NASA Contractor Report 189703 Air-Breathing … · NASA Contractor Report 189703, Air-Breathing...

NASA Contractor Report 189703,

Air-Breathing Hypersonic Vehicle Guidance and Control Studies:An Integrated Trajectory/Control Analysis MethodologyPhase II

Philip D. HattisHarvey L. Malchow

(_iASA-C_-I_7n3) AIR-BREATHiNG

HYPER3JN[C VEHICLE GUTOANCE AND

CCNT_gL STU_IE_: AN INTFGRATED

TPAJECT_Y/CONTPOL ANALYSIS

METPO_OLOGY, PHASE 2 Pinal Report

(qraper (Charles Stark) Lab.) So pG3/08

N93-12_13

Unclas

0129302

The Charles Stark Draper Laboratory, Inc.Cambridge, Massachusetts

Contract NASI-18565October 1992

Nalional Aeronautics a;_d

Space Adminislration

Langley Research Center

Hampton, Virginia 23665-5225

https://ntrs.nasa.gov/search.jsp?R=19930003225 2018-09-04T20:01:29+00:00Z

__ 1_m __

Table of Contents

SectionI, Task Synopsis 1

II. Introduction (Background on Task) 2

A. Problem Overview 2

B. Summary of Phase I Progress 2

III. List of Symbols and Acronyms 4

IV. Research Goals 7

A. Phase II Task Focus 7

1. Completion of Analysis Using Phase I Vehicle Model 7

2. Identification of Issues Resulting from Use of Phase I Imple- 7mentation

3. Algorithm Evaluation with Alternative Phase II Vehicle Model 8

B. Longer Term Research Objectives 8

1. Treatment of Overlapping Air-Breathing/Rocket Propulsive 8Phases

2. Treatment of Vehicle Thrust Direction and Pitch Moment 8

Changes with Angle of Attack

3. Incorporation of Thermal Constraints into Trajectory Results 8

4. HSV Guidance and Control (G&C) Performance Robustness 8Studies

V. Progress for April 1991 - March 1992 10

A. Analysis with Phase I Vehicle Models 10

1. Summary of Case Run 10

2. Numerical Orbit Integration Issues Resulting from Polar Orbit 10Target

3. Overview of Trajectory/Control Analysis Results 10

4. Optimization Stability Problems Due to Use of Angle of Attack 12as a Control

Paae

Table of Contents (Cont.)

SecUonv

_B,Alaor!thm and Model Improvements/Simplifications 12Due to Phase I Model Results

1. Change of Control Variable 12

2. Replacement of Propulsion Model 13

3. Elimination of Elevon Trim Calculations 13

4. Change of Orbit Targets 13

5. Updated Analysis Algorithm Formulation 14

6. Updates to Model Equations Required for Algorithm Analysis 17

a. Vehicle State Dynamics Models 17

b. The Boundary Conditions 19

c. Cost Function Specification 20

d. Relevant Partial Derivatives 20

C, Analysis Demonstration Results with Phase II Model 21

1. Summary of Case Run 21

2. Overview of Trajectory/Control Analysis Results 21

3. Effects of Propulsion Model Discontinuity at Mach 2 22

4. A Look at Dynamic Pressure History Stability 23

5. Sensitivity of the Solution Trajectories to the Propulsion 23Model

D. Analysis Observations Affecting G&C Design 24

E. Progress Tqward AIPS Requirements Specification 24

VI, Conclusions 25

27VII. ReferQnces

Paae

TablQ of Contents (Cont.)

Appendix A: Partial Derivatives for the ExampleVehicle Model with Respect to States

Appendix B: Partial Derivatives for the ExampleVehicle Model with Respect to Controls

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Table of Figures

Description

Coordinate systems for dynamics equations

Altitude vs time - phase I model

Mach number vs time - phase I model

Q vs time - phase I model

Q vs Mach number - phase I modelFirst 500 seconds of flight

Phi vs time - phase I model

Phi vs Mach number - phase I modelFirst 500 seconds of flight

Alpha vs time - phase I model

Alpha vs Mach - phase I modelFirst 500 seconds of flight

Delta vs time - phase I model

Delta vs Mach number - phase I modelFirst 500 seconds of flight

Gamma vs Mach number - phase I modelFirst 500 seconds of flight

Inertial acceleration vs time - phase I model

Inertial acceleration vs Mach number - phase I modelFirst 500 seconds of flight

Mass vs time - phase I model

Mass flow vs time - phase I model

Mass flow vs Mach number - phase I modelFirst 500 seconds of flight

Q vs time - phase I modelStarting 100 seconds after takeoff

Altitude vs time - phase I modelStarting 1000 seconds after takeoff

Propulsion model spline data output: Thrust coefficient vs Mach number -Low speed phase

Propulsion model spline data output: Thrust coefficient vs Mach number -High speed phase

Propulsion model spline data output: Specific impulse vs Mach number -Low speed phase

Propulsion model spline data output: Specific impulse vs Mach number -High speed phase

Altitude vs time - phase II model

Mach number vs time - phase U model

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Eiggr Description

Table of Figures (Cont.)

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Q vs time - phase II model

Q vs Mach number - phase II modelFirst 500 seconds of flight

Phi vs time - phase II model

Phi vs Mach number - phase II modelFirst 500 seconds of flight

Theta vs time - phase II model

Theta vs Mach number - phase II modelFirst 500 seconds of flight

Alpha vs time - phase II model

Alpha vs Mach number - phase II modelFirst 500 seconds of flight

Gamma vs time - phase II model

Gamma vs Mach number - phase II modelFirst 500 seconds of flight

Gamma vs time - phase II modelStarting 250 seconds after takeoff

Inertial acceleration vs time - phase II model

Inertial acceleration vs Mach number - phase II modelFirst 500 seconds of flight

Thrust vs time - phase II model

Thrust vs Mach - phase II modelFirst 500 seconds of flight

Inertial velocity vs time - phase II model

Mass vs time - phase II model

Mass Flow vs time - phase II model

Mass flow vs Mach number - phase II modelFirst 500 seconds of flight

Q vs time - phase II modelFrom 250 to 2000 seconds after takeoff

Paae

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V

I, Task Synopsis

Air-breathing hypersonic vehicles, with fully integrated airframe and propulsion systems,are expected to face the most complicated design, trajectory, and control coupling challengesofany class of flight vehicles ever considered. Assessment of appropriate trajectory managementand control strategies must be accomplished during the conceptual phases of vehicle design toassure a high probability that the fully integrated system will accomplish mission objectives. Also,a valid mathematical basis is needed to rate the relative performance of alternative vehicle designconcepts and/or subsystem technologies against a common mission goal. To address theseconcerns, a generically applicable methodology has been developed under phase I of this taskto generate and evaluate single-stage-to-orbit air-breathing hypersonic vehicle flight profiles.Capable of developing trajecto_/control histories in an integrated manner, the formulationincludes direct treatment of physically derived inequality constraints and permits determinationof optimal propulsive mode phasing.

During the phase II effort reported here, the air-breathing flight analysis capabilities of theintegrated analysis algorithm have been assessed by generation of near-minimum-fuel trajectoriesand desired control strategies using two different vehicle models and orbit targets. Both modelsused tabulated winged-cone accelerator vehicle representations that are combined with a mul-tidimensional cubic spline data smoothing routine. Near-fuel-optimal, horizontal takeofftrajectories, imposing a dynamic pressure limit of 1000 psf, were developed.

The initial analysis methodology demonstration involved a polar orbit track with a targetair-relative velocity of 25,000 ft/s. This case used an existing air-breathing propulsion model andincluded the dynamic effects of using elevens to maintain longitudinal trim. Analysis of resultsindicated problems with the adequacy of the propulsion model and highlighted dynamic pres-sure/altitude instabilities when using vehicle angle of attack as a control variable. Similar insta-bilities were encountered by Langley representatives in in-house trajectory studies for hypersonicvehicles. The Langley studies showed that use of pitch attitude relative to the local vertical as acontrol variable instead of angle of attack eliminates dynamic pressure/altitude instabilities, andsuggested use of pitch attitude as control in the trajectory optimization algorithm. Application ofthe pitch control variable in this study showed a continued tendency toward dynamtc pressur-e/altitude oscillation, but without divergent instability. Another observed effect was the magnitudeof computed eleven deflections to maintain trim which suggests a need for an alternative pitchmoment management strategy.

The second analysis methodology demonstration case reformulated the vehicle model andassociated derivatives to use vehicle pitch attitude relative to the local vertical as the controlvariable. A new, more realistic, air-breathing propulsion model was incorporated. Also, to furthersimplify subsequent analysis, pitch trim calculations were dropped and an equatorial orbit wasspecified. A target altitude of 400,000 ft was established with states that institute a 100 NM apogeeby 0 NM (sea level) perigee orbit. Changes in flight characteristics resulting from the effects ofthe new propulsion model were identified. Flight regimes demanding rapid attitude changes werecharacterized. Also, some issues regarding design of closed-loop controllers are ascertained.

The demonstration cases provided examples of the kind of information that the analysismethodology can develop for a wide range of HSV designs and mission applications. Near-fuel-optimal trajectories that satisfy the applied constraints were determined. Previouslyunspecified propulsive discontinuities were located. Flight regimes demanding rapid attitudechanges were identified, dictating control effector and closed-loop G&C design requirementsbased on the required control response. Also, inadequacies in vehicle model representationsand specific subsystem models with insufficient fidelity were determined.

After completion of analysis of the information resulting from use of the integrated trajec-tory/control methodology, closed-loop guidance and control design requirements can beestablished. This information must precede specification of on-board advanced informationprocessing system design characteristics.

