Near-optimal guidance scheme for a Mars trajectory

Pergamon www.elsevier.com/locate/actaastro

Near-Optimal

PII:

Acto Astronautica Vol. 51, No. 1-9, pp. 351-378,2002 Q 2002 Published by Else&r Science Ltd Printed in Great Britain SOO94-5765(02)00039-S 0094-5765102 $ - see front matter

Guidance Scheme for a Mars Trajectory

A. Miele and T. Wang

Rice University, Houston, Texas, USA Abstract. This paper deals with a near-optimal guidance scheme for a spacecraft flying from a

low Earth orbit (LEO) to a low Mars orbit (LMO). The importance of guidance can be understood from the following consideration: a small error in the launch velocity at LEO can induce a very large position error on arrival at LMO due to error propagation associated with the large flight time and distance. Indeed, the uncertainties and errors in the thrust, system equations, and measurements constitute a special challenge to the designer of the guidance and control for a Mars mission. On account of the large changes in the gravitational field near a planet, the guidance design has three phases: near-Earth phase, interplanetary phase, and near-Mars phase. The focus of this paper is on guidance for the interplanetary phase. We assume that chemical engines are used and that the control is implemented by changing the magnitude and/or direction of the thrust via propellant consumption. A detector-predictor-corrector scheme is developed for guidance and control. The detector computes the spacecraft trajectory from the current measured state, estimating the trajectory error in the neighborhood of the low Mars orbit. The predictor is activated when the trajectory error exceeds a specified threshold. The task of the predictor is to generate a velocity correction, hence a thrust input, such that the target trajectory (the trajectory after the velocity correction) can rendezvous with the low Mars orbit. The corrector is a feedback control scheme to implement the control commands from the predictor. The key to the guidance design is the predictor. The maximum control margin concept is used in the predictor design to enhance the guidance robustness. Application of this concept means that the target trajectory for rendezvous with the low Mars orbit is to be achieved with

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minimum propellant consumption. The sequential gradient restoration algorithm (SGRA) is used in conjunction

with system decomposition to achieve the real-time implementation of- the

guidance, and the target trajectory is computed via a two-step process, which includes target restoration followed by velocity restoration. The objective of the target restoration is to generate a trajectory leading from the current position to rendezvous with the low Mars orbit with the minimum velocity change w.r.t. the current velocity; the objective of the velocity restoration is to achieve the required velocity change via proper thrust input to the spacecraft. Considerable enhancement of SGRA for robustness, convergence, and speed can be achieved via this decomposition. Preliminary

numerical

results

are

encouraging.

They

show

that

the

above

detector-predictor-corrector scheme can achieve safely the rendezvous with the low Mars orbit, while containing the propellant consumption due to the errors and uncertainties in the thrust, system, and measurements. 0 2002 Published by Elsevier Science Ltd.

1. Introduction A Mars mission is characterized by high cost, long travel time, and considerable risk. A fundamental issue of trajectory optimization for a Mars mission is the containment of both the departure mass and mission time. For previous research on suboptimal and optimal trajectories for spacecraft powered via chemical propulsion, see Refs. 1-9; for spacecraft powered via electrical propulsion, see Refs. 10-13. Once a baseline trajectory is determined via optimization, a guidance and control scheme is needed so that the spacecraft can achieve the mission goals safely. In this paper, we assume that guidance and control are being developed for a spacecraft powered by chemical engines; in subsequent papers, we will extend the guidance and control to a spacecraft powered by hybrid engines (chemical engines for planetary flight and electrical engines for interplanetary flight). The guidance design for a Mars mission must account for the following factors:

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(i) Sensitivity to Errors. For a minimum energy trajectory from LEO to LMO, covering about 600 million km in 258 days, a one per million error in the launch velocity at LEO induces a position error of 1000 km on arrival at LMO if no guidance and control is applied. (ii) Control Cost. Consumption of one pound of propellant in the outgoing trip of an interplanetary voyage requires the addition of about 40 pounds to the takeoff mass from the Earth surface. (iii) Environmental and System Complexity. The spacecraft guidance and control is subject to various errors and uncertainties associated with the thrust, system, and measurements. To meet the challenges posed by the above factors, we employ the maximum control margin concept proposed by the authors in Refs. 14-15. The essence of the maximum control margin concept is that, to enhance robustness, the trajectory control should not be determined by aiming only at the desired final state; instead, the control is to be determined by maximizing the control margin while aiming at the desired final state; in turn, the maximized control margin will assure the satisfaction of the final conditions in the presence of uncertainties and errors in the system and environment. Because the spacecraft is nominally in coasting flight along a large portion of an interplanetary voyage, the control margin can be measured in terms of the available propellant. Hence, the correction should be executed with minimum propellant consumption, while aiming at the desired final condition, namely, rendezvous with the low Mars orbit. The sequential gradient restoration algorithm (SGRA) can be employed to carry out the maximum control margin computation. For SGRA details, see Refs. 16-17. Note that, in view of the system complexity, direct use of SGRA is not recommended for real-time implementation of the guidance scheme. Indeed, decomposition of a complex system into simpler systems is desirable to enhance the robustness, convergence, and speed of SGRA. 1.1. Algorithm. SGRA is an iterative technique which involves a sequence of two-phase cycles, each cycle including a gradient phase and a restoration phase. In the gradient phase, the augmented performance index (performance index augmented by the constraints weighted via

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appropriate Lagrange multipliers) is decreased, while avoiding excessive constraint violation. In the restoration phase, the constraint error’is decreased, while avoiding excessive change in the variables. In a complete gradient-restoration cycle, the performance index is decreased, while the constraints are satisfied to a preselected accuracy. Thus, a succession of feasible suboptimal solutions is generated, each new solution being an improvement over the previous one from the point of view of the performance index. The sequential gradient-restoration algorithm was developed by Miele et al during the period 1968 to 1986 forboth mathematical programming problems and optimal control problems. It has proven to be a powerful tool for solving optimal trajectory problems of atmospheric and space flight. Applications and extensions of this algorithm have been reported in the US, Japan, Germany, and other countries around the world; in particular, a version of this algorithm is used at NASA-JSC under the code name SEGRAM, developed by McDonnell Douglas Technical Service Company (Ref. 18).

