OPTIMUM LAUNCH TRAJECTORIES for the ATS-E MISSION by Omer

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OPTIMUM LAUNCH TRAJECTORIES FOR THE ATS-E MISSION

by Omer F. Spurlock and Fred Teren

Lewis Research Center

Cleveland, Ohio

TECHNICAL PAPER proposed for presentation at

Astrodynami cs Conferen ce sponsored by the American Astronautical Society and the American Institute of Aeronautics and Astronautics

Santa Barbara, California, August 20-21, 1970

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION OPTIMUM LAUNCH TRAJECTORIES FOR THE ATS-E MISSION

by Omer F. Spurlock and Fre_i Teren

Lewis Research Center National Aeronautics and Space Administration Cleveland, Ohio !

Abstract Intuitively, it seems reasonable that this conventional profile is near optimum if the burn and Optimum trajectories for the Applications coast durations may be varied to maximize the mass Technology Satellite (ATS)-E mission are obtained. at the end of each burn. However, if the total Analysis, procedure, and results are presented. impulse of the final burn is fixed at less than The trajectories are numerically integrated from the optimum value, the conventional trajectory launch to insertion into the final orbit. As a must be modified to yield maximum payload to the result of a much smaller than optimum apogee motor, final orbit. In particular, the parking orbit is these trajectories, unlike conventional synchro- noncircular, the perigee radius of the transfer nous orbit trajectories, require non-clrcular park- orbit increases, and the second burn removes more ing orbits and large amounts of inclination reduc- than a minor part of the inclination. The optimi- tion before the solid motor burn at apogee. Con- zation problem is to find the best combination of straints on parking orbit perigee radius and dura- these changes and other less important ones to tion are included. Figures are presented deserlb- yield maximum payload to circular synchronous equa-' ing the results. torial orbit.

Introduction Optimization of the conventional trajectory to circular synchronous equatorial orbit has been f., The Applications Technology Satellite (ATS)-E treated by several authors. Hoelker and Silber _±)

mission is a circular synchronous equatorial orbit present a detai_ analysis of the conventional mission. The ATS program has the objective of problem. Rider _ _ considers the problem of chang- advancing technology in areas which may have appli- ing the plane and also the radius of a circular cation to future spacecraft. The experiments which orbit. These and other similar studies treat the are conducted are spacecraft, con_nunlcation, and problem as one of changing the plane and radius of science orlented. a circular orbit, ignoring the ascent to the first (parking) circular orbit. This is satisfactory The spacecraft-orlented experiments on the for the conventional case. However, an unconven- ATS-E provide information on power supply and con- tional trajectory is more complex since the parking trol systems, a gravity-gradient s!abilization orbit is in general noncircular. The ascent must system, resistojet and ion mlcropound thrusters, Be included as part of the optimization problem. and synchronous environment. The scientific- Therefore, a more sophisticated optimization pro- oriented experiments gather data on the particle cedure is required for unconventional trajectories. (electron and proton) distribution and flux and Additionally, the references mentioned above are the character of the electric and magnetic fields general and consequently are not concerned with at synchronous altitude. constraints which may alter the acceptability of a given trajectory, such as limitations on coast time The launch vehicle for the ATS-E mission was or the minimum perigee radius of the noncircular an Atlas-Centaur and the solid apogee motor was a parking orbit. part of the spacecraft system. The apogee motor total impulse was sized for the early ATS missions The problem of optimizing trajectories to on the Atlas-Agena launch vehicle, which has less circular synchronous equatorial orbit may be con- payload capability than the Atlas-Centaur. The sidered as a multistage launch vehicle optimization. apogee motor, although smaller than optimum for Several analyses have been performed to optimize the larger vehicle, remained unchanged. multistage launch vehicles, including one by the authors of this report _#. For optimizing the un- For an optimally sized apogee motor, a con- conventional trajectory, the analysis in Ref. 3 ventional trajectory to circular synchronous was expanded to three dimensions and also, to in- equatorial orbit is near optimum. A conventional clude a constraint on the parking orbit perigee trajectory consists of five consecutive phases radius. The perigee radius constraint must be as shown in Fig. 1. The first phase is an ascent included to limit aerodynamic heating on the space- from the launch site to a circular parking orbit. craft. To maximize the mass in orbit, a 90 ° launch azimuth is used, which results in a parking orbit The Applications Technology Satellite (ATS)-E inclination equal to the launch site latitude. mission on the Atlas-Centaur vehicle requires an This inclination must be removed during the tra- unconventional trajectory to achieve maximum pay- Jectory. The second phase is a coast arc to the load. The final burn is performed by a solid proximity of the equator. A small portion of the motor which is part of the spacecraft system. That required plane change is removed by the second motor is significantly smaller than optimum. burn, the third phase. Much more importantly, the There are spacecraft and launch vehicle constraints second burn must place the vehicle in a transfer on the trajectory which must be incorporated into orbit whose apogee is over the equator and equal the solution. The perigee radius and the parking to synchronous altitude. The vehicle coasts to orbit coast duration are limited. The results for apogee in the fourth phase. The fifth phase con- this mission are presented. sists of a final burn that removes the major portion of the inclination and circularizes the orbit. 1 rI'MX-52836 Ax_l_vs is Calculus of Variations Solution

