Acta Astronautica Vol. 15, No. 6/7, pp. 399411, 1987 Printed in Great Britain. All rights reserved
00945767/87 $3.00+0.00 © 1987 PergamonJournals Ltd
OPTIMAL TRAJECTORIES FOR AEROASSISTED, NONCOPLANAR ORBITAL TRANSFER*t A. MIELE~, V. K. BASAPUR¶ and W. Y. LEE§ AeroAstronautics Group, Rice University, Houston, Tex., U.S.A. (Received 15 December 1986)
AbstractThis paper considers both classical and minimax problems of optimal control which arise in the study of noncoplanar, aeroassisted orbital transfer. The maneuver considered involves the transfer from a high planetary orbit to a low planetary orbit with a prescribed atmospheric plane change. An example is the HEOtoLEO transfer of a spacecraft with a prescribed plane change, where HEO denotes high Earth orbit and LEO denotes low Earth orbit. In particular, HEO can be GEO, a geosynchronous Earth orbit. The basic idea is to employ the hybrid combination of propulsive maneuvers in space and aerodynamic maneuvers in the sensible atmosphere. Hence, this type of flight is also called synergetic space flight. With reference to the atmospheric part of the maneuver, trajectory control is achieved by modulating the lift coefficient (hence, the angle of attack) and the angle of bank. The presence of upper and lower bounds on the lift coefficient is considered. Within the framework of classical optimal control, the following problems are studied: (Pl) minimize the energy required for orbital transfer; (P2) minimize the time integral of the heating rate; (P3) minimize the time of flight during the atmospheric portion of the trajectory; (P4) maximize the time of flight during the atmospheric portion of the trajectory', (P5) minimize the time integral of the square of the path inclination; and (P6) minimize the sum of the squares of the entry and exit path inclinations. Problems (PI)(P6) are Bolza problems of optimal control. Within the framework of minimax optimal control, the following problems are studied: (Q 1) minimize the peak heating rate; (Q2) minimize the peak dynamic pressure; and (Q3) minimize the peak altitude drop. Problems (QI)(Q3) are Chebyshev problems of optimal control, which can be converted into Bolza problems by suitable transformations. Numerical solutions for Problems (P1)(P6) and Problems (Q1)(Q3) are obtained by means of the sequential gradientrestoration algorithm for optimal control problems. The engineering implications of these solutions are discussed, and it appears that the energy solution (P1) and the nearlygrazing solution (P5) are superior to the remaining solutions. While the nearlygrazing solution (P5) requires more energy than the energy solution (P1), at the same time it involves less integrated heating rate, less peak heating rate, less peak dynamic pressure, and less peak total acceleration. Therefore, it is felt that both solutions (P1) and (P5) should be considered as candidates for flight operations, and hence candidates for spacecraft design.
i. INTRODUCTION This paper considers both classical and minimax problems of optimal control which arise in the study of noncoplanar, aeroassisted orbital transfer. The maneuver considered involves the transfer from a high planetary orbit to a low planetary orbit with a prescribed atmospheric plane change. An example is the HEOtoLEO transfer of a spacecraft with a prescribed plane change, where HEO denotes high Earth orbit and LEO denotes low Earth orbit. In particular, H E O can be GEO, a geosynchronous Earth orbit. *This paper, a condensation of the investigation reported in [1], was presented at the 37th Congress o f the International Astronautical Federation, Innsbruck, Austria, 411 October 1986 (Paper No. IAF86229). tThis research was supported by the Jet Propulsion Laboratory, Contract No. 956415. The authors are indebted to Dr K. D. Mease, Jet laropulsion Laboratory, for helpful discussions. ~Professor of Astronautics and Mathematical Sciences. ¶Graduate Student. ~Graduate Student.
The basic idea is to employ the hybrid combination of propulsive maneuvers in space and aerodynamic maneuvers in the sensible atmosphere. Hence, this type of flight is also called synergetic space flight. With reference to the atmospheric part of the maneuver, trajectory control is achieved by means of lift modulation and bank angle modulation. The presence of upper and lower bounds on the lift coet~cient is considered. Previous research. Previous research on the topics covered in this paper can be found in[238]. For an overview of theoretical calculus of variations and optimal control, see[23]. For an analysis of the performance of hypervelocity vehicles, see[45]. For a discussion of optimal atmospheric trajectories, see[6]. For a discussion of optimal space trajectories, see[7]. Concerning the algorithms useful for solving optimal control problems of the Bolza type, see[819]. In particular, firstorder methods (gradient methods) are presented in[815] and secondorder methods (quasilinearization methods) are presented in[16]. A m o n g
