The 1st John V. Breakwell Memorial Lecture

Pergamon PII:

Acto Astronautica Vol. 38, No. 10, pp. 747-168, 1996 Copyright 0 1996Elsevier Science Ltd Printed in Great Britain. All rights reserved SOOW-5765(%)00076-8 0094-5765/96 $15.00+0.00

THE 1st JOHN V. BREAKWELL

MEMORIAL

LECTURE

RECENT ADVANCES IN THE OPTIMIZATION AND GUIDANCE AEROASSISTED ORBITAL TRANSFERSI_$

OF

A. MIELE$ Aero-Astronautics

Group, MS-322, Rice University, Houston, TX 77251-1892, U.S.A. (Received 12 January 1996)

Abstract-Aeroassisted orbital transfer (AOT) combines space maneuvers and atmospheric maneuvers so as to achieve propellant savings vis-&is purely propulsive maneuvers, albeit at the expense of high heating rates during the atmospheric pass. Of critical importance is the study of optimal trajectories; equally important is the development of guidance schemes in view of the precision requirements of AOT trajectories and the need for coping with various sources of dispersion effects. In this lecture, optimal trajectories are reviewed with reference to angle of bank control. Trajectories are optimized. from the point of view of performance and control margin. Compromise trajectories are generated from a trade-off between performance and control margin. The ultimate goal is to arrive at practical AOT guidance trajectories exhibiting many of the desirable characteristics of AOT optimal trajectories. Guidance advances are reviewed with reference to angle of bank control, in particular, safety zone exit guidance and complete entry/exit guidance. Continuation of research efforts in the area of guidance is important, indeed essential, for the future development of AOT vehicles. In particular, the use of parallel computation for real-time guidance is worthy of investigation. Copyright 0 1996 Elsevier Science Ltd

DEDICATION I first met John V. Breakwell in 1956. At that time, I was consultant to the Guided Missiles Division of Douglas Aircraft Company and used to travel to

Santa Monica, California several times a year. During a visit, my friends at Douglas Aircraft asked me to make a presentation at the Los Angeles Section of the Institute of the Aeronautical Sciences. It was on that occasion that I met John. This was the beginning of an amicable and rewarding professional relationship that spanned 35 years. This relationship continued in time at meetings in the U.S.A. and all over the world, and it intensified through our joint editorial work on the Journal of Optimization Theory and Applications. It made me acutely aware of his enormous intelligence and his unusual way of looking at mathematical and engineering problems. In spite of his genial and extroverted participation at social gatherings, my perception of John was and remains that he was basically a shy and reserved person. Therefore, I was totally surprised, one tPaper IAF-94-A.2.010 presented at the 45th International Astronautical Congress, Jerusalem, Israel, 9-14 October

1994. IThis research was supported by Jet Propulsion Laboratory, NASA Marshall Snace Fliaht Center. Boeing Military Airplane Company, and Tixas Advanced Tichnology Program. $A. J. Foyt Professor Emeritus of Engineering, Aerospace Sciences, and Mathematical Sciences.

afternoon at the 1982 IAF Congress in Paris, when John grabbed me by the arm and informed me of his problems and fears. This was at the beginning of his long fight with cancer that ultimately caused his death nine years later in 1991. The 1982 episode in Paris has remained clearly etched in my mind for a most personal reason. One year later, in 1983, I had my own cancer scare, which fortunately proved to be unfounded. While laboratory tests were being made over several weeks, reminding me daily of my own mortality, I realized that the only way to cope with the resulting mental pressure was to do exactly the same as John did in that 1982 afternoon in Paris: to talk with trusted friends about my problems and fears. This work is lovingly dedicated to the memory of my friend John V. Breakwell, world-class scientist, distinguished applied mathematician, and foremost aerospace engineer. 1. INTRODUCTION

The field of hypervelocity flight covers the problems arising from the motion of bodies at extremely high velocities in the sensible atmosphere. The speeds considered are much larger than the speed of sound (hence, hypersonic); they extend to the upper hypersonic regime and include subcircular, circular, and supercircular velocities. The fluid regimes covered include the continuous flow regime, 747

748

A. Miele

free-molecular flow regime, and intermediate transition regime. Historically speaking, the field of hypervelocity flight was initiated in Europe with the important work of Sanger and Bredt[l]. It was continued in the U.S.A. with the fundamental studies of Allen and Eggers[Z], Eggers, Allen, and Neice[3], and Chapman [4-51. See also the textbooks by Miele [6, Chapters 13-141, Loh[7], and Vinh, Busemann, and Culp[8]. For more recent developments, see the survey paper by Walberg[9] and the special issue edited by Miele[lO]. Examples of past applications of hypervelocity flight concepts include ballistic missiles, orbiting capsules (Mercury and Gemini), lunar return vehicles (Apollo), and more recently the Space Shuttle. Examples of future applications of hypervelocity flight concepts include the European version of the Space Shuttle (Hermes), vehicles for transportation between orbits (aeroassisted orbital transfer vehicles, AOTV), vehicles servicing the Space Station (crew emergency return vehicles, CERV), vehicles returning from lunar missions, vehicles for Mars missions, and transatmospheric vehicles (national aerospace plane, NASP). 1.1. Aeroassisted orbital transfer

A particular case of hypervelocity flight is that of aeroassisted orbital transfer (AOT). It consists of the combination of propulsive maneuvers in space and aerodynamic maneuvers in the atmosphere with the intent of achieving propellant savings us-ci-yis purely propulsive maneuvers; hence, the alternative name of synergetic maneuvers. AOT maneuvers can be used in Mars exploration/return vehicles, lunar return vehicles, and also near-Earth vehicles. Historically speaking, it appears that the field of aeroassisted orbital transfer was initiated by London with a paper dealing with the plane change problem[l 11. Among early contributions, we mention a paper by Rossler, also dealing with the plane change problem[l2]. But aeroassisted orbital transfer can also be used advantageously in planar or nearly planar flight from a high Earth orbit (HEO) to a low Earth orbit (LEO). Indeed, it is known that, for a GEO-to-LEO planar or nearly planar transfer (GE0 stands for geosynchronous [13] Earth orbit), AOT maneuvers yield about 60% propellant savings vis-his Hohmann transfer maneuvers. Clearly, for a properly designed AOT vehicle, there are potential benefits in propellant savings and payload increase, albeit at the expense of high heating rates during the atmospheric pass and complex guidance schemes. The guidance complexity is due to the superposition of the following factors: (a) precision requirements arising from the planned rendezvous with other vehicles, such as the Space Shuttle or the Space Station; (b) severe environmental conditions, specifically the need for coping with dispersion effects due to navigation errors, atmos-

pheric density errors, and aerodynamic coefficient errors; (c) the fact that the control resources of AOT vehicles are limited: if the angle of attack is held constant at the trim position, the only control is the bank angle. Over the past two decades, there has been a considerable effort in AOT analyses for both the planar and nonplanar cases. In the U.S.A, this effort has involved a number of laboratories (NASAJohnson, NASA-Langley, NASA-Marshall, Jet Propulsion Laboratory, Draper Laboratory) and universities (Georgia Tech, Princeton University, Rice University, University of Texas at Austin, University of California at Irvine, University of Michigan). This has led to a considerable literature on the subject and related topics[14-991, which is impossible to review in its totality. Given the above premises, this paper is centered on the AOT research done by the Aero-Astronautics Group of Rice University[68-991. However, for completeness, an extensive bibliography has been included concerning the work done at other organizations[l4-671. 1.2. Outline For simplicity, this paper treats only planar or nearly-planar flight. In Section 2, starting from Hohmann transfer considerations, we supply a justification of the potential advantages accrued via AOT techniques. In Section 3, recognizing the fact that an Earth-fixed system is useful for the atmospheric portion of the trajectory and an inertial system is useful for the space portion of the trajectory, we supply equations of motion and associated boundary conditions. In Section 4, we introduce decomposition techniques decoupling the complete system of order six into two subsystems of order three: the longitudinal motion subsystem and the lateral motion subsystem. This is useful in guidance. In Section 5, we review the optimization criteria: performance, control margin, or compromise between performance and control margin. In Section 6, we review the work done on optimal trajectories. This leads to segmented trajectories including subarcs along which the bank angle is constant. The five-subarc sequence of the compromise optimal trajectory becomes the nominal trajectory for guidance. In Sections 7 and 8, we present predictor-corrector algorithms useful for the guidance problem. Starting from the optimal trajectory properties, we subdivide the complete guidance into an entry phase and an exit phase, separated by the switch time. In the entry phase, the main goal is performance; in the exit phase, the main goal is controllability. The best switch time is obtained by minimizing a quadratic functional requiring the guidance trajectory to be close to the optimal trajectory from the viewpoints of both performance and controllability. With the

Recent advances

switch time known, the exit phase makes use of safety zone guidance; here, the intent is to satisfy the correct boundary conditions at atmospheric exit, while completing the atmospheric pass with maximum controllability. Numerical examples show the robustness of safety zone guidance and complete entry/exit guidance vis-ci-vis dispersion effects due to navigation errors, system errors, atmospheric density errors, and aerodynamic coefficient errors. Section 9 contains some closing considerations plus recommendations for continuing AOT research. Appendix A contains the transformations leading from Earth-fixed quantities to inertial quantities, and vice versa. Append:ix B contains the relations leading from inertial quantities to some derived quantities (orbital inclination, longitude of the ascending node) and further derived quantities (out-of-plane position angle, out-of-plane velocity angle, wedge angle). Finally, Appendix C summarizes the data used in the numerical experiments.

