Minimum-Time Trajectory Optimization of Multiple Revolution Low-Thrust Earth-Orbit Transfers

Transcription

1 Minimum-Time Trajectory Optimization o Multiple Revolution Low-Thrust Earth-Orbit Transers Kathryn F. Graham Anil V. Rao University o Florida Gainesville, FL Abstract The problem o determining high-accuracy minimum-time Earth-orbit transers using low-thrust propulsion is considered. The optimal orbital transer problem is posed as a constrained nonlinear optimal control problem and is solved using a variable-order Legendre-Gauss-Radau quadrature orthogonal collocation method. Initial guesses or the optimal control problem are obtained by solving a sequence o modiied optimal control problems where the inal true longitude is constrained and the mean square dierence between the speciied terminal boundary conditions and the computed terminal conditions is minimized. It is ound that solutions to the minimum-time low-thrust optimal control problem are only locally optimal in that the solution has essentially the same number o orbital revolutions as that o the initial guess. A search method is then devised that enables computation o solutions with an even lower cost where the inal true longitude is constrained to be dierent rom that obtained in the original locally optimal solution. A numerical optimization study is then perormed to determine optimal trajectories and control inputs or a range o initial thrust accelerations and constant speciic impulses. The key eatures o the solutions are then determined, and relationships are obtained between the optimal transer time and the optimal inal true longitude as a unction o the initial thrust acceleration and speciic impulse. Finally, a detailed post-optimality analysis is perormed to veriy the close proximity o the numerical solutions to the true optimal solution. Nomenclature a = Semi-major Axis, m e = Eccentricity = Second Modiied Equinoctial Element g = Third Modiied Equinoctial Element g e = Sea Level Acceleration Due to Earth Gravity, m/s 2 H = Optimal Control Augmented Hamiltonian h = Second Modiied Equinoctial Element i = Inclination, deg or rad (i r, i θ, i h ) = Rotating Radial Coordinate System J 2 = Second Zonal Harmonic J 3 = Third Zonal Harmonic J 4 = Fourth Zonal Harmonic k = Fith Modiied Equinoctial Element Ph.D. Student, Department o Mechanical and Aerospace Engineering. kschuber@ul.edu. Associate Proessor, Department o Mechanical and Aerospace Engineering. anilvrao@ul.edu. Associate Fellow, AIAA. Corresponding Author. 1

2 L = Sixth Modiied Equinoctial Element (True Longitude), rad or deg m = Mass, kg n = Mean Motion P k = Legendre Polynomial o Degree k p = First Modiied Equinoctial Element (Semi-Parameter), m R e = Radius o the Earth, m T = Thrust, N t = Time, s or d u = Control Direction u r = Radial Component o Control u θ = Tangential Component o Control u h = Normal Component o Control = Spacecrat Speciic Force, m s 2 λ = Optimal Control Costate µ = Optimal Control Path Constraint Lagrange Multiplier µ e = Earth Gravitational Parameter, m3 s 2 ν = True Anomaly, deg or rad Ω = Longitude o Ascending Node, deg or rad ω = Argument o Periapsis, deg or rad 1 Introduction Low-thrust propulsion systems are typically characterized by high speciic impulses and small initial thrust accelerations (thrust-to-initial-mass) on the order o O(1 4 ) m s 2. The use o low-thrust propulsion has been studied extensively or orbital rendezvous, orbit maintenance, orbit transer, and interplanetary space mission applications. While the eiciency o low-thrust propulsion is highly appealing, the resulting trajectory design problem is particularly challenging to solve. For example, the high speciic impulse o a low-thrust engine combined with the small engine speciic orce leads to computational challenges due to the long duration o the orbital transer. In addition, the trajectory design problem is particularly problematic when the initial and terminal orbits are widely spaced resulting in a trajectory that requires a large number o orbital revolutions in order to complete the transer. Then, even i a solution is obtained, it is highly likely that the trajectory is not the global optimal solution. 1 Low-thrust trajectory optimization has been the subject o much previous research. In Res. 1 8 numerical optimization techniques were used or to optimize interplanetary space trajectories. In Re. 9 a variation o parameters approach was employed to solve a minimum-uel time-ixed rendezvous problem, while in Re. 1 Pontryagin s minimum principle 11, 12 was used to determine the optimal thrust acceleration or an orbit maintenance study. In Re. 13 optimal control theory was used to solve minimum-time, circleto-circle, constant thrust orbit raising and simple graphical and analytical tools were used that related vehicle design parameters to orbit design parameters. In addition, a variety o approximation methods have been developed to overcome the computational challenge associated with the large number o orbital revolutions typical o a low-thrust orbital transer. One o the most common approximation techniques is orbital averaging where simple approximations are derived to express incremental changes in the orbital elements or each orbital revolution. Using orbital averaging, in Re. 14 the problem o minimum-uel powerlimited transers between coplanar elliptic orbits was studied, while in Re. 15 near-optimal, minimum-time low-earth orbit (LEO) to geostationary orbit (GEO) and geosynchronous transer orbit (GTO) to GEO transers were examined. In addition, in Re. 16 a parameterized control law was employed together with orbital averaging in order to solve three common near-optimal, minimum-time Earth-orbit transers. More recently, in Re. 5 an orbital averaging approach was developed in conjunction with hybrid control ormulations to solve LEO to GEO and GTO to GEO transers. Next, in Re. 17 a 1-revolution LEO to GEO coplanar transer was solved using direct collocation paired with a Runge-Kutta parallel-shooting 2

