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An optimal guidance approximation for quasi-circular orbital rendezvous
An Optimal Guidance Approximation for Quasi-circular Orbital Rendezvous H. J. KELLEY and J.c. DUNN Summary
Three-dimensional guidance about a time-optimal rendezvous flight path is examined within the framework of a quasi-circular orbital dynamics assumption. The guidance scheme, optimal in the same sense as the nominal trajectory, is based formally upon the approximate construction of a field of neighbouring optimal rendezvous paths, following a method developed by Kelley in an earlier paper!. For purposes of illustration, a class of reference trajectories, generated by a low magnitude, circumferentially directed thrust vector, has been adopted for the guidance analysis. Such trajectories are time optimal, under appropriate circumstances, for transfer between neighbouring co-planar circular orbits. The application considered permits an analytical representation of the extremal field which in turn leads to a closed form linear feedback control solution with time-varying gains. Some suggestions are also given for possible modifications which might enhance system accuracy and the range of operability during practical implementation of low-thrust rendezvous guidance. Sommaire
On examine le probleme du guidage dans les trois dimensions au voisinage d'une trajectoire de rendez-vous optimale par rapport au temps, dans le cadre de la dynamique des orbites quasi-circulaires. Le schema de guidage, qui est optimal dans le meme sens que la trajectoire nominale, est formellement base sur la construction approximative d'un champ de trajectoires de rendez-vous optimales voisines, suivant la methode exposee par M . Kelley dans un article recent. Comme exemple, on a adopte pour l'analyse du guidage une classe de trajectoires de reference produite par un vecteur circonferential de faible poussee. De telles trajectoires sont, dans des circonstances appropriees, optimales par rapport au temps pour le transfert entre des orbites circulaires voisines dans le meme plan. L'application consideree autorise une representation analytique du champ extremal qui, a son tour, mene a une solution de commande sous forme lineaire fermee a reaction, avec ga ins variant dans le temps. En conclusion, on presente quelques suggestions, qui ont, pour objet d'ameliorer la precision du systeme et le domaine de fonctionnement dans la realisation pratique du guidage de rendez-vous a faible poussee. Zusammenfassung
Das Problem der 3-dimensionalen Lenkung, urn eine zeitoptimale Flugbahn flir Begegnungen von Raumfahrzeugen (Rendezvous-Technik) zu erzielen, wird im Rahmen einer quasi-kreisformig angenommenen (dynamischen) Bahn untersucht. Der Lenkvorgang, im gleichen Sinne optimal wie die Bezugsbahn, stiitzt sich formal auf die angenaherte Konstruktion eines Bereiches (Biindel) von nebeneinanderliegenden optimalen Begegnungsbahnen; diese Methode wurde in einem friiheren Aufsatz von Kelley entwickelt. Zur Erlauterung wurden eine Reihe von Begegnungsbahnen, die ein in Richtung der Peripherie zeigender Schubvektor von geringem Betrag erzeugt, fiir die Untersuchung der Lenkung angenommen. Solche Flugbahnen sind zeitoptimal, sie eignen sich unter entsprechenden Bedingungen zum Uberwechseln zwischen zwei nebeneinanderliegenden, in der gleichen Ebene befindlichen kreisformigen Bahnen.
274
Die betrachtete Anwendung laBt eine analytische Darstellung des extremen Bereiches zu, was auf einen Regelkreis mit zeitveranderlicher Verstarkung flihrt. Der Aufsatz enthalt einige Vorschlage flir mogliche Abanderungen, die die Genauigkeit und den Operationsbereich wahrend der prak· tischen Lenkung mit geringem Schub erhohen konnen.
Introduction The second-order guidance approximation scheme employed in this paper has been developed in an earlier publication 1 . Essentially, the idea is to select a flight path, optimized in some appropriate sense, as a nominal trajectory, and then to base guidance upon a family, or 'field', of optimal trajectories approximated in the vicinity of the nominal. The investigation, as is shown later, applies this scheme to the guidance problem for orbital rendezvous. The method is tractable for a fairly wide class of problems, although in general it must be carried through numerically. However, in the present case an analytical treatment becomes feasible because of the recent availability of a particularly simple optimal transfer manoeuvre suitable for use as a nominal trajectory. The nominal manoeuvre, as it appears in this paper, is a direct outgrowth of a co-planar circular orbit transfer analysis conducted by Hinz2 . The problem posed by Hinz, although phrased somewhat differently with regard to coordinate systems and assumptions employed in deriving the equations of motion, is mathematically equivalent to the nominal transfer manoeuvre problem investigated herein. Both cases yield to an analytical treatment of the boundary value problem whenever the manoeuvre duration is an integral multiple of a reference orbit's period. The analysis of three-dimensional rendezvous guidance for the class of trajectories discussed above leads directly to a synthesis of a linear feedback control solution with timevarying gains given in closed form. Some suggestions are also given for possible modifications which might enhance system accuracy and the range of operability during practical implementation of low-thrust rendezvous guidance. The Differential Equations of Powered Flight Considerations begin with the differential equations of threedimensional powered flight in a central inverse-square force field (Figure 1): i'- reos 2 tf;(J2 -
r!jJ2 + ~ =~. veos p sin Cl. r m
. re
.. .
F 8+2--2 tan tf; tf;8= tf;' veos peos Cl. r mreos
(1)
AN OPTIMAL GUIDANCE APPROXIMATION FOR QUASI-CIRCULAR ORBITAL RENDEZVOUS
Finally, if note is made of the fact that kl R5w~ = I, then eqns (1) can be put into the following reduced first-order form:
.. fti; 1 . F t/t+2-+-2 sin2 t/t 82 = - ' vsinfJ r
n1r
. vF m=--'
2' 2 1 F* . =(1+I1)(l+u) cos t/t-(l+11)2+(i+OvcosfJsm:x
,..
U
C'
where F is the maximum thrust level of the reaction engine; C is the propellant exhaust velocity; p is a throttle variable; and III is the instantaneous vehicle mass. The difficulty in obtaining any sort of particular solution for these equations needs no comment here, except that it provides a motive for the simplifications which are now introduced. The object is to devise certain assumptions which will allow the replacement of eqns (1) by an approximate set of differential equations which are linear in the state variables r, 0, and t/t, and their time derivatives and, preferably, separable in the control variables P, eX, and {J. (Simplifications of this kind are required to make flight path optimization and guidance problems analytically tractable.) To be specific, it is preferable that these approximate differential equations describe low-thrust acceleration transfer trajectories between neighbouring circular orbits.
, 2u(1+u) u =-----+2w(1+v)tant/t 1+11 F* + (1 + 11)( 1 + ~) v cos f3 cos 'l. , 2 uw 1 ( · 2·, F* . f3 w = - 1 + 11 - T 1 + u) sm 2 ifJ + (1 + 11)( 1 + 0 v sm
11' =U
(2)
8'=V
,//=w F*
z
~'=--v
C*
where F* = FlmoRo 11'6 is the reduced maximum thrust acceleration; C* = Cl Ro Wo is the reduced exhaust velocity; and the superscribed prime denotes differentiation with respect to the reduced time, T. Now let us assume that F*,~, u, v, 11',11, and t/t are all terms of order 6 or smaller (0 < < I). Under these circumstances, one would therefore anticipate that all terms of order 02 in eqns (2) will become negligible with respect to terms of order O. Thus the following simplified differential equations are arrived at:
x
u' = 2 v + 311 + F*v cos fJ sin 'l.
y
Figure 1. Coordinate geometry
v' = - 2 u + F*\' cos fJ cos Cl.
