Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

A98-37225 AIAA-98-4255

AN ALTERNATIVE ENTRY GUIDANCE SCHEME FOR THE X-33

Ping Lu*Iowa State UniversityAmes, IA 50011-3231

John M. Hanson"''Greg A. Dukeman*

NASA Marshall Space Flight CenterAlabama, 35812

Swati Bhargava§Iowa State UniversityAmes, IA 50011-3231

Abstract1 The entry guidance design for the X-33 involves trajectory optimization, generation of a drag ac-

celeration profile as the nominal, closed-loop trajectory control and update, and satisfaction of trajectoryconditions at the end of entry flight. This paper reviews the unique challenges encountered in the X-33 entryguidance and discusses methods used for each of the aforementioned tasks. Monte Carlo simulations areperformed to evaluate the effectiveness of the guidance scheme.

1. Introduction After a short transition phase following the mainengine cut-off (MECO) at an altitude of about

The X-33 Advanced Technology Demonstrator, 190'000 ft> the X~33 is Suided b^ the entrv Suid-a half-scale prototype for reusable launch vehicle ance system toward the landinS site untl1 the vehlcle

(RLV), is an unmanned autonomous vehicle. The velocity reduces to Mach 2'5' which is at about 19

primary technical goals of the X-33 program are nm to the headin§ alignment cylinder (HAC). Thusto develop and test key technologies needed for a maJ°r Portion of the fli§ht is under the contro1 of

the next-generation of single-stage-to-orbit RLV, in- the entry S^dance system. The effectiveness of theeluding structure, thermal protection system, linear entrv Stance scheme is a key factor in assuring theaerospike engine, aerodynamic prediction, and con- success of the fli§ht tests' Be^ause of the umclue

trol and guidance. Figure 1 shows the configuration trajectory pattern of the X-33 (i.e., entry flight im-of the X-33. The X-33 will be launched vertically mediately follows ascent), the entry flight is directlyfrom Edwards Air Force Base, California, and land couPled with ascent' and anv trajectory dispersionshorizontally at Michael Army Air Field in Utah. The in the asceDt Phase wil1 be P^pagated to the entryflight trajectories will cover a range of Mach num- trajectory. Therefore not only should the X-33 entrybers for various test objectives such as maximum §uldance be accurate in a nominal situation, but alsoentry catalytic heating, maximum entry integrated ^^ adaptive and robust in off-nominal cases,heat load and maximum delay of transition to tur- While the entry guidance concept for the X-33bulent flow. follows the framework of the Space Shuttle entry

"Associate Professor, Department of Aerospace Engineering and Engineering Mechanics, Associate Fellow AIAA. Email:plu@iastate.edu

t Chief, Flight Mechanics, Guidance, Navigation & Control Systems Branch* Aerospace Engineer, Flight Mechanics, Guidance, Navigation & Control Systems Branch5 Graduate Student, Department of Aerospace Engineering and Engineering MechanicsCopyright ©1998 by the American Institute of Aeronautics and Astronautics. All Rights Reserved.

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Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

guidance,1 significant differences exist in methodol-ogy, algorithms and implementation.2'3 The entryguidance scheme studied in this paper is not intendedto represent the actual X-33 guidance algorithms.Rather, this research is carried out as a parallel in-vestigation. As such, there are similarities as well asnontrivial differences between the two sets of algo-rithms.

The main body of this alternative X-33 entryguidance scheme has been developed and reported inRefs. 4 and 5. This paper reviews the guidance de-sign methods on trajectory optimization, generationof a drag acceleration profile as the nominal, closed-loop trajectory control, and reference drag profile(on-board) update. However the emphases of thispaper are on: (1) a final entry guidance phase to sat-isfy a strict heading condition of the X-33 at the ter-minal area energy management (TAEM) interface;(2) evaluation of the effectiveness and robustness ofthe entry guidance algorithms under off-nominal con-ditions. To this end, the guidance scheme is evalu-ated by Monte Carlo simulations in which completetrajectories from liftoff to TAEM are simulated witha number of uncertainties involving propulsion sys-tem, propellant loading, and aerodynamics.

