(Preprint) AAS 2-66 LYAPUNOV-BASED ELLIPTICAL TO CIRCULAR PLANAR ORBIT TRANSFERS IN LEVI-CIVITA COORDINATES Sonia Hernandez, Maruthi R. Akella, and Cesar A. Ocampo INTRODUCTION We consider planar orbit transfers around a central body using Lyapunov stability theory. The model used is the Levi-Civita transformation of the planar two-body problem. The advantage of working in these transformed coordinates is that the solution to the unperturbed equations of motion is that of a simple harmonic oscillator, so the analytical solution is known at all times during coast phases. We design a closed-loop guidance scheme for an orbit transfer from any initial elliptical orbit to a final circular orbit, utilizing a spacecraft with thrust-coast capabilities. The proposed Lyapunov functions give rise to asymptotically stabilizing control laws. The algorithm designed is robust, computationally fast, and can be used for both low- and high-thrust problems. Our new guidance solutions are seen to compare quite favorably to optimal orbit transfers using low thrust engines. The problem of computing both low- and high-thrust orbit transfers around a central body has been studied for many decades., 2 Some of the earliest work was done by Edelbaum on lowthrust orbit transfers using variational techniques to find the optimal transfer solution based on small changes in the orbital elements. 3 Much work has concentrated on optimal transfers using both direct and indirect methods. 4, 5 However, these solutions are computationally expensive and do not often result in closed-form solutions. Some attention has been given to heuristic models where feedback control laws are designed based on candidate Lyapunov functions. 6 8 The advantage of using these types of controllers is that the computation time is much smaller than that for optimization, even though the solutions may be suboptimal. In this paper we take on a new approach to the problem of orbit transfers by working in a transformed model to design Lyapunov-based control laws. The idea of using a transformed model has been looked at in Ref. 9 to solve the minimum-time transfer between coplanar circular orbits. The model we use is the classical Levi-Civita regularization transformation of the planar two-body problem. The advantage of working in these transformed coordinates is that the solution to the unperturbed equations of motion is that of a simple linear harmonic oscillator, where the frequency of oscillation is a function of a, the semi-major axis of the orbit. The problem at hand is to design a closed-loop guidance scheme for a coplanar orbit transfer from any initial elliptical orbit ( e < ) with a given a, to a final circular orbit (e f = ) Graduate Student, Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX, 7872, USA, sonia.hernandez@utexas.edu. Associate Professor, Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX, 7872, USA, makella@mail.utexas.edu. Associate Professor, Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX, 7872, USA, cesar.ocampo@mail.utexas.edu.
with a specified a f, where e is the eccentricity, utilizing a spacecraft with thrust-coast capabilities. The convergence to the desired orbit is performed in two steps. The first step involves converging the spacecraft orbit to the desired semi-major axis a f, even though the eccentricity at this stage may not be the desired one, by using a Lyapunov analysis that gives rise to an asymptotically stabilizing control law. Once a f has been reached to within a specified tolerance, the second step of the algorithm involves circularizing the orbit (while maintaining the specified target a f ), using the same control law as before, but with an added on/off switching mechanism for coasting during certain intervals. The algorithm designed is robust, computationally fast, and can be used for both low- and high-thrust problems. Our low-thrust solutions are seen to be qualitatively similar in fuel consumption to Edelbaum s optimal solution. The paper is organized as follows. We begin by deriving the Levi-Civita equations of motion of a spacecraft which is allowed to thrust, in a two-body force model. The next section shows the guidance scheme designed to target any desired semi-major axis a f (regardless of the final eccentricity e f ) from