399 A.A. 15/b7H
400
A. MIELEet al.
the firstorder methods,[812] deal with the primal formulation and[1315] deal with the dual formulation. Both firstorder methods and secondorder methods require the solution of a linear, twopoint boundaryvalue problem at each iteration. In this connection, one of the most effective techniques is the method of particular solutions, discussed in[1719]. Concerning optimal control problems of the Chebyshev type (minimax problems), see[2027]. While Chebyshev problems are not covered by the Boiza formulation, they can be brought into the Bolza format by various transformations. These transformations are discussed in[2022]. Aerospace applications of minimax optimal control are presented in[2327]. Finally, concerning aeroassisted orbital transfer, see[2438]. General aeroassisted orbital transfer is discussed in[2831]. In particular, the topics of optimization and guidance of trajectories of AOT vehicles are discussed in[3238] as well as in[2427]. Present research. This paper deals with noncoplanar, aeroassisted orbital transfer under the following assumptions: (i) the initial and final orbits are circular; (ii) three impulses are employed, one at HEO, one at LEO, and one at the exit from the atmosphere; (iii) the prescribed plane change is performed entirely in the atmospheric part of the trajectory; and (iv) the gravitational field is central and is governed by the inverse square law. The maneuver is performed partly in outer space and partly in the sensible atmosphere. The following terminology is employed: r e is the radius of the Earth; ra is the radius of the outer edge of the sensible atmosphere; r0o is the HEO radius; and r~l is the LEO radius. The four key points of the maneuver are these: point 00 (exit from HEO); point 0 (entrance into the atmosphere); point 1 (exit from the atmosphere); and point 11 (entrance into LEO). The maneuver starts with a tangential propulsive burn, having characteristic velocity A V00, at point 00; here, the spacecraft exits from the high Earth orbit and enters into an elliptical transfer orbit, connecting the points 00 and 0. At point 0, the spacecraft enters into the atmosphere; after traversing the upper layers of the atmosphere, it exits from the atmosphere at point 1; during the atmospheric pass, the velocity of the spacecraft is reduced, due to the aerodynamic drag; in addition, the plane change is performed. At point 1, the spacecraft exits from the atmosphere; right at the exit, a tangential propulsive burn takes place, having characteristic velocity A VI; then, the spacecraft enters into an elliptical transfer orbit connecting the points i and 11; this elliptical transfer orbit is such that its apogee occurs at rtl. The maneuver ends with a tangential propulsive burn, having characteristic velocity A V~, at point 11; here, the spacecraft enters into the low Earth orbit, in that the magnitude of AV~ is such that the desired circularization into LEO is achieved. For the AOT maneuver described above, optimal
trajectories were investigated by the senior author and his associates in[2426], under the additional assumption that (v) the flight time is given. Here, assumption (v) is removed, and the flight time is optimized simultaneously with the lift coefficient history and the bank angle history. Therefore, this paper extends to noncoplanar transfer the approach employed in[27] for coplanar transfer. Specifically, within the framework of classical optimal control, the following problems are studied: (PI) minimize the energy required for orbital transfer; (P2) minimize the time integral of the heating rate; (P3) minimize the time of flight during the atmospheric portion of the trajectory; (P4) maximize the time of flight during the atmospheric portion of the trajectory; (P5) minimize the time integral of the square of the path inclination; and (P6) minimize the sum of the squares of the entry and exit path inclinations. Problems (PI)(P6) are Bolza problems of optimal control. Within the framework of minimax optimal control, the following problems are studied: (Q1) minimize the peak heating rate; (Q2) minimize the peak dynamic pressure; and (Q3) minimize the peak altitude drop. Problems (Q1)(Q3) are Chebyshev problems of optimal control, which can be converted into Bolza problems by suitable transformations. Numerical solutions for Problems (PI)(P6) and Problems (Q1)(Q3) are obtained by means of the sequential gradientrestoration algorithm for optimal control problems. The engineering implications of these solutions are discussed in order to determine, among all of the solutions, the best candidates for flight operations, and hence the best candidates for spacecraft design. Outline. In Section 2, we state the basic equations to be solved for aeroassisted, noncoplanar orbital transfer using lift modulation and bank angle modulation. In Section 3, we review the performance indexes of interest for AOT maneuvers. In Sections 4 and 5, we summarize the numerical experiments performed and present the results obtained using the sequential gradientrestoration algorithm for optimal control problems. Finally, in Section 6, we present the conclusions. 2. BASIC EQUATIONS
In this section, we refer to the atmospheric portion of the trajectory of an AOT vehicle, and we state the basic equations to be solved. We assume that the atmospheric pass is made with engine shutoff; hence, in this portion of the flight, the AOT vehicle behaves as a particle of constant mass. In addition, we neglect Coriolis acceleration terms and transport acceleration terms. Finally, we evaluate the aerodynamic forces using the inertial velocity, rather than the relative velocity. The symbols employed are as follows: r is the local radius; r00 is the HEO radius; r H is the LEO radius;
401
Optimal trajectories for aeroassisted, noncoplanar orbital transfer re is the radius of the Earth; r, is the radius of the outer edge of the sensible atmosphere; and H is the height of the sensible atmosphere. Also, h is the altitude above sea level; V is the velocity; D is the drag; L is the lift; g is the local acceleration of gravity; m is the mass; and z is the flight time. In addition, y is the path inclination, the angle between the velocity vector 17 and the local horizon; is positive if I7 is directed upward with respect to the local horizon; 0 is the longitude, measured along the equator, starting from the fundamental meridian; 0 is positive eastward; q~ is the latitude, measured along the local meridian, starting from the equatorial plane; ~b is positive northward; ~Ois the heading angle, the angle between the projected velocity vector 17eand the local parallel; 17p is obtained by pr~ecting V on the local horizon, and qJ is positive if Vp is directed inward with respect to the local parallel; finall~ a is the bank angle, the angle between lift vector L and the (L 17) plane; a is positive if the spacecraft is banked to the left. Differential system. With the above assumptions, and upon normalizing the flight time to unity, the equations of motion are given by /~ = z[V sin),],
O~
(lb)
= z [ ( L / m V ) c o s a + (V/r  g / V ) c o s y],
0~
0~
q~=~[VcosTsin~b/r],
R = h/hR,
h R = H,
(5a)
= r/rR,
rR = G,
(5b)
I7 = V/VR,
VR = x/(#/r~),
(5c)
f = z/z~,
zR = 2rtGx/(G/IU),
(5d)
P = P/PR,
PR = P~,
(5e)
where the subscript R denotes a reference value. In (5), VR is the circular velocity at r = ra and Pb is the air density at the altitude hb = 40 km. Also, we introduce the dimensionless constants k = rJH,
(6a)
CoR = 2C~,
(6b)
CLR = ~/(COO/K),
(6C)
E , ~ = ( L / D ) ~ , = CLR/CoR = l/~/(4CooK),
(6d)
A = (pRSH/2m)Coo,
(6e)
B = (PR S H / 2 m ) CLR.
(6f)
Finally, we introduce the normalized lift coefficient
(la) 0~
12=r[D/mgsin~,],
Normalized differential system. Next, we introduce the following normalized variables:
(lc) (ld) (le)
which is one of the two control variables of the normalized differential system. Then, upon dropping the tilde, we rewrite the differential system (1) in the form /~ = (2nkz)[V sin 7],
0 ~< t ~< 1,
0~
ksinT/(kl+h)2],
In the above equations, r=r~+h,
,(2)
0~
L = (1/2)CLpSV 2,
(3)
where Co is the drag coefficient, CL is the lift coefficient, p = p ( h ) is the air density, and S is a reference surface. Under extreme hypersonic conditions, the dependence of the aerodynamic coefficients on the Mach number and the Reynolds number is disregarded, and the following parabolic relation is postulated: Co = Coo + K C ~,
(4)
where Coo denotes the zerolift drag coefficient, K C ~ the induced drag coefficient, and K the induced drag factor. In the system (1), the independent variable is the normalized time t, 0 <~ t <~ 1. The dependent variables include six state variables (h, V, ~,, 0, 4~, ~0), two control variables ( Q , a), and one parameter (z).