2. Entry Trajectory Optimization

Because the entry flight for the X-33 follows rightafter the ascent flight, the design of ascent and entrytrajectories are carried out simultaneously.6 The ref-erence entry drag profile is extracted from this tra-jectory. In this study, however, the 3-DOF entrytrajectory is separately obtained mainly to serve asa benchmark. The reference drag profile for on-boardguidance is computed in a different way as describedin the next section.

The point-mass dimensionless equations of 3-Dmotion of the X-33 over a spherical, rotating Earthare

V cos 7 sin tbrcos<?>

V cos 7 cos ̂

V = -D- Q2r cos <2>(sin 7 cos <j>

— cos 7 sin </> cos i

(1)

(2)

(3)

(4)

7 = —[Lcosv- 2 l\ cos7V2-- r

+ Q2r cos </i>(cos 7 cos <b + sin 7 cos ifr sin

1 rLsin<7 V2 . .= —I ———— + —— cos 7 sin ip tan

V cos 7 r— 2£lV(tan 7 cos if; cos <j> — sin <j>)

+cos 7

sin ib sin 6 cos <j>] (6)

where r the radial distance from the center of theearth to the X-33, normalized by the radius of theearth R0 = 20,925,673 (ft). The longitude and lat-itude are d and 6, respectively. The earth-relativevelocity V is normalized by -\/goRo with go = 32.185ft/sec2. D and L are aerodynamic accelerations ing's. Q is the rotation rate of the earth normalizedby ^/go/Ro- 7 is the flight path angle and a thebank angle. The velocity azimuth angle ip is mea-sured from the North in a clockwise direction. Thedifferentiation is with respect to the dimensionlesstime T = i/^/Ro/go- The main reason for using thedimensionless form is for better numerical condition-ing of the trajectory optimization problem discussedbelow.

The conditions at the end of the transition phaseare determined by the ascent trajectory and transi-tion guidance, and are considered the given initialconditions for the entry trajectory. The terminalconditions for entry flight are specified at the TAEMinterface. For trajectory optimization purposes, thelocation of a target TAEM point and other condi-tions at this point are given

< 7(r/) <

where 77 is at an altitude of about 82,000 ft (25 km)and Vf corresponds to Mach 2.5. The azimuth ifr/ isdetermined by requiring the velocity at the TAEMpoint to direct toward the HAC. In addition, the fol-lowing trajectory constraints are imposed

|Lcosa + £>sina| <

9 <•Q, < Qr,

(8)(9)

(10)

where Eq. (8) is a constraint on the acceleration inthe body-normal direction; Eq. (9) is on dynamicpressure q; and Eq. (10) is on heat rate Qs ata stagnation point. Multiple heat rate constraintsfor several stagnation points can be imposed, al-though only one is used in this work. In this study,n*»« = 2-5 (g), qma* = 11,970 N/m2 (250 psf), andQmax = 431,259 W/m2 (38 BTU/sec-ft2) are used.

In the design of the nominal entry trajectory,the angle of attack a is scheduled as a function ofMach number, beginning at large value (40-45 de-grees ) and gradually reducing to 15 degrees at the

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Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

TAEM point. The bank angle a is modulated tocontrol the trajectory. Therefore in this section <r isparametrized as a piecewise linear function of time.The nodal values of the parametrization of & and theflight time r/ are found to satisfy the TAEM condi-tion (7), the inflight constraints (8-10), and minimizea performance index.

J r*=Jo

'dr (11)

where p = 0.033994e-*°(r-1)/22000 (slug/ft3) is areasonable approximation to the 1976 U.S. standardatmosphere in the altitude range of interest to entryflight. The performance index (11) is proportional tothe accumulated heat load per unit area at the stag-nation point. Other performance indices may alsobe selected,6 but we have found that the impact ofdifferent performance indices on the entry trajectoryis not significant once the initial entry conditions aregiven. For more detail on the trajectory optimizationalgorithm, the reader is referred to Ref. 4.

In the trajectory optimization program, the X-33tabulated aerodynamic data are fitted with smoothanalytical functions of Mach and angle of attack. Inaddition to above terminal and inflight constraints(7-10), the piecewise linear parametrization of thebank angle enables us to enforce easily a rate con-straint on the bank angle history

< 10 (deg/sec) (12)

This constraint is imposed because of the practicallimits on the X-33 flight control system.

