# Rodney L AndersonCalifornia Institute of Technology | CIT · Jet Propulsion Laboratory

Rodney L Anderson

PhD

## About

104

Publications

7,011

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1,024

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Citations since 2017

Introduction

Rodney L. Anderson is a Technologist in the Mission Design and Navigation section at the Jet Propulsion Laboratory where he has worked since 2010. He is the PI on multiple research tasks focusing on applying dynamical systems techniques to solving astrodynamics problems. He is a recipient of JPL’s early career Lew Allen award and coauthored the book “Low-Energy Lunar Trajectory Design.” He has been a Lecturer at Caltech, CU Boulder, and USC, and he is an Associate Editor for JAS.

Additional affiliations

December 2005 - July 2009

March 1998 - April 1999

**FDC/NYMA at NASA Langley Research Center**

Position

- Trajectory Analyst

Description

- Modified and tested the new version of the Program to Optimize Simulated Trajectories (POST II). Performed 3 DOF and 6 DOF trajectory analyses for the reentry portion of the Genesis mission.

Education

August 2001 - December 2005

May 1999 - July 2001

May 1995 - December 1997

## Publications

Publications (104)

Multi-objective reinforcement learning is used to uncover a subset of the multi-objective solution space for low-thrust transfers between two [Formula: see text] southern halo orbits in the Earth–Moon circular restricted three-body problem. Multiple policies are trained in this scenario to recover transfers while maximizing their reward functions t...

A fundamental problem in spacecraft mission design is to find free-flight paths from one place to another that satisfy various design criteria. We explore the geometry of free-flight paths between departure and arrival points for Kepler's problem. Newton showed that these paths are conic arcs. We find the parameters for all conic paths between a de...

When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, families of unstable periodic orbits break up into whiskered tori, with most tori persisting into the perturbed system. In this study, we (1) develop a quasi-Newton method which simultaneously solves for the tori and their center, stable, and unstable directions; (...

View Video Presentation: https://doi.org/10.2514/6.2022-1887.vid Multi-Reward Proximal Policy Optimization (MRPPO) is a multi-objective reinforcement learning algorithm used to train multiple policies to uncover solutions within a multi-objective solution space. MRPPO is used in this paper to train policies to construct low-thrust transfers for a S...

Multi-Reward Proximal Policy Optimization (MRPPO) is a multi-objective reinforcement learning algorithm used to train multiple policies to uncover solutions within a multi-objective solution space. MRPPO is used in this paper to train policies to construct low-thrust transfers for a SmallSat from the vicinity of L2 to an L5 short period orbit in th...

Multi-Reward Proximal Policy Optimization (MRPPO) is a multi-objective reinforcement learning algorithm used to construct low-thrust transfers between periodic orbits in multi-body systems. Previous implementations of MRPPO have relied on a predefined reference transfer to successfully train each policy. In this paper, an algorithmic modification l...

When the planar circular restricted 3-body problem is periodically perturbed, most unstable periodic orbits become invariant tori. However, 2D Poincar\'e sections no longer work to find their manifolds' intersections; new methods are needed. In this study, we first review a method of restricting the intersection search to only certain manifold subs...

Many unstable periodic orbits of the planar circular restricted 3-body problem (PCRTBP) persist as invariant tori when a periodic forcing is added to the equations of motion. In this study, we compute tori corresponding to exterior Jupiter-Europa and interior Jupiter-Ganymede PCRTBP resonant periodic orbits in a concentric circular restricted 4-bod...

In recent years, stable and unstable manifolds of invariant objects (such as libration points and periodic orbits) have been increasingly recognized as an efficient tool for designing transfer trajectories in space missions. However, most methods currently used in mission design rely on using eigenvectors of the linearized dynamics as local approxi...

Placing a small satellite into a high-inclination orbit with respect to the ecliptic plane may offer a low-cost option for opportunistic and targeted observations of the polar regions of the Sun or the zodiacal dust cloud of the solar system. In this paper, dynamical systems theory and hybrid optimization techniques are integrated into a cohesive f...

When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, families of unstable resonant periodic orbits break up into whiskered tori, with most tori persisting into the perturbed system. In this study, we 1) develop a quasi-Newton method which simultaneously solves for the tori and their center, stable, and unstable direc...

