AIAA-93-3713-CP

TRAJECTORY OPTIMIZATION FOR A NATIONAL LAUNCH SYSTEM VEHICLE

Eugene S. Chen, Frederick W. Boelitz, Jeanne M. Sullivan

The Charles Stark Draper Laboratory 555 Technology Square Cambridge, MA 02 139

ABSTRACT

A trajectory design process is presented for a proposed National Launch System (NLS) vehicle configuration. A software trajectory planner was developed to generate near optimal pitch plane trajectories for variable mission plans, variable launch site selections, and last minute wind conditions at the time of launch. In addition, the trajectory planner performs in-flight near optimal trajectory redesign in the event of an engine-out or unexpected wind dispersions. The trajectory planner iteratively simulates trial trajectories defined by variables corresponding to a parameterized trajectory shape. The trajectory planner optimizes these trajectory parameters according to the fuel used to realize the trajectories. For prelaunch scenarios, the trajectory planner generated near optimal trajectories of consistent fuel performance irrespective of the winds encountered at the launch site. In-flight trajectory redesigns were also shown to be effective in fulfilling the mission plan and preserving the structural integrity of the vehicle. The in-tlight redesign results indicate that closed-loop guidance in the early stages of ascent is highly effective in accommodating engine failures or wind dispersions. All trajectory designs were verified using a 6DOF simulation with tlight controllers.

INTRODUCTION

In focusing on the current high costs for lifting payloads to orbit. an important cost factor in present launch systems is the large amount of mission preparation required for every launch. Typically, the entire trajectory must be custom designed for each particular mission, depending on the payload weight and specified mission constraints. This preparation requires a large amount of lead time before launch and makes current planning systems inflexible to last minute changes in launch conditions. In light of this fact, proponents for the development of a new, low cost, highly reliable, unmanned (but possibly man-ratable) National Launch System (NLS) have highlighted the possibility of reducing launch costs through the use of automated mission planning software systems.' A proposed application of automated software planning systems is the design of optimal trajectories to minimize propellant requirements.'

An autonomous software trajectory planner is developed to design near optimal pitch plane trajectories for prelaunch and in-tlight launch scenarios. The vehicle design studied in this paper is a proposed National Launch System (NLS) configuration. The trajectory design approach is based upon

a parameterized trajectory model shape definition. The fuel usage performance for a trajectory defined by these shape parameters is evaluated with a 3DOF predictive simulation. The trajectory parameters are optimized using a conjugate gradient optimization method. Capability to accommodate variable mission plans and variable launch site selections has been built into the trajectory planner in order to demonstrate its applicability to realistic launch situations. A 6DOF simulation is used to verify the performance of optimal trajectories designed with this planner.

In this paper, the ALS-L vehicle configuration and environmental models are foremost defined. The parameterized trajectory model and guidance design used i n the trajectory planner are then discussed. The developnlent of a 6DOF simulation with tlight controller designs is explained. The 6DOF simulation is used to verify "optimal" trajectories generated by the trajectory design process. A 3DOF simplified simulation is used to predict the fuel usage for a given trial trajectory. The 3DOF simulation is then incorporated into the framework of a con.jugate gradient optimization technique to complete the trajectory planner. Prelaunch and in-flight optimal trajectory design results for various wind patterns are then presented. In addition, a full launch sequence scenario is examined to characterize the utility of the trajectory planner. Finally, conclusions are presented for the trajectory design technique explained in this paper.

VEHICLE / ENVIRONMENTAL MODELS

Bowter Module

7 LHILOA Engines

- Core Module

3 LHILOX

Figure 1. ALS-L Vehicle Configuration.

Copyright 01993 by The Charles Stark Draper Laboratory, Inc. Publishcd by the American Institute of Aeronautics and Astronautics, Inc. with permission.

The vehicle chosen for this study is an National Launch System (NLS) program vehicle designed by General Dynamics. This ALS-L vehicle consists of a col-e and a

booster module. The liquid hydrogentliquid oxygen (LHILOX) booster section is powered by seven low-cost, refurbishable engines. The core module has three engines which are identical to those in the booster section. The payload is attached above the core section of the vehicle. The booster module rides on top of the core and payload modules during the flight to orbit. A diagram of the ALS-L vehicle configuration is shown in Figure 1.

The major environmental factor that affects the vehicle is the aerodynamic effects caused by winds at the launch site. Therefore, wind dispersions were the only environmental perturbations modeled in this study. To develop a wind model, a piecewise linear wind profile was fitted to actual wind measurement data from the often windy Vandenberg Air Force Base (VAFB). The wind measurement data used in this study is classified as the #69 Vandenberg profile, and is expressed as a function of wind speed versus altitude. Since this study concentrates on pitch plane trajectory design. the wind profiles were taken to be headwinds or tailwinds of various strengths in the pitch plane direction, and assumed to be parallel to the Earth-relative horizontal.

to both a prespecified pitch attitude, Of, and an aerodynamic loading constraint limit (Qa) at the conclusion of Phase 11. The Q a limit is defined as the product of the dynamic pressure. Q, and the angle of attack, a.

PITCH ATTITUDE 1

R

TRAJECTORY PARAMETER SET:

[hl , Of, "2. ""d

Qa LOADING

PHASE I

PHASE I1

TIME

Figure 3. Parameterized Trajectory Shape for Phases I & 11. TRAJECTORY DESIGN MODEL

The trajectory model developed in this study is based on a parameterized standard trajectory shape. The justification for assuming a standard trajectory shape comes from a study by Corvin' of optimally generated trajectories for a single- stage-to-orbit (SSTO) winged boost vehicle. Although the vehicle studied by Corvin differs from the ALS-L vehicle, the aerodynamic loading effects and mission plan for both vehicles are similar. Figure 2 illustrates the general flight history of the ALS-L vehicle from Earth to orbit. Figures 3 and 4 show the parameterized trajectory model shape used to design the vehicle guidance.