Figure 2 shows the altitude-versus-velocity vari-ation along the optimized entry trajectory with theinitial conditions specified for the maximum catalyticheating mission. This flight trajectory will reach apeak Mach number of 10. Also plotted in Fig. 2 arethe constraint boundaries for Eqs. (8)-(10). Theentry trajectory must lie above all the constraintboundaries in the velocity-altitude space. Clearly inthis case the only active constraint in Eqs. (8-10) isthe normal load constraint. Figure 3 shows the bankangle history of the same entry trajectory, and Fig.4 depicts the drag acceleration variation versus thedimensionless specific energy

1 V2

3. Reference Drag Profile

(13)

While the optimized trajectories provide the nec-essary information for oil-line analysis, the closed-loop entry guidance of the X-33 is accomplished by

tracking a reference drag-versus energy profile8 withthe bank angle as the primary mean of trajectorycontrol. The reference drag profile is obtained byapproximating the drag acceleration variation suchas in Fig. 4 with a piecewise linear discretization.Alternatively, a different approach is suggested inRefs. 4 and 5 for generation of the reference dragprofile. In this approach, only a parameter optimiza-tion problem with analytical objective function andconstraint needs to be solved without involving in-tegrations. This process uses three to four orders ofmagnitude less time as compared to the trajectoryoptimization approach. Consequently, many scenar-ios can be examined in a short period of time for off-line analysis, and potential on-board design of thereference drag profile is feasible.

We briefly review this method in this section.Complete development and detail can be found inRef. 5. Let eo and e/ be the prescribed energylevels at the beginning and the end of entry trajec-tory, respectively. Divide the interval [eo,e/j inton — I subintervals by the points {ei, e2,..., en} withei = e0 and en = e/. In each interval [c,-,e,-+i],i= 1,..., n — 1, let the reference drag acceleration beparametrized by a linear function of e

wherea,- =

D(e)=ai(e-ei)+bi

A+i-A- ,.._,

(14)

(15)

values of Di 's are to be determined. The con-straints (8-10) can be expressed in the D-e space asthe constraints on Di

Dmin(ei) < D{ < i = 2, ...,n- I (16)

If the variation of CD is ignored, the performanceindex Eq. (11) can be shown to be proportional to

n-l

(17)

where the parametrization (14) has been used, andAJi's are analytical functions of A+i and Di theexpressions of which can be found in Ref. 5. Thedownrange distance traveled by following the dragprofile (14) is accurately predicted by

e

..(18)

where

191

aO 1* (A+i/A), a,' + 0

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

The requirement of reaching the TAEM point(d(rj) = df and (/>(rf) = (j>f) is replaced by the re-quirement

n-l= sto_go (20)

with Sto-go being the range-to-go from the beginningof the entry to the TAEM interface.

The design of a reference drag profile now is for-mulated as a parameter optimization in which theobjective function is

n-l r JT-I/ \dJJ(e)I *

.den —1

(21)

where e > 0 is a small constant (on the order of10~6). The additional integral is a regularizationterm that makes the resulting drag profile flyable(see Ref. 5 for detail). The optimization parame-ters are D,-'s which are to be found to minimize Jand satisfy the equality constraint (20) and inequal-ity constraints (16). This optimization problem canbe readily solved much more efficiently than the tra-jectory optimization problem. It has been demon-strated in Ref. 4 that the drag profile thus obtainedis remarkably close to the one obtained from trajec-tory optimization.

4. Trajectory Control

During a major portion of the entry flight of theX-33, the downrange motion will be controlled bytracking the reference profile with bank angle modu-lation. A nonlinear proportional-plus-integral-plus-derivative (PID) control law for the bank angle isused. The control law can be derived by using input-output feedback linearization2'9 or a nonlinear pre-dictive control method.5 In this alternate entry guid-ance design we use the latter which is reviewed brieflyin the following.

