Automatica 45 (2009) 1668–1678

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Automatica

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On-line robust trajectory generation on approach and landing for reusablelaunch vehiclesI

Zhesheng Jiang ∗, Raúl Ordóñez 1Department of Electrical and Computer Engineering, University of Dayton, Dayton, OH 45469-0232, USA

a r t i c l e i n f o

Article history:Received 10 September 2007Received in revised form9 November 2008Accepted 19 March 2009Available online 29 April 2009

Keywords:Reusable launch vehicleMotion primitiveNeighboring optimal controlAdaptive techniquesRobustificationNeighboring feasible trajectory existencetheoremTrajectory robustness theorem

a b s t r a c t

A major objective of next generation reusable launch vehicle (RLV) programs includes significantimprovements in vehicle safety, reliability, and operational costs. In this paper, novel approachesthat can deliver an RLV to its landing site safely and reliably are proposed. Trajectory generation onapproach/landing (A&L) for RLVs usingmotion primitives (MPs) and neighboring optimal control (NOC) isfirst discussed. In this stage, the proposed trajectory generation approach is based on an MP scheme thatconsists of trims andmaneuvers. From an initial point to a given touchdown point, all feasible trajectoriesthat satisfy certain constraints are generated and saved into a trajectory database. An optimal trajectorycan then be found off-line by using Dijkstra’s algorithm. If a vehicle failure occurs, perturbations areimposed on the initial states of the off-line optimal trajectory, and it is reshaped into a neighboring feasibletrajectory on-line by using an NOC approach. If the perturbations are small enough, a neighboring feasibletrajectory existence theorem (NFTET) is then investigated and its proof is provided as well. The approachgiven in the NFTET shows that a vehicle with stuck effectors can be recovered from failures in real time.However, when the perturbations become large, for example, in severe failure scenarios, the NFTET is nolonger applicable and often the vehicle cannot be recovered from such failures. A new method is thenused to deal with this situation. The NFTET is now extended to the trajectory robustness theorem (TRT).According to the TRT and its proof, a robustifying term is introduced to compensate for the effects ofthe linear approximation in the NFTET. The upper bounds with respect to input deviation are adaptivelyadjusted to eliminate their uncertainty. In order to obtain best performance, σ -modification is employed.The simulation results verify the excellent robust performance of this method.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The increased demand for commercial and military utilizationof space is a substantial driver for the development of newtechnologies to improve space vehicle economics. Reusable launchvehicles (RLVs) have the potential to increase space launchefficiencies far beyond those achieved by current systems.Second generation (and future generation) RLVsmay eventually

take the place of the space shuttles, but not before scientists perfect

I This work was supported with AFRL/AFOSR grant No. F33615-01-2-3154.In addition the work was supported by DAGSI and the University of DaytonGraduate SchoolDean’s Summer Fellowship. Thematerial in this paperwaspartiallypresented at the 31st Annual Dayton–Cincinnati Aerospace Science Symposium,2007 American Control Conference, and 46th IEEE Conference on Decision andControl. This paper was recommended for publication in revised form by AssociateEditor Shuzhi Sam Ge under the direction of Editor Miroslav Krstic.∗ Corresponding author. Tel.: +1 937 829 6279; fax: +1 937 229 4529.E-mail addresses: zhesheng9@gmail.com, jiangzhz@notes.udayton.edu

(Z. Jiang), ordonez@ieee.org (R. Ordóñez).1 Tel.: +1 937 2293183; fax: +1 937 2294529.

0005-1098/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2009.03.017

the technologies that make RLVs safer, more reliable, and lessexpensive than the shuttle fleet. To achieve this goal, a varietyof RLV trajectory design approaches have recently been proposed.Generally, an RLVmission is composed of four major flight phases:ascent, re-entry, terminal area energy management (TAEM), andapproach and landing (A&L). Some results on trajectory generationin ascent and re-entry phases were presented in Doman (2004).Some methods of trajectory planning for TAEM were presentedin Burchett (2004), Grantham (2003) and Hull, Gandhi, andSchierman (2005). A few trajectory design approaches at theA&L phase were discussed in Barton and Tragesser (1999), Cox,Stepniewski, Jorgensen, Saeks, and Lewis (1999), Hull et al. (2005),Kluever (2004), Schierman, Hull, and Ward (2002), Schierman,Hull, and Ward (2003) and Schierman et al. (2004).A&L is a critical flight phase that brings the unpowered vehicle

from the end of the TAEM phase to runway touchdown. Hullet al. (2005) addressed the on-line trajectory reshaping problemfor RLVs during the TAEM and A&L phases of re-entry flight.In Kluever (2004) a guidance scheme that employs a trajectory-planning algorithm was developed for the A&L phase of anunpowered RLV. The trajectory planning scheme computed a

Z. Jiang, R. Ordóñez / Automatica 45 (2009) 1668–1678 1669

reference flight profile by piecing together several flight segmentsthat were defined by a small set of geometric parameters. Afeasible reference profile that brings the vehicle from its currentstate to a desired landing condition was obtained by iteratingon a single geometric parameter. The flight-path angle at thestart of the flare was selected as the iteration variable. Open-and closed-loop guidance commands were readily available oncethe reference trajectory was obtained. In Schierman et al. (2002)the authors proposed an optimum-path-to-go (OPTG) algorithm,which was a general framework to perform the on-line trajectory-command generation task. The methodology was applied to theX-33 RLV for the A&L, and a Monte Carlo simulation analysiswas used to demonstrate the benefits of the approach. However,this approach did not effectively adjust to drastically alteredvehicle dynamics caused bymore serious problems, such as lockedcontrol surfaces or vehicle damage. In Schierman et al. (2003),an improvement on the OPTG algorithm was presented. Thenew OPTG approach eliminated these problems by referencinga database of precomputed, verified solutions. During flight, theOPTG routine chose a feasible trajectory for the failed vehiclewhile an adaptive guidance system made corrections for errors ordisturbances. Schierman et al. (2004) presented a fault-tolerantautonomous landing system for a re-entry RLV with flight-testresults of the new OPTG methodology. The results indicated thatfor more severe, multiple control failures, control reconfiguration,guidance adaptation, and trajectory reshaping were all needed torecover the mission. An autolanding trajectory design for the X-34 Mach 8 vehicle was presented in Barton and Tragesser (1999).The techniques facilitate rapid design of reference trajectories.The trajectory of the X-34 based on the shuttle approach andlanding design were from steep glideslope, circular flare, andexponential flare to shallow glideslope. In Cox et al. (1999),a neural network autolander for the X-33 prototype RLV wasdeveloped. The autolander was based on a new linear quadraticadaptive critic algorithm. It was implemented by an array ofFunctional Link Neural Networks and was trained by a modifiedLevenberg–Marquardt method.The goal of this paper is to develop new approaches that can