II. Introduction (Back oround on Task)

II.A. Problem Overview

Hypersonic air-breathing vehicles (HSVs) are under consideration for a range of advancedflight applications including horizontal single-stage-to-orbit (SSTO) launch operations. Relyingon aerodynamic lift and air-breathing propulsion, the performance of these vehicles will be verysensitive to the trajectory management strategy. Additionally, all proposed HSV concepts havephysical coupling between propulsion performance, aerodynamics, and trajectory dynamics toa degree that greatly exceeds previous flight vehicle experience. Even under ideal circumstances,much of the HSV ascent is accomplished by engines which produce only slightly more thrustthan the overall vehicle drag. To achieve adequate acceleration, HSVs must operate much ofthe time near structural and/or thermal limits of the airframe, necessitating explicit treatment ofthose constraints in the trajectory management strategy. Multiple distinct air-breathing propulsivemodes are likely, and a rocket capability is required to complete flight from subsonic takeoffconditions to the target orbit state. A poor guidance and control (G&C) implementation can resultin reduced propulsive performance and excessive aerodynamic losses, possibly preventingsuccessful accomplishment of mission objectives. Therefore, successful HSV trajectory man-agement will be dependent on the development of a G&C system that efficiently manages thevehicle ascent path while accommodating the coupled system dynamics.

With currently anticipated technology, the HSV SSTO mission is at best marginally feasible.To achieve the SSTO capability with a significant payload, it is essential that near-minimum-fuel-to-orbit trajectories be flown. The relative merits of alternative vehicle configurations, sub-system technologies, and G&C design concepts can only be assessed in a valid manner if thecomparative performance standard assures consideration of efficient trajectories for each case.To accomplish this, a generic tool that would generate near-minimum-fuel-to-orbit trajectorysolutions for a wide variety of HSVs is required. The methodology should account for effects ofkey physical design constraints, the timing of propulsive phasing, and the principal sources ofdrag losses in order to provide an accurate indication of system performance and to evaluatedesired G&C strategies.

Upon completion of development of integrated generic methodologies that determinenear-optimal trajectory management and control strategies for HSVs, it is necessary to demon-strate the ability of the resulting tools to contribute to system performance analysis and designsensitivity studies. These capabilities are needed to support closed-loop G&C designrequirements definition.

Application of the capabilities discussed above will help enable successful development ofHSVs meeting a wide range of operational objectives.

ll.B. Summary_ of Phase I Pro_oress

During phase I of this task (performed from June 1990 to March 1991), a generic meth-odology to permit integrated trajectory/control analysis for generic HSVs was implemented [Ref.1]. Including the effects of physically derived constraints, the methodology was successfullyadapted from previously developed Draper Laboratory analysis tools [Refs. 2-5] to the assess-ment of single-stage-to-orbit HSVs with multi-mode propulsion systems. Both an overall trajectoryevaluation algorithm and a preliminary near-fuel-optimal trajectory development tool weremathematically formulated and implemented for use in a work station computational environment.The methodology is capable of assessing the relative performance of alternative vehicles and/orsubsystem technologies with respect to a single mathematically valid measure. To assure goodnumerical behavior of the analysis algorithms, a cubic spline based data smoothing routine wasdeveloped and included in the resulting tool to permit use of multidimensional tabulated vehiclemodels.

2

Efforts to demonstrate the generic analysis tool were initiated. A seven state vehicledynamics model which included treatment of incremental forces due to maintenance of vehiclelongitudinal moment trim was constructed. Angle of attack and fuel/air equivalence ratios wereutilized as control variables based on a desire to permit some intuitive analysis to be applied.The model was implemented on a work station including all relevant partial derivatives.

A specific preliminary test case of the hypersonic vehicle integrated trajectory/controlanalysis methodology was defined, performed, and evaluated to get an early assessment of thecapabilities of the algorithm. To limit the computational effort to derive some results during phaseI of the task, the trajectory was derived by sequential computation of several trajectory segmentsfrom takeoff to high Mach flight. The computational procedures were based on methods devel-oped under the task to construct suitable initial representations of the trajectory needed for overallintegrated optimization. The performance data utilized by the example vehicle model was basedon an early version of an air-breathing accelerator, winged-cone vehicle database [Ref 6]. Per-formance from horizontal takeoff to 20,000 ft/s relative airspeed on a polar orbit track wasdetermined. A dynamic pressure constraint of 1000 psf was enforced. The following listsummarizes some of the most significant phase I conclusions:

• Three near-fuel-optimal flight segments were derived that satisfy the applied constraints.They successively cover flight from horizontal takeoff to Mach 2, Mach 2 to 10,000 ft/s,and 10,000 ft/s to 20,000 ft/s (where velocity was specified in terms of relative air speed).Segment boundary condition matching criteria were enforced to permit eventual merger(during phase II) of the individual segment results into a single full trajectory usable foroverall optimization.

• The analysis methodology was able to identify previously unspecified discontinuities and

phasing characteristics in the vehicle propulsion model. This information was used totdentify appropriate model improvements for phase II studies.

• Flight regimes with strong performance sensitivities and inadequate model fidelity werehighlighted.

• Data to determine lower bound control effector response times and the remaining effectorcontrol authority after accounting for longitudinal moment trim was generated.

• Drag losses due to required elevon activity was shown to be very high, demonstratinga need to consider alternate longitudinal moment management strategies and/or effectorsfor the vehicle.

When analysis results were considered collectively, they showed the potential of themethodology to provide much of the information needed to define guidance and control systemdesign requirements. Based on these results, more complete demonstration of the methodologywas planned under phase II of the task.

3

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III. List of Symbols end Acronyms

the atmospheric speed of sound (a function of r)

advanced information processing system

the vehicle drag coefficient (a function of M t and a)

a coefficient to establish the desired weight given to cost improvement contributionsto variation terms

the vehicle lift coefficient (a function of M t and cz)

the dynamic pressure constraint cost weighting term

the vehicle air-breathing engine thrust coefficient derived from the Langley data

base with dimension ft2 (a function of 4)_, and M/)

a diagonal matrix of constant coefficients to establish the desired weights given to

equality constraint violation improvements in variation terms

the drag force acting on the vehicle

the derivative of the state vector x with respect to time

the derivative of x, with respect to time

the partial derivative of the vector [ with respect to the control vector u

the partial derivative of the vector f with respect to the state vector x

the LVLH vector of forces acting on the vehicle

the gravity force acting on the vehicle

the force component parallel to the vehicle free stream velocity vector

the force component normal to the vehicle free stream velocity vector, pointing

down in the vehicle plane of symmetry

the force component normal to F 2and F 3, pointing toward increasing I-L

the force component parallel to the equatorial plane, pointing eastward,

normal to F 3

the force component parallel to r - points toward Earth

a constant representing the number of pounds mass per slug (approx. 32.2)

guidance and controlvehicle altitude

the Hamiltonian

the partial derivative of the Hamiltonian H with respect to the control vector u

hypersonic vehiclea scalar influence function of cost

the air-breathing engine specific impulse derived from the Langley data base with

dimension sec (a function of _a and M/)

a vector Influence function of constraints and cost

a vector influence function of constraints and cost for the ith trajectory segment

4

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a matrix influence function of constraints

a mathematical cost imposed on the system

a heuristically determined specified cost improvement

the distributed mathematical cost term

the partial derivative of the distributed mathematical cost term L with respect to the

control vector u


state vector x

the distributed mathematical cost term

the lift force acting on the vehicle


control vector u


control vector x

Langley Research Centerthe local vertical/local horizontal reference frame

the vehicle mass

the air-breathing engine fuel mass

the vehicle mass at takeoff

the free stream Mach number

nautical miles

pounds per square foot

the dynamic pressure

the desired dynamic pressure constraint bound

the vehicle distance from the center of the Earth

the radius of the Earth

the target circular orbit altitude

the vehicle distance from the center of the Earth at takeoff

the vehicle reference wing area

single-stage-to-orbittime

the thrust force acting on the vehicle

the control vector

the control for the air-breathing engine

the ith element of the control vector

a step function which changes from 0 to 1 at a functional value designated inbrackets

a diagonal matrix of functions of time to weight different elements of the controlvariation vector

the free stream velocity magnitude at the desired atmospheric exit conditions

5

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the vehicle velocity magnitude relative to the free stream

the free stream velocity component normal to u2 and u3, pointing toward

increasing I_

the northerly component of the takeoff velocity

the free stream velocity component parallel to the equatorial plane, pointing

eastward, normal to u3

the easterly component of the takeoff velocity

the vehicle free stream velocity component parallel to r (points toward Earth)

the desired vehicle free stream velocity component at takeoff parallel to r at atmo-

spheric exit (points toward Earth)

the vehicle free stream velocity component at takeoff parallel to r (points toward

Earth)the state vector

the ith component of the state vector x

the vehicle state vector at takeoff

the vehicle angle .of attack

the vehicle flight path angle

the eleven deflection angle

the vehicle pitch attitude relative to the Earth's surface

the costate vector

a matrix influence function of equality constraints

latitude, measured positive northward from the equator

Earth's gravitational constant

the takeoff latitude

longitude, measured eastward

the takeoff longitude

the atmospheric density (a function of r)

the terminal time of the trajectory

the terminal cost term

the partial derivative of the terminal cost term ¢ with respect to the state vector x-

the air-breathing engine equivalence ratio

the flight azimuth angle

tne equality constraint vector

the derivative of the equality constraint vector _Ywith respect to the control vector x-

the Earth's rotation rate (.00418 deg/s)

the state integration cutoff function

the derivative of the cutoff function _2with respect to the state vector x

when used in figures, indicates a negative number

6

IV. Research Goals

A number of research goals were set for this task. Some of them have been the focus ofthe phase II work done between Aprl11991 and March 1992, whlle the remainder have establishedguidelines for work under posslble future efforts.

IV.A. Phase II Task Focus

SSTO HSVs are likely to have limited performance margins. Their delivered payload capacitywill be very adversely affected by non-ideal flight conditions and/or G&C strategies. The mostefficient trajectories, control strategies, and propulsive phasing histories that satisfy all relevantperformance constraints must be identified and utilized. New methodologies to determineefficient, constrained HSV ascent trajectories are required.

Initial phase I activity on this task focussed on the implementation and preliminary dem°onstration of a novel generic HSV integrated trajectory/control analysis methodology. Theresulting tool can determine HSV ascent paths that maximize payload performance while satisfyingrelevant constraints. Data derived using the methodology can also support development andassessment of G&C strategies for managing the flight path and propulsive phasing.