3 scheme, while in Re. 18 a single shooting method was combined with a homotopic approach to solve a minimum-uel transer rom a low, elliptic, and inclined orbit to GEO. In order to increase accuracy over orbital averaging techniques, in Re. 19 sequential quadratic programming (SQP) was used with direct collocation to solve a minimum-uel low-thrust near-polar Earth-orbit transer with over 578 revolutions. In Re. 2 an anti-aliasing method utilizing direction collocation was developed to obtain solutions to simple low-thrust trajectory optimization problems. Finally, in Re. 21 a minimum-time LEO to high-earth orbit (HEO) transers was solved using direct collocation with a single speciic impulse value. While a great deal o progress has been made in low-thrust trajectory optimization, much o this work ocuses on determining near-optimal solutions and very little work has been done to veriy the optimality o the solutions obtained. The contribution o this research is on determining high-accuracy solutions to minimum-time low-thrust trajectory optimization problems or a wide-range o initial thrust accelerations and speciic impulse values. Speciically, in this paper a variable-order Gaussian quadrature orthogonal collocation method is used to determine minimum-time optimal trajectories o two common low-thrust Earth-orbit transers. As a result, solutions to the optimal control problem are obtained without having to replace the equations o motion with averaged approximations over each orbital revolution such as in using an orbital averaging technique. Using the aorementioned collocation method, an initial guess generation method is used together with a simple search method to determine the solution that has the lowest cost amongst a range o locally optimal solutions. Numerical solutions are generated or a range o initial thrust acceleration values and speciic impulse values that are typical o a low-thrust propulsion system. Then, in a manner similar to that o Re. 21, regression analyses are perormed to determine the transer time as a unction o the initial thrust acceleration and the speciic impulse and to determine the inal true longitude as a unction o the transer time. From these regressions it is possible to estimate the transer time and inal true longitude or dierent initial thrust accelerations and speciic impulses without having to re-solve the optimal control problem. A post-optimality analysis is then perormed to veriy the optimality o the solutions obtained in this study. This paper is organized as ollows. Section 2 describes the minimum-time low-thrust orbit transer trajectory optimization problem solved in this research. Section 3 describes the direct collocation method used to solve the optimal orbital transer problem. Section 4 describes the numerical results obtained in this study and includes a post-optimality analysis to veriy the optimality o the solutions obtained. Finally, Section 5 provides conclusions on this work. 2 Low-Thrust Earth-Orbit Transer Optimal Control Problem Consider the problem o transerring a spacecrat rom an initial Earth-orbit to a inal Earth-orbit using low-thrust propulsion. The objective is to determine the minimum-time trajectory and control that transer the spacecrat rom the speciied initial orbit to the speciied terminal orbit. The low-thrust optimal control problem or the orbit transer is now described. 2.1 Equations o Motion The dynamics o the spacecrat, modeled as a point mass, are described using modiied equinoctial elements together with a ourth-order oblate gravity model and a continuous thrust propulsion system. The state o the spacecrat is comprised o the modiied equinoctial elements (p,, g, h, k, L) 3 together with the mass, m, where p is the semi-parameter, and g are modiied equinoctial elements that describe the eccentricity o the orbit, h and k are modiied equinoctial elements that describe the inclination o the orbit, and L is the true longitude. The control is the thrust direction, u, where u is expressed in rotating radial 3

4 coordinates as u = (u r, u θ, u h ). The dierential equations o motion o the spacecrat are given as dp p dt = 2p µ e q θ F p, d p (sin dt = L r + 1 ( ) q (q + 1) cos L + θ g ( ) ) q h sin L k cos L h F, µ e dg p ( dt = cos L r + 1 ( ) µ q (q + 1) sin L + g θ + ( ) ) q h sin L k cos L h F g, e dh p dt = s 2 cos L h F h, µ e 2q dk p dt = s 2 sin L h F k, µ e 2q dl p dt = ( ) h sin L k cos L h + ( ) q 2 µ e p F L, µ e p dm dt = T F m, g e I sp (1) where q = 1 + cos L + g sin L, r = p/q, α 2 = h 2 k 2, s 2 = 1 + h 2 + k 2. In this research, time is replaced as the independent variable in avor o the true longitude, L, because L provides a more intuitive understanding o the transer. Since the spacecrat moves rom an orbit close to the Earth to GEO, more true longitude cycles will be completed in a given amount o time near the start o the transer than will be completed near the terminus o the transer. Using the true longitude as the independent variable, the dierential equation or L is replaced with the dierential equation dt dl = 1 F L = F 1 L G t, (3) while the remaining six dierential equations or (p,, g, h, k, m) that describe the dynamics o the spacecrat are given as dp dl = F 1 L F p G p, d dl = F 1 L F G h, dg dl = F 1 L F g G g, (4) dh dl = F 1 L F h G h, dk dl = F 1 L F k G k, dm dl = F 1 L F m G m. Next, the spacecrat acceleration, = ( r, θ, h ), is modeled as = g + T, (5) where g is the gravitational acceleration due to the oblateness o the Earth and T is the thrust speciic orce. The acceleration due to Earth oblateness is expressed in rotating radial coordinates as g = Q T r δg, (6) (2) 4

5 where Q r = [ i r i θ i h ] is the transormation rom rotating radial coordinates to Earth-centered inertial coordinates and whose columns are deined as i r = r r, i h = Furthermore, the vector δg is deined as where i n is the local North direction and is deined as r v r v, i θ = i h i r. (7) δg = δg n i n δg r i r (8) i n = e n (e T ni r )i r e n (e T ni r )i r (9) and e n = (,, 1). The oblate earth perturbations are then expressed as δg r = µ e r 2 4 (k + 1) k=2 δg n = µ e cos φ r 2 4 k=2 ) k P k (sin φ)j k, r (1) ) k P k r k, (11) where R e is the equatorial radius o the earth, P k is the k th -degree Legendre polynomial, P k is the derivative o P k with respect to sin φ, and J k represents the zonal harmonic coeicients or k = (2, 3, 4). Next, the thrust speciic orce is given as ( Re ( Re Finally, the physical constants used in this study are given in Table 2. T = T u. (12) m Table 2: Physical Constants. Quantity Value g e m s 2 µ e m 3 s 2 R e J 2 J 3 J m Boundary Conditions and Path Constraints The boundary conditions or the orbit transer are described in terms o both classical orbital elements and modiied equinoctial elements. The spacecrat starts in either a near circular inclined low-earth orbit (LEO) or a geostationary transer orbit (GTO) at time t =. The initial orbit is speciied in terms o classical orbital elements 31 as a(l ) = a, Ω(L ) = Ω, e(l ) = e, ω(l ) = ω, i(l ) = i, ν(l ) = ν, where a is the semi-major axis, e is the eccentricity, i is the inclination, Ω is the longitude o the ascending node, ω is the argument o periapsis, and ν is the true anomaly. Equation (13) can be expressed equivalently (13) 5