(8c)
61 = (2nkr)[V cos ~, cos dg/(k  1 + h)cos ~b],
where # denotes the Earth's gravitational constant. In addition, D = ( I / 2 ) C o p S V 2,
(8b)
= (2nkz)[BpV2 cosa + V c o s v / ( k  1 + h ) kcosT/V(kl+h)2],
g = l z / r 2,
(8a)
I2 = ( 2 n k z ) [  A p V Z ( l + 2 2)
= z [ ( L / m V ) s i n a/cos  ( V / r ) c o s 7 cos qJ tan ~b], 0 ~< t ~< 1. (If)
(7)
2 = CL/CLR,
0~
(8e)
(J = (2ukz )[BpV2 sin a/cos ~, 
V cos 7 cos ~, tan (o/(k  1 + h)], 0 ~< t ~< 1.
(8f)
In the system (8), the independent variable is the normalized time t, 0 ~
(9a)
402
A. MIELEet al. (2

2 2  2roo + V 2 cos 2 70 = 0, Vo)roo
(9b)
0o = 0,
(9c)
q~0= 0,
(9d)
~P0= 0,
(9e)
and the exit conditions are represented by h I = 1,
(10a)
+ (Vl + ~) 2cos2 71 = 0,
(10b)
cos q~l cos ~'l  cos(Ai) = 0.
(10c)
[2  (Vl + ~)2]r ~  2rll
Equations (9a) and (10a) are a consequence of the normalization scale for the altitude. Equations (9b) and (10b) are derived from energy conservation and angular momentum conservation applied to the HEOtoentry elliptical transfer orbit and the exittoLEO elliptical transfer orbit, respectively; in particular, equation (10b) accounts for the fact that a tangential propulsive burn takes place at the exit of the atmosphere, incrementing the exit velocity from Vl to V~ + ~, where ~ is a parameter. Equations (9c), (9d), (9e) assume a particular type of entry, with the entry velocity contained in the equatorial plane. Finally, equation (10c) constitutes a relationship between the exit latitude, the exit heading angle, and the prescribed plane change Ai under the assumption that the plane change is performed entirely in the atmospheric part of the trajectory. Note that equation (10c) is derived from the general relation cos tp cos tp  cos i = 0,
0~
(11)
after observing that Ai=illioo=ilio=ilO=i
t.
(12)
To ensure that the vehicle enters into the atmosphere at point 0 and exits from the atmosphere at point 1, the path inclinations 70, 71 must satisfy the inequality constraints
The twosided inequality constraint (16) can be converted into an equality constraint by means of a trigonometric transformation of the type 2 = (1/2)[().a + ).b) + ().b  ,~a)sin fl ],
(17)
where /t(t) denotes an auxiliary control. Equation (17) can be employed to eliminate 2(t) from (8). Therefore, the old control 2(t) is replaced with the new control fl (t). R e m a r k . To sum up, the equations governing the atmospheric pass include the differential system (8), the boundary conditions (9) and (10), the supplementary boundary conditions (14), and the twosided inequality constraint (16), converted into an equality constraint by means of the trigonometric transformation (17). In this formulation, the independent variable is the time t, 0~
Subject to the previous constraints, different AOT optimization problems can be formulated, depending on the performance index chosen. The resulting optimal control problems are either of the Bolza type [see Problems (P1)(P6) below] or of the Chebyshev type [see Problems (Q1)(Q3) below]. P r o b l e m ( P I ) . Minimum Energy. It is required to minimize the energy required for orbital transfer. A measure of this energy is the total characteristic velocity AV, the sum of the initial characteristic velocity A V00, associated with the propulsive burn from HEO, the final characteristic velocity AV H , associated with the propulsive burn into LEO, and the intermediate characteristic velocity AVe. associated with the propulsive burn at the exit of the atmosphere. Clearly, (18a)
70 ~<0,
(13a)
I = A V = AV~o + AVII + A V 1.
?t > / 0 .
(13b)
The characteristic velocities A Voo, A V~I, A V~ are given by
These inequality constraints can be converted into equality constraints by means of the Valentine transformations 70 + q2 = 0,
(14a)
?t  ~2 = 0,
(14b)
where q, ( denote supplementary parameters. C o n t r o l bounds. The presence of upper and lower bounds on the lift coefficient is necessary if realistic solutions are to be obtained. Therefore, we assume
AVoo  x/(1/roo)  (l/roo) Vocos 70,
(18b)
AVII
(18c)
~ N/(l/rll)
0~
(15)
with the implication that 2~ ~< ). ~<2b,
0~
(16)
~)cos 7,,
AVI = ~.
(18d)
P r o b l e m ( P 2 ) . Minimum Integrated Heating Rate. It is required to minimize the time integral of the heating rate at a particular point of the spacecraft, for instance, the stagnation point. Except for a proportionality constant, the performance index is given by
that Cz~CL<~CLb,
 (1/r,l)(V~ +
I
=f0' z "/Pv3°8dt.
(19)
P r o b l e m ( P 3 ) . Minimum Time. It is required to minimize the flight time during the atmospheric pass.
Optimal trajectories for aeroassisted, noncoplanar orbital transfer Here, the performance index is given by I = z.
(20)
Problem (P4). Maximum Time. It is required to maximize the flight time during the atmospheric pass. Here, the performance index is given by I = z.
(21)
Problem (P5). NearlyGrazing Trajectory. It is required to minimize the time integral of the square of the path inclination. Here, the performance index is given by I =
j"
~72 dt.
(22)
0
Problem (P6). NearlyGrazing Trajectory. It is required to minimize the sum of the squares of the entry and exit path inclinations. Here, the performance index is giyen by I=7~+7~
(23)
Problem (Q 1 ). Minimum Peak Heating Rate. It is required to minimize the peak value of the heating rate at a particular point of the spacecraft, for instance, the stagnation point. Except for a proportionality constant, the performance index is given by I = max (x/p V3°"),
0~
(24a)
This problem can be reformulated as that of minimizing the integral performance index[22]
J =
f0 z [ , ~
V3°8]q dt,
(24b)
for large values of q (for example, q = 5, 10, or 20).