Phase II completed demonstration of the analysis methodology using two combinations ofvehicle models and trajectory target conditions. Specific trajectory results were derived underrealistic flight constraints. Interpretation of the resulting trajectory/control profiles has attemptedto charactedze the type of information that can be gleaned using the analysis methodology. Thefollowing subtasks were pursued:

IV.A.1. Completion of Analysis Using Phase I Vehicle Model

An HSV vehicle model which included the aerodynamic effects of maintaining longitudinalitch trim with elevon was incorporated into the analysis tool as part of the phase I task work.sing angle of attack as a control, incorporating an early version of the propulsion model, targeting

for a polar orbit, and constraining dynamic pressure to an upper bound near 1000 psf, a sequentialseries of flight segments were generated to assess expected vehicle performance to 20,000 ft/srelative air speed. Under phase II, an integrated optimization was performed for a single flightsegment from Mach 2 (1950 ft/s) to 25,000 ft/s relative air speed. A separate fli_]ht segment fromhorizontal takeoff to Mach 2 was separately computed for reasons discussed in Ref. 1.

IV.A.2. Identification of Issues Resulting from Use of Phase I Implementation

Careful review during phase II of results derived from use of the phase I vehicle model wasperformed. The focus was to identify any state equation, algorithm, and model deficiencies thatrequired attention to permit effective subsequent use of the analysis algorithm. Several significantproblems were identified. These are summarized below, with their resolution discussed insubsequent sections of the report:

• The polar orbit state equations were found to have a singularity at the Earth's poles thatwould could cause the trajectory integration to break down if approached within a degree.The trajectory under study came within one degree of the pole at a relative air speed ofabout 21,000 ft/s.

• Use of angle of attack as a longitudinal control variable was found to produce a openloop instability during trajectory optimization. The effect was to produce a weaklydivergent dynamic pressure oscillation at high Mach numbers around the upper bound(due to associated small altitude fluctuations). Since the period was about 200 seconds,the problem only became apparent when flight segments under analysis grew wellbeyond 1000 seconds duration.

7

• The phase I propulsion model manifested somein .co.nsistencieswhen used inthe analysisalgorithm. The model specified separate calculations of specific impulse and thrustcoefficient (which in fact are functionally related) causing great sens_ to the effectsof indlvldual, small database errors, uontrary (and incorrect) performance trends for thetwo engine performance variables sometimes resulted. Unreasonable jumps in systemperformance as a functlon of Mach number were Implied which caused convergenceproblems for the Integrated analysis optlmlzatlon algorithm.

IV.A.3. Algorlthm Evaluatlon Wlth Alternatlve Phase II Vehlcle Model

The integrated analysis algorithm was demonstrated over an entire trajectory, from hori-zontal takeoff to atmospheric exit, using an updated phase II vehicle model. Problems identifiedusing the phase I vehicle model were resolved, and subseguently some additional modelsimplifications were incorporated to expedite results. A new attitude control variable was intro-duced, the propulsion model was improved, treatment of elevon effects was removed, and thetrajectory target was changed to an equatorial orbit. Results were evaluated to assess the kindsof Informatlon available from application of the analysis algorithm.

IV.B. Lon_aer Term ResearQh Oblectlves

The task statement of work identifies several investigation paths which are intended asdirections for research under possible future efforts. These are summarized below:

IV.B.I. Treatment of Overlepplng Alr-BreathlnglRocket Propulslve Phases

The Integrated analysis methodology includes features that permit determination of theo_timal tlming of discrete propulsive mode transitions (if explicitly modeled). With the inclusion

a rocket propulsion system in the vehicle model, the basis for its best application can bedetermined Including possible overlapping operations with the air-breathing engine. The potentialfor slgnificant HSV operational performance improvements could be explored using this algorithmfeature.

IV.B.2. Treatment of Vehicle Thrust Direction and Pitch Moment Changes with Angle ofAttack

The initial example HSV models included propulsion systems that represented thrustmagnitude as Independent of angle of attack and forced thrust direction to always be in thelongitudinal body axis. More realistic vehicle models specify the thrust magnitude and directionas functions of angle of attack, which also implies associated longitudinal moment variations.This subtask would characterize the changes mnHSV trajectory management and control strat-egies that result from explicit consideration of angle of attack effects on propulsion performance.

IV.B.3. Incorporation of Thermal Constraints into Trajectory Results

The integrated trajectory/control analysis methodology supports treatment of inequalityconstraints which are functionally dependent on both states and controls. However, the initialdemonstration cases were restricted to studying the effects of dynamic pressure inequalityconstraints which are only state dependent. Treatment of thermal constraints would address animportant physical performance limitation that has both state and control dependencies. Thissubtask would Incorporate a vehicle stagnation point thermal flux constraint and assess itsqualitative effects on mission performance and G&C strategies.

IV.B.4. HSV G&C Performance Robustness Studies

Uncertainty in the environment model and expected vehicle dynamics are important desiQnconsiderations for HSV G&C systems. The atmosphere can experience siQnificant densityvariations atthe higher HSV air-breathing altitudes. The aerodynamic and propulsmonperformanceof HSVs may not be accurately modelled before there is significant flight experience and may

8

vary between vehicles or for a given vehicle on different flights. The likelihood of successfullyaccomplishing specific mission objectives depends on understanding the effects of theseuncertainties and designing a G&C system that is robust enough to handle them. This subtaskwould evaluate the performance of the near-fuel-optimal HSV trajectories and candidateclceed-loop G&C systems in the presence of aerodynamic variations, propulsion modellinguncertainties, and atmospheric disturbances. These uncertainty effects would be evaluated fora range of applicable inequality constraint bounds.

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V. Prooress for April 1991 - March 1992

The phase II technical effort under this task was performed between April 1991 and March1992. During this period the analysis methodology assessment using the example HSV modelprovided by the government during phase I was completed. Some issues regarding the modelcharacteristics end the selection of control variables were identified. Also, some features of thevehicle model and orbit target states were found unessential to further algorithm assessment.An improved vehicle model was provided by the government, control vector changes were made,and new analysis simplifications were incorporated. Based on the new model and assumptions,additional algorithm assessment studies were accomplished. Details on these phase II studiesfollow.

V.A. Analvsls wlth Phase I Vehicle Modelsv

During phase I studies analysis of a near-fuel-optimal polar trajectory from horizontal takeoffto 20,000 ft/s relative air speed was achieved by combining several sequentially derived trajectorysegments. The same vehicle model was used during phase II to perform an integrated trajec-tory/control analysis on a single segment spanning flight from Mach 2 to 25,000 ft/s relative airspeed. (Reference 1 explains whyMach 2 was selected as an initial condition.) The followingsubsections detail the case run and results of subsequent analysis.

V.A.1. Summary of Case Run

An HSV model implemented during] the phase I work represented a winged-cone config-uration with elevon use required to maintain longitudinal moment trim. Aerodynamics wererepresented as a combination lift and drag contributions from the airframe and elevons withdependency on Mach number, angle of attack, and deflection angle of elevons. The air-breathing

opulsion model represented specific impulse and thrust coefficient separately as functions ofach number, dynamic pressure, and fuel/air equivalence ratio. Angle of attack and equivalence

ratio were selected as time varying controls. An upper bound of 1000 psf was imposed ondynamic pressure. For the phase II study, the trajectory was based on horizontal takeoff from28.5 deg latitude with a polar orbit target. Results of phase I studies were used to initialize thephase II analysis of a trajectory going from Mach 2 (1950 ft/s relative air speed) to 25,000 ft/srelative air speed.

V.A.2. Numerical Orbit Integration Issues Resulting from Polar Orbit Target

A significant numerical problem was encountered as a result of the polar orbit target. Theselected trajectory went very close (within 0.8 deg) of the north pole at about 21,000 ft/s relativeairspeed. In initial analysis efforts, the trajectory integration numerically diverged at the closest

approach to the pole. The problem was traced to calculation of the flight azimuth angle, _ ,whichis derived from the arc tangent function of the ratio of easterly airspeed to the northerly airspeedvelocity (components 2 and 1 respectively in figure 1). The arc tangent function range of appli-cability is -90 deg to +90 deg. Calculations over this range asymptotically diverge at the northpole (+90 deg). Mathematical manipulation of the form of the dynamics equations to exclusivelyutilize inverse trigonometric functions applicable over a range from 0 deg to 180 deg (e.g. arccotangent) resolved the problem.

V.A.3. Overview of Trajectory/Control Analysis Results

Various state, control, and effector variable histories were generated by the trajecto-/control analysis algorithm. Data were generated for the flight segment from Mach 2 to 25,000s relative air speed. For completeness, when plots were generated for specific variables, the

separately derived results for takeoff to Mach 2 were included. The flight from takeoff to Mach 2took 92.2 seconds. Transients seen where the two flight segments merge at Mach 2 are discussedin reference 1.

10

Altitude, Mach Number, and Dynamic Pressure

Figures 2 and 3 plot altitude and Mach number vs time respectively. On the scale of theplot, both variables are seen to be virtually monotonic, with their rate of increase declining afterthe first 300 seconds of flight. The dynamic pressure, Q, is seen in figures 4 and 5 to reach thebound of 1000 psf within that rapid climb interval, around Mach 2. The dynamic pressure limitis closely tracked for the remainder of flight. A weakly divergent oscillatory instability in the dynamicpressure can be seen in figure 4 after the first 500 seconds of flight (discussed in more detaillater).

Controls

The equivalence ratio control variable (phi), is plotted in figures 6 and 7, and the angle ofattack control history is plotted In figures 8 and 9. Significant equivalence ratio peaks are seenin the low Mach regime and near Mach 6. The low Mach peak correlates to a reduced angle ofattack while getting through the high drag transonic regime. The Mach 6 equivalence ratio peakcorrelates to steep changes in propulsion model data which demands corresponding sizeable,short duration changes in the an_le of attack. Other sudden changes in equivalence ratio areseen at higher Mach numbers (wtth some associated angle of attack transients near Mach 8).During high Mach flight, the angle of attack declines relatively smoothly due to reduced liftrequirements as orbital speed is approached. The discontinuous equivalence ratio is evidenceof inadequacies in the pnase I propulsion model which helped motivate a subsequent modelchange.