Problem Description A Calculus of Variations formulation was used to miximize the payload to circular synchronous A conventional trajectory to circular syn- equatorial orbit without resorting to a parametric chronous equatorial orbit launched from the Eastern search. The optimization of the atmospheric por- Test Range consists of five phases. They are: tion of the trajectory is omitted from the variations/ analysis since the steering is constrained i. Ascent to parking orbit. by factors other than optimizing performance, such 2. Parking orbit coast. as aerodynamic loading and heating limitations. 3. Second impulse. The analysis considers the problem from the point 4. Transfer orbit coast. in the trajectory that the atmosphere can be ne- 5. Third impulse or apogee burn. glected to insertion into the final orbit. In addition to optimizing the steering, the durations Figure 1 shows the planar characteristics of of any unspecified burns and coasts are optimized the conventional trajectory. The nonplanar charac- while maintaining the specified perigee radius of teristics are shown in Fig. 2. The vehicle is the parking orbit. The analysis is presented in launched at an azimuth of 90 ° in order to maximize appendix B. It is derived in three dimensional the vehicle mass in parking orbit and to minimize rectangular coordinates in a manner similar to tile inclination of the parking orbit. The circular Ref. 4. The equations for optimum burn and coast parking orbit altitude is as low as aerodynamic duration are obtained from an analysis similarto heating constraints will allow, usually about 165 that used by the authors in Ref. 3. It is nece§- to 185 kilometers. The parking orbit coast time sary to extend the analysis to include an inter- is usually about fifteen minutes - the time re- mediate boundary condition which specifies the quired to coast from orbit insertion to the first perigee radius of the parking orbit at the end of equator crossing. The third phase places the the ascent. Additionally, the oblate earth model vehicle in a transfer orbit whose apogee and peri- must be added to the variations/ analysis. The gee are over the equator. The apogee altitude is effect of oblateness is not negligible in trajec- about equal to the required altitude for a circular tories to circular synchronous equatorial orbit. synchronous orbit. The transfer orbit coast time Trajectories to that orbit arelong, minimally is about five and one-half hours. The third im- around six hours. Oblateness is the major perturb- pulse, the apogee burn, occurs at apogee of the ing force during most of a trajectory. Because of transfer orbit. Apogee is designed to occur at the the large change in inclination required to perform equator and at the proper altitude for injection the mission, any perturbation in the inclination, into the final orbit. A small part of the inclina- thus increasing or decreasing the amount of plane tion is removed by the second impulse with the re- change required of the propulsion systems, affects mainder being removed by the apogee burn, In this the final mass and should be considered in the conventional method, the final conditions at the analysis. end of each burn are known and the mass can easily be maximized progressively phase by phase if the The trajectories are numerically integrated second and third impulse sizes are unspecified. to incorporate a nonimpulsive vehicle model and to include the effects of oblateness and small thrusts Now suppose that the total impulse of the over long periods of time which cannot be conven- third burn is fixed. Then the transfer orbit must iently treated impulsively. be constructed such that the &V available from the third impulse is exactly that required to place The analysis presented in appendix B requires the vehicle in circular synchronous equatorial the solution of a two point boundary value problem. orbit. If the &V available from a fixed third The solution to the two point boundary value prob- total impulse is less than that required to cir- lem for the circular synchronous orbit problem with cularize and equatorialize the orbit for the mass a fixed apogee burn and parking orbit coast time re- available from a conventional ascent and second quires satisfaction of a minimum of eight final con- impulse, then the trajectory to transfer orbit ditions with an equal number of initial conditions. insertion must be altered to reduce the _V required The number and specific initial and finalconditions of the third impulse. This can be done by reducing are explained in appendix B. the required plane change and the AV required for circularization. For reasons described at length Procedure in the Results and Discussion section, the unconventional trajectory needed to reduce the &V re- A simple Newton Raphson iteration scheme was quired of the apogee motor varies in many respects used to solve the two point boundary value problem. from the conventional profile. The most dramatic This scheme was used successfully with as many as changes are a noncircular parking orbit, nontrivial twelve iteration variables. For further explana- inclination reduction by the second impulse, and a tion of the iteration scheme, see Ref. 5. significantly nonequatorial latitude for the second impulse. As is desired, the changes result in The partial derivatives required for the iter- lowering the &V required of the fixed apogee motor. ation scheme were obtained by integrating the ad- However, in this unconventional profile, the final Joint equations. These were obtained as in Ref. 4. conditions required at the end of the ascent and Solutions were initially obtained by using a spher- second impulse are unknown. They might be deter- ical earth model for the adjoint equations, but it mined by varying those final conditions parametri- was found that including the oblateness terms im- cally until the optimum is obtained. However, proved the convergence properties of the problems. because of the number of variables, this process In some problems of this type, it was found that is clumsy and time consuming. including the oblateness terms was necessary to obtain convergence. It wasdifficult to obtainsolutionsto these ventlonal transfer orbit than the apogee motor can problemsbecauseof thehighdegreeof nonlinearity place in circular synchronous equatorial orbit. of manyof the derivativesaswell asthediffi- Therefore, an unconventional trajectory is required culty of guessingat the initial valuesof the to lower the AV required of the apogee burn. thrust anglein pitch andyawandtheir rates. A techniquewasdevisedto systematicallyproceed An optimum unconventional trajectory was ob- froma simple,easilyconvergedproblemto