deliver an RLV to its landing site safely and reliably, recover thevehicle from some failures, and avoid mission abort as muchas possible. A motion primitive (MP) scheme is proposed togenerate A&L trajectories off-line under nominal condition. Afeasible trajectory database under nominal conditions for the RLVat the A&L phase is constructed. When the vehicle experiences afailure, neighboring optimal control (NOC) is then used to generatea neighboring feasible trajectory in real time to recover it from thefailure. Failures considered in this paper correspond to the caseof a stuck effector. When the perturbations applied during NOCimplementation are small enough and hence linear approximationis applied, a neighboring feasible trajectory existence theorem(NFTET) is investigated and its proof is provided as well. Whenthe perturbations become large, for example, in severe failurescenarios, NFTET is no longer applicable and often the vehiclecannot be recovered from such failures. A new method is usedto deal with this situation and the robustness of the RLV systemis then enhanced. The NFTET is now extended to the TRT —trajectory robustness theorem. Its proof will also be providedlater. This approach has at least three advantages over existingtrajectory planning methods. First, it uses MP to generate an off-line feasible trajectory database that will be easily to create a newdatabase for a new initial point. Second, it finds an off-line optimaltrajectory using Dijkstra’s algorithm instead of the traditionaloptimal control method so that NOC promises to find an on-line neighboring feasible trajectory. Third, a novel robustificationapproach is introduced to enhance the robustness of the systemand hence an RLV with severe failure might be recovered in realtime.

Section 2 introduces a motion primitive (MP) scheme andshows how to generate feasible trajectories in the A&L phasebased on the MP scheme. Section 3 shows that the trajectoryretargeting can recover the vehicle from some failures in real timeby using an NOC approach. Section 4 discusses how to enhancethe robustness of NOC using adaptive boundswith respect to inputdeviation. Some results and discussions are given in Section 5. Theconclusions are drawn in Section 6.

2. Trajectory generationonapproach and landingusingmotionprimitives

This section will first introduce the point-mass equations ofmotion for the A&L problem of an RLV and then briefly describe theMP scheme. After that, how to generate feasible trajectories usinganMP scheme is discussed (Jiang & Ordóñez, 2007; Jiang, Ordóñez,Bolender, & Doman, 2006).

2.1. Point-mass equations of motion

For an unpowered RLV during A&L, the discussion is restrictedonly to flight in the longitudinal plane. The gliding flight in avertical plane of symmetry is then defined by the following point-mass equations (Jiang et al., 2006; Kluever, 2004; Miele, 1962):

V =(−DW− sin γ

)g, (1)

γ =

(LW− cos γ

)gV, (2)

h = V sin γ , (3)

x = V cos γ , (4)

where V is the vehicle velocity, γ is the flight-path angle, h is thealtitude, x is the downrange, g is the gravitational acceleration,W is the vehicle weight, L = qSCL is the lift, and D = qSCD isthe drag, where q is the dynamic pressure, S is the aerodynamicreference area of the vehicle, and CL and CD are the lift and dragcoefficients, respectively. The dynamic pressure q = ρV 2/2,wherethe air density ρ at altitude h is approximated using an exponentialmodel ρ = ρ0e−βh, where ρ0 is the air density at sea level and β isthe atmospheric density scale.Generally, the lift coefficient CL is a linear function of α, where

α is the angle of attack and the drag coefficient CD is a quadraticfunction of CL, namely, CL = CL0 + CLαα and CD = CD0 + KC2L ,where CL0 is the lift coefficient at zero angle of attack, CLα is the liftslope coefficient, CD0 is the drag coefficient at zero lift, and K is acoefficient relative to induced drag. Substituting the CL expressioninto CD gives it as a function of α, namely, CD = kD0+kD1α+kD2α2,where kD0, kD1, and kD2 are resulting coefficients with respect to α.The constraints at touchdown are

hTDmin ≤ hTD ≤ hTDmax, (5)VTDmin ≤ VTD ≤ VTDmax, (6)

where hTD is the sink rate at touchdown, hTDmin and hTDmaxare its minimum and maximum values, respectively, VTD is thetouchdown velocity, and VTDmin and VTDmax are its minimum andmaximum values, respectively.In the state Eqs. (1)–(4), V , γ , h, and x are the four state variables

and α is the control variable.

2.2. Motion primitives

A motion plan consists of two classes of MPs (Frazzoli, 2002).The trajectory generation in this paper involves transitioning fromone class of MPs to the other. The first class of MPs is a special classof trajectories, known as trims. A trim is a steady-state or quasi-

1670 Z. Jiang, R. Ordóñez / Automatica 45 (2009) 1668–1678

Fig. 1. A&L geometry and phases.

steady flight trajectory that includes a set of steep glideslopesand a set of shallow glideslopes on A&L. The second class of MPsconsists of transitions between trims. These trajectory segmentsare known as maneuvers. A maneuver is an ‘‘unsteady’’ trajectorythat includes pull-up and flare maneuvers on A&L. A pull-upmaneuver transitions a steep glide to a shallow glide and a flaremaneuver is added after the shallow glide to dissipate the energyof the impact at landing to touchdown with an acceptable sinkrate for the vehicle (Walyus & Dalton, 1991). In this paper, atrajectory basically means a downrange-altitude trajectory thatconsists of these four segments (please see Fig. 1). From an initialpoint to a given touchdown point, all feasible trajectories thatsatisfy certain constraints are generated and saved into a trajectorydatabase (Jiang et al., 2006).