Elevon Deflection

The elevon deflection angle (delta), used to maintain vehicle longitudinal moment trim, istted in figures 10 and 11. The trim angle bottoms out above Mach 10 at a value near -17 deg.h a +_20 deg stop on the elevons, little attitude transient control capability remains. Also,

sustained high elevon deflection introduces high drag penalties. The elevon deflection decreasesnear orbital speed as the angle of attack declines. The elevon deflection history suggests thatalternate moment control strategies are required. This was an issue that was not pursued inphase II due to lack of resources.

Flight Path Angle

The flight path angle (gamma) is plotted against Mach number for the initial 500 secondsof flight in figure 12. The early peak is consistent with the rapid climb in flight shortly after takeoff.The very small flight path an_le at high Mach numbers (gamma near 0) is consistent with the lowrate of climb as orbital veloc0ties are approached.

Acceleration

Figures 13 and 14 illustrate the inertial acceleration history of the phase I vehicle model.Significant, rapid chan_es are seen near specified propulsion model data points through Mach8. Generally, acceleration levels off above about Mach 10.

Mass and Mass Flow

Figures 15-17 illustrate vehicle mass and mass flow histories. The mass history is nearlylinear with time, unlike exponential decay curves for conventional rockets. The mass flow is highlyvariable, but consistent with the trends seen in vehicle acceleration, including a leveling off at highMach.

Key Observations Using the Phase I Model

Completion of the analysis using the phase I model for a single flight segment from Mach2 to 25,000 ft/s relative air speed leads to similar control concerns as discussed in reference 1.

11

• The elm/on deflection angle history reflects large changes in vehicle trim characteristics

due to Mach number dependent aerodynamic effects and vehicle mass chan_es. Thecombination of the vehicle daslgn and control effector placement lead to high dragpenalties and limited control authority over extended flight intervals.

• Some steep perturbations in the angle of attack can result from propulsion model variations(as seen below Mach 6). These effects put upper limits on attitude change responsetimes.

• Significant problems exist in the phase I propulsion model which result in unrealisticequivalence ratio transients near propulsion model data points. An improved (morerealistic) model should be used for subsequent studies.

V.A.4. Optimization Stability Problems Due to Use of Angle of Attack as a Control

The government supplied winged-cone vehicle model aerodynamic data base [Ref. 6]anddefines lift drag coefficients as functions of Mach number and angle and anl_le of attack (as

well as aerosurface deflections when their incremental contributions are cons,dered). Thesedependencies make angle of attack a natural selection as a control variable. However, when theexample analysis case was completed using the phase I vehicle model, the use of angle of attackwas found to present significant problems.

Figure 18 plots dynamic pressure, Q, vs time starting 100 seconds after takeoff. Illustratedon a magnified scale, it can be seen that a weakly divergent oscillation with a period of about 200seconds is propagated around the 1000 psf upper bound. The last cycle of the oscillation at theend of the trajectory shows evidence of going unstable. Figure 19 plots the altitude history starting1000 seconds after takeoff. Also illustrated on a magnified scale, slight altitude fluctuations aroundan otherwise monotonic climb can be seen as the cause of the dynamic pressure oscillations.Similar instabilities were encountered by Langley representatives ,n in-house trajectory studiesfor hypersonic vehicles.

At least one previous study of air-breathing HSVs found similar problems with the use ofangle of attack as a control [Ref. 7]. While the dynamics are very nonlinear, an approximatelinearized eigenvalue analysis shows that perturbations to angle of attack control histories derivedfrom an optimization algorithm can induce instability.

Upon completion of the example trajectory/control analysis case using the phase I vehiclemodel, a number of changes were made to improve and expedite subsequent phase II studies.Problems noted in the initial analysis were addressed. Additional simplifications were incorporatedwhen warranted by results. The details of the major changes follow.

V.B.1. Change of Control Variable

In reference 7, a linearized analysis of the dynamics of an early HSV concept suggestedthat use of vehicle pitch attitude as a control instead of angle of attack could eliminate dynamicpressure/altitude instabilities. Recent in-house Langley studies of hypersonic vehicles alsoshowed that use of pitch attitude relative to the local vertical as a control variable instead of angleof attack eliminates dynamic pressure/aititude instabilities. The vehicle control vector and theassociated dynamics model (including all required partial derivatives) were reformulated to usea local vertical vehicle pitch angle the attitude control variable. This change is reflected in theupdated model equations presented later in this report.

12

V.B.2. Replacement of Propulsion ModelThe phase I vehiclepropulsion model separated calculation of specific impulse and thrust

coefficient. Its use in the example trajectory/control analysis case resulted in numerous jumpsand/or transients in the equivalence ratio history which also affected the angle of attack profile.Careful analysis of the data base indicated some inconsistencies in the trends of the specificimpulse andthrust coefficient data at each of the Math regimes of concern. NASNLangleyrepresentatives derived an improved propulsion model based on the actual functional relationshipat specific impulse and thrust coefficient. For simplicity, the very weak dependence on dynamicpressure was also removed.

The phase II propulsion model is separated into two propulsive modes. The low speedmodel has a number of data points spanning Mach numbers from 0.3 to 2.0 and equivalenceratios from 0.5 to 2.0. The high speed model has a number of data points spanning Mach numbersfrom 2.0 to 25.0 and equivalence ratios from 0.25 to 5.0. As incorporated for further demonstrationof the integrated trajectory/control algorithm, a discrete change is assumed to be made from thelow speedmodel to the high speed model at Mach 2.

As was done for the aerodynamic data base and the phase I propulsion model duringphase I studies, the phase II propulsion model implementation is processed through a cubicspline fit routine to assure continuous behavior of the relevant functions and their first derivatives.The smoothing of the data is performed to avoid trajectory/control optimization algorithm insta-bilities that could result from gradient calculations near data discontinuities. To illustrate the phaseIlpropulsion model characteristics, outputs of the spline fitted data have been plotted for an arrayof Mach numbers and equivalence ratios. Figures 20-21 illustrate some cross sections of thethrust coefficient for the low speed and high speed phase respectively, and figures 22-23 illustratesimilar cross sections of the phase II model's specific impulse. The spline data points werecalculated at Mach number intervals of 0.5 which accounts for the segmented appearance of theplots. Data jumps actually occur in the propulsion model when the data base (the engine phase)changes discretely at Mach 2. The plotted data represents the low speed data at Mach 2, andthe high speed data at Mach 2.5 with the entire set of plots drawn in a piecewise linear manner.

V.B.3. Elimination of Elevon Trim Calculations

Inclusion of elevon usage for longitudinal moment trim in the phase I vehicle model providedsignificant insight into pitch control authority restrictions and established upper bounds onrequired aerosurface response times. However, subsequent use of the example winged-conevehicle for trajectory control/studies with the improved propulsion model was not expected toprovide new insight into aerosurface control issues. Therefore, the longitudinal moment calcu-lations and associated elevon trim deflection determination were removed for phase II modelstudies in order to reduce computational burden and complexity.

V.B.4. Change of Orbit Targets

Phase II integrated trajectory/control studies using phase I vehicle models aimed for polarflight paths with near orbital velocities. However, as work progressed, interest in polar orbit (oncean objective of the NASP program) waned. Consideration instead of equatorial launch toequatorial orbit offered additional computational simplicity without compromising the qualitativealgorithm evaluation objectives of phase II task studies. The new boundary conditions for studiesusing phase II vehicle models were defined as horizontal, equatorial takeoff with a target state atatmospheric exit on the way to low, circular, equatorial orbit. Equation formulation of theseboundary conditions and associated derivatives are provided later in a section detailing the phaseII model.

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V.B.5. Updated Analysis Algorithm Formulation

The trajectory optimization algorithm uses a_..generalized steepest descent gradient two-potnt boundary value problem .aolutlo.nt.ech.nlque, tne most _leneral form of the Draper developedmethodology as¢_peble oTsoM.ng simu_aneously for vehicle c0nnguration, trajectory, dynamicdlacontinuity times, ano contro=s [Refs. 2-5]. It can accommodate equality and Inequality con-stralnta while optimizing with respect to a Performancs Index of states, vehicle design parameters,consols., and time. In the phase I! app.licaflon, vehicle g._m .et:_.is assumed fixed and propulsivecontlnumas are not explicitly moaeleo, criminating consioeration of design parameters and dis-continuity times. The optimization performance index J as applied to this problem has the form:

/o*d('r)-' ,(x(T)) + L(x(t),u(t))dt (Eq. J)

To raduce the state dimensionality for the phase II studies, rotation dynamics are not tracked.The resulting state vector is:

V

g

x - u: (Eq. 2)

U2

The applicable control vector for phase II only includes explicit treatment of air-breathingpropulsion and vehicle pitch attitude:

u = = (Eq. 3)u 2 0

The relationship between the first control vector element and the actual propulsion systemthrottle variable is:

_,_. (ua) 2 (Eq.4)

A comte like function A is adjoined to the problem to explicitly treat a vector of applicable

equality constraints functions called W. The formulation for A is given in Eqs. 11-12. W'rththeinitial take-off conditions given, the equality constraints are needed to enforce the target orbitconditions at time 1:. One smoothly varying target orbit state equality constraint is needed to

determine 1:. The target orbital velocity is used in this application and is therefore excluded from

the adjoined function list. This trajectory integration cutoff function called Q helps determine

boundary conditions on the costate function _., while the remaining equality constraints determine

boundary conditions used in the adjoined function A. The formulation for _. is given in Eqs. 8-9.