the tained for the ATS-E mission to circular synchro- final solution. Thistechniqueis describedat nous equatorial orbit. The AV required of the lengthin appendixC. Othertechniques,suchas apogee motor is reduced by decreasing each of the gradientmethods,mightavoidsomeof thediffi- two components which together make up the total &V - culties associatedwith the NewtonRaphsontechni- that needed to circularize the orbit and to reduce que. However,themethoddescribedin the appendix the inclination to zero. The AV for circulariza- is convenient,straightforward,andadequate. tion is.reduced by increasing the horizontal velo- After obtainingonesolution,proceedingto others city at apogee of the transfer orbit without adding in theregionof interest is not difficult. radial velocity. Any radial velocity would have to be removed by the apogee burn. Increasing the Results and Discussion horizontal velocity at a fixed apogee radius is equivalent to raising the perigee radius of the I. Launch Vehicle transfer orbit - thereby decreasing the ellipticity of the transfer orbit. The spacecraft is launched by an Atlas- Centaur, a two-and-a-ha/f stage vehicle. The Atlas The AV required at apogee for reducing th_ is propelled by two booster engines and one sus- inclination to zero is decreased by lowering the talner engine. The booster engines are Jettisoned inclination of the transfer orbit. However, at a predetermined acceleration level. The sus- raising the velocity at apogee increases the AV tainer engine continues to burn (sustainer solo). required for inclination removal at a fixed trans- The Centaur insulation panels and then the payload fer orbit inclination. Therefore, the combination fairing are Jettisoned in this phase. The sus- of the two methods represents a compromise which tainer solo ends at propellant depletion and the is optimized as part of the total problem. Atlas stage is Jettisoned. After about ten seconds, the Centaur engines, burning hydrogen and oxygen, In order to obtain the modified transfer orbit, ignite and burn until the desired parking orbit is the trajectory to insertion into that orbit is reac_ed. During the parking orbit, a hydrogen modified. Most of the inclination reduction is peroxide propulsion system is used to maintain a performed by the second burn near the equator. very small acceleration for propellant retention Only a small part of the inclination change to and for attitude control. At the end of the park- transfer orbit insertion is accomplished in the ing orbit, the _entaur engines burn again until ascent to parking orbit. the proper transfer orbit is achieved. After engine shutdown, the Centaur control system acquires The characteristics of the optimum parking the proper orientation for the spacecraft burn, orbit are changed from the conventional profile to the Centaur and the spacecraft separate, and the increase the perigee radius of the transfer orbit. spacecraft is spun up for stability. The space- An elliptical rather than circular parking orbit craft coasts up to the proper altitude maintaining is used to raise the altitude of the second burn. the separation attitude. The spacecraft motor The perigee radius of the optimum parking orbit burns to place the spacecraft in the final orbit. remains limited by aerodynamic heating considera- The spacecraft apogee motor has thrust of 22 240 tion at some acceptable value. Since injection nevtons and an effective specific impulse of into the parking orbit occurs near perigee, the 279.1 seconds. The total impulse available from vehicle must coast along the ellipse to a higher the motor is 950 900 newton-seconds, which corre- radius. Due to limitation of the coast duration sponds to a propellant load of 347 kilograms. for the ATS-E mission, the Second burn was required to occur near the first equator crossing. (From ii. Tra_ectp.r_ Description tracking or other considerations, a second (or greater) equator crossing could be chosen for the The trajectory starts with a short vertical second burn, which would increase the parking or- rise, followed by a rapid pitchover phase in the bit coast time by a half period (or more)). The desired azimuth direction. The amount of pitch- latitude of the second burn is no longer equatorial over determines the amount of lofting during the as in the cohventional case since the optimum posi- atmospheric portion of the trajectory. The re- tion for raising the perigee radius and decreasing mainder of the atmospheric phase (which is assumed the inclination is dependent on radius and velocity to end at booster stage Jettison), is flown with a as well as latitude. The parking orbit coast time near-zero angle of attack steering program (des- is greater for this unconventional profile since cribed in Ref. 5), to minimize vehicle heating and the time to the equator is greater for an ellipti- aerodynamic loads. The thrust direction is con- cal than for a circular parking orbit and addition- strained to be parallel to the launch azimuth ally, the second burn occurs significantly south plane, which is established at launch. of the equator. Optimum true anomalies are found for the beginning and end of the parking and trans- Since the ATS-E spacecraft motor has a fixed fer orbit coasts. In addition, the optimum com- propellant load, the trajectory must be designed bination of the changes Just described as charac- such that the &V required at apogee of the transfer terizing the unconventional profile is selected. orbit is exactly that required to place the spacecraft in the desired final orbit. As mentioned The desired final inclination for the ATS-E earlier, the ATS-E motor is much smaller than opti- mission is not exactly zero. The perturbations of mum. The Atlas-Centaur can put more mass in a con- the sun, moon, and oblateness of the earth cause a spacecraftto drift fromanexactlyequatorial radius of the transfer orbit decreases as the alti- orbit. Sincezeroinclination is nota stable tude of the second burn decreases. The apogee al- condition,a final orbit inclinationyieldingthe titude of the transfer orbit is almost constant at smallestaverageinclination overthelifetime of synchronous altitude, hence as perigee decreases, thespacecraftis desired. Smallfinal inclina- so does apogee velocity. tions with theproperinertial ascendingnodeare foundto yield acceptableinclination overthe Figures 5 and 6 also indicate that more £V is lifetime of the satellite. Theparticular combi- required of the apogee motor as final inclination nationsof final orbit inclination andascending decreaseS. It can be seen that both the plane nodearefunctionsof thepositionsof the sunand change and circularization AV are