an orbit with an initial semi-major axis a. With the same guidance scheme, we are able to show in the next section that targeting a final eccentricity e f = is possible when a = a f at the beginning of the transfer. By allowing the spacecraft to have thrust-coast capabilities, we then derive the control algorithm required for a general orbit transfer from any initial elliptical orbit ( e < ) with a given a, to a final circular orbit (e f = ) with a specified a f. Several solutions are presented, including a low-thrust example which is compared to Edelbaum s optimal solution, and a high-thrust solution which is compared to its equivalent optimal control transfer. The last section discusses the conclusions and our future work. PERTUBED LEVI-CIVITA EQUATIONS OF MOTION In this section, the Levi-Civita equations of motion are derived from the governing equations of two bodies. The equations of motion of a spacecraft in a perturbed planar two-body model are [ ẍ = µ [ [ x ax ÿ r 3 + () y where µ is the gravitational parameter of the central body, the position vector r = [x y T has magnitude r = x 2 + y 2, and a = [a x a y T is the perturbing force. Using polar coordinates r and θ (where θ is measured counter-clockwise from the positive x-axis), the position can be written as x = r cos θ y = r sin θ Introducing two new varialbes ξ and η, we make the following transformation x = ξ 2 η 2 ξ = [ [ [ r cos θ 2 x ξ η ξ y = 2ξη η = r sin θ = (2) y η ξ η 2 ( ) Note that r = ξ 2 + η 2 and θ = 2 tan η ξ. Also, the following time transformation will be used to change from t units to s units, We refer to d dt ( ) = ( ) and d ds ( ) = ( ). dt ds = a y r µ (3) 2
Rewriting the governing equations in Eq. () with the new coordinates ξ and η yields [ ẍ = µ [ [ [ ξ η ξ ax ÿ r 3 + η ξ η Taking the time-derivative of Eq. (2), [ ẋ = 2 [ [ µ ξ η ξ ẏ r η ξ η = [ ξ η a y = 2 µ [ ξ η η ξ [ ẋ ẏ Next, since r = ξ 2 + η 2 from Eq. (2), its derivative in s coordinates is r = 2ξξ + 2ηη. Therefore, µ ṙ = r r = 2 µ ( ξξ + ηη ) r Using the result for ṙ and taking the time-derivative of Eq. (2) twice, yields after performing the necessary algebra [ ẍ = 2µ [ ξ ÿ r 3 (ξ2 + η 2 ) 2 η 2 + 2µ [ [ ξ η ξ 2ξη r 2 η ξ η (6) Equating Eq. (4) and (6) and solving for ξ and η yields [ ξ η = [ [ ξ 2(ξ 2 + η 2 ) 2r η = α [ [ ξ aξ + 4 η a η a η + r 2µ where the accelerations in the transformed coordinates are [ aξ = r [ [ ξ η ax 2µ η ξ and the parameter α is a y [ ξ η [ ax η ξ a y (4) (5) (7) (8) α = 2 r v2 µ = a where a is the semi-major axis. Note that we have used the fact that [ ξ T [ ξ ξ 2 + η 2 = η η = [ T [ [ ẋ ξ η ξ η 4µ ẏ η ξ η ξ where v 2 = ẋ 2 + ẏ 2. Taking the time derivative of Eq. (9) yields α = 2 r 2 ṙ 2 µ (ẋẍ + ẏÿ) = 2 r µ Using Eq. (5) and the time transformation in Eq. (3) [ ξ T [ ξ η [ x y T [ ax a y [ ẋ ẏ = rv2 4µ (9) α = 4 µ η η = 8 ( ξ a ξ + η ) a η r ξ [ ax a y () Summarizing, the equations of motion in the Levi-Civita coordinates for ξ, η, and α are given by Eq. (7) and Eq. (). 3
Unperturbed Equations of Motion In the case when there is no perturbation, that is, a ξ = a η =, the equations of motion in Eq. (7) become de-coupled linear harmonic oscillators and α = from Eq. (). The analytical solution is therefore known, with a frequency of oscillation of ω = α 2. Its position in s time-coordinates is given by and its velocity by ξ(s) = ξ cos(ωs) + ξ ω sin(ωs) η(s) = η cos(ωs) + η ω sin(ωs) () ξ (s) = ξ ω sin(ωs) + ξ cos(ωs) η (s) = η ω sin(ωs) + η cos(ωs) where ξ, η, ξ, and η are the initial conditions at s. Finite Thrust Maneuvers Suppose that the perturbation now comes from a finite thrust engine, with constant thrust magnitude T, specific impulse c, and variable thrust angle φ, such that [ ax a y [ cos φ = A sin φ The acceleration magnitude A = T/m and m is the mass of the spacecraft at any point in time, which varies according to ṁ = T/c. In the Levi-Civita coordinates, this corresponds to [ aξ a η = Ar3/2 The equations of motion reduce to the form where we have defined 2µ ξ = α 4 ξ + Ar3/2 2µ η = α 4 η + Ar3/2 2µ [ cos(φ θ/2) sin(φ θ/2) α = 4Ar/2 ξ 2 + η 2 µ m = r T µ c sin λ = cos(φ θ/2) sin(φ θ/2) ξ ξ 2 + η 2 and cos λ = sin(φ + λ θ/2) (2) η ξ 2 + η 2 4