Denote u = CL cos cr/Co > ignoring the rotationof the Earth, CD and CD, we can express

D = ap + (22)

where C > 0 and wn > 0 are two constants. At anyinstant r the influence of u(r) on z(r + T) for a timeincrement T > 0 can be predicted by a first-orderTaylor series expansion

z(r+T)bD (T)U(T) + 2Cwn AD(r)

(24)

where for the piecewise linear parametrization of D*with respect to e, we have in the interval [e,-, e,-+i]

D* = ai(DV + DV) (25)

where a; is from Eq. (15). Note that for accuratetracking of D* , we desire that z -> 0. To find thecontrol u for this purpose, consider the minimizationof the performance index

(26)

at an arbitrary T E. [0,ry). Replacing Z(T + T) byEq. (24) and setting dJ/du(r) = 0 give a continu-ous, nonlinear feedback control law

(27)

This is a nonlinear PID control law when z is re-placed by its definition (23). Globally asymptoti-cally stable tracking of £>* for any T > 0 under thiscontrol law can be seen by substituting Eq. (27) intothe equation for z to arrive at

1(28)

Thus z —> 0 exponentially with a time-constant T,and z —> 0. From the definition of z, z = 0 leads to

AD + 1&n AD + = 0 (29)

where OD and &D are functions of r, V, 7 and D,which can be readily obtained from the definition ofD and Eqs. (1) and (4). Let AD = D-D' with D*representing the reference drag acceleration. Definean auxiliary variable

„ rz(r) = AD + 2&nAD + wl AD(p.)dfj. (23)

Jo192

Therefore, AD -» 0 with a damping ratio of C andnatural frequency of wn • For the X-33 applications,we have chosen £ = 0.7, T = Q.QlyRo/go (sec), andun = 0.041^/^0/501 or l/wn corresponds to 25 sec inreal time. The magnitude of the commanded bankangle <rcom for trajectory tracking is computed fromCOS (Team = uCo/CL-

In the implementation, the computed acom is alsoconstrained by the following amplitude, rate, and ac-celeration limits

<85 (deg), \crcon

< 2 (deg/s2)< 10 (deg/s

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

where the rate and acceleration are computed by fi-nite differences.

The cross range is controlled by orientation of thebank angle. A simple dead-zone criterion for revers-ing the sign of the bank angle, similar to the one em-ployed by the Space Shuttle,1 is used in this scheme.Suppose that tp* is the azimuth pointing from thecurrent position toward the target point (the centerof the HAC). When the magnitude of the azimutherror \i/> — i{>*\ exceeds a dead-band limit of 10 deg,the bank angle is reversed to the opposite direction.Note that due to the rate constraint in Eq. (30), thebank reversals cannot be achieved instantaneously.A more elaborate bank reversal algorithm is given inRef. 2.

5. Drag Profile Update

The reference drag profile D* is designed basedon the downrange distance requirement (20) at thebeginning of the entry trajectory. Because of off-nominal dispersions and cross-range motion of thevehicle, the reference drag profile will need to be up-dated periodically to null the downrange errors. TheShuttle uses a first-order approximation approach toadjust one segment of the drag profile at a time.1The piecewise linearity of the parametrization of D*for the X-33 makes it a simple matter to update theentire D* by scaling it with appropriate coefficients.

Define the range-to-go by

(31)

where Shac is the downrange distance along the greatcircle from the current point to the HAC, and staemis a constant bias term, taken to be 24 nm. Sup-pose that Sprd is the predicted downrange distanceby Eq. (18) with the current values of the nodes D*.Then the drag profile update is done to each of theremaining node2

Di = (32)o— go

Given the piecewise linearity of the drag profile, thisupdate is equivalent to scaling the entire referencedrag profile by a factor of sprd/Sto-go, therefore thepredicted downrange distance determined by the up-dated drag profile from Eq. (18) is now exactlySfo—ga-

in addition to update (32), it is realized that thediscrepancy in the initial energy level of the trajec-tory also contributes to range errors. Hence a one-time compensation to this effect is desired. At theinitiation of the closed-loop guidance, let the actualenergy level be CQ, and the nominal energy at the

same point be ej$. Note that CQ and ej$ may be dif-ferent because of the uncertainties and dispersionsassociated with the ascent. The final energy e/ isfixed. Thus the very first update should be

_ (Jprd

Sto-go / \e/ ~ eo

-eo (33)

This update procedure represented in Eqs. (32)-(33) is found to be one of the important factorsthat enhance noticeably the robustness of the per-formance of the entry guidance algorithms in thepresence of significant trajectory dispersions. For thesimulations reported in Section 7, the update is doneonce every second. It should be noted, though,' thatcaution must be exercised when the reference dragprofile lies close to the boundaries of the entry flightcorridor specified by constraints (8-10). In such acase the scaling of the drag profile by (32) (or (33))could result in the update drag profile violating someof these constraints.