We present a simple justification of the potential advantages of aeroassisted orbital transfer, starting from Hohmann transfer considerations. We refer to planar or nearly-planar IIight and consider three cases: (i) descending transfer; (ii) ascending transfer; (iii) combined descending/ascending transfer. We employ the following assumptions: (Al) the terminal orbits are circular and coplanar; (A2) the gravitational field is central and is governed by the inverse square law; (A3) velocity impulses are applied at the apogee/perigee points of the transfer trajectories. We introduce two reference velocities. One is a constant reference velocity, the circular velocity at r = r,, r, = r, + h,,

(14

where pc is the Earth gravitational constant, r, the Earth radius, r, the radius of the outer edge of the atmosphere, and h, the thickness of the atmosphere. The other is a variable reference velocity, the circular velocity associated with any radius r, VC=,/;;;7;,

r=r,+h,

certain basic features of a Hohmann aeroassisted orbital transfer.

transfer and an

2.1. Descending transfer First, we consider the transfer of a spacecraft from a high Earth orbit (HEO) of radius rm to a low Earth orbit (LEO) of radius rll (Fig. 1). Under Assumptions (Al)-(A3), the resulting space maneuver is called Hohmann transfer; it is characterized by an elliptical transfer trajectory 00 4 11, which is bitangent to HE0 and LEO. Indeed, points 00 and 11 are the apogee and perigee of the transfer trajectory. The total characteristic velocity and its components are given by (3a)

A&, = I’.,/&&

- -1,

ASH= V&&r,lr,

(W

-

11,

r0l

=

Ok0

+

rd/2

AP = 3.881 km/set,

(4a)

Avm = 1.464 km/set,

(4b)

ApI, = 2.417 km/set,

(4c)

(lb)

(2a)

where V. is given by (la), or via u’ = T//vE,

(2b)

where V, is given by (lb). If the normalization (2b) is used, we call the instantaneous velocity subcircular if u’< 1, circular if li = 1, and supercircular if t2 > 1. This terminology is of considerable help in describing

(34

is the arithmetic average of rm and rlI . Both A rw and A piI are braking velocity impulses. Because of the braking impulse (3b), the velocity at rw changes from circular to subcircular. As the spacecraft descends, the velocity increases becoming circular at rorand supercircular at rll. Finally, because of the braking impulse (3c), the velocity at rll changes from supercircular to circular. As an example, consider a HEO-LEO Hohmann transfer with boo= 35,786 km (hence, HE0 = GEO) and h,, = 330 km. For rm = 42,164kn-i and rll = 6708 km, use of eqns (3) yields

where h is the altitude above the sea level. Note that V, reduces to V. for r = r,. Let the tilde superscript denote a quantity evaluated in an inertial system. If P is the spacecraft velocity, it can be normalized in one of the two ways, either via z?= P/v.,

(3c)

where V. is the reference velocity (la) and

2. WHY AEROASSISTED ORBITAL TRANSFER

V* = ym,

749

in the optimization and guidance of orbital transfers

Fig. 1. HE-LEO

Hohmann transfer.

750

A. Miele implying that 62.3% of the characteristic velocity is expended in the accelerating action at LEO and 37.7% is expended in the accelerating action at HEO. Note that the characteristic velocity of the ascending transfer is the same as that of the descending transfer. 2.3. Descending/ascending transfer HE0

Fig. 2. LE&HEO Hohmann transfer. implying that 37.7% of the characteristic velocity is expended in the braking action at HE0 and 62.3% is expended in the braking action at LEO. 2.2. Ascending transfer Next, we consider the complementary case of a spacecraft being transferred from a low Earth orbit (LEO) of radius rW to a high Earth orbit (HEO) of radius r,, (Fig. 2). Under Assumptions (Al)-(A3), the resulting space maneuver is also a Hohmann transfer; it is characterized by an elliptical transfer trajectory 00 + 11, which is bitangent to LEO and HEO. Indeed, points 00 and 11 are the perigee and apogee of the transfer trajectory. The total characteristic velocity and its components are given by AP = A~w + API,,

Next, we consider the transfer of a spacecraft from a high Earth orbit (HEO) of radius rw to a low Earth orbit of radius r,, via a sub-low Earth orbit (SLEO) of radius r22(Fig. 3). Under Assumptions (Al)-(A3), the resulting space maneuver involves two Hohmann transfers: a descending transfer 00 + 22, which is bitangent to HE0 and SLEO, and an ascending transfer 22 + 11, which is bitangent to SLEO and LEO. These are incomplete Hohmann transfers in that the arrival of SLEO is not followed by circularization and the departure from SLEO is not preceded by circularization; the velocity impulse at SLEO merely reduces the supercircular arrival velocity of the descending transfer to the less supercircular departure velocity of the ascending transfer. For the descending branch, the points 00 and 22 are the apogee and perigee of the transfer trajectory; for the ascending branch, the points 22 and 11 are the perigee and apogee of the transfer trajectory. The total characteristic velocity and its components are given by (7a)

(5a)

(7d)

API,= V*&[l

- -1,

(54

where

where r02

r01= (ro0+ r11)/2.

=

Oh0

+

rd2,

(74

(5d)

Both ApW and AT,, are accelerating velocity impulses. Because of the accelerating impulse (5b), the velocity at roechanges from circular to supercircular. As the spacecraft ascends, the velocity decreases becoming circular at rol and subcircular at rll . Finally, because of the accelerating impulse (SC), the velocity at rll changes from subcircular to circular. As an example, consider a LEO-HE0 transfer with boo= 330 km and h,, = 35,786 km (hence, HE0 = GEO). For roe= 6708 km and rll = 42,164 km, use of eqns (5) yields AP = 3.881 km/set,

(6a)

ApW = 2.417 km/set,

(6b)

A p,, = 1.464 km/set,

(6~)

r21 =

G-22 + rd/2.

Fig. 3. HESSLEO-LEO

Hohmann transfer.

(7f)

Recent advances in the optimization and guidance of orbital transfers

Note that AvW and Arzz are braking velocity impulses, while A PI;, is an accelerating velocity impulse. Because of the braking impulse (7b), the velocity at r00 changes from circular to subcircular. As the spacecraft descends, the velocity increases becoming circular at ro2and supercircular at r22. Because of the braking impulse (7c), the velocity at r22changes from supercircular to a less supercircular value. Subsequently, as the spacecraft ascends, the velocity decreases, becoming circular at rzl and subcircular at rll . Finally, because of the accelerating impulse (7d), the velocity at rll changes from subcircular to circular. As an example, consider a HEO-SLEO-LEO transfer with boo= 35,786 km (hence, HE0 = GEO), ha = 75 km, h,, ==330 km. For rm = 42,164 km, rD = 6453 km, rll := 6708 km, use of eqns (7) yields AP = 3.981 km/set,

(8a)

A VW= 1.490 km/set,

(8b)

AT== 2.416 km/set,

(8~)

Atf , = 0.075 km/set.

(8d)

Comparison of the results (4) and (8) shows that: (i) in terms of the total characteristic velocity, an indirect HEO-LEO transfer via SLEO is slightly more expensive than a direct HEO-LEO transfer; (ii) the arrival at LIE0 occurs with nearly-circular velocity, so that only a minor velocity impulse is needed; (iii) the largest velocity impulse is Ar22, which takes place at SLEO and is nearly 60.7% of the total velocity impulse. Assume now that the altitudes ho0and h,, are fixed at the previous HE0 and LEO values, and consider various SLEO altitudes h22in the range 0 Q /zn < h,, . Use of eqns (7) allows one to compute the characteristic velocity and its components as functions of h22. These functions are shown in Fig. 4, which is valid in the absence of a planetary atmosphere. This figure suggests that, in the absence of a planetary atmosphere, the observations (i) to (iii) made for hZ2= 75 km are qualitatively valid for any value of h22in the range 0 < hz2& h,,. 2.4. Aeroassisted

orbital transfer

Next, we consideer the presence of a planetary atmosphere and examine the following question: can the aerodynamic forces be employed to slow down the spacecr.aft in such a way that the concentrated velocity impulse at SLEO is no longer needed? In other words, can the aerodynamic forces be employed to generate a distributed velocity change equivalent to the concentrated velocity impulse at SLEO? To answer the above question, let h. = 120 km denote the thickness of the atmosphere, and let us AA38/10--8

751

HEO+SLE!O+LEO

IO

b22DonI Fig. 4. HEWSLEO-LEO Holunann transfer:characteristic velocity and its components versus SLEO altitude.

subdivide the interval 0 < h22< h,, into three smaller ranges, 1.0 < h&

< 2.75,

(9a)

0.5

<

(9b)

<

h22lk

1.0,

0.0< hn/ha < 0.5

(9c)

In the range (9a), 120 < hz2d 330 km, the air density is too small; it is physically impossible to generate aerodynamic forces sufficiently intense to adequately slow down the spacecraft. In the range (SC), 0 < hu & 60 km, the air density is too large; because the traversing of the planetary atmosphere is done at supercircular velocities, this leads to undesirable values of the peak dynamic pressure and peak heating rate. However, in the range (9b), 60 < h22< 120 km, an engineering compromise can be achieved in the following sense: aerodynamic forces sufficiently intense can be generated so that the concentrated velocity impulse Av22 can be replaced by a distributed velocity change, while simultaneously avoiding excessive values of the peak dynamic pressure and peak heating rate. See the shaded area in Fig. 4. Because the atmospheric pass is done at supercircular velocities, special aerodynamic designs are needed; indeed, to avoid an early skipping from the atmosphere, negative lift must be generated. One such design is the aeroassisted flight experiment (AFE) spacecraft, which generates a negative lift coefficient CL = - 0.381 at the trim angle of attack a = 17.0 deg (Fig. 5). For the AFE spacecraft and for the terminal conditions of Section 2.1, a typical AOT trajectory is qualitatively shown in Fig. 6, where 0 and 1 denote the atmospheric entry and exit points. The

A. Miele

752

Fig. 5. AFE spacecraft, raked cone configuration. characteristic velocity and its components by A P = 1.562 km/set,

are given (lOa)

Arm = 1.490 km/set,

(lob)

API, = 0.072 km/set.