6 in terms o the modiied equinoctial elements as p(l ) = a (1 e 2 ), h(l ) = tan(i /2) sin Ω, (L ) = e cos(ω + Ω ), k(l ) = tan(i /2) cos Ω, g(l ) = e sin(ω + Ω ), L = Ω + ω + ν. (14) For both cases considered, the spacecrat terminates in a geostationary orbit (GEO). The GEO terminal orbit is speciied in classical orbital elements as a(l ) = a, Ω(L ) = Free, e(l ) = e, ω(l ) = Free, i(l ) = i, ν(l ) = Free. (15) Equation (15) can be expressed equivalently in terms o the modiied equinoctial elements as p(l ) = a (1 e 2 ), ( 1/2 2 (L ) + g 2 (L )) = e, ( h 2 (L ) + k 2 (L )) 1/2 = tan(i /2). (16) Finally, during the transer the thrust direction must be a vector o unit length. Thus, equality path constraint u 2 = u 2 r + u 2 θ + u2 h = 1 (17) is enorced throughout the orbital transer. 2.3 Optimal Control Problem The goal o this study is to determine solutions to the ollowing constrained nonlinear optimal control problem. Determine the trajectory (p(l), (L), g(l), h(l), k(l), m(l), t(l)) and the control (u r (L), u θ (L), u h (L)) that minimize the cost unctional J = αt, (18) subject to the dynamic constraints o Eqs. (3) and (4), the initial conditions o Eq. (14), the terminal conditions o Eq. (16), and the path constraints o Eq. (17). Finally, it is noted that α = 1/864 is the conversion actor rom units o seconds to units o days. 3 Numerical Solution o Low-Thrust Optimal Control Problem The minimum-time low-thrust optimal control problem described in this paper was solved using the optimal control sotware GPOPS II. 29 GPOPS II is a MATLAB sotware that transcribes the optimal control problem to a nonlinear programming problem (NLP) that implements the variable-order Legendre-Gauss- Radau quadrature collocation method described in Res. 26, 27, and 32 together with an hp adaptive mesh reinement method (see Res. 28, 33, and 34). In this study the NLP arising rom the LGR collocation method is solved using the open-source NLP solver IPOPT 35 with analytical irst and second derivatives obtained using the open-source algorithmic dierentiation package ADiGator whose method is described in Re. 36. The remainder o this section is organized as ollows. First, an approach is described or generating initial guesses or solving the optimal control problem. Second, because the solutions obtained rom the NLP are only locally optimal, a motivation is provided or developing a simple search method to obtain solutions that are closer to the global optimal. Finally, the developed simple search method is described and is applied to the low-thrust trajectory optimization problem. 6

7 3.1 Initial Guess Generation In order to solve the low-thrust orbital transer optimal control problem described in Section 2, it was necessary to provide initial guesses rom which the NLP solver would converge to a solution. Because this research is ocused on solving a problem whose solution will result in a large number o orbital revolutions, the initial guess must itsel contain a number o orbital revolutions that is reasonably close to the actual number o orbital revolutions o the solution obtained by the NLP solver. In this paper, an initial guess procedure was devised where a sequence o optimal control sub-problems were solved. The goal o each sub-problem was to determine the state and control that transer the spacecrat rom the initial orbit to the terminal conditions that minimize the ollowing mean square relative dierence: [ ] p(l ) p 2 [ d 2 (L ) + g 2 (L ) e 2 ] 2 d J = p d 1 + e 2 + d [ ] h 2 (L ) + k 2 (L ) tan 2 ( i 2 d 2 ). (19) 1 + tan 2 ( i d 2 ) In other words, the objective o the optimal control sub-problem is to attain a solution that is as close in proximity to the desired terminal semi-parameter, p d, eccentricity, e d, and inclination, i d. Each sub-problem is evaluated at most over one true longitude cycle using the terminal state o the previous sub-problem as the initial state o the current sub-problem. The continuous-time optimal control sub-problem is then stated as ollows. Minimize the cost unctional o Eq. (19) subject to the dynamic constraints o Eqs. (3) and (4), the path constraint o Eq. (17), and the boundary conditions p (r) (L (r) ) = p(r 1) (L (r 1) ), p (r) (L (r) ) = Free, (r) (L (r) ) = (r 1) (L (r 1) ), (r) (L (r) ) = Free, g (r) (L (r) ) = g(r 1) (L (r 1) ), g (r) (L (r) ) = Free, h (r) (L (r) ) = h(r 1) (L (r 1) ), h (r) (L (r) ) = Free, k (r) (L (r) ) = k(r 1) (L (r 1) ), k (r) (L (r) ) = Free, m (r) (L (r) ) = m(r 1) (L (r 1) ), m (r) (L (r) ) = Free, L (r) = L (r 1), L (r) L (r 1) + 2π, (2) or r = 1,..., R where R represents the total number o true longitude cycles. The initial conditions or the irst cycle, when r = 1, are simply the initial conditions stated in Eq. (14). Once the desired terminal conditions, as stated in Eq. (16), are obtained within a user speciied tolerance, the sub-problem solutions are then combined to orm the initial guess. The initial mesh is comprised o intervals based on the total number o true longitude cycles and an arbitrarily chosen number o collocation points assigned to each interval. 3.2 Search Method to Assist in Obtaining Globally Optimal Solution It is generally the case that gradient-based optimization methods converge to locally optimal solutions as opposed to globally optimal solutions. In the case o the low-thrust orbital transer problems solved in this research, it was ound that the optimal solution typically contained the same number o true longitude cycles as that o the initial guess due to the act that the initial guess was very close to satisying the terminal constraints. Thus, while the NLP solver converges with the initial guess provided, the solution is usually not the global optimal. In order to obtain a solution with a number o true longitude cycles dierent rom that o the initial guess, the inal true longitude was bounded within a speciied cycle (that is, to lie within a speciied interval o 2π). By bounding the inal true longitude, the NLP solver is orced to deviate rom the initial guess and potentially move closer to a global solution. Figure 1 shows the cost obtained when the inal true longitude is bounded as described above and when the inal true longitude is ree or the GTO to GEO orbit transer with (T/m, I sp ) = ( m s 2, 3 s). It is seen rom 7