PHASF l V - -

POWERED EXPLICIT Gl!IDANCE

STAGIKG PHASE I l l

CON5TRAIKT OBSFRVED

PHASE I1

CORE SHUTDOWN - CARGO

SEPARATIOK

477171 [IF. PITCHOVER \*AX Qa REACHED MAKFI 'LER

CCAFS LAUNCH SITE

ALTITI!DE h l

PHAYE I

A - LIFTOFF il

Figure 2. Flight Phases for the ALS-L Vehicle.

Phase I is the vertical liftoff phase for the ALS-L vehicle. The transition point from Phase I to Phase I1 occurs when the vehicle reaches a certain altitude, h l . Phase I1 involves an attitude pitchover maneuver specified by a sinusoidal pitch attitude rate schedule which is calculated as an analytical function of time. The proper time duration of the pitch rate schedule is determined by a golden section numerical search routine and is designed to bring the vehicle

Phase I11 is initiated when the Q a limit is reached at the end of Phase 11. During Phase 111, the Q a limit is held constant to preserve the structural integrity of the vehicle. Initially, this results in a decreasing angle of attack which compensates for the still rising dynamic pressure, Q. As the vehicle emerges from the atmosphere and the dynamic pressure begins to fall, the angle of attack rises until i t reaches a value of a2. The angle of attack is held constant at a2 from this point forward until the booster module has staged.

PARAMETER SET.

A Qa LOADING ANGLE OF CONSTRAINT] ATTACK u CORVIN

PARAMETERIZED REFERENCE TRAJECTORY TRAJECTORY

/

+ PHASE IV

PHASE 111

Figure 4. Parameterized Trajectory Shape for Phase 111.

Phase IV is designated as the flight from booster staging to orbit. During this flight phase, aerodynamic effects have been found to be negligible with respect to thrust and gravitational forces. Consequently, a modified version of Powered Explicit Guidance (PEG)4-6 is employed during Phase IV to optimally bring the ALS-L vehicle from the booster staging point to orbit. The PEG routine uses a calculus of variations approach under the assumption of no aerodynamic forces to solve for optimal vehicle control commands. Employed by the Space Shuttle, the PEG algorithm has been verified as an effective optimal guidance

software routine for exoatmospheric maneuvers. The PEG routine outputs the time until main engine cutoff (MECO) at orbital insertion as well as the optimal thrust vector direction history for satisfying the mission plan.

The parameters which define the parameterized trajectory shape are: 1 ) h , , the altitude which determines the end of the Phase I liftoff. 2 ) 8,. the pitch attitude achieved by the vehicle at the end of Phase 11. and 3) a?, the constant angle of attack tlown at the end of Phase 111. The final tlight parameter is the vehicle-dependent aerodynamic loading con\traint Qa. Varying these parameters changes the iehizle trajectory and affects the amount of fuel used to reach orbit for the given trajectory parameter set.

For trajectory redesign scenarios. the design guidance trajectory shapes remain the same for each flight phase, with the exception of a trajectory redesign that occurs during Phase 11. Such a trajectory redesign typically occurs in the middle of the original Phase I1 pitchover maneuver. Consequently, the sinusoidal pitch rate schedule is reconfigured to account for the fact that the redesigned maneuver is started with initial pitch attitude and pitch rate values that do not coincide with those at the end of Phase I.

6DOF SIMULATION / CONTROLLER DESIGN

A full 6DOF simulation was developed to follow a tr-a.jecto~-y specified by a given set of trajectory parameters. The 6DOF simulation feeds open-loop flight guidance commands, derived from the trajectory design model and uarameters, into flight controllers that guide the vehicle along the predesigned trajectory. Two types of steering controllers were developed for the 6DOF simulation. Pitch attitude steering control is used for tlight Phases I, 11. and IV, while a variation of acceleration-direction steering control i \ employed in tlight Phase 111.

For t h i \ study. all feedback flight parameters used in the controllers. with the exception of the acceleration-direction of the vehicle, are assumed to be perfectly known. In addition. the engine nozzle servos were idealized so that the commanded nozzle deflection was perfectly achieved. Correspondingly, engine nozzle dynamics were not modeled and are not included in the controller block diagrams. However, i n developing the 6DOF simulation, a rate limitation of I03/second is placed upon the engine nozzle deflection in order to ensure reasonable operating behavior.

Although this study assumes perfect estimation for most of the feedback variables, note that estimators for these variable\ \\ere previously designed by Boelitz7 and irnplernented for the same Phase I. 11. and I11 controllers pre~ented i n this paper. Specifically, a first order complernenta~-y filter approach was employed to form an

angular rate estimate. h . An angle of attack estimate, 6 , u.as derived using a second order complementary filter

design. Finally, a dynamic pressure estimate, Q, was formed from estimates of the local air density and the air- relative velocity of the vehicle. The estimated air density was detet-mined from a standard atmospheric model and the IMU-I-epol-ted altitude of the vehicle. The air-relative

velocity was approximated using the angle of attack estimate and the IMU-sensed earth-relative velocity.

Phase I and I1 Controller Design

Figure 5 shows the control block diagram for the pitch attitude controller used in tlight Phases I and 11. A similar pitch attitude controller for Phase IV is discussed later. The open-loop flight guidance commands consist of the commanded pitch attitude. Q,, and commanded pitch rate. o,. In tracking the pitch attitude commands, 8,. a pitch attitude error signal, ee, is formed as the difference between the pitch attitude command and the actual pitch attitude, 8. This signal is then sent through proportional-integral compensation. The resulting signal is then summed with the pitch rate command, a,, that is previously scaled by a feedforward gain. The sum of these signals is the net pitch rate command, o',, which is differenced with the actual pitch rate, o, to form the pitch rate error, e,.

Figure 5 . Pitch Attitude Controller for Phases I & 11.