Problem (Q2). Minimum Peak Dynamic Pressure. It is required to minimize the peak value of the dynamic pressure. Except for a proportionality constant, the performance index is given by l=max(pV2),
0~
(25a)
403
Remark. Problems (P1)(P6) are standard problems of optimal control of the Bolza type. Problems (Q1)(Q3) are nonstandard problems of optimal control of the Chebyshev type, if treated in the format (24a), (25a), (26a); however, Problems (Q1)(Q3) become standard problems of optimal control of the Bolza type, if treated in the format (24b), (25b), (26b). Inspection of (24a) and (25a) shows that the expressions being minimaximized are the products of the powers of two quantities, the density p and the velocity V. The density p is a fast variable, in that it changes by a factor of order /> l0 a during the atmospheric pass. The velocity V is a slow variable, in that it changes by a factor of less than 2 during the atmospheric pass. From this, it follows that, in order to minimize the peak heating rate or the peak dynamic pressure, one must act principally on the density (hence, the altitude). This suggests the idea of minimizing the peak altitude drop during the atmospheric pass, and hence Problem (Q3). 4. D A T A
FOR
THE
EXAMPLES
The data used in the numerical experiments are summarized in Tables 14. Table 1 refers to the spacecraft data. It shows the mass per unit reference surface m/S, the zerolift drag coefficient Coo, the induced drag factor K, the lift coefficient for maximum lifttodrag ratio CLR, and the maximum lifttodrag ratio Em~. Table 1 also shows the coefficients A, B appearing in the normalized differential system (8). In addition, Table 1 shows the bounds C~, CLb on the lift coefficient CL and the bounds 2a, 2b on the normalized lift coefficient 2, which is one of the two control variables of Problems (PI)(P6) and (Q1)(Q3). Table 2 supplies the major physical constants appearing in Problems (P1)(P6) and (QI)(Q3). This includes the radius of the Earth re, the radius of
t
This problem can be reformulated as that of minimizing the integral performance index[22]
J =
:01
z[pv2]qdt,
(25b)
for large values of q (for example, q = 5, 10, or 20).
Problem (Q3). Minimum Peak Altitude Drop. It is required to minimize the peak value of the altitude drop. Here, the performance index is given by /=max(Ih),
0~
(26a)
T a b l e 1. S p a c e c r a f t d a t a Quantity
m/S Coo K CLR Ema x A B Ct~ Ct, ).~ 2h
Value
Units
0.3000E + 03 0.1000E + 00 0.1110E+01 0.3002E + 00 0 . 1 5 0 1 E + 01 0 . 7 9 9 1 E  01 0.2399E + 00 0.9000E + 00 +0.9000E + 00  0 . 2 9 9 8 E + 01 + 0 . 2 9 9 8 E + 01
kgm 
2
t
This problem can be reformulated as that of minimizing the integral performance index[22] J =
f0lz[1  hlqdt,
(26b)
for large values of q (for example, q = 5, 10, or 20).
T a b l e 2. P h y s i c a l c o n s t a n t s Quantity re ra H /~ k
Value 0.6378E + 0.6498E + 0.1200E + 0.3986E+ 0.5415E +
Units 07 07 06 15 02
m m m m3s 
2
404
A. MIELEet al. where P is the norm squared of the constraint error and Q is the norm squared of the error in the optimality conditions. An upper bound was imposed on the total number of iterations, namely,
T a b l e 3. Reference values Quantity hA r~ VR zR PR
Value 0.1200E 0.6498E 0.7832E 0.5212E 0.3996E
+ + + + 
Units 06 07 04 04 02
m m m s 1 s kg m 3
N ~< 100.
T able 4. T e r m i n a l orbits Quantity
Value
Units
r~
0 . 1 2 9 9 6 E + 08 0 . 6 5 5 8 0 E + 07
m m
roo/r R rjj/r R
0 . 2 0 0 0 0 E + 01 0 . 1 0 0 9 3 E + Ol

i00
0 . 0 0 0 0 0 E + 00 0 . 2 0 0 0 0 E + 02
deg deg
r00
ill
the outer edge of the sensible atmosphere r,, the height of the sensible atmosphere H, the Earth's gravitational constant #, and the ratio k = ra/H. Table 3 supplies the reference values employed for the altitude hR, the radius r R, the velocity VR, the flight time zR, and the density PR. Table 4 presents the radii r00 and r~ of the terminal orbits considered, in both dimensional form and dimensionless form. It presents also the terminal orbital inclinations io0 and i~l. Clearly, roo/rjl ~ 2 and Ai = 20 deg. The atmospheric model assumed is that of the U.S. Standard Atmosphere, 1976 (see[39]). 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).