6. Final Heading Control

The TAEM interface for the X-33 is only about19 nm from the HAC. Therefore there is virtually notime to correct the trajectory if the velocity headingat the interface is not pointed toward the HAC tan-gency. The drag-profile-tracking trajectory controland bank reversals described in Section 4 cannot en-sure the alignment of the heading angle to the HAC.In fact, Monte Carlo simulations of 100 trajectoriesshowed that at the TAEM interface, the heading er-rors to the HAC had a mean value of about 15 deg,and quite a few trajectories ended with 20 to 25 degof heading errors, which is not acceptable. There aretwo major contributing factors to this phenomenon:(1) in general the heading error will continue to in-crease for some time due to the vehicle dynamicsafter the bank angle command is reversed when theheading error exceeds 10 deg; (2) when the vehiclegets close to the HAC, the variation of the azimuthi/>* to the HAC becomes faster. Consequently a bankreversal is always called for near the TAEM interface.But the bank rate and acceleration constraints in Eq.(30) severely limit the speed of the bank reversal,thus resulting further overshoot of the heading errorat the TAEM interface. Using a smaller heading er-ror dead-band does not seem to solve this problemeffectively, but increases the number of bank rever-sals unnecessarily. This challenge necessitates theuse of a different guidance strategy before the vehi-cle reaches the TAEM interface.

A proportional-navigation (PN) guidance law10

is our choice. PN guidance has been an exten-sively researched guidance method in the literature

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because of its simplicity and effectiveness. The ap-plications have been mainly in short-range intercep-tion. In almost all the analytical investigations ofthe PN guidance, a key assumption is constant ve-locity for the interceptor. In a recent work,10 thePN guidance method is extended to intercept of anonmoving target by an interceptor with arbitrarytime-varying velocity. It has been shown that insuch a scenario, any navigation constant of greaterthan one will lead to intercept, and any navigationconstant of greater than two will result in the in-terceptor to directly head toward the target (directcollision course). These developments appear to fitthe current situation perfectly well: the velocity ofthe X-33 cannot be regarded as constant; the targetpoint (HAC) is nonmoving; and the heading of theX-33 needs to be pointed to the HAC.

Again, let ip* be the azimuth angle from the cur-rent position to the HAC. The PN guidance law isthen

t/> = Xip* (34)

where a navigation constant of A > 2 should be cho-sen. If the self-rotation of the earth is ignored, thebank angle control law corresponding to (34) is ob-tained from Eq. (6)

V2

sin <7 = (XVTJJ* — — cos 7 sin if? tan <{>) cos 7/L (35)

The guidance algorithm is switched to this PN guid-ance law at a distance do from the HAC. For theX-33 trajectory to Michael AAF, d0 is selected tobe 35 nm where the nominal velocity is about Mach3.6. This gives a range of about 16 nm for the PNguidance to align the heading to the HAC before theX-33 reaches the TAEM interface where the entryguidance stops. A relatively large navigation con-stant of

A = 5

is used to quickly orient the heading to the HAC inthis period. An added advantage of this approach isthat once the vehicle is heading toward the HAC, nosignificant trajectory maneuvers are needed, thus thebank angle at the TAEM interface will be relativelysmall.

It should be mentioned that a bank angle controllaw based on the feedback linearization approach canalso be derived from the heading dynamics Eq. (6)to steer ^ —> TJJ* . The performance of such a guid-ance approach is found to be comparable to that ofthe above PN guidance. However, the PN guidanceis regarded more favorably because of its simplicity.