A fundamental problem in spacecraft mission design is to find a free flight path from one place to another with a given transfer time. This problem for paths in a central force field is known as Lambert's problem. Although this is an old problem, we take a new approach. Given two points in the plane, we produce the conic parameters for all conic pa...

The endgame scenario that was explored in this analysis consisted of the part of the trajectory starting at the last Ganymede flyby and ending at the final Europa approach. The basic design components included computing the phasing for the final Ganymede encounter, computing the required intermediate Europa flybys, determining the required maneuver...

A cell-mapping approach is implemented and parallelized to analyze three-body problem orbits in the vicinity of icy moons (Europa and Enceladus). The cell-mapping method is developed for studying nonlinear dynamics with periodic motions. The method does not require previously known solutions as inputs, which is an essential requirement of continuat...

Multi-Reward Proximal Policy Optimization, a multi-objective deep reinforcement learning algorithm, is used to examine the design space of low-thrust trajectories for a SmallSat transferring between two libration point orbits in the Earth-Moon system. Using Multi-Reward Proximal Policy Optimization, multiple policies are simultaneously and efficien...

In recent years, stable and unstable manifolds of invariant objects (such as libration points and periodic orbits) have been increasingly recognized as an efficient tool for designing transfer trajectories in space missions. However, most methods currently used in mission design rely on using eigenvectors of the linearized dynamics as local approxi...

Earlier work focused on the computation of isolating blocks around the libration points in the circular restricted three-body problem and the use of these isolating blocks to compute the stable and unstable asymptotic sets of the isolated invariant set around the libration points. In this study, the isolated invariant set, or the invariant three-sp...

Asteroids and comets often capture and sometimes transit near a planet by traveling through the \(\hbox {L}_1\) and \(\hbox {L}_2\) libration point gateways, and these regions are therefore key to understanding the mechanism by which captures, transits, and some potential impacts of these bodies occur. Isolating blocks have recently been used to pr...

Uncertainty quantification is a valuable methodology in the field of astrodynamics, but approaches are often computationally expensive or rely on assumptions that do not account for non-Gaussian posteriors or the nonlinear evolution of the a priori uncertainty. Separated representations accurately estimate posterior distributions when using a sampl...

In this study, the characteristics of petal rotation trajectories are explored in both the two-body problem and the circular restricted three-body problem (CRTBP). Petal rotation trajectories alternate long- and short-period nonresonant transfers between one or more gravity assist bodies to rotate the line of apsides relative to the central body. T...

The pitch motion of spacecraft in the planar elliptic restricted three-body system is studied. Previous studies laid the foundation for spacecraft stability analysis with a small perturbation to the zero pitch motion. In this study, a cell-mapping approach that combines analytical and numerical techniques is used to study the global behavior of the...

In this study, a cell-mapping approach is applied to various systems in the circular restricted three-body problem to obtain a rapid understanding of the global dynamics. The method is generic for various classes of problems including non-autonomous systems and different types of periodic solutions. The cell-mapping method also does not require pre...

A previous study introduced the use of clustering as a tool to support mission designers in organizing sets of ballistic orbits for small-body missions. In that study, periodic orbits generated in simplified dynamical models were examined, leading to the identification of attractive characterization orbits. This paper furthers that study by analyzi...

A comprehensive tour of the complex outer planet systems is a central goal in space science. However, orbiting multiple moons of the same planet would be extremely prohibitive using traditional propulsion and power technologies. In this paper, a new mission concept, named Magnetour, is presented to facilitate the exploration of outer planet systems...

Isolating blocks may be used as computational tools to search for the invariant manifolds of orbits and hyperbolic invariant sets associated with libration points while also giving additional insight into the dynamics of the flow in these regions. We use isolating blocks to investigate the dynamics of objects entering the Earth-Moon system in the c...

Orbital dynamics around small bodies are as varied as the shapes and dynamical states of these bodies. While various classes of orbits have been analyzed in detail, the global overview of relevant ballistic orbits at particular bodies is not easily computed or organized. Yet, correctly categorizing these orbits will ease their future use in the ove...