increasing. How- moon,whichare in turn functions of launch time ever, Fig. 4 shows that the ignition mass of the and date. Therefore, data were obtained for pay- fixed solid moto_ is decreasing, which increases load to circular synchronous orbits as a function the AV capability of the apogee motor. of final inclination. Negative inclinations are included in the data. This convention indicates Figure 13 shows the percentage of the Centaur that the node has been switched approximately propellant used in the first burn. The figure 180 ° by the apogee burn. shows that as the final inclination increases, the first burn duration increases as the apogee alti- The Atlas-Centaur has a twenty-flve minute tude increases (Fig. 12). limitation on parking orbit coast time for the m_sion. Therefore, inclusion of that constraint The final longitude as a function of final is necessary for realistic determination of vehicle inclination is shown in Fig. 14. It shows that capability. However, optimizing the coast time longitude decreases as final inclination increases. provides a more dramatic and obvious demonstration The satellite remains at the longitude indicated of the optimization procedure. Launch azimuth was only when the inclination is zero, the orbit cir- not optimized along with the other trajectory cular, and the altitude synchronous. For other parameters. The effect of launch azimuth was inclinations, the position (latitude and longitude) investigated parametrically to determine its ef- of the satellite subpoint describes a figure eight fect on separated spacecraft mass. on the surface of the rotating earth. The longitudes indicated in Fig. 14 are injection longitudes, Figure 3 presents separated spacecraft mass not necessarily the longitude at which the equator as a function of launch azimuth for final inclina- crossing occurs. For small inclinations, the lon- tions of (-)2 ° and 5.25 ° for both optimum and gitude does not vary greatly during the period of twenty-five minute parking orbit coast times. the orbit. Separated spacecraft mass is the mass of the spacecraft when it is separated from the Centaur Now consider the limitation of parking orbit vehicle. This figure shows that the separated coast time. Twenty-five minutes is less than spacecraft mass is rather insensitive to launch optimum for all the final inclinations considered, azimuth. Hence, for simplicity, launch azimuth as seen in Fig. 10. The differences in separated is fixed at 90°for the remaining figures. spacecraft mass are shown in Fig. 4. As seen from these figures, as the difference between the optl- Figure 4 shows the separated spacecraft mass mum and limited coast times decreases, the loss in as a function of final inclination. The separated payload due to coast time limitation decreases alsG spacecraft mass decreases as final inclination decreases. Figures 5 and 6 show the effect of The coast time limitation reduces the advan- final inclination on the transfer orbit inclina- tage of raising the apogee of the parking orbit as tion and inertial velocity at apogee. As might final inclination increases. The energy required be expected, as the final inclination decreases, to raise apogee does not yield the payload in- so does the transfer orbit inclination. creases available with optimum coast time since the altitude cannot be acquired as efficiently in As might not be expected, the velocity at the shorter coast time. The energy is better apogee also decreases as final inclination de- spent by the second burn to reduce the inclination creases. Figures 7 through 12 show why this of the transfer orbit. This is reflected in sev- occurs. Figure 7 shows the latitude of the second eral of the figures. In Fig. 5, the transfer or- Centaur engine start as a function of final in- bit inclination for the coast limited case lies clination. Since the second burn is required to well below the optimum case. The lower second remove more inclination as final inclination de- burn altitude is reflected in the lower velocity creases, it is advantageous to move the burn at apogee of the transfer orbit, as seen in Fig. 6. nearer the equator for more efficient plane chang_ Because the parking orbit characteristics do not Figure 8 shows that the longitude of second burn vary greatly with final inclination, the latitude start also decreases as final inclination de- and longitude of the second burn and the true creases. These trends decrease the parking orbit anomaly at second Centaur cut-off are nearly con- coast arc as final inclination decreases. This stant. These may be seen in Figs. 7, 8, and 9. is reflected in a decrease in the true anomaly at The conclusions which may be drawn for the per- second Centaur cut-off, as seen in Fig. 9. Fig- centage of Centaur propellant used in the first ures lO, ll, and 12 also show additional effects burn and final longitude (Figs. 13 and 14), are of moving the second burn nearer the equator. It similar to those for the optimum coast case. decreases the parking orbit coast time, the altl- tude of the second burn, and the apogee altitude Summary of Conclusions of the parking orbit. These all occur as a result of the decrease in parking orbit coast arc. These Analysis and results are presented for tra- figures show why the apogee velocity is decreasing Jectories to circular synchronous equatorial orbit as final inclination decreases. The perigee where the apogee motor is fixed at a smaller than optimumtotal impulse.Theresults for thesmall unit vector pointing at north pole, N.D. apogeemotorcasewereobtainedfor theATS-E mission,whichusedtheAtlas-Centaurlaunchve- mass flow rate, kg/sec hicle. E Jump factor Theresults showsomeof the characteristics of optimumtrajectoriesfor launchvehicle-apogee Lagrange multiplier, kg/sec motorcombinationswheretheapogeemotoris smaller thanoptimum.Moreimportant,theresults Lagrange multiplier, kg-sec/m demonstratethat optimumtrajectoriesto circular synchronousequatorialorbits maybe obtainedwith Lagrange multiplier, kg/m detailedandhencecomplicatedvehiclemodelsfor unconventional(smallapogeemotor)trajectory Lagrange multiplier, N.D. profiles. Theseresults maybeobtainedwithout resortingto exoticmathematicalproceduresfor pitch attitude, keg. solvingthetwopoint boundaryvalueproblem. Theseresults wereobtainedwith a simpleNewton- yaw attitude, deg. Raphsoniteration scheme.Thepartial derivatives wereobtainedby integratingthe adJointequations. Superscripts Thesimpleiteration schemewith the integrated partial derivativesis ableto obtainsolutionsto time derivative thehighly nonlineartwopointboundaryvalue problemevenwhenthenumberof initial andfinal - vector conditionsreachestwelve. A unit vector Appendix A f final