(1Oc)

Comparison of the results of (4) and (10) shows that, vis-d-vis Hohmann transfer techniques, the saving in characteristic velocity due to the use of AOT techniques is 59.7%. This value is close to that predicted in Section 2.3 (60.7%) even though in that section the presence of a planetary atmosphere was disregarded!

3. SYSTEM DESCRIPTION

The motion of the AOT spacecraft takes place partly in space and partly in the atmosphere. For the purposes of this paper, the trajectory begins at HE0 and ends at LEO. It includes a preatmospheric branch, an atmospheric branch, and a postatmospheric branch. We assume that HE0 and LEO are coplanar circular orbits. The key points of the maneuver are these: point 00, exit from HEO; point 0, atmospheric entry; point 1, atmospheric exit; point 11, entry into LEO. Point 00 is the apogee of the preatmospheric transfer orbit OO+ 0; point 11 is the apogee of postatmospheric

transfer orbit 1 + 11. Propulsive impulses are applied twice: a braking impulse at point 00 to deorbit from HE0 and an accelerating impulse at point 11 to circularize the motion into LEO. See Fig. 6. For the atmospheric portion (h < h,) of the trajectory of the AOT spacecraft, we employ an Earth-fixed system; for the space portion (h > h,) of the trajectory, we employ an inertial system: here h, = 120 km denotes the thickness of the atmosphere. For h < h,, we compute the air density using the 1976 US Standard Atmosphere[l9]; for h > h,, we assume that the air density is zero. For both the atmospheric and space portions of the trajectory, we neglect the effects due to the oblateness of the Earth; we assume that the gravitational field is central and obeys the inverse square law. With reference to the atmospheric portion of the trajectory of the AOT vehicle, the following additional hypotheses are employed: (Bl) the atmospheric pass is made with the engine shut off; hence, the AOT spacecraft behaves as a particle of constant mass; (B2) under extreme hypersonic conditions, the dependence of the aerodynamic coefficients on the Mach number and Reynolds number is disregarded; (B3) the sideslip angle is zero; hence, the side force component of the aerodynamic force is zero; (B4) the angle of attack is constant; (B5) the AOT spacecraft is controlled via the angle of bank. 3.1. Dt@erential system With the above assumptions and upon using an Earth-tied system, the equations of motion[83, 841 include the kinematical equations for the longitude 0, latitude 4, and radius r, I = V cos y cos x/r cos 4,

(114

4 = - V cos y sin X]r,

(lib)

i=

(llc)

Vsiny,

and the dynamical equations for the velocity V, path inclination y, and heading angle x, p=

-D/m-gsiny + 02r(sin y cos24 + cos y sin x cos 4 sin f$),

(1W

*j= (L/m V)cos p + (V/r - g/ V)cos y + 2w cos x cos 6

+ (w’r/V)(cos y cos “C#J -sin y sin x cos 4 sin $J),

Wb)

t = (L/mV)sin p/cos y + (V/r)cos y cos x tan 4 + 2 o(sin (b + tan y sin x cos 4) Fig. 6. HEO-LEO aeroassisted orbital transfer.

+ (o’r/V)cos x cos f#~sin 4/cos y.

(12c)

Recent advances in the optimization and guidance of orbital transfers In the dynamical equations, D is the drag, L the lift, m the spacecraft mass, p the bank angle, w the Earth angular velocity; terms linear in w are due to the Coriolis acceleration; terms quadratic in w are due to the transport acceleration. Also in the dynamical equations, the local acceleration of gravity is given by

must be consistent with eqn (15b), where V. denotes a reference velocity, the circular velocity at r = r, [see eqn (WI.

If the entry path inclination conditions take the form r0

g = &r2,

=

is fixed, the entry

r.,

(164

given,

(L6b)

Wa)

where H denotes the Earth gravitational constant. In addition, the aerodynamic forces are given by D = (1/2)Co(z)p(h)SY2,

W)

L = (1/2)CL(a)p(h)SVZ,

(13c)

where CDis the drag coefficient, Cr the lift coefficient, S a reference area, and p the air density. The aerodynamic coefficients depend on the angle of attack a (hence, they are constant if a is constant); in turn, the air density depends on the altitude h, with h=r-r,,

(134

where r, is the Earth radius. For given initial conditions and controls a = const and p = p(f), eqns 1111)and (12) subject to eqns (13) can be integrated forward in time over the interval 0 < t ,< r. Here, the initial time I = 0 corresponds to atmospheric entry and the final time I = r corresponds to atmospheric exit. 3.2. Inequality constraint The bank angle time rate is subject to the inequality -A
753

+A,

(14)

VO =

j$ = given,

(1W

& = given,

(16d)

& = given,

(16e)

,& = given.

(16f)

Note that the values chosen for p0 and j$, must be. consistent with (15b). Regardless of whether the entry conditions have the form (15) or (16), the fact that the longitude, latitude. and heading angle are prescribed implies that the orbital inclination i0 and longitude of ascending node & are given. Also, by definition, 6b = & = t&l= 0,

(17)

where 6, L, TVdenote the out-of-plane position angle, out-of-plane velocity angle, and wedge angle. For the computation of i, R, 6, 6, q, see Appendix B. 3.4. Exit conditions For the AOT vehicle, the desired exit conditions are given by

where A is a prescribed constant bound.

rl = r.,

3.3. Entry conditions Two main cases are of interest, depending on whether the entry path inclination is free or given. In both cases, an inertial system is employed: for the transformations leading from the Earth-fixed system to the inertial system, see Appendix A. If the entry path inclination is free, the entry conditions take the form r0 = r.,

(L5a)

2(r& - roor.)Vt -t-(ri cos 70 - rj,,) G = 0,

(15b)

$ = given,

(15c)

f#~~ = given,

(15d)

;iO= given.

(15e)

Equation (15a) states that the radial distance at entry equals the value corresponding to the outer edge of the atmosphere. Equation (15b) arises from energy conservation and angular momentum conservation applied to the preatmospheric transfer orbit 00 --, 0 connecting HE0 with atmospheric entry; while po and $, are not prescribed individually, their values

2(rf, - r,lr.)Vt+(r~cos2$

(184 -rf,)c=O,

i, = &,

(18b) (W

Clearly, the state variable rl is given; the state variables p, and y, must be chosen consistently with (18b); and the state variables 8,, c$,,2, must be chosen consistently with (1%~) and (18d). See Appendix B. Equation (18b) arises from energy conservation and angular momentum conservation applied to the postatmospheric transfer orbit 1 --* 11 connecting the atmospheric exit with LEO. Equations (18~) and (18d) imply that the exit orbital plane is identical with the entry orbital plane; hence, s, = i, = q, = 0.

(19)

3.5. Summary The atmospheric pass is governed by the differential system (11) and (12), the inequality constraint (14). the entry conditions (15) or (16), and the exit

A.

154

conditions (18). In the boundary conditions, the inertial quantities are related to the Earth-fixed quantities via eqns (68) and (69), Appendix A; also, the inertial quantities are related to the orbital elements via eqns (72), Appendix B, and to the vehicle instantaneous position, velocity, and orbital plane relative to the entry orbital plane via eqns (76) Appendix B. In this formulation, the independent variable is the time t, 0 < t < T. The dependent variables include six state variables [e(t), 4(t), r(t), V(t), y(t), x(t)], one control variable [p(r)], and one parameter [T]. Note that 7 represents the duration of the atmospheric pass.

or the form r0

=

ra,

(244

V.

=

given,

(24b)

y. = given.

(24c)

Equations (23) hold if the entry path inclination y. is free; eqns (24) hold if the entry path inclination if fixed, with the provision that the given values of V0 and y. be chosen consistently with (23b). For both cases, the exit conditions take the form rl - ra = 0, 2(r:, - rllr.)V? + 2(r,’ - rf,)w,V,

4. REDUCED SYSTEM DESCRWMON

(254 cos yI cos i.

+ (r,‘cos *y, - rf,)fl

For an AOT vehicle, the conceptual understanding of the trajectory properties can be enhanced via decomposition techniques decoupling the longitudinal motion from the lateral motion. The decoupling is possible if the following approximations are introduced in the equations of motion (11) and (12): ig iO,

Miele

(204

slCz&,

(2Ob)

cl9 z 0.

(2Oc)

= 0.