The following set of dependent and independent variable transformations will prove useful for our purposes. Let
and
w' =
-
r(t)=Ro [1 +11(t)]
11' =
11
o(t) = r (t) + B (t)
s'=v
m(t)=mo[1+~(t)]
(3)
t/t'=w
where Ro is the radius of a circular reference orbit situated in the t/t = 0 plane (Figure 1); mo is some reference mass; and T is a fictitious angle defined by the differential expression
dr dt =wo,
t/t + F*v sin fJ
r(O)=O
where Wo is the reference orbit's period. Furthermore, since T(t), as defined above, is a monotonic time-like parameter, it is permissible to change the independent variable in eqns (I) from t to T. This can be accomplished by simply relating t derivatives to T derivatives as follows:
d dr d d dt=dt dr=wodr
It may be said that, to the first order of small quantities, eqns
(3) * are descriptive of quasi-circular flight for the following reason; if, as has been assumed, 1I, v, I}, etc. are of order b, then it follows at once that
where E and h are specific energy and angular momentum respectively. Consequently, the energy-momentum images of
* These equations are identical in form to the differential equations of Wheelon 3 and Anthony4. However, the dependent variables and thrust vector steering angles are not subject to the same interpretation. In particular, the quantity E in eqns (3) is not required to be small-an important point in the subsequent development. 275
H. J. KELLEY AND J. C. DUNN
trajectories which are adequately described by eqns (3) should everywhere be close to the locus of circular orbits in the E-/z phase plane (Figure 2). Clearly, the validity of the quasi-circular differential equations will be compromised when the parameters F*, F* / C*, and T exceed certain critical values. Just precisely what these critical values are cannot be determined until the nature of the control schedule peT), .x(T), and (3(T), is specified. The reader is advised to bear this in mind in the sequel.
./
oH
(6)
A·=--
,
oX i
X;=~~,
(7)
[eqns(3)]
together with the corresponding natural boundary and transversality conditions, (Sa) (6a)
~-----------------.h
XiO=o,
E=- ~
HI = F*V( Al cos (3 sin a+ A2 cos (3 cos a + A3 sin (3 -
E
Figure 2. Energy-momentum phase diagram
Optimal Transfer Between Neighbouring Circular Orbits The optimal orbit transfer problem may be stated as follows: given two neighbouring circular orbits, find the steering angles .x(T) and (3(T) and the throttle schedule peT) which produce a transfer between the two orbits in minimum time. For present purposes, the case for which the terminal orbits are co-planar will be selected, a class of optimal transfer paths within the framework of the quasi-circular dynamics assumption derived, and later, these paths employed as nominal trajectories for the three-dimensional rendezvous guidance analysis. To reiterate, if the subscripts 0 and f denote initial and final conditions respectively, a search is made for a set of control functions .x (T), (3 (T), and peT) which minimize Th produce a state transition which evolves in accord with eqns (3), and which satisfies the circular orbit boundary conditions, viz: at T = 0, U = v = W = t) = (' = 1/1 = 0, ~ = 0; at T = Tt u = W = 2 v + 3}) = 1/1 = 0, 1} = K (const.). The problem so stated is a Mayer variational problem with bounded control variables. The necessary conditions to be satisfied by its solution are well known and are written here for the present application without further comment: Let Xi = state variables, u, v, W, 1}, E, 1/1, ~ y" = control variables P, iX, (3 }'i = multiplier functions / l; = undetermined constant multipliers P = function to be extremized = Tj + Al (Uj) + .112
=
Hamiltonian function
=
(8)
However, note that the final mass, /no (1 +~j) does not appear in the pay-off, i.e., the final mass is left open. However, only those trajectories for which l~f I is of order 0 are admissible because of assumptions implicit in eqns (3). Therefore, eqns (6) and (6a) imply that }'7 == O. Consequently, eqn (8) simplifies to:
HI =F*V(Al cosj1sina+A2 cosj1cosa+}'3 sin (3)
(8a)
The requirements on the control variables iX, (3, and l' are determined by reasoning as follows. Since iX is unbounded, oHdC)(X = 0 and o2HI/O(X2 :2 0 at the minimum of HI and hence,
siniX= -Al /(}ci+A~)t,
cosiX= - A2/u i+A~}t
(9)
HI reduces to:
Hl=F*v[ -V,i+A~)tcosf3+;'3sinf3J
(10)
which can be written in the form
HI
=
F*v v .i + ;,~ + A~yt sin (f3 -
.
Sill
-1
(Ai+A~)t
(Ai + A~ + A~r}
( 11)
By virtue of the fact that sin qJ :2 0, it follows that the principal value of
and HI reduces still further to:
i
Then the following equations and inequalities must be satisfied,
i.e., the optimal control y minimizes the function H .
~:)
( - ~2 -
~ }'iX;'
H(y)~H(Y)
(7 a)
Now, a minimum of H is attained at a minimum of HI' where HI is that part of H which depends on the control variables, y. For the problem here,
Reference orbit A O-neighbourhooc of the point (EO.h O)
H
i=1, ... ,7
Xl,=x 3,=2x2 ,+3 X4J=X6,=0, x 4,=K
Locus of circular orbits
(5)
HI
= -
F*v (Ai + A~ + I.~)t
(3)
* Notice that Hl is also stationary with respect to this value of (3, i.e., (OHI! 0(3) (3 = o. 276
AN OPTIMAL GUIDANCE APPROXIMATION FOR QUASI·CIRCULAR ORBITAL RENDEZVOUS
From eqn (13) it follows immediately that whenever
'V =
where A is a matrix of constant coefficients and g is a vector whose elements depend upon the control variables iX, {J, and 'V. Solutions for eqn (17) can therefore usually be phrased in terms of a fundamental solution matrix tJ> (T f, T) and superposition integrals, e. g. :
1 minimizes HI
(14) Furthermore, it can be verified [by solving eqns (6)] that + A~ + A~ i' 0 except at a finite number of points on any T interval of length 2n. Thus the indeterminate values of 'V corresponding to Ai + A~ + A~ = 0 form a set of measure zero and our problem is therefore well behaved. In summary, the optimal control variables depend upon the multipliers }'l' .12 , and }'3 in the following manner:
Ai
x(rI)=rJ:>Crj,O)x(O)+
f
rj
rJ:>(rI,r)g[y(r)]dr
0
Eqn (19)
(18)
*
,
where
(15)
g2=F*vcos{Jcos'Y.=
A2F*
---.=----
(Ai + A~ + }.~yl:
The A's in turn depend on the unknown Lagrange multipliers, .1i through eqn (6), and the solution of the boundary value problem turns upon one's capability to solve for these undetermined constants. However, before attention is directed to the boundary value problem, it is worth while to emphasize two points. First, it is easily demonstrated that the optimal control variables given by eqns (15) satisfy a strong form of eqn (5), i.e.,
g3=
F*
. {J vsm = -(12
)'3 F *
12
12)t
(20)
/'1 +'"-2+'"-3
(16) But eqn (16) together with the linear character of eqns (3), are sufficient conditions for a strong relative minimum of P. Thus it is assured that the control law of eqns (15) will provide a time-optimal transfer between two co-planar circular orbits which is unique whenever the boundary value equations possess a unique solution. Second, it should be noted that the result expressed by the last of eqns (15) is independent of the multiplier functions Ai, and is therefore insensitive to the boundary conditions. Thus a full throttle operational mode is a characteristic of the entire extremal field of quasi-circular transfer trajectories for the problem of minimum time transfer with final mass open.
As already pointed out, the }:s depend on the undetermined constants Ai in accord with eqn (6a). Furthermore, it is noted that in matrix notation, eqns (6) have the form
(21) and thus are adjoint differential expressions for eqn (17) [Le., eqns (3)]. Consequently, their solution is determined when tJ> (Tt> T) is known, i.e.,
T
},(r)=1> ).(r/)=1>
T{ap} ~
(22)
X'j
The Orbital Transfer Boundary Value Problem-a Special Class of Solutions for Rendezvous
where
The differential equations, eqns (3), when written in a symbolic matrix notation, have the following structure:
x' =Ax+ g(y)
* Eqn
(19):
cosr -2sin r 0 rJ:> (rI' r)= sin r -2(1-cosr) 0 0
2sin r -(3-4cosr) 0 2(1-cosr) -(3r-4sinr) 0 0
(17)
0 0 cosr 0 0 sin r 0 277
In view of these considerations, the co-planar circle-to-circle transfer boundary conditions [see eqn (7)] are:
3 sin r -6(1-cosr) 0 (4-3cosr) -6(r-sin r) 0 0
0 0 0 0 1 0 0
0 0 -sinr 0 0 cosr 0
~l
0 0 0 0 1
(19)
H. J. KELLEY AND J. C. DUNN
(23)
transfer between two neighbouring co-planar circular orbits differing in altitude so that I(R j - RA)/RAI = 4nn P. This result was first obtained by Hinz 2 in an analysis only slightly different from that presented here. The relatively simple optimal orbit transfer manoeuvre just described can be used as a reference rendezvous trajectory for the guidance analysis, if one imagines that the transfer is initiated at the proper time. For example, if, as in Figure 3, a target at point B in orbit 2 leads the vehicle initially at point A in orbit 1 by an angle ETo, then it is necessary that ETo = 3rJl/8F* in order to effect a rendezvous at point C. The nature of the transfer is such that points Band C are symmetrically deployed with respect to the initial line OA.