o initial C first integral of Euler-Lagrange equations, kg/sec Subscripts

eccentricity, N.D. i,j,k, g,m,n stage numbers energy per unit mass, m2/sec 2 f final unit thrust direction, N.D. o initial functional defined by equation (3) kg/sec d desired

g intermediate boundary equation pk parking orbit

gravity acceleration, m/sec 2 Operators

GI, G2 components of oblate gravity accelera- dot product tion, m/sec 2

X cross product spherical earth gravity constant, m3/sec 2 differential h angular momentum per unit mass, m2/sec gradient with respect to J functional to be minimized, kg _( ) partial derivative

m _S S, kg Appendix B N total number of stages, N.D. Derivation of Optimum Control P semi-latus rectum, m As mentioned in the Analysis Section, the optimization of a trajectory to a circular synchro- r radius, m nous equatorial orbit may be considered as the problem of optimizing a multi-stage launch vehicle to perigee radius, m rp a particular final orbit. The optimization problem to be considered here begins at booster Jettison, S variational switching function, N.D. which is assumed to be a fixed position and velo- t time, sec city. The sustainer portion of the Atlas continues until propellant depletion. The sustainer is Jet- tisoned and a few seconds later the first Centaur T thrust, N burn begins. Its duration is variable and must be optimized. The perigee radius of the parking orbit V velocity, m/sec is fixed. The duration of the parking orbit X state variable may or may not be optimized. The parking orbit is not a true coast since a small acceleration is maintainedfor propellantretention. Theduration The o_tlmum thrust direction _ is nbtained by com- of thesecondCentaurburnmustbe optimized,fol- bining equations (ld) and (5d) and using the lowedbyanoptimumtransferorbit coastCatrue Welerstrass E-test. This procedure results in coast),anda final burnof fixed total impulse. Theanalysispresentedin this appendixto solve } = _, (6) this problemis a specialcaseof the analysis derivedin Ref.3, with anadditionalconstraint- Integrals of the Motion parkingorbit perigeeradius. Since F does not explicitly depend on time, Thevariationalproblemto be solvedis to an integral of the motion is find the steeringprogramandvariousstagedura- tions whichmaximizethepayloadcapabilityof a multi-stagelaunchvehicleto a specifiedfinal C + k • G + _ • v + --Ti k - o9i = 0 (7) orbit. Thetrajectory mustsatisfy certainini- m tial, final, andintermediateconditionsonthe When a s_herical gravity model is assumed (i.e., statevariables. Thethrust, propellantflowrate, _(r-) _ G_/r 5 r-), three additional integrals of andJettisonweightfor eachstageare assumedto the motion exist which are given by be constant.Theequationsof motionandcon- straints for eachstagemaybewrittenas Xv + _ X r = constant

• Ti ^ Since _, _, r, and _ are all continuous except v -C-(3) -_ f =E (ia) where an intermediate boundary condition is im- posed (as will be shown later), the three integrals are constant across staging points where continuity r - v = 0 (lb) holds. However, for the oblate gravity model used in this analysis, only a single component of the + _i = 0 (ic) above vector integral is constant, as can be verified by differentiation with respect to tim& f" _-i:0 (Id) (_ X v + _ × _) • z = constant (8) where f is the unit thrust direction $n_ G (_) is the oblate?earth gravity acceleration [6), Transversalit_ E_uation which may also be written

^ A A The transversality equation for this problem _(_) = Gl(r , _ • z) r + G2(r , _ • z)z (2) is