(25b)

Equations (23b) and (25b) are a restatement of eqns (15b) and (18b) after transforming inertial quantities into Earth-fixed quantities via eqns (69), Appendix A, and accounting for the approximations (20). 4.2. Lateral motion In the Earth-fixed system, the subsystem governing the lateral motion is given by tj = V cos y cos x/r cos 4,

Approximations (20a) and (20b) mean that the instantaneous orbital plane is nearly identical with the entry orbital plane and imply that

(264

q5 = - V cos y sin X/r,

(26b)

i = (L/m V)sin p/cos y + (V/r)cos y cos x tan f#~

+ 2o(sin 4 + tan y sin x cos d), cos i0 z cos q5cos x,

(214

sin(6 - 4) z cot &tan 4.

(21b)

and includes the state variables e(t), 4(t), x(t). For this subsystem, the entry conditions are given by

Approximation (2Oc) means that terms quadratic in o (transport acceleration terms) are negligible with respect to terms linear in w (Coriolis acceleration terms) and terms not containing o. 4.1. Longitudinal motion

e. = given,

(27a)

+. = given,

(27b)

,y0= given,

(27~)

and the exit conditions

In the Earth-fixed system, the subsystem governing the longitudinal motion is given by i=

Vsiny,

pi= - D/m - g sin y,

(22a) (22b)

3 = (LlmV)cos p

(26~)

are given by [see (2111

cos i. = cos 0, cos xl,

VW

sin(0, - a,) = cot i0 tan 4,.

(28b)

The precision of the exit conditions can be enhanced if inertial quantities replace Earth-fixed quantities in (28). 4.3. Remark

+ (V/r - g/V)cos y + 2w cos i0,

(22~)

and includes the state variables r(r), V(t), y(t). For this subsystem, the entry conditions take either the form r. - r. = 0, (2W 2(& - hr.) V? + 2(ra - r&)0x0 V, cos y. cos i0 + (r.’ cos *yo- r&,)G = 0,

(23b)

For the sake of discussion, assume that the entry conditions are prescribed for the subsystems (22) and (26). Assume that the parameter 7 is given and that the control r(r) is prescribed over the time interval 0 d f d 7. Then, the subsystem (22) can be integrated forward in time independently of (26) to yield the state variables of longitudinal motion. Subsequently, the subsystem (26) can be integrated forward in time to yield the state variables of lateral motion.

Recent advances in the optimizatio In and guidance of orbital transfers 4.4. Stability analysis

Having decoupled the system (11) and (12) into the longitudinal motion subsystem (22) and the lateral motion subsystem (26), a stability analysis is facilitated by the fact that each subsystem has order three. For longitudinal motion, one must compute the Jacobian matrix containing the partial derivatives of the right-hand sides of eqns (22) with respect to the triad r, V, y. For lateral motion, one must compute the Jacobian matrix containing the partial derivatives of the right-hand sides of eqns (26) with respect to triad 0, 4, x. With the Jacobian matrices known, the associated eigenvalues can be computed. The computation must be done at various points of the interval of integration, assuming that the entry conditions are given and that the control p(t) is prescribed over the time interval 0 < t < r. 4.5. Stability results

With reference to the AFE spacecraft (see Appendix C), a stability analysis was carried out particular entry assuming conditions ( p0 = 10.308 km/set, A = - 4.130 deg) and further assuming that the angle of bank is set at p(t) = 5.14deg; see eqn (37) Section 6. For the longitudinal motion subsystem (22), the largest eigenvalue real part was nearly +25, which is indicative of strong intrinsic instability. For the lateral motion subsystem (26) the real part of each eigenvalue was almost zero, which is indicative of almost neutral stability. Of course, the above computation was made assuming a particular scaling for the variables elf the problem and assuming that the control function p(t) is fixed. While stability can be artificially induced via feedback control, the effectiveness of a feedback control scheme depends on control margin iavailability (see Section 5); in turn, this enables the spacecraft to cope with dispersion effects due to various sources of error: navigation errors, system errors, atmospheric density errors, and aerodynamic coefficient errors (see Sections 7 and 8).

755

of a reference value. The associated performance index is given by PI = Ar/Ar..,

where Ap is the characteristic trajectory, _ _

(29a) velocity of the AOT

AV = AV, + AV,,,

(29W

Avm = V*&

- (r,/roo)pOcos yO,

(29~)

Ap,, = V.&K

- (r&I) VI cos 5,

(294

and ATo is the characteristic velocity of the Hohmann transfer trajectory, given by eqns (3) of Section 2. The constants in eqns (29) are r, = 6498 km, V. = 7.832 km/set. Peak heating rate

An alternative to (29) is the peak value of the heating rate, which can be normalized in terms of a reference value. Therefore, PI = max (HRIHR.),

0 G t < z,

t

(3Oa)

where HR = C, m(

V/ V.)3.07,

HR. = C,.

(3Ob) (3Oc)

In eqns (30) based on a nose radius of 30.5 cm, HR is the stagnation point heating rate for any altitude-velocity pair; HR. is a reference value of HR, corresponding to h = h. and V = V.. The constants in eqns (30) are h. = hJ2 = 60 km, pa = 0.3097E-03 kg/m’, V* = 7.832 km/set, C, = 0.2832 kW/cm?. Peak dynamic pressure

A further alternative to (29) or (30) is the peak value of the dynamic pressure, which can be normalized in terms of a reference value. Therefore, PI = max (DPIDP.),

I

0 < t < z,

(3W

where 5. PERFORMANCE AND CONTROL MARGIN INDEXES

Optimal trajecto.ries can be generated by extremizing a functional measuring performance, control margin, or a combination of performance and control margin. 5.1. Performance lqdexes

The main performance indexes of interest in AOT flight are the characteristic velocity, peak heating rate, peak dynamic pressure, and peak altitude drop. Characteristic ve/ocity

The characterist:ic velocity is a measure of the propellant consumed and can be normalized in terms

DP = G@IP)(V/V*)*, DP. = Cz,

(31b) (31c)

with Cz = (1/2)p. V?. In eqns (31), DP is the dynamic pressure for any altitude-velocity pair; DP. is a reference value of DP, corresponding to h = h. and V = V.. The constants in eqns (31) are h. = h./2 = 60 km, p = 0.3097E-03 kg/m’, V. = 7.832 km/set, CZ = 9.498 kPa. Peak altitude drop

A further alternative to (29), (30), or (31) is the peak value of the altitude drop, which can be normalized in terms of a reference value. Therefore, PI = max (Ah/Ah.),

I

0 6 t < T,

(32a)

756

A. Miele

with h, = 120 km.

during the exit phase of the atmospheric pass so as to increase the ability of the AOT spacecraft to meet the desired exit conditions. Comparison of (33b) and (33~) shows that CM. can be obtained by maximizing CM w.r.t. the control p(t); this is achieved for /I = f n/2.

Remark

5.3. Combined indexes

For planar or nearly-planar AOT flight, the optimal trajectory analyses[78] have shown that there is a remarkable consistency between the performance indexes (29)-(32) in the following sense: the control p(r) that minimizes one of the performance indexes (29)-(32) is the same or nearly the same as the control p(t) minimizing each of the remaining performance indexes. This being the case, it is proper to regard (29) or (30) as representative of the group of performance indexes (29)-(32).

A combined linear index can be derived starting from the performance index PI and control margin index CMI,

where Ah = h. - h,

(32b)

Ah. = h,.

(32~)

5.2. Control margin index The analyses of Section 4.5 have shown the existence of strong intrinsic instability in longitudinal motion and almost neutral stability in lateral motion. While stability can be artificially induced via feedback control, the effectiveness of a feedback control scheme depends on control margin availability; in turn, this enables the spacecraft to cope with dispersion effects due to navigation errors, system errors, atmospheric density errors, and aerodynamic coefficient errors. In light of the above situation, an alternative to the performance indexes (29x32) is the control margin index, which is defined below. Let the vertically projected lift L, be defined by L, = L cos p, where L is the lift and p the bank angle. For the AOT spacecraft to be able to maneuver in the longitudinal sense via the angle of bank, cos p should be away from the extreme values + 1. This leads to the following definition of control margin index: CMI = CM/CM., where CM is the control reference value for CM, CM =

margin

‘~(1 - cos2p) dt =

CM. =

s

‘wdt.

0

(34a)

where 0 < K < 1. Note that CLI=

PI,

if K=O,

CLI=l-CMI,

W)

ifK=l.

(34c)

Thus, K = 0 corresponds to minimizing the performance index PI; K = I corresponds to minimizing the complement of the control margin index 1 - CMI, and hence corresponds to maximizing the control margin index CMI. Alternatively, one can consider the combined quadratic index CQI = [(l - K)PI]* + [K(l - CMI)]‘,

(35a)

where 0 < K < 1. Note that CQI = (PI)2, CQI=

if K = 0,

(1 - CMI)l,

if K=

(35b)

1.

(35c)

As a further alternative to (35), one can consider the modified combined quadratic index MCQI = [(l - K)(PI/PI.

- l)]’

+ [K(CMI/CMI.

- I)]‘,

(W

if K = 0,

(W

where 0 < K < 1. Note that MCQI = (PI/PI. - l)‘,

(33a) and CM. is a

‘w sin ?p dt, (33b)

MCQI = (CMI/CMI.

- 1)2, if K = 1.

(36c)

Here, PI. and CMI. are prescribed values of the performance index and control margin index. The combined linear index (34) is of interest for optimization studies, while the combined quadratic index (35) and its modification (36) are of interest for guidance studies.