The pitch rate error, e,, is passed into the inner loop where the signal is scaled by a proportional gain, K8/KV. The resulting signal, 8, is the commanded engine nozzle deflection angle. The pitch rate error signal, e,,,. is compensated by the gain Ks/Kv in order to keep the total forward gain of the inner loop constant during tlight. The gain Ks is set to be constant in the controller. The gain Kv is approximately

olnes where T = Thrust force produced by the en,' xCg = Location of the vehicle's center of gravity as

measured from the vehicle base I,, ., = Rotational inertia of the vehicle in the pitch

plane direction

Phase 111 Controller Design -

The Phase 111 controller design used in the 6DOF simulation employs acceleration-direction steering subject to a control override based on an aerodynamic Qa loading limit. This control scheme combines the best features of 1) velocity-direction steering with angle of attack limiting (used by Corvin3), and 2) acceleration-direction steering with add-on load relief (traditional). Specifically. the acceleration-direction steering with angle of attack control

override combines the fast load relief characteristics of the velocity-direction steering approach with the fast steering response of traditional acceleration-direction steering. In this way. the acceleration-direction steering is unimpaired by the add-on load relief, and load relief via angle of attack limiting will take place only when needed. This dual mode control scheme was first developed by B ~ s h n e l l . ~

The acceleration-directionlangle of attack tlight controller has two modes of operation. In the primary mode, the vehicle employs cross-product steering so as to null the difference between the vehicle's sensed acceleration-direction

vector, 0, . and the stored time-based acceleration-direction

vector command, O A , ( t ) . The sensed acceleration-direction

vectol- is defined as the vehicle acceleration due to thrust and aerodynamic forces. and is estimated from the changes in inertial velocity as measured by the Inertial Measurement Unit (IMU) on board the vehicle. The effects due to gravity are not included i n the acceleration-direction. The error signal, e , , for this control mode is generated by

normalizing the vectors O A and OA, , and taking the cross

product C between these two normalized vectors, call them

IJ, and tii\, respectively.

Then the angle between these two vectors is

Thu5, the vector UF has components which correspond to the error angle\ In roll, p~tch and yaw between the vectors

0 \ and OAi . The error \ignal, e ~ . is simply the pitch

component of the error vector U,

The secondary mode, or Q a limiting mode, is activated when the aerodynamic loading limit for the vehicle is exceeded. The loading limit on the vehicle is characterized by a QCX metric that is defined as the dynamic pressure, Q. of the vehicle multiplied by the angle of attack, a . To monitor the loads on the vehicle during flight, the predicted angle of attack. apred, is compared to an angle of attack limit, C X ~ ~ , , , , ~ , defined by the quotient of the Q a limit and the dynamic pressure Q. When the vehicle has a predicted angle of attack that exceeds the angle of attack limit. the secondary Q a control mode activates and tries to eliminate the difference between these two angle of attack signals.

where a = Angle of attack

ei, = Acceleration-direction error

Figure 6. Acceleration-DirectionIQa Controller (Phase 111).

A control block diagram for acceleration-direction steering is shown i n Figure 6. The mode switching i n this control design results in either an acceleration-direction error signal. e,,, or an angle of attack error signal, e,, being provided as the guidance command to the control system. In either control mode, the error signal is sent through proportional- integral compensation in the controller. The resulting net pitch rate command, a',, is the compared to the actual pitch rate of the vehicle, w. This pitch rate error. e,,,, is passed into the inner loop where it is scaled by a proportional gain. KdKv, to produce an engine nozzle deflection command. 6. for the vehicle. Again, as in the pitch attitude controller design for Phases I and 11, the pitch rate error, e,!,. in the Phase I11 controller is multiplied by the gain KdKl. to keep the total inner loop forward gain constant during flight.

Note that this two mode controller features a transient-free switch when changing modes. At the switchover. the control integration error is reset so that smooth. continuous nozzle deflection commands are sent to the engine set-vos. Since each mode of the Phase IT1 controller ernploy\ a different guidance command. the closed-loop system exhibits different dynamics in each case. Consequently, the control gains must differ for each mode, and have been identified by the different subscripts i n the gains shobvn in Figure 6 ( K P I L , K I I 1 , and K f i l L ) . These gains uese

computed via stability analyses of linearized system models.

Phase IV Cont~oller Design

The pitch attitude controller designed for tlight Phase IV is similar to that i n Phases I and I1 except that the feedforward signal is omitted and a derivative gain is placed i n the feedback path for the pitch rate (see Figure 7) . The feedforward term is not needed in this controller since the PEG-generated pitch attitude guidance commands. and consequently the vehicle pitch rate, do not change rapidly. Therefore, the controller can track the pitch attitude guidance

commands with good performance using the chosen design. As with the Phase I and I1 controller design, linearization of the vehicle dynamics was used to generate a transfer function between the pitch attitude, 0 , and the nozzle deflection angle, 6. The controller gains were determined using this dynamic model in an LQR design framework.

Figure 7. Pitch Attitude Controller for Phase IV.

PREDICTIVE SIMULATION

The optimization scheme used in the trajectory planner repetitively simulates trial flight trajectories and evaluates them based upon fuel usage. Thus, the simulation used in the optimization process must be computationally efficient i n order to reduce the amount of time required for the optimization. With this objective in mind, a faster, but less accurate, 3DOF simplified simulation9 has been developed as an alternative to the 6DOF simulation.

The 3DOF predictive simulation estimates the fuel required to follow a trajectory from Earth to orbit for a given set of flight trajectory parameters. The assumptions of the predictive simulation include a flat Earth model,9 a non- constant, directional gravitational model,I0 and idealized flight c ~ n t r o l . ~ These assumptions simplify the calculations needed to execute the predictive simulation, thereby making the 3 D O F s imulat ion more computationally efficient than employing a full 6DOF simulation. In addition to these modeling simplifications, the 3DOF simulation employs a computationally efficient state propagation scheme so as to reduce execution time.1°

Modeling Assumptions

In order to increase computational efficiency, several modeling simplifications were made in developing the 3DOF simulation. The predictive simulation includes assumptions of a flat Earth model, a non-constant, directional gravitational model, and idealized flight control.

The flat Earth assumption is justified since the down-range distance traveled by the vehicle from Earth to orbital insertion is quite small relative to the Earth's circumference. To account for different launch sites,1° the predictive simulation transforms the initial launch velocity imparted by the Earth's rotation to the vehicle into a corresponding effect i n the tlat Earth reference frame.

The non-constant, directional gravitational model is developed in order to more effectively characterize the effects of gravity that a constant, unidirectional gravitational model cannot resolve. Perfect flight control is also assumed at each time step of the simulation. The use of idealized control is justified since the simulation serves to predict the

mission fuel usage, not to resolve controllability issues.