Unstable resonant orbits in the circular restricted three-body problem have increasingly been used for trajectory design using optimization and invariant manifold techniques. In this study, several methods for computing these unstable resonant orbits are explored including flyby maps, continuation from two-body models, and grid searches. Families o...

This paper presents the mission design for the proposed MoonRise New Frontiers mission: a lunar far side lander and return vehicle, with an accompanying communication satellite. Both vehicles are launched together, but fly separate low-energy transfers to the Moon. The communication satellite enters lunar orbit immediately upon arrival at the Moon,...

In this study, the final approach to a moon or other body from resonance is explored and compared to the invariant manifolds of unstable periodic orbits. It is shown that the stable manifolds of planar Lyapunov orbits can act as a guide for the periods or resonances that are required for the final approach in both the planar and spatial problems. P...

We present the methods and results of the Jet Propulsion Laboratory team in the 5th Global Trajectory Optimization Competition. Our broad-search strategy utilized several recently developed phase-free metrics for rapidly narrowing the search options. Two different, adaptive, branch-and-prune strategies were employed to build up asteroid sequences u...

In this work, the approach phase from the final resonance of the endgame scenario in a tour design is examined within the context of invariant manifolds. Previous analyses have typically solved this problem either by using numerical techniques or by computing a catalog of suitable trajectories. The invariant manifolds of a selected set of libration...

In this study, the characteristics of petal rotation trajectories are explored in both the two-body and circular restricted three-body problem (CRTBP) models. Petal rotation trajectories alternate long and short period non-resonant transfers between one or more gravity assist bodies to rotate the line of apsides relative to the central body. They a...

The purpose of this book is to provide high-level information to mission managers and detailed information to mission designers about low-energy transfers between the Earth and the Moon. This book surveys thousands of trajectories that one can use to transfer spacecraft between the Earth and various locations near the Moon, including lunar libratio...

Operations Executive Summary Operations Introduction Launch Sites Launch Vehicles Designing a Launch Period Navigation Spacecraft Systems Design

Executive Summary Introduction Direct Transfers Between Earth and Low Lunar Orbit Low-Energy Transfers Between Earth and Low Lunar Orbit Transfers Between Lunar Libration Orbits and Low Lunar Orbits Transfers Between Low Lunar Orbits and the Lunar Surface

Executive Summary Introduction for Transfers to the Lunar Surface Methodology Analysis of Planar Transfers between the Earth and the Lunar Surface Low-Energy Spatial Transfers Between the Earth and the Lunar Surface Transfers Between Lunar Libration Orbits and the Lunar Surface Transfers Between Low Lunar Orbits and the Lunar Surface Conclusions Re...

Purpose Organization Executive Summary Background The Lunar Transfer Problem Historical Missions Low-Energy Lunar Transfers

Executive Summary Introduction Direct Transfers Between Earth and Lunar Libration Orbits Low-Energy Transfers Between Earth and Lunar Libration Orbits Three-Body Orbit Transfers

Introduction Setting up the System Triangular Points Collinear Points Algorithms

Methodology Introduction Physical Data Time Systems Coordinate Frames Models Low-Energy Mission Design Tools

In this study, the use of low-energy trajectories computed from the invariant manifolds of unstable periodic orbits is explored as a means to transfer from low-Earth orbit to Near Earth Objects. Previous work estimated the maneuvers required to reach these asteroids using planar assumptions, and various other studies have focused on returning an as...

A simple strategy is identified to generate ballistic transfers between the Earth and Moon, i.e., transfers that perform two maneuvers: a translunar injection maneuver to depart the Earth and a lunar orbit insertion maneuver to insert into orbit at the Moon. This strategy is used to survey the performance of numerous transfers between varying Earth...

This study examines the effects of density model time-delay errors by focusing on the Challenging Minisatellite Payload spacecraft data over the 2003-2007 time period. The data are first analyzed to determine the magnitudes of time delays encountered when using a density model. The analysis is initiated by examining the effects of the expected dela...

This paper offers an early examination of the challenges of navigating a crewed vehicle, with all of the associated unmodeled accelerations that arise from the crew's activities, in an orbit about the Earth-Moon L2 point. The combination of the unstable nature of libration orbits with the lack of acceleration knowledge makes the station keeping str...