(33c) 6. OPTIMAL TRAJECTORIES

Here, w is a weighting function, w = pt,

CLI = (1 - K)PI + K(l - CM),

(334

with p the density and t the time. The presence of p stresses the fact that the lift is nearly a linear function of p; indeed, the lift coefficient is constant and velocity changes are small ois-his density changes during the atmospheric pass. The presence of t stresses the fact that more controllability is desired

In this section, we summarize the numerical studies made on optimal trajectories for two cases: (i) entry path inclination free; (ii) entry path inclination fixed. The optimization involved a functional measuring performance, control margin, or a combination of performance and control margin. For the planar or near-planar case, it was found that each of the functionals (29x32) produces the same or nearly the same control distribution; this being the case, among

Recent advances in the optimization

and guidance

of orbital

transfers

A weakness of this trajectory margin index is nearly zero.

757

is that the control

Worst performance

---

z 3 zt.

I

0.0

d.2

6.4

tW1

6.6

6.8

i.0

This trajectory maximizes the characteristic velocity [performance index (29)] for A free, hence minimizes the complement of (29). The results show the entry path inclination has the value $ = - 5.39 deg. The bank angle history consists of a single subarc p = const (see Fig. 7),

1.0

p = 174.7 deg.

100 0 i

-‘*he

ttksccl .

(38)

This trajectory has two weaknesses: high characteristic velocity and control margin index nearly zero. Best control margin

___,

i.0 d.8 d.6 t we4 Fig. 7. Optimal control versus time, entry path inclination free. 0.0

d.2

d.4

the performance functionals, we refer only to the characteristic velocity. The computations presented here refer to a HEO-LEO transfer, with HE0 = GEO, and to the AFE spacecraft for the following conditions: (a) atmospheric entry in the Space Shuttle orbital plane with velocity and path inclination consistent with the prescribed HE0 conditions; (b) atmospheric exit in the Space Shuttle orbital plane with velocity and path inclination consistent with the prescribed LEO conditions. For details, see Appendix C.

This trajectory maximizes the control margin index (33) for $, free, hence minimizes the complement of (33). This is the same as minimizing the combined linear index (34) for K = 1. The results show that the entry path inclination has the value A = - 4.46 deg. The bank angle history consists of three subarcs p = const (see Fig. 7),

6.1. Entry path inclination free For this case, the following engineering interest

trajectories

have

Best performance

This trajectory minimizes the characteristic velocity [performance Index (29)] for jtOfree. Note that minimizing (29) is the same as minimizing the combined linear index (34) for K = 0. The results show that the entry path inclination has the value j& = - 4.13 deg. The bank angle history consists of a single subarc p = const (see Fig. 7), 11= 5.14 deg.

p, = 90.0 -+ pZ = - 90.0 -+ p3 = 90.0 deg.

This trajectory has a control margin index CMI = 0.93. The fact that there is a 7% loss in control margin with respect to the maximum value CM1 = 1.00 can be explained as follows: the transition from one subarc to another does not occur instantaneously, but takes place over a 12 set time interval, since inequality (14) is taken into account with A = 15.0 deg/sec. For the above trajectories, summary results can be found in Table I. The data pertaining to best performance are shown in the left column; those pertaining to worst performance are shown in the center column; those pertaining to best control margin are shown in the right column.

(37)

6.2. Entry path inclination fixed Based on the results of Section 6.1, the value of the entry path inclination was set at the level of the best control margin trajectory for entry path inclination free, y,, = - 4.46 deg. This level is good for tolerance

Table 1. Results for ?o fra Trajectory

!A$ Aplq) Max HR Max DP Min h r PI

c-if/

Best performance

(39)

worst performann

Best

control margin

- 4.130

- 5.389

- 4.46 I

I S62 I.490 0.072 0.135 1.042 78. I 0.844 0.402 0.01

1.684 I.493 0.191 0.243 3.918 68.5 0.256 0.434 0.01

1.580 I.490 0.090 0.169 1.770 74.2 0.490 0.407 0.93

Units deg km/xc km/se-c km/set kW/cm’ kPa km ksec

tiiele

-100 I

0.0

I

0.2

I

0.4

tB.W

0.6

0.8

I

1.0

For K = 0, one obtains the best performance trajectory, characterized by two subarcs p = const; see (40). For K = 1, one obtains the best control margin trajectory, characterized by three subarcs ~1= const; see (41). For intermediate value of K, one obtains the best compromise trajectory, characterized by five subarcs p = const. One such trajectory includes the sequence (see Fig. 8) p, = 190.0 -+ /l* = 0.0 -

/l, = 90.0

-+ P4 = - 90.0 -

Fig. 8. Optimal control versus time, entry path inclination fixed. to navigation errors. The following trajectories have engineering interest. Best performance

This trajectory minimizes the characteristic velocity [performance index (29)] for $, = - 4.46 deg. Note that minimizing (29) is the same as minimizing the combined linear index (34) for K = 0. The results show that the bank angle history consists of two subarcs p = const (see Fig. 8), pI = 190.9 + p2 = 5.0 deg. A weakness of this trajectory margin index is nearly zero.

(40)

is that the control

Best control margin

This trajectory maximizes the control margin index (33) for $ = - 4.46 deg, hence minimizes the complement of (33). This is the same as minimizing the combined linear index (34) for K = 1. The results are identical with those of the best control margin trajectory of Section 6.1. This is not surprising, since j$ = - 4.46 deg was obtained from the best control margin trajectory of Section 6.1. Therefore, the three-subarc control distribution is given by (see Fig. 8) pl = 90.0 -+ p2 = - 90.0 + ~6 = 90.0 deg. (41) Best compromise

A satisfactory compromise between high performance and high controllability can be achieved by minimizing the combined linear index (34) for assigned values of the parameter K. As K changes, a one-parameter family of best compromise trajectories is generated, and the results can be parametrized in terms of K or CMZ.

p5 = 90.0 deg,

(42)

with the switch times adjusted in such a way that the boundary conditions are satisfied. For this trajectory, the values of the performance index and control margin index are intermediate between those of the best performance trajectory and those of the best control margin trajectory. In particular, the control margin index has the satisfactory value CMZ = 0.73. Comparison of (40), (41) and (42) shows that (42) arises by employing nearly (40) and (41) in sequence and explains how the best compromise trajectory works: the goal of good performance is achieved in the entry phase; the goal of good controllability is achieved in the exit phase. In the entry phase, the intent is to arrive at the desired minimum altitude. Recall that a high minimum altitude is consistent with low characteristic velocity, low peak heating rate, and low peak dynamic pressure. The bank angle is set first at p1 = 190.0 deg, then switched to p2 = 0.0 deg. In the exit phase, the intent is to achieve the required orbital inclination and longitude of the ascending node at the exit, while ensuring the largest control margin enabling one to achieve the specified LEO apogee following the atmospheric exit. The bank angle is set first at II, = 90.0 deg, then switched to p4 = - 90.0 deg, then switched to p5 = 90.0 deg. For the above trajectories, summary results are shown in Table 2. The data pertaining to best performance are shown in the left column; those pertaining to best control margin are shown in the right column; those pertaining to best compromise between performance and control margin are shown in the center column. It is worthwhile to remark that the best compromise trajectory of this section is the basis for the development of the guidance schemes of Sections 7 and 8. 4.3. Remark The optimal trajectories of Section 6.1 and 6.2 were computed using the complete differential system (11) and (12), whose order is six. If the decomposition techniques of Section 4 are employed, the system (11) and (12) decouples into two smaller subsystems of order three: the longitudinal motion subsystem (22) and the lateral motion subsystem (26). A property of the longitudinal motion subsystem (22) is that the state variables r(t), V(t), y(t) remain unchanged if the control distribution p(t) is replaced

Recent advances in the optimization and guidance of orbital transfers

by -p(t) for either a portion of the interval of integration or the entire interval of integration. In turn, this has the following implications concerning the longitudinal m.otion: (i) the best control margin trajectory of Section 6.1 reduces to a single subarc p = const, p = 90.0 deg;

(43)

(ii) the best control margin trajectory of Section 6.2 reduces to a single subarc p = const, p = 90.0 deg;

(44)

(ii) the best compromise trajectory of Section 6.2 reduces to the three-subarc sequence p, = 190.0 -+ p2 = 0.0 -+ ~3 = 90.0 deg.

guidance. The objective of longitudinal guidance is to achieve the specified apogee after exiting the atmosphere; the objective of lateral guidance is to achieve the specified orbital plane at atmospheric exit. Because the angle of attack is held constant at the trim position during the atmospheric pass, the only control is the bank angle ,u. Thus, the modulus of the bank angle is used to control longitudinal motion, while the sign of the bank is used to control lateral motion. Concerning lateral motion, we assume that the sign of the bank angle is determined via the out-of-plane velocity angle i. Whenever 2 violates some prescribed altitude-dependent upper and lower bounds,

(45)

These results are useful in guidance; see Sections 7 and 8. 7. SAFETY ZONE GUIDANCE The objective of the atmospheric pass guidance is to deplete velocity so that, after exiting the atmosphere, the vehicle can ascend to the specified apogee in the spec:ified orbital plane; of course, the peak heating rate during the atmospheric pass must be contained within acceptable levels; furthermore, the AOT vehicle must be able to execute safely the atmospheric pass, even in the presence of severe environmental conditions such as navigation errors, system errors, at.mospheric density errors, and aerodynamic coefficient errors. To achieve the above goals, based on the optimal trajectory results of Section 6, we decompose the atmospheric pass guidance into an entry phase and an exit phase. The objective of the entry phase is to contain the peak heating rate within acceptable levels. The objective of the exit phase is to steer the vehicle with high control margin to the specified apogee in the specific orbital plane. The switch time t, from entry phase to exit phase is to be determined as a compromise between the peak heating rate of the entry phase and the control margin of the exit phase. This is done in Section 8. In this section, assuming that the switch time &has already been determined, we focus our attention on exit phase guidance. The latter can be further decomposed into longitudinal guidance and lateral