State Propagation Method

Various state propagation methods were considered for improving the execution speed of the 3DOF s i m ~ l a t i o n . ' ~ In determining the most appropriate state propagation method for the 3DOF simulation, a tradeoff between accuracy and execution speed was considered. Prospective methods were evaluated on the basis of the amount of error in the on-orbit mass predictions as well as the time taken to simulate a trajectory. A predictor-corrector scheme was selected for its speed and accuracy characteristics. A simulation time step of h = 0.2 seconds was employed.

The method used in the trajectory planner is a Draper Laboratory (DL) simulation integration technique.ll The DL scheme predicts a new function value, y,,,, based on the second most recent state value, y,.], and the current state derivative value, y',. The derivative is then evaluated at the predicted function value, P,,.

The corrector step uses the current function value, y,,, the current derivative value, y',, and the newly evaluated derivative value, P',, to generate a corrected value, C,,.

This corrected function value is combined with the predicted function value to yield the new function value, y,,,.

F l i ~ h t Guidance Command Output

In addition to predicting the fuel required to satisfy a launch mission plan, the predictive simulation also outputs flight guidance information corresponding to the identified optimal trajectory parameter set. As described earlier, the four vehicle flight phases are defined by the trajectory parameters { h , , O f , or2, and Qor } . The predictive simulation stores flight guidance commands generated during each of these flight phases. These guidance commands are to be used during an actual launch and can be verified by the closed loop flight controllers developed in the 6DOF simulation.

Phase I flight guidance output consists of the altitude transition point, h, . For Phase 11, parameters defining the sinusoidal pitch attitude schedule are written to an output file. During Phase 111, the inertial acceleration-direction history7 of the vehicle is stored as a function of time, and is used as flight guidance for the Phase I11 flight controllers in the 6DOF simulation. No flight guidance is stored for Phase IV because the PEG algorithm generates the optimal guidance solution based only upon the vehicle state at the end of Phase 111 and the desired terminal tlight conditions. Consequently, during execution of the 6DOF simulation,

the PEG algorithm is used on-line to calculate the guidance solution according to the vehicle state at the end of Phase 111 and the desired orbital establishment conditions.

Note that since the acceleration-direction information is calculated with respect to the 3DOF simulation inertial reference frame (3DOF IRF), the acceleration-direction data must be translated into the 6DOF inertial reference frame (6DOF IRF) in order to be used by the flight controllers in the 6DOF simulation. This transformation is accomplished using rotation matrices that are formed from the launch azimuth and the launch site declination and longitude."

OPTIMIZATION SCHEME

With the computational speed of the 3DOF predictive simulation, it becomes possible to combine a conjugate gradient optimization routine with the 3DOF predictive simulation in order to optimize trajectory parameter sets for the ALS-L ~ e h i c l e . ~ The 3DOF simulation essentially becomes a function generator whose inputs are the trajectory parameters { h , , Of, a 2 } , and whose output value is the fuel required to reach orbit. It is this function value, or fuel usage, that the conjugate gradient method minimizer;.

In this study. the conjugate gradient technique as described i n Press1' is used to optimize the flight parameters { h l , Of, a ? ] defined in the trajectory model. The dynamic loading constraint Q a is held constant during the optimization process since it is specified for each particular vehicle studied. For the ALS-L vehicle, the structural Q a loading constraint is assumed to be 3,500 Ibs*deg/ft2. Each iteration of the conjugate gradient technique modifies the trajectory parameters {h , , Of, a 2 ) , and determines the fuel needed to follow the resultant trajectory using the predictive 3DOF simulation. The fuel usage is essentially an objective function value that is minimized within a defined tolerance. The fact that trajectories are defined by the compact parameter set { h , , O f , a 2 } results in a computational advantage in the optimization process since the dimension of the problem is small compared to those of traditional trajectory solutions involving, for example, time-dependent pitch rate histories.

RESULTS

The launch site used in the trajectory designs was Cape Canaveral Air Force Station (CCAFS). Cape Canaveral is located at a declination of 28.3389355' and a longitude of -80.0," The vehicle is assumed to have a launch azimuth of 86.2336223.' The vehicle is required to fulfill a mission plan of an 80 x 150 Nmi. orbit with insertion at perigee. Prelaunch optimal trajectory designs were generated for various wind patterns of varying strengths. In addition, in- tlight trajectory redesigns were performed for engine-out and unexpected wind gust scenarios. Lastly, a full launch sequence scenario was examined to demonstrate the utility and effectiveness of the trajectory planner.

Prelaunch Trajectory Design Results

The effects of different strengths of various wind patterns on optimal prelaunch trajectory designs were examined.

Specifically, optimal trajectories were generated under #69 Vandenberg headwind or tailwind conditions with scaled magnitudes of 10.2, 0.5, 0.8, 1 .O} of the original linearized wind profile. An optimal trajectory was designed under a no-wind condition and used as a baseline for comparison. The Q a design limit was set at 3,000 Ibs*deg/ft2 for all trajectory design scenarios. This value was obtained from a safety factor assessment of the structura! Q a loading constraint of 3,500 lbs*deg/ft"or the ALS-L vehicle. The prelaunch trajectory design results are shown in Table 1.

Head I 1 .0 1 (89.82", 400.0', 7.49") 1 131 5 Tail 1 0.2 1 (82.82", 400.0', 8.52") 1 1409

Wind Direction

None Head He2d Head

Tail 1 0.8 1 (80.29", 400.0', 6.48") 1 1364 Tail I 1.0 1 (79.54", 400.0', 5.86") 1 1350

Table 1. Optimal Trajectory Designs under Various #69 Vandenberg Wind Conditions.

Wind Strength

0.0 0.2 0.5 0.8

From the data in Table 1 , the fuel remaining after mission completion decreases as the strength of the winds increase. This demonstrates the restrictive effect that winds impose on trajectory performance. Specifically, the winds reduce the freedom of the trajectory path and thereby increase the amount of fuel needed to fulfill the mission plan. Nevertheless, notice that the trajectory fuel usage performance over all the design cases varies by less than 100 slugs of fuel from the baseline no-wind case. This supports the fact that the trajectory planner is able to design trajectories that generate relatively consistent fuel performance regardless of various wind conditions present.