In this study, transfer trajectories from the Earth to the Moon that encounter the Moon at various flight path angles are examined, and lunar approach trajectories are compared to the invariant manifolds of selected unstable orbits in the circular restricted three-body problem. Previous work focused on lunar impact and landing trajectories encounte...

A targeting scheme is presented to build 21-day launch periods for low-energy trajectories from any specified Earth parking orbit to any specified low lunar orbit, using up to two mid-course maneuvers. A total of 288 launch periods are constructed for transfers to a variety of different targeted low lunar orbits, arriving at different times during...

It is increasingly desirable to incorporate multi-body effects in tour design early in the process to make use of these effects and potentially discover new solutions. Flybys have previously been designed using the heteroclinic connections of resonant orbits in the circular restricted three-body problem (CRTBP), but tour design often requires the c...

This paper examines the benefits of navigating a crewed vehicle at the Moon using both ground tracking and satellite-to-satellite tracking, where the tracking satellite is stationed in a lunar halo orbit. Linked Autonomous Interplanetary Satellite Orbit Navigation (LiAISON) is a new technique that has been shown to dramatically improve the navigati...

Autonomous orbit determination for departure stages of interplanetary trajectories is conducted by means of realistic radiometric observations between the departing spacecraft and a satellite orbiting the first lunar libration point. Linked Autonomous Interplanetary Satellite Orbit Navigation (LiAISON) is used to estimate the orbit solution. This p...

This paper examines the benefits of navigating a crewed vehicle between the Earth and the Moon using both ground tracking and satellite-to-satellite tracking. Linked Autonomous Interplanetary Satellite Orbit Navigation (LiAISON) is a new technique that has been shown to dramatically improve the navigation of lunar satellites, libration orbiters, an...

Evaluation of Initial Guess Strategies for Low-Energy Resonant Flybys

Linked Autonomous Interplanetary Satellite Orbit Navigation (LiAISON) is a new technique that takes advantage of the asymmetrical gravity field present in a three-body system in order to perform absolute tracking of satellites using only relative satellite-to-satellite observations. Previous studies have demonstrated LiAISON's practical application...

In this study techniques are developed which allow an analysis of a range of different types of transfer trajectories from the Earth to the lunar surface. Trajectories ranging from those obtained using the invariant manifolds of unstable orbits to those derived from collision orbits are analyzed. These techniques allow the computation of trajectori...

Tour design has traditionally relied on the use of two-body patched
conic techniques in combination with differential correction or
optimization algorithms. Recently, three-body effects have increasingly
been included more directly in the design and analysis problem using a
variety of different methods. In previous work it was shown that
ballistic,...

Demands on numerical integration algorithms for astrodynamics applications continue to increase. Common methods, like explicit Runge-Kutta, meet the orbit propagation needs of most scenarios, but more specialized scenarios require new techniques to meet both computational efficiency and accuracy needs. This paper provides an extensive survey on the...

A targeting scheme is presented to build trajectories from a specified Earth parking orbit to a specified low lunar orbit via a low-energy transfer and up to two maneuvers. The total transfer ΔV is characterized as a function of the Earth parking orbit inclination and the departure date for transfers to each given low lunar orbit. The transfer ΔV c...

In this study, transfer trajectories from the Earth to the Moon that encounter the Moon at various flight path angles are examined, and lunar approach trajectories are compared to the invariant manifolds of selected unstable orbits in the circular restricted three-body problem. Previous work focused on lunar impact and landing trajectories encounte...

The invariant manifolds of libration point orbits (LPOs) in the Sun-Earth/Moon system are used to construct low-energy transfers from Low Earth Orbits (LEOs) to geosynchronous orbits. A maneuver is performed in LEO to insert onto a stable manifold trajectory of an LPO. The spacecraft travels to the host LPO and then follows an unstable manifold tra...

In this analysis, resonant flybys were explored within the context of the circular restricted three-body problem using dynamical systems theory. The first step in this process involved the construction of a flyby trajectory continuously transiting between 3:4 and 5:6 resonances in the Jupiter-Europa circular restricted three-body problem. An examin...