- B(h) < i c + B(h),

Trajectory & AP

Ai’w API, Max HR Max DP Min h r PI

&I

(46)

the bank angle changes sign. The function B(h) decreases with the altitude, being of order 0.50 deg at lower altitudes and 0.05 deg at atmospheric exit. Note that the switch from a positive value to a negative value of the bank angle, and vice versa, cannot be done instantaneously but must be consistent with the inequality constraint -A
(47)

where A is a constant. With the above understanding, the generation of the bank angle value is done in three steps: (i) the modulus of the bank angle pp is obtained via longitudinal motion control; (ii) the command bank angle pr is obtained from p,, after the sign of the bank angle is determined via the lateral motion control [see (46)]; (iii) the real bank angle p is obtained from pc in light of the bank angle time rate constraint [see (47)l. 7.1. Predictor-corrector

algorithm

In this section, we present the predictor-corrector algorithm governing exit phase guidance in one version: safety zone guidance (SZG). For alternative versions [constant bank angle guidance (BAG), constant climb rate guidance (CRG), and constant path inclination guidance (PIG)], see [95, 961. With SZG, one aims at the specified apogee while maintaining maximum control margin during the atmospheric pass; with BAG, CRG, PIG, one aims at

Table 2. Results for W &en Best performance

759

Best compromise

Best control margin

- 4.461 1.563 I .490 0.073 0.144

- 4.461 1.577 1.490 0.087 0.149

- 4.461 1.580

I.128 77.8

I .438 75.4

1.770 74.2

0.821

0.529

0.401

0.406

0.02

0.73

0.490 0.407 0.93

1.490

0.090 0.169

units deg km/set km/set km/set kW/cm2 kPa km ksec

760

A. Miele

the specified apogee without consideration of control margin. Because use of the complete equations of motion is expensive in terms of CPU time, we employ decomposition techniques decoupling the longitudinal motion from the lateral motion; see Section 4. The decoupling is possible if approximations (20) are introduced in the equations of motion. Specifically, (20a) and (20b) mean that the instantaneous orbital plane is nearly identical with the entry orbital plane; (20~) means that transport acceleration terms are being neglected in the equations of motion. Control law With the above understanding the longitudinal motion is described by eqns (22) which must be integrated forward in time over the interval I, Q t < T. Here, the initial time r = t, corresponds to the present position of the spacecraft and the final time t = r corresponds to atmospheric exit. The integration must be carried out for a = const in connection with the control law (SZG) corresponding

p = n/2,

(48a)

cos /J = 0.

(48b)

to (SZG)

If the initial value of the bank angle p, is different from (48a), the transition from p, to (48a) must be performed at the maximum permissible time rate consistent with (47), hence /i=

fA.

(49)

Two-point boundary-value problem (TPBVP)

For the longitudinal motion subsystem (22), the initial conditions (time I = I,) are as follows: (SZG)

r, = given,

V, = given,

y, = free.

(50)

The final conditions (time t = T) require the satisfaction of eqns (25a) and (25b), which determine the values of the parameters 7, and 7. For a given value of y,, eqns (22) must be integrated forward in time subject to the control law (48) to yield the functions r(r), V(r), v(f) over the time interval r, < I < 7; the final time 7 must be determined so that (25a) is satisfied. The procedure must be repeated by changing the guess for 7, until a pair (y,, 7) is found such that (25b) is satisfied simultaneously with (25a). This requires a trial-and-error procedure, which can be speeded up via a process of linearization and/or bisection. Details are omitted for brevity. Feedback control

In the predictor-corrector algorithm, the TPBVP is solved every Ar seconds. Therefore, a feedback control law is needed for the time interval I, < t < I, + At between successive solutions of the TPBVP.

In safety zone guidance, let the functions r(t), V(t), y(r) obtained from the solution of the TPBVP be regarded as parametric equations of the trajectory. For the time interval 1, < f < t, + At, elimination of the parameter t from the functions V(f) and y(t) yields the function y.(v), which constitutes the safety zone during this time interval. In other words, with reference to the velocity-path inclination domain, the function y.(Y) is the centerline of the region from which the atmospheric pass can be executed with maximum control margin; hence, the name safety zone guidance. This leads to the following feedback control law: (SZG)

cos pp = cos p. + k tan(Ay), Ay = ‘r’- y.( I’), cos /I.

=

0,

(51a) (51b)

(514

where k is the gain coefficient and Ay is the deviation of the real path inclination 7 from the computed safety zone path inclination y.(v). Note that the function y.(v), obtained by solving the TPBVP, must be updated every At seconds. Also note that, for small deviations, the nonlinear law (51a) reduces to the linear law cos p,, = cos /.I. + kAy . (51d) Of course, the values of cos pp predicted (5la)-(51d) are subject to the inequality -l
+I.

via (5le)

7.2. Dispersion effects In real AOT flight, there are dispersion effects due to navigation errors, system errors, atmospheric density errors, and aerodynamic coefficient errors. Let the subscript n denote a nominal condition; let the subscript e denote an estimated condition; and let the absence of a subscript denote a real condition. With this understanding, the following dispersion factors can be defined:

F, = t,jt,.,

Fp= dh)i~dh). F. = CD/CD,,

FL = CL/CL,,

F, = twit,.,

(52b)

Fcx= pdh)/pn(h),

(52~)

FD, = CkiC'Dnr

(524

FLC= C,.iC,,.

We)

In (52). the left-column dispersion factors are the ratios of real quantities to nominal quantities; they are used in the integration of the complete differential system (11x12) simulating the real environment. The right-column dispersion factors are the ratios of estimated quantities to nominal quantities; they are used in the integration of the reduced differential system (22) employed in the predictor+orrector algorithm. Each dispersion factor is such that its

Recent advances in the optimization and guidance of orbital transfers

value is 1.0 if the real or estimated condition is identical with the nominal condition; its value is not 1.0 if the real or estimated condition is different from the nominal condition. Clearly, the goodness of a guidance scheme IS measured by the range of values of the dispersion factors (52) for which the AOT spacecraft can safely execute the atmospheric pass. Indeed, the wider the range, the better the guidance scheme. 7.3. Evaluation criteria In studying the effects of the dispersion factors (52) on the atmospheric pass, a variety of physical quantities must be considered. Here, we restrict our attention to apogee altitude and postatmospheric characteristic velocity. Apogee altitude

Note that (54) and (55) do not include the HE0 characteristic velocity A?m. This is due to the fact that, in the numerical experiments of Section 7.4, the comparison of guidance schemes is done for fixed entry conditions; hence, for each given entry path inclination pO, the characteristic velocity component Arm must be regarded as a constant, independent of the particular guidance scheme being considered. 7.4. Numerical experiments Systematic numerical experiments were carried out for the AFE spacecraf1[95]. The assumed nominal conditions were & = - 4.46 deg,

(56a)

I,. = 137.0 set,

(56b)

p.(k) = density of the 1976 US

Because the guidance

is not perfect, the apogee radius following the atmospheric pass is generally different from the LEO radius. Use of the energy and angular momentum conservation laws for the transfer orbit connecting the atmospheric exit point (radius ra) and the apogee point (radius r~ generally not equal to r,,) leads to the following relationship: rzz = r, I? cos ‘7, /[V? - JK

E)* cos *j$ + v! sin y,],

with the implication

(53a)

that h22 =

r22 -

r,.

VW

The resulting error in apogee altitude is h2! - hi, =

r22 -

(53c)

rll,

where rll is the desired LEO radius. If the guidance is hypothetically perfect, the differences in (53~) vanish, which is the same as stating that (53a) is satisfied with r2 replaced by rll.

Standard Atmosphere,

Assuming that r2z2 rll, this includes two parts: the velocity impulse LLp2*at apogee and the velocity impulse Av,, at perigee (LEO), namely, AIf= Apz,, + A?,,,

CD. = 1.355,

(56d)

C,. = - 0.381.

(56e)

Each of the dispersion factors (52) was subject to ample variations vis-ti-vis the nominal value of 1.0. With reference to the apogee altitude kU and postatmospheric characteristic velocity A p, it was found that these quantities are nearly constant for SZG, while they exhibit much wider variations for BAG, CRG, PIG. Therefore, the goodness of SZG vis-ci-vis BAG, CRG, PIG was clearly established. The explanation for the above behavior lies in the fact that SZG is constructed so as to maximize controllability, while completing the atmospheric pass and aiming at the desired LEO; on the other hand, BAG, CRG, PIG are constructed by aiming only at the desired LEO, while ignoring the controllability issue.

One particular group of tests was made assuming F, = F, = 1.000, 0.927, 0.854, 0.781, 0.708, F, = 0.8,

(54a)

with (57a) corresponding A?22 = V*Jm

rJr22r28 -

(r&22)6

A P,, = f’*~~[~~

cos

71,

- 11,

622 +

(54c)

The remaining dispersion factors were set equal to 1.0. The results are shown in Figs 9 and 10 and indicate that, for SZG, the values of the apogee altitude hn and postatmospheric characteristic velocity AP are roughly independent of the switch time.

denotes the average of ru and rll. perfect, If the guidance is hypothetically rz = rll = r2,; the right-hand side of (54c) vanishes; the postatmospheric characteristic velocity reduces to V.,,m

- (r,/r,,)f?

cos ?I.