2.3. Trajectory planning under nominal conditions

2.3.1. TrimsA steady or quasi-steady approach is applied during trims.

Since the flight-path angle γ is a constant, dh/dx = tan γcan be easily calculated. In this stage, however, the velocity(and downrange) will still require numerical integration of theirdifferential equations. If the altitude h is chosen to be theindependent variable during trims, and noting that the gliding isin a constant flight-path angle, (1)–(4) now become (Jiang et al.,2006)

dVdh=(−D/W − sin γ )g

V sin γ, (7)

dxdh=

1tan γ

. (8)

From γ = 0, (2) becomes L = W cos γ . Noting that L = qSCL, CL isthen given by CL =

W cos γqS and the control α is finally obtained by

α =CL − CL0CLα

. (9)

2.3.2. ManeuversFor the maneuvering flights, a nonsteady approach has to be

used. For convenience, time is not chosen to be the independentvariable. Instead, γ can be used as the independent variable duringpull-up. For the pull-upmaneuver, (1)–(4) nowbecome (Jiang et al.,2006)

dVdγ=V (−D/W − sin γ )L/W − cos γ

, (10)

dhdγ=

V 2 sin γg(L/W − cos γ )

, (11)

dxdγ=

V 2 cos γg(L/W − cos γ )

. (12)

Since the load factor n = L/W is fixed during the pull-up, the liftcoefficient CL can be calculated by CL = nW

qS . The control α is thenobtained by (9).The flare maneuver is to dissipate the energy to achieve an

acceptable sink rate hTD. Therefore, it becomes important to obtainan altitude function in this stage (see Fig. 1). A cubic polynomialmodel gives such a function with downrange as the independentvariable (Jiang et al., 2006), h(x) = ax3 + bx2 + cx + d, wherea, b, c , and d are constants to be determined from the desiredinitial altitude of the flare and the amount of downrange that isdesired for the maneuver. Without loss of generality, it is assumedthat the maneuver starts at x = 0. Letting x3 be the downrangeat which the flare starts (recall that x3 = 0), it turns out thath3 = h(x3) and dh3/dx3 = tan γ2, where γ2 is the flight-pathangle for shallow glide. Given h(xTD) = 0 and dhTD/dxTD = tan γTD,where γTD is the flight-path angle at touchdown, the coefficients ofthe cubic polynomial are a = [2h3 + (tan γ2 + tan γTD)xTD]/x3TD,b = −[3h3 + (2 tan γ2 + tan γTD)xTD]/x2TD, c = tan γ2, and d = h3.Similarly, x is chosen to be the independent variable during

flaremaneuver. Based on the above equations, (1)–(4) nowbecome(Jiang et al., 2006)

dVdx=g(−D/W − sin γ )

V cos γ, (13)

dγdx= cos2 γ

d2hdγ 2= cos2 γ (6ax+ 2b), (14)

dhdx= tan γ = 3ax2 + 2bx+ c. (15)

Eqs. (2) and (4) give dγdx =g(L/W−cos γ )V2 cos γ

. Hence, L = W cos γ (1 +V2gdγdx ), where dγ /dx is already known from (14). Noting that L =

qSCL, the control α is again obtained by (9).At each of four trajectory segments, corresponding equations

are used to calculate that segment with different independentvariables. Once the entire A&L trajectory meets the constraints (5)and (6), that trajectory is considered to be a feasible one for aspecified initial point. Since an entire trajectory has four segments,there exist a few feasible trajectories for that initial point whichsatisfy constraints (5) and (6) at touchdown and the transitionsbetween two adjacent segments have to be chosen carefully sincethe equations at different segments having different independentvariables. All feasible trajectories for that initial point are thenstored in an off-line feasible trajectory database. The feasibletrajectories for other initial points are generated in the same wayand saved into different trajectory databases, respectively.

3. On-line trajectory retargeting using neighboring optimalcontrol

When a failure significantly affects the forces on an RLV,or modifies the flight envelope constraint boundaries, trajectoryretargeting may be used to recover it. Failures considered herecorrespond to the case of a stuck effector so that the nominalstates and controls can be used to reshape the trajectory (Fahroo& Doman, 2004). An off-line optimal trajectory is always assumedto be found from the feasible trajectory database. When a failureoccurs, NOC is used to determine its neighboring feasible trajectoryon-line. If such a neighboring feasible trajectory exists, the RLV canthen be recovered from that failure in real time.

3.1. Off-line trajectory optimization

A shortest-path problem in computer graph and algorithmtheories is used for the off-line optimal trajectory search. To findthe shortest path, several algorithms can be used (Bellingham,

Z. Jiang, R. Ordóñez / Automatica 45 (2009) 1668–1678 1671

Kuwata, & How, 2003). Dijkstra’s algorithm (Dijkstra, 1959) is areasonable choice since it is a best-first search and uses greedystrategy. It is guaranteed to find an optimal path by repeatedlyselecting the vertex with the minimum shortest-path estimate orcost. The cost function or weight is defined by the mean valueof α − α∗, where α∗ is a nominal value of angle of attack α.For example, α∗ = 5 corresponds to the base pitching momentcoefficient Cm0 = 0 due to the wing body.

3.2. On-line trajectory optimization using NOC

Based on the off-line optimal trajectory, a small perturbation isimposed on each state variable and new control can be obtained byNOC. The RLV can be recovered from the aforementioned failureswith stuck effectors.