5. N U M E R I C A L S O L U T I O N S
'Problems (P1)(P6) and (Q1)(Q3) were solved employing the sequential gradientrestoration algorithm (SGRA). This algorithm was programmed in F O R T R A N IV; a F O R T R A N GI compiler was used; the numerical results were obtained in doubleprecision arithmetic. Computations were performed at Rice University using an NASAS9000 computer. The interval of integration was divided into 100 steps. The differential systems were integrated using Hamming's modified predictorcorrector method, with a special RungeKutta starting procedure[40]. Definite integrals were computed using a modified Simpson's rule. Linear algebraic systems were solved using a standard Gaussian elimination routine. For both the gradient phase and the restoration phase of SGRA, the linear twopoint boundaryvalue problem was solved using the method of particular solutions[ 1719]. For each problem, the functional being minimized was suitably scaled. The following stopping conditions were employed for the sequential gradientrestoration algorithm: e ~< E08,
Q ~< E08,
(27)
(28)
As noted in Section 1 and[15], SGRA is available in both the primal formulation (PSGRA) and the dual formulation (DSGRA). For the present problems, some test runs indicated that PSGRA is superior to D S G R A in terms of stability and ability to achieve a solution. Hence, the systematic runs were made with PSGRA. Summary results for the numerical solutions are shown in Table 5. This matrix table presents the values of all the performance indexes (i 8)(26) for all of the solutions of Problems (P1) (P6) and (Q1)(Q3). Clearly, a consistency check requires the minimal values to occur along the principal diagonal of the square matrix table. This is precisely the case, as shown in Table 5. The following comments are pertinent. Solutions ( P I ) and (P2). From the solution (P1), we see that the minimum total characteristic velocity is A V = 1869ms t; on the other hand, from the solution (P2), we see that the total characteristic velocity is AV = 6767 m s 1; therefore, the total characteristic velocity of (P2) is 3.6 times that of (Pl). We note that the solution (P2) is characterized by a much deeper penetration into the atmosphere, Ah = 87 km, than the solution (PI), Ah = 71 km. This explains why the peak heating rate of (P2) is 1.6 times that of (PI), the peak dynamic pressure of (P2) is 5.0 times that of (P1), and the peak total acceleration of (P2) is 18 times that of (PI). The only favorable comment on the solution (P2) concerns the integrated heating rate: that of the solution (P2) is 18% of that of the solution (PI). One surmises that, from the point of view of" total characteristic velocity, peak heating rate, peak dynamic pressure, and peak total acceleration, the solution (P1) is to be preferred to the solution (P2). The latter solution is undesirable for flight operations. It is of interest to compare the total characteristic velocity of the solution (PI), AV = 1869 m s ~, with the corresponding quantity for the twoimpulse Hohmann transfer (AV = 3030m s ~). Clearly, the total characteristic velocity of the minimum energy solution (P1) is 62% of the corresponding quantity for the twoimpulse Hohmann transfer. Solutions (P3) and (P4). From the solution (P3), we see that the minimum flight time is z = 80 s; on the other hand, from the solution (P4), we see that the maximum flight time is r = 734 s. Therefore, the maximum flight time is 9.1 times the minimum flight time. We note that the solution (P3) is characterized by a much deeper penetration into the atmosphere, Ah = 94 km, than the solution (P4), Ah = 71 km. This
Optimal trajectories for aeroassisted, noncoplanar orbital transfer
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~
405
explains why the peak heating rate of (P3) is 4.6 times that of (P4), the peak dynamic pressure of (P3) is 22 times that of (P4), and the peak total acceleration of (P3) is 54 times that of (P4). In addition, the total characteristic velocity of (P3) is 3.7 times that of (P4). The only favorable comment on the solution (P3) concerns the integrated heating rate: that of the solution (P3) is 19% of that of the solution (P4). One surmises that, from the point of view of total characteristic velocity, peak heating rate, peak dynamic pressure, and peak total acceleration, the solution (P4) is to be preferred to the solution (P3). The latter solution is undesirable for flight operations. Solutions (P5) and (P6). These nearlygrazing optimal trajectories extend to noncoplanar orbital transfer the properties of the analogous trajectories introduced in[26, 27] for coplanar orbital transfer. Comparing the solution (P5) with the solution (P6), we see that the solution (P5) is characterized by lower values of the total characteristic velocity, the integrated heating rate, the peak heating rate, and the peak dynamic pressure. One surmises that the solution (P5) is to be preferred to the solution (P6) for flight operations. Solutions (QI), (Q2), (Q3). The solutions (Q1) and (Q2) must be discarded, because their total characteristic velocities [ A V = 3 6 2 1 m s 1 for (Q1) and A V = 3187 m s~ for (Q2)] are higher than the total characteristic velocity of the twoimpulse Hohmann transfer (A V = 3030 m s l). Also, the solution (Q3) must be discarded, because its total characteristic velocity [A V = 2946 m s 1] is nearly the same as that of the twoimpulse Hohmann transfer. Comment. The detailed analysis of the solutions (P1)(P6) and (Q1)(Q3) has led to the discarding of the solutions (P2), (P3), (P6) and (QI), (Q2), (Q3) for a variety of technical reasons. Therefore, we are left with three solutions: the minimum energy solution (PI), the maximum time solution (P4), and the nearlygrazing solution (P5). Upon comparing the solutions (P1), (P4), (P5), we are led to the conclusions below. (i) From the energy viewpoint, we see that the total characteristic velocity of the solution (P4) is 28% higher than that of the solution (P1); also, the total characteristic velocity of the solution (P5) is 25% higher than that of the solution (P1). (ii) From the integrated heating rate viewpoint, we see that the integrated heating rate of the solution (P4) is 38% higher than that of the solution (P1); on the other hand, the integrated heating rate of the solution (P5) is 8% lower than that of the solution (PI). (iii) From the peak heating rate viewpoint, we see that the peak heating rate of the solution (P4) is 26% lower than that of the solution (P1); also, the peak heating rate of the solution (P5) is 33% lower than that of the solution (P1). (iv) From the peak dynamic pressure viewpoint,
406
A . MIELE et al.