(57b)

to

t, = tr = 137, 127, 117, 107,97 sec.

(546)

r11)/2

F, = 0.9,

(57a)

WW

where r!l =

(56c)

Switch time errors

Postatmospheric characteristic velocity

AP=

761

(55)

8.

(58)

COMPLETE ENTRY/EXIT GUIDANCE

In Section 7, we supplied the predictor-corrector algorithm for the exit phase of the atmospheric pass. The algorithm requires the specification of the switch

A. Miele

162

300

and subject to inequality (47). The forward integration yields a family of solutions r(t), V(t), y(t) depending on the triad (t,,, tQ, 7). In particular, for a given t,, , the pair (ts2,T) must be determined so that the boundary conditions (25) are satisfied. The procedure must be repeated by changing the guess &I; this leads to parametric solutions of the form 0.8

0.7

0.9

S

1.0

Fig. 9. Safety zone guidance: apogee altitude versus switch time dispersion factor. time 1, from entry phase to exit phase. It is based on the control law (48) in connection with the reduced system (22) describing the longitudinal motion. In this section, we consider the proper determination of the switch time 1%as a compromise between the peak heating rate of the entry phase and the control margin of the exit phase. Once more, we make use of the reduced system (22). This being the case, we replace the five-subarc bank angle sequence (42) with the three-subarc bank angle sequence (45), rewritten here for convenience as follows: ~1,= 190.0 deg,

0 S r < h,

VW

PI =

0.0 deg,

cl G r $ I,?,

(59b)

1,

90.0 deg,

r,z < t < T.

(59c)

=

Here, r,, is the switch time from cc, to p2, tti is the switch time from /.L?to p3, and T is the final time of the atmospheric pass. Clearly, the solution of the reduced system depends on the three parameters f,~, I~?, T. Of course, the transition from one subarc to another must be consistent with inequality (47). 8.1. Predictor-corrector

algorithm

Consider now the current time instant r,, with 0 < t, < I,~. Assume that h,, V,, y, are given. Then, integrate the differential system (22) forward in time subject to the bank angle time history

[,I = t,,

(61a)

ta = tr&r),

(61b)

t = S(L),

(61~)

which can be computed via two successive one-dimensional searches. Details are omitted for brevity and can be found in [96]. With relations (61) known, the switch time t, from entry phase to exit phase is determined by minimizing the modified combined quadratic index (36), rewritten here for convenience as follows: MCQI = [(I - K)(PI/PI. - l)]’ + [K(CMI/CMI.

- l)]‘.

(62)

Here, K is a preselected constant, 0 < K f 1; PI is the performance index (30), based on the peak value of the stagnation point heating rate; PI. is a preselected value of PI; CMI is the control margin index (33); and CMI. is a preselected value of CMI. The fact that the switch time t, is chosen so as to minimize the modified combined quadratic index (62) means that we intend to approximate the optimal trajectory with a guidance trajectory close to the optimal trajectory in both the values of peak heating rate and control margin. 8.2. Numerical experiments Systematic numerical experiments were carried out for AFE spacecraft[96]. The assumed nominal conditions were represented by (56). The dispersion factors (52) were subject to ample variations vis-ci-vis the nominal value of 1.0. Navigation errors

pl = 190.0 deg, ~2 =

0.0 deg,

p3 = 90.0 deg.

r, < t < trl, t,[ G t Q tr2, t,2G t < T,

NW (Mb) (6oC)

To study the effect of navigation errors, we assume the following values for the path inclination dispersion factor:

F: = F, = 1.045, 1.022, 1.000, 0.978, 0.955,

(63a)

implying that j&= A = - 4.66, - 4.56, - 4.46, - 4.36, - 4.26 deg,

(63b)

and that A$ = Aj& = - 0.20, - 0.10, 0.00, + 0.10, + 0.20 deg. 6.8

6.9

i.0

pr Fig. IO. Safety zone guidance: postatmospheric characteristic velocity versus switch time dispersion factor.

(63~)

The remaining dispersion factors (52) are assumed to be equal to one, except (52b). which are set at a value dictated by the optimization of the modified

Recent advances in the optimization and guidance of orbital transfers

763

% P 3M

rnrn.4 0.6

Fig. 11. Complete entry/exit guidance: apogee altitude versus entry path inclination dispersion factor.

combined quadratic index (62). The results show that apogee altitude /I;~ and postatmospheric characteristic velocity Ap are nearly invariant with respect to changes in F7; see Figs 11 and 12. Atmospheric densil’y errors

To study the efiect of atmospheric density errors, we assume the following values for the density dispersion factor: FP = FP = 0.6, 0.8, 1.0, 1.2, 1.4,

(64a)

implying that APIP. = Apcl~n = - 0.4, - 0.2,0.0,+ 0.2,+ 0.4.

(64b)

The remaining dispersion factors (52) are assumed to be equal to one, except (52b), which are set at a level dictated by the optimization of the modified combined quadratic index (62). The results show that apogee altitude hz! and postatmospheric characteristic velocity AT are nearly invariant with respect to changes in F,; see Figs 13 and 14. Aerodynamic coeficient errors

To study the effect of aerodynamic coefficient errors, we assume the following values for the lift coefficient dispersion factor: FL = Fb = 0.6,0.8,1.0,1.1, 1.2,

(65a)

implying that AC,/C,. = A&/&n = - 0.4., - 0.2, 0.0, + 0.1, + 0.2.

(65b)

1.0

1.2

5 Fig. 13. Complete entry/exit guidance: apogee altitude versus density dispersion factor.

The remaining dispersion factors (52) are assumed to be equal to one, except (52b), which are set at a level dictated by the optimization of the modified combined quadratic index (62). The results show that apogee altitude h22 and postatmospheric characteristic velocity AP are nearly invariant with respect to changes in FL; see Figs 15 and 16. 9. SUMMARY, CONCLUSIONS,AND RECOMMENDATIONS

We have reviewed the multi-year effort of the Aero-Astronautics Group of Rice University in the development of optimal and guidance trajectories for aeroassisted orbital transfer. The main assumption is that the AOT vehicle flies at constant angle of attack and is controlled via the angle of bank. We have presented general equations of motion for flight in a 3-D space and then simplified equations decoupling the longitudinal motion from the lateral motion. While the general system has order six, the reduced subsystems resulting from decomposition have order three. An eigenvalue analysis of the reduced subsystems shows that, for fixed controls, the longitudinal motion subsystem is highly unstable, while the lateral motion subsystem is almost neutrally stable. While stability can be artificially induced via feedback control, the effectiveness of a feedback control scheme depends on control margin availability; in turn, this enables the spacecraft to cope with dispersion effects due to various sources of errors: navigation errors, system errors, atmospheric density errors, and aerodynamic coefficient errors. We have optimized trajectories from the viewpoint of performance, control margin, or combination of performance and control margin. The results show that the optimal trajectories are made up of segments

_.___ ,

i.4 i.2 5 Fig. 14. Complete entry/exit guidance: postatmospheric characteristic velocity versus density dispersion factor. 0.6

Fig. 12. Complete entry/exit guidance: postatmospheric characteristic velocity versus entry path inclination dispersion factor.

0.8

Li.s

i.0

164

A. Miele

rnrn.* 0.6

FL

Fig. 15. Complete entry/exit guidance: apogee altitude versus lift coefficient dispersion factor. (subarcs) along which the angle of bank is constant. The number of subarcs is n = 1 for a performance optimal trajectory with entry path inclination free, n = 2 for a performance optimal trajectory with entry path inclination fixed, n = 3 for a control margin optimal trajectory with entry path inclination free, and n = 5 for a compromise optimal trajectory with entry path inclination fixed. In turn, the best compromise trajectory is the basis for the development of effective guidance schemes. For the AFE spacecraft, the best compromise trajectory involves the sequence of bank angles 190, 0, 90, -90, 90 deg. The subsequence 190, 0 deg characterizes the entry phase and takes care of the performance problem by ensuring that the correct minimum altitude is reached. The subsequence 90, - 90, 90 deg characterizes the exit phase and ensures that the correct apogee is reached in the correct orbital plane, while retaining maximum control margin during the atmospheric pass. Complete entry/exit guidance involves two predictor-corrector algorithms operating in sequence, separated by the switch time t, from entry phase to exit phase. The first algorithm refers to the entry phase and determines the best switch time as a compromise between the peak heating rate of the entry phase and the control margin of the exit phase. The second algorithm refers to the exit phase and constitutes safety zone guidance: for each altitudevelocity pair, the best value of the local path inclination is computed by maximizing the control margin of the exit phase, subject to the requirement that the desired apogee is reached after exiting the atmosphere in the prescribed orbital plane.