2. System Description The objective of the guidance and control system is to fly the spacecraft safely from the low Earth orbit (LEO) to the low Mars orbit (LMO). We employ the restricted four-body scheme (spacecraft, Earth, Mars, Sun) and the following assumptions: (Al) the Sun is fixed in space; (A2) Earth and Mars are subject only to the Sun gravity force; (A3) the spacecraft is subject to the gravity forces of Earth, Mars, and Sun along the entire trajectory; (A4) the eccentricity of the Earth and Mars orbits around the Sun is neglected, implying circular planetary motions; (AS) the inclination of the Mars orbital plane vis-a-vis the Earth orbital plane is neglected, implying planar spacecraft motion; (A6) LEO and LMO are circular orbits; (A7) if needed,

midcourse

corrections

are applied along the

interplanetary voyage. Having adopted the restricted four-body scheme, five motions must be considered: the

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inertial motions of Earth, Mars, and spacecraft with respect to the Sun; the relative motions of the spacecraft with respect to Earth and Mars. To study these motions, we employ three coordinate systems: Sun coordinate system (SCS), Earth coordinate system (ECS), and Mars coordinate system (MCS). In this paper, the inertial motions of Earth, Mars, and spacecraft are described in Sun coordinates, while the boundary conditions are described in Earth coordinates for LEO and Mars coordinates for LMO. In each coordinate system, a position vector is defined via the radial distance r and phase angle 0, while a velocity vector is defined via the velocity modulus V and local path inclination y. Let E, M, S denote the centers of Earth, Mars, Sun; let P denote the spacecraft; let t denote the time. For the coordinate transformations leading from inertial coordinates to relative-to-planet coordinates, and viceversa, see Refs. 4-5. 2.1. System Equations. Below, we give the system equations for Earth, Mars, and spacecraft. Earth. In view of Assumption (A4). the motion of Earth (subscript E) around the Sun is

described by the following equations: rE = const , 8, =

W,(t - 4)),

(la)

q?-=

J(F&>

9

(lb)

V, = J(cL&) .

UC)

YE=09

(14

where os is the angular velocity of Earth around the Sun. Note that e,(b) = 0, meaning that the x-axis of the Sun coordinate system is identical with the line connecting the Sun with the Earth initial position E(t,). Mars. Also in view of Assumption A4, the motion of Mars (subscript M) around the Sun is

described by the following equations: rM= const ,

(24

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356 8,

v,

= %jl(t

- to) + %g(t,)

=J@&j)

9

OM = +/&)

1

I’

PC)

yM==o,

whereo,

(2b)

(W

is the angular velocity of Mars around the Sun.

Spacecrufi. Subject to the gravitational attractions of the Sun, Earth, and Mars along the

entire trajectory, the motion of the spacecraft (subscript P) around the Sun is described by the following differential equations for the radial distance r, , phase angle Or,, velocity VP , path inclination yp with respect to the local horizon, and spacecraft mass mp drp/dt = V,siny, ,

W

d$/dt

= W@p) cosyp ,

WI

dvddt

= CJ- tp&$Mnrp +@M/‘&)

+ (I.+&)

co~WpE-~p~

cos(vpM-yP) + (BT/qd

drpldt = C,UV&+-WV&)cosy,

cosd

)

(3c)

+ (pLENp&)sinWpn-yp)

+ (pflpi&)sin(WpM-y,)P) + (BT/mpvp) sinol] . dmJdt= - $T/g,I, .

(34 W

The first two equations are the kinematic equations along the radial and transversal directions. The next two equations are the dynamic equations along the tangent and normal to the flight path. The last equation governs the change in the spacecraft mass due to the application of thrust via an engine having maximum thrust T, power setting p, and specific impulse I,. Note that, in Eqs. (3c)-(3e), the maximum thrust can be written as T = CrT,.,,

Cr=I+$-,

(da)

where C, and ~7 measure the deviation of T from the corresponding nominal value T,. If the maximum thrust is exactly equal to its nominal value, C-r = 1 and t+ = 0; otherwise, C, f 1 and % f 0, Also note that the power setting p is subject to upper and lower bounds,

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(4b) Finally note that, in the dynamic equations (3c)-(3d), the totality of system uncertainties and errors is expressed via the single multiplicative coefficient C,=l+&,, where C, and

(4c) E,

measure the deviation of the actual tangential and normal accelerations from the

corresponding nominal values. If there are no uncertainties and errors in the dynamic system, C, = 1 and E, = 0; otherwise, C, f 1 and E, f 0. In the dynamic equations, ps, pn, pM are the gravitational constants of Sun, Earth, and Mars; rpn and rpMare the radial distances of the spacecraft from Earth and Mars; V’pEand \vpi,,,are the inclinations of the Earth and Mars gravity directions PE and PM with respect to the local horizon. The distances rpn and rpMare given by rPE= J[(xp - xE)2 + (YP - YE)2l .

