Aas 20-572 Characterization and Optimization of Efficient

AAS 20-572

CHARACTERIZATION AND OPTIMIZATION OF EFFICIENT CHANCE-CONSTRAINED NAVIGATION STRATEGIES FOR LOW-ENERGY TRANSFERS

Riley M. Fitzgerald,∗ Philip D. Hattis,† Kerri L. Cahoy,‡ and Richard Linares§

Low-energy transfers expose a spacecraft to long periods of unstable dynamics, and navigation along them currently relies on frequent tracking by expensive ground infrastructure. This paper characterizes minimum-cost navigation strategies for these trajectories, subject to constraints on the probability of transfer success. First, stochastic augmented state dynamics are developed to model the be- havior of a spacecraft with a given measurement and correction strategy. Secondly, Monte-Carlo simulations of a sample correction schedule over an example transfer are presented, demonstrating the characteristic ∆v and estimator error proﬁles. Finally, a method for chance-constrained optimization is presented, resulting in the minimum-cost tracking windows and correction times guaranteeing a speci- ﬁed probability of success.

INTRODUCTION Low-energy transfers are a promising method to enable future lunar exploration; while generally longer and more complicated than traditional trajectories, they have the potential to drastically reduce the overall ∆v required for transfer to the moon. Additionally, in some cases they can almost entirely eliminate the need for orbit insertion ∆v at the target body and achieve ballistic capture,1,2 enabling rendezvous missions even for spacecraft with greatly reduced propulsion capability. While their dynamics has been studied extensively,3–5 one often-ignored disadvantage of these low-energy transfers is their sensitivity. Figure1 shows how small perturbations to the initial state (in this example, normally-distributed with σx = 1 km and σv = 1 m/s) can lead to drastically different end states after the duration of the transfer. Due to the long transfer times and the nonlinear, chaotic dynamics small errors in the position or velocity tend to grow quickly and become unacceptably large. This tendency must be mitigated by some means of orbit determination and correction. Ground-based tracking using an RF array such as the Deep Space Network (DSN) is typically performed at regular intervals in order to provide an estimate of the spacecraft state and thereby plan correction maneuvers. For example, Zuiani et al.6 and Vetrisano et al.7 describe a typical tracking scheme proposed for the European Student

∗Doctoral Candidate, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, and Draper Fellow, The Charles Stark Draper Laboratory, Inc. Cambridge, MA 02139. †Laboratory Technical Staff, Space and Mission Critical Systems, The Charles Stark Draper Laboratory, Inc. Cambridge, MA 02139. ‡Associate Professor, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139. §Charles Stark Draper Assistant Professor, Department of Aeronautics and Astronautics, Massachusetts Institute of Tech- nology, Cambridge, MA 02139.

1 Figure 1: An example transfer (in black), and 50 variations with normally-distributed (σx = 1 km and σv = 1 m/s) perturbations to the initial condition. The black circle represents the orbit of the moon, and the sun is ﬁxed in the −x direction.

Moon Orbiter (ESMO). You et al.8 present the details of a similar scheme, successfully used for the GRAIL mission. This method is effective, but relies critically on large, expensive ground-based tracking infrastructure. For larger missions, this added operational cost may not be a problem, but it can prohibit smaller-budget missions from utilizing these trajectories. For example, if one eight-hour contact is performed per day using a 34-meter dish (as was required during the GRACE mission cruise phase8), each will cost approximately $15,000.9 With a transfer period over 100 days, costs for tracking alone would be in the millions of dollars. If it were not for this complication, low-energy transfers would be perfect for enabling lunar missions with low-cost spacecraft. This paper aims to develop a framework for characterizing, and reducing, the ground tracking necessary to accomplish a low-energy transfer to the moon. First, the consisted dynamics and state representation are examined, developing transformations of the augmented state corresponding to propagation, continuous measurement, and discrete trajectory correction. Second, numerical methods for integrating the stochastic equations of motion are described, and the gradients used for optimization are considered. Third, the transfer scenarios are deﬁned, and results using a reference correction schedule are given. Finally, optimization is performed in order to reduce the required measurement cost while assuring transfer success.

CONSIDERED STATE AND DYNAMICS Circular-Restricted Four-Body Problem For this work, orbital motion according to the coplanar Circular-Restricted Four-Body Problem (CRFBP) is assumed. This is based on the following assumptions:

(1) The Earth and moon follow unperturbed circular orbits about their barycenter.

2 Table 1: Sun/Earth/Moon CRFBP Parameters

Parameter Value [unitless] +2 d¯s 3.8917 × 10 −5 µ¯e 3.0035 × 10 −8 µ¯m 3.6959 × 10

(2) The Earth/moon barycenter follows an unperturbed circular orbit about the sun. (3) The orbits of the Earth, moon, and sun are all coplanar.