8 Fig. 1 that the solution obtained using the provided initial guess with a ree terminal true longitude does not have the lowest cost solution. Instead, the lowest cost lies somewhere between 8 and 9 true longitude cycles. Based on the structure shown in Fig. 1, the ollowing simple search method is used to identiy the approximate location o the globally optimal solution. First, an initial guess is generated as described in Section 3.1 and the inal true longitude that is obtained rom this initial guess, denoted L (), is the starting point or the search method. An iteration on the inal true longitude is then perormed as ollows, where K is the iteration number and K = corresponds to L (). The optimal control problem is solved or a inal true longitude L [L (K) 2π, L (K) ] = I (K) l and the cost obtained rom this solution is denoted J (K) l. Next, in order to determine which direction to search or a lower cost solution, the optimal control problem is solved again or a inal true longitude L [L (K), L (K) + 2π] = I r (K) and the cost associated with this solution is denoted J r (K) ollowing iterative process: Set K K + 1. Case 1 (minimum lies to the right o the initialization): L (K 1) + 2π, I (K) l = I r (K 1), J (K) l control problem again or L I (K) r The lowest cost is J (K) l. The cycle that contains the lowest cost is then obtained using the with L I (K) l. = J r (K 1), and I r (K) = [L (K) I J (K 1) l, L (K) and the cost obtained is denoted J (K) r Case 2 (minimum lies to the let o the initialization): I J (K 1) l 2π, I (K) r = I (K 1) l, J r (K) problem again or L I (K) l lowest cost is J (K) r = J (K 1) l, and I (K) l with L I (K) = [L (K) 2π, L (K) and the cost obtained is denoted J (K) l r. > J r (K 1), then set L (K) = + 2π]. Then solve the optimal. Repeat until J (K) l < J (K) r. < J r (K 1), then set L (K) = L (K 1) ]. Then solve the optimal control. Repeat until J (K) r < J (K) l. The 4 Results and Discussion The GTO to GEO Earth-orbit transer problem was solved using the ollowing initial orbit: a(l ) = km, Ω(L ) = deg, e(l ) =.725, ω(l ) = deg, i(l ) = 7 deg, ν(l ) = deg, (21) while the LEO to GEO Earth-orbit transer problem was solved using the ollowing initial orbit: a(l ) = 6656 km, Ω(L ) = deg, e(l ) =.1, ω(l ) = deg, i(l ) = 28.5 deg, ν(l ) = deg. (22) Both the GTO to GEO and LEO to GEO Earth-orbit transer problems were solved using the ollowing terminal orbit: a(l ) = km, Ω(L ) = Free, e(l ) =, ω(l ), = Free, i(l ) = deg, ν(l ) = Free. The minimum-time GTO to GEO and LEO to GEO transers were solved with initial thrust acceleration values o T/m = (2., 1.,.667,.5,.4,.333,.286,.25,.222,.2) 1 3 m s 2 and speciic impulse values o I sp = (5, 1, 3, 5) s. 8 (23)

9 68 67 L Bounded L Free 66 J = t (days) L /(2π) Figure 1: Locally optimal solutions obtained with ree and bounded inal true longitude vs. L /(2π) or GTO to GEO transer with (T/m, I sp ) = ( m s 2, 3 s). The results o this study are divided into six sections. First, the key eatures o the GTO to GEO transer are described and analyzed. Second, the key eatures o the LEO to GEO transer are examined. Third, the relationship between the transer time, initial thrust acceleration, and speciic impulse is identiied through regression techniques. Fourth, regression techniques are utilized urther to identiy the relationship between the inal true longitude, transer time, and speciic impulse. Fith, the coeicients o determination or the previously deined relationships are shown to validate the it o the regressions. Finally, a post-optimality analysis o the solutions is provided to veriy the optimality o the solutions obtained. 4.1 Key Features o Optimal GTO to GEO Transers Figure 2 shows a view in Earth-centered inertial Cartesian coordinates (x, y, z) o a typical optimal GTO to GEO trajectory or the case (T/m, I sp ) = ( m s 2, 3 s). The optimal solution has a 7.78% change in mass, a duration o 74 days, and nearly 13 orbital revolutions. It is seen rom Fig. 3a that the semi-major axis increases nearly linearly throughout the entire transer, and this rate o increase varies only slightly as a unction o T/m. Furthermore, Fig. 3b shows that the eccentricity decreases slowly near the start o the transer and decreases more rapidly starting rom approximately one third o the way into the transer and beyond. Also, Fig. 3c shows that the inclination decreases at an approximately linear rate throughout the entire transer. 9

10 4 2 z (km 1 3 ) y (km 1 3 ) x (km 1 3 ) Figure 2: GTO to GEO transer trajectory or (T/m, I sp ) = ( m s 2, 3 s). 1

11 a (km 1 3 ) T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s ν/(2π) (a) a vs. ν/(2π). e T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s ν/(2π) (b) e vs. ν/(2π). i (deg) T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s ν/(2π) (c) i vs. ν/(2π). Figure 3: a, e, and i vs. ν/(2π) or GTO to GEO transer with various values o T/m and I sp = 3 s. 11