A closer examination of the results in Table 1 reveals the physical significance of these optimal trajectory design solutions. In the case of headwinds, the optimal trajectory parameter Of tends to increase as the headwind strength rises. This corresponds to the vehicle taking a more vertical tlight path during the launch maneuver. The result of this Of adjustment is that the vehicle will enter the Q a limiting phase earlier than for a smaller headwind scenario.

Optimal Set 1 % h , , a21

(83.41°, 400.0', 10.51") (84.72", 400.01, 10.31") (86.50°, 400.01, 8.92") (88.07", 400.01, 9.09")

When the vehicle enters the QCX "bucket," the angle of attack is decreased to observe loading constraints, and the pitch attitude rotates towards alignment with the flight path angle. This pitch attitudelflight path angle alignment behavior is seen in all trajectory design cases because of the presence of the Q a constraint. In the case of headwinds, the pitch attitude is forced to rotate faster than a no-wind case because the headwinds increase the effective angle of attack on the vehicle. Thus, a faster rotation of the pitch attitude will induce a faster rotation of the tlight path angle.

Fuel Left (slugs) 1429 1419 1386 1374

The danger in not entering the Q a bucket with a high enough pitch attitude Of is that in observing the Q a limit, the vehicle may pitch over too quickly, and the vehicle will

not build up enough vertical velocity to escape the atmosphere. The limiting case that results from increasingly lower parameter values is a diving trajectory that eventually brings the vehicle crashing back down to Earth. Thus. in the case of headwinds, the vehicle will assume higher 8, parameter values in order to ensure that the flight path angle during the Qa "bucket" is directed with enough loft to allow the vehicle to escape the atmosphere.

Conversely, i n the tailwind cases. the parameter €4 drops as the strength of the tailwinds increases. For increasing tailwinds. the vehicle does not require as high a pitch attitude 0, to enter the Qa bucket since the tailwinds serve to reduce the effective angle of attack on the vehicle. Consequently. the pitch attitude will not rotate as quickly during the Qa bucket region, and the resulting decreases in pitch attitude angle and tlight path angle are less than those experienced i n the headwind cases. This reduction in the pitch attitudelflight path angle rates results in enough trajectory loft that the vehicle escapes the atmosphere.

For the headwind cases, the angle of attack parameter a2 varies inversely to the parameter 8,. This behavior is logical since a higher O f parameter value seeks to preserve loft of the vehicle trajectory during the QCX bucket region. For increasing headwind strength, the parameter rises, and the effective loft of the vehicle increases. In order to correct height deviations due to the loft of the trajectory, the angle of attack parameter u2 decreases for increasing values of 8,-. A lower angle of attack a? forces the pitch attitude of the vehicle to stay more closely aligned with the flight path angle during the latter portion of Phase 111. As a result. the vehicle flight path angle will rotate more quickly and thereby correct any deviations in the loft of the trajectory experienced up to this point.

For tailwind cases, the a> parameter decreases as the pitch attitude parameter Of decreases. Again, lower Qf parameter \.dues are chosen in stronger tailwinds because tailwinds reduce the angle of attack experienced by the vehicle. During the Qa bucket region, the vehicle pitch attitude rotates at a slower rate. and the effect is a loftier trajectory. T h u ~ to correct for deviations caused by loftier trajectories, the parameter a? decreases as the tailwinds increase.

For the test cases in Table I . the initial guess for the flight parameter h l was set at 400 feet and allowed to vary thereafter under the optimization method. Looking at the resulting optimal trajectory designs, the optimization process appears to be rather insensitive to the parameter h l . To further examine this, another set of trajectory design cases was generated under the identical wind scenarios. However, the parameter h l was initially set at 3,000 feet instead of 400 feet. The design results are listed in Table 2.

The results i n Table 2 indicate a small change, on average, of the remaining fuel amounts after orbital insertion. This maybe due to the new starting value of the parameter h l in the trajectory design process. Nevertheless, sensitivity of the optimization to the parameter h , is still indiscernible in this set of trajectory designs. These results suggest that the pal-arneter h l could be eliminated from the trajectory

parameter set without much loss of generality i n the trajectory model shape. However, the appropriate choice of a constant h, altitude for Phase I must be ju\tified

None I 0 .0 1 (72.49", 3000.08, 14.61 O) 1 1426 Hed 1 0.2 1 (73.6g0, 3000.0', 14.39") ( 1435

Tail 1 0.8 1 (64.56". 3000.0'. 10.73') 1 1418 Tail 1 1.0 1 (64.3S0, 3000.0t, 10.82") 1 1429

Wind Direction

Table 2. Second Set of Optimal Trajectory Designs under #69 Vandenberg Winds.

Optimal Set (El,-, h l , a?+

Wind Strength

If the altitude hl is fixed at a high value, as i n this second set of test cases, the resulting generality of the trajectory shape may be con~promised. That is, placing the initial guess for the parameter h, at higher altitudes eliminates any variability of the trajectory tlight path until Phase I1 is initiated. On the other hand, if the parameter hl is set at a low value. such as 400 feet, the trajectory tlight path can react to environmental conditions much sooner. Furthermore, if the optimal height h l is indeed higher than 400 feet. the vehicle can compensate for this during Phase 11 by assuming a very high pitch attitude O f . The resulting trajectory path would, in effect, closely resemble a Phase I vertical liftoff where the parameter h, is a high value.

Fuel Left ( s l u ~ s i

Thus, in order to preserve generality and responsiveness of the trajectory design shape, the parameter h l was set at a constant value of 400 feet for the remaining trajectory design cases, and removed from the trajectory parameter set. The height of 400 feet is chosen so that the vehicle may safely clear the launch tower and thereafter perform the Phase I1 launch maneuver.

In-Flight Traiectorv Redesign Results

ANGLE OF ATTACK a

PHASES I & 11

HASE 111 - I PHASE IV

? 17 +11

I TIME

t - t-limt t = t-staging (Qtr = Qa limit)

Figure 8. Designated Trajectory Redesign Times.