For this work, the dynamics are treated in a Euclidean frame (eˆ1, eˆ2, eˆ3) centered at the Earth/moon barycenter and rotating about the eˆ3 axis with angular rate equal to the orbital rate of the Earth/moon system about the sun. The sun is stationary on the negative eˆ1 axis, and the Earth and moon follow circular orbits about the origin in the eˆ1, eˆ2 plane. Distance are normalized by the Earth/moon distance, time is normalized such that the angular rate of the Earth/moon system in the rotating coordinate frame is unity. In this formulation, the Circular-Restricted Four-Body problem equations of motion for position vector r are given by

2 X r − rb(t) r¨(t, r, r˙ ) =ω ¯ E12 r +µ ¯sr¯seˆ1 + 2¯ω(r˙ × eˆ3) − GM µ¯b 3 (1) kr − rb(t)k | {z } | {z } b∈{s,e,m} Centrifugal Term Coriolis Term | {z } Gravitational Terms where E12 = eˆ1 eˆ2 0 represents a projection into the eˆ1, eˆ2 plane, the constant sun position is rs(t) = rs = −r¯seˆ1, and the time-dependent Earth/moon positions re(t) and rm(t) are given by r (t) = µ¯e [cos(t + φ )eˆ + sin(t + φ )eˆ ] (2a) m µ¯e+¯µm 0 1 0 2 r (t) = − µ¯m r (2b) e µ¯e e

The initial phase φ0 speciﬁes the orientation of the Earth/moon system at t = 0. Only three physical parameters are required to fully specify these dynamics: r¯s, the distance from the Earth/moon barycenter to the sun (normalized by the Earth/Moon distance), and any two of µ¯s, µ¯e, µ¯m, giving the mass of the sun, Earth, and moon respectively (normalized by the total mass of all three bodies). The third mass fraction is found by µ¯s +µ ¯e +µ ¯m = 1, while ω¯ and GM are deﬁned by −1 √ −2 p 3 −3/2 ω¯ = r¯s (¯µe +µ ¯m) − 1 , GM = r¯s − µ¯e +µ ¯m (3)

The deﬁning parameters for the sun/Earth/moon system are speciﬁed in Table1. For these values, we have time unit TU = 4.6930 days, and distance unit DU = 384,399 km. The position/velocity state vector for a spacecraft orbiting according to these dynamics is denoted x, with xT = rT r˙ T. The noise-free evolution of this state is denoted in the usual state-space dynamics function form, x˙ (t) = f(t, x), with r˙ f(t, x) = (4) r¨(t, r, r˙ )

3 ∂f We will denote the Jacobian matrix of this function as F (t, x) = ∂x (t, x), and the Hessian tensor ∂2f of this function as F (t, x) = (∂x)2 (t, x). To include process noise, and thereby account for unmodeled dynamics and unpredictable perturbations, by including process noise. Therefore, we assume that the state evolves according to the Stratonovich stochastic differential equation (SDE) dx(t) = f(t, x) dt + G(t, x) ◦ dw (5) where w represents a multidimensional Weiner process, and G(t, x) is the matrix of diffusion co- efﬁcients. (In this context, the dynamics function f(t, x) gives the vector of drift coefﬁcients.) Assuming that the process noise consists only of unmodeled forces with intensity covariance matrix Q(t), then w(t) is a three-dimensional process and the diffusion matrix can be represented as 0 G(t, x) = G(t) = (6) chol Q(t) where chol · denotes the Cholesky decomposition. We note that by taking process noise intensity Q(t) as time-dependent only, the diffusion matrix G(t, x) = G(t) is independent of the state, and ¯ ¯ ¯ the Stratonovich SDE is equivalent to the corresponding Itoˆ SDE.

Augmented State Dynamics The spacecraft position-velocity state x is sufﬁcient to capture the natural dynamics of the system, but cannot account for the measurements taken, estimates performed, and trajectory correction maneuvers applied along the transfer. To fully capture the dynamics, we consider the onboard state of the spacecraft. The augmented state x, deﬁned by ¯ xT = xT xˆT pT c (7) ¯ where, x represents the true position/velocity state, xˆ represents the estimated position/velocity state, p = vech P represents half-vectorized estimator covariance matrix P , and c represents some running cost of correction (for example, spent ∆V ). We consider the evolution of the extended state x according to the augmented version of the Stratonovich SDE given in Eq.5: ¯ dx(t) = f(t, x) dt + G(t, x) ◦ dw(t) (8) ¯ ¯ ¯ ¯ ¯ ¯ The underbars indicate quantities that represent or operate on the augmented state. For a segment governed only by orbital dynamics, the extended state drift function f and diffusion matrix G are given by: ¯ ¯  f(t, x)  G(t)  f(t, xˆ)   0  f(t, x) =   , G(t, x) =   (9) ¯ ¯ A(t, xˆ)p + q(t) ¯ ¯  0  0 0 Note that, since noise does not affect any states other than x, we have that the augmented noise process w = w is still three-dimensional. The third entry of the vector in Eq.9 represents the half- ¯ vectorized Lyapunov equation for the estimator covariance evolution; qT(t) = 0T vechTQ(t) represents the half-vectorized process noise covariance, and A(t, xˆ) is given by A(t, xˆ) = D†(I ⊗ F (t, xˆ) + F (t, xˆ) ⊗ I)D (10)

4 which is derived by Axelsson and Gustafsson.10 The matrix D and D† represent the elimina- tion matrix and its pseudoinverse respectively, deﬁned such that vech P = D vec P and vec P = D†vech P . For later use, the augmented state drift function Jacobian F (t, x) is given by ¯ ¯ F (t, x) 0 0 0  0 F (t, xˆ) 0 0 F (t, x) =   (11) ¯ ¯  0 p˙ xˆ A(t, xˆ) 0 0 0 0 0 † p˙ xˆk = D (I ⊗ Fk(t, xˆ) + Fk(t, xˆ) ⊗ I)Dp (12)

th ∂2f ∂ where Fk refers to the k matrix entry of the Hessian tensor of f, that is Fk = = F . ∂x∂xk ∂xk The augmented state drift function time partial derivative ft is ¯   ft(t, x)  ft(t, xˆ)  ft(t, x) =   (13) ¯ ¯ At(t, xˆ)p + qt(t) 0 where At and qt represent the time partial derivatives of A and q, respectively.