@a)

‘[txp- xM>2 + (YP- YM)21

rpM =

0)

9

with xp= r,cos 8,,

XE = rE cos

yp= r,sin 0,,

YE = rE

8,

,

xM = rM cos

sin 8, ,

yM = rM

8M ,

sin 8, .

@a)

(6b)

The gravitational inclinations VpE and VpMare given by cos“

(7a)

=

-7KQ - k,

\IlPM

=

-a

kPE

=

sign[xp(yp

-yE)-yp(xp-@I 1

@a)

kPM

=

sign[xP(yP

-YM)-yp(xP

@b)

- kPM CW’

I [x&p

-xE)+Yp(Yp-yE)&+E) 1

\vPE

i rXdXP

-xM)+YdYP

-YM)&rPM

1,

Vb)

with

Remark.

-xM)l

.

In Eqs. (l)-(8), the O-angles are measured counterclockwise starting from the

x-axis, assumed to be identical with the line connecting the Sun S and the Earth initial position

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The y-angles and v-angles are measured clockwise starting from the local horizon. 2.2. Boundary Conditions . The spacecraft mission is defined via the initial conditions for

departure from LEO and the final conditions for arrival to.LMO. Dqxzrrure from LEO. In the Earth coordinate system, the spacecraft conditions at departure from LEO (time t = b) are given by r&a) = rLEo , 8,,(b) =

b(b) Yp&J

Relative

t0

@a)

free,

= v,

9

Pb) bo.=

PC)

J(k~rLEo).

W

=0 * Earth rpE, 8pE, V,,

inclination of the spacecraft; V,

ypE are the radial distance, phase angle, velocity, and path is the circular velocity, in the low Earth orbit, before the

application of the accelerating velocity impulse AVLEo(tc). Anival to LMO. In the Mars coordinate system, the spacecraft conditions at arrival to LMO (time t = tr) are given by

rPMW= rmO

WW

y

%M(tf) = free,

y&~ YpM&)

= VLMO .

(lob) vLMO = %&LMO)

,

= o .

(1~)

(W

Relative to Mars rpM$%pM9VpM,,ypM are the radial distance, phase angle, velocity, and path inclination of the spacecraft; VLMOis the circular velocity in the low Mars orbit, after the application of the braking velocity impulse AVLMO(tr).

3. Guidance Scheme The guid,ance, scheme is partitioned into three phases: near-Earth phase, interplanetary phase, and near-wars phase. We focus on guidance for the interplanetary phase.

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Due to the uncertainties and errors in the launch velocity, system, and measurements, the actual trajectory will deviate from the computed baseline trajectory. This might cause the spacecraft to lose its target, and hence fail to rendezvous with the low Mars orbit. Therefore, a correction must be executed via application of thrust in such a way that the spacecraft can achieve its final target (rendezvous with LMO) with minimum propellant consumption. To save propellant, the correction is performed in discrete format rather than in continuous format. With this understanding, the guidance scheme for the interplanetary phase is made of three parts: detector, predictor, and corrector. The detector discovers the possible presence of a trajectory error; the predictor obtains a target trajectory correcting the trajectory error with minimum propellant consumption; the corrector is a feedback control scheme implementing the commands received from the predictor. 3.1. Detector. The task of the detector is to compute the trajectory leading from the current measured state to the final state and determine the target error, which is defined as the minimum distance between the spacecraft and the low Mars orbit. Let 6 denote current time, and let tr denote the time at which spacecraft achieves the minimum distance from the desired low Mars orbit. The trajectory computed by the detector employs the following initial conditions, system equations, and final conditions. Initial Conditions. Let the current state variables for the spacecraft and Mars be

r&J, @&J9 Vr4tC)9y&J 1

(114

and q&J,

$AJ.

V&h

h&J

.

(lib)

For simplicity of analysis, we assume that the quantities (11) are obtained from errorless measurements, with the exception of the phase angle Cl&,), which is computed from er(t3 - %A) = &&) - t&.&k

+ 6 rPM(tC)/rPM(t&

(12)

where the nominal phase angle difference between the spacecraft and ,Mars is subject to an additive error. In the additive term, the angle 6 is the assumed largest error, which is scaled

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according to rPM(tc)/rPM(tc,), the ratio between the current spacecraft/Mars distance and the initial spacecraft/Mars distance. This ratio has the value 1 at t=t, and tends to 0 at t=&. System Equations. The system equations (3) used for detector computations are employed with the following understanding: (a) the computation is started from the current time t, and is terminated at the final time tr ; (b) for the time interval t, 5 t 5 4 between the current point and the final point, the spacecraft is coasting in interplanetary flight, T =0, and the system uncertainty and error coefficient is set to C, = 1. Final Condition. In the Mars coordinate system, the final condition used for detector

computations is given by (13a)

Yp&) = 0.

Equation (13a) is a tangency condition: at the final point, the velocity of the spacecraft in the Mars coordinate system is tangent to the local horizon. The detector computation is terminated whenever Eq. (13a) is satisfied, resulting into a target error given by TE=

(13b)

Ih&)-r~0I.