3.2.1. NOC approach descriptionThe NOC approach is discussed in Bryson and Ho (1975), Speyer

and Bryson (1968) and Seywald and Cliff (1994). However, not onlythe states but also the system parameters are under considerationin this paper and it is necessary to make some changes.Consider the following general nonlinear system:

x(t) = f (t, x(t), u(t), λ(t)), x(t0) = x0, (16)

where x(t) ∈ C are state variables, u(t) ∈ Ω are control inputs,λ(t) ∈ P contains system parameters, and C ⊂ Rn, Ω ⊂ Rm,and P ⊂ Rp. With the convex cost function J(u) = φ(tf , xf , λf ),where the subscript ‘‘f ’’ denotes the final time, it is assumed thatthe optimal solution (x∗(t), u∗(t)) can be found, where the optimalcontrol is u∗(t) = g(t, x∗(t), λ(t)) under the state constraints andcontrol constraints

xmin ≤ x∗(t) ≤ xmax, umin ≤ u∗(t) ≤ umax. (17)

When small perturbations δx(t0) and small variations δλ areimposed on initial states x0 and system parameters λ, respectively,the neighboring feasible trajectory z(t) and new optimal controlv(t) can then be determined by δx(t0), δλ, and (16). Since systemparameters are considered in the NOC approach of this paper, thefollowing theorem is then investigated.

3.2.2. Neighboring feasible trajectory existence theorem (NFTET)It requires the following assumptions.

Assumption 1. For the nonlinear system (16), the following mustbe satisfied. (a) f is continuous in (t, x, λ); (b) f is locally Lipschitzin x and uniformly Lipschitz in t and λ; (c) C ⊂ Rn, Ω ⊂ Rm, andP ⊂ Rp are convex, connected sets.

Assumption 2. The open-loop optimal solution x∗(t) exists andoptimal control u∗(t) is well defined for system (16).

Theorem 1 (NFTET). (1) For an initial condition z(t0), which can beequal to x0, there exist neighboring feasible state trajectories z(t) andneighboring optimal control v(t), satisfying

supt‖z(t)− x∗(t)‖ ≤ K , (18)

where K is an upper bound.(2) When the system parameters vary from λ1 to λ2 (λ2 =

λ1 + δλ), the neighboring feasible state trajectories z(t) and theneighboring optimal control v(t) still exist, where v(t) is determinedby v(t) = u∗(t)+ G(t)(Z(t)− X∗(t)) or

v(t) = u∗(t)+ G(t)∆X(t), (19)

where Z(t) =[z(t)λ2

], X∗(t) =

[x∗(t)λ1

], and∆X(t) is a deviation vector

of the new augmented state vector Z(t) from the open-loop optimal

(nominal) state trajectory X∗(t). G(t) is the feedback gain matrix,which is given by

G(t) = δU(t)δX−1(t), (20)

where δU(t) are the control errors caused by the perturbationsimposed on the augmented state vector X i(t) and δX(t) is aperturbationmatrix due to perturbations in the initial states xi(t0) andinitial system parameters λi1, where i = 1, 2, . . . , n+ p.

Proof. According to the Assumptions 1 and 2, the optimal solutionof the initial value problem (IVP) (16) does exist and Part(1) has already been proved when no system parameters areconsidered (Rampazzo & Vinter, 1999).Based on the consequence of Part (1), Part (2) can be proved

as follows. For convenience, the independent variable t inparentheseswill be neglected in the subsequent notations. Assumethat the control u is of the form u = g(t, x, λ), where the systemparameters λ are included. The system (16) then becomes anunforced system x = f (t, x, λ), x(t0) = x0, where x ∈ C , λ ∈ P ,and f = f in general. Under the above conditions, a controllermustexist satisfying the constraints of (17).First, we prove the existence of the optimal feasible solution of

the IVP (16) with the new control input v of (19). By Theorem 3.5in Khalil (2002), all the conditions of the NFTET are satisfied;therefore, there exists a unique solution of (16) defined on [t0, tf ].Now that the solution of the IVP (16) exists and its optimal

solution (x∗, u∗) can be found according to Assumption 2, theperturbations can be imposed on (x∗, u∗). In the neighborhood ofx∗, the new state vector z can be defined as z = x∗+δx, where δx isthe resulting perturbation vector from the optimal state variablesdue to perturbations in the initial states x0. Recall that the systemparameters change fromλ1 toλ2, i.e.,λ2 = λ1+δλ. By Taylor seriesexpansion, the real input control v∗ at time t is now

v∗ = g(t, z, λ2) = g(t, x∗ + δx, λ1 + δλ)

= g(t, x∗, λ1)+∂g∂xδx+

∂g∂λδλ+ H.O.T.

= u∗ +∂g∂x(z − x∗)+

∂g∂λ(λ2 − λ1)+ H.O.T.

= u∗ +[∂g∂x

∂g∂λ

]([zλ2

]−

[x∗

λ1

] )+ H.O.T., (21)

where H.O.T. denotes the higher order terms. If the perturbationis small enough, i.e., δx is small enough or z is in the feasibleneighborhood of x∗, in other words, (18) is satisfied, then thehigher order products of δx and δλmust be small enough as well.Moreover, δλ is basically small and then its higher order productsare small enough. Therefore, the H.O.T. in (21) can be neglected atthis point. After the linearization, the new controller is of the formv = u∗ + G(Z − X∗), where Z ∈ Rn+p and X∗ ∈ Rn+p are defined inthe theorem. For the closed-loop system, the perturbations δx andδλ are actually themeasured deviations. Therefore, v = u∗+G(Z−X∗) is equivalent to (19), where ∆X denotes a deviation vector ofthe closed-loop state vector from the open-loop optimal (nominal)state trajectory and the gain matrix G is defined by

G =[∂g∂x

∂g∂λ

]=∂g∂X∈ Rm×(n+p). (22)

Second, a numerical method is used to evaluate the gain matrixG at every time t ∈ [t0, tf ] so that no partial derivatives areevaluated. From u = g(t, x, λ) and X := [xT, λT]T, the totaldifferential of u is du = ∂g