12 Further insight into the behavior o the optimal GTO to GEO transers is obtained by examining the control u = (u r, u θ, u h ) along dierent segments o the optimal solution. The typical overall behavior o u is shown in Fig. 4 or (T/m, I sp ) = ( m s 2, 3) s. A closer examination o u reveals that the ollowing our segments identiy the key eatures o the optimal control: (1) the irst ew orbital revolutions o the transer, (2) the region where u r, (3) the region where u h becomes 1, and (4) the inal revolutions o the transer. First, the eect o the control on the optimal trajectory in the irst ew orbital revolutions, can be explained via the ollowing dierential equations or the semi-major axis, eccentricity, and inclination: 37 da dt de dt di dt 2e sin ν = nx u r + 2ax nr u θ, (24) = x sin ν na u r + x ( ) a 2 x 2 na 2 r u θ, (25) e r r cos(ν + ω) = na 2 u h, (26) x where n is the mean motion and x = 1 e 2. It is seen rom Eq. (24) that the semi-major axis will increase when the control points either in the positive u θ -direction, radially outward near ν = π/2 (halway between periapsis and apoapsis), or radially inward near ν = 3π/2 (halway between apoapsis and periapsis). Furthermore, this cyclic behavior o u r increases apoapsis and decreases periapsis when ν [, π] and decreases apoapsis and increases periapsis when ν [π, 2π]. Equivalently, thrusting radially in this manner increases both the semi-major axis and the eccentricity. On the other hand, rom Eq. (25) the eccentricity will decrease when the control points either in the positive u θ -direction, radially inward near ν = π/2, or radially outward near ν = 3π/2. It is seen rom Fig. 5a that u r is positive near ν = π/2 and negative near ν = 3π/2, while u θ 1 in both cases. Even though u r increases the semi-major axis, it simultaneously increases eccentricity. This small eect o u r increasing eccentricity, however, is negated by the act that u lies predominantly in the positive u θ -direction, thereby resulting in an overall increase in semi-major axis and decrease in eccentricity. Lastly, it is seen rom Eq. (26) that di/dt is most negative when cos(ν + ω) u h is most negative. Examining Fig. 6a, it is seen that u h is most positive and cos(ν + ω) = 1 when the spacecrat is at apoapsis, thereby resulting in the largest negative slope in di/dt as seen in Fig. 6b. Next, Fig. 5b shows u in the segment o an optimal GTO to GEO transer where u r. For every orbital revolution on the optimal solution beyond where u r becomes zero (that is, all values beyond ν/(2π) = 38.5 as shown in Fig. 5b), u points radially inward near ν = π/2 such that apoapsis decreases and periapsis increases when ν [, π] and points radially outward near ν = 3π/2 such that apoapsis increases and periapsis decreases when ν [π, 2π]. Thrusting radially in this manner decreases both the semi-major axis and eccentricity. Although u r decreases the semi-major axis, the thrust direction lies predominantly in the positive u θ -direction, thereby increasing the semi-major axis and decreasing the eccentricity. Finally, because u h is most positive near ν = π and cos(ν + ω) = 1, the inclination decreases most rapidly near apoapsis. Next, Fig. 5c shows u in the segment o an optimal GTO to GEO transer where u h drops to 1. It is seen in this segment that u r continues to point inward near ν = π/2 and outward near ν = 3π/2, decreasing both the semi-major axis and the eccentricity. Moreover, u θ no longer dominates the thrust direction, reaching its most positive value at apoapsis and a gradually decreasing value at periapsis. Thrusting in this manner in the u r - and u θ -directions raises periapsis when ν [, π] and lowers apoapsis when ν [π, 2π]. Consequently, the semi-major axis increases while eccentricity decreases. Lastly, rom Fig. 6c, u h attains its most negative value near ν = π and cos(ν + ω) = 1 and near ν = 2π and cos(ν + ω) = +1. While the inclination continues to decrease signiicantly near apoapsis o the transer, Fig. 6d shows that di/dt is also negative near periapsis. Finally, Fig. 5d shows u during the inal ew revolutions o an optimal GTO to GEO transer. It is seen that u r is negative near ν = π/2 and positive near ν = 3π/2, while u θ is positive near ν = 2π and is negative near ν = π. Consequently, by thrusting in this manner periapsis increases and apoapsis decreases, 12

13 thereby resulting in a larger semi-major axis and a smaller eccentricity. Finally, u h is negative near ν = 2π and is positive near ν = π. Because the orbit is nearly circular near the end o the transer, the rate at which inclination decreases is essentially the same near periapsis and apoapsis. 1 ur ν/(2π) (a) u r vs. ν/(2π). uθ ν/(2π) (b) u θ vs. ν/(2π). uh ν/(2π) (c) u h vs. ν/(2π). Figure 4: u vs. ν/(2π) or GTO to GEO transer with (T/m, I sp ) = ( m s 2, 3 s). 13

14 u u -.5 u r u θ u h Periapsis Apoapsis ν/(2π) (a) u vs. ν/(2π) during the irst ew revolutions. -.5 u r u θ u h Periapsis Apoapsis ν/(2π) (b) u vs. ν/(2π) when u r u u -.5 u r u θ u h Periapsis Apoapsis ν/(2π) (c) u vs. ν/(2π) when u h becomes u r u θ u h Periapsis Apoapsis ν/(2π) (d) u vs. ν/(2π) during the last ew revolutions. Figure 5: u vs. ν/(2π) or GTO to GEO transer with (T/m, I sp ) = ( m s 2, 3 s). 14

15 1. u h cos(ν + ω) uh and cos(ν + ω) i (deg) ν/(2π) (a) u h and cos(ν + ω) vs. ν/(2π) during the irst ew revolutions ν/(2π) (b) i vs. ν/(2π) during the irst ew revolutions. 1. u h cos(ν + ω) uh and cos(ν + ω) i (deg) ν/(2π) (c) u h and cos(ν + ω) vs. ν/(2π) when u h becomes ν/(2π) (d) i vs. ν/(2π) when u h becomes 1. Figure 6: u h, cos(ν + ω), and i vs. ν/(2π)) or GTO to GEO transer with (T/m, I sp ) = ( m s 2, 3 s). 15