Continuous Measurement To account for measurements taken over the course of the transfer, additional terms must be added to Eqn.8. Over a time window where continuous measurements are taking place, we have

dx(t) = f(t, x) dt + G(t, x) ◦ dw(t) + m(t, x) dt + N(t, x) ◦ du(t) (14) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯| {z } | {z } Natural Dynamic Terms Continuous Measurement Terms

Say h(t, x) deﬁnes a function taking in a time and state and returning an m-dimensional vector of measurements. Then, u(t) is an m-dimensional Weiner process, and the additive continuous measurement drift and diffusion are given by

 0   0  K(t, x)[h(t, x) − h(t, xˆ)] + χ K(t, x) chol R(t) m(t, x) =  ¯  ,N(t, x) =  ¯  (15) ¯ ¯  −D†[I ⊗ K(t, x)H(t, xˆ)]Dp  ¯  0  ¯ 0 0

∂h where H = ∂x is the Jacobian of the measurement function, and K is the usual extended Kalman ﬁlter (EKF) gain given by

vec K(t, x) = vec P (p)HT(t, xˆ)R−1(t) = R−1(t)H(t, xˆ) ⊗ ID p (16) ¯ | {z } (vec K)p

The term χ = 0 speciﬁes the Stratonovich iterpretation of the noise terms in the continuous extended Kalman ﬁltering equations, reasonable if u is considered the limit of band-limited noise. If the Itoˆ interpretation of the EKF is desired, then a non-zero χ (the form of which is given in the appendix) must be included to make the above Stratonovich SDE valid. χ = 0 is taken here,

5 but the alternative is made available; in either case, the divergence between the two is small if the measurement function is nearly linear in the neighborhood considered. Later, we will make use of the measurement drift function Jacobian

 0 0 0 0 KH(t, x) −KH(t, xˆ) − VW 0 M(t, x) =   (17a) ¯ ¯  0 −D†X −D†Y 0 0 0 0 0 T −1 Vcol k = Kxˆk ∆h = P (p)Hk (t, xˆ)R (t)∆h (17b)

Wcol ` = Kp` ∆h = [(vec K)p,col `]n×m∆h (17c)

Xcol k = vec [Kxˆk H(t, xˆ) + KHk(t, xˆ)]P (p) (17d) Y = [I ⊗ K(t, x)H(t, xˆ)]D + [P (p)HT(t, xˆ) ⊗ I](vec K) (17e) ¯ p where ∆h = h(t, x) − h(t, xˆ), and [v]n×m refers to column-major devectorization of mn-element v into a n × m matrix. The partial derivative of the drift function with respect to time is

 0   Kt∆h + K[ht(t, x) − ht(t, xˆ)]  mt(t, x) =  †  (18a) ¯ ¯ −D vec [KtH(t, xˆ) + KHt(t, xˆ)]P (p) 0 T −1 T −1 Kt = P (p)[Ht (t, xˆ)R + H (t, xˆ)Rt ] (18b)

Trajectory Correction Maneuvers

A trajectory correction meaneuver (TCM) is treated as a discrete, discontinuous event interrupting the normal SDE evolution. Taken at time t, we have stochastic transformation τ of the augmented ¯ state given by

x0(t) = τ (t, x; ∆t, x∗) + s (19) ¯ ¯ ¯ ¯ where x0(t) refers to the augmented state immediately after time t, and s is a zero-mean stochastic ¯ TCM error with covariance S. The TCM is deﬁned by parameters ∆t (the target-ahead interval) ¯ and x∗ (the targeted state). Only the spacecraft velocity can be changed at any point in time, but both position and velocity need to be managed. The assumed stategy is to adjust the current velocity such that, after elapsed time ∆t, the current state and targeted state converge in position (at which time velocity errors can be easily canceled by another TCM). The necessary velocity change is calculated based on the current state estimate, and then applied to both the true state (via thrusters) and the estimated state (via an update). Let φ(t, x; ∆t) represent the function which takes in a time and position-velocity state and returns the state x(t + ∆t), propagated forward according to the natural orbital mechanics deﬁned by f. Given a state change ∆xˆ, the estimated state error at the target-ahead time is given by

ˆ(∆xˆ) = φ(t, xˆ + ∆xˆ; ∆t) − φ(t, x∗; ∆t) (20)

6 The desired state change ∆xˆ is the solution to the following set of equations:

ˆi(∆xˆ) = 0 ∀i ∈ {1, 2, 3} (21a)

∆ˆxi = 0 ∀i ∈ {1, 2, 3} (21b)

This is solved numerically as follows:

∗ ∗ ∂φ(t,x∗) 1. Calculate φ(t, x ; ∆t) and its Jacobian Φ = ∂x∗ . The Jacobian is equivalent to the state d transition matrix Φ(t + ∆t) integrated over the targeted trajectory according to ds Φ(s) = F (s, x)Φ(s), starting with Φ(t) = I. ∗ 2. Let the initial estimate for ∆xˆ be ∆xˆ(0), deﬁned using Φ as follows:

∗ ∗ 0 ∗ Φ11 Φ12 ∆xˆ(0) = ∗−1 ∗ ∗ , Φ = ∗ ∗ (22) − Φ12 Φ11 I (xˆ − x ) Φ21 Φ22

3. Perform the followng Newton-Raphson iteration: " # 0 ∆xˆ(s+1) = ∆xˆ(s) + −1 (23) −Φ(s),12ˆ1 ∆xˆ(s)

where Φ(s) refers to the state transition matrix integrated over the estimated trajectory using ∆xˆ(s), with the partitioning subscripts as above. When ˆ1 ∆xˆ(s) is sufﬁciently small, let ∆xˆ = ∆xˆ(s).