MATCHING SEMI-MAJOR AXIS The first step in the two-step process to transfer from an initial orbit to a final specified orbit involves targeting the semi-major axis of the desired orbit. Suppose that the semi-major axis of the target orbit is denoted by a f, and therefore, its inverse is α f = /a f. At any point in time (in s coordinates) the spacecraft, whose motion is governed by Eq. (2), has a semi-major axis a(s), and correspondingly α(s) = /a(s). The difference between the current α(s) and the target α f is denoted by e α = α α f. The quadratic potential V = 2 e2 α (3) is minimized when the semi-major axis of the spacecraft matches that of the desired orbit. The time-derivative of V along solutions of (2) is given by V = e α e α = (α α f )α = (α α f ) 4A r ξ 2 + η 2 µ sin (φ + λ θ/2) Choosing where e α = e α (s ), results in which ensures that V is non-increasing. φ = λ + θ/2 + sin (Ke α ), K = / e α > (4) r V = 4A ( ξ 2 + η 2) Ke 2 α (5) µ Theorem Let a spacecraft with a constant available maximum thrust T and initial mass m be on a closed orbit with initial semi-major axis a(s ) = a. All solutions governed by the force model (2) with control (4) and A = T/m converge to any prescribed final semi-major axis a f. An example trajectory under the gravitational pull of Earth (µ = 3986 km 3 /s 2 ) using control (4) is shown in Figure. A final semi-major axis a f =, km is targeted from an initial circular orbit with a = 7, km. The integration of Eq. (2) is performed in dimensionless units, in order to avoid errors caused by e α = α α f, which can be a very small number when working in units of km. The dimensionless units are distance unit DU = a f =, km, mass unit MU = m =, kg, and time unit T U is chosen such that µ = DU 3 /T U 2. The trajectory in x, y coordinates is shown in Figure (a), which has been transformed using Eq. (2) from the ξ, η results of the integration. Note that the semi-major axis has been met to within km precision, but this has resulted in an the eccentricity e f =.3. The next section deals with targeting any circular orbit with e f = once the desired semi-major axis a f has been met. 5
y (DU).5.5.5.5.5 x (DU) (a) Position x vs y Earth Orbit Orbit f sma match t t f mass (MU).8.6.4.2 2 3 4 5 time (hours) (b) Mass m vs time.95.9.4.3 a (DU).85.8.75.7 2 3 4 5 time (hours) (c) Semi-major axis a vs time e.2. 2 3 4 5 time (hours) (d) Eccentricity e vs time Figure. Semi-major axis matching using control (4) from an initial orbit with a = 7, km, e =. to a f =, km using T = 2 N, Isp = 4 s, m = kg. The semi-major axis is targetted to within km, and the distance unit used DU = a f. MATCHING ECCENTRICITY Once the desired semi-major axis a f has been met, the next step in the process is to target any desired eccentricity e f. In this paper we restrict our results to target circular orbits with e f = from any initial orbit with e <. We assume in this section that a = a f at s. The angular momentum vector expressed in the direction ˆk normal to the orbit plane is given by h = r v = (xẏ yẋ) ˆk [ T [ [ x ẋ h = h = y ẏ = 2 [ µ T [ [ [ x ξ η ξ r y η ξ η = 2 [ T [ [ ξ ξ µ η η = 2 µ(ξη ηξ ) (6) 6
Recall that α = /a and the semi-latus rectum p = a( e 2 ) = h 2 /µ. Therefore, the eccentricity can be written as e 2 = α µ h2 = 4α(ξη ηξ ) 2 (7) The quadratic potential W = 2 e2 = ( αµ ) 2 h2 = 2 2α ( ξη ηξ ) 2 (8) is minimized when the spacecraft is in a circular orbit, i.e. e(s f ) = e f =. The time-derivative in s-coordinates of W along solutions of (2) is given by W = ee = 2α (ξη ηξ ) 2 4α(ξη ηξ ) [ ξη ηξ = 2α (ξη ηξ ) 2 4α(ξη ηξ ) Ar3/2 2µ [ξ sin(φ θ/2) η cos(φ θ/2) Choosing the same control law as before (Eq. (4)) and since the initial semi-major axis of the orbit is already the desired one, a = a f e α = α α f =. φ = λ + θ /2 + sin (Ke α )) = λ + θ /2 Therefore, from Eq. (2), α = s and a = a(s) = a f s. The control law can be written more compactly as The time derivative of the potential function with control (9) is φ = λ + θ/2 (9) W = 2α (ξη ηξ ) 2 4α(ξη ηξ ) Ar3/2 [ξ sin( λ) η cos( λ) 2µ = 2α (ξη ηξ ) 2 4α Ar 3/2 ( + ξη ηξ ) ( ξξ + ηη ) ξ 2 + η 2 2µ = α Ahr 3/2 ( ξξ ξ 2 + η 2 µ 2 + ηη ) (2) Note that (ξη ηξ ) = h/(2µ) >. Therefore, the only term that needs to be regulated in order to ensure that W is non-increasing is (ξξ + ηη ). We do this by adding a switching function such that A = { +T/m if (ξξ + ηη ) < T/m if (ξξ + ηη ) (2) which ensures W. Theorem 2 Let a spacecraft with a constant available maximum thrust T and initial mass m be on a closed orbit with a semi-major axis a = a f and an eccentricity e <. All solutions governed by the force model (2) with control (9) and switching function (2) converge to a final circular orbit with semi-major axis a f and eccentricity e f =. 7
