Aas 20-488 Motion Primitives Summarizing Periodic Orbits And

AAS 20-488 MOTION PRIMITIVES SUMMARIZING PERIODIC ORBITS AND NATURAL TRANSPORT MECHANISMS IN THE EARTH-MOON SYSTEM Thomas R. Smith∗ and Natasha Bosanacy Rapid trajectory design in multi-body systems often leverages individual arcs along natural dynamical structures that exist in an approximate dynamical model. To reduce the complexity of analysis during this process, motion primitives are con- structed as a set of arcs that represent the finite geometry, stability, and energy characteristics exhibited by a family of trajectories. In the absence of generalizable analytical criteria for extracting these representative solutions, clustering is employed. In this paper, this clustering-based approach to constructing motion primitives is improved and applied to spatial periodic orbit families and hyperbolic invariant manifolds in the Earth-Moon circular restricted three-body problem. 1 INTRODUCTION Rapid trajectory design strategies are crucial to cislunar exploration, particularly during mission concept development and in real-time contingency scenarios. In the absence of generalizable analytical expressions, one current approach to rapid trajectory design in multi-body systems is to first generate a large library of fundamental solutions, finely discretized along families of periodic and quasi-periodic orbits.1 Specialized design tools are then used to explore the library and se- lect suitable arcs to construct an initial guess for a complex trajectory.2 However, this approach currently relies heavily on a human-in-the-loop for analysis and decision-making and may be time- consuming. Inspired by solutions in Big Data, one approach to reducing the analytical workload of the trajectory designer relies on data mining techniques. Data analysis techniques such as clustering have previously been used to summarize and analyze the solutions present in nonlinear dynamical systems for which analytical expressions are unavail- able. For instance, spectral clustering has been employed by Hadjighasem, Karrasch, Teramoto, and Haller to identify coherent Lagrangian vortices from the flow of Lagrangian trajectories to develop a simplified description of the vortices in the underlying dynamical system.3 In astrodynamics, the partition-based clustering algorithm k-means has been used by Nakhjiri and Villac to identify bounded motions in a specific region of a Poincare´ map and by Villac, Anderson, and Pini to group periodic orbit solutions based on the locations of apses and orbital period in the augmented Hill’s three-body problem.4,5 In addition, Bosanac has applied hierarchical density-based clustering methods to planar Poincare´ maps in the circular restricted three-body problem (CR3BP) to group trajectories with similar geometries; the result is a reduced representative dataset that reduces the ∗Graduate Research Assistant, Colorado Center for Astrodynamics Research, Smead Department of Aerospace Engineer- ing Sciences, University of Colorado Boulder, Boulder, CO 80303. yAssistant Professor, Colorado Center for Astrodynamics Research, Smead Department of Aerospace Engineering Sci- ences, University of Colorado Boulder, Boulder, CO 80303. 1 complexity of visualization and facilitates analysis in the trajectory design process.6 Bonasera and Bosanac use a similar approach to simplify the analysis of higher-dimensional Poincare´ maps in both the natural CR3BP and a low-thrust enabled CR3BP.7 Each of these clustering applications demonstrate the value of using data mining techniques to summarize the solution space of trajectories in chaotic dynamical systems in an unsupervised manner. Recently, Smith and Bosanac have used clustering algorithms to construct a set of motion primitives that summarize a family of planar periodic orbits in the CR3BP, with the goal of reducing the analysis workload for a trajectory designer.8 A motion primitive is the most representative solution among a group of similar solutions and is a well-known concept in robotics; motion primitives offer a summary of the solution space as well as a mechanism for rapid path planning.9–11 However, there is no explicit analytical criteria for extracting a set of motion primitives to summarize the solution space within a multi-body system. In Smith and Bosanac, clustering is used to extract a set of fundamental motion primitives that represent the finite characteristics of a family of planar periodic orbits in an unsupervised manner.8 Using this procedure, a variety of feature sets and clustering methods effectively recovered motion primitives that summarize the family of Distant Prograde Or- bits (DPOs) in the Earth-Moon system. These results motivate further application of the motion primitive construction process to more complex dynamical structures within the spatial CR3BP. The focus of this paper is to apply and extend our previous work on the motion primitive construction process to families of spatial periodic orbits and hyperbolic invariant manifolds in the CR3BP. Generating a set of motion primitives to summarize families of spatial trajectories and hyperbolic invariant manifolds