(All symbols are defined in appendix A.) Suppose N ti that each stage of the vehicle is numbered con- dJ= i--_ (C dt + _ • dv + _ • d_ + _ dm)t i'l secutively starting with the booster, For analysis purposes a stage change occurs when the thrust and/or propellant flow rate changes and/or a mass - ¢mf (9) is jettisoned. A Bolza formulation of the variational problem is used [7), and the functional to which is set equal to zero for an o__timal solution. be minimized is written as in Hcf. 3 as Reference 3 shows that: i) _ and _ are continuous everywhere if there are no intermediate boundary conditions. If the intermediate boundary condition J = -mf + _N f ti F i dt (3) (assumed to occur at a staging point) is expressed i=2 _i-1 as g(_,_)= o (lO) where the functional F i for each stage is

reference 8 shows that the discontinuities in and _are e_-_ and e_-g, respectively. The variable e is used as an initial condition in the two point boundary value problem to satisfy the inter- + _(i + _i) ÷ _(_ " _ - l) (4) mediate boundary condition (eq. (i0)). 2) The equations that must be satisfied to optimize the The resulting Euier-Lagrange equations are duration of the powered and coast stagesare derived in Ref. 3. The applicable results are presented here. Let j be the first optimized powered • _ ^ ^ stage. Then for constant jettison weight the equation for optimizing stage _ is + G I [ + ([ • r) _rr GI - GI([ " r)_

t-i ^ (s_- s_+I)= o (ill + (% " z) _rr G2 = _ (Sb) i=J T where o and f refer to Initla/ and final values o-_" _:o (Se) and the S functions are defined as

_" G +U " v +-fix ^ T i C Si=l_" a= - _i ' _i _ o (l_a) A si - o, _i _ o (12b) (_gx_+_rgX_O .,--o (17) where the right side of equation (12a) is obtained It will be shown later that equation (17) is by using equation (7). For coasting stages satisfied for all functions g used herein. (_i _ Ti _ O) to be optimized, the equation The calculation of _ proceeds as follows: ci= (l" G+_. v-)=o (i3) (_xv+_x:) • z= (_ xv) • z must be satisfied for maximum payload. 3) For free initial or final state variable x, (_ x • z - (k x ' z = o the required or final condition for maximum payload (Ref. 4) is ?:=: (_x:-_x^ r) "z^ (_ x :) • z (i8)

_+ _ • :ix= o Computing _ with equation (18) guarantees that equation (16a) will be satisfied. Initial Conditions Intermediate Conditions If the initial position and velocity are s?_c!fied__the initial values of any five of the As explained earllerj it is necessary to six h and _ may be used as variable initial constrain the perigee radius at injection into conditions in order to satisfy the required final the first parking orbit. Otherwise, the optimum conditions of the two point boundary value pro- solution would result in the parking orbit injec- blem. In order to eliminate the difficulty asso- tion and/or the equator crossing occurring at very ciated with guessing at values of the multipliers, low altitudes, thus violating spacecraft heating the values of h and _ can be ex_0ressed in te.rms of constraints. Therefore, the intermediate con- pitch and yaw attitude (_ and@) and rates (_ straint is and _). These equations ma_ be fo_und in Ref. 4, appendix C. The values of k and _ are then cal- g(:, 7) = rp - rp, d = 0 (19) culated from: where the desired value corresponds to the perigee altitude. By using equations found in Refo 9, equation (19) can be written as = - _ - _ (lSb)

- rp, d = 0 The value of % can be set equal to unity without i + e (2o) loss of generality. The initial value of _ can be calculated in closed form, as will be shown where by the following development. h • h (semi-latus rectum) (21a) Final Conditions

Final conditions for both the conventional and unconventional synchronous equatorial orbit e = _ l + _ (eccentricity) (2lb) mission require a circular orbit at synchronous Gm orbit altitude with prescribed inclination. If G* the required inclination is non-zero, both the E = E-_ - __ (energy per unit mass) (21c) longitude of the ascending node and the injection point in the final orbit are free for optimization. As shown in Ref. 4, the corresponding h = r x v (angular momentum per unit (21d) auxiliary variational final conditions are m_ss)

(:xv+_ x:) • z=O (i6a) The required gradients are calculated to be and h(_ x r - r2 _) _- g = P (22a) (ixv+_ x_ • Cry7)=0 (iSb) v eG m r 2 If the desired inclination is zero, equations !(Vx£) -_-: (16a) and (16b) degenerate into one equation G * r 2 (zero inclin_tlo_ is equlvalen_ to two final conditions, r " z = 0 and v • z = 0), and only equation (16a) must be satisfied. It is easily shown that equation (17) is satisfied Since equation (16a) is a constant of the for the gradients in equation (22a) and (22b). motion (eq. 8), it may be satisfied at the beginning of the trajectory, and used to calculate In fact, equation (17) is satisfied for ar_ . However, it must first be verified that Jump function g of r, v, h and r • v. For such a discontinuities in :and _ at intermediate boun- function g, dary points do not change the value of the constant. This requires that _v g = v + (h X :) + 8(: • :) It should be recognized that there may be any number of fixed stages between tA and tk, etc. _r ^ _ _g •. V (e3) Also, the last three final conditions are evaluated g = r + (_ x _) + $(r • v) at intermediate points in the trajectory. and Appendix C 5Vvg xv + V_rg x r-- _vV xv Two Point Boundarz Value Problem

+_(hxT) xV+ 3---_---rxv The following technique was devised to syste- _(r" _) matically proceed from a simple, easily converged problem to the solution of the two point boundary value problem for a circular synchronous equatorial orbit.