Table 6A. Results for A O T , Problem (Pl), m i n i m u m energy Quantity
Dimensionless value
Initial characteristic velocity Exit characteristic velocity Final characteristic velocity Total characteristic velocity
0.1321E 0.1043E 0.2314E 0.2386E
Integrated heating rate Flight time
0.1344E  01 0.9090E  01
0.1817E + 11 0.5212E + 04
0.2443E + 09 0.4738E + 03
W sm 2 s
0.6239E + 00
0.3487E + 07
0.2176E + 07
W m:
0.3175E + 00
0.1226E + 06
0.3893E + 05
N m 2
0.5938E + 00
0.1200E + 06
0.7126E + 05
m
0.4062E + 00
0.1200E + 06
0.4874E + 05
m
0.2602E + 01
0.9810E + 01
0.2553E + 02
ms
2
0.3999E + 01
0.9810E + 01
0.3923E + 02
ms
2
Peak heating rate (t = 0.33) Peak dynamic pressure (t = 0 . 3 5 ) Peak altitude drop (t = 0.36) M i n i m u m altitude (t = 0.36) Peak tangential acceleration (t = 0.35) Peak normal acceleration (t = 0.34) Peak total acceleration (t = 0.34) Plane change
+ + +
00 00 02 00
Reference value 0.7832E 0.7832E 0.7832E 0.7832E
+ + + +
04 04 04 04
Dimensional value
Units
0.1034E 0.8166E 0.1812E 0.1869E
m m m m
+ + + +
04 03 02 04
s ~ s t s ~ s t
0.4770E + 01
0.9810E + 01
0.4679E + 02
ms 2
0.3491E + 00
0.5730E + 02
0.2000E + 02
deg
Table 6B. Results for A O T , Problem (PI), m i n i m u m energy Quantity Entry Entry Entry Entry Entry Entry Exit Exit Exit Exit Exit Exit
altitude velocity path inclination longitude latitude heading angle
altitude velocity path inclination longitude latitude heading angle
Dimensionless value
Reference value
Dimensional value
Units
0.1000E 0.1154E 0.7740E 0.0000E 0.0000E 0.0000E
+ + + + +
01 01 01 00 00 00
0.120OE 0.7832E 0.5730E 0.5730E 0.5730E 0.5730E
+ + + + + +
06 04 02 02 02 02
0.1200E 0.9035E 0.4435E 0.0000E 0.0000E 0.0000E
+ + + + + +
06 04 01 00 00 00
m m s deg deg deg deg
0.1000E 0.8981E 0.4359E 0.5509E 0.1156E 0.3301E
+ + + + +
01 00 06 00 00 00
0.1200E 0.7832E 0.5730E 0.5730E 0.5730E 0.5730E
+ + + + + +
06 04 02 02 02 02
0.1200E 0.7034E 0.2497E 0.3156E 0.6626E 0.1891E
+ + + + +
06 04 04 02 01 02
m m s deg deg deg deg
we see that the peak dynamic pressure of the solution (P4) is 17% lower than that of the solution (P1); also, the peak dynamic pressure of the solution (P5) is 41% lower than that of the solution (P1). (v) From (i) to (iv), we see that the solution (P5) is consistently better than the solution (P4). As a consequence, we are left with two solutions: the minimum energy solution (PI) and the nearlygrazing solution (P5). While the solution (P5) requires 25% more energy than the solution (P1), at the same time it involves less integrated heating rate (8%), less peak heating rate (33%), and less peak dynamic pressure (41%). In addition, the solution (P5) requires less peak total acceleration (23%) than the solution (P1). Therefore, it is felt that both solutions (P1) and (P5) should be considered as candidates for flight operations, and hence candidates for design. Solutions (P1) and (P5). Because the best solutions are the energy solution (P1) and the nearlygrazing solution (P5), we present here these solutions in more detail. For the remaining solutions, see[l]. Summary data for the solutions (P1) and (P5) are presented in Tables 6 and 7. Each table refers to a different optimization problem and contains the following quantities: the initial characteristic velocity,
the exit characteristic velocity, the final characteristic velocity, the total characteristic velocity; the integrated heating rate, the flight time; the peak heating rate, the peak dynamic pressure, the peak altitude drop, the minimum altitude; the peak tangential acceleration, the peak normal acceleration, the peak total acceleration; the atmospheric plane change; the entry values of the altitude, the velocity, the path inclination, the longitude, the latitude, and the heading angle; and the exit values of the altitude, the velocity, the path inclination, the longitude, the latitude, and the heading angle. Finally, plots of the solutions (P1) and (P5) are shown in Figs 1 and 2. Each figure contains eight parts: the altitude history h(t), the velocity history V(t), the path inclination history y (t), the longitude history O(t), the latitude history ~b(t), the heading angle history if(t), the lift coefficient history 2 (t)= Q(t)/CLR, and the bank angle history cr(t). 6. C O N C L U S I O N S
This paper considers both classical and minimax problems of optimal control which arise in the study of noncoplanar, aeroassisted orbital transfer. The
Optimal trajectories for aeroassisted, noncoplanar orbital transfer
407
Table 7A. Results for AOT, Problem (P5), minimumtime integral of the square of the path inclination Dimensionless Reference Dimensional Quantity value value value Units Initial characteristic velocity Exit characteristic velocity Final characteristic velocity Total characteristic velocity
0.1311E 0.1657E 0.2314E 0.2992E
Integrated heating rate Flight time
0.1236E  01 0.1021E + 00
0.1817E + 11 0.5212E + 04
0.2247E + 09 0.5321E + 03
W s m 2 s
0.4192E + 00
0.3487E + 07
0.1462E + 07
W m 2
0.1863E + 00
0.1226E + 06
0.2284E + 05
N m 2
0.5656E + 00
0.1200E + 06
0.6787E + 05
m
0.4344E + 00
0.1200E + 06
0.5213E + 05
m
0.2114E + 01
0.9810E + 01
0.2074E + 02
ms
2
0.3025E + 01
0.9810E + 01
0.2967E + 02
ms
2
0.3690E + 01
0.9810E + 01
0.3620E + 02
ms 2
0.3491E + 00
0.5730E + 02
0.2000E + 02
deg
Peak heating rate (t = 0.51) Peak dynamic pressure (t = 0.52) Peak altitude d r o p (t = 0.54) M i n i m u m altitude (t = 0.54) Peak tangential acceleration (t = 0.52) Peak normal acceleration (t = 0.52) Peak total acceleration (t = 0.52) Plane change
+ + +
00 00 02 00
0.7832E 0.7832E 0.7832E 0.7832E
+ + + +
04 04 04 04
0.1027E 0.1298E 0.1812E 0.2343E
+ + + +
04 04 02 04
m m m m
ss t ss
Table 7B. Results for A O T , Problem (P5), m i n i m u m time integral o f the square of the path inclination Quantity Entry Entry Entry Entry Entry Entry Exit Exit Exit Exit Exit Exit
altitude velocity path inclination longitude latitude heading angle
altitude velocity path inclination longitude latitude heading angle
Dimensionless value
Reference value
Dimensional value
Units
0.1000E 0.1154E 0.5943E 0.0000E 0.0000E 0.0000E
+ + + + +
01 01 01 00 00 00
0.1200E 0.7832E 0.5730E 0.5730E 0.5730E 0.5730E
+ + + + + +
06 04 02 02 02 02
0.1200E 0.9038E 0.3405E 0.0000E 0.0000E 0.0000E
+ + + + + +
06 04 01 00 00 00
m m s i , deg deg deg deg
0.1000E 0.8366E 0.0000E 0.6243E 0.9866E 0.3354E
+ + + + +
01 00 00 00 01 00
0.1200E 0.7832E 0.5730E 0.5730E 0.5730E 0.5730E
+ + + + + +
06 04 02 02 02 02
0.1200E 0.6552E 0.0000E 0.3577E 0.5653E 0.1922E
+ + + + + +
06 04 00 02 01 02
m m s ] deg deg deg deg