Detector Sfrafegy. The propellant consumption

depends on the number of midcourse

corrections and also on the time interval between two consecutive midcourse corrections, due to accumulation and propagation of the effects of the uncertainties and errors in the thrust, system, and measurements. For effective saving of propellant, four midcourse corrections might be desirable: the first correction is applied shortly after departure (time ti) to contain the effects of a thrust error at launch from LEO; the fourth correction is applied toward the end of the interplanetary voyage (time tJ so as to create favorable conditions for guidance in near-Mars space; the second and third corrections (times 5 and ts) are used to contain the propagation of the effects due to errors and uncertainties. The number of corrections might be increased or decreased depending on the satisfaction or dissatisfaction of the following inequality by the trajectory error (13b): K,Qdt#-PM(b)

S m S K$p&/rpM&h

(14)

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where the dimensional constants K,, K, depend on the particular baseline trajectory being considered (minimum energy trajectory or fast transfer trajectory). Note that the constants K,, K, are scaled according to the ratio rPM(tc)/rPM(to). This ratio has the value 1 at t = tu and tends to 0 at t = 4. Hence, the bounds in Ineq. (14) tighten as the spacecraft progresses from the initial point to the final point. There are three possible detector regimes: (Rl) The lower bound of Ineq. (14) is violated. This implies that the trajectory error at the target is small; there is no need for a correction at the current time t, and the subsequent predictor-corrector algorithm is bypassed. (R2) The upper bound of Ineq. (14) is violated. This implies that the trajectory error at the target is large; a correction

is needed at the current time tC , and the subsequent

predictor-corrector algorithm is invoked. (R3) Both bounds of Ineq. (14) are satisfied1 This implies that the trajectory error at the target is neither small nor large; while there is no need of an immediate correction at the current time &, prudence demands that a correction be executed at the next time among the preselected correction times ti, t2, t3, t. The specification of the bounds K,, K2 and the preselected correction times t,, b, t3, tJ is done in Section 5, dealing with examples of the proposed guidance scheme. 3.2. Predictor:

Target Restoration.

To achieve a feasible implementation in real time, it

is appropriate to decompose the action of the predictor into two phases: target restoration and velocity restoration. In this section, we consider the target restoration phase; the velocity restoration phase is treated in Section 3.3. Let tCbe the current time, and let t,be the final time. In the Sun coordinate system, let V,(G) and V,(t,) be the values of the velocity before and after the velocity change, assumed to occur instantaneously; also in the Sun coordinate system, let yP(tC) and ^jp(tJ be the corresponding values of the path inclination. Let vector[AVr,(tJ] denote the difference

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vector[AV&)] = vector[T&)]

- vector[Vp(tC)],

(15)

and let AVp(tC) be the modulus of the vector (15). With this understanding, the following minimization problem can be formulated: min AVPW, WOJ

=

UW

JI%OcJ

+ %:cr>

Because the quantities V&J

- 2Vp(t&o,cOs&OJ

-

rP(4)I 1.

(16b)

and y&d computed before the velocity change are known, the

unknowns in the performance index (16) are the quantities ?&,) and -$(tJ computed after the velocity change. These quantities are not free, but subject to the constraints below. Initial Conditions. In the Sun coordinate system, the initial conditions (time tJ employed in the target restoration phase are the same as the initial conditions employed in the detector, albeit with the following difference: in the detector, the initial velocity VP(&)and path inclination yp(tC) are known from measurements; in the predictor target restoration, the initial velocity VP(C) and path inclinationqr(tc) are unknowns to be optimized. Sysrem Equations. In the Sun coordinate system, the system equations employed in the predictor target restoration are the same as the system equations (3) employed in the detector. Final Conditions. In the Mars coordinate system, the final conditions are given by

(17a)

Yp&) = 0, rpMtt) - rm0

= 0.

(17b)

Equation (17a) is a tangency condition: at the final point, the velocity of the spacecraft relative to Mars must be tangent to the low Mars orbit. Equation (17b) means that the spacecraft has reached the low Mars orbit. Comment. To sum up, the target restoration problem can be formulated as a mathematical

programming

problem or as an optimal control problem. In either case, the sequential

gradient-restoration algorithm can be used, the mathematical programming format being simpler. The quantities qp(tC) and -jp(tc) must be determined such that the performance index (16) is

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minimized subject to the constraints represented by the initial conditions

at tC (current

conditions), the system equations, and the final conditions at t, (target conditions). 3.3. Predictor: Velocity Restoration. Once more, let tc be the time at the current point. We assume that the target restoration phase has been completed, leading to the optimized values of the velocity Vp(t,) and path inclination -jr(&). The task of the velocity restoration phase is to actually bring the spacecraft velocity and path inclination from the current values VP(&)and y&) to the optimized values VP(&)= V&t,+@ and lyp(&)= yp(&+z). This should be done quickly and with small propellant consumption. With this understanding, the velocity restoration phase can be formulated as an optimal control problem involving the following performance index, system equations, initial conditions, and final conditions. Performance Index. The performance index of the velocity restoration phase is a linear

combination of the propellant mass and time required for velocity restoration. Hence, the problem is min J,

(lga)

J = m,(t,) - m&+z) + Kz,

(lgb)

where the weight coefficient K must be chosen as a compromise between minimizing the propellant mass and minimizing the time employed in the velocity restoration maneuver. System Equarions. In deep interplanetary space, we rewrite the system equations (3) by neglecting the gravitational attractions of Earth and Mars and by setting the uncertainty/error coefficients at the values C, = C, = 1. Therefore, drddt = V,sinY, ,

WW

d0+dt = (VP/r,) cosyp ,

(19b)

dV,/dt = - (p&)sinYp + @T/m,) cosa 1

(19c)

dY+dt =(Vp/ri&Vy2)c0sY~

(19d)

dm/dt= -PT/g,I,, ,

+ (PT/mpVp) sina v

(1%

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where T is the maximum thrust, ~1 is the thrust inclination, and l3 is power setting, which is subject to the inequality OCgSl.

(20)

Initial Condirions. The initial conditions at time 6, defined by the current state r&), %(t,), VP(t,), &(t,), m&J, are known from measurements. Final Conditions. At the final time fc+r, the values of the velocity and path inclination of the spacecraft should be equal to the optimized value obtained from the target restoration phase, Hence, VP&+@ = q&).