The vehicle state equation is defined as follows:

dx

dr f ( x ' u) (Eq. 5)

14

A forward Integrationcutoff condition is defined (e.g.target orbital velocity):

o (Eq. 6)

A Hamiltonian is constructed after defining a costate vector ;_ which is determined bybackwards integration:

H = L + X, T .[ (Eq. 7)

where:

d _ (Eq. 8)T;k- L_=== _.fx

dt

and where the backwards integration boundary condition is evaluated at the forward integrationcutoff time -i::

1o ,An equality constraint vector is constructed to accommodate the terminal state boundary

conditions defined at the forward integration cutoff time -c:

_(x(T)) = 0 (Eq. 10)

An additional adjoint variable, also determined by backwards integration, is defined to factorIn the equality constraint functions (boundary conditions) at the terminal time:

d A (Eq. 11)TA

dt

where the backwards integration boundary condition is evaluated at the forward integration cutofftime 1::

( _ xf_x ) (Eq. 12)Ar(_c) = V,, _x,f I_(._,_c_)

It is helpful to introduce the following variables for subsequent notational simplicity whereUis a diagonal matrix of coefficients used to weight the relative size of perturbations to the controls:

fO _ 7' TIv_, = A ,[,,//[_ ArH (Eq. 13)

lvj = A ,,UH,,dt (Eq. 14)

fo _ T dt (Eq. 15)Ijj = !-i _,U I-I,,

15

The rasultlng algorithm variational equation for the control vector u is:

6u= U(.f_AI_,_C_,q/+Cj(H_ T -- " -.fu A I v_'! vj)) (Eq. 16)

whore:

dJ,C j - - _ (Eq. 17)

Ijj I_jl;,_l_,j

With specification of d J, and C v, the variational equations can be used to perturb the

desired control history on successive iterations through use of Eq. 19 to eventually achieve vehicleperformance optimality while satisfying all applicable constraints. The scalar d J, reflects the

specified cost improvement sought on each optimization algorithm iteration. The diagonal matrixC _ weights the relative size of equality constraint improvements on each optimization algorithm

iteration. Proper selection of these weightlngs as well as U is critical to the numerical stability andconvergence rate of the optimization process.

There are several types of constraints applicable to HSV trajectory optimization. As is truefor any two-point boundary value problem, it is necessary to apply equality constraints at the

trajectory final target condition. These are treated by adjoining the variable A, therebyIncorporating the influence of the boundary conditions.

There are constraints on some of the controls. An example would be a lower bound (fuelflow off) on throttle setting. Often such constraints can be treated by nonlinear mapping] into acontrol space without bounds. Care must be taken when using nonlinear mapping to avoid initialsolution guesses that specify mapped control values in insensitive regions of the actual controlspace.

Inequality constraints need to be applied as a result of physical design constraints such asdynamic pressure and thermal limits. For this methodology, penalty functions are introduced

through the Integral performance index L. This approach requires periodic adjustment of penaltyfunction gains as the optimal solution is approached.

As is generally true when two-point boundary problem optimization algorithms are used,some application specific considerations are required to assure good algorithm performance forthe HSV problem. These include the careful choice of the integration cutoff condition as well asspecial approaches to balancing the perturbations to boundary value constraint violations againstperturbations to controls needed to improve performance. Failure to carefully treat these issuescan prevent acceptable algorithm numerical behavior.

Unlike a rocket ascent to orbit, HSVs are more likely to follow a trajectory at low Machnumbers akin to supersonic fighter aircraft minimum time to climb profiles. Often this results ina "zoom maneuver" in the transonic range, where the vehicle uses already acquired potentialenergy (altitude), combined with thrust, to quickly transit the maximum drag environment nearMach 1 [Ref. 8]. The resulting ascent can involve a negative vertical velocity component for ashort interval. The nonmonotonic behavior of this velocity component makes it an unacceptableparameter for algorithm integration cutoff. Either horizontal velocity or orbital energy state providesmuch greater likelihood of achieving acceptable state conditions at integration cutoff.

Successful convergence of the optimization algorithm to the desired minimum cost solutionrequires that the effort to drive the violation of the boundary conditions to zero be balanced againstthe effort to find an optimal control history. Each parameter requiring perturbation must bessperetaly weighted, with the emphasis that each receives chosen to stress those variables thatcurrently promlse the most improvement without numerically destabilizing the algorithm. (Badchoices could Induce highly nonlinear steps.) As the interim trajectory solution approaches

16

optimality, the relative sensitivity of the perturbed parameters change. Techniques must beapplied to determinecurrent sensitivityand modify the weightings as the algorithm pro_]resses.In some instances,this may requireacceptanceof perturbationsthat do not uniformly ,mproveall boundary value violations. A composite measure of proposed algorithm step acceptabilitycan be constructed that takes precedence over individual variation effects, as discussed in ref.[Ref. 5].

Given an initial control history, the algorithm iteratively computes perturbations to thecontrols to improve the specified performance index while maintaining boundary conditions.Additive perturbation contributions are derived to improve the cost as well as to drive downviolations of the equality constraints. Weighting functions balance the relative contributions ofeach additive perturbation term.

V.B.6. Updates to Model Equations Required for Algorithm Analysis

Changes were made in the example vehicle model during phase II studies to improve thequality of the propulsion system representation and to eliminate consideration of elevon use forlongitudinal moment trim. In addition, when the new vehicle representation was applied, thealgorithm demonstration case boundary conditions were updated from those used previously.The motivation for these changes was discussed in previous sections. The formulation detailsfollow.

V.B.6.a. Vehicle State Dynamics Models

A seven state vehicle dynamics model can be written as follows in state equation formconsistent with the frame of reference given in figure 1 :

_X 6

xs

X j COSX 3

x" 4

x" I

x4x" e- (Xs) 2tan x 3 F,- 2oo,x ssin x 3- x, (00,) 2sin x 3Cosx 3 +--

X I x" 7

(xe + x+ tanx3) Fzx s * 2_.(x4sin x3+ xecosx3)+--

Xl. x7

-((x+)_+ (x_) _ ) F_- 2uo.xscosx 3- x,(cu,cosx3) z+-

Xl X7

l (QC,

.f=

(Eq. 18)

Using an assumption of constant zero bank angle, the following force equations in the vehicleLVLH frame result assuming an arc tangent definition over the range from -90 deg to +90 deg:

F= F 2 = sign(vl)sin_(FTcos¥+ FNsin¥) (Eq. 19)

F 3 -FTSin ¥ + FNCOS Y + F_

where sign (v _) has a value of 1 for v _ > 0, and a value of -1 for v _ < O. Also note that only two

force components remain in the body frame (no side force)"

17

F r = T co s a - D (Eq. 20)

F N = -(Tsina + L) (Eq. 21)

Also, the following equations are used to relate flight angles to states:

¥ = tan_l( -X6 )+(Eq. 22)

(Eq. 23)

The lift, drag, and engine thrust can be written in the following form involving coefficientfunctions specified in the example vehicle model data tables:

L = CtQS,o/ (Eq. 24)

D = C o Q S r./ (Eq. 25)

T = Q C r (Eq. 26)

where:

v t = ) a ) a ) 2 (Eq. 27)

When using the example vehicle model data base, M / and czare required, where v f is used

along with r in an atmosphere model to compute M t • A determination of the value of a is madefrom the following relationship:

a = 0 - y (Eq. 28)

The gravity force acting on the vehicle can be written in the following form:

t-t_X7F - (Eq. 29)

Finally, the following equation is used to transform the throttle control on the propulsion toan unbounded control vector component. It is based on the assumption that _,, _>0.

4:'a = ( u _ ) 2 (Eq. 30)

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V.B.6.b.The Boundary Conditions

Initial Conditions: The initial conditions assumed for the example vehicle in the analysis dem-onstration are set immediately after horizontal takeoff. They correspond to due east launch froman equatorial location. Note that initial longitude (v o) has no effect on the problem and is therefore

arbitrarily set at 0 deg.

f o\V o

UIXO= o

U2 o

U3 o

/' 20,909,723.ft

Odeg

O cleg

O ft/s

446.473 ft/s

-6"46676.ft/sj297,382 lb

(Eq. 31)

Target Atmospheric Exit Conditions: The target condition corresponds to atmospheric exit at400,000 ft altitude ( h ) for a transfer orbit that has an apogee altitude of 100 NM and a perigeealtitude of sea level. Assuming a near impulsive orbit change, a rocket burn of 184 ft/s is requiredat apogee to circularize the orbit. With an equatorial circular orbit as the objective, latitude andlongitude are unconstrained at the target condition. Also, since equatorial launch to equatorialorbit is to be computed, the northerly Velocity component ( u _ ) remains zero without application

of a boundary condition. A target transfer orbit velocity is not a suitable forward integration cutoffcondition since the air-breathing engine power drop that results from the decline in G late in ascentcombines with drag and gravity loss effects to produce a non-monotonic vehicle velocity history.Therefore the trajectory cutoff condition is specified in terms of an atmospheric exit altitude thatwill only be crossed once on the way to orbital apogee. This results in a forward integration cutoffrepresented as:

.(2(1:) = i- - rtar_ (Eq. 32)

The two remaining terminal time state boundary condition constraints are atmospheric exit targetvelocity and the radial component of the velocity.

"tt[ = ( (U2)2 + (U3)2- (U D)2 )u3_U3D (Eq. 33)

The orbital parameters at atmospheric exit (assumed to be at h -- 400,000 ft altitude) result in

an inertial exit velocity of 25,645 ft/s and a heading angle (flight path angle in an inertial frame) of0.958 deg (based on formulas found in Refs. 9-10). Horizontal and vertical velocity componentsare derived and the horizontal component is converted to a free stream value by including Earthrotation (c_ _) effects. The results are the following values of the atmospheric exit target conditions:

19

= | 24, 091.6 ft/s\ - 429. ]fi /

(Eq. 34)

V.B.6.c. Cost Function Specification

The terminal cost term ¢ is used to minimize the consumed fuel mass.

_ = rno- rrt(_ ) (Eq. 35)

The integral function L treats inequality constraints. In this application where dynamic

pressure is constrained, L is a function only of states since dynamic pressure is a function of

velocity states and density, and density is a function of r when using a standard atmospheremodel.

where:

L= CQ(Q-Q_)_Uo[Q-Qo] (Eq. 36)

p(v/) _Q - (Eq. 37)

2

and where u o[ Q - Q o ] is a unit step function that changes from zero to one at Q = Q o.

V.B.6.d. Relevant Partial Derivatives

Boundary Condition Derivatives:boundary condition functions.

_x=(1 0 0

The following equations are the required derivatives of the

0 0 0 O) (Eq. 38)

v =(: o o o 2v_ 2v_ o)0 0 0 0 l 0(Eq. 39)

Cost Function Derivatives:

that are required:The following equations are the derivatives of the cost functions J

_x=(O 0 0 0 0 0 -l) (Eq. 40)

Lx = OifQ < Qo, and when Q_> Qo :

aq (Eq. 41)

Lx= 2Ko(q-qo)a-- _

The Hamiltonian derivative with respect to controls can be expressed by the relationship

H, = L,, + ;kT f, which requires a partial of the integral cost function with respect to the control

vector. For the functions of L selected in the example here, L,, has zero value.