∂X dX +∂g∂t dt . For a known perturbation

instance, its perturbation vectors are of the form δX = Xperturbed −X∗ ∈ Rn+p and δu = uperturbed − u∗ ∈ Rm. Using a linear

1672 Z. Jiang, R. Ordóñez / Automatica 45 (2009) 1668–1678

approximation and noting that δt = 0 at every time t ∈ [t0, tf ],δu can be obtained by

δu =∂g∂XδX +

∂g∂tδt = GδX . (23)

In order to evaluate G, an (n+ p)× (n+ p) square matrix of δX isrequired. From n+ p known perturbation instances, two matricesare constructed by

δX :=[δX1 . . . δXn+p

]∈ R(n+p)×(n+p), (24)

δU :=[δu1 . . . δun+p

]∈ Rm×(n+p), (25)

where δX i, i = 1, 2, . . . , n+ p and δui, i = 1, 2, . . . , n+ p are theith known perturbation instances of δX and δu respectively. Thematrix expression of (23) is then

δU = GδX . (26)

When the known perturbation instances are chosen to be linearlyindependent yet still small enough, the invertibility of δX can beguaranteed at almost every time point. In case δX is singular at atime ts ∈ [t0, tf ], that point is simply excluded in order that theremaining δX(t) is nonsingular where t ∈ [t0, tf ] and t 6= ts.From (26), the gain matrix G can then be evaluated by (20).

3.2.3. Application of the NFTET to the RLV trajectory reshapingAccording to the NFTET, it can be easily shown that the

neighboring feasible trajectory for an RLV system exists whenapplying the NOC method to the RLV system at A&L phase.In this system, V , γ , h, and x are the four state variables

and α is the control variable; CL and CD are two systemparameters. If assuming CL and CD are two special state variablesand the variations of CL and CD between nominal and thosefailed conditions are viewed as perturbations as well, then theneighboring feasible trajectory does exist if all the perturbationson the six state variables are small enough according to the NFTET.Since the off-line optimal (nominal) controller α∗ has alreadybeen obtained by Dijkstra’s algorithm from the feasible trajectorydatabase using theMP scheme, the new controller αperturbed is thenobtained by αperturbed = α∗ + G(Z − X∗), where G, Z and X∗ havethe same meanings as described in the NFTET.To evaluate G, some perturbations are imposed on X∗. Note that

the altitude h is chosen to be the independent variable (instead oftime t) so that the problem becomes a fixed ‘‘final time’’ problem(because hTD ≡ 0). Since the dimension of Z is now five, fiveperturbation instances of V , γ , x, CL, and CD are used. Similarly,there are five corresponding instances of α. Let δX := [[V 1 −V ∗, γ 1 − γ ∗x1 − x∗C1L1 − C

∗

L1C1D1 − C

∗

D1]T . . . [V 5 − V ∗γ 5 − γ ∗x5 −

x∗C5L1 − C∗

L1C5D1 − C

∗

D1]T] and δα := [α1 − α∗, . . . , α5 − α∗], where

superscript i denotes the ith perturbation instance, i = 1, . . . , 5.The gain matrix G can then be evaluated by G(h) = δαδX

−1.

Letting Z = [V , γ , x, CL2, CD2]T and X∗ = [V ∗, γ ∗, x∗, C∗L1, C∗

D1]T,

where the subscript ‘‘1’’ of CL and CD denotes nominal conditionand ‘‘2’’ denotes failure condition, the new controller αperturbedunder a certain failure can then be determined easily. Finally, theneighboring feasible trajectories are determined by the same Eqs.(1)–(4) with the independent variable h.

4. Robustness enhancement on NOC using adaptive boundswith respect to input deviation

In the proof of the NFTET, the real control input v∗(t) for thegeneral nonlinear system is obtained by (21), i.e., v∗(t) = u∗(t)+G(t)∆X(t)+ H.O.T.

If the perturbations are small enough, H.O.T. can be neglectedand the NFTET is applicable. The neighboring feasible controllerwith linear approximation v(t) can then be evaluated by (21).When the RLV experiences a failure that causes small changes ofCL and CD, NOC can recover it from the failure and so demonstratesgood robust performance in small perturbations (Jiang & Ordóñez,2007).When the perturbations become large such as in a severe

failure case, the H.O.T. cannot be neglected. In order to compensatefor the effects of the linear approximation, a robustifying orstabilizing term vs(t) is introduced to approximate theH.O.T. (Jiang& Ordóñez, 2007), and v(t) is now re-defined as

v(t) = u∗(t)+ G(t)∆X(t)+ vs(t), (27)

where vs(t) is to be determined.For simplicity, only single input systems (i.e., m = 1) are

discussed. In this case, H.O.T. will denote the value of the higherorder terms and hence becomes a scalar in all the subsequentnotations. Extension to multiple input systems is left to futureresearch.Since the H.O.T. must include a ∆X(t) term, it is reasonable to

assume that they satisfy

|H.O.T.| ≤ B∗T |∆X(t)|, (28)

where B∗ ∈ Rn+p is a constant column vector where B∗i,1 ≥0, i = 1, . . . , n + p and the | · | operation is performed element-wise. The bound B∗ is not explicitly known. Note that B∗ is notunique, since any B∗i,1 > B∗i,1 for i = 1, . . . , n + p satisfiesinequality (28). To avoid confusion, for analytical purposes, B∗is defined to be a constant vector with smallest non-negativeelements such that (28) is satisfied (Polycarpou & Ioannou, 1996).The estimate of B∗ can then be found adaptively. To robustify NOC,a trajectory robustness theorem (TRT) is now investigated.