is critical to capturing the natural transport mechanisms that are often used in trajectory design.12–15 This paper presents an updated approach for motion primitive construction, improved to accommodate spatial trajectories and an updated feature vector definition. In addition, input parameters for each clustering algorithm are selected automatically through the use of relative validity criteria, reducing the burden on a human analyst. This updated and expanded procedure is used to construct motion primitives for the L1, L2, and L3 northern halo, axial, and vertical families as well as trajectories along the unstable manifold associated with an L1 Lyapunov orbit in the Earth-Moon CR3BP. The resulting primitives effectively summarize each family of solutions and may, potentially, be useful in exploring the solution space in a guided, efficient, and rapid manner, thereby reducing the analytical workload of the trajectory designer. 2 BACKGROUND: DYNAMICAL MODEL The CR3BP approximates the motion of a spacecraft under the gravitational influence of two primaries, assumed to travel along circular orbits. The spacecraft is assumed to possess a negligi- ble mass compared to the larger primary body, the Earth, with constant mass M1 and the smaller primary body, the Moon, with constant mass M2. Additionally, both primary bodies are modeled as point masses.16 Following application of these assumptions, a rotating reference frame x^y^z^ is defined: the x^-axis is directed from the Earth to the Moon, the z^-axis is aligned with the orbital angular momentum of the primary system, and the y^-axis completes the right-handed triad. Then, the length, mass, and time quantities describing the state of the spacecraft are nondimensionalized using the characteristic parameters l∗, m∗, and t∗, respectively. Typically, l∗ is set equal to the distance between the Earth and Moon, m∗ corresponds to the total mass of the system, and t∗ is defined such that the nondimensional period of the primary system is equal to 2π. The state of the spacecraft is then written as a nondimensional vector ~x = [x; y; z; x;_ y;_ z_]T relative to the barycenter of the system and in the rotating frame. Using these assumptions and definitions, the nondimensional 2 equations of motion for the spacecraft in the CR3BP are written in the rotating frame as (1 − µ)(x + µ) µ(x − 1 + µ) x¨ = 2y _ + x − 3 − 3 r1 r2 (1 − µ)y µy y¨ = −2x _ + y − 3 − 3 (1) r1 r2 (1 − µ)z µz z¨ = − 3 − 3 r1 r2 where µ = M2 is the mass ratio of the system, r = p(x + µ)2 + y2 + z2 is the distance M1+M2 1 p 2 2 2 between the spacecraft and the Earth, and r2 = (x − 1 + µ) + y + z is the distance between the spacecraft and the Moon. In this dynamical system, an integral of motion, commonly labeled the Jacobi constant, is conserved along natural trajectories and is equal to 2 2 2(1 − µ) 2µ 2 2 2 CJ = (x + y ) + + − x_ − y_ − z_ (2) r1 r2 At a single value of the Jacobi constant, a variety of natural dynamical structures such as periodic orbits, quasi-periodic orbits, and hyperbolic invariant manifolds exist in families throughout the phase space.16, 17 Hyperbolic invariant manifolds govern the natural flow of trajectories that asymptotically approach or depart an unstable periodic orbit: trajectories that asymptotically approach a periodic orbit lie on the stable manifold, while trajectories that asymptotically depart the periodic orbit form the unstable manifold. Each family of periodic orbits or trajectories along a hyperbolic invariant manifold corresponds to a continuous set of solutions that tend to exhibit a finite number of distinct characteristics. However, there are often no clear and generalizable analytical expressions for grouping solutions along a family according to these characteristics. 3 BACKGROUND: CLUSTERING Clustering is an unsupervised learning process for separating the members of a dataset into groups based on a defined set of features.18 Data in the same cluster possess similar properties while data in different clusters possess dissimilar properties in the prescribed feature space; therefore, a quantita- tive feature space description must be defined. Each of the n objects in a dataset is described by a m- dimensional feature vector f~. A clustering algorithm is then be applied to the (n×m)−dimensional dataset to uncover the natural groupings within the set. There are many factors that may impact the resulting clusters, such as: the type of clustering method and associated algorithm used to perform the clustering, the feature space description of each object in the dataset, and the input parameters used in the algorithm. Smith and Bosanac supply a detailed overview of common partition, hierarchical, and density-based clustering algorithms including: k-means, spectral clustering, mean shift, agglomerative clustering, Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN), and affinity propagation.8 In this paper, each of these algorithms are leveraged to apply the motion primitive construction process to spatial periodic orbit families and hyperbolic invariant manifolds in the Earth-Moon CR3BP.

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