Once the estimate change ∆xˆ has been calculated, the augmented state transformation can be speciﬁed by

 x + ∆xˆ  I 0  xˆ + ∆xˆ  0 x (t) =   +   s (24) ¯  p + vech S  0 c + ∆c(∆xˆ) 0 | {z } | {z } τ (t,x;∆t,x∗) s ¯ ¯ ¯ The cost increment ∆c is a function of the applied state change, and for this work is assumed to be ∆c(∆xˆ) = k∆xˆk = ∆v. The error term s has covariance matrix S, and represents a random error in the application of the desired state change. The matrix ∆T represents the Jacobian of the correction ∆xˆ with respect to the input estimated state, and is deﬁned by

0 0 ∆T = −1 (25) −Φ12 Φ11 −I where the submatrices of Φ are taken from the ﬁnal Φ of the above iteration. The Jacobian of the augmented state transformation is given by

0 ∆T 0 0 0 ∆T 0 0 T(t, x) = I +   (26a) ¯ ¯ 0 0 0 0 0 ∆c∆xˆ ∆T 0 0

7 and the time partial derivative of τ is ¯   ∆xˆt  ∆xˆt  τ t(t, x) =   (27a) ¯ ¯  0  ∆c∆x(∆xˆ)∆xˆt 0 ∆xˆt = −1 ∗ (27b) −Φ12 [φt(t, xˆ + ∆xˆ; ∆t) − φt(t, x ; ∆t)]1

The time partial derivatives φt = xt(t + ∆t), Φt = Φt(t + ∆t) can be calculated by adding the following condition to the ﬁnal integration of φ:

d xt(t) = 0, ds xt(s) = ft(s, x) + F (s, x)xt(s) (28)

∗ ∗ The target partial derivative φt(t, x ) depends on the speciﬁcation of x :

( ∗ ∗ ∗ xt (t + ∆t) (via Eqn. 28) if constant x provided, φt(t, x ; ∆t) = (29) f(t + ∆t, x∗(t + ∆t)) if trajectory x∗(t) provided

The ∆t partial derivative τ (t, x; ∆t, x∗) follows the same structure as Eqns. 27, replacing all t ¯ ∆t ¯ subscripts with ∆t, and using φ∆t = x˙ (t + ∆t) = f(t + ∆t, x(t + ∆t)). Finally, it should also be noted that a modiﬁed TCM transformation is also allowed, wherein the velocity is corrected to match the velocity of the target state at the current time. This is notated as

x0(t) = τ˜(t, x; x∗) + s (30) ¯ ¯ ¯ ¯ and the state change ∆xˆ is given simply by the current velocity difference (with the position ele- ments all zero as before).

NUMERICAL METHODS AND DYNAMICS SIMULATION The natural dynamics are assumed to be invariant, and we can only change the mission profile by defining measurement segments and trajectory correction maneuvers. We denote a measurement by m = (ton, toff , h,R), specifying its start time, stop time, measurement function, and noise. A TCM ∗ is denoted c = (ttcm∆t, x ,S), specifying the time, look-ahead interval, target state, and noise (denoting a modified velocity-targeting TCM by ∆t = 0). A set of measurements and corrections, specified by some parameter vector θ, determines the schedule S:

Nm Nc S(θ) = {m}i=1 ∪ {c}i=1 (31)

Such a schedule deﬁnes a set of critical times, which are those times when either (1) the scenario begins or ends, (2) a instantaneous event occurs, such as a TCM, or (3) the dynamics change dis- continuously, such as when a measurement arc begins or ends. Let these times be the set T∗(θ), deﬁned and enumerated as

N N [m [c T∗(θ) = {t0, tf } ∪ {ton,i, toﬀ,i} ∪ {ttcm,i} = {t0, t1, t2, ...} (32) i=1 i=1

8 where the numbering tk is chosen such that tk+1 ≥ tk ∀k. This defines a set of phases (tk, tk+1) over which the equations of motion are to be integrated (with TCMs occuring at the phase boundaries where specified). For those phases which contain no measurement, the noisy augmented CRFBP dynamics are integrated alone; for those which overlap with some m, the measurement dynamics are also added. In order to integrate the Stratonovich SDEs defining the equations of motion, noise must be generated. Consider first the Weiner process w required by Eqn.8; this is approximated by a process w˜ with a finite correlation time δtw. The dynamics integrated accordind to such a process will approach the true Stratonovich dynamics as δtw → 0. Each independent element w˜i of the −1 process is generated by first sampling normal random variables zi,n with variance δtw at times nδtw, and then interpolating the result with a piecewise cubic Hermite interpolating spline (PCHIP):