4.1. Trajectory robustness theorem (TRT)

Theorem 2 (TRT). For the system (16) satisfying (28)with uncertainupper bound B∗, there exists a robustifying term vs(t) that canenhance the robustness of NOC. The robustifier vs(t) is of the form

vs(t) = −BT(t)|∆X(t)|sgn(BT(t)∆X(t)). (29)

The adaptation law is defined by

˙B(t) = γb[|∆X(t)| − σ(B(t)− B(0))], (30)

where γb > 0 and σ > 0 are constant parameters,∆X(t) is the sameas the definition in the NFTET, B(t) is the estimate of B∗, and B(0)is an initial guess of B∗ which is set to B(0) = 0. By definingB(t) = B(t) − B∗ and the adaptation law (30), B(t) converges toa neighborhood of zero and its upper bound can be chosen by thedesigner.

Proof. For convenience, the independent variable t will again beneglected in the subsequent notation.Whether the error dynamics goes to zero does notmatter at this

point (we are only concernedwith the controller), so the Lyapunovcandidate VL is simply chosen to be

VL =12γbBTB, (31)

where γb > 0 and B = B − B∗, where B is the estimate of B∗ withnon-negative elements.Recalling that B∗ is a constant column vector and from (28), the

derivative of VL is then

Z. Jiang, R. Ordóñez / Automatica 45 (2009) 1668–1678 1673

VL =1γbBT ˙B

≤

vs − H.O.T.+

1γbBT ˙B, if vs > H.O.T.

H.O.T.− vs +1γbBT ˙B, if vs ≤ H.O.T.

≤

B∗T |∆X | +

1γbBT ˙B+ vs, if vs > H.O.T.

B∗T |∆X | +1γbBT ˙B− vs, if vs ≤ H.O.T.

= B∗T |∆X | +1γbBT ˙B+ vssgn(vs − H.O.T.), (32)

where the signum function can be defined in many different ways.Considering the RLV application later, one such definition is givenby

sgn(y) =1, if y > 0−1, if y ≤ 0. (33)

Choosing the adaptation law

˙B = γb|∆X |, (34)

(32) becomes

VL ≤ BT|∆X | + vssgn(vs − H.O.T.). (35)

If the robustifier or stabilizing term vs is of the form

vs = −BT|∆X |sgn(vs − H.O.T.), (36)

(35) becomes

VL ≤ 0, (37)

which means that VL is negative semidefinite. This guarantees that|B| is bounded.If vs > H.O.T., then sgn(vs − H.O.T.) = 1. Since B(0) is set

to B(0) = 0 and all elements of B(t) are monotonically non-decreasing due to the choice (34), BT|∆X | ≥ 0 and hence vs ≤ 0from (36). Similarly, vs ≥ 0 if vs ≤ H.O.T. Therefore, there are onlytwo cases under consideration if vs is chosen to be (36), namely,(a) 0 ≥ vs > H.O.T. and (b) H.O.T. ≥ vs ≥ 0. The two cases canbe expressed in the same form: |vs| ≤ |H.O.T.| (This is just for theconvenience of later discussion. More accurately, |vs| < |H.O.T.|for case (a).)However, it can be noticed that vs is included in the sgn(·)

function in (36), so it has to be re-defined. From (28) and (36), bothvs and the H.O.T. include a ∆X term, and therefore the followingequality must hold:

vs − H.O.T. = ET∆X, (38)

where E is a deviation column vector with respect to ∆X . Sinceboth B and E are small, it is possible to replace E with B in the sgn(·)function on the right side of (36). Hence (36) becomes (29).Next, it is needed to verify if (37) still holds when vs is of the

form (29) instead of (36). As discussed above, there are only cases(a) and (b) based on the choice of (34) and (36). Therefore, onlythese two cases need to be verified.(i) If BT∆X > 0, then vs = −BT|∆X | ≤ 0 from (29). This is Case

(a), i.e., H.O.T. < vs ≤ 0. When vs > H.O.T., vssgn(vs − H.O.T.) =−BT|∆X | in (35) and hence (37) still holds.(ii) If BT∆X ≤ 0, then vs = BT|∆X | ≥ 0 from (29). This is Case

(b), i.e., H.O.T. ≥ vs ≥ 0. When vs ≤ H.O.T., vssgn(vs − H.O.T.) =−BT|∆X | in (35) and hence (37) still holds.

Table 1Some constant parameters.

Parameter Value

Reference area S(ft2) 41.45Vehicle massm (slug) 89Gravitational acceleration g(ft/s2) 32.2Density coefficient β(1/ft) 4.348e–5Air density at sea level ρ0(slug/ft3) 0.002377

Table 2Constraints.

Constrained variable Value

Minimum touchdown velocity VTDmin(ft/s) 260Maximum touchdown velocity VTDmax(ft/s) 300Minimum sink rate hTDmin(ft/s) −10Maximum sink rate hTDmax(ft/s) 0

By (i) and (ii), (29) and (36) are equivalent when |vs| ≤ |H.O.T.|.If H.O.T. is positive (H.O.T. is a scalar for a single input system andwill never be zero by its definition), vs will reach H.O.T. from zero(since B(0) = 0, vs(0) = 0 by (29)) after a finite time. At thattime, the effect of H.O.T. has been compensated completely (notethat vs might become larger than H.O.T. later if there is no furthermodification). If it is negative, vs will not reach H.O.T. from zeroand will never be beyond H.O.T., i.e., |vs| < |H.O.T.|, or vs > H.O.T.(but the effect of H.O.T. has not been compensated completely. If itis required to do so, simply changing the definition of the signumfunction will work).In order to ensure that |vs| ≤ |H.O.T.|, it is necessary to limit

the increase of B and further modification on the adaptation law isthenmade. Now ˙B is further chosen by using a σ -modification suchthat VL < 0 (Ioannou & Sun, 1996; Polycarpou & Ioannou, 1996). If˙B is defined by (30), where σ > 0 and B(0) is an initial guess of B∗

which is set to B(0) = 0, it will result in

VL ≤ −σ BT(B− B(0))

= −σ BT(B+ B∗ − B(0))