16 4.2 Key Features o Optimal LEO to GEO Transers Figure 7 shows a three-dimensional view in Earth-centered inertial Cartesian coordinates (x, y, z) o a typical optimal LEO to GEO trajectory or the case (T/m, I sp ) = ( m s 2, 1 s). The optimal trajectory has a 44.73% change in mass, takes approximately 152 days, and contains nearly 1,23 revolutions. The semi-major axis and inclination or all values o T/m and I sp = 1 s are shown in Figs. 8a and 8b, respectively. It is seen that the semi-major axis increases at a slower rate at the beginning o the transer and increases more rapidly towards the end o the transer. The inclination decreases at a slower rate at the beginning o the transer and decreases more rapidly towards the end o the transer. The eccentricity is roughly zero throughout the entire transer. 25 z (km 1 3 ) y (km 1 3 ) x (km 1 3 ) -5 Figure 7: LEO to GEO transer trajectory with (T/m, I sp ) = ( m s 2, 1 s). 16

17 45 35 a (km 1 3 ) T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s L/(2π) (a) a vs. L/(2π). i (deg) T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s L/(2π) (b) i vs. L/(2π). Figure 8: a and i vs. L/(2π) or LEO to GEO transer using various values o T/m and I sp = 3 s. 17

18 The structure o the optimal LEO to GEO transers is examined in greater detail by studying the components o the control along the optimal solution. The typical overall behavior o the control u = (u r, u θ, u h ) is shown in Fig. 9 or (T/m, I sp ) = ( m s 2, 1 s). As expected, u r stays near zero, whereas u θ and u h are non-zero throughout the entire transer. Greater insight into the structure o the optimal control is now obtained by examining the control near the start and the terminus o the transer. Figure 1b shows u as a unction o ν near the start o the transer (where ν = L Ω ω L because L, ω, and Ω, are approximately zero). It is seen rom Fig. 1b that u θ points in the positive u θ -direction to increase the semi-major axis (see Eq. (24)), while u h attains its most positive value near apoapsis and its most negative value near periapsis to decrease the inclination (see Eq. (26)). Near the terminus o the transer, ν = L Ω ω L+3π/2 since Ω approaches an approximate value o 3π/2 (see Fig. 1a) and ω is assumed to be zero. Fig. 1c shows u as a unction o ν near the end o the transer where it is seen that u θ points in the positive u θ -direction and u h attains its most positive value near apoapsis and its most negative value near periapsis. Thus it is clear throughout the LEO to GEO transer that the optimal thrust direction increases the semi-major axis and decreases the inclination while the eccentricity remains relatively unchanged near zero. 18

19 1 ur L/(2π) (a) u r vs. L/(2π). 1 uθ L/(2π) (b) u θ vs. L/(2π). 1 uh L/(2π) (c) u h vs. L/(2π). Figure 9: u vs. L/(2π) or LEO to GEO transer with various values o T/m and I sp = 1 s. 19

20 π/2 Ω (rad) π 3π/2 2π L/(2π) (a) Ω vs. L/(2π) u u -.5 u r u θ u h Periapsis Apoapsis ν/(2π) (b) u vs. ν/(2π) during the irst ew revolutions. -.5 u r u θ u h Periapsis Apoapsis ν/(2π) (c) u vs. ν/(2π) during the last ew revolutions. Figure 1: Ω vs. L/2π and u vs. ν/(2π) or LEO to GEO transer with (T/m, I sp ) = ( m s 2, 1 s). 2

21 4.3 Estimation o Minimum-Time Transer Time A key eature o the results is the ability to estimate the optimal transer time as a unction o initial thrust acceleration and speciic impulse. Figure 11 shows the inal time o the orbit transer as a unction o the speciic impulse or each o the initial thrust acceleration values examined. For each value o T/m, t increases slightly as I sp increases in a manner similar to that o a power unction t = AI B sp + C, (27) where the coeicients A, B, and C are unctions o T/m because each value o T/m has an associated power unction expression or t in terms o I sp. The coeicients A, B, and C are determined as ollows. Figures 12a and 12b show the coeicient A as a unction o T/m or the GTO to GEO and LEO to GEO transers, respectively. It is seen that the relationship between A and T/m has the orm A = a 1 (T/m ) b 1, (28) where a 1 and b 1 are constant coeicients. Figures 12c and 12d show the coeicient B as a unction o T/m or the GTO to GEO and LEO to GEO transers, respectively. Because B has no signiicant change as a unction o T/m, it is assumed that B is constant, and or any particular orbital transer this constant is the average value o B over all values o T/m and I sp or that transer. Figures 12e and 12 show the coeicient C as a unction o T/m or the GTO to GEO and LEO to GEO transers, respectively. It is seen that the relationship between C and T/m is given as C = a 2 (T/m ) b 2, (29) where a 2 and b 2 are constants. Thereore, the estimated transer time, ˆt, can be written as a unction o both I sp and T/m and is given as ˆt = a 1 (T/m ) b 1 (I sp ) B + a 2 (T/m ) b 2. (3) Values or the coeicients a 1, b 1, B, a 2, and b 2 are shown in Table 3. Equation 3 makes it possible to estimate the inal transer time, t, or values o I sp that are dierent rom those obtained in this study without having to re-solve the optimal control problem. 21

22 12 t (d) T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s I sp (s) (a) GTO to GEO transer t (d) T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s 2 T/m = m s I sp (s) (b) LEO to GEO transer. Figure 11: t vs. I sp. 22