−1 zi,n ∼ N (0, δtw ) (33a) dw˜i dt (t) = pchip t; {(nδtw, zi,n)}n (33b) The samples can be generated on the ﬂy, and the spline extended, in order to prevent storing large quantities of random values. The resulting derivative dw/dt˜ can then be used allong with the diffusion matrix to add a stochastic forcing term to the dynamics. The SDE given in Eqn.8 is approximated by the ODE

d x(t) = f(t, x) + G(t, x) dw˜ (t) (34) dt ¯ ¯ ¯ ¯ dt The stochastic differential equation including measurement (Eqn. 14) can be integrated similarly, provided an additional spline-based process u˜ with sample time δtu. The process u˜ is only required during measurement phases, so for efficiency a shorter process (with length equal to the total measurement time) is generated, and referenced witha lag only during measurement phases. Over each phase defined by the schedule, the noisy equations of motion are integrated using an explicit Runge-Kutta 5(4) method with adaptive time stepping. Unlike fixed-step SDE solvers, this method allows the numerical integration step to be adjusted to assure accuracy, which is highly important in the CRFBP; the time steps required for accuracy in the vicinity of the Earth and moon are quite small, while the slow cruise phase of the transfer allows large time steps.

SCENARIO AND MONTE-CARLO RESULTS The mathematical framework has been determined, and the scenario details can now be provided. The test cases are a set of example transfers from the Earth to lunar orbit in the coplanar CRFBP; the example transfers I-VI are shown in Fig.2. The parameters describing these orbits are given in Table2. For this work, we consider that the only measurements available are range and range-rate from a ground station on Earth. The time-varying position of the ground station is given by

rgs(t) = re(t) + Re Rot3(ψ0 + ψt˙ ) Rot2(ε) Rot3(λ0 + λt˙ ) Rot2(ϕ) eˆ1 (35)

−1 where ϕ is the latitude, λ = λ0 + λt˙ is inertial longitude (λ˙ = 2π/day = 29.568 TU ), ε = 0.4084 is the obliquity of the ecliptic, and ψ = ψ0 + ψt˙ describes the location of Earth in its orbit (φ = 0 at northern-hemisphere winter solstice, and ψ˙ = 2π/sidereal year = 8.0729 × 10−1 TU−1).

9 Figure 2: The six example Earth-moon transfers in the CRFBP

Table 2: Parameters for Example Transfers I-VI, shown in Figure2

Transfer φ0 [rad] r1 [km] r2 [km] r3 [km] r˙1 [km/s] r˙2 [km/s] r˙3 [km/s] I 4.9184 5719.953 3492.873 0 1.717664 10.681767 0 II 4.2744 7054.109 8057.271 0 -5.731627 9.192684 0 III 5.8642 2361.892 567.255 0 2.129476 10.600866 0 IV 4.2993 7233.234 8404.896 0 -6.615337 8.581563 0 V 1.2639 4786.521 -7157.963 0 4.342317 9.923197 0 VI 1.8617 4691.930 1397.413 0 -9.395759 5.372372 0

−2 Re = 6371.0 km = 1.6574×10 DU is the mean radius of the Earth, and Roti(θ) gives a rotation matrix of angle θ about axis i. The range and range-rate components of h are given by

hr(t, x) = kr − rgs(t)k (36a)

r − rgs(t) hr˙(t, x) = [r˙ − r˙ gs(t)] · (36b) kr − rgs(t)k and the corresponding measurement error covariance matrix R is 2 × 2. For simplicity, it will be considered known and constant. ∗ If the desired trajectory has initial state x (t0), then it is assumed that the intial augmented state is a random variable distributed according to  ∗  x (t0) ∗  x (t0)  x(t0) ∼ N µ0, Σ0 , µ0 =   , Σ0 = blockdiag Σ0, 0, 0, 0 (37) ¯ ¯ ¯ ¯ vech Σ0 ¯ 0

10 Table 3: Fixed Parameters of the Simulation

Param. Description Formula and/or Value(s)

t0 Scenario initial epoch 0 2 2 Σ0 Initial state covariance blockdiag[I3 · (1 km) ,I3 · (1 m/s) ] −9 2 3 Q(t) Process noise intensity I3 · 1 × 10 m /s R(t) Meas. noise intensity diag[(3 m)2, (1 mm/s)2] · (60 s) 2 Sj TCM error covariance blockdiag[03×3,I3 · (1 mm/s) ] δtw Proc. noise sample 24 hours δtu Meas. noise sample 5 minutes ◦ ◦ (λ0, ϕ) Ground station lat./lon. (35 , 0 ) based on Deep Space Network ◦ ψ0 Initial earth orbit phase 0 for consistency

where Σ0 is some known covariance at the initial time, indicating (for example) the uncertainty after trans-lunar injection by the launch vehicle. To determine the probability of transfer success, the set of acceptable ending states must be known. Let ρ = r − rm and ν = r˙ − r˙ m +ω ¯eˆ3 × (r − rm) respectively indicate the position and velocity relative to the moon in an inertial frame, and let µm = GMµ¯m be the gravitational parameter of the moon. Then, the acceptable set A is deﬁned as follows: ¯ tˆp = t : ρˆ(t) · νˆ(t) = 0 Estimated time of Perilune (38a)

r µm νˆ(tˆp) νp = ν(tˆp) − νˆ(tˆp) + True velocity after LOI (38b) kρˆ(tˆp)k kνˆ(tˆp)k 2 kν k2 −1 a = − p Orbit Semi-major Axis (38c) kρ(tˆp)k µm 1/2 kρ(tˆ ) × ν k2 e = 1 − p p Orbit Eccentricity (38d) µma A = {x : ρ < a(1 ± e) < ρ , c < c } Set of Acceptable States (38e) ¯ ¯ lb ub ub This corresponds to those augmented states which, if a lunar orbit insertion (LOI) burn were to be made at the estimated perilune of the trajectory based on the estimated velocity, the resulting true orbit would have a radius bounded in (ρlb, ρub). Additionally, the total TCM cost must be less than cub. Table3 the various parameters which are set or assumed for the remaining analysis in this work.