≤ −σ

2BTB+

σ

2(B− B(0))T(B− B(0))

= −σγb

(12γbBTB

)+σ

2(B− B(0))T(B− B(0))

= −k1VL + k2, (39)

where k1 = σγb > 0 and k2 = σ2 (B − B(0))

T(B − B(0)) ≥ 0are bounded constants. From Lemma2.1 in Spooner,Maggiore, andOrdóñez (2002),

VL =12γbBTB ≤

k2k1+

[VL(0)−

k2k1

]e−k1t . (40)

From the above equation, it turns out that

limt→∞‖B‖ ≤

√2γbk2k1= ε. (41)

Therefore, B converges to a neighborhood of zero whose upperbound is ε.Now, |vs| ≤ |H.O.T.| is guaranteed by (36). Application of vs

to the NOC scheme will result in increased tolerance to severedeviations from the off-line optimal trajectory. This is because theeffects of the H.O.T. are no longer ignored, thereby rendering therobustified control v into a closer approximation of (21).

1674 Z. Jiang, R. Ordóñez / Automatica 45 (2009) 1668–1678

Fig. 2. Trajectory reshaping using NOC without robustifier under the Failure Case of a speedbrake fixed at 50 and a body flap at−10 .

Table 3CL and CD .

Nominal case Failure case Severe failure case 1 Severe failure case 2

CL0 0.04798 0.05975 0.1761 −0.1575CLα(1/deg) 0.07487 0.06683 0.05804 0.06159kD0 0.1446 0.2103 0.2851 0.2075kD1(1/deg) −5.051e-005 −0.0003217 −0.001623 −0.004345kD2(1/deg2) 8.157e-004 0.0007856 0.0007283 0.0007568

4.2. Application of the TRT to the on-line RLV trajectory generation onA&L

According to the TRT, the new robustified controllerαrobustified isthen obtained by αrobustified = α∗+ G(Z − X∗)+ αs, where α∗, G, ZandX∗ have the samemeanings as described in Section 2 (also Jiangand Ordóñez (2007)), while αs is the robustifying term. Since thealtitude h is chosen to be the independent variable instead of timet as described in Jiang et al. (2006) and Jiang and Ordóñez (2007),the total number of state variables for the RLV system reducesto five from six. The dimension of B∗ and B reduces to five fromsix, accordingly. The gain matrix G is obtained in the same way asbefore while the robustifying term αs is determined by (29) exceptthat the independent variable is h.Unlike the MP scheme, the new control variable αrobustified is

evaluated by the above unique equation. Also note that dB/dh(instead of ˙B) is used in the adaptation law of (30) since theindependent variable is h, where dB/dh = ˙B/h.

5. Results and discussions

Given initial and touchdown conditions for an RLV, a feasibletrajectory database is built under the nominal condition. The initialpoint for the trajectory generation in all cases has the followingvalues: V0 = 465.7 ft/s, γ0 = −21 deg, h0 = 10066 ft,and x0 = −25000 ft. Tables 1 and 2 list the parameters andconstraints used in the MATLAB simulation when constructing the

feasible trajectory database. Under all nominal conditions for theRLV system (1)–(4), the lift and drag contributions only dependon angle of attack α. Its lift and drag coefficients in the nominalcondition are given in Table 3.When a failure occurs for the vehicle, trajectory reshaping can

be used to recover it. To observe the results of trajectory reshapingusing NOC and adaptive bounds with respect to input deviation,one failure case and two severe failure cases for the RLVwith stuckeffectors are illustrated based on the lift and drag characteristics ofthe vehicle. The lift and drag coefficients CL and CD for these failurecases after a curve fit are also listed in Table 3.The first failure case is the speedbrake fixed at 50 and the body

flap at−10. The two severe failure cases are: (1) both right rudderand left rudder at−4 and speedbrake at 70; (2) left flap at−16

and speedbrake at 55.Fig. 2 indicates the trajectory reshaping without the robustifier.

In the plot of flight-path angle, it can be seen that the first segmentis a steep glide with γ = −21, the second segment is a pull-upwith n = 1.1, the third segment is a shallow glide with γ = −15,and the fourth segment is a flare with VTD = 298.7 ft/s andhTD = −9.890 ft/s. It can also be observed that the RLV is recoveredfrom the first failure case in real time because the touchdownconstraints (5) and (6) are met. Fig. 3 shows that the trajectoryreshaping with the robustifier recovers the RLV in real time fromthe same failure. The touchdown velocity VTD and sink rate hTDare now 282.5 ft/s and −5.370 ft/s, respectively. In the MATLABsimulation, γb = 0.0000148, σ = 5, and B(0) = [0, 0, 0, 0, 0]T.

Z. Jiang, R. Ordóñez / Automatica 45 (2009) 1668–1678 1675

Fig. 3. Trajectory reshaping using NOC with robustifier under the Failure Case of a speedbrake fixed at 50 and body flap at−10 .

Fig. 4. Trajectory reshaping using NOC without robustifier under Severe Failure Case 1.

The control energy is calculated by∫ hTDh0

α2 dh/(hTD − h0) and theelapsed time is the system execution time from the initial pointto touchdown in the same computer environment (using ‘‘tic’’and ‘‘toc’’ functions in MATLAB, running on a COMPAQ Presario2199US computer on Windows XP and a 2.12 GHz processor). Theenergies consumed and times for the trajectory reshaping withoutthe robustifier andwith the robustifier are 30.96, 1.662 s and 38.84,

2.394 s, respectively. The latter consumes more energy and needsmore time, as expected.Figs. 4 and 5 show the trajectory reshaping without and with

the robustifier under Severe Failure Case 1, respectively. The fourtrajectory segments are labeled on the plot of the flight-path anglein Fig. 4. The trajectory reshaping without the robustifier fails torecover the vehicle fromSevere Failure Case 1 since the touchdownconstraint of hTD = −39.41 ft/s is not met. After applying the

1676 Z. Jiang, R. Ordóñez / Automatica 45 (2009) 1668–1678

Fig. 5. Trajectory reshaping using NOC with the robustifier under Severe Failure Case 1.