Initiattarget position
where
Al =
A4[~~ COSi+(I- ~:)Sin iJ
L/
,Reference line rotating with cons.tant angular velocity -eA
'W,eA _---r--_
Target orbit -
C
Rendezvous point "'. Transfer path
A2=2A4[~~ sin i-(l- ~:)COSi+lJ and
(24)
Lead angle for .· rendezvous """""'"
The natural boundary condition on H [see eqn (5)] determines the magnitude of the scaling constant A4 e. g., Figure 3. Rendezvous /fight-path geometry
The Optimal Rendezvous Guidance Approximation
and therefore
1 A I 41 =
[(
F*
~~)2 + (~: )2 + 4 (~:)2J-!-
(25)
The sign of A4 will depend upon the sign of K. When eqns (24) and (25) are substituted in eqns (23), a set of five boundary value conditions transcendental in five unknowns, A 1/A 4, A 2/A 4 , A3/A4' A5/A4' and Tj are obtained. In general, it is not possible to arrive at an analytical solution of the boundary value problem so stated for arbitrary values of K. However, the following observation is made: if one sets / 11/A4 = A 3/A 4 = A5//14 = 0, and A2/A4 = I, then eqns (15), (24) and (25) indicate that:
sin
p=O,
cos
-
cosp= 1
sgn (A4) (26)
which conditions in turn prescribe an in-plane circumferentially directed thrust vector. Furthermore, it should be noted that the first four of eqns (23) are identically satisfied by such a steering programme whenever Tj = 2nn, and the fifth is satisfied for this value of final time when K = - 4nn F* sgn (A4)' Thus. in view of the assumptions, and the sufficiency argument connected with eqn (16), it is concluded that full-throttle in-plane, circumferentially directed thrust is time-optimal for ascending (sgn ,14 = - I) or descending (sgn ; 14 = + I) thrust-limited 278
With a class of time-optimal rendezvous trajectories available for reference flight paths, one is now in a position to consider optimal rendezvous guidance. Presumably, the guidance scheme selected should preferably be optimal in the same sense as the nominal trajectory. Small disturbance assumptions should also be invoked, thereby obtaining a relatively simple guidance law. However, because of the stationary character of an optimal flight path, an immediate conflict of interest arises. It is found that, to the first order of small control variations, all guidance schemes which satisfy the final boundary conditions of the original problem are equally attractive, i.e. , performance is insensitive to small control variations. Thus, if one wishes to discriminate among various modes of guidance, it is necessary to consider second-order control effects at least. Because of the anticipated difficulties associated with secondorder analyses, and the apparent needlessness for such a refinement (when, for example, the expected initial condition errors are truly small), most of the effort expended in the past has been directed at finding any simple correction law which satisfies the prescribed final end conditions. A common approach involves the construction of an artificial variational problem with some attendant quadratic 'loss' functional to be minimized, the object being to obtain a simple control law which results in the restoration of final boundary values. Recently, KelleyI, and Breakwell and Bryson5 , have experimented with another approach to the guidance problem. Their considerations have been based on the theory of the second variation and have led to a treatment formally similar to that
AN OPTIMAL GUIDAN~E APPROXIMATION FOR QUASI·CIRCULAR ORBITAL RENDEZVOUS
of Jacobi's accessory minimum problem in classical variational theory. The method is rational in the sense that it selects a guidance scheme on the basis of the original criterion for optimality to an approximation correct as far as second-order control variation effects are concerned. Although this new approach will ordinarily lead to a numerical attack on the control law synthesis problem, the method's effectiveness in an analytical treatment of time-optimal rendezvous guidance for the class of nominal trajectories derived in the previous section is now demonstrated. A general theoretical discussion of the guidance theory may be found in Kelley's workI . Here the necessary equations pertinent to our application are simply written down and the analysis is carried through to a closed-form solution. If (jp, (j(X, and ofJ represent small deviations from the nominal (optimal) control schedule, then the time histories of these quantities, consistent with control bounds, which restore final end conditions and minimize transfer time are searched for. These time histories are assumed to commence from some 'initial' time To (not necessarily the initial point T = 0 on the nominal trajectory) at which excursions from the reference flight path, OXi (TO) are detected. Ultimately, the control increments (jv, (j(X, and ofJ will be expressed as functions of the OXi (T) as T ranges from 0 to Tf (feedback form). First, it is noted that within the limits of a second-order approximation, all guided trajectories are required to belong to the same time-optimal extremal field as the reference flight paths. However, it has already been pointed out that full throttle operation (v = 1) is characteristic of this field and therefore OP == O. Thus, only two control variations remain for timeoptimal rendezvous guidance, viz., O(X and ofJ. It is observed that for every member of the extremal field, HI is stationary with respect to (X and fJ, i. e., min HI -+ oHj"o(X = = oHjofJ = 0 [see the discussion immediately preceding eqns (9), and the footnote relating to eqn (12)]. Consequently, our second-order approximation [see Ref. 7] demands that:
b
G:)
blc l =cos i bA l , - 2 sin i bA21 + sin i 6}'41 + 2 (1- cos i) 6..1. 51 and
(29) The undetermined (jAif's are interrelated by a linearized version of the natural boundary condition for H [eqn (5a)] viz: (
Xl 6..1.1 + A2 bAz + X3 bA3) = 0
JXi+A~+A~
(30)
J
or since
Al==A3==0 (31)
6)'2/=0
In any event, when eqns (28) and (29) are combined, and coefficients of like terms collected, the final result is:
(32) where the C;'s are undetermined independent constants which parametrize the guidance boundary value problem. A formulation of the rendezvous guidance boundary conditions requires the solution of the state equations, eqns (3), linearized in the steering angles (X and fJ, i. e.;
6u' = 3 611 + 2 6v + F* bet 6v'= -26u bw'= -6t/J+F*6{3
6et< < I
61'1' =6u
6{3< < 1
(33)
6e' =6v
== F* [(blc l cos iX - bAl sin iX) cos J5
- 0:1 sin iX + Az cos iX) betJ = 0 b ( ~; ) ==F*{-(bAlsiniX+bA2cosiX)sinJ5+bA3cosJ5
(27)
where Oll, ov, etc., are excursions from the nominal state. An approximate but consistent solution is easily obtained for these equations and the boundary conditions can be written in the following form:
Ilu J= bUJ+ F*
- [(Xl sin iX+ Az cos iX) cos J5 + A3 sin J5J bp} =0
JTI
cos! (C 1 cos i + Cl sin i + C 3) d, =0
TO
where the superscribed bars denote nominal quantities, and the (j)./s in the present problem satisfy a set of differential equations identical in form to eqn (21). For the class of reference flight paths which are considered here, sin (X = sin fJ = 0, cos (X = sgn K, cos fJ = 1, and ,12 = - 1j F* sgn K [see eqns (24), eqns (26) and the accompanying text]. Therefore if, without loss in generality, guidance about an ascending rendezvous path (sgn K = + 1) is examined, then eqns (27) imply that: and
(28) Furthermore, since the (j,1/s evolve in accord with eqn (21) it follows at once from eqns (22) and (19) that: 279
-
Ilv J =6vJ-F* JTI 2sini(Clcosi+Clsini+C 3)d,
Ilw J=6w J +F*
J\
co,f(C,co,f+C, ,in f)
d,~:F' "f~O
~
Il11J=611J+F*f~1 sin i(C l cos! + Cl sin i+C 3)d,=0 TO
(~
H. J. KELLEY AND J. C. DUNN
Considerations on Rendezvous Guidance Implementation