Monte-Carlo Results for an Example Schedule over Transfer I This section will present Monte-Carlo results for a sample schedule executed over example Trans- fer I. One simple schedule consists of equally-spaced tracking windows of constant length, each of which is followed by a TCM with fixed ∆t. An example schedule of (nearly) this type is described in Table4, and illustrated in Figure3; the first measurement is of double length, and the first TCM is slightly modified in length. Additionally, the final measurement has no associated TCM. Monte-carlo simulation of this mission shows that the proposed schedule is quite successful. Figure4 shows the mean orbit altitude (a − Rm) and eccentricity (e) of the post-LOI lunar orbit,

11 Figure 3: A basic, equally-spaced tracking schedule illustrated along Transfer I. The red sections indicate tracking windows, while the red markers indicate the positions of TCMs. The dotted red lines indicate the targeted future location for each TCM.

Table 4: Example Schedule for Transfer I

3 ton [10 s] 0 1479 2979 4479 5979 7479 3 toﬀ [10 s] 33 1497 2997 4497 5997 7497

3 ttcm [10 s] 36 1500 3000 4500 6000 ∆t [103 s] 1464 1500 1500 1500 1500

assuming that an LOI burn is performed as the estimated perilune. Of the 1000 runs, none fell outside of a 15-kilometer range centered on the expected value, and all had eccentricity less than 5 × 10−4. Additionally, as shown in Fig. 5a, the total ∆v necessary for the TCMs was consistently less than 15 m/s, and always less than 25 m/s. It is worth noting that most of this comes from the ﬁrst one or two corrections, which cancel the initial errors present from launch or TLI; Fig. 5b shows the distribution of the total TCM budget among the ﬁve maneuvers. Finally, Fig. 6a and 6b show the estimator covariance and true position/velocity errors over the transfer. It can be seen that the estimator is consistent with the range of errors observed.

SCHEDULE OPTIMIZATION

Here, a method for optimization of the tracking schedules subject to constraints on probability of success is presented.

12 Figure 4: Post-LOI mean altitude and eccentricty, assuming LOI is performed at estimated perilune.

Derivatives with Respect to the Parameters

Important for the optimization is the Jacobian of each sample with respect to the parameters. This can be calculated along each run, using the individual Jacobians of each dynamic and transforma- ∂x(tk) tion. We deﬁne the matrix Jk = ¯∂θ , initialized by J0 = 0. We represent propagation of the augmented state from critical time t to t = t + ∆t by x = φ(t , x ; ∆t ). Then we k k+1 k k ¯ k+1 k ¯ k k have: ¯

J = J(t ) + f (t , x )[t − t ] (39a) k+1 k+1 k k+1 ¯ k+1 k+1,θ k,θ d ¯ J(s) = F (s, x)J(s) + ft(s, x)tk,θ + fθ(s, x),J(tk) = Jk (39b) ds ¯ ¯ ¯ ¯ ¯ ¯ where Eqn. 39b should be integrated along during the evaluation of the phase. For measurement arcs, the above equation holds with the addition of the measruement terms and associated derivatives. Note that here the Jacobians and partial derivatives of the dynamics ignore the noise terms of the equations; if desired, the various derivative terms representing these should be included, letting, ∂ dw˜ for example, F be replaced by F + ∂x [G(t, x) dt (t)]. ¯ ¯ ¯ ¯ ¯ The derivatives of the critical times with respect to the schedule parameters are generally simple identities, except where the ﬁnal time is allowed to vary with the state. This is the case with the scenario described in the next section, where the end condition is the estimated perilune. In this case, we have

T T T − 0 αˆ 0 0 0 I tf,θ = J(tf ), αˆ = xˆ(tf ) − xm(tf ) (40) T αˆ xˆ˙ (tf ) − x˙ m(tf ) I 0 where xm(t) represents the position-velocity state of the moon.

At time tk representing a discontinuous TCM, J is transformed as follows:

J 0 = T(t , x ; ∆t)J + τ (t , x )t + τ (t , x )∆t (41) k ¯ k ¯ k k ¯t k ¯ k k,θ ¯∆t k ¯ k θ

13 (b) ∆v used for each TCM, showing median, inner (a) Total TCM ∆v used over the transfer quartiles, and ±1.5 IQR for 1000 runs

Figure 5: ∆v used for trajectory correction maneuvers over 1000 runs of transfer I.