s.5 t t f.5 s f y (DU) (DU).5 Earth Orbit Orbit f.5.5 ecc match x (DU).5 (a) Position x vs y.5 Earth Orbit Orbit f (DU) ecc match (b) Position ξ vs η Switch term ( + ).4.2.2 A = T/m A = +T/m mass (MU).9.8.7.6.5.4 2 3 time (hours) (c) Switching scheme A vs time.4 2 3 time (hours) (d) Mass m vs time..8 a(du).8.6.4.2 2 3 time (hours) (e) Semi-major axis a vs time e.6.4.2 2 3 time (hours) (f) Eccentricity e vs time Figure 2. Eccentricity matching using control (4) and the switching scheme (2) from an initial orbit with a = a f =, km, and e =.7 to a final orbit with e f = using T = 5 N, Isp = s, m = kg. Figure 2 shows the orbit transfer from an orbit with a semi-major axis a = a f =, km, initial eccentricity e =.7, and initial spacecraft location at θ = 8 o, to a final orbit with eccentricity e f =. using the guidance scheme (4) and switching function (2). The initial 8
spacecraft mass is m = kg, the engine parameters are T = 5 N and Isp = s. The results have been scaled using DU = a f and MU = m. The transfer takes about 28 hours and consumes 5% of its mass in fuel (Figure 2(d)). Note how in Figure 2(e) the semi-major axis remains constant throughout the entire transfer. ELLIPTICAL TO CIRCULAR ORBIT TRANSFER The previous two sections deal with either targeting a desired semi-major axis a f (disregarding the final eccentricity e f ) or targeting a final circular orbit e f = when the semi-major axis to begin with is already the desired one. This section addresses the general orbit transfer problem from any elliptical orbit to any final circular orbit by coalescing both methods described. Assume that a spacecraft in an initial orbit with semi-major axis a and eccentricity e < thrusts using the guidance scheme (4) from s s int until it reaches a desired semi-major axis a(s int ) a f (to within a specified tolerance), but the eccentricity e(s int ) e f. At this point, the second part of the orbit transfer begins, from s int s s f, where the quadratic potential W in Eq. (8) is again minimized when e(s f ) = e f =. Choosing the same control law as in Eq. (4), and since a(s int ) a f e α, makes the derivative of α, α at s = s int. φ = λ + θ/2 + sin (Ke α ) λ + θ/2, K = / e α > The time derivative of the potential function (8) with control (4) is W = 2α (ξη ηξ ) 2 + 4α Ar 3/2 ( ξη ηξ ) ( ξξ + ηη ) (22) ξ 2 + η 2 2µ α Ahr 3/2 ( ξξ ξ 2 + η 2 µ 2 + ηη ) where h is given by Eq. (6). Note that the switching scheme in Eq. (2) cannot be used to ensure W. Because α precisely (α f has been matched only to within a specified tolerance), using A = T/m if (ξξ + ηη ) results in the time derivative of the quadratic potential V in Eq. (5) to be positive definite, V >. This results in the difference between the desired and actual semi-major axis e α to grow, which in turn causes α (in fact, it may grow enough to make the first term in Eq. (22) dominant, which cannot guarantee W to be negative definite). The new potential switching function, which enables thrust-coast capabilities, is designed such that A = { +T/m if (ξξ + ηη ) < if (ξξ + ηη ) (23) This ensures that α s and therefore W. In fact, it is interesting that when the spacecraft is allowed to thrust, the control scheme is the same as the semi-major axis targeting (Eq. (4)). Therefore, even if the desired semi-major axis a f has not been achieved at s int, the semi-major axis of the transfer orbit will still approach the desired one during the eccentricity matching process, a(s) a f as s s f. Coast-Arc Durations During the coast times, A =, so neither the semi-major axis nor the eccentricity vary, and the Levi-Civita equations of motion become de-coupled. The analytical solution is given by Eq. () 9