@la)

Yr&+@ = %OJ.

@lb)

Problem Simplification. To speed up the computation, the above optima1 control problem can be simplified by assuming that (i) maximum power setting is employed and that (ii) the thrust inclination 01is constant, since the velocity restoration time is relatively short. For P=I,

a,=const,

(22)

the behavior of the differential system (19) depends on only the parameters a and ‘c. These two parameters must be determined so that the final condition (21) are satisfied. As a result, there are no degrees of freedom left in the optimal control problem, straightfonvard

nonlinear two-point boundary-value

problem,

which degenerates

into a

solvable with the modified

quasilinearization algorithm employed in conjunction with the method of particular solutions (Refs. 19-20). Comments. It should be noted that the time covered within the velocity restoration phase is in the order of seconds, while the flight time covered within the target restoration phase is in the order of months. Therefore, by comparison with the target restoration phase, the velocity restoration phase can be regarded as being executed instantaneously. 3.4. Corrector:

Feedback

Control.

The velocity restoration phase of the predictor is

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implemented via feedback control in the corrector. Since the time interval ‘5 for velocity restoration is very short, the relative changes of r-r and mp are very small. Hence, we can employ the approximations. rP = const, mp = const.

(23)

For quick execution of the velocity restoration, we assume that the power setting is p = 1; therefore, the only control is the thrust direction a. With this understanding, the dynamic equations (19c)-( 19d) can be simplified to dV,/dt = - @&)sinyp + (T/m,) cosa ,

Wa)

dyddt = (VtJr,psN$p)cosyp

Wb)

+ (T/m,V,) sina.

Let VP&), ypN(t), apN(t) be nominal values derived from the velocity restoration phase, and let VP(t), yp(t), a&) be the actual values. The differences between the actual values and the nominal values take the form V&t) = VP(t) - VP&) I

(25a)

ye(t) = YpO)- YPN(t)3

(25b)

q(t) = o+(t) - o&t) *

(25c)

Linearizing the simplified system equations (24) with respect to the nominal values, we obtain the differential relations d,a, ,

dVc/dt = cllVC+q2yti dy&t

=

c21Vc+c22~c+d2ac v

(264 (26b)

where

(274

crt =o, cl2 = -(k&9 cal =

cw,

,

(llrp+psN~p)cosyp - (T/m,VZ) sina ,

c22 = - Wp~rp-kPp$siny,~

(27b) (27~) (27d)

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d, =- (T/m,) sina, d, = (T/m,V,) cosa

We) .

(27f)

Then, the feedback control law of the corrector can be written as q(t) = -kv V,(t) - q 7/C(t).

(28)

Here, kv is the gain coefficient for velocity correction Vc(t) and $ is the gain coefficient for path inclination correction ye(t). Well-established feedback control procedures can be used to derive the proper values of kv and q ( see Ref. 21).

4. Data for the Numerical Computations 4.1. Planetary Data. The gravitational constants for Sun, Earth, and Mars are given by ps = 1.327Ell ,

pn = 3.986EO5 ,

pt,, = 4.283EO4 km3/s2 .

(2%

Earth and Mars travel around Sun along orbits with average radii r, = 1.496EO8 ,

rM =

2.279EO8 km.

(3W

The associated average translational velocities and angular velocities are given by V, = 29.78 ,

v,

on = 0.986,

+,,.,= 0.524 deg/day .

=

24.13 km/s,

(30b) (3Oc)

In particular, the angular velocity difference between Earth and Mars is Ao = o, - wr+,, = 0.462 degiday .

(31)

4.2. Orbital Data. For the outgoing trip, the spacecraft is to be transferred from a low Earth orbit to a low Mars orbit. The radii of the terminal orbits are rLEO= 684 1 ,

rLMO = 3597 km,

Wd

hJLMo=200km,

Wb)

corresponding to the altitudes hLEO= 463 ,

since the Earth and Mars surface radii are given by

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R, = 6378 ,

lqq

=3391 km.

367

(32~)

The circular velocities (subscript c) at LEO and LMO are given by (VJLEO = 7.633,

(Vc)LMO =

3.451 km/s ,

(324

and the corresponding escape velocity (subscript *) are (V,),,

= 10.795 ,

4.3. Spacecraft

('*)LMO =

4.880 km/s .

We)

Data. A chemical engine is used for flight from LEO to LMO. The

assumed values for the specific impulse and the structural factor are I,,= 45os,

E=

0.10 .

(33)

For the flight form LEO to LMO, we assume that a two-stage rocket is used. The first stage of the rocket is jettisoned after the spacecraft launch from LEO. The spacecraft is coasting in interplanetary flight if no midcourse correction is applied. The second stage of the rocket is jettisoned after the spacecraft arrival to LMO. To carry out effectively the midcourse correction, one additional, smaller propulsive system might be required.