23 Table 3: Regression coeicients or ˆt. Transer a 1 b 1 B a 2 b 2 GTO to GEO LEO to GEO Estimation o Minimum-Time Final True Longitude Another key eature o the results is the ability to approximate L(t ) as a unction o the inal transer time and speciic impulse. Figure 13a shows L as a unction o t. It is seen or each value o I sp that L(t ) increases linearly as a unction o t, that is, L = Dt + E, (31) where D and E are unctions o I sp. Expressions or coeicients D and E are determined as ollows. Figures 14a and 14b show the coeicient D as a unction o I sp or the GTO to GEO and LEO to GEO transers, respectively. It is seen that as I sp increases, D decreases in a manner similar to that o an exponential unction D = a 3 e b 3I sp + c 3, (32) where a 3, b 3, and c 3 are constant coeicients. Figures 14c and 14d show the coeicient E as a unction o I sp or the GTO to GEO and LEO to GEO transers, respectively. Is is seen that as I sp increases, E is small. For example, E is no more than hal a true longitude cycle or both the GTO to GEO and LEO to GEO transers. Consequently, E is treated as a constant and or any orbital transer, this constant is the average value o E over all values o T/m and I sp or that transer. Using the derived expressions or the coeicients D and E, L can be written as a unction o both I sp and t and is given by ˆL = ( a 3 e b 3I sp + c 3 ) t + E (33) where a 3, b 3, c 3, and E are the regression constants and ˆL denotes the estimate o L. Values or the regression constants are shown in Table 4. Together with Eq. 3, Eq. 33 makes it possible to quickly estimate L at points or which the results were not obtained. Table 4: Regression coeicients or ˆL. Transer a 3 b 3 c 3 E GTO to GEO LEO to GEO Coeicient o Determination to Assess Quality o Regressions To demonstrate the quality with which the regression models or ˆt and ˆL it the observed data, the coeicients o determination, R 2, are calculated. 38 The coeicient o determination is a value between and 1, where a value o unity indicates a perect it. The coeicient o determination or ˆt is calculated as ollows. Let t i and ˆt i be the observed and predicted values, respectively, o the i th value o t. Then, the sum o the squares o the error, denoted S, is calculated as S = n ( ) 2 ti ˆt i. (34) i=1 23

24 Next, let S yy denote the total sum o squares S yy = n ( ) 2 ti t, (35) i=1 where t is the mean o the observed values o t. Finally, the coeicient o determination is calculated using R 2 = 1 S S yy. (36) The coeicient o determination or ˆL is calculated in a similar manner to ˆt. All coeicients o determination are shown in Table 5, where it is seen that R 2 is close to unity in all cases. Table 5: Coeicient o determination R 2 or ˆt and ˆL. Transer ˆt ˆL GTO to GEO LEO to GEO

25 -2-4 A ( 1 3 ) -4 A ( 1 3 ) T/m (m s ) (a) GTO to GEO transer T/m (m s ) (b) LEO to GEO transer B B T/m (m s ) (c) GTO to GEO transer T/m (m s ) 4 (d) LEO to GEO transer C 8 C T/m (m s ) (e) GTO to GEO transer T/m (m s ) () LEO to GEO transer. Figure 12: Regression coeicients A, B, and C vs. T/m. 25

26 L/(2π) I sp = 5 s I sp = 1 s I sp = 3 s I sp = 5 s t (days) (a) GTO to GEO transer L/(2π) I sp = 5 s I sp = 1 s I sp = 3 s I sp = 5 s t (days) (b) LEO to GEO transer. Figure 13: L /(2π) vs. t. 26

27 D 1.42 D I sp (s) (a) GTO to GEO transer I sp (s) (b) LEO to GEO transer E E I sp (s) (c) GTO to GEO transer I sp (s) (d) LEO to GEO transer. Figure 14: Regression coeicients D and E vs. I sp. 27

28 4.6 Post-Optimality Analysis It is known rom previous research (see Res. 26 and 27) that the irst-order optimality conditions o the nonlinear programming problem arising rom discretization o a continuous optimal control problem via the LGR collocation method are a discrete approximation o the irst-order calculus o variations optimality conditions o the optimal control problem. Moreover, the costate o the optimal control problem can be obtained via a simple linear transormation o the Lagrange multipliers o the NLP arising rom the LGR collocation. In addition to the equivalence between the NLP and calculus o variations optimality conditions, it has also been proven that the solution obtained using the variable-order (hp) LGR collocation method converges exponentially (that is, the state, control, and costate associated with the LGR collocation method all converge) at the convergence rate given in Re. 39. Consequently, by solving the NLP arising rom the LGR collocation method on an appropriate mesh, an accurate approximation to the solution o the optimal control problem is obtained to both the primal variables (that is, the state and control) and the dual variable (that is, the costate). Thereore by solving the variable-order LGR NLP on a suiciently accurate mesh, it is possible to veriy the extremality o the solutions without having to solve the Hamiltonian boundary-value problem that arises rom the calculus o variations. In other words, by obtaining the solution to the variable-order LGR NLP on an appropriate mesh the optimality o the solution can be veriied without having to resort to solving the optimal control problem using an indirect method. In this study the proximity o the numerical solutions to the true optimal solutions is investigated by examining various aspects o the irst-order calculus o variations conditions. In this analysis the irstorder variational conditions are presented in terms o the classical orbital elements (as opposed to the modiied equinoctial elements which were used to solve the optimal control problem), where the irstorder optimality conditions are obtained in terms o the classical orbital elements as ollows. First, the discrete approximation o the costate in terms o the modiied equinoctial elements are obtained using the transormation o the NLP Lagrange multipliers as described in Res. 26 and 27 (where it is noted that GPOPS II perorms this costate computation ater the NLP is solved). Next, the costate approximation in terms o the modiied equinoctial elements obtained rom the LGR collocation method is transormed to the costate in terms o classical orbital elements using the relationship between the modiied equinoctial element costate and the classical orbital element costate as derived in the Appendix. Then, using the act that the costate is the sensitivity o the cost with respect to the state along the optimal solution, the costate in terms o classical orbital elements at the initial time is also approximated by solving the optimal control problem at a perturbed initial orbital element and taking the ratio o the change in cost to the change in the orbital element o interest (or example, i it is interested in computing the costate associated with the eccentricity, then the ratio o the change in the cost to a perturbation in the eccentricity at the initial point is computed). The costates associated with the classical orbital elements o interest were veriied by resolving the problem with a small perturbation in the initial semi-major axis, initial eccentricity, and initial inclination. For a perturbation in the initial semi-major axis, the change in cost rom the optimal cost is approximated as [ ] J J δ J ( + a δ (L ) a (L ) ) (37) a(l ) where J δ and J denote the cost on the perturbed and optimal solutions, respectively. Thereore, the estimated semi-major axis costate at L = L is approximated by [ ] J a(l ) J δ J a δ (L ) a (L ) = J a which is then compared to the derived costate value, λ a(l ). For a perturbation in the initial eccentricity or initial inclination, the estimated costate value is calculated in a similar manner (that is, replace the semi-major axis, a, with either the eccentricity, e, or the inclination, i, in Eqs. (37) and (38)). In this analysis, the perturbations in the initial semi-major axis, initial eccentricity, and initial inclination were (38) 28