A trajectory is obtained to a slightly elliptical (parking) orbit with the desired perigee + __ x r = o (24) radius without plane change with a 90 ° launch azi- _(_. v) muth. This problem converges easily. Then the ascent burn time is fixed at the value obtained ilence the value of Z x v + _ x r is unaffected and a variable length parking orbit coast, a fixed by the Jump in _ and _ resulting from a function parking orbit perigee radius and a second burn are g as defined above. added. This problem is targeted to the desired apogee and 180 ° argument of perigee for first Boundar_ Value Problem equator crossing second burn. An inclination decrease of about two degrees is then added to these For the ATS-E mission, both fixed and final conditions and the problem is retargeted to optimum parking orbit coast times were considered. the augmented final conditions. Now the transfer The transfer orbit coast time was always opti- orbit coast (variable) and apogee burn (fixed or mized, however, along with the durations of the variable) are added. This trajectory is integrated first and second Centaur burns. Based on the to the end with the converged initlal guesses from preceding discussion of the transversality equa- the last step. The final conditions achieved will tion, the initial and final conditions for the frequently be far from a circular synchronous equa- two point boundary value problem are as follows torial orbit. However, specify the final condi- for the case where the parking orbit coast time tions actually achieved as the desired ones, and was optimized: optimize the problem. The parking orbit coast, second burn, and transfer orbit coast durations Initial Conditions Final Conditions will change. Now alter the achieved final conditions toward the desired ones Judiciously in steps, E d retargeting at each step. In this manner, the desired final orbit conditions may be obtained. r d Now the ascent burn duration may be optimized. Any sizable change in a constraint or final condi- 5d tion is best achieved by proceeding in steps. The problem is quite nonlinear. Attempts to plot initial conditions as functions of the final condi- rp, d (Parking orbit) tions for extrapolation purposes were made. They

£ (r'z) = 0 were generally unsuccessful.

(v'z) = o References tj (First Centaur Burn)

i. Hoelker, R. F. and Silber, R., "Injection k-i f o tk (Parking Orbit Coast) Schemes for Obtaining a Twenty-Four Hour i_ (Si-Sl+l) = 0 Orbit," Aerospace .Engineerlng, Vol. 20, No. l, Jan. 1961, pp. 28-29, 76-84. t-I (sf_s ° 2. Rider, L., "Characteristic Velocity Require- ti (Second Centaur Burn) Z - i i+l" = 0 i=J ments for Impulsive Thrust TransfersBetween Non Co-Planar Circular Orbits," American t m (Transfer Orbit Coast) (_ " G + _ " v) = 0 Rocket Societ_Jo_nal, Vol. 31, No. S, Mar. 1961, pp. 345-351. (2s)

3. Teren, F. and Spurlock, O. F., "Payload O_ti- If the desired final inclination is non-zero, mization of Multistage Launch Vehicles, then (7"_)and TND-3191, 1966, NASA, Cleveland, Ohio. (v?'_) and=x 0_) are= O.replacedIf the bYpar_ingia (7 x _ + 7, x v) orbit coast time is fixed, then an initial and 4. Teren, F. and Spurlock, O. F., "Optimal Three final condition are removed. These are t k and Dimensional Launch Vehicle Trajectories with Attitude and Attitude Rate Constraints," k-i f o TND-5117, 1969, NASA, Cleveland, Ohio. (Si-Si+l) = 0 i=J 5. Spurloek,0. F. andTeren,F., "A Trajectory 7. Bliss, G. A., Lectures on the Calculus of Codefor MaximizingthePayloadof Multistage Variations, University of Chicago Press, 1946. .... Launch Vehicles," TN D-4729, 1968, NASA, " Cleveland, Ohio. 8. Pontr_agin , L. S., et 8/., The Mathematical : Theory of Optimal Processes, Interscience, i. 6. Clarke, V. C., Jr., "Constants and Related Data New York, 1962. Used in Trajectory Calculations at the Jet Propulsion Laboratory," JPL-TR-32-273, May 9. Dobson, W. F., Huff, V. N., and Zimmerman, A. V., 1982, Jet Propulsion Lab., California Inst. "Elefuents and Parameters of the Osculating 1! Tech., Pasadena, Calif. Orbit and Their Derivatives, TN D-1106, 1962, NASA-Cleveland, Ohio. APOGEEBURN ,NA,,/ / ',,

/ ASCENTTO k \ tO t_. I I PARKING'ORBIT-, / "'-TRA??FEoRAST \

/ PARKING ;_,_ _// / \ ORBITCOAST-"\'_2" /

(PLANECHANGESNOTSHOWN)

Figure l. - Conventional trajectory to circular synchronous equatorial orbit.