20

State Equation Derivatives: The partials of the vehicle dynamics equation / are required.Because of the complexity of its specification, details ere given in the Appendix A.

The partials of the dynamics equation f with respect to controls are required. Because ofthe complexity of its specification, details are given in Appendix B.

V.C. Analysis Demonstration Results with Phase II Model

The integrated trajectory/control analysis algorithm was further demonstrated using thephase II vehicle model. Incorporating the changes discussed previously, the example casespanned the entire atmospheric flight phase for a sin_lle-stage-to-orbit HSV. Details of thedemonstration case that was accomplished and conclusions from resulting analysis follow.

V.C.1. Summary of Case Run

A near-fuel-optimal trajectory was determined for the phase II HSV model which spannedhorizontal, equatorial takeoff to atmospheric exit (400,000 ft altitude) on the way to a 100 NMcircular equatorial orbit. At atmospheric exit, apogee was set at 100 NM and perigee at sea level.Air breathing propulsion was active throughout flight, with its fuel/air equivalence ratio setting andthe vehicle local vertical pitch attitude time histories determined by the trajectory/control optimi-zation algorithm. The thrust coefficient and the functionally related specific impulse were calculatedas functions of Mach number and equivalence ratio. Lift and drag forces of the airframe werecalculated as functions of Mach number and angle of attack. Dynamic pressure was constrainedto be near, or below 1000 psf.

V.C.2. Overvlew of Trajectory/Control Analysis Results

As was done for theprevious analysis algorithm demonstration example, data for a numberof variables were output for the case run using the phase II vehicle model. Included were allrelevant state and control variables.

Altitude, Mach Number, end Dynamic Pressure

Figures 24 and 25 plot altitude and Mach number respectively. The altitude profile ismonotonic, with a rapid climb at low speed, a much slower climb while accelerating through highMach numbers, and finally a atmospheric pull out maneuver (starting around 2000 seconds) togo to the target orbit. The Mach number profile follows the trend of the altitude profile throughoutatmospheric acceleration. However, due to temperature variations in the atmosphere model atextreme altitudes, the Mach profile peaks and then drops during atmospheric climb out. The

dynamic pressure Q, is seen in figures 24 and 25 to reach the bound of 1000 psf within the rapidclimb interval, around Mach 1.2. While the dynamic pressure bound is effective for the remainderof atmospheric acceleration, the change in propulsion system model at Mach 2 causes sometransients in the dynamic pressure history around the bound for about 200 seconds (a periodconsistent with the dynamic pressure oscillation frequency seen in the phase I model analysiscase). Finally, the dynamic pressure drops rapidly during atmospheric climb out of the vehscle.

Controls and Angle of Attack

The equivalence ratio control variable (phi) is plotted in figures 28 and 29, the local horizontalpitch attitude control variable (theta) is plotted in figures 30 and 31, and the resulting angle ofattack (alpha - not a control variable in this case) is plotted in figures 32 and 32. Equivalenceratio undergoes rapid variations at low speed, it drops well below unity during subsonic accel-eration after takeoff, then climbs to a peak near 1.9 during transonic acceleration. During highspeed acceleration it gradually declines to near unity at Mach 5, and then very slowly declines toslightly below unity for the remainder of flight. The pitch angle has a sharp peak dunng the rapidlow speed climb seen in figure 24. It then rapidly drops to about 5 degrees followed by a slowdecline during high Mach number acceleration. It necessarily increases during atmospheric climbout to provide the aerodynamic lift necessary for the maneuver. The angle of attack starts at a

21

high value to get the lift necessaryfor..t_:.eoff:drp to a minimum durin_ .bran.sonic flig.ht(keeping drag low when the drag coemc_ent is nlgn), rises ouring ear_/nign Mecn night (up toMech 10) and the declines as o.rbit_, speed is approached (.el.rice r.equired lift drops to zero.atorbital speed). The rate of decline in angle of attack is oras1:lC_lW oecreased, however, as meaerodynamic climb out maneuver IS initiated.

Flight Path Angle

Figures 34-36 plot the fight path angle !gamm.a.).on v.ar.io_. scales to !!lustrate a range oftrends. A very sharp peak (45 dog) is seen to result n'om tne lOW s .1:_. climb ..with the .an_lesubsequently dropping off to a very smal/ValUe (about 0.1 deg] aDo.,ve Mach 5 for me remainderof high Mach number acceleration. The nignt path angle ne_,y increases oack to about 1degree to provide the rapid altitude dee achieved during atmospnenc climb out as seen in figure24,

Acceleration, Thrust, end Velocity

Inertial acceleration is plotted in figures 37 and 38, air-bre .athin_ en_line thrust.is plotted. !nfigures 39 and 40, and Inertial velocity is plotted !n ngure 41. Accemrauon orops in subsonic nightdudng takeoff, and climbs through the transonic region, with a jump at Mach..2 to a peak of 45ft/sec z when the propulsion model is changed. A decline to about 10 ft/sec z occurs by Mach10 which is then sustained until atmospheric climb out. While the thrust shows no real decline inthe subsonic flight regime, it follows the trends in the acceleration profile in all other respects.The differences In the acceleration and thrust trends during subsonic flight can be attributed tothe varying drag effects due to the angle of attack profile seen in figure 33 (drag is a function ofMach number and angle of attack). Inertial velocity climbs quickly during the high accelerationat low Mach numbers, climbs more slowly during the lower acceleration experienced at high Machnumbers, and experiences a gradual drop during atmospheric climb out due to an exchange ofkinetic energy (speed) for potential energy (altitude).

Mass and Mass Flow

Figures 42-44 illustrate the mass and mass flow histories. During air-breathing acceleration,the mass follows a nearly linear decline (relatively constant mass flow) except at low speed whena peak in mass flow (andacceleration) occurs. Asmall jump in mass flow is seen at Mach 2 whenthe engine model changes.

V.C.3. Effects of Propulsion Model Discontinuity at Mach 2

The phase II vehicle model has two separate propulsion data bases. There is a low speedmodel applicable from takeoff to Mach 2 and a high speed model applicable from Mach 2 toorbital velocity. The transition between models results in a discontinuity in thrust coefficient andspecific impulse at Mach 2. This in turn causes discontinuities in acceleration (figure 38), thrustforce (figure 40), and mass flow (figure 44) at Mach 2.

The treatment of the propulsion system in the phase II vehicle model does not accommodatean analysis feature available in the integrated analysis algorithm (as documented in reference 1).That feature permits explicit treatment of the timing of propulsive discontinuities as performanceoptimization variables. However, the performance of the alternative engine modes must bemodelled for ovedapping operational ranges so that the optimum propulsive mode transition timeand state can be determined as a function of those models and the vehicle aerodynamics (all thesources of vehicle forces). By contrast, the phase II vehicle model predetermines the transitionas a particular state value (Mach 2 - a function of velocity and temperature).

Some transient effects appear inthe neer-fuel-optimal trajectory/control solution when usingthe phase II propulsion model. Angle of attack reverses its decline at Mach 2 and undergoes asmall jump in desired value. The equivalence ratio starts a decline from the transonic peak atMach 2. Also , the dynamic pressure magnitude passes through a cusp a Mach 2. Between Mach1.5 and 2 the dynamic pressure declines from a local maximum slightly above 1000 psf. BetweenMech 2 and 6 (spanning a flight time of about 200 seconds) the dynamic pressure undergoes

22

an erratic transient initiated with a sudden change in gradient at Mach 2. During the 200 secondinterval, dynamic pressure values remain a little below the 1000 psf bound. The correlation ofthe time of the dynamic pressure transient initiated at Mach 2 and the period of subsequentdynamic pressure oscillations at high Mach numbers is probably not coincidental.

V.C.4. A Look at Dynamic Pressure History Stability

An weakly divergent instability in the dynamic pressure was observed in the demonstrationexample using the phase I model, as plotted in figure 18. This behavior motivated a change inthe defined control variable for the phase II vehicle model from angle of attack to local verticalpitch angle. Figure 45 provides a plot of the dynamic pressure history for the phase II modeldemonstration example that is on scales similar to figure 18. An oscillation still exists in the phaseII model example, with a period remaining close 200 seconds (though somewhat variable). Inthe phase II example the oscillation seems to be near neutral stability rather than divergent.However, the presence of any significant oscillation adversely affects the convergence of theanalysis algorithm to a near-fuel-optimal trajectory/control solution. The change of control variablemade for the phase II model was helpful, but not fully effective at eliminating the undesired behavior.Future studies ought to consider alternatives to simple vehicle pitch orientation variables ascontrols in order to fully remove dynamic pressure oscillation effects.

V.C.5. Sensitivity of Solution Trajectories to the Propulsion Model

The character of HSV near-fuel-optimal trajectory solutions at hypersonic Mach numbersis sensitive to details of the vehicle propulsion model implementation. Close inspection of thesecond example vehicle propulsion model data suggests that small modeling differences suchas alternative data interpolation strategies may be capable of causing substantial change in thehigh Mach number control history andfiight profile while causing relatively small perturbation tothe overall solution performance. Variations in the air-breathing propulsion throttle settings thatin turn affect total time spent accelerating to orbital velocities are possible. Features peculiar toHSV dynamics models are at the root of these sensitivities. HSV acceleration in the high Machregime results from the application of a thrust which is larger, but of the same order, as net vehicledrag. However, typically HSV models represent the propulsion system characteristics withsparsely populated data tables. The tables must be curve fit, and the resulting propulsionapproximations must be differenced with the aerodynamics to produce a net force for thedynamics model.

Figure 23 plots specific impulse vs Mach number for a range of phi as curve fit for thesecond example vehicle model when operating in the high speed propulsion phase. Scrutiny ofthe plot near Mach 6 shows that comparatively little specific impulse is lost by increasing theequivalence ratio above 1.0, while comparatively large improvements in specific impulse arepossible for the equivalence ratio below 1.0 at Mach numbers above 7. However, flying at lowphi in the high Mach regime reduces acceleration, increases flight times, and results in highercumulative drag losses. As a result, small disparities in the curve fit for thrust and drag modelcharacteristics can have a much amplified relative impact on total performance gradients thatdetermine near-fuel-optimal trajectories and control histories. If different curve fit approximationscause further reduction of specific impulse variation in the Mach 6 regime, then desired equivalenceratio will be increased. Also, slightly larger residuals of differenced thrust and drag estimates dueto alternative curve fits will encourage lower phi and longer flight times in the high Mach regime.