5. Numerical

Results

5.1. Basic Trajectories.

The numerical results presented in this section refer to the outgoing

branches of two basic round-trip trajectories, already considered in Ref. 8: the minimum energy trajectory (MET) and the fast transfer trajectory (FIT). For MET, the optimization results of Ref. 8 show that the flight time from Earth to Mars is 257.8 days; the total characteristic velocity is AV = 5.65 km/s, which includes an accelerating velocity impulse on departure AV,, and a braking velocity impulse on arrival AV,,,

=3.55 km/s

= 2.10 km/s. For FIT, the optimization results

of Ref.8 show that the flight time from Earth to Mars is 207.7 days; the total characteristic velocity is AV = 6.34 km/s, which includes an accelerating velocity impulse on departure AV,, = 3.62 km/s and a braking velocity impulse on arrival AVLMo= 2.72 km/s. Figure 1 shows the geometry of the minimum energy trajectory and fast transfer trajectory

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in inertial coordinates. Clearly, MET is nearly a Hohmann transfer trajectory, while FIT deviates considerably from the Hohmann transfer behavior. Note that the values of the characteristic velocity AV given above refer only to the major velocity impulses on departure and arrival. They do not include the minor velocity impulses needed for guidance, which are considered next. 5.2. Correction Schedule. As explained in Section 3, the correction schedule is based on the experimental selection of the four basic times t,, t,, t3, t4 at which corrections are made. For the trajectories under consideration, the schedule is given in Table 1, where the times are measured in days. The time t 1 is within 5 days of the departure from LEO and represents the beginning of the interplanetary phase. Here, the main objective is to contain the propagation of errors due to a thrust error at launch, hence to a velocity error at launch. The time td is within 30 days of the arrival on LMO and represents the end of the interplanetary phase. Here, the main objective is to create favorable conditions for the subsequent guidance in near-Mars space. The times b and ts are intended to limit the time interval between consecutive corrections to no more than 80 days for MET and no more than 60 days for FIT. In so doing, we limit the propagation of the effects due to system/measurement errors and uncertainties. Even though the times t,, 5, t3, t, are preselected, the total number of corrections can be larger or smaller than four. This depends on the continuous computation at each current time & of the target error TE [see (13b)] and on the satisfaction or dissatisfaction of the detector Ineq. (14). This leads to three possible detector regimes: (Rl) If the lower bound of Ineq. (14) is violated, there is no need for a correction at the current time t,; the predictor-corrector algorithm is bypassed. (R2) If the upper bound of Ineq. (14) is violated, a correction must be undertaken at the current time t,; the predictor-corrector algorithm is invoked. (R3) If both bounds of Ineq. (14) are satisfied, there is no need of an immediate correction

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at the current time &; however, prudence demands that a correction be executed at the next time among the preselected correction times t,, t2, ts, td. Table 2 gives the values of the constants appearing in Ineq. (14). 5.3. Uncertainty and Error Description. The spacecraft guidance is complicated by the presence of uncertainties and errors in the thrust, system, and measurements. For convenience and simplicity of analysis, we look at a sample of possible factors. Thrust. A thrust error is represented via the multiplicative coefficient C, in Eq. (4a), C,=l+&r,

(34a)

where I+ = (T - TN)&,, = AT/T,.

(34b)

In particular, upon linearization, we see that a thrust error at launch induces a proportional error in the velocity impulse at launch. Dynamic System. Uncertainties and errors in the dynamic equations (3~) and (3d) are represented via the multiplicative coefficient C,, where c, = 1 + E,.

(35)

Measurement Errors. Uncertainties and errors in the spacecraft position are represented by the additive term on the right -hand side of Eq. (12). Because the position of Mars is precisely known, Eq. (12) can be rewritten as A%($) = %0J - [%&>lN= 6 ~PMt~Y~PM(~)9

(36)

where 6 is the assumed largest error in the spacecraft phase angle. In the simulations presented here, the parameter values (34)-(36) are given in Table 3. The first row (error scale 1.O) refers to relatively strong uncertainties and errors, while the second row (error scale 0.2) refers to relatively weak uncertainties and errors. 5.4. Preliminary Tests. Based on the correction schedule of Table 1, the assumptions of Table 2 for the detector test, and the assumptions of Table 3 for the uncertainties/errors in the

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thrust, system, and measurements, the detector-predictor-corrector algorithm was run for the basic trajectories of Ref. 8, namely, the minimum energy trajectory (MET) and the fast transfer trajectory (FIT). The numerical results are shown in Tables 4-5 for MET and Tables 6-7 for FIT. While Tables 4 and 6 are computed for error scale 1.0 (see Table 3). Tables 5 and 7 are computed for error scale 0.2 (see Table 3). In these tables, Column 1 shows the current time t,; Column 2 shows the current spacecraft-Mars distance rrt&); Column 3 shows the target error TE = IrPr,,,(tJ- r,,ol; Column 4 shows the velocity correction modulus AV&); Column 5 shows the ratio m(tJ/m(t,) of current mass to initial mass, abbreviated to q/m,. Minimum Energy Trajectory (Tables 4-5). The results show that the total characteristic velocity for guidance is 0.168 km/s with error scale 1.0 and 0.035 km/s with error scale 0.2. The corresponding ratio of propellant mass to initial mass is 0.0143 with error scale 1.0 and 0.0031 with error scale 0.2. Therefore, a reduction of the error scale by a factor of 5 causes a nearly proportional reduction in the characteristic velocity for guidance and corresponding propellant mass. The total flight time from Earth to Mars reduces from the nominal value of 257.8 days for the optimal trajectory to 255.3 days for the guidance trajectory with error scale 1.0 and to 257.2 days for the guidance trajectory with error scale 0.2. Fast Transfer Trajectory (Tables 6-7). The results show that the characteristic velocity for guidance is 0.190 km/s with error scale 1.0 and 0.039 km/s with error scale 0.2. The corresponding ratio of propellant mass to initial mass is 0.0160 with error scale 1.0 and 0.0033 with error scale 0.2. Once more, a reduction of the error scale by a factor of 5 causes a nearly proportional reduction in the characteristic velocity for guidance and corresponding propellant mass. The total flight time from Earth to Mars reduces from the nominal value of 207.7 days for the optimal trajectory to 203.2 days for the guidance trajectory with error scale 1.0 and to 206.3 days for the guidance trajectory with error scale 0.2.