where: (a) TO is the point in time at which corrective action is initiated; (b) the QXi/S are dispersions which would occur at the terminus of the nominal trajectory in the absence of corrective guidance; (c) btt is a variation in the final time, Tt «(ht < < 1); and (d) the coefficients of QTt in each equation are to be interpreted as state variable closing rates, i. e., differences between vehicle and target state variable derivatives on the nominal path at T = Tt (notice that only v has a non-zero closing rate). A total of six equations linear in the six unknowns Cl' ... , C5 , and QTt are at our disposal. Their solution may be obtained most conveniently by first combining the second and fourth of eqns (34) to obtain the result,
bVf+2bl'/f= -F* bTf
In this section, several ways to enhance system accuracy and range of operability in practical implementation of lowthrust rendezvous guidance are suggested. First of all it is noted that real time generation of the system gains and the state variables along the nominal trajectory will be required in practice. In the case of command guidance, these would presumably be calculated 'on the fly' by a ground-based computer, whereas if a vehicle-borne computer were employed, these quantities would more likely be stored in polynomial approximation. Now, with regard to the computation of gains, it is worth calling attention to the advisability, in either case, of employing double precision arithmetic in the calculation of the determinants of the Appendix, since they tend rapidly toward zero as time-to-go approaches zero. This, of course, does not necessarily imply the need for a highly precise representation of the gains themselves. As concerns the generation of the state variables along the nominal trajectory, it should be pointed out that over the course of many revolutions there will be an accumulation of error stemming from higher-order effects if the linear equations are used. This error will place an additional burden upon the guidance system. The situation may be alleviated somewhat by introducing second-order corrections to the nominal in the following approximate fashion. Second-order terms of the first two equations of the system (5), which are significant along the nominal, are p= -31'/2+2I'/v+V2
(35)
which may then be used to eliminate ('iLt. The remaining equations can then be written in matrix notation (see Appendix), e. g. :
(36)
and
(37) where A (TO) and B (To) are matrices whose elements depend upon To and J and K are constant matrices. But the QX i! are related to the QXi (lO) through eqn (19). Therefore the constants Ci may be expressed as functions of the 'initial' excursions at time TO, i.e.:
I
Cll C 2 =A-l(TO) C3
]f]Jl(To)
jbUl !~J &
detjAj#O
q= -2uv+2ul'/-F*1'/
(40)
If p (t) and q (t) are estimated by insertion of the nominal
values of u, v, and 'Yj as computed via the Iinearized equations, corrections QUe, QV e, QT)e and oee, corresponding to the forcing terms p and q, may then be calculated by means of the influence functions presented earlier. The results take a particularly simple form if the integrals are evaluated at l' = 2mT, m an integer, 0::;; m::;; n, namely
(36a)
<0
bUj(2 mn)=22F*2mn
(37a) where f]Jl and f]J2 are submatrix elements of the complete state transition matrix [eqn (19)]. Eqns (32), (36a), and (37 a) then provide the open loop solution for the steering angle corrections, viz:
8"
~
{cod, ,in ;,1} [ ' ('0) J
I~~) l&
and
b{3t= {cos i, sin i} B- 1 (To) K f]J2 (To)
{bW} blj;
(38) to
(39) <0
TO
bv;(2 mn)=F*2 (68 m 2 n 2 - 80 mn) 151'/;(2 mn)=F*2 (-16 m 2n 2+40 mn) &;(2 mn)=F*2(36 n+8 m 3 n 3 )
(41)
If these are employed as corrections to the nominal values with 2mn replaced by 1', the result is incorrect by the omission of certain oscillatory effects; however, it is thought that these will be unimportant in comparison with the secular terms which are properly accounted for. In any event, the idea is to provide the closed loop control system with some anticipation for errors accruing on account of non-linearity. One of the most significant limitations arising from the approximations made in the foregoing analysis is the restriction that the change in arrival time from the nominal be small compared with the orbital period. Shift in arrival time is directly proportional to the difference between actual and nominal values of the linearized energy parameter, v + 21). Furthermore, analysis reveals that relatively large steering corrections are associated with errors in this variable. Since the energy parameter is a monotonicaIIy increasing function of time, it therefore appears reasonable to consider this
280
AN OPTIMAL GUIDANCE APPROXIMATION FOR QUASI-CIRCULAR ORBITAL RENDEZVOUS
parameter as a candidate for independent variable in the mechanization of the system. According to this scheme, the system gains and the state variables along the nominal would be generated (or stored) as functions of energy-to-be-gained, the difference between the terminal and instantaneous values of v + 2YJ. In an analysis conducted with an independent variable having fixed terminal value, complexities arising from shifts in terminal time would be avoided and there would be one less approximation required. There is another point to which scant attention has been paid in the present investigation, which is essentially a feasibility study, and this is the choice of reference orbit and the position of the target vehicle in relation to the reference axis system. We have tacitly assumed the target to be moving in a perfectly circular orbit and chosen our axis system location for analytical convenience as being at the initial point of the nominal orbit transfer trajectory which terminates at the target. It seems clear for a number of reasons that the appropriate choice in actual implementation of a system would be a reference circular orbit having the period of the target's orbit and a reference axis system moving along this circular orbit in the vicinity of the target. One advantage of such a choice is the facilitation of guidance in terms of relative position and velocity measurements. Finally, it is pointed out that by regarding F* as an auxiliary state variable (as opposed to a parameter) defined by the differential equation, F*' = 0, it then becomes possible, with little real
increase in complexity, to derive an additional term for the guidance law whose function would be to counteract the effects of instantaneous fiuctuatinns in the reduced thrust-acceleration level. The procedure for treating system parameters as auxiliary state variables is discussed at greater length by Kelleyl. Conclusions The low-thrust quasi-circular orbital rendezvous example, presented in this paper, illustrates a possible practical application of an optimal guidance scheme developed in a previous publication. The closed-form result obtained for guidance corrections should be safely applicable for correctional manoeuvres during the last several revolutions of a low-thrust ascending (or descending) approach to rendezvous. Some numerical computations designed to establish the range of validity of the guidance approximation are currently in progress. The authors extend their gratitude to Mr. Frank Sobierajski and Mrs. Agnes Zevens of Grumman's Research Department: to the former, for assistance in preparing and verifying several of the foregoing analytical results; to the latter, for the preparation of the figures appearing ill this paper. This research was partially supported by the USAF Office of Scientific Research under Contracts AF 29(600)-2671 and AF 49(638)-1207.
Appendix The matrices appearing in eqns (36) through (39) are:
1-1 F*
J=2.
0 0 0
0 0
~1
0
-1 0
[~ ~J
1 K=--
F*
[
(2nn- Lo) _ (sin 2 LO)J .
2
(l-COS2LO) ------
4
(1- cos 2 LO)
A=
-sin LO
4
[
4
(2nn- Lo) 2
+
(SiI12Lo)J 4 (4 nn-2 LO +2sin LO)
[2nn-LoJ-2sinLoJ
B=
e-C~S2LO )
ct>l=
(2nn-Lo 2
+
Sin2Lo) 4
COS Lo
-2sin LO
3 sin LO
2 sin LO
-(3-4coS'r 0 )
- 6 (I-cos LO)
-sin LO
2(1-cos Lo)
(4-3coSLO)