29 a = 1 m, e =.1 and i = rad (=.1 deg), respectively. Tables 6a and 6b show the costate approximations (λ a(l ), λ e(l ), λ i (L )) alongside the ratios o the cost to the perturbations, ( J/ a, J/ e, J/ i) in the orbital elements or the GTO to GEO case with T/m = m s 2 and or the LEO to GEO case with T/m = m s 2. For both cases the LGR costate approximations closely match the estimated change in cost due to a perturbation in the classical orbital element o interest, and the costate approximations are consistent with the expected behavior (or example, increasing the initial semi-major axis or either orbit transer decreases the cost, while increasing the initial eccentricity increases the cost). Moreover, it is seen that perturbing the initial eccentricity signiicantly increases the cost or the GTO to GEO case but increases the cost much less or the LEO to GEO case. Also, in all cases increasing the initial inclination increases the cost. Finally, in all cases the magnitude o cost sensitivity increases as the speciic impulse increases. This last result is consistent with the act that the eiciency o the engine increases as the speciic impulse increases. As a urther veriication o the close proximity o the numerical solutions to the true optimal solution, the inal column o Tables 6a and 6b show the maximum absolute value o H u H/ u = (H ur, H uθ, H uh ) on L [L, L ], that is, Tables 6a and 6b show max ( H u r, H uθ, H uh ), L [L,L ] where H is computed as given in the Appendix using the costate approximation obtained rom the LGR collocation method as described in Re. 27. Because the control lies on the interior o the allowable control set or the problem studied in this paper, it is known theoretically that H u is zero along the optimal solution. Commensurate with this known value o H u, Tables 6a and 6b show that H u is extremely small, urther substantiating the close proximity o the numerical solution to the true optimal solution. Table 6: Post-optimality results or GTO to GEO and LEO to GEO transers. (a) GTO to GEO post-optimality results or T/m = m s 2. I sp λ a(t ) J/ a λ e(t ) J/ e λ i (t ) J/ i max L [L,L ] ( H u r, H uθ, H uh ) (b) LEO to GEO post-optimality results or T/m = m s 2. I sp λ a(t ) J/ a λ e(t ) J/ e λ i (t ) J/ i max L [L,L ] ( H u r, H uθ, H uh )

30 5 Conclusions The problem o high-accuracy low-thrust minimum-time Earth-orbit transers has been studied. The optimal orbital transer problem is posed as a constrained nonlinear optimal control problem. It is solved using a variable-order Legendre-Gauss-Radau (LGR) quadrature orthogonal collocation method paired with a search method that helps the NLP solver determine the best locally optimal solution. A numerical optimization study has been conducted to determine optimal trajectories and controls or a range o initial thrust accelerations and constant speciic impulses. The key eatures o the solutions have been identiied and relationships have been obtained that relate the optimal transer time to the optimal number o true longitude cycles as a unction o the initial thrust acceleration and speciic impulse. Finally, a postoptimality analysis has been perormed that veriies the optimality o the solutions that were obtained in this study. Acknowledgments The authors grateully acknowledge support or this research by the NASA Florida Space Grant Consortium under grant NNX1AM1H, by Space Florida under grant 11-17, and by the National Aeronautics and Space Administration under grant NNX14AN88G. Reerences [1] Dachwald, B., Optimization o Very-Low-Thrust Trajectories Using Evolutionary Neurocontrol, Acta Astronautica, Vol. 57, No. 2-8, 25, pp [2] Tang, S. and Conway, B. A., Optimization o Low-Thrust Interplanetary Trajectories Using Collocation and Nonlinear Programming, Journal o Guidance, Control, and Dynamics, Vol. 18, No. 3, May-June 1995, pp [3] Rauwol, G. A. and Coverstone-Carroll, V. L., Near-Optimal Low-Thrust Orbit Transers Generated by a Genetic Algorithm, Journal o Spacecrat and Rockets, Vol. 33, No. 6, November-December 1996, pp [4] Coverstone-Carroll, V., Hartmann, J. W., and Mason, W. J., Optimal Multi-Objective Low-Thrust Spacecrat Trajectories, Computer Methods in Applied Mechanics and Engineering, Vol. 186, No. 2-4, June 2, pp [5] Falck, R. D. and Dankanich, J. D., Optimization o Low-Thrust Spiral Trajectories by Collocation, AIAA/AAS Astrodynamics Specialist Conerence, No , Minneapolis, MN, August 212. [6] Petropoulos, A. and Longuski, J. M., Shaped-Based Algorithm or the Automated Design o Low- Thrust, Gravity Assist Trajectories, Journal o Spacecrat and Rockets, Vol. 41, No. 5, September- October 24, pp [7] Starek, J. A. and Kolmanovsky, I. V., Nonlinear model predictive control strategy or low thrust spacecrat missions, Optimal Control Applications and Methods, Vol. 35, No. 1, January-February 214, pp [8] Bartholomew-Biggs, M. C., Dixon, L. C. W., Hersom, S. E., and Maany, Z. A., The solution o some diicult problems in low-thrust interplanetary trajectory optimization, Optimal Control Applications and Methods, Vol. 9, No. 3, July-September 1988, pp [9] Kechichian, J. A., Optimal Low-Thrust Transers Using Variable Bounded Thrust, Acta Astronautica, Vol. 36, No. 7, 1995, pp