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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,y Kerri L. Cahoy,z and Richard Linaresx Low-energy transfers expose a spacecraft to long periods of unstable dynam- ics, and navigation along them currently relies on frequent tracking by expensive ground infrastructure. This paper characterizes minimum-cost navigation strate- gies for these trajectories, subject to constraints on the probability of transfer suc- cess. 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 trans- fer 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 re- duce 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 ex- ample, 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. yLaboratory Technical Staff, Space and Mission Critical Systems, The Charles Stark Draper Laboratory, Inc. Cambridge, MA 02139. zAssociate Professor, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139. xCharles 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 infras- tructure. 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 (e^1; e^2; e^3) centered at the Earth/moon barycenter and rotating about the e^3 axis with angular rate equal to the orbital rate of the Earth/moon system about the sun. The sun is stationary on the negative e^1 axis, and the Earth and moon fol- low circular orbits about the origin in the e^1; e^2 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_ ) =! ¯ E12 r +µ ¯sr¯se^1 + 2¯!(r_ × e^3) − GM µ¯b 3 (1) kr − rb(t)k | {z } | {z } b2fs;e;mg Centrifugal Term Coriolis Term | {z } Gravitational Terms where E12 = e^1 e^2 0 represents a projection into the e^1; e^2 plane, the constant sun position is rs(t) = rs = −r¯se^1, and the time-dependent Earth/moon positions re(t) and rm(t) are given by r (t) = µ¯e [cos(t + φ )e^ + sin(t + φ )e^ ] (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 !¯ and GM are defined by −1 p −2 p 3 −3=2 !¯ = 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 per- turbations, 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 sys- tem, 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 x^T pT c (7) ¯ where, x represents the true position/velocity state, x^ 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: ¯ ¯ 2 f(t; x) 3 2G(t)3 6 f(t; x^) 7 6 0 7 f(t; x) = 6 7 ; G(t; x) = 6 7 (9) ¯ ¯ 4A(t; x^)p + q(t)5 ¯ ¯ 4 0 5 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.
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