maneuver considered involves the transfer from a high planetary orbit to a low planetary orbit with a prescribed atmospheric plane change. An example is the HEOtoLEO transfer of a spacecraft with a prescribed plane change, where HEO denotes high Earth orbit and LEO denotes low Earth orbit. In particular, HEO can be GEO, a geosynchronous Earth orbit. The basic idea is to employ the hybrid combination of propulsive maneuvers in space and aerodynamic maneuvers in the sensible atmosphere. Hence, this type of flight is also called synergetic space flight. With reference to the atmospheric part of the maneuver, trajectory control is achieved by modulating the lift coefficient (hence, the angle of attack) and the angle of bank. The presence of upper and lower bounds on the lift coefficient is considered. Within the framework of classical optimal control, the following problems are studied: (P1) minimize the energy required for orbital transfer; (P2) minimize the time integral of the heating rate; (P3) minimize the time of flight during the atmospheric portion of the trajectory; (P4) maximize the time of flight during the atmospheric portion of the trajectory; (P5)
minimize the time integral of the square of the path inclination; and (P6) minimize the sum of the squares of the entry and exit path inclinations. Problems (P1)(P6) are Boiza problems of optimal control. Within the framework of minimax optimal control, the following problems are studied: (Ql) minimize the peak heating rate; (Q2) minimize the peak dynamic pressure; and (Q3) minimize the peak altitude drop. Problems (Q1)(Q3) are Chebyshev problems of optimal control, which can be converted into Bolza problems by suitable transformations. Numerical solutions for Problems (PI)(P6) and Problems (Q1)(Q3) are obtained by means of the sequential gradientrestoration algorithm for optimal control problems. The engineering implications of these solutions are discussed, and it appears that the energy solution (P1) and the nearlygrazing solution (PS) are superior to the solutions (P2), (P3), (P4), (P6), (Q1), (Q2), (Q3). While the nearlygrazing solution (PS) requires 25% more energy than the energy solution (P1), at the same time it involves less integrated heating rate (8%), less peak heating rate (33%), less peak dynamic pressure (41%), and less peak total acceleration (23%). Therefore, it is felt
408
A. MIEI.E et al.
1.0
1.2
H
v
/
0.5
1.0 ......__..,_..___.
0°0
0.8 0.0
0.5
1.0
Fig. 1A. Problem (P1), altitude h versus time t.
0.0
0.5
T
1.o
Fig. IB. Prob]em(Pl),velocity Vversustimet
0.2
1
OAMMA 0.o
THETA j
~//f'~
f
f...

l
f
o
!
0,2 0.0
T
0.5
o .o
t.O
Fig. IC. Problem (PI), path inclination ~,, versus time t.
PHI
o .5
T
t .o
Fig. 1D. Problem (P1), longitude 0 versus time t.
PSI
o
t
_ f
o
1
1
0.0
0.5
T
0.0
1.0
Fig. 1E. Problem (P1), latitude q5 versus time t.
0 .S
T
1.0
Fig. IF. Problem (PI), heading angle ~k versus time t.
4
3
LAMBOA
f
".
51GMR 2
0
3 0.0
0 .S
T
1.0
Fig. IG, Problem (P1), lift coefficient 2 versus time t.
o.o
o.s
T
~.o
Fig. 1H. Problem (P1), bank angle a versus time t.
409
Optimal trajectories for aeroassisted, noncoplanar orbital transfer
,.2
.
f
0.5
'~"~
~
v
j
I .0
,,,
0.0
0,8 0.0
T
0.5
•o
1.0
Fig. 2A. Problem (P5), altitude h versus time t.
0.2]
o .5
T
t ,o
Fig. 2B. Problem (P5), velocity V versus time t.
'1
ORMMR
THETR
/
//~"~
o.o
1
0.2 0.0
O.B
T
o .o
1.0
Fig. 2C. Problem (P5), path inclination ), versus time t.
1
PHI
PSI
o
o
o .o
o.5
I.o
T
Fig. 2E. Problem (PS), latitude 4~ versus time t.
T
I.o
Fig. 2D. Problem (P5), longitude 0 versus time t.
1
1
o .s
1 o.o
0.5
T
~.o
Fig. 2F. Problem (P5), heading angle q versus time t.
4
LRMSD3R~
SG I MQ
~~'~
2
o
.,./
o
o.o
o.s
T
1.o
Fig. 2G. Problem (P5), lift coefficient it versus time t.
o.o
o.s
T
z .o
Fig. 2H. Problem (P5), bank angle a versus time t.
410
A. MmLE et al.
that both solutions (P1) and (P5) should be considered as candidates for flight operations, and hence candidates for spacecraft design. In closing, we note that the present analysis of noncoplanar orbital transfer is based on the assumption that the prescribed plane change is performed entirely in the atmospheric part of the trajectory[l]. The modifications of the analysis arising when the plane change is performed partly in outer space and partly in the atmosphere [41,42] will be discussed in a subsequent paper. REFERENCES
1. A. Miele, V. K. Basapur and W. Y. Lee, Optimal trajectories for aeroassisted, noncoplanar orbital transfer, Part I. Rice University, AeroAstronautics Report No. 197 (1985). 2. A. E. Bryson Jr and Y. C. Ho, Applied Optimal Control. Blaisdell, Waltham, Mass. (1969). 3. G. Leitmann, The Calculus of Variations and Optimal Control. Plenum, New York (1981). 4. A. Miele, Flight Mechanics, Vol. 1. Addison Wesley, Reading, Mass. (1962). 5. N. X. Vinh, A. Busemann and R. D. Culp, Hypersonic and Planetary Entry Flight Mechanics. The University of Michigan Press, Ann Arbor, Mich. (1980). 6. N. X. Vinh, Optimal Trajectories in Atmospheric Flight. Elsevier, New York (1981). 7. J. P. Marec, Optimal Space Trajectories. Elsevier, Amsterdam (1979). 8. A. Miele, R. E. Pritchard and J. N. Damoulakis, Sequential gradientrestoration algorithm for optimal control problems. J. Optim. Theory Applic. 5, 235282 (1970). 9. A. Miele, J. N. Damoulakis, J. R. Cloutier and J. L. Tietze, Sequential gradientrestoration algorithm for optimal control problems with nondifferential constraints. J. Optim. Theory Applic. 13, 218255 (1974). 10. A. Miele, Recent advances in gradient algorithms for optimal control problems. J. Optim. Theory Applic. 17, 361430 (1975). 11. S. Gonzalez and A. Miele, Sequential gradientrestoration algorithm for optimal control problems with general boundary conditions. J. Optim. Theory Applic. 26, 395425 (1978). 12. A. Miele, Gradient algorithms for the optimization of dynamic systems. In Control and Dynamic Systems, Advances in Theory and Application (Edited by C. T. Leondes), Vol. 16, pp. 152. Academic Press, New York (1980). 13. A. Miele and T. Wang, Primaldual properties of sequential gradientrestoration algorithms for optimal control problems, Part 1, Basic problem. In Integral Methods in Science and Engineering (Edited by A. HajiSheikh), pp. 577607. Hemisphere, Washington, D.C. (1986). 14. A. Miele and T. Wang, Primaldual properties of sequential gradientrestoration algorithms for optimal control problems, Part 2, General problem. J. Math. Anal. Applic. 119, 2154 (1986). 15. A. Miele, T. Wang and V. K. Basapur, Primal and dual formulations of sequential gradientrestoration algorithms for trajectory optimization problems. Acta Astronautica 13, 491505 (1986). 16. A. Miele, Y. M. Kuo and E. M. Coker, Modified quasilinearization algorithm for optimal control problems with nondifferential constraints and general boundary conditions, Parts 1 and 2. Rice University, AeroAstronautics Reports Nos 161 and 162 (1982).