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Arrival to LMO Minimum energy

Arrival to LMO

Fig. 1. Minimum

transfer

energy and fast transfer trajectories from Earth to Mars.

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Table 1. Correction schedule (times in days) for LEO-to-LMO

transfer.

Time

MET

t1

to+5

to+ 5

t2

t,, +

85

to+65

t3

t,, +

155

to+ 120

4

G-30

Table 2. Dimensional

tf- 30

constants ( lo6 km)

in Ineq.( 14). Constant

MET

Kl

0.30

0.25

K2

3.00

2.50

Table 3. Parameter values employed in the simulations. Error scale

CT

cs

1.0

1.005

1.005

-0.10

0.2

1.001

1.001

-0.02

@W

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Table 4. Correction history, minimum energy trajectory, error scale 1.O.

tc

-IF

[lO%rn]

kW1

AV,&)

[Msl

o.o+

159817

0

0.0

0.3830

5.0-

152530

1506

0.0

0.3830

5.0+

152530

0

0.067

0.3772

85.0-

57626

538

0.0

0.3772

85.0+

57626

0

0.023

0.3753

155.0-

22786

363

0.0

0.3753

155.0+

22786

0

0.033

0.3725

225.3-

6735

125

0.0

0.3725

225.3+

6735

0

0.045

0.3687

4 = 255.3 days,

XAV, = 0.168 km/s,

A(m/m,) = 0.0143

Table 5. Correction history, minimum energy trajectory,

tc

rPM(tc)

[lO%n]

TE

error scale 0.2.

&4,)

ti%

[ lO%m]

[kmw

159817

0

0.0

0.3846

5.0

152553

436

0.0

0.3846

5.0

152553

0

0.014

0.3834

85.0

57617

126

0.0

0.3834

85.0

57617

0

0.005

0.3829

155.0

23015

90

0.0

0.3829

155.0

23015

0

0.007

0.3824

227.2

6772

26

0.0

0.3824

227.2

6772

0

0.009

0.3815

[WI 0.0

4 = 257.2 days,

CAV, = 0.035 km/s,

A(m/m,) = 0.0031

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Table 6. Correction history, fast transfertrajectory, error scale 1.O. k

rPM(tc)

TE

AVP&)

m/ml

[ lO%m]

[ lO%m]

o.o+

162184

0

0.0

0.3772

5.0-

154806

1328

0.0

0.3772

5.0+

154806

0

0.056

0.3725

65.0-

76530

603

0.0

0.3725

65.0+

76530

0

0.052

0.368 1

120.0-

33081

317

0.0

0.368 1

120.0+

3308 1

0

0.037

0.3650

173.2-

10149

137

0.0

0.3650

173.2+

10149

0

0.045

0.3612

[day1

4 = 203.2 days,

ZAV, = 0.190 km/s,

[Msl

A(m/rq,) = 0.0160

Table 7. Correction history, fast transfer trajectory, error scale 0.2.

tc

rPM(t3

TE

fW4J

wmo

[day1

[lO%TiJ

[ lO%rn]

O.O+

159817

0

0.0

0.3788

5.0-

1621 84

262

0.0

0.3788

5.0+

154828

0

0.012

0.3778

65.0-

76637

149

0.0

0.3778

65.0+

76637

0

0.010

0.3769

120.0-

33356

106

0.0

0.3769

120.0-F

33356

0

0.007

0.3763

176.3-

9793

28

0.0

0.3763

176.3+

9793

0

0.010

0.3755

4 = 206.3 days,

&Iv, = 0.039 km/s,

[km/S1

A(m/mJ = 0.0033

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6. Conclusions This paper deals with the development of a near-optimal guidance scheme for a spacecraft flying from a low Earth orbit to a low Mars orbit. We assume that the spacecraft is powered by a chemical engine and that the guidance and control is implemented by changing the magnitude and/or direction of the thrust via propellant consumption. The focus of the paper is on guidance for the interplanetary phase. The main conclusions are given below: (i) A detector-predictor-corrector

scheme is developed for guidance and control. The

detector computes the spacecraft trajectory from the current measured state and determines the trajectory error in the neighborhood of the low Mars orbit. The predictor is activated when the trajectory error exceeds a specified threshold. The predictor task is to compute a target trajectory, leading from the current state to rendezvous with the low Mars orbit via proper thrust input to the spacecraft. The corrector is a feedback control scheme implementing the control commands from the predictor. (ii) The maximum control margin concept is used in the predictor design to enhance the guidance robustness. Key to the guidance is the design of the target trajectory generated by the predictor, based on an application of the above concept. Because the spacecraft is in coasting flight along a large portion of the interplanetary trajectory, the control margin is measured by the amount of available propellant. Therefore, application of the maximum control margin concept means that the target trajectory should be designed so that the spacecraft can rendezvous with the low Mars orbit with minimum propellant consumption. (iii) The sequential gradient-restoration algorithm (SGRA) in conjunction with system decomposition is used to achieve the real-time implementation of the guidance: the target trajectory is generated via a two-step process, which includes target restoration followed by velocity restoration. The objective of the first step is to obtain a target trajectory starting from the current spacecraft position with rninimum change of the velocity magnitude; the objective of the second step is to achieve the required velocity change via proper thrust input to the spacecraft.