For the in-flight trajectory redesign cases, the vehicle is assumed to encounter an engine out or detect an unexpected wind gust at the chosen test times o f (5, 15, 40, 80, 120) seconds. Figure 8 shows the times at which the redesign cases are examined. These times were selected in order to test the capabilities o f the trajectory redesign process in different operating flight regions o f the trajectory path.

ANGLE OF ATTACK a

I PHASE 111 PHASE l V ----.t

- TIME

Figure 9. Q a Design and Flight Limits for Phase I11

During flight Phase 111, the trajectory planner is set to generate trajectory designs and redesigns that correspond to a Qcc de.rign limit o f 3,000 Ibs-deg/ft2. The vehicle is then supposedly flown along this Q a design limit using acceleration-direction commands. However, i f an engine out occurs or unexpected wind gusts are present, the vehicle will deviate from the designed trajectory flight path, and the vehicle will incur Q a values higher than that accounted for in the design. As the Q a value rises, the Phase I11 acceleration-direction control mode is designated to switch over to the Qa-limiting control mode when the Qaf l ight limit o f 3,250 Ibs*deg/ft2 is exceeded. Acceleration- direction steering is allowed to resume when the Q a value falls below the flight limit. The difference between the Q a d e s i g n and f l i gh t limits is shown in Figure 9. This distinction becomes important when evaluating the performance o f in-flight trajectory redesigns.

For the case o f an engine out or an unexpected wind dispersion, the vehicle is assumed to initially fly an optimal trajectory design generated for a no-wind condition and a mission plan o f 80 X 150 Nmi. orbit with insertion at perigee. At various times in the flight { t = 5, 15, 40, 80, 120 seconds), the trajectory is redesigned to address an engine out occurrence or the presence o f unexpected wind gusts not accounted for in the prelaunch design.

In the case o f an engine out, either a single booster or core engine failure was simulated at designated times during the ascent. Given the state o f the vehicle at the engine failure occurrence, the trajectory was redesigned according to the loss in thrust so as to optimally satisfy the mission plan. A no-wind condition was maintained for both the pre-failure flight and the subsequently redesigned tlight path. The results from the engine out trajectory redesign scenarios are listed in Tables 3 and 4. The original prelaunch trajectory

design for the no-wind condition was (83.65". 10.47")

Selected Fuel Redesigned Fuel Time 1 e 1 e f t w o 1 Trajectory 1 e f t w 1 o f I Redesign Parameters Redesign Engine

Booster

Table 3. In-Flight Trajectory Redesign\ for an Engine-Out.

Core Core Core Core

Table 3 lists in-flight trajectory redesign solutions for an engine out occurring prior to booster staging. Trajectory performance corresponding to the redesigned trajectory i s compared to the fuel performance i f no redesign were employed after the engine failure. The disparity between the trajectory fuel performance with and without redesign tends to become worse for earlier engine failure times. This is expected since the loss o f an engine earlier in flight restricts the capabilities o f the vehicle to satisfy the mission plan. Specifically, an engine out decreases the fuel expenditure rate as well as the thrust-to-weight ratio. A reduced thrust- to-weight ratio implies that the vehicle will ultimately need to extend the burn time and expend more fuel in order to boost the vehicle to the required orbit insertion conditions.

(slugs)

Crash

In the case o f a core engine out condition, the vehicle is unable to reach orbit, regardless o f trajectory redesign. for core failures occurring at 5, 15. or 40 seconds into the flight. However, for a core failure at 80 or 120 seconds. the vehicle is able to reach the desired orbit in both the redesign and no redesign cases. For the 120 second case, the trajectory performance is better when redesigned. However, the redesigned trajectory fuel performance for the 80 second core engine failure case is actually worse than i f the trajectory were not redesigned.

Crash Crash 855 1084

When an engine out occurs and the tra.jectory is not redesigned, the vehicle will continue to follow the original trajectory guidance commands, and will usually breach the Qcc flight limit. Subsequently, the vehicle controller will guide the vehicle along this Q a flight limit which is higher than the Q a design limit but less than the structural Q a loading limit for the vehicle (see Figure 9). Thus, the added performance o f the vehicle due to flying at the higher Q a tlight limit allows the vehicle to use less fuel to achieve the proper orbit. However, the downside is that the vehicle is forced to withstand higher loading effects while operating along the Q a flight limit, thus exposing the vehicle to greater risk o f structural failure. This compromise o f better fuel performance at the expense o f structural safety margins applies to all o f the engine-out and unexpected wind gust trajectory redesign scenarios. Thus, in comparing trajectory performance for the redesign and no redesign cases. the added

{of, a;?> (84.69", 10.99")

No Solution No Solution

( N / A , 1 1 S3") (,N/A, 16.21")

(slue;)

905

~ailure

5 sec.

0 0

726 1 167

15 sec. 40 sec. 80 sec. 120 s .

fuel performance realized from the higher Qa flight limit o f the no redesign trajectory must be considered.

For a booster engine failure that occurs before Phase 111, the vehicle is able to optimally readjust both flight parameters { O , , a?] in order to satisfy the mission plan. Since the original no-wind trajectory design was (83.65", 10.47"), it can be seen that the vehicle attempts to maintain an optimal trajectory path by increasing the pitch attitude parameter Of in compensation for the loss o f thrust. Furthermore, the angle o f attack parameter a2 is also increased in order to redirect the vehicle towards the optimal trajectory path which is now difficult to achieve due to reduced thrust-to- weight ratio conditions. Notice that without the trajectory redesign. the vehicle flight will result in a diving trajectory that eventually crashes. This result is extremely important in that the trajectory redesign process is able to save the mission from total failure, thus demonstrating the utility o f the trajectory planner in realistic launch situations.

When a booster engine out condition occurs during Phase 111, the remaining trajectory design parameter a2 is increased from the original trajectory design value. Similar to the booster failure cases in Phases I and 11, the vehicle attempts to maintain proper loft o f the trajectory, given reduced thrust conditions, by decreasing the amount o f pitch attitude rotation towards the flight path angle. In this way, an acceptable trajectory path will be achieved by the end o f Phase I11 so that the core module can still maneuver into the desired orbit once staging occurs and the defective booster module is jettisoned. From the table data, an earlier engine out in Phase I11 is compensated by larger increases in the parameter a2.