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Aerodynamic Heating of Missiles Entering the Earth’s Atmosphere at High Supersonic Speeds, Technical Note

I

4047. NACA (1957). 3. A. J. Eggers, H. J. Allen and S. E. Neice, A Comparative

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OMS-

Analysisof the Performance of Long-Range Hypervelocitv Vehicles. Technical Note 4046. NACA (1957). 4. DP.R. Chapman, An Approximate Analytical ;Methbdfor Studying Entry into Planetary Atmospheres, Technical

0.050, 0.6

0.8

FL

1.0

1.2

Fig. 16. Complete entry/exit guidance: postatmospheric characteristic

Systematic numerical experiments have been carried out for the AFE spacecraft. The effects of the dispersion factors associated with navigation errors, system errors, atmospheric density errors, and aerodynamic coefficient errors have been studied. Experiments show that complete entry/exit guidance enables the spacecraft to safely execute the atmospheric pass even in the presence of relatively large disturbances. For the exit phase, safety zone guidance appears to be much more reliable than alternative schemes such as constant bank angle guidance, constant climb rate guidance, and constant path inclination guidance. From the above summary and conclusions, it is clear that, over the past decade, considerable progress has been achieved in the optimization and guidance of aeroassisted orbital transfers. In spite of the advances, there remain problem areas which must be addressed. While the results concerning longitudinal motion are technically satisfactory, this is not yet the case with the results concerning lateral motion. Typically, at atmospheric exit, the orbital inclination and longitude of the ascending node are precise to within 0.1 deg, due to the empirical nature of the thresholds governing the switch from positive values of the bank angle to negative values, and vice versa [see (46)]. In light of this situation, we suggest the development of a coupled longitudinal-lateral guidance scheme via continuous bank angle control, based on the use of the complete differential system (1 l), (12), which is of order six. Alternatively, it might be of interest to replace (1 l), (12) with the decomposed subsystems (22) and (26) which are of order three. Finally, consideration should be given to the use of parallel algorithms to speed up the computation associated with predictorcorrector algorithms. In turn, this will require the development of appropriate parallel algorithms, exemplified by Ref. [93]. Research in this promising area is important, indeed essential.

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Pcosysini-

VcosysinX=O,

P sin 7 - V sin y = 0.

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(67~)

B= e + wt,

(68u)

6-4,

(68b)

i=

r,

(68c)

and P = J[ v2 + 2wr V cos y cos x cos 4 + (wr cos dr)*], tan?=

(69a)

Vsinyl

J[( V cos y)* + 2wr V cos y cos x cos 4 + (wr cos r$)*], (69b)

tan 2 = V cos y sin x/( V cos y cos x + wr cos 4).

(69~)

The inverse relations are B=B-Ult,

(1992).

41 (2), 139-163 (1993).

(67b)

These equations can be solved explicitly to obtain the inertial quantities starting from Earth-fixed quantities (direct relations), and vice versa (inverse relations). See Refs (83, 841 for details. The direct relations are

92. A. Miele and T. Wang, Nominal trajectories for the aeroassisted flight experiment. Journal of the Astronautical Sciences,

767

(7Oa)

4 = $9

(70b)

r = i,

(7Oc)

and V = J[ P - 2wPP cos jr cos 2 cos f$ + (oi cos&*],

(7 1a)

tany = Psiny/ J[( P cos j)l - 2wiP cos 7 cos x’cos l$ + (OP cos f&2], (71b) tan x = P cos p sin f/( P cos p cos i - or’ cos 4).

(7lc)

APPENDIX B: DERIVED QUANTITIES Once the state variables are known in either the Earth-fixed system or the inertial system, one can compute some derived quantities such as the orbital inclination i and longitude of the ascending node R. In the inertial system, these quantities are defined through the relations cos i = cos I$ cos x’,

(72a)

sin(8 - a) = cot i tan 6.

(72b)

In the Earth-fixed system, the relations analogous to (72) are given below, cos i = cos 4 cos x. (73n) sin@ - G) = cot i tan 4.

(73b)

Further derived quantities

&-q5=0,

(66b)

Other quantities of interest in the flight of an AOT spacecraft are the out-of-plane position angle S (angle between the instantaneous radius vector and entry orbital plane), out-of-plane velocity angle 6 (angle between the instantaneous velocity vector and entry orbital plane), and wedge angle r~ (angle between the instantaneous orbital plane and entry orbital plane). In the inertial system, these quantities are defined through the relations

r’-r=O,

(66c)

sin S = :r’a’ 0 ,

W4

sin E = &J6,

(74b)

cos ri = M?,

(74c)

APPENDIX A: TRANSFORMATION

RELATIONS

General transformation relations connect the Earth-fixed quantities and inertial quantities. Specifically, the position transformations are

8--fl-wt=o,

and the velocity transformations Pcos~cos~-

(664

are

Vcosycosx-wrcos4=0,

(67a)

A. Miele

768

in which d is the unit vector in the radial direction, 6 is the unit vector in the velocity direction, and c’is the unit vector normal to the instantaneous orbital plane. The superscript T denotes transposition of vector or matrix; the tilde superscript refers to quantities computed in the inertial system; the superscript 0 denotes entry condition. The 3 x 1 vectors d, 6, ?, are given by

cos Bcos I$ cc 1 sinBcos$ [ sin 6

,

(754

COST sin 2 co9 B sin 6 cosp sin2 sing sin& - cos F sin 2 cos $ I [

b’=

+

G, = 0.000,

(79b)

Physical constants

The radius of the Earth is r. = 6378 km; the radius of the outer edge of the atmosphere is r, = 6498 km; the thickness of the atmosphere is h. = 120 km; the Earth gravitational constant is fi = 0.3986E + 06 km’@+ the circular velocity at r = r, is V. = 7.832 km/set; the angular velocity of the Earth is o = 0.7292E-04 rad/sec. Atmospheric model

-COST cosf sinfT+siny’ cost7 cosf$ cosf cosff cosO+siny sin& cosq , (75b) sin F sin 6 1 [ e=

E = 0.281,

where CO is the drag coefficient, CL the lift coefficient, E the lift-to-drag ratio modulus, and CH the moment coefficient. As explained, the spacecraft is controlled via the angle of bank.

(75c)

[ 2SJL].

In particular, eqns (74)-(75) imply sin s’ = cos 6 cos iosin@& - 8) + sin 4 cos in, (76a) sin r = (cos 1’sin ff sin 6 + sin p cos &sin L sin& - t7)

In the Earth-fixed system, the heating rate is computed with eqns (30b)--(3Oc), where CI = 0.2832 kW/cm2; the dynamic pressure is computed with eqns (3 1b)-(3 lc), where C2 = 9.498 kPa. Transfer data

conditions are as follows: the altitude corresponding to the radius rw = 42,164 km; the path inclination is lim= 0.00 dep. The LEO conditions are as follows: the altitude is hll = 310 km, corresponding to the radius rlr = 6708 km; the path inclination is 71, = 0.00 deg. is

(76b)

cos fj = sin i sin i0cos@ - Q0) + cos i cos 6.

Heating rate and dynamic pressure

The

+ (sin 7 sin 6 - cos f sin 2 cos &cos io - cos p cos ff sin iocos(&%- 8),

The assumed atmospheric model is that of the 1976 US Standard Atmosphere[l9]. In this model, the values of the density are tabulated at discrete altitudes. For intermediate altitudes, the density is computed by assuming an exponential fit for the function p(h).

(76~)

HE0

hm = 35,786 km,

Remark

Entry conditions

Inspection of eqns (72) leads to the conclusion that the orbital inclination and longitude of the ascending node can be represented via functional relations of the form

In the inertial system, the given entry conditions are as follows: the longitude is 8, = - 134.52 deg; the latitude is 40 = - 4.49 deg; the altitude is ho = 120 km, corresponding to the radius ro = 6498 km; the heading angle is i = - 28.13 deg; the orbital inclination is & = 28.45 deg; the longitude of the ascending node is 0 = - 126.19 deg; the out-of-plane position angle is b, = 0.00 deg; the out-of-plane velocity angle is b = 0.00 deg; the wedge angle is r&= 0.00 deg.

i = i(&, f),

(774

d = 6(8, $5, x’).

(77b)

Analogously, inspection of eqns (76) leads to the conclusion that. if the entry orbital mane is given (hence, if L and I% are known), the out-of-plane po&ion angle, .out-of-plane velocity angle, and wedge angle can be represented via functional relations of the form (78a)

VW

ri= a@, 6, 2).

(78~)

The inertial quantities on the right-hand sides of (77)-(78) can be obtained from the solutions of the differential system (I l)-(12) upon using the direct transformation relations (68~(69). APPENDIX C: EXPERIMENTAL

DATA

The following data are used in the numerical experiments described in this paper. Spacecraft dara

The AFE configuration The mass of the AFE reference surface area is ation, the angle of attack angle of attack,

is shown in Fig. 5 of Section 2. spacecraft is m = 1678 kg; the S = 14.31 m*. For this configuris constant, a = 17.0 deg. At this

Co = 1.355, CL = - 0.381,

(79a)

Exit conditions

In the inertial system, the desired exit conditions are as follows: the altitude is hl = 120 km, corresponding to the radius r, = 6498 km; the orbital inclination is G = 28.45 dee: the longitude of the ascending node is fir = - 126.19 de;; the out-of-plane position angle is $1 = 0.00 deg; the out-of-plane velocity angle is ir = 0.00 deg; the wedge angle is $ = 0.00 deg. Inequalities

The angle-to-bank time rate is subject to inequality (14) or (47), with A = 15.0 deg/sec. The out-of-plane velocity angle is subject to inequality (46), in which the function B(h) is represented by linear segments between the following reference points: B(60) = 0.52, (8Oa) B(75) = 0.52,

(gob)

B(84) = 0.08,

(8Oc)

B(95) = 0.05,

(8Od)

B(120) = 0.05.

(8Oe)

In eqns (80), the values of h are in kilometers and the values of the threshold B are in degrees.