When the bank angle control is switched fromdrag-profile-tracking to heading control, the longitu-

194

dinal motion tends to produce too shallow a trajec-tory at the TAEM interface with high altitude andsmall flight path angle.2 To compensate for this un-desirable dispersion from the required TAEM condi-tions, in the period when the heading control (35) isin effect, the angle of attack a is modulated accord-ing to

a = oPe/-1.5(7-7*) (36)where aref is the nominal value of the angle of at-tack, and 7* is a constant value determined on-boardas follows: Let ho be the altitude at the distanced0 from the HAC (where the final heading controlphase is initiated), htaem the specified TAEM. alti-tude (82,000 ft), and dhac the nominal distance fromthe TAEM interface to the HAC (115,446 ft, or 19nm). Then

ho — htaemtan 7* = (37)do — dtaem

Evidently, the meaning of 7* is such that if the ve-hicle flies a constant flight-path-angle trajectory at7 = 7*, it would reach the TAEM interface at thealtitude of htaem. Note that h0 is different alongdifferent trajectories, thus 7* should be determinedon-board. The modulation of a is limited by theconstraints

a - aref | < 5 (deg), \a\<5 (deg/sec) (38)

7. Monte Carlo Trajectory Simulations

The algorithms described in Sections 3-6 are im-plemented for the X-33 in simulation. The vehicletrajectory simulation is performed by a computerprogram called Marshall Aerospace Vehicle Repre-sentation in C (MAVERIC), developed at the NASAMarshall Space Flight Center. The trajectory simu-lation is from liftoff through entry. In this study theTAEM interface is defined by the value of the spe-cific energy corresponding to an altitude of 82,000 ftand Mach 2.5. The simulation terminates when theenergy reaches this value. The entry guidance com-mand update cycle is one second. Figure 5 showsthe drag acceleration variation during entry alongthe nominal maximum catalytic heating trajectory,and Fig. 6 gives the entry bank angle command andFig. 7 the angle of attack versus Mach history. FromFig. 6 it is seen that two bank reversals took place.Notice the similarity between Figs. 4 and 5, and be-tween Figs. 3 and 6. The final heading control phasestarted at t = 400 seconds from liftoff. At the endof the nominal entry trajectory, the distance to theHAC is 19.5 nm, the heading error to the HAC is0.22 deg, the altitude is 84,641 ft and velocity 2426ft/s.

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

To evaluate the performance of the entry guid-ance scheme in the presence of uncertainties inpropulsion, navigation and aerodynamics, MonteCarlo simulations were performed. The propulsionsystem uncertainties include ±1% in Isp (specific im-pulse), ±1% in propellant utilization, 404.1 Ibm inloaded LOX and 54.96 Ibm in loaded LH2. Theaerodynamic coefficient uncertainty model is basedon comparison of predicted versus actual flight datafrom past lifting body programs. MAVERIC canrandomly generate the uncertainties in the specifiedranges and the corresponding trajectory.

A total of 100 trajectories were first generatedto demonstrate the need for tighter heading controlat the TAEM interface. These trajectories did notuse the final heading control strategy discussed inSection 6. Figure 8 shows the flight path angle andaltitude at the end of the trajectories. The TAEMaltitude is generally well controlled around the nom-inal value of 82,000 ft. But Fig. 9, which containsthe range to the HAC and heading error to the HAC,reveals significant heading deviations from the HAC.The average heading error is about 15 deg and themaximum error is as high as 25 deg. Figure 10shows the values of bank angle and angle of attack atTAEM interface. It is seen the bank angle is nearlyuniformly distributed in the [—85°, 85°] range.

Next, when the final heading control strategy isemployed, another 120 trajectories were simulated.Figure 11 illustrates the dispersions of heading errorand range-to-HAC. Now it is evident that the head-ing control markably improves the heading-to-HAC.The majority of the trajectories had heading errorswithin ±2 deg, and all but two trajectories met the±5 deg heading error specification. The range-to-HAC had somewhat larger dispersions as comparedto Fig. 9. The average value of range-to-HAC is19.636 nm with a standard deviation of 2.489 nm.The ±5 nm error specification on the range-to-HACis still satisfied by most of the trajectories.

Figure 12 shows the altitude and flight path an-gle dispersions. The trajectories generally end higherthan the nominal value with the altitude having anaverage value of 84,378 ft and a standard deviationof 1,117 ft. Yet all trajectories met the ±5,000 ftaltitude dispersion requirement.

Figure 13 depicts the bank angle and angle ofattack at the TAEM interface. The a-modulation(36) in most cases reduces the angle of attack inan attempt to lower the altitude. Without the a-modulation (36), the trajectories would terminate ataltitudes as high as 93,000 ft. Because the trajec-tories headed directly toward the HAC in a nearlystraight line under the PN guidance law (34), as a

result the bank angle was relatively small, mostlyconfined between ±25 deg, which is in sharp con-trast to Fig. 10.