EARTHROTATIONCL_

// //ORBIT PLANE" _ TRANSFER\ \ ( PARKING _ ORBIT PLANE \l

\\ ORBIT PLANEJ-\_ l/ \ /

Figure 2. - Circular synchronous equatorial orbit ascent profile. 960 -- 25 MIN COAST OPTIMUM COAST

c_ 920-- _E _m_J_ "'-INCLINATION : 5.25° t,,,, t...) l.n_l (_) 880-- %x It'} ¢C (__ "HNCLINATION: 5.25° tO !

840-- ,,-INC LINATION = -2°

r_ w t_ '_INCLINATION = -2° 8oo I I I I I 84 88 92 96 I00 104 LAUNCH AZIMUTH, DEG

Figure3. - Separatedspacecraftmass asa functionof launchazimuth.

(D 1000-- 25MIN COAST ct-} OPTIMUM COAST _E

< 900-- t-,,, fj O

r, 800-- t-,, LLI

ID_ l.a.l 700 I I I I -L -2 0 2 4 FINAL INCLINATION,DEG

Figure 4. - Separatedspacecraft mass as a function of final inclination. 26 m 25MIN COAST

..... OPTIMUM COAST J J J 24-- f J

ILl f J J z" J O 22-- J J f

Z J .--I f J Ok] Z f J tO 20-- J

mm f I O

Q_ 18-- Z

I--

16--

14 I I I I -Z -2 0 2 4 FINAL INCLINATION,DEG

Figure 5. - Transfer orbit inclination as a function of final inclination.

2100n 25MIN COAST s-" OPTIMUM COAST s s _ ILl s s O C..._ 2000 -- Q_ I._ ,,o s .,mS ,,,,,.SS s w 1900 -- sS" Z

m,," •---I O 1800 W Z

1700 I I I I -4 -2 0 2 4 FINAL INCLINATION,DEG

Figure 6. - Inertial velocity at apogeemotor ignition as a function of final inclination. __

Z i

Z -I0 -- 25MIN COAST 0 - ..... OPTIMUM COAST l..l.J

_C,,,

Z -20 -- ° (-q

I"- I.-- U') ! p-I -50 I I I I -4 -2 0 2 4 FINAL INCLINATION,DEG

Figure 7. - Latitude at second main engine start as a function of final inclination.

70-- -- 25MIN COAST OPTIMUMCOAST 60-- s s re,- s_ S l.- 50-- i

z iJ.J s z s i s .:c 40--

Z 0 ¢0 l.l.J cP) 50--

i---,

i (__ Z 20-- 0

I0 I I I I -L -2 0 2 4 FINAL INCLINATION,DEG

Figure 8. - Longitude at second main engine start as a function of final inclination. 8O

(D Z O

j,,#p _s 6O s,, ,P _s ~~- C..) dpw_' ,._J 25MIN COAST~~

~~.G.1 OPTIMUM COAST 1'O O(..9 ZZ 4O LO _:: ,,, ! ,,'Z~~

~~2O I I I I I -L -2 0 2 4 6 FINAL INCLINATION, DEG~~

~~Figure 9. - True anomaly at second main engine cut-off as a function of final inclination.~~

~~70B~~

~~J 25MIN COAST s s ..... OPTIMUM COAST sJ S Z 60_ t S =E S S J J :E S m I-- jS I-.-- S 50-- d S O S S S S S s J~~

~~O 40-- t..D Z v"~~

~~r_ 30--~~

~~2C I I I I -4 -2 0 2 4 FINAL INCLINATION,DEG~~

Figure 10. - Parking orbit coast time as a function of final inclination. 12000 -- 25MIN COAST ..... OPTIMUM COAST / S P S r_ I0000 -- S S S S i,1 $ g S m (.9 S Z / ILl 8 000 -- S S g E S S J J S S Z S • Oa O 6 000 ['O S S ILl

~~! s S I---~~

~~4000--~~

~~I-- .-.I~~

~~2 000 I I I I -4 -2 0 2 4 FINALINCLINATION,DEG~~

~~FigureII.- Altitudeatsecondmain enginestartas a functionoffinalinclination.~~

~~16000 j#,J 25MIN COAST S S ..... OPTIMUM COAST S S~~

~~S 14000 S S S S S S ,,-,, / 12000 I S J S $ ILl S I,I S S O S m, I0 000 S S S S S ,,',m, S m," O (.,9 8 000 g~~

~~,"w"~~

~~(=!- 6 000~~

~~4 000 -t -2 0 2 4 6 FINAL INCLINATION,DEG~~

~~Figure 12. - Parking orbit apogeealtitude as a function of final inclination. m,.,, 100-- 25MIN COAST OPTIMUM COAST~~

~~o___ 0 -- ILl LI- .,i,..-. "J 1'O b- If) ,-_.z ! Z p_ 80 -4 -2 0 2 4 FINAL INCLINATION, DEG~~

~~Figure 13. - Percentage of Centaur propellant burned during first burn as a function of final inclination.~~

~~100--~~

~~¢D ILl~~

~~Z O ..-I 80 m 25MIN COAST Z u OPTIMUM COAST~~

~~70 I I I I I -4 -2 0 2 4 FINAL INCLINATION, DEG~~

~~Figure 14. - Final longitude as a function of final inclination.~~

~~NASA-Lewis~~