and is known as long as there is no thrusting. The question is to determine how long to coast for before (ξξ + ηη ) < and the thrust is turned back on (Eq. (23)). To determine the coast time s for each subinterval we solve ξ(s )ξ(s ) + η(s )η(s ) = (24) using the analytical solution for ξ, η, ξ, and η in Eq. (). Solving algebraically, s = s (ξξ +ηη )= = ( ) ω cos c 2 (25) 2 c 2 + 4c2 2 where ω = α 2 c = ξ ξ + η η c 2 = ω (ξ2 + η 2 ) ω(ξ 2 + η) 2 To obtain the equivalent time t that the spacecraft is coasting, the integral of Eq. (3) is taken, which can be done by computing the area under the curve of r/ µ for s s s. Note, as can be seen in Figure 5(c), the switching function during a coast arc satisfies q. = (ξξ +ηη ), which is a known sinusoidal function. Once the thrust is turned back on, (ξξ +ηη ) changes sign and becomes negative; however the term q might start immediately increasing again, become greater than zero too fast, and potentially result in chattering. This is why Eq. (23) is for practical implementation purposes not a good candidate, even though it is theoretically valid. The switching scheme is modified so that at the end of a coast arc, the term (ξξ + ηη ) reaches a minimum. This way the thrust arc will have a long enough duration before the term q changes signs again, q. The two-step process is. s s s int : Semi-major axis Matching A(s) = +T/m while a(s) a f tol a (26) 2. s int s s f : Eccentricity Matching { until (ξξ A(s) = + ηη ) is min at s min +T/m while (ξξ + ηη (27) ) < and s > s min where s min = s + π/4 and tol a can specified by the user. Eq. (26) matches the semi-major axis to a close-enough value to the desired one, while Eq. (27) deals with matching the eccentricity during a coast-arc solution. The control in Eq. (4) and the switching scheme in Eq. (26 27) still make the time-derivative to the potential W in Eq. (22), and therefore ensures convergence to any desired circular orbit. Theorem 3 Let a spacecraft with a constant available maximum thrust T and initial mass m be on a closed orbit with an initial semi-major axis a and an eccentricity e <. All solutions governed by the force model (2) with control (4) and switching function (26 27) converge to a final circular orbit with semi-major axis a f and eccentricity e f =.
Figure 3 shows a low-thrust (T = N, Isp = 3 s) orbit transfer from an initial semi-major axis a = 6878 km and eccentricity e =.6 to a final orbit at GEO a f = 4264 km and e f =. The distance unit DU = a f km and mass unit MU = m = kg. The process is performed in two steps. The first one increases the semi-major axis until it reaches a(s int ) a f, which is when the eccentricity coast-arc solutions begin. Note that even during the eccentricity matching step, the semi-major axis continues to converge to the desired a f = DU. The transfer takes 4 days and uses a propellant mass m prop = 2 kg. y (DU).5.5.5.5 x (DU) Earth Orbit Orbit f sma match ecc match coast Switch term ( + ).2.. Coast.2 5 5 (a) Position x vs y (b) Switching scheme A vs time mass (MU).9.8 a (DU).8.6.4.2 e.8.6.4.2.7 5 5 (c) Mass m vs time. 5 5 (d) Semi-major axis a vs time 5 5 (e) Eccentricity e vs time Figure 3. Orbit matching using control in Eq. (4) and the switching scheme in Eq. (26 27) from an initial orbit with a = 6978 km, and e =.6 to a final orbit with a f = 42, 64 km and e f = using T = N, Isp = 3 s, m = kg. Another low-thrust solution is shown in Figure 4. In this case it is a circle to circle transfer from a = 6978 km to a f = 4264 using T = N and Isp = 3 s. In this guidance solution the only required control is to match the desired semi-major axis, since the thrust is low and the eccentricity stays small throughout the transfer. This solution is compared to Edelbaum s optimal solution for circle to circle transfers, 3 which is derived assuming a constant acceleration magnitude and a low eccentricity (e <<.) during the transfer. The comparison is shown in Table. A high-thrust solution with T = N and Isp = 3 s is shown in Figure 5. The initial orbit has a = 2, km and e =.6 while the final oribt has a f = 4, km and e f =. The distance unit DU = a f and the mass unit MU = m =, kg. The transfer is performed such that the eccentricity never exceeds e max =.7. The transfer orbit using control algorithm in Eq. (4) and the switching scheme in Eq. (26 27) (Figure 5(a)) is compared to the optimal control The Levi-Civita equations of motion have been modified in this example to allow for a constant acceleration magnitude, to allow us to compare our solution to Edelbaum s.