Finally, Fig. 14 contains the final velocity andaltitude information. The average value of the Machnumber at the TAEM interface is 2.497, with a stan-dard deviation of 0.02.

8. Conclusions

The unconventional trajectory pattern of the X-33, i.e., entry immediately following ascent, posessome unique challenges to the design of the entryguidance algorithms for the X-33. In addition togood nominal performance, the entry guidance sys-tem must be sufficiently robust and adaptive be-cause of the direct influence of any off-nominal con-ditions during the ascent on the entry flight. Thesechallenges are met by employing some of the re-cent research achievements in trajectory optimiza-tion, nonlinear control theory and guidance method-ology. While the classical entry guidance conceptoriginally developed for the Space Shuttle is still thefoundation, many significant new developments andimprovements have been made. This paper offers adifferent perspective and approach on the design ofthe entry guidance scheme for the X-33. Extensivesimulations demonstrate that this scheme is quitesatisfactory in meeting the stringent demands of theX-33 entry flight.

Acknowledgment

This research has been supported by NASA Mar-shall Space Flight Center under Grant NAG8-1289.

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Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

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50.0

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Figl: The X-33 vehicle

-100.0

-50.0 •

100.0 200.0t(sec)

Fig. 3: Open-loop bank angle history

300.0

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Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

100.0

50.0

~ 0.0

-50.0

0.90 0.92 0.94 0.96 0.98energy

Fig. 4: Drag variation during entry (open-loop)

-100.0200.0 300.0 400.0

t(sec)

Fig. 6: Closed-loop bank angle history

500.0

2.5

2.0

1.0

0.50.90 0.92 0.94 0.96 0.98

eFig. 5: Drag variation during entry (closed-loop)

50.0

40.0

30.0

20.0

10.02.0 4.0 6.0

Mach8.0

Fig. 7: Angle of attack history

10.0

197

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

o**.u

83.0

1, 82.0.c

81.0

ftn n

0 °

o

"o o* 0 o

O O -Q Q

° <^°o ^ o° o o g°oo i9 o o o -^ j^

o o o ° }̂-P ^ oo _ ^ o

O o

8 ° ooo

o

-20.0

15.70 r

-15.0 -10.0gamma (deg)

Fig. 8: Altitude and flight path angleat TAEM interface without heading control

-5.0

15.60 r

15.50

15.40 -

n 0§ §

l0 00 ° 0

o00 o

oo

-100.0 -50.0 50.00.0bank (deg)

Fig. 10: Bank angle and angle of attackat TAEM interface without heading control

100.0

££.U

20.0

^^18? 18.00(bo

i

16.0

0O o

00

° & ° °0 o^lfl^o^0

o §> <®o°Cb'o <£> <•£O C) ^ r\ ^O

O OO 0 __ Q

O & 0o0 0

o oo

nn nOU.U

25.0

^^

— '^I 20.0od)c2

15.0

14.0 ——————————————————————————————————————————-10.0 0.0 10.0 20.0 30.0 1nn

Oo

o oo @o 0 8 . ,0 00§ZO 0<* 0

O n rtSc^ O OOo % <^8 o^=sojco o

o "^b^30^o IE °

° °° o°S ooo

§

, , ,

heading error (deg)Fig. 9: Heading-to-HAC error and range-to-HAC

at TAEM interface without heading control

.-10.0 -S.O 5.0 10.00.0

heading error (deg)Fig. 11: Heading-to-HAC error and range-to-HAC

at TAEM interface with heading control

198

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

88.0

87.0 -

86.0 -

85.0 -

84.0 -

83.0 -

82.0 -

81.0 •

80.0

25.0

20.0

-20.0 -18.0 -16.0 -14.0 -12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0gamma (deg)

Fig. 12: Altitude and flight path angleat TAEM interface with heading control

10.0-50.0 50.00.0

bank (deg)Fig. 13: Bank angle and angle of attackat TAEM interface with heading control

100.0

88.0

87.0

86.0

85.0

84.0

83.0

82.0

81.0

80.0

b°o oo oO _,,£ .0-•

2400.0 2420.0 2440.0 2460.0 2480.0V(ft/s)

Fig. 14: Velocity and altitude atTAEM interface with heading control

2500.0

199