This paper presents a method to construct optimal transfers between unstable periodic orbits of differing energies using invariant manifolds. The transfers constructed in this method asymptotically depart the initial orbit on a trajectory contained within the unstable manifold of the initial orbit and later, asymptotically arrive at the final orbit...

This study examines the effects of density model time-delay errors by focusing on the CHAMP spacecraft data over the 2003-2007 time period. The data is first analyzed to determine the magnitudes of time delays encountered when using a density model. The analysis is initiated by examining the effects of the expected delays of one to three hours on a...

Tour designs using flybys have traditionally been studied using two-body patched conic methods. Previous work has shown that trajectories designed using these techniques and with optimization methods follow the invariant manifolds of unstable resonant orbits as they transition between resonances. This work is continued here by computing heteroclini...

We introduce an automated approach to design efficient low-energy trajectories by extracting initial solutions from dynamical channels formed by invariant manifolds, and improving these solutions through variational optimal control. We consider trajectories connecting two unstable periodic orbits in the circular restricted 3-body problem (CR3BP). U...

In this study techniques are developed which allow an analysis of a range of different types of transfer trajectories from the Earth to the lunar surface. Trajectories ranging from those obtained using the invariant manifolds of unstable orbits to those derived from collision orbits are analyzed. These techniques allow the computation of trajectori...

Libration point orbits in the three-body problem play an important role in mission design. Their size, shape, location, and dynamical features are attractive for many different science and exploration missions, with objectives ranging from the collection of solar observations and measurements, to a search for the first massive black holes, to the d...

In this analysis, the relationship between a planar Europa Orbiter trajectory and the invariant manifolds of resonant periodic orbits is studied An Understanding of this trajectory with its large impulsive maneuvers should provide basic tools that can be extended to cases that approximate low thrust with many small maneuvers This study therefore re...

Techniques from dynamical systems theory have been applied to the construction of transfers between unstable periodic orbits
that have different energies. Invariant manifolds, trajectories that asymptotically depart or approach unstable periodic orbits,
are used to connect the initial and final orbits. The transfer asymptotically departs the initia...

The invariant manifolds of libration point orbits (LPOs) in the Sun-Earth/Moon system are used to construct low-energy transfers from Low Earth Orbits (LEOs) to geosynchronous orbits. A maneuver is performed in LEO to insert onto a stable manifold trajectory of an LPO. The spacecraft travels to the host LPO and then follows an unstable manifold tra...

This paper summarizes the first stage in a research effort designed to obtain a quantitative knowledge and deeper understanding of how prediction error depends on density variability. Initial numerical simulations are performed to obtain a detailed understanding of the theoretical effects of density variations on satellite orbit predictions. The le...

A method is developed for constructing locally optimal transfer trajectories between libration point orbits with different energies. The unstable manifold of the first orbit is connected to the stable manifold of the second orbit by the execution of two or more maneuvers. Two-body parameters define the selection of the unstable and stable manifold...

This research presents a method of using invariant manifolds to connect libration point orbits. The method presented is applicable to constructing transfers between planar or three-dimensional orbits that have different initial energies. Two deterministic maneuvers are used to connect an unstable manifold trajectory of the first orbit to a stable m...

In this analysis the relationship between a planar Europa Orbiter trajectory and the invariant manifolds of resonant periodic orbits was studied. An understanding of this trajectory with its large impulsive maneuvers should provide basic tools which can be extended to cases that approximate low thrust with many small maneuvers. This study therefore...

This paper summarizes the first stage in a research effort designed to obtain a quan - titative knowledge and deeper understanding of how prediction error depends on density variability. Initial numerical simulations were performed to obtain a detailed understand- ing of the theoretical effects of density variations on satellite orbit predictions....

An initial study of techniques to be used in understanding how invariant manifolds are involved in low thrust trajectory design for the Jovian moon missions was performed using a baseline trajectory from the Europa Orbiter (EO) studies. Poincare sections were used in order to search for unstable resonant orbits. The unstable manifolds of these reso...

Previous research has focused on locating trajectories in the three-body problem useful for mission design using quasi-periodic orbits and their invariant manifolds. Although these techniques have been very successful, the question arises as to whether other useful trajectories exist which cannot be found using the current methodology. In this stud...