Table. Comparison of a low-thrust solution orbit transfer with Edelbaum s optimal solution with parameters shown in Figure 4. Time Transfer (days) Mass Propellant (kg) Eccentricity Edelbaum 3 5.89 4.34.9 Lyapunov Control 59.88 6.2.48 solution, 2 (Figure 5(b)). The comparison (shown in Table 2) clearly indicates that, even though robust, this new guidance scheme does not produce a transfer that is close to the optimal one when dealing with high-thrust engine capabilities. Table 2. Comparison of a high-thrust solution orbit transfer with the optimal control solution with parameters shown in Figure 5. Time Transfer (days) Mass Propellant (kg) Optimal Control.29 45.63 Lyapunov Control 3.8 369.8 CONCLUSION In this paper we take on a new approach to the problem of orbit transfers by working in a transformed model to design Lyapunov-based control laws. The model we use is the classical Levi-Civita regularization transformation of the planar two-body problem. The advantage of working in these transformed coordinates is that the solution to the unperturbed equations of motion is that of a simple linear harmonic oscillator, where the frequency of oscillation is a function of a, the semi-major axis of the orbit. We design a closed-loop guidance scheme for a coplanar orbit transfer from any initial elliptical orbit ( e < ) with a given a, to a final circular orbit (e f = ) with a specified a f utilizing a spacecraft with thrust-coast capabilities. The convergence to the desired orbit is performed in two steps. The first step involves converging the spacecraft orbit to the desired semi-major axis a f, even though the eccentricity at this stage may not be the desired one, by using a Lyapunov analysis that gives rise to an asymptotically stabilizing control law. Once a f has been reached to within a specified tolerance, the second step of the algorithm involves circularizing the orbit (while maintaining the specified target a f ), using the same control law as before, but with an added on/off switching mechanism for coasting during certain intervals. The algorithm designed is robust, computationally fast, and can be used for both low- and high-thrust problems. Several examples are given for various initial and final parameters as well as different engine capabilities. Our low-thrust solutions are seen to be qualitatively similar in fuel consumption to Edelbaum s optimal solution, while the higher-thrust problems, though robust, do not compare as well to orbit transfers using optimal control theory. The optimal control problem has been solved by defining a cost function to minimize the amount of propellant used in the transfer and solving for the costate equation and transversality conditions, which come from defining the Hamiltonian and Bolza functions. 2 The end solution involves solving for 5 equality constraints with 5 unknowns, and are therefore found using a nonlinear solver. 2
Earth Orbit Trajectory Orbit f.5 Orbit f sma match.5 Orbit y (DU) y (DU).5.5.5.5 x (DU) (a) Lyapunov Solution x vs y.5.5 x (DU) (b) Edelbaum Optimal Solution x vs y mass (MU).95.9.85 Lyapunov Control Edelbaum m f at t f a (DU).8.6.4.2 Lyapunov Control Edelbaum e..8.6.4.2 Lyapunov Control Edelbaum.8 2 4 6 (c) Mass m vs time 2 4 6 (d) Semi-major axis a vs time 2 4 6 (e) Eccentricity e vs time Figure 4. Low-thrust orbit matching using control Eq. (4) and the switching scheme in Eq. (26 27) from an initial circular orbit with a = 6978 km to a final circular orbit with a f = 42, 64 km using T = N, Isp = 3 s, and m = kg. The solution is compared to Edelbaum s optimal low-thrust solution. Future Work Part of the reason why our high-thrust solutions use a high