23

V.D. Analvala Observations Affectlna G&C Deslon

The demonstration cases of the integrated trajectory/control analysis methodology illustrateits power to identify important issues affecting HSV G&C system design. Some specific examplesfollow.

• Significant oscillation in the dynamic pressure profile can result from the use of vehiclelongitudinal orientation angle related control variables. In the first demonstration case,use of angle of attack led to a slowly divergent oscillation in dynamic pressure and altitudeas seen in figures 18 and 19. In the second demonstration case, the use of local horizontalpitch attitude as a control variable, instead of angle of attack, reduced the instability topersistent, somewhat chaotic oscillations as seen in figure 45. While the behavioralchange in the second case was an improvement, it indicates that the appropriate controlvariable choice needs further consideration.

• The inclusion of elevon deflection histories in the first demonstration case highlights thelimits on real control authority that may apply to these effectors. Figure I0 shows thatfor hundreds of seconds, nearly 17 degrees of elevon deflection is needed for vehiclelongltudlnal moment trim. Given that these effectors have a typical deflection limit of 20degrees, very little authority remains for actual vehicle attitude control. Furthermore, largedeflectlons of aerosurfaces over extended intervals can have significant adverse impacton vehicle payload capacity due to incremental drag effects.

• Significant rapid changes in vehicle angle of attack may be essential to efficient vehicletrajectory management. In both demonstration cases such variations in desired angleof attack are apparent. They appear between Mach 5 and 8 for the first case (see figure9) and near Mach 5 in the second case (see figure 33). These desired variations set alower bound on control system response times. A real vehicle design would also haveto be responsive to disturbances due to wind effects and other atmospheric disturbances.

V.E. Prooress Toward AIPS Reaulrements SDeclflcatlon

The HSV integrated trajectory and control analysis methodology developed under this taskprovides information that is essential to development of closed-loop G&C system designrequirements. The demonstration winged-cone vehicle results provide enough trajectory man-agement strategy and performance sensitivity information (as reviewed in this report and reference1) to initiate development of a strawman G&C requirements specification. A complete G&Cspecification would permit characterization of a closed-loop G&C algorithm, including input/outputprocessing rates and redundancy requirements. Completion of all these steps, to accommodateassessment of expected overall G&C related throughput and computational processing rates, isa mandatory part of the AIPS requirements specification [11].

24

VI. Concluslons

Air-breathing hypersonic vehicle designs demand complex integration of airframes andpropulsion systems. As a result, there is a strong dependence between vehicle configurationdetails, the feasible mission performance, and trajectory management and control strategies. Toassure that an overall vehicle design has a high probability of achieving mission objectives, it isessential to have a means to perform an early assessment of how specific vehicle designs andtheir guidance and control system characteristics interact.

A gelenericmethodology to permit integrated trajectory/control analysis, including the effectsof physically derived constraints, has been applied to the analysis of example air-breathingsingle-stage-to-orbit hypersonic vehicles. The resulting algorithm includes features that allow anassessment of the relative performance of alternative vehicles and/or subsystem technologieswith respect to a single mathematically valid measure. Also, since the available vehicle modelsused to represent aerodynamics and propulsion characteristics exist as multidimensional tabu-lated data bases, a cubic spline data smoothing routine has been included in the software packageto process the information. The splines assure good numerical behavior of the analysis algorithm.

The methodology has been demonstrated using two vehicle dynamics models. The caseswere selected to evaluate the power of the algorithm to derive information of critical importanceto developers of guidance and control systems. Based on a winged-cone vehicle concept bothmodels were combined with the algorithm including representations of all relevant partial deriv-atives. Use of the trajectory/control tools with the vehicle models produced near-minimum-fueltrajectories and desired control strategies from takeoff to near-orbit targets. Throttle setting limitsand a dynamic pressure inequality constraint of 1000 psf were explicitly included in the demon-strations. The first case, using an early propulsion model, considered fi=ght on a polar orbit trackup to 25,000 ft/s relative air speed. It included the aerodynamic effects of using elevons to maintainlongitudinal trim throughout flight. The second case, using an improved propulsion model butremoving the elevon aerodynamics, aimed for a atmospheric exit at 400,000 ft on track to a 100NM circular orbit. The following list summarizes some of the most significant conclusions:

• The selection of control variables can significantly affect the stability and performance ofthe analysis algorithm. Use of angle of attack can induce dynamic pressure/altitudeinstability duringconvergence to a desired reference trajectory. Replacement of angleof attack with vehicle local vertical pitch attitude eliminates the instability, but retains someundesired oscillations in the dynamic pressure profile. Additional investigation of suitablecontrol variables is warranted.

• Small inconsistencies in propulsion model features and/or discontinuities in the propulsionmodels that are not explicitly treated as optimization parameters can introduce transientsin the vehicle dynamics that adversely affect analysis algorithm performance.

• Aerosurface effectors can have significant impact on vehicle drag losses and limited actualcontrol authority after accounting for deflection required to trim vehicle moments. As aresult, design and placement of aerosurfaces needed for vehicle attitude control cannotbe separated from the overall hypersonic vehicle design integration problem.

• A characteristic of all air-breathing hypersonic vehicle models is a large vehicle drag termas compared to thrust throughout hypersonic flight. This phenomenon was consideredin combination with a close inspection of the curve fitted second example vehicle modelpropulsion data. The results suggest that small model changes, such as those resultingfrom alternative curve fit strategies for sparse tabulated model data, may induce significantdifferences in analysis algorithm solutions.

25

The demonstration cases provided much information about the analysis methodologycapabilitiesand limitationsas wellas Insightintokey guidanceand control system design issues.If existing extensions of the analysis methodology are applied, important physical effects notconsidered in the demonstration cases can be treated, such as accommodating thermal con-stralnts and determining efficient propulsive mode phasing. These extensions would provideadditional Important input into air-breathing hypersonic vehicle guidance and control designrequirements.

26

VII. References

_Co]Philip D. Hattis and Harvey L. Malchow, "Air-Breathing Hypersonic Vehicle Guidance andontrol Studies; An Integrated Trajectory/Control Analysis Methodology: Phase I", NASA

ntractor Report 187623, September 1991.

[2] Hattis, Philip D., "An Optimal Design Methodology for a Class of Air-Breathing Launch Vehicles",Doctor of Philosophy Thesis, Massachusetts Institute of Technology, June 1980.

_] Hattis, Philip D., "Optimal Air-Breathing Launch Vehicle Design", Journal of Guidance andontrol, Vol. 4, No. 5, Sept. - Oct. 1981, pp. 543-550.

4] National Aerospace Plane Configuration and Trajectory Assessment, C. S. Draper Laboratoryseal Year 89 IR&D Technical Plan, Report CSDL-R-2087, Project Number 236.

[5] Adams, Neil J. and Philip D. Hattis, "An Integrated Configuration and Control Analysis Tech-nique for Hypersonic Vehicles", presented at the 1989 American Control Conference, Pittsburgh,Pennsylvania, June 21-23, 1989, pp. 1105-1110 of the proceedings.

6] John E. Shaughnessy, S. Zane Pinckney, John D. McMinn, Christopher I. Cruz, and Marie-ouise Kelley, "Hypersonic Vehicle Simulation Model: Winged-Cone Configuration", NASA TM

102610, November 1990.

7] Lawrenca H. Stein, Malcolm L. Matthews, and Joel W. Frenk, "STOP - A Computer Programr Supersonic Transport Trajectory Optimization", Final Report, Boeing Aerospace Group,

Seattle, Washington, January 1967.

8] A.E. Bryson and W.F. Denham, "A Steepest-Ascent Method for Solving Optimum Programmingroblems", Journal of Applied Mechanics, June 1962, pp. 247-257.

_ Richard H. Battin, "An Introduction to the Mathematics and Methods of Astrodynamics",erican Institute of Aeronautics and Astronautics, Inc., New York, 1987.

[10] William Tyrrell Thomson, "Introduction to Space Dynamics", John Wiley & Sons, New York,1961.

[11] J.H. Lala, R.E. Harper, K.R. Jaskowiak, G. Rosch, LS. Ai@er, A.L Schor, "Advanced Infor-matlon Processing System for Advanced Launch System: Avionics Architecture Synthesis", NASACR-187554, 1991.

27

Annendix APartial Derivatives for the Example Model with Respect to States

The matrix .f,, for the example application is too large to print as a single equation, so it is

specified here on a column by column basis. The equations for.f ,,are simplified bythe knowledge

that the following derivatives have zero value: _o _U/. _q. _M/. _Q eU/_" ; _', ' _U ' T;" ' _,,, ; e,,, • Simplifications are also

made by utilizing the knowledge that M t is a function only of r and u/.

0

--X 5

(X1)2COSX3

--X 4

(Xl) _

(x_)2tanxa-x,x6 (oo,)2sin 2Xa 1 eF_-- .4-

(x_) 2 2 xTax|

-xs(x6+ x4tanxa) 1 aFzD-

(XI) 2 x?ex !

(x4) a+ (xs) a 1 eFa)z+

(x_) 2 - (_.cosxa xzex,

go I .,,oe,¢ l ,,. eM, - )=eMT,la-iT,c, j

(Eq. A.1)

e/ e/ex2 ev

f 0

0

0

1 eFl

XzeX2

1 eF2

XzeX2

1 eFa

k Xz3X20

(Eq. A.2)

28

t

0

xssinxa

x,(cosx3) _

0

-(x_) _

x i(cosx3)2- 2°°,xsc°sx_- x j(o%) 2 cos2x_ +--

XsX+ l

xl(cosxa) 2 + 2cu.(x 4cosx 3- x 6sin x a)+ _--

2c_,x ssin x a + x l(oo,) 2sin 2x 3 +----

xs tan xa

X!