-2 (I-cos LO) - (6nn - 3 LO + 4sin Lo)-(12 nn - 6 LO + sin LO) 281
o o o 1
H. J.
4>2=
COS T O
sin To
-sin TO
COS To
[
KELLEY AND
1
J. C.
DUNN
References 1
also if 2
2 nn-To x= 2 ,O::<;;x::<;;nn
3
4
det IAI =4x 3 -4 sin 2 x(3 -sin 2 x) x +8 sin 3 x cos x #0 for x>O 5
282
KELLEY, H. J. Guidance theory and extremal fields, Inst. Radio Engrs. Nat. Aerospace Electronics Conf, Day ton, Ohio (May 1962); Trans. Inst. Radio Engrs., AC (October 1962) HINZ, H. K. Optimal low-thrust near-circular orbital transfer, AIAA Jnl. (in the press) WHEELON, A. D. Midcourse and terminal guidance, Space Technology. 1959. New York; Wiley CLOHESSY, W. H. and WILTSHIRE, R. S. Terminal guidance system for satellite rendezvous, J. Aerospace Sci. Vo!. 27, No. 9 (September 1960) BREAKWELL, J. V. and BRYSON, A. E. Neighboring-optimum terminal control for multivariable nonlinear systems, Meeting Soc. Industr. Appl. Math., Cambridge, Massachusetts (November 1962)
Three-dimensional guidance about a time-optimal rendezvous flight path is examined within the framework of a quasi-circular orbital dynamics assumption. The guidance scheme, optimal in the same sense as the nominal trajectory, is based formally upon the approximate construction of a field of neighbouring optimal rendezvous paths, following a method developed by Kelley in an earlier paper!. For purposes of illustration, a class of reference trajectories, generated by a low magnitude, circumferentially directed thrust vector, has been adopted for the guidance analysis. Such trajectories are time optimal, under appropriate circumstances, for transfer between neighbouring co-planar circular orbits. The application considered permits an analytical representation of the extremal field which in turn leads to a closed form linear feedback control solution with time-varying gains. Some suggestions are also given for possible modifications which might enhance system accuracy and the range of operability during practical implementation of low-thrust rendezvous guidance. Sommaire
On examine le probleme du guidage dans les trois dimensions au voisinage d'une trajectoire de rendez-vous optimale par rapport au temps, dans le cadre de la dynamique des orbites quasi-circulaires. Le schema de guidage, qui est optimal dans le meme sens que la trajectoire nominale, est formellement base sur la construction approximative d'un champ de trajectoires de rendez-vous optimales voisines, suivant la methode exposee par M . Kelley dans un article recent. Comme exemple, on a adopte pour l'analyse du guidage une classe de trajectoires de reference produite par un vecteur circonferential de faible poussee. De telles trajectoires sont, dans des circonstances appropriees, optimales par rapport au temps pour le transfert entre des orbites circulaires voisines dans le meme plan. L'application consideree autorise une representation analytique du champ extremal qui, a son tour, mene a une solution de commande sous forme lineaire fermee a reaction, avec ga ins variant dans le temps. En conclusion, on presente quelques suggestions, qui ont, pour objet d'ameliorer la precision du systeme et le domaine de fonctionnement dans la realisation pratique du guidage de rendez-vous a faible poussee. Zusammenfassung
Das Problem der 3-dimensionalen Lenkung, urn eine zeitoptimale Flugbahn flir Begegnungen von Raumfahrzeugen (Rendezvous-Technik) zu erzielen, wird im Rahmen einer quasi-kreisformig angenommenen (dynamischen) Bahn untersucht. Der Lenkvorgang, im gleichen Sinne optimal wie die Bezugsbahn, stiitzt sich formal auf die angenaherte Konstruktion eines Bereiches (Biindel) von nebeneinanderliegenden optimalen Begegnungsbahnen; diese Methode wurde in einem friiheren Aufsatz von Kelley entwickelt. Zur Erlauterung wurden eine Reihe von Begegnungsbahnen, die ein in Richtung der Peripherie zeigender Schubvektor von geringem Betrag erzeugt, fiir die Untersuchung der Lenkung angenommen. Solche Flugbahnen sind zeitoptimal, sie eignen sich unter entsprechenden Bedingungen zum Uberwechseln zwischen zwei nebeneinanderliegenden, in der gleichen Ebene befindlichen kreisformigen Bahnen.
274
Die betrachtete Anwendung laBt eine analytische Darstellung des extremen Bereiches zu, was auf einen Regelkreis mit zeitveranderlicher Verstarkung flihrt. Der Aufsatz enthalt einige Vorschlage flir mogliche Abanderungen, die die Genauigkeit und den Operationsbereich wahrend der prak· tischen Lenkung mit geringem Schub erhohen konnen.
Introduction The second-order guidance approximation scheme employed in this paper has been developed in an earlier publication 1 . Essentially, the idea is to select a flight path, optimized in some appropriate sense, as a nominal trajectory, and then to base guidance upon a family, or 'field', of optimal trajectories approximated in the vicinity of the nominal. The investigation, as is shown later, applies this scheme to the guidance problem for orbital rendezvous. The method is tractable for a fairly wide class of problems, although in general it must be carried through numerically. However, in the present case an analytical treatment becomes feasible because of the recent availability of a particularly simple optimal transfer manoeuvre suitable for use as a nominal trajectory. The nominal manoeuvre, as it appears in this paper, is a direct outgrowth of a co-planar circular orbit transfer analysis conducted by Hinz2 . The problem posed by Hinz, although phrased somewhat differently with regard to coordinate systems and assumptions employed in deriving the equations of motion, is mathematically equivalent to the nominal transfer manoeuvre problem investigated herein. Both cases yield to an analytical treatment of the boundary value problem whenever the manoeuvre duration is an integral multiple of a reference orbit's period. The analysis of three-dimensional rendezvous guidance for the class of trajectories discussed above leads directly to a synthesis of a linear feedback control solution with timevarying gains given in closed form. Some suggestions are also given for possible modifications which might enhance system accuracy and the range of operability during practical implementation of low-thrust rendezvous guidance. The Differential Equations of Powered Flight Considerations begin with the differential equations of threedimensional powered flight in a central inverse-square force field (Figure 1): i'- reos 2 tf;(J2 -
r!jJ2 + ~ =~. veos p sin Cl. r m
. re
.. .
F 8+2--2 tan tf; tf;8= tf;' veos peos Cl. r mreos
(1)
AN OPTIMAL GUIDANCE APPROXIMATION FOR QUASI-CIRCULAR ORBITAL RENDEZVOUS
Finally, if note is made of the fact that kl R5w~ = I, then eqns (1) can be put into the following reduced first-order form:
.. fti; 1 . F t/t+2-+-2 sin2 t/t 82 = - ' vsinfJ r
n1r
. vF m=--'
2' 2 1 F* . =(1+I1)(l+u) cos t/t-(l+11)2+(i+OvcosfJsm:x
,..
U
C'
where F is the maximum thrust level of the reaction engine; C is the propellant exhaust velocity; p is a throttle variable; and III is the instantaneous vehicle mass. The difficulty in obtaining any sort of particular solution for these equations needs no comment here, except that it provides a motive for the simplifications which are now introduced. The object is to devise certain assumptions which will allow the replacement of eqns (1) by an approximate set of differential equations which are linear in the state variables r, 0, and t/t, and their time derivatives and, preferably, separable in the control variables P, eX, and {J. (Simplifications of this kind are required to make flight path optimization and guidance problems analytically tractable.) To be specific, it is preferable that these approximate differential equations describe low-thrust acceleration transfer trajectories between neighbouring circular orbits.
, 2u(1+u) u =-----+2w(1+v)tant/t 1+11 F* + (1 + 11)( 1 + ~) v cos f3 cos 'l. , 2 uw 1 ( · 2·, F* . f3 w = - 1 + 11 - T 1 + u) sm 2 ifJ + (1 + 11)( 1 + 0 v sm
11' =U
(2)
8'=V
,//=w F*
z
~'=--v
C*
where F* = FlmoRo 11'6 is the reduced maximum thrust acceleration; C* = Cl Ro Wo is the reduced exhaust velocity; and the superscribed prime denotes differentiation with respect to the reduced time, T. Now let us assume that F*,~, u, v, 11',11, and t/t are all terms of order 6 or smaller (0 < < I). Under these circumstances, one would therefore anticipate that all terms of order 02 in eqns (2) will become negligible with respect to terms of order O. Thus the following simplified differential equations are arrived at:
x
u' = 2 v + 311 + F*v cos fJ sin 'l.
y
Figure 1. Coordinate geometry
v' = - 2 u + F*\' cos fJ cos Cl.