Fig. 6. Robustifier (the independent variable is the altitude).

robustifier, the trajectory reshaping recovers the vehicle in realtime from that failure as VTD = 282.5 ft/s and hTD = −6.688 ft/snow. The energies consumed and times for the trajectory reshapingwithout the robustifier and with the robustifier are 37.92, 1.662 sand 38.48, 2.163 s, respectively. Again, the latter consumes moreenergy and needsmore time, as expected. Note that all the relevantparameters and hardware in the simulation are the same as thefirst failure case since all the parameters only need to be tuned onceand will be applied to all cases.From Fig. 5 it can also be seen that the x − h trajectory with

the robustifier is very close to the off-line optimal trajectory,which means that the robustifier αs does compensate for theeffects of the linear approximation. By reducing the sensitivityof system performance with respect to system uncertainty, themethodpresented appears to greatly enhance the robustness of theoriginal NOC approach.Fig. 6 shows the robustifier αs in the Severe Failure Case 1. It

smooths the instantaneous jumps of α by compensating the effect

of linear approximation. Similar αs curves can be found for otherfailure cases. The plots are not drawn here due to limited space.Fig. 7 gives an example where the trajectory reshaping both

with and without the robustifier fails to recover from SevereFailure Case 2. It can still be observed that the trajectory reshapingwith the robustifying term demonstrates better performance.The touchdown conditions with the robustifier become closer tothe desired range since VTD and hTD are changed to 373.3 ft/sand −51.97 ft/s from 421.4 ft/s and −246.4 ft/s after applyingrobustification, respectively. In a case like this, the failure tofind a feasible trajectory would indicate that the vehicle cannotbe recovered and that the mission should be aborted. Crucially,this information would be available in a couple of seconds,thereby allowing time to take life-savingmeasures and helping thevehicle’s crew initiate bail-out procedure before complete disasterstrikes.

6. Conclusions

Under nominal condition for an RLV, a feasible trajectorydatabase is built by using the MP scheme. The database is welldefined under the nominal condition. Under some failures suchas stuck effectors, trajectory reshaping can recover the RLV.When those failures are not severe, the NOC approach does helprecover the vehicle in real time. The neighboring feasible trajectoryexistence theorem (NFTET) has been proved in this paper and itguarantees that the RLV can be delivered to its landing site safelyand reliably under those failures by reshaping the trajectory usingthe NOC approach. Therefore, it indicates good robust performanceof NOC approach.When a severe failure occurs, however, the robustness of NOC

has to be enhanced in order to improve the chances of recoveringthe vehicle in real time. The trajectory robustness theorem (TRT)has also been proved in this paper and a robustifier has beenapplied to compensate for the effects of the linear approximationemployed in the NFTET. The robustifying term is adaptivelydetermined by the adaptation law with respect to the input

Z. Jiang, R. Ordóñez / Automatica 45 (2009) 1668–1678 1677

Fig. 7. Trajectory reshaping fails using NOC both with and without the robustifier under Severe Failure Case 2.

deviation bound. The simulation results show that this approachgreatly improves the robust performance of NOC and allows theRLV to be delivered to its landing site safely and reliably even undercertain severe failures. Moreover, even when the method fails togenerate a feasible trajectory for landing, it yields this negativeresult in real time, thereby allowing the vehicle’s crew to bail outbefore complete disaster strikes.As mentioned in Section 4, the TRT is limited to single input

systems. Extension to multiple input systems is left to futureresearch. As a future direction, the TRT can be extended to morefailure scenarios when wind effects are considered.

Acknowledgements

The authors wish to acknowledge Dr. Michael A. Bolenderand Dr. David B. Doman in the Air Force Research Laboratory atWright–Patterson Air Force Base for providing helpful suggestionsand enlightening ideas on the research. The authors would alsolike to acknowledge Dr. George Doyle in the Mechanical andAerospace Engineering Department at the University of Dayton forhis comments on this paper.

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Zhesheng Jiang is a Ph.D. candidate in the Electricaland Computer Engineering Department at the Universityof Dayton. He received his B.S. and M.S. degrees fromZhejiang University and the Chinese Academy of Sciences,respectively. He is working on trajectory generationand planning for reusable launch vehicles. His researchinterests also include adaptive control, robust control,optimization, and algorithm design. Most of his researchworks were sponsored by an AFRL/AFOSR grant, DAGSIscholarship, and University of Dayton Graduate SchoolDean’s Summer Fellowship.

Raúl Ordóñez received his M.S. and Ph.D. in electricalengineering from the Ohio State University in 1996 and1999, respectively. He spent two years as an assistantprofessor in the department of electrical and computerengineering at Rowan University, and then joined the ECEdepartment at the University of Dayton, where he hasbeen since 2001 and is now an associate professor. Hehas worked with the IEEE Control Systems Society as amember of the Conference Editorial Board of the IEEEControl Systems Society since 1999; Publicity Chair forthe 2001 International Symposium on Intelligent Control;

member of the Program Committee and Program Chair for the 2001 Conference onDecision and Control; and Publications Chair for the 2008 IEEE Multi-conferenceon Systems and Control. Dr. Ordóñez is also serving as Associate Editor for thecontrol journal Automatica. He is a coauthor of the text book ‘‘Stable AdaptiveControl and Estimation for Nonlinear Systems: Neural and Fuzzy ApproximatorTechniques’’ (2002). Between 2001 and 2007, he worked in the research teamof the Collaborative Center for Control Science (CCCS), funded by AFRL, AFOSRand DAGSI at the Ohio State University. His research in the CCCS has focused oncoordination and control of UAV teams, aswell as on real-time trajectory generationfor reusable hypersonic launch vehicles. Dr. Ordóñez received a Boeing WelliverFaculty Fellowship in 2008.