(a) Estimator position/velocity covariance maximum standard deviation over the ideal transfer I trajectory

(b) True estimator position/velocity error, krˆ − rk and krˆ˙ − r˙ k, over 1000 runs of transfer I

Figure 6: State estimate error covariance and true error over transfer I

14 Optimization Methodology

We seek to minimize the cost of navigation, subject to constraints on mission success and schedule feasibility. The desired chance-constrained optimization can be expressed as

N Xm minimize g(θ) = (toﬀ − ton)j, (42a) θ j=1 subject to x θ; ω ∈ A ≥ p∗, (42b) P ¯ f ¯ Cθ d (42c) where p∗ is the minimum allowable probability of success. The matrix C and vector d represent the practical schedule parameterization constraints, such as keeping critical times within the scenario time range and preventing negative target-ahead intervals. The problem is now slightly relaxed, and treated by a sampled linearization about a local schedule. Let ω(i) represent some realization of the probability space, including all random initial states and stochastic noise terms. Let x(i) = x θ; ω(i)) be the augmented state at the estimated perilune, ¯ p ¯ p as propagated according to schedule parameters θ and noise realization ω(i). Then, the vector of constrained values b and its derivative A with respect to the parameters are given by

(i) T (i) ∂b (i) b(θ; ω ) = c −a(1 − e) a(1 + e) (i) A(θ; ω ) = Jp(θ; ω ) (43) xp(θ;ω ) ∂x ¯ ¯ p

(1) (n) where Jp is the derivative of the perilune states with respect to θ. Given some set Ω = {ω , ... , ω } of probability-space realizations (for example, a subset of the Monte-Carlo samples), we construct the matrix A(θ; Ω) and vector b(θ; Ω):

A(θ; ω(1)) b(θ; ω(1))  .   .  A(θ; Ω) =  .  , b(θ; Ω) =  .  (44) A(θ; ω(n)) b(θ; ω(n)) which are used to linearize the optimization problem. The set Ω is chosen by the following heuristic, in order to increase computational efﬁciency and bound the optimization:

1. A large number N of Monte-Carlo samples is run for the initial schedule, without calculating gradients. 2. At most (1 − p∗)N outlier samples (those with large deviations from the acceptable set) are rejected, indicating missions for which we accept failure. 3. Some number n of the most extreme remaining samples (those outside of or close to the boundary of the acceptable set) are chosen for Ω.

In this way, Ω consists of the n most-likely points to exit the acceptable set (ignoring an acceptable number of outliers). These samples are then re-run to calculate the gradients, and the A and b matrices can be calculated.

15 Table 5: 10-Variable Parameterized Schedule for Transfer I

3 ton [10 s] 0 θ6 θ7 θ8 θ9 θ10 3 toﬀ [10 s] θ1 − 3 θ2 − 3 θ3 − 3 θ4 − 3 θ5 − 3 7497

3 ttcm [10 s] θ1 θ2 θ3 θ4 θ5 3 ∆t [10 s] θ2 − θ1 θ3 − θ2 θ4 − θ3 θ5 − θ4 tf − θ5

Finally, the optimization problem in Eqns. 42 is linearized with these samples to give a linear program (LP):

minimize gT∆θ (45a) ∆θ A(θ; Ω) 1 ⊗ b∗ − b(θ; Ω) subject to ∆θ (45b) C d − Cθ

∗ T T ∂g The vector b = [cub − ρlb ρub] represents upper-bound values on b, and g = ∂θ . This LP can be solved by any available solver to yield the optimal parameter increment ∆θ∗. This is iterated, taking θ ← θ + η∆θ∗ (for some fractional step size η) and then re-calculating the gradients after each solution is obtained. Optionally, a batch of such linear program can be solved, for sample sets Ω1, ... , Ωm, in order to reduce the size of each linear program; in this case, a weighted combination of each resulting step ∆θ is applied.

Results for Example Transfer

In this section, the above optimization method is applied, starting with evenly-spaced transfers like those pictured in Fig.3 and speciﬁed in Table4 The 10-variable θ parameterization used is given in Table5. The initial value of θ is taken as

3 θ0 = 36 1500 3000 4500 6000 1479 2979 4479 5979 7479 [10 s] which closely represents the example in Table4. For the optimization, this yields the linear cost function

5 10 X X T g(θ) = θj − θj, g = 1 ··· 1 −1 · · · −1 (46) j=1 j=6 with the constraint Cθ d formulated to specify that

(1) tracking windows must have length of at least half an hour, (2) all TCMs must have a positive target-ahead interval, and

(3) no critical times can exist outside the time range (t0, tf ).

Additionally, the following mission bounds are imposed: cub = 25 m/s, ρlb = Rm + 235 km, ρub = Rm + 250 km. 1000 Monte-Carlo samples were taken, and Ω was chosen as the closest 50

16 Figure 7: Evolution of the durations of measurements m1, ... , m6 over the course of the linear program iteration described in Eqns. 45. (Step fraction η is set to 1.)

to the bounds of the acceptable region. (No outliers were rejected.) 200 iterations of θ updates by linear program Eqns. 45 yields the following improved schedule: θ = 18.43 1493.69 2999.55 4498.42 5995.12 ··· 1479.08 2979.06 4478.95 5979.13 7491.19 [103 s] This schedule has a total measurement time of 7.98×104 seconds, a 35% reduction from 1.23×105 seconds for the initial schedule θ0. Figure7 shows the progress made by the iterations. Upon re-running the 1000 initial samples with the new schedule, a success rate of 100.0% was observed; all samples, not just those used for the optimization, fall within the boundaries of the acceptable region.

CONCLUSION In this work, the framework for an analysis of navigation along low-energy transfers to the moon was presented. In the context of the circular-restricted four-body problem, generalized augmented- state dynamics were developed, including natural orbital motion, continuous measurement, and impulsive trajectory correction maneuvers. The derivatives with respect to all states and time were presented, enabling the calculation of end-state sensitivities to tracking and correction schedule parameters.