17. A. Miele, Method of particular solutions for linear, twopoint boundaryvalue problems. J. Optim. Theory Applic. 2, 260273 (1968). 18. A. Miele and R. R. Iyer, Modified quasilinearization method for solving nonlinear, twopoint boundaryvalue problems. J. Math. Anal. Applic. 36, 674692 (1971). 19. A. Miele and R. R. Iyer, General technique for solving nonlinear, twopoint boundaryvalue problems via the method of particular solutions. J. Optim. Theory Applic. 5, 382399 (1970). 20. G. J. Michael, Computation of Chebyshev optimal control. A1AA J. 9, 973975 (1971). 21. W. F. Powers, A Chebyshev minimax technique oriented to aerospace trajectory optimization problems. AIAA J. 10, 1291 1296 (1972). 22. A. Miele and T. Wang, An elementary proof of a functional analysis result having interest for minimax optimal control of aeroassisted orbital transfer vehicles. Rice University, AeroAstronautics Report No. 182 (1985). 23. A. Miele and P. Venkataraman, Minimax optimal control and its application to the reentry of a space glider. In Recent Advances in the Aerospace Sciences (Edited by L. Casci), pp. 2140. Plenum4 New York (1985). 24. A. Miele and P. Venkataraman, Optimal trajectories for aeroassisted orbital transfer. Aeta Astronautiea ll, 423433 (1984). 25. A. Miele and V. K. Basapur, Approximate solutions to minimax optimal control problems for aeroassisted orbital transfer. Acta Astronautiea 12, 809818 (1985). 26. A. Miele, V. K. Basapur and K. D. Mease, Nearlygrazing optimal trajectories for aeroassisted orbital transfer. J. Astronaut. Sci. 34, 3 18 (1986). 27. A. Miele, V. K. Basapur and W. Y. Lee, Optimal trajectories for aeroassisted, coplanar orbital transfer. J. Optim. Theory Applic. 52, 124 (1987). 28. H. S. London, Change of satellite orbit plane by aerodynamic maneuvering. J. Aerospace Sci. 29, 323332 (1962). 29. J. P. Paine, Some considerations on the use of lifting reentry vehicles for synergetic maneuvers. J. Spacecraft Rockets 4, 698700 (1967). 30. M. Rossler, Optimal aerodynamicpropulsive maneuvering for the orbital plane change of a space vehicle. J. Spacecraft Rockets 4, 1678 1680 (1967). 31. G. D. Walberg, A survey of aeroassisted orbit transfer. J. Spacecraft Rockets 22, 318 (1985). 32. B. K. Joosten and B. L. Pierson, Minimum fuel aerodynamic orbital plane change maneuvers, Paper No. AIAA810167. AIAA 19th Aerospace Sciences Meeting, St Louis, MO. (198l). 33. J. A. Kechichian, M. I, Cruz, N. X. Vinh and E. A. Rinderle, Optimization and closedloop guidance of drag modulated aeroassisted orbital transfer, Paper No. AIAA832093. AIAA Flight Mechanics Conference, Gatlinburg, Tenn. (1983). 34. M. I. Cruz, Trajectory optimization and closedloop guidance of aeroassisted orbital transfer, Paper No. AAS83413. AAS/AIAA Astrodynamics Specialist Conference, Lake Placid, N.Y. (1983). 35. N. X. Vinh and J. M. Hanson, Optimal aeroassisted return from high Earth orbit with plane change. Paper No. IAF83330. 34th Congress of the International Astronautical Federation, Budapest, Hungary (1983). 36. N. X. Vinh and J. M. Hanson, Optimal aeroassisted return from high Earth orbit with plane change. Acta Astronautica 12, 1125 (1985). 37. K. D. Mease and N. X. Vinh, Minimumfuel aeroassisted coplanar orbit transfer using lift modulation. J. Guidance Control Dynamics 8, 134141 (1985).
Optimal trajectories for aeroassisted, noncoplanar orbital transfer 38. D. G. Hull, J. M. Giltner, J. L. Speyer and J. Mapar, Minimumenergyloss guidance for aeroassisted orbital plane change. J. Guidance ControlDynamics 8, 487493 0985). 39. NOAA, NASA and USAF, U.S. Standard Atmosphere, 1976. U.S. Government Printing Office, Washington, D.C. (1976). 40. A. Ralston, Numerical integration methods for the solution of ordinary differential equations. In Mathematical Methods for Digital Computers, Vol. 1 (Edited
411
by A. Ralston and H. S. Wilf), pp. 95109. Wiley, New York (1960). 41. A. Miele, W. Y. Lee and K. D. Mease, Optimal trajectories for aeroassisted, noncoplanar orbital transfer, Part 2. Rice University, AeroAstronautics Report No. 204 (1986), 42. A. Miele, W. Y. Lee and K. D. Mease, Optimal trajectories for aeroassisted, noncoplanar orbital transfer, Part 3. Rice University, AeroAstronautics Report No. 205 (1986),