_o Isp c)X4 _ I

f

2xs tan xa

0

1 ,gF_

xzc)x3

,_Fz

Xz_X3

1 3F3

xzo_x3

o

o

1

Xi

x6 1 OF 1

xi Xzc)X4

+ 2c_,sin xa +

2X4 1 3F3

Xl XTC)X 4

O act

sot) M/

o

1

l c)F2

2x 5

xl

+ I

XzSX4

(I_o.)2,3Ms ax4

1 3F_- 2oo,sin x3 +

Xl Xz_X 5

X6 + x4taIlX3 1 OF2÷

Xl XT_X 5

1 3F32_eCOSX 3 +

Xzc)Xs

0 C_CT QCTC)Ispa'_c_M,'_

:..<#Ms ( I :_> ) 2 DM t J -_-_xs J

(Eq. A.3)

(Eq. A.4)

(Eq. A.5)

29

af afax6 ava

aft)x?

-1

0

0

x4 1 3F 1

Xl Xz t)X6

Xs_+ 2UOoCOSXa +Xl

1 aFa

XTaX6

1 aF2

XTaX6

aQ ( Q aCT QCr c)]spo_c)Mt)

;Z÷ j-E-_x,jj

aIam

o00

1 c)FI FI

xzax7 (xT) 2

1 aFz F2

XTaX7 (xT) 2

1 aFa Fa

_x_ax_ (x_) 2o )where:

• aq/aF.__Z= stgn_[-sin_(Frcos¥ + FNstnV)_XaX

a'y

+ cos_--£x(-Frsin Y

[aFt aFN )+c°sVk'-_-x c°sY+ a-x siny

+ FNCOSy)]

aFr aFN .bF___.__2s_gn_[cos_/(Frcos_t + FNsin,c)_._x + sin_ -_-x cosx + -_--x smx)_x

av+sin_x(-Frsiny+ FNcosv)]

aF3 aFT aFN aFt ay.... siny+_cosy÷ax ax ax ax ax

(FTCOSy+ FNsiny)

-- X4 0 O)a__ xs )2 (x)2+(x_)2Tx 0 0 0 (x_)2+(x s

(Eq. A.6)

(Eq. A.7)

(Eq. A.8)

(Eq. A.9)

(Eq. A.IO)

(Eq. A. 11 )

3O

3a 3Dc)FT 3T ,Tsin a_-.___-= _cosc_- _ "3x 3x3x 3x

c)a 3 L

3F u _______3Tsin a- (T cosa)_-_- -_-_3x 3x

o00

X4X6

XsX6

(v/) _0

__( -2_t_x! 0 0 0 0 0

,_x _,-(x_) _

3Co 3Q3 D_ = QS ref'----" + C oS ret-_X3x 3x

3Ct 3Q3 L Q S r.t _ + C LS _./ -_xc)--_ = 3 x

OCT C) Q

aT Q + C r _"_3x 3x

¢)x

3Co3a3Co 3CD 3Mt + .._.__._ii.iii. _ ,_.,_.iiii..i..--im--i_

3x 3a 3x3x 3Mr

/

(x,) =)

acLaaacL 3CLaM/+._____

3x aM/ 3x 3a 3x

(Eq. A.12)

(Eq.A.13)

(Eq.A.14)

(Eq. A.15)

(Eq. A.16)

(Eq. A.17)

(Eq.A.I8)

(Eq. A.19)

(Eq. A.20)

(Eq. A.21)

31

c_C r C)C T _ M /.... (Eq. A.22)_x _M f _x

o/M / = -- (Eq. A.23)

a

au I atl t au !(Eq. A.24)

To complete the evaluation of .f x, the following partial derivatives are derived from the

_CD 8CD C)CL C]CL 8CT t]IsPa t]a

example vehicle model data tables: _u---_;_ ;_u _ ; _= ;_u j; au _ • Also a and _ are derived from

a standard atmosphere model.

32

APPendix BPartial Derivatives for the Exarnole Model with Respect to Controls

All the equations required to compute the matrix f ,, for the example application are derivedbelow.

where:

.fu =

f o o0 0

0 0

£7 au_ x7 au2

X7 i X7 c)U2

go&U l Isp. 0

(Eq. B.1)

OF, (OFt _FN )-_-_u, = sign,cos, -_-_,u-ucosy+ 5u_ siny(Eq. B.2)

3F2 (OFT bFN )- sign_sin_ _cosy+_sinybu, 3u, bu,

(Eq. B.3)

_)F3 3Fr 3FN.... siny+3u_ 3u, 3u_

cosy (Eq. B.4)

,(cT)2uo( cTcT ,s a) (Eq. B.5)

3Fr 3T 3a 3D-- = --cosa- Tsina3u, 3u, 3u, 3u_

(Eq. B.6)

c)F N _ (aT --+--3(13L)5u_ _--u/sinc_ + Tc°setau, 3u,(Eq. B.7)

33

Also:

--=(o 1)3u

(_= o qs_._au_)

aT( aC_ )= 2Qu,a¢_ 0

3C D 3C z_ 3a

3u2 3a 3u2

(Eq. B.8)

(Eq. B.9)

(Eq. B.IO)

(Eq. B.11)

(Eq. B.12)

,)CL 3CL3(z- (Eq. B.13)

_U 2 _(l O_U2

To complete the evaluation of /=, the following partial derivatives are derived from the

C)CT . c)l=P a. ¢)CD . C)CL

example vehicle model data tables:_4_a s _0 a P _a ' _a"

34

X,x

Y

1

3 ; 2

_t

Z

Z

X-Y-Z:

x-y-z:

1-2-3:

Earth centered inertial frame

Earth fixed frame (rotates with Earth)

Vehicle fixed LVLH frame

Figure 1. Coordinate systems for dynamics equations

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REPORT DOCUMENTATION PAGE oM8 No.070,,-0188, ., | , |m ,

I'ubl. f_f3_)¢11ng hclrdPn for thl_ c,)ll¢_"flO_ 0t li_f(lf_a[l_lll i_, ._|li_r_a|_ tO aP4eraqe | huui r _,1- f_.%p.gn%t_, tn,iludin 9 the time Ior rpvr-wln_l insti'uctlOnS_ searchin 9 ex_stln<J data r_0urce_,

q_iLh_fif1_}."_md_1a_n_n_theda_neede_d_ar_d_mphq_¢_J_dfev_t_w_ngth_h._n_iIdUrm_1t_ %end comment0, rPgaldlng this burden eStlf1_ate o¢ ¢1n¥ other aspect of t h1$

toilet lion (_f irH_rmattOO. =ncludllrlg _ucJcJc"_[Iuns for rPdu_tn 9 tJ.% burden, tu W_%h.nc]t_Jn Ht'adquJcte¢% $ervl(e_. DlteEtOrate ¢o¢ Information O_ratlon_ and Reports, 1215 Jefferson

Daw_, H.Jhw,sy. _,u¢[e 1704, Arlinqlon, VA 22202-4]0?...,d [o [hi= [)tfhe ol Ma.a,.jemPnt 41_d RudCjet, PaPerwork RPclucllo_n Pro Pc[ (0/04 0188), Wa_J'.ngton, DC ?0503.

'1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE i. REPORT TYPE AND DATES COVERED

October 1992 Contractor Report

_.TITREAND SUBTITLE .... "S FUNDING NUMBERS

Air-Breathing }lyt_rsontc Vehicle Guidance and Control

Studies: An Integrated Trajectory/ControlMethodology - Phase II

6. AUTHOR(S)

Philip D. Harris and Harvey L. Malchow

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

The Charles Stark Draper Laboratory, Inc.

555 Technology Square

Cambridge, MA 02139-3563

"9. SPONSORiNG/MONITORII_G AGENCY NAME(S) AND A'ODRESS(ES) ......

National Aeronautics and Space Administration

Langley Research Center

Hampton, VA 23665-5'125

|1. SUPPLEMI_NTARY NOTES ....

C NASI-18565

WU 505-70-04-01

8. PERFORMING ORGANIZATIONREPORT NUMBER

_mlysis

10. SPON$ORIN(_ / MONITORING

AGENCY REPORT NUMBER

NASA CR-189703

Langley Technical Monitor: John D.

Final Report - Task 13

_2a. DIST'RmUTION/AVAILABILITVSTATEMENT

Unclasslfied-Unlimlted

Subject Category 08

J3. ABSTRACT (Maximum 200 words)

Shaughnessy

'Ub.O,sTem'uho,_CODE

An integrated trajectory/control analysis algorithm has been used to generate trajectorias anddasired control strategies for two different hypersonic alr-breething vehicle models and orbittargets. Both models used cubic spline curve fit tabulated win.ged-cone accelerator, vehiclerepresentations. Near-fuel-optimal, hodzomal takeoff trajectories, imposing a aynamicpressure limit of 1000 psf, were developed. The first model analysis c..ase involved a polar orbitand included the dynamic effects of using elevons to maintain _ongituoinal trim. Anal/sis resultsindicated problems with the adequacy of the propulsion mocl.el and hi.ghl!ght._..dyn .amicpressure/altitude Instabilities when using vehicle angm of attack as a contro_ vanaom. PJso,the magnitude of computed el(won deflections to maintain trim suggested a need for alternativepitch moment management strategies. The second analysis case was reformulated to usevehicle pitch attitude relative to the local vertical as the control variable. A new, more realistic,air-breathing propulsion model was incorporated. Pitch trim calculations were dropped andan equatodal orbit was specified. Chan_las in flight characteristics due to the new propulsionmodel have been identified. Flight regimes demanding rapid attitude changes have beennoted. Also, some issues that would affect design of closed-loop controllers were ascertalnea.

14. SUBJECT TERMS

Alr.brNthtng _nic vehicles; Constrained trajectorlas;Data smootfllr_; Generic hypersonic guidance _ control;New.minimum_fuel trajectories; Single-stage-to-orbit

15. NUMBER OF PAGES

8516. PRICE CODE

A05

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECU'RI'rY CLASSIFICATION

OF REPORT OF THIS PAGE OF ABSTRACT

Unclassified Unclassified

NsN 7540-01-280-5500 Standard Form 298 (Rev 2-89)Pf_cflb_N_l by ANSI Std Z39-18

298.102

20. LIMITATION OF ABSTRACT

NASA Contractor Report 189703 Air-Breathing … · NASA Contractor Report 189703, Air-Breathing...

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