The following set of dependent and independent variable transformations will prove useful for our purposes. Let
and
w' =
-
r(t)=Ro [1 +11(t)]
11' =
11
o(t) = r (t) + B (t)
s'=v
m(t)=mo[1+~(t)]
(3)
t/t'=w
where Ro is the radius of a circular reference orbit situated in the t/t = 0 plane (Figure 1); mo is some reference mass; and T is a fictitious angle defined by the differential expression
dr dt =wo,
t/t + F*v sin fJ
r(O)=O
where Wo is the reference orbit's period. Furthermore, since T(t), as defined above, is a monotonic time-like parameter, it is permissible to change the independent variable in eqns (I) from t to T. This can be accomplished by simply relating t derivatives to T derivatives as follows:
d dr d d dt=dt dr=wodr
It may be said that, to the first order of small quantities, eqns
(3) * are descriptive of quasi-circular flight for the following reason; if, as has been assumed, 1I, v, I}, etc. are of order b, then it follows at once that
where E and h are specific energy and angular momentum respectively. Consequently, the energy-momentum images of
* These equations are identical in form to the differential equations of Wheelon 3 and Anthony4. However, the dependent variables and thrust vector steering angles are not subject to the same interpretation. In particular, the quantity E in eqns (3) is not required to be small-an important point in the subsequent development. 275
H. J. KELLEY AND J. C. DUNN
trajectories which are adequately described by eqns (3) should everywhere be close to the locus of circular orbits in the E-/z phase plane (Figure 2). Clearly, the validity of the quasi-circular differential equations will be compromised when the parameters F*, F* / C*, and T exceed certain critical values. Just precisely what these critical values are cannot be determined until the nature of the control schedule peT), .x(T), and (3(T), is specified. The reader is advised to bear this in mind in the sequel.
./
oH
(6)
A·=--
,
oX i
X;=~~,
(7)
[eqns(3)]
together with the corresponding natural boundary and transversality conditions, (Sa) (6a)
~-----------------.h
XiO=o,
E=- ~
HI = F*V( Al cos (3 sin a+ A2 cos (3 cos a + A3 sin (3 -
E
Figure 2. Energy-momentum phase diagram
Optimal Transfer Between Neighbouring Circular Orbits The optimal orbit transfer problem may be stated as follows: given two neighbouring circular orbits, find the steering angles .x(T) and (3(T) and the throttle schedule peT) which produce a transfer between the two orbits in minimum time. For present purposes, the case for which the terminal orbits are co-planar will be selected, a class of optimal transfer paths within the framework of the quasi-circular dynamics assumption derived, and later, these paths employed as nominal trajectories for the three-dimensional rendezvous guidance analysis. To reiterate, if the subscripts 0 and f denote initial and final conditions respectively, a search is made for a set of control functions .x (T), (3 (T), and peT) which minimize Th produce a state transition which evolves in accord with eqns (3), and which satisfies the circular orbit boundary conditions, viz: at T = 0, U = v = W = t) = (' = 1/1 = 0, ~ = 0; at T = Tt u = W = 2 v + 3}) = 1/1 = 0, 1} = K (const.). The problem so stated is a Mayer variational problem with bounded control variables. The necessary conditions to be satisfied by its solution are well known and are written here for the present application without further comment: Let Xi = state variables, u, v, W, 1}, E, 1/1, ~ y" = control variables P, iX, (3 }'i = multiplier functions / l; = undetermined constant multipliers P = function to be extremized = Tj + Al (Uj) + .112
=
Hamiltonian function
=
(8)
However, note that the final mass, /no (1 +~j) does not appear in the pay-off, i.e., the final mass is left open. However, only those trajectories for which l~f I is of order 0 are admissible because of assumptions implicit in eqns (3). Therefore, eqns (6) and (6a) imply that }'7 == O. Consequently, eqn (8) simplifies to:
HI =F*V(Al cosj1sina+A2 cosj1cosa+}'3 sin (3)
(8a)
The requirements on the control variables iX, (3, and l' are determined by reasoning as follows. Since iX is unbounded, oHdC)(X = 0 and o2HI/O(X2 :2 0 at the minimum of HI and hence,
siniX= -Al /(}ci+A~)t,
cosiX= - A2/u i+A~}t
(9)
HI reduces to:
Hl=F*v[ -V,i+A~)tcosf3+;'3sinf3J
(10)
which can be written in the form
HI
=
F*v v .i + ;,~ + A~yt sin (f3 -
.
Sill
-1
(Ai+A~)t
(Ai + A~ + A~r}
( 11)
By virtue of the fact that sin qJ :2 0, it follows that the principal value of
and HI reduces still further to:
i
Then the following equations and inequalities must be satisfied,
i.e., the optimal control y minimizes the function H .
~:)
( - ~2 -
~ }'iX;'
H(y)~H(Y)
(7 a)
Now, a minimum of H is attained at a minimum of HI' where HI is that part of H which depends on the control variables, y. For the problem here,
Reference orbit A O-neighbourhooc of the point (EO.h O)
H
i=1, ... ,7
Xl,=x 3,=2x2 ,+3 X4J=X6,=0, x 4,=K
Locus of circular orbits
(5)
HI
= -
F*v (Ai + A~ + I.~)t
(3)
* Notice that Hl is also stationary with respect to this value of (3, i.e., (OHI! 0(3) (3 = o. 276
AN OPTIMAL GUIDANCE APPROXIMATION FOR QUASI·CIRCULAR ORBITAL RENDEZVOUS
From eqn (13) it follows immediately that whenever
'V =
where A is a matrix of constant coefficients and g is a vector whose elements depend upon the control variables iX, {J, and 'V. Solutions for eqn (17) can therefore usually be phrased in terms of a fundamental solution matrix tJ> (T f, T) and superposition integrals, e. g. :
1 minimizes HI
(14) Furthermore, it can be verified [by solving eqns (6)] that + A~ + A~ i' 0 except at a finite number of points on any T interval of length 2n. Thus the indeterminate values of 'V corresponding to Ai + A~ + A~ = 0 form a set of measure zero and our problem is therefore well behaved. In summary, the optimal control variables depend upon the multipliers }'l' .12 , and }'3 in the following manner:
Ai
x(rI)=rJ:>Crj,O)x(O)+
f
rj
rJ:>(rI,r)g[y(r)]dr
0
Eqn (19)
(18)
*
,
where
(15)
g2=F*vcos{Jcos'Y.=
A2F*
---.=----
(Ai + A~ + }.~yl:
The A's in turn depend on the unknown Lagrange multipliers, .1i through eqn (6), and the solution of the boundary value problem turns upon one's capability to solve for these undetermined constants. However, before attention is directed to the boundary value problem, it is worth while to emphasize two points. First, it is easily demonstrated that the optimal control variables given by eqns (15) satisfy a strong form of eqn (5), i.e.,
g3=
F*
. {J vsm = -(12
)'3 F *
12
12)t
(20)
/'1 +'"-2+'"-3
(16) But eqn (16) together with the linear character of eqns (3), are sufficient conditions for a strong relative minimum of P. Thus it is assured that the control law of eqns (15) will provide a time-optimal transfer between two co-planar circular orbits which is unique whenever the boundary value equations possess a unique solution. Second, it should be noted that the result expressed by the last of eqns (15) is independent of the multiplier functions Ai, and is therefore insensitive to the boundary conditions. Thus a full throttle operational mode is a characteristic of the entire extremal field of quasi-circular transfer trajectories for the problem of minimum time transfer with final mass open.
As already pointed out, the }:s depend on the undetermined constants Ai in accord with eqn (6a). Furthermore, it is noted that in matrix notation, eqns (6) have the form
(21) and thus are adjoint differential expressions for eqn (17) [Le., eqns (3)]. Consequently, their solution is determined when tJ> (Tt> T) is known, i.e.,
T
},(r)=1> ).(r/)=1>
T{ap} ~
(22)
X'j
The Orbital Transfer Boundary Value Problem-a Special Class of Solutions for Rendezvous
where
The differential equations, eqns (3), when written in a symbolic matrix notation, have the following structure:
x' =Ax+ g(y)
* Eqn
(19):
cosr -2sin r 0 rJ:> (rI' r)= sin r -2(1-cosr) 0 0
2sin r -(3-4cosr) 0 2(1-cosr) -(3r-4sinr) 0 0
(17)
0 0 cosr 0 0 sin r 0 277
In view of these considerations, the co-planar circle-to-circle transfer boundary conditions [see eqn (7)] are:
3 sin r -6(1-cosr) 0 (4-3cosr) -6(r-sin r) 0 0
0 0 0 0 1 0 0
0 0 -sinr 0 0 cosr 0
~l
0 0 0 0 1
(19)
H. J. KELLEY AND J. C. DUNN
(23)
transfer between two neighbouring co-planar circular orbits differing in altitude so that I(R j - RA)/RAI = 4nn P. This result was first obtained by Hinz 2 in an analysis only slightly different from that presented here. The relatively simple optimal orbit transfer manoeuvre just described can be used as a reference rendezvous trajectory for the guidance analysis, if one imagines that the transfer is initiated at the proper time. For example, if, as in Figure 3, a target at point B in orbit 2 leads the vehicle initially at point A in orbit 1 by an angle ETo, then it is necessary that ETo = 3rJl/8F* in order to effect a rendezvous at point C. The nature of the transfer is such that points Band C are symmetrically deployed with respect to the initial line OA.