17 A numerical scheme for adaptive-step integration of the stochastic models was presented, and used to perform Monte-Carlo simulation of an example transfer and schedule. The results demon- strate the effectiveness of the schedule, and also that of the model presented. Finally, a methodology for the optimization of tracking and correction schedules was presented, based on local approximation with a linear program; this method ensures that constraints on sample success rate are met, solving the relaxed chance-constrained optimization problem. Results for an example schedule are presented, and analyzed to give intuition for the observed trends. Future work plans to continue the development of this model to include more realistic tracking error effects (bias, stability, etc.), as well as different measurement methodologies such as optical tracking. Additionally, the optimization methods will continue to be improved, and applied to a broader selection of low-energy transfer types. It is hoped that this will form the basis for a consistent and efﬁcient navigation schedule design methodology for these transfers, and enable the expansion of their use in the future.

ACKNOWLEDGMENT

We would like to thank the Charles Stark Draper Laboratory, Inc. for its generous funding of this research through the Draper Fellow Program.

NOTATION a a scalar ◦ Stratonovich integral xˆ Estimate of x a a vector ⊗ Kronecker product x˙ Time derivative of x A a matrix † Pseudo-inverse x Augmented version of x ¯ A a tensor [·]× Cross-product matrix xa Partial der. of x w.r.t. a

APPENDIX: DERIVATIVES OF THE CRFBP

For the methods required here, the derivatives of the CRFBP dynamics function f(t, x) are required. The Jacobian matrix F is described by

0 I F (t, x) = (47a) Fvr −2¯ω[eˆ3]×  2 ω¯2 i = j = 1 X δijkρbk − 3ρb,iρb,j 2 [Fvr]ij = −GM µ¯b 5 + −ω¯ i = j = 2 (47b) kρbk b∈{s,e,m} 0 otherwise where [ · ]× represents the cross-product matrix, δij is the Kronecker delta function, and ρb = r−rb. The time partial derivative ft(t, x) is given by

" 0 # 2 ft(t, x) = P 3(ρb·r˙ b)ρb−kρbk r˙ b (48) −GM µ¯b 5 b∈{e,m} kρbk where r˙ b is given by the time derivatives of Eqns.2.

18 APPENDIX: MEASUREMENT ITO-TO-STRATONOVICHˆ CORRECTION As mentioned, the term χ in Eqn. 15 is assumed to be zero, corresponding to the Stratonovich interpretation of the noise terms in the extended Kalman ﬁltering equations. However, if the other interpretation is desired, it is still possible to write the equations in Stratonovich form for solution with the integration method used in this work. In this case, the Ito-to-Stratonovichˆ correction term χ is non-zero, and given by

χ = − 1 K (t, x) ··· K (t, x) vec R(t)KT(t, x) (49) 2 xˆ1 ¯ xˆn ¯ ¯ This follows from the multivariate correction term formula provide by Blowey et al.11

REFERENCES [1] E. Belbruno and J. K. Miller, “Sun-perturbed Earth-to-moon transfers with ballistic capture,” Journal of Guidance, Control, and Dynamics, Vol. 16, No. 4, 1993, pp. 770–775, 10.2514/3.21079. [2] F. Topputo and E. Belbruno, “Earth–Mars transfers with ballistic capture,” Celestial Mechanics and Dynamical Astronomy, Vol. 121, Apr 2015, pp. 329–346, 10.1007/s10569-015-9605-8. [3] C. C. Conley, “Low Energy Transit Orbits in the Restricted Three-Body Problem,” SIAM Journal on Applied Mathematics, Vol. 16, No. 4, 1968, pp. 732–746. [4] E. Belbruno, “Lunar capture orbits, a method of constructing Earth–Moon trajectories and the lunar GAS mission,” 19th International Electric Propulsion Conference, 1987. [5] W. S. Koon, M. W. Lo, J. E. Marsden, and S. D. Ross, “Low Energy Transfer to the Moon,” Celestial Mechanics and Dynamical Astronomy, Vol. 81, Sep 2001, pp. 63–73, 10.1023/A:1013359120468. [6] F. Zuiani, A. Gibbings, M. Vetrisano, F. Rizzi, C. Martinez, and M. Vasile, “Orbit determination and control for the European Student Moon Orbiter,” Acta Astronautica, Vol. 79, 2012, pp. 67 – 78, https://doi.org/10.1016/j.actaastro.2012.03.031. [7] M. Vetrisano, W. V. d. Weg, and M. Vasile, “Navigating to the Moon along low-energy transfers,” Celestial Mechanics and Dynamical Astronomy, Vol. 114, Oct 2012, pp. 25–53, 10.1007/s10569-012- 9436-9. [8] T. You, P. Antreasian, S. Broschart, K. Criddle, E. Higa, D. Jefferson, E. Lau, S. Mohan, M. Ryne, and M. Keck, “Gravity Recovery and Interior Laboratory Mission (GRAIL) Orbit Determination,” Interna- tional Symposium on Space Flight Dynamics, 10 2012. [9] NASA, “Space Communications and Navigation (SCaN) Mission Operations and Commu- nications Services (MOCS),” https://www.nasa.gov/sites/default/files/atoms/ files/mission_operations_and_communications_services.pdf, 2016. Accessed: June 13, 2019. [10] P. Axelsson and F. Gustafsson, “Discrete-Time Solutions to the Continuous-Time Differential Lyapunov Equation With Applications to Kalman Filtering,” IEEE Transactions on Automatic Control, Vol. 60, No. 3, 2015, pp. 632–643. [11] J. Blowey, J. P. Coleman, and A. W. Craig, Theory and Numerics of Differential Equations. Springer, Berlin, Heidelberg, 2001, 10.1107/978-3-662-04354-7.