Head-0.4 1 1404 1 (86.36",8.46") 1 1364 1 15 sec. Head-0.4 1 1404 1 (NIA.26.14") 1 1284 1 40 sec.

Wind Direction

& Strength

Head-0.4 1 1404 1 (N/A , 14.17') 1 1449 1 120 sec. Tail-0.4 1 1417 1 (81.74",8.68') 1 1404 1 15 sec.

Fuel Left w/o Redesign

(slugs)

Table 4 . In-Flight Trajectory Redesigns for Unexpected Wind Gust Scenarios.

Tail-0.4 Tail-0.4 Tail-0.4

The in-flight trajectory redesign results for unexpected wind dispersion cases are shown in Table 4. In these scenarios, the vehicle flies according to a prelaunch trajectory design based upon wind measurements that no longer match the true wind gusts present at the time o f launch. As a result, the vehicle trajectory path deviates from the optimal path once launched. At designated times in the simulation { t = 15, 40. 8 0 , 120], the trajectory was redesigned in order to account for the winds that were not commensurate with those used in the prelaunch trajectory design. For these results. the prelaunch trajectory was designed under a no-

Redesigned Trajectory Parameters

(of3 a21

wind condition. The actual wind gusts were set to vary 40% from the no-wind condition in both the headwind and tailwind directions. The original trajectory parameters were designed to be (83.65", 10.47"). When the original trajectory is flown under headwinds without redesign, the vehicle has 1404 slugs o f fuel at the end o f the mission. For the tailwind case, the fuel remaining is 1417 slugs.

141 7 141 7 141 7

For the unexpected wind gust cases, the trajectory planner was able to generate redesigned trajectories that has fuel usage performance comparable to that o f the nominal no- wind case. Remember though, that the reason the no redesign trajectory performance can match the redesigned trajectory performance is that the vehicle flies the higher Qa flight limit during Phase I11 rather than the Qa design limit. As a result, the fuel performance is inflated and does not reflect the added loading penalty incurred by the vehicle in flying the no redesign trajectory. Thus, the redesigned trajectories of fer comparable performance with the added safety margin o f flying along the Qa design limit rather than the Qa flight limit.

Fuel Left w/

Redesign (slugs)

For redesigns during Phase 11, the new trajectories exhibit changes similar to those in prelaunch design cases. In the headwind case, the vehicle maintains a higher Of parameter value in order to accommodate for the additional angle o f attack e f fects imposed by the headwinds, and thereby maintain proper trajectory loft for exiting the atmosphere. Correspondingly, the a2 parameter drops to correct for the additional loft achieved by the vehicle during the Qa bucket region. For the Phase I1 tailwind redesign case, the vehicle reduces the parameter Of since the amount o f rotation during the Qa bucket region will be less due to a reduction o f the angle o f attack from the tailwinds. The a2 parameter again drops from the original design value o f 10.47" to correct for the additional trajectory loft attained from the tailwind effect on the pitch attitudeblight path angle rotation.

Time o f Redesign

(sec)

(N/A,6.78") (N/A,10.47") (NIA, 10.47")

For trajectory redesigns during Phase 111, it appears that the fuel performance o f the redesigned trajectories increases when the redesign is initiated at later times in the flight. This might suggest that it is not wise to redesign at an early point in the flight, even i f the presence o f unexpected wind gusts are detected early enough. However, it must be considered that later redesign times imply that the vehicle flies the Qa flight limit up until the time o f redesign. As a result, trajectory redesigns initiated at later times tend to produce superior fuel performance. This is because flying along the Qa flight limit rather than the Qa design limit improves fuel performance at the cost o f reducing the structural safety margin on the vehicle. Therefore, the choice to redesign the trajectory becomes a tradeoff between safety and performance needs.

Looking at the in-flight wind gust redesigns during Phase 111, it can be seen that the redesigned a2 parameter values differ more from the original parameter value o f 10.47" when the trajectory redesign is initiated at earlier times. When the vehicle flies the Qa flight limit for a long time before redesign, the additional flight performance gained up to the point o f redesign will reduce the degree to which the vehicle will need to perform radical maneuvers thereafter to optimally satisfy the mission plan. Consequently, the a2

1372 1437 1441

40 sec. 80 sec. 120 sec.

parameter will approach the original a 2 design value for later redesign times.

Full Launch Sequence Scenario

A full launch sequence scenario for the ALS-L vehicle is examined in this section in order to highlight the areas during a launch sequence that the trajectory planner can be of utility. All assumptions as to the nature of the launch sequence are chosen based on availability of information. Prior to a launch date, an optimal trajectory design is assumed to be formulated according to a seasonal wind pattern that is set to be a #69 Vandenberg headwind of 1.0 magnitude. The launch site for this scenario is CCAFS, and the mission plan is the establishment of an 80 x 150 Nmi. orbit with insertion at perigee. On the day of launch, wind measurements are taken just prior to liftoff and are found to differ from the seasonal wind patterns. Specifically, the measured winds are assumed to be a #69 Vandenberg headwind of 0.2 magnitude.

At this point, the trajectory planner could be employed to redesign the trajectory in order to maximize fuel performance. Launching with the trajectory design incorrectly based upon seasonal wind patterns could very possibly result in decreased fuel performance and, in the extreme case, a mission failure. The trajectory redesigned for the true wind measurements is compared to the trajectory resulting from launching the vehicle according to the mismatched trajectory design (see Figures 10 and 11). The amounts of fuel left for the redesigned and the mismatched prelaunch trajectory designs are 1404 slugs and 1257 slugs respectively. The fuel usage for the redesigned trajectory is significantly less than that of the trajectory designed according to seasonal headwinds. This is because the seasonal wind trajectory design tended to loft the vehicle too high. The PEG guidance algorithm corrects this deviation during Phase IV by guiding the vehicle along a more shallow trajectory. This compensates for the additional loft so that the vehicle may establish the proper orbit.

Vehicle Altitude vs. Time

100000

0 50 100 150 200 250 300 350

Time (sec)

Figure 10. Vehicle Altitude for Mismatched and Correct Prelaunch Trajectory Designs.