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The decomposition is effective because the time required for velocity restoration is in the order of seconds, while the flight time covered within the target restoration phase is in the order of months. Considerable enhancement of SGRA for robustness, convergence, and speed can be achieved by this decomposition. (iv) Preliminary numerical results show that the detector-predictor-corrector

scheme can

achieve the rendezvous with the low Mars orbit while containing the propellant consumption in the presence of errors and uncertainties in the thrust, system, and measurements. (v) The guidance scheme developed in this paper is robust. However, the fact is that large errors and uncertainties induce a severe penalty on propellant consumption. Efforts to reduce the measurement errors, improve the accuracy of the system equations, and improve the accuracy of the thrust representation are indispensable to the reduction of the propellant consumption for guidance.

References 1. Walberg, G., How Shall We Go to Mars?

A Review

of Mission

Scenarios,

Journal of

Spacecraft and Rockets, Vol. 30, No. 2, pp. 129-139, 1993. 2. Wercinski, P. F., Mars Sample Return:

A Direct

and Minimum-Risk

Design,

Journal of

Spacecraft and Rockets, Vol. 33, No. 3, pp. 381-385, 1996. 3. ONeil, W. J., and Cazaux, C., The Mars Sample Return Project, Acta Astronautica, Vol. 47, Nos. 2-9, pp. 453-465,200O. 4. Miele, A., and Wang, T., Optimal Transfers from an Earth Orbit to a Mars Orbit, Acta Astronautica, Vol. 45, No. 3, pp. 119-133, 1999. 5. Miele, A., and Wang, T., Optimal Trajectories Missions,

and Mirror Properties

for Round-Trip

Mars

Acta Astronautica, Vol. 45, No. 11, pp.655-668, 1999.

6. Miele, A., and Mancuso,!?, Design Feasibility via Ascent Optimality for Next-Generation Spacecraji,

Acta Astronautica, Vol. 45, No. 12, pp. 705-715, 1999.

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7. Miele, A., Wang, T., and Mancuso, S., Optimal Free-Return and Mars Missions,

Journal of the Astronautical

Trajectories for Moon Missions

Sciences, Vol. 48, Nos. 2-3, pp. 183-206,

2000. 8, Miele, A., Wang, T., and Mancuso, Missions,

S., Fundamental

Paper IAA-00-IAA.13.1.09,

Issues of Orbital Transfers for Mars

51st International

Astronautical

Congress,

Rio de

Janeiro, Brazil, 2000. 9. Miele, A., Wang, T., and Mancuso, Human Mars Excursions,

10. Ganapathi,

S., Assessment

Acta Astronautica,

Vehicle Advances

to Enable

Vol. 49, No. 11, pp.563-580,200l.

C. S., Performance

G. B., and Engelbrecht,

Space One, Journal of Spacecraft

of Launch

of the Xenon Feed

System on Deep

and Rockets, Vol. 37, No. 3, pp. 392-398, 2000. , Mars Missions Using Solar Electric Propulsion,

11. Williams, S. N., and Coverstone-Carroll,V.

Journal of Spacecraft and Rockets, Vol. 37, No.1, pp. 7 l-77,2000. 12. Chang-Diaz, F. R., Hsu, M. M., Braden, E., Johnson, I., and Yang, T. F., Rapid Mars Transits with Exhaust-Modulated

Plasma Propulsion, NASA Technical Paper 3539, 1995.

13. Vadali, S. R., Nah, R., Braden, Trajectories

E., and Johnson, I. L., Fuel-Optimal

Using Low-Thrust Exhaust-Modulated

Propulsion,

Planar Earth-Mars

Journal of Guidance, Control,

and Dynamics, Vol. 23, No. 3, pp. 476-482,200O. 14. Miele, A., and Wang, T., Near-Optimal

Highly Robust

Transfer, Journal of Guidance, Control, and Dynamics,

15. Miele, A., and Wang, T., Robust Predictor-Corrector Transfer, Journal of Guidance, Control, and Dynamics,

16. Miele, A., Pritchard, R. E., and Darnoulakis, for Optimal Control Problems,

Guidance

Orbital

Vol. 19, No. 3, pp. 549-556, 1996. Guidance

for Aeroassisted

Orbital

Vol. 19, No. 5, pp. 1134-l 141, 1996.

I. N., Sequential

Journal of Optimization

for Aeroassisted

Gradient-Restoration

Algorithm

Theory and Applications,

Vol. 5, No.

4, pp. 235-282, 1970. 17. Miele, A., Wang, T., and Basapur, Gradient-Restoration

Algorithms

for

V.K., Primal and Dual Trajectory

Formulations

Optimization Problems,

of Sequential

Acta Astronautica,

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Vol. 13,No. 8, pp. 491-505,1986. 18. Rishikof, B. H., Mccormick, B. R., Pritchard, R. E., and Sponaugle, S. J., SEGRAM: A Practical and Versatile Tool for Spacecraf

Trajectory Optimization, Acta Astronautica, Vol.

26, Nos. 8-10, pp. 599-609, 1992. 19. Miele, A., and Iyer, R. R., Modified Quasilineariza:ion Method for Solving Nonlinear Two-Point Boundary-Value Problems, Journal of Mathematical Analysis and Applications,

Vol. 36, No. 3, pp. 674-692,197l. 20. Miele, A., Naqvi, S., Levy, A. V., and Iyer, R. R., Numerical Solutions of Nonlinear Equations and Nonlinear

Two-Point Boundary-Value

Problems, Advances

in Control

Systems, Edited by C. T. Leondes, Academic Press, New York, NY, Vol. 8, pp. 189-215, 1971. 21. Franklin,G. F., Powell, J. D. and Emami-Naeini, A., Feedback Control of Dynamic Systems, Addison-Wesley, Reading, Massachusetts, 1994.