propellant mass might be because of how we have implemented the thrust-coast arc times. As explained before, and can be seen in Figure 5(c), we allow the thrusting arc to last until the term q. = (ξξ + ηη ) reaches a minimum. This might not be the optimal way of solving the orbit transfer problem. We are looking at a more optimal way of quantifying when to switch from a coast arc to a thrust arc. We have constrained the results of this paper to target only final circular orbits (e f = ). However, the guidance scheme shown can be modified to allow for orbit transfers to any desired eccentricity e f < and semi-major axis a f. This will involve taking into account the orientation of the final desired orbit, which can be solved for by looking at the final argument of periapsis. We would also like to generalize the guidance schemes to allow for rendezvous problems, which involve targeting a final location on the desired orbit at some final time. Also for future work, we plan on expanding these solutions to a 3D model, via the regularization problem. ACKNOWLEDGEMENTS The authors would like to thank Prof. David G. Hull from the University of Texas at Austin and Mr. Brent W. Barbee from GSFC NASA for the insights exchanged with them on optimal guidance schemes for low-thrust orbit transfer applications. 3
REFERENCES [ D. F. Lawden, Optimal Transfers Between Coplanar Elliptical Orbits, AIAA Journal of Guidance, Control, and Dynamics, Vol. 5, May-June 992, pp. 788 79. [2 T. N. Edelbaum, Optimum Power-Limited Orbit Transfer in Strong Gravity Fields, AIAA Journal, Vol. 3, 965, pp. 92 925. [3 T. N. Edelbaum, Optimum Low-Thrust Rendezvous and Station-Keeping, Proc. AIAA Journal of Spacecraft and Rockets, Vol. 4, 23, pp. 96 965. [4 W. E. Wiesel and S. Alfano, Optimal Many-Revolution Orbit Transfer, AIAA Journal of Guidance, Control, and Dynamics, Vol. 8, Jan-Feb 985, pp. 55 57. [5 H. J. Oberle and K. Taubert, Existence and Multiple Solutions of the Minimum-Fuel Orbit Transfer Problem, Journal of Optimization Theory and Applications, Vol. 95, November 997, pp. 243 262. [6 A. E. Petropoulos, Low-Thrust Orbit Transfers Using Candidate Lyapunov Functions with a Mechanism for Coasting, AIAA/AAS Astrodynamics Specialist Conference, Providence, RI, August 24. [7 D. E. Chang, D. F. Chichka, and J. E. Marsden, Lyapunov-Based Transfer Between Elliptic Keplerian Orbits, Discrete and Continuous Dynamical Systems-Series B, Vol. 2, February 22, pp. 57 67. [8 M. R. Ilgen, Low Thrust OTV Guidance Using Lyapunov Optimal Feedback Control Techniques, Advances in the Astronautical Sciences, Vol. 85, 98, pp. 527 545. [9 J. D. Thorne and C. D. Hall, Minimum-Thrust Orbit Transfers Using the Kustaanheimo-Stiefel Transformation, AIAA Journal of Guidance, Control, and Dynamics, Vol. 2, 997, pp. 836 838. [ R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series, 999. [ D. G. Hull, Optimal Control Theory for Applications. Springer, Mechanical Engineering Series, 2. [2 B. A. Conway, Spacecraft Trajectory Optimization. Cambridge University Press, 2. 4
y (DU).5 Earth Orbit Orbit f sma match coast ecc match.5.5.5.5 x (DU) (a) Position x vs y (b) Optimal Control Transfer Switch term ( + ).4.2.2 Coast mass (MU).9.8.7 Optimal Control Soln.4 2 3 4 (c) Switching scheme vs time.6 2 3 4 (d) Mass m vs time..9.8.6 a (DU).8.7 e.4.6.2.5 2 3 4 (e) Semi-major axis a vs time 2 3 4 (f) Eccentricity e vs time Figure 5. Orbit matching using control in Eq. (4) and the switching scheme in Eq. (26 27) from an initial orbit with a = 2, km, and e =.6 to a final orbit with a f = 4, km and e f = using T = N, Isp = 4 s, m = kg. The solution is compared to the optimal control solution. 5