Initiattarget position
where
Al =
A4[~~ COSi+(I- ~:)Sin iJ
L/
,Reference line rotating with cons.tant angular velocity -eA
'W,eA _---r--_
Target orbit -
C
Rendezvous point "'. Transfer path
A2=2A4[~~ sin i-(l- ~:)COSi+lJ and
(24)
Lead angle for .· rendezvous """""'"
The natural boundary condition on H [see eqn (5)] determines the magnitude of the scaling constant A4 e. g., Figure 3. Rendezvous /fight-path geometry
The Optimal Rendezvous Guidance Approximation
and therefore
1 A I 41 =
[(
F*
~~)2 + (~: )2 + 4 (~:)2J-!-
(25)
The sign of A4 will depend upon the sign of K. When eqns (24) and (25) are substituted in eqns (23), a set of five boundary value conditions transcendental in five unknowns, A 1/A 4, A 2/A 4 , A3/A4' A5/A4' and Tj are obtained. In general, it is not possible to arrive at an analytical solution of the boundary value problem so stated for arbitrary values of K. However, the following observation is made: if one sets / 11/A4 = A 3/A 4 = A5//14 = 0, and A2/A4 = I, then eqns (15), (24) and (25) indicate that:
sin
p=O,
cos
-
cosp= 1
sgn (A4) (26)
which conditions in turn prescribe an in-plane circumferentially directed thrust vector. Furthermore, it should be noted that the first four of eqns (23) are identically satisfied by such a steering programme whenever Tj = 2nn, and the fifth is satisfied for this value of final time when K = - 4nn F* sgn (A4)' Thus. in view of the assumptions, and the sufficiency argument connected with eqn (16), it is concluded that full-throttle in-plane, circumferentially directed thrust is time-optimal for ascending (sgn ,14 = - I) or descending (sgn ; 14 = + I) thrust-limited 278
With a class of time-optimal rendezvous trajectories available for reference flight paths, one is now in a position to consider optimal rendezvous guidance. Presumably, the guidance scheme selected should preferably be optimal in the same sense as the nominal trajectory. Small disturbance assumptions should also be invoked, thereby obtaining a relatively simple guidance law. However, because of the stationary character of an optimal flight path, an immediate conflict of interest arises. It is found that, to the first order of small control variations, all guidance schemes which satisfy the final boundary conditions of the original problem are equally attractive, i.e. , performance is insensitive to small control variations. Thus, if one wishes to discriminate among various modes of guidance, it is necessary to consider second-order control effects at least. Because of the anticipated difficulties associated with secondorder analyses, and the apparent needlessness for such a refinement (when, for example, the expected initial condition errors are truly small), most of the effort expended in the past has been directed at finding any simple correction law which satisfies the prescribed final end conditions. A common approach involves the construction of an artificial variational problem with some attendant quadratic 'loss' functional to be minimized, the object being to obtain a simple control law which results in the restoration of final boundary values. Recently, KelleyI, and Breakwell and Bryson5 , have experimented with another approach to the guidance problem. Their considerations have been based on the theory of the second variation and have led to a treatment formally similar to that
AN OPTIMAL GUIDAN~E APPROXIMATION FOR QUASI·CIRCULAR ORBITAL RENDEZVOUS
of Jacobi's accessory minimum problem in classical variational theory. The method is rational in the sense that it selects a guidance scheme on the basis of the original criterion for optimality to an approximation correct as far as second-order control variation effects are concerned. Although this new approach will ordinarily lead to a numerical attack on the control law synthesis problem, the method's effectiveness in an analytical treatment of time-optimal rendezvous guidance for the class of nominal trajectories derived in the previous section is now demonstrated. A general theoretical discussion of the guidance theory may be found in Kelley's workI . Here the necessary equations pertinent to our application are simply written down and the analysis is carried through to a closed-form solution. If (jp, (j(X, and ofJ represent small deviations from the nominal (optimal) control schedule, then the time histories of these quantities, consistent with control bounds, which restore final end conditions and minimize transfer time are searched for. These time histories are assumed to commence from some 'initial' time To (not necessarily the initial point T = 0 on the nominal trajectory) at which excursions from the reference flight path, OXi (TO) are detected. Ultimately, the control increments (jv, (j(X, and ofJ will be expressed as functions of the OXi (T) as T ranges from 0 to Tf (feedback form). First, it is noted that within the limits of a second-order approximation, all guided trajectories are required to belong to the same time-optimal extremal field as the reference flight paths. However, it has already been pointed out that full throttle operation (v = 1) is characteristic of this field and therefore OP == O. Thus, only two control variations remain for timeoptimal rendezvous guidance, viz., O(X and ofJ. It is observed that for every member of the extremal field, HI is stationary with respect to (X and fJ, i. e., min HI -+ oHj"o(X = = oHjofJ = 0 [see the discussion immediately preceding eqns (9), and the footnote relating to eqn (12)]. Consequently, our second-order approximation [see Ref. 7] demands that:
b
G:)
blc l =cos i bA l , - 2 sin i bA21 + sin i 6}'41 + 2 (1- cos i) 6..1. 51 and
(29) The undetermined (jAif's are interrelated by a linearized version of the natural boundary condition for H [eqn (5a)] viz: (
Xl 6..1.1 + A2 bAz + X3 bA3) = 0
JXi+A~+A~
(30)
J
or since
Al==A3==0 (31)
6)'2/=0
In any event, when eqns (28) and (29) are combined, and coefficients of like terms collected, the final result is:
(32) where the C;'s are undetermined independent constants which parametrize the guidance boundary value problem. A formulation of the rendezvous guidance boundary conditions requires the solution of the state equations, eqns (3), linearized in the steering angles (X and fJ, i. e.;
6u' = 3 611 + 2 6v + F* bet 6v'= -26u bw'= -6t/J+F*6{3
6et< < I
61'1' =6u
6{3< < 1
(33)
6e' =6v
== F* [(blc l cos iX - bAl sin iX) cos J5
- 0:1 sin iX + Az cos iX) betJ = 0 b ( ~; ) ==F*{-(bAlsiniX+bA2cosiX)sinJ5+bA3cosJ5
(27)
where Oll, ov, etc., are excursions from the nominal state. An approximate but consistent solution is easily obtained for these equations and the boundary conditions can be written in the following form:
Ilu J= bUJ+ F*
- [(Xl sin iX+ Az cos iX) cos J5 + A3 sin J5J bp} =0
JTI
cos! (C 1 cos i + Cl sin i + C 3) d, =0
TO
where the superscribed bars denote nominal quantities, and the (j)./s in the present problem satisfy a set of differential equations identical in form to eqn (21). For the class of reference flight paths which are considered here, sin (X = sin fJ = 0, cos (X = sgn K, cos fJ = 1, and ,12 = - 1j F* sgn K [see eqns (24), eqns (26) and the accompanying text]. Therefore if, without loss in generality, guidance about an ascending rendezvous path (sgn K = + 1) is examined, then eqns (27) imply that: and
(28) Furthermore, since the (j,1/s evolve in accord with eqn (21) it follows at once from eqns (22) and (19) that: 279
-
Ilv J =6vJ-F* JTI 2sini(Clcosi+Clsini+C 3)d,
Ilw J=6w J +F*
J\
co,f(C,co,f+C, ,in f)
d,~:F' "f~O
~
Il11J=611J+F*f~1 sin i(C l cos! + Cl sin i+C 3)d,=0 TO
(~
H. J. KELLEY AND J. C. DUNN
Considerations on Rendezvous Guidance Implementation
where: (a) TO is the point in time at which corrective action is initiated; (b) the QXi/S are dispersions which would occur at the terminus of the nominal trajectory in the absence of corrective guidance; (c) btt is a variation in the final time, Tt «(ht < < 1); and (d) the coefficients of QTt in each equation are to be interpreted as state variable closing rates, i. e., differences between vehicle and target state variable derivatives on the nominal path at T = Tt (notice that only v has a non-zero closing rate). A total of six equations linear in the six unknowns Cl' ... , C5 , and QTt are at our disposal. Their solution may be obtained most conveniently by first combining the second and fourth of eqns (34) to obtain the result,
