ż, z 3, ż 3 89 Endpoint constaints ht f = z t f Re 48 t f = ż t f + z t f ż t f ż3 t f cos z t f γt f = actan 4 Numeical esults ż t f z t f ż + ż 3 cos z 49 5 In this section we gie numeical esults using the poposed methodology Details about the initialisation and conegence ae gien Accuacy of the method is discussed and compaisons with efeence esults ae gien 7

8 ht f ft 6 t f ft/sec 396 γt f deg θt f deg 38 Figue : Initial guess teminal alues and cost function alue 4 Numeical setup 4 Initial guess The system was initialized with contol aiables set to α = deg fo the angle of attack, and βt = 75 + t/t f fo the bank angle in deg Afte a caeful integation pefomed with Matlab ode3, the coesponding tajectoy was found to gie the data gien in Figue Fom these tajectoies the unknown coefficients wee computed though a least squae B-spline appoximation Of couse the esults depend on the numbe of coefficients, the ode of the B-splines and the multiplicity of thei knots and the fitting mesh Then we ecomputed the contol histoies fom the B- splines epesentation of the outputs z, z, z 3 using the fomulas gien in Section 35 Finally we eintegated the system dynamics fom the same initial condition as befoe while using the latest contol histoies Results ae gien fo a typical case with 4 inteals 44 coefficients pe aiable, 6 points mesh, 4 th ode B-Splines with multiplicity of 3 h 4 6 t f h guess t f = 5544 ft, 4 6 t f guess t f = 7559 ft/sec, γ 4 6 t f γ guess t f = 66 deg Results ay with the numbe of coefficients and Results ae gien fo a typical case with inteals 4 coefficients pe aiable unknown aiables, points mesh, 4 th ode B-Splines with multiplicity of 3 h t f h guess t f = 395 ft, t f guess t f = 5795 ft/sec, γ t f γ guess t f = 6 deg In these two cases the mesh was efined aound the two boundaies of the domain, to limit the side effects of least squae appoximation In fact, a linealy spaced mesh would poduce much lage eos With the inteals and the points linealy spaced mesh the same test gies h l t f h guess t f = 4 ft, l t f guess t f = 749 ft/sec, γ l t f γ guess t f = 4 deg We wee inestigating wethe the B-Splines wee able to poide us with a high degee of accuacy as equied fo ou application The aboe numeical inestigation suggests ht f ft 88 t f ft/sec 4753 γt f deg -579 θt f deg Figue : Teminal alues and cost function alue afte optimisation that they ae well suited poided a sufficiently lage numbe of coefficient is chosen Also the choice of the mesh mattes In the est of the epot we conduct the tests with a mesh efined aound the two ends of the time inteal 4 Soling the optimal contol poblem All the tests wee conducted using Matlab 65 with the collocation outines fom the Splines toolbox and the fmincon outine fom the Optimisation toolbox No analytical gadients wee poided, neithe fo the cost no fo the constaints This has an impact on the computation times Scalings wee used fo the cost function and the constaints This helped the nonlinea pogamming outine to find appopiate seach lines Also nonlinea equality constaints oe the time inteal due to the paameteization wee elaxed to help conegence Eentually the optimisation pocedue was estated once with the peious solution as an initial guess and moe stingent alues fo the elaxation paamete We used 4 inteals 44 coefficients pe aiable and a 65 nonlinealy spaced points mesh With a fist un elaxation paamete set to e- 4 the obtained solution gae ht f = 7996 ft, t f =7537 ft/sec, γt f =-476 deg, θt f = deg This fist poblem was soled using 4 iteations of fmincon, which used 47 F-count and took appoximatiely minutes on a Pentium III 3 GHz Windows XP based compute These esults wee eentually impoed using a new elaxation paamete of e-5 Final esults ae gien in Figue The coesponding tajectoy is detailed in Figue 3 and Figue 4 This un used iteations of fmincon, which used 673 F-count and took appoximatiely 5 minutes on the same compute 5 Conclusions The numeical esults must be compaed to the solution gien in [] that gies θt f =344 deg, a highe alue The esult pesented hee wee obtained by a much diffeent technique It seems we coneged to a diffeent solution Also it seems that the accuacy could be impoed futhe using moe coefficients fo the B-splines epesentation and well adapted meshes It should be noted that only a simple nonlinea sole was used in this study and that the use of moe complex, yet less conenient fo implementation, soles such as NPSOL [5] with analytic gadients could help too 8

9 x Altitude feet Velocity feet/sec Longitude deg Latitude deg Flight Path deg Azimuth deg 5 5 Figue 3: Reenty state aiables Optimal solution plain and initialisation dotted 3 Angle of attack deg Bank angle deg Figue 4: Reenty contol aiables Optimal solution plain and initialisation dotted 9

10 Acknowledgement This wok was sponsoed by CNES though a ARMINES contact Refeences [] J T BETTS, Pactical Methods fo optimal contol using nonlinea pogamming, SIAM, [] A E BRYSON, Dynamic optimization, Addison Wesley, 999 [3] N PETIT, M B MILAM, AND R M MURRAY, Inesion based constained tajectoy optimization, in 5th IFAC symposium on nonlinea contol systems, [4] P ROUCHON, Necessay condition and geneicity of dynamic feedback lineaization, J Math Systems Estim Contol, 5 995, pp [5] H SEYWALD, Tajectoy optimization based on diffeential inclusion, J Guidance, Contol and Dynamics, 7 994, pp [3] M FLIESS, J LÉVINE, P MARTIN, AND P ROU- CHON, Flatness and defect of nonlinea systems: intoductoy theoy and examples, Int J Contol, 6 995, pp [4], A Lie-Bäcklund appoach to equialence and flatness of nonlinea systems, IEEE Tans Automat Contol, , pp [5] P GILL, W MURRAY, M SAUNDERS, AND M WRIGHT, Use s Guide fo NPSOL 5: A Fotan Package fo Nonlinea Pogamming, Systems Optimization Laboatoy, Stanfod Uniesity, Stanfod, CA 9435, 998 [6] C HARGRAVES AND S PARIS, Diect tajectoy optimization using nonlinea pogamming and collocation, AIAA J Guidance and Contol, 987, pp [7] A ISIDORI, Nonlinea Contol Systems, Spinge, New Yok, nd ed, 989 [8] P MARTIN, R M MURRAY, AND P ROUCHON, Flat systems, in Poc of the 4th Euopean Contol Conf, Bussels, 997, pp 64 Plenay lectues and Mini-couses [9] M B MILAM, K MUSHAMBI, AND R M MUR- RAY, A new computational appoach to eal-time tajectoy geneation fo constained mechanical systems, in IEEE Confeence on Decision and Contol, [] M B MILAM AND N PETIT, Constained ta jectoy geneation fo a plana missile, tech epot, Califonia Institute of Technology, Contol and Dynamical Systems, [] M B MILAM, N PETIT, AND R M MURRAY, Constained tajectoy geneation fo mico-satellite fomation flying, in AIAA Guidance, Naigation and Contol Confeence,, pp [] R M MURRAY, J HAUSER, A JADBABAIE, M B MILAM, N PETIT, W B DUNBAR, AND R FRANZ, Softwae-Enabled Contol, Infomation technology fo dynamical systems, Wiley-Intescience, 3, ch Online contol customization ia optimizationbased contol, pp 49 74