I 1

Prelaunch

60 Devgn

Mimatched Prelaunch Dehign

Time (sec)

Figure 11. Flight Path Angle for Mismatched and Correct Prelaunch Trajectory Designs.

The vehicle is now launched using the redesigned prelaunch trajectory. No change in the winds is assumed to occur between the time of redesigning the prelaunch trajectory and the liftoff. At 15 seconds into the launch, a booster engine fails, and a trajectory redesign is employed. A comparison of the redesigned and the no-redesign trajectories is shown in Figures 12 and 13. The no-redesign trajectory causes the vehicle to crash. This is because the vehicle enters the Qa loading region with a low pitch attitude. As the angle of attack is reduced, the pitch attitude and the tlight path angle decrease too quickly. The result is a diving trajectory where the vehicle crashes. The redesigned trajectory guides the vehicle into the Qa region with a higher pitch attitude. During the Qa bucket, the pitch attitude and tlight path angle decrease slowly enough so as to the loft the vehicle to altitudes where the air density drops off rapidly. The vehicle is able to escape the atmosphere and complete the mission.

500000 Vehicle Altitude vs. Time /-

Time (sec)

Figure 12. Vehicle Altitude of Prelaunch v. Redesigned Trajectories for Booster Failure.

1 SO 100 150 200 250 300 350

Time (sec)

Figure 13. Flight Path Angle of Prelaunch v. Redesigned Trajectories for Booster Failure.

CONCLUSIONS AND RECOMMENDATIONS

In this paper, a software program to automate the trajectory design process for a National Launch System (NLS) vehicle configuration was presented. Tests results demonstrate that the trajectory planner is a valuable tool because it offers tlexibility in formulating a wide range of trajectory design scenarios. The trajectory planner is able to generate near optimal prelaunch trajectory designs, as well as near optimal in-tlight trajectory redesigns for the cases of an engine out or an unexpected wind dispersion. The applicability of the trajectory planner extends beyond this by supporting the definition of variable mission plans, variable launch site locations, and alterable wind conditions. With this tlexibility, the trajectory planner can be used not only for designing optimal trajectories, but also for studying the effects of various mission plans and launch sites on trajectory design results. Although the results presented i n this paper cover a wide range of trajectory design cases. the most significant example is the full launch sequence scenario. This test scenario demonstrated the capabilities of the software trajectory planner in a realistic prelaunch and in-tlight trajectory design scenario. The performance gains realized from its application were significant, confirming the capability of the technique to generate meaningful trajectory optimization results.

Due to the execution time required in the iterative optimization process, the trajectory planner may not be highly effective as an on-line in-flight trajectory redesign technique. Nevertheless, results from the planner may still provide insights into the optimal trajectory design process for many different launch scenarios. That is, the trajectory planner lends itself readily as a analysis tool for examining the underlying trends i n optimal trajectory designs for varlous launch scenarios. For this study, the trajectory planner results ~ndicated that closed-loop guidance of some form may be applied early i n the ascent so as to effectively accommodate for an engine-out or unexpected wind dispersions. Future work may be to improve the execution

speed of the parameterized trajectory design optimization approach if on-line implementation is desired. In addition, the optimality of the trajectory designs generated by the trajectory planner should be compared to results computed by other trajectory design algorithms. Algorithms such as the Program to Optimize Simulated Trajectoriesl"POST) may provide valuable benchmarks for the technique presented here.

REFERENCES

Scott, W., "ALS Cost, Efficiency to Depend Heavily on Process Improvements," Aviation Week & S ~ a c e Technologv, October 23, 1989, pp. 41-43. Shackelford 111, J., "Adaptive Guidance. Navigation. and Control for the Advanced Launch System," AIAA GN&C Conference Proceedings, New Orleans, 199 1 . Corvin, M., "Ascent Guidance for a Winged Boost Vehicle," MIT M. S. Thesis, CSDL-T- 10 12, 1988. Brand, T. , Brown, D., Higgins, J., "Unified Powered Flight Guidance." Shuttle G&N Equation Document, No. 24/2, The C.S. Draper Laboratory, C-4108, 1974. Fill, T., "Introduction to Bi-Linear Tangent Steering For Shuttle Ascent and Aborts," The C. S. Draper Laboratory, EGB-89- 108, SHUTTLE-89-022, 1989. Shepperd, S., "Conic State Extrapolation," Space Shuttle GN&C Equation Document No. 25, Rev. I , The C. S. Draper Laboratory, 1974. Boelitz, F., "Guidance, Steering, Load Relief and Control of an Asymmetric Launch Vehicle," MIT Master of Science Thesis, CSDL-T- 1036, 1989. Bushnell, G., "Guidance, Steering and Control of a Three Stage Solid Propellant Boost Vehicle," MIT Master of Science Thesis. CSDL-T- 10 12, 1989. Sullivan, J . , "Trajectory Optimization for an Asymmetric Launch Vehicle," MIT Master of Science Thesis. CSDL-T- 1062. 1990.

10. Chen, E., "Trajectory Optimization for a National Launch System Vehicle," MIT Master Science Thesis, CSDL-T- 1 184, 1993.

11. Dogan, P., Santarelli, M., Sklar, S . , "Simulation of the Deep Submergence Rescue Vehicle," Report R-67 1 . The C. S. Draper Laboratory, June 1972, pp. 18 1 - 185.

12. Press, W., Flannery, B., Teukolsky, S., Vetterling, W.. Numerical Recipes. Cambridge, University Press. Cambridge, 1986.

13. Bauer, G., Cornick, D., Harper, A., Peterson, F . , Stevenson, R., "Program to Optimize Simulated Trajectories (POST)," Vols. I, 11, and 111, NASA CR- 132960, April 1975.

ACKNOWLEDGEMENTS

This paper was prepared at The C. S. Draper Laboratory, Inc., under CSR #109. Some concepts in this study were developed under Task Order 74 from the NASA Langley Research Center (LaRC), under Contract NAS9-18 147 with NASA JSC. Aerodynamic data for the ALS-L vehicle and wind profile data from VAFB were provided by LaRC.