bVf+2bl'/f= -F* bTf
In this section, several ways to enhance system accuracy and range of operability in practical implementation of lowthrust rendezvous guidance are suggested. First of all it is noted that real time generation of the system gains and the state variables along the nominal trajectory will be required in practice. In the case of command guidance, these would presumably be calculated 'on the fly' by a ground-based computer, whereas if a vehicle-borne computer were employed, these quantities would more likely be stored in polynomial approximation. Now, with regard to the computation of gains, it is worth calling attention to the advisability, in either case, of employing double precision arithmetic in the calculation of the determinants of the Appendix, since they tend rapidly toward zero as time-to-go approaches zero. This, of course, does not necessarily imply the need for a highly precise representation of the gains themselves. As concerns the generation of the state variables along the nominal trajectory, it should be pointed out that over the course of many revolutions there will be an accumulation of error stemming from higher-order effects if the linear equations are used. This error will place an additional burden upon the guidance system. The situation may be alleviated somewhat by introducing second-order corrections to the nominal in the following approximate fashion. Second-order terms of the first two equations of the system (5), which are significant along the nominal, are p= -31'/2+2I'/v+V2
(35)
which may then be used to eliminate ('iLt. The remaining equations can then be written in matrix notation (see Appendix), e. g. :
(36)
and
(37) where A (TO) and B (To) are matrices whose elements depend upon To and J and K are constant matrices. But the QX i! are related to the QXi (lO) through eqn (19). Therefore the constants Ci may be expressed as functions of the 'initial' excursions at time TO, i.e.:
I
Cll C 2 =A-l(TO) C3
]f]Jl(To)
jbUl !~J &
detjAj#O
q= -2uv+2ul'/-F*1'/
(40)
If p (t) and q (t) are estimated by insertion of the nominal
values of u, v, and 'Yj as computed via the Iinearized equations, corrections QUe, QV e, QT)e and oee, corresponding to the forcing terms p and q, may then be calculated by means of the influence functions presented earlier. The results take a particularly simple form if the integrals are evaluated at l' = 2mT, m an integer, 0::;; m::;; n, namely
(36a)
<0
bUj(2 mn)=22F*2mn
(37a) where f]Jl and f]J2 are submatrix elements of the complete state transition matrix [eqn (19)]. Eqns (32), (36a), and (37 a) then provide the open loop solution for the steering angle corrections, viz:
8"
~
{cod, ,in ;,1} [ ' ('0) J
I~~) l&
and
b{3t= {cos i, sin i} B- 1 (To) K f]J2 (To)
{bW} blj;
(38) to
(39) <0
TO
bv;(2 mn)=F*2 (68 m 2 n 2 - 80 mn) 151'/;(2 mn)=F*2 (-16 m 2n 2+40 mn) &;(2 mn)=F*2(36 n+8 m 3 n 3 )
(41)
If these are employed as corrections to the nominal values with 2mn replaced by 1', the result is incorrect by the omission of certain oscillatory effects; however, it is thought that these will be unimportant in comparison with the secular terms which are properly accounted for. In any event, the idea is to provide the closed loop control system with some anticipation for errors accruing on account of non-linearity. One of the most significant limitations arising from the approximations made in the foregoing analysis is the restriction that the change in arrival time from the nominal be small compared with the orbital period. Shift in arrival time is directly proportional to the difference between actual and nominal values of the linearized energy parameter, v + 21). Furthermore, analysis reveals that relatively large steering corrections are associated with errors in this variable. Since the energy parameter is a monotonicaIIy increasing function of time, it therefore appears reasonable to consider this
280
AN OPTIMAL GUIDANCE APPROXIMATION FOR QUASI-CIRCULAR ORBITAL RENDEZVOUS
parameter as a candidate for independent variable in the mechanization of the system. According to this scheme, the system gains and the state variables along the nominal would be generated (or stored) as functions of energy-to-be-gained, the difference between the terminal and instantaneous values of v + 2YJ. In an analysis conducted with an independent variable having fixed terminal value, complexities arising from shifts in terminal time would be avoided and there would be one less approximation required. There is another point to which scant attention has been paid in the present investigation, which is essentially a feasibility study, and this is the choice of reference orbit and the position of the target vehicle in relation to the reference axis system. We have tacitly assumed the target to be moving in a perfectly circular orbit and chosen our axis system location for analytical convenience as being at the initial point of the nominal orbit transfer trajectory which terminates at the target. It seems clear for a number of reasons that the appropriate choice in actual implementation of a system would be a reference circular orbit having the period of the target's orbit and a reference axis system moving along this circular orbit in the vicinity of the target. One advantage of such a choice is the facilitation of guidance in terms of relative position and velocity measurements. Finally, it is pointed out that by regarding F* as an auxiliary state variable (as opposed to a parameter) defined by the differential equation, F*' = 0, it then becomes possible, with little real
increase in complexity, to derive an additional term for the guidance law whose function would be to counteract the effects of instantaneous fiuctuatinns in the reduced thrust-acceleration level. The procedure for treating system parameters as auxiliary state variables is discussed at greater length by Kelleyl. Conclusions The low-thrust quasi-circular orbital rendezvous example, presented in this paper, illustrates a possible practical application of an optimal guidance scheme developed in a previous publication. The closed-form result obtained for guidance corrections should be safely applicable for correctional manoeuvres during the last several revolutions of a low-thrust ascending (or descending) approach to rendezvous. Some numerical computations designed to establish the range of validity of the guidance approximation are currently in progress. The authors extend their gratitude to Mr. Frank Sobierajski and Mrs. Agnes Zevens of Grumman's Research Department: to the former, for assistance in preparing and verifying several of the foregoing analytical results; to the latter, for the preparation of the figures appearing ill this paper. This research was partially supported by the USAF Office of Scientific Research under Contracts AF 29(600)-2671 and AF 49(638)-1207.
Appendix The matrices appearing in eqns (36) through (39) are:
1-1 F*
J=2.
0 0 0
0 0
~1
0
-1 0
[~ ~J
1 K=--
F*
[
(2nn- Lo) _ (sin 2 LO)J .
2
(l-COS2LO) ------
4
(1- cos 2 LO)
A=
-sin LO
4
[
4
(2nn- Lo) 2
+
(SiI12Lo)J 4 (4 nn-2 LO +2sin LO)
[2nn-LoJ-2sinLoJ
B=
e-C~S2LO )
ct>l=
(2nn-Lo 2
+
Sin2Lo) 4
COS Lo
-2sin LO
3 sin LO
2 sin LO
-(3-4coS'r 0 )
- 6 (I-cos LO)
-sin LO
2(1-cos Lo)
(4-3coSLO)
-2 (I-cos LO) - (6nn - 3 LO + 4sin Lo)-(12 nn - 6 LO + sin LO) 281
o o o 1
H. J.
4>2=
COS T O
sin To
-sin TO
COS To
[
KELLEY AND
1
J. C.
DUNN
References 1
also if 2
2 nn-To x= 2 ,O::<;;x::<;;nn
3
4
det IAI =4x 3 -4 sin 2 x(3 -sin 2 x) x +8 sin 3 x cos x #0 for x>O 5
282
KELLEY, H. J. Guidance theory and extremal fields, Inst. Radio Engrs. Nat. Aerospace Electronics Conf, Day ton, Ohio (May 1962); Trans. Inst. Radio Engrs., AC (October 1962) HINZ, H. K. Optimal low-thrust near-circular orbital transfer, AIAA Jnl. (in the press) WHEELON, A. D. Midcourse and terminal guidance, Space Technology. 1959. New York; Wiley CLOHESSY, W. H. and WILTSHIRE, R. S. Terminal guidance system for satellite rendezvous, J. Aerospace Sci. Vo!. 27, No. 9 (September 1960) BREAKWELL, J. V. and BRYSON, A. E. Neighboring-optimum terminal control for multivariable nonlinear systems, Meeting Soc. Industr. Appl. Math., Cambridge, Massachusetts (November 1962)