Information about Automated Patch Point Placement for Spacecraft Trajectory Targeting

2014 AAS/AIAA Space Flight Mechanics Meeting

Introduction Problem Summary Targeting in dynamically sensitive regimes benefits from multi- phase algorithms, which simultaneously operate on a startup arc divided into multiple patch states (e.g. nodes) Gradient-based targeting algorithms (optimal or sub-optimal) are sensitive to the quality of the initial guess. Arbitrary placement of patch states (e.g. nodes) can negatively impact algorithm response in dynamically sensitive regimes Solution Approach Develop an automated patch state / node selection strategy, suitable for onboard guidance processes, that can intelligently select patch state sets. Seek algorithm that reduces impact of dynamical sensitivities to improve overall algorithm response. January 27th, 2014 Harden, Haapala, Howell, & Marchand 2

Lyapunov Exponents January 27th, 2014 Harden, Haapala, Howell, & Marchand Characterize the rate of separation of two infinitesimally close trajectories as they evolve in time, and given by: where . If , the full Lyapunov exponent spectrum is characterized by , one for each linearly independent fundamental direction. In general, there is no analytical means of identifying the Lyapunov exponents. They must be approximated numerically over a finite horizon. 3

Finite-Time Lyapunov Exponents and Local Lyapunov Exponents January 27th, 2014 Harden, Haapala, Howell, & Marchand The term Finite Time Lyapunov Exponents is often used to refer to the full spectrum of Lyapunov exponents over a specific finite time horizon. A reasonable approximation of the local growth rate is determined by considering only the largest exponent in the set, or Local Lyapunov Exponent (LLE): Here, t denotes the selected time horizon. Note that if the trajectory spans over , then: 4

Visualization of LLE Contours as a Function of Normalization Time January 27th, 2014 Harden, Haapala, Howell, & Marchand The relation between the time along the trajectory and the horizon (or normalization time) is : A large LLE value is indicative of a high degree of dynamical sensitivity at that specific location along the arc. The dark regions in the contour are associated with local minima of the LLE value, while the highest intensity corresponds to local maxima. (Clearly sometimes embracing the dark side is a good thing ) 5

LLE Contour Dependence on Horizon Time January 27th, 2014 Harden, Haapala, Howell, & Marchand 6 Note that the LLE contour for a given arc will change according to the normalizing factor selected

Patch Point Placement on an LLE Surface January 27th, 2014 Harden, Haapala, Howell, & Marchand 7

Patch Point Placement on an LLE Surface January 27th, 2014 Harden, Haapala, Howell, & Marchand 8

Patch Point Placement on an LLE Surface January 27th, 2014 Harden, Haapala, Howell, & Marchand 9

Patch Point Placement on an LLE Surface January 27th, 2014 Harden, Haapala, Howell, & Marchand 10

Patch Point Placement on an LLE Surface January 27th, 2014 Harden, Haapala, Howell, & Marchand 11

Patch Point Placement on an LLE Surface January 27th, 2014 Harden, Haapala, Howell, & Marchand 12

Automated Patch State Selection: Motivation Previous research reveals that patch states placed at local minima along the LLE contour offer the best convergence for targeting and optimization algorithms. This observation suggests a patch state selection strategy that automatically identify candidate points, based on the LLE criteria, is desirable. To develop an automated patch state placement algorithm, it is useful to establish a simple metric by which to systematically and autonomously compare the “quality” of a given patch state set against another. January 27th, 2014 Harden, Haapala, Howell, & Marchand 13

Automated Patch State Selection: Evaluation Metric (1) All multi-phase targeting algorithms presented operate on the initial guess by modifying a set of control parameters: in an effort to satisfy a set of linearized constraint equations: The vector of control parameters varies depending on the exact targeting algorithm selected. January 27th, 2014 Harden, Haapala, Howell, & Marchand 14

Automated Patch State Selection: Evaluation Metric (2) To evaluate the impact of varying a specific patch state set, on the constraint error, we seek a simple scalar expression that Relates the norm of the constraint vector as a function of the norm of the patch state errors. Captures the impact of our “confidence” on the quality of the patch states. January 27th, 2014 Harden, Haapala, Howell, & Marchand 15

By leveraging the properties of the expected value, and some properties of the trace, this expression reduces to: For the specific targeting algorithm selected, this reduces to: and ultimately to Automated Patch State Selection: Evaluation Metric (3) January 27th, 2014 Harden, Haapala, Howell, & Marchand 16

Automated Patch State Selection: Computational Process Having established the metric for comparison, the computation of a patch state set proceeds as follows: Start with one patch state at the beginning of the trajectory, and at any scheduled maneuver points, iteratively. For a specific segment, identify a set of candidate states, any one of which could represent the new patch state. Each candidate must satisfy the constraint between duration and normalization (i.e. select points along diagonals of the LLE contour). Select a reasonably representative number of candidates to properly characterize the options along that diagonal. For each candidate state, evaluate the approximate error metric and identify which is associated with the smallest error. January 27th, 2014 Harden, Haapala, Howell, & Marchand 17

Motivating Example #1: Altitude Targeting Near Earth Fix initial position, target final position. Target final position vector aligned with initial guess, but seeks change in altitude. Compare candidate multi-phase targeter performance using evenly-spaced vs. automatically-selected nodes. January 27th, 2014 Harden, Haapala, Howell, & Marchand 18

Two-Stage Corrector: Performance Comparison Across Patch State Selection Strategies January 27th, 2014 Harden, Haapala, Howell, & Marchand 19

Motivating Example #2: Orion trans-Earth Trajectory 2x2BP Initial Guess in Earth-Moon 3BP January 27th, 2014 Harden, Haapala, Howell, & Marchand One Earth-centered arc One moon-centered of arcs LLO to Apogee raise seg. Apogee to Inc. change seg. Inc. change to Trans-Earth seg. Segments patch points. 2BP patch points 3BP Discontinuities between segments and @ interface 20

Motivating Example #2: Orion trans-Earth Trajectory Converged Solution in Earth-Moon 3BP January 27th, 2014 Harden, Haapala, Howell, & Marchand Target entry altitude via 3-maneuver sequence Poor initial guess quality degrades performance of Linear targeting Targeting performance Equally spaced patch states: DNC Automated patch state selection: 8 iterations 21

Conclusions Preliminary results indicate automated patch point placement algorithm improves response of multi-phase targeting algorithms. The initial error prediction model considered offers a simple effective metric by which to compare the quality of candidate patch state sets. January 27th, 2014 Harden, Haapala, Howell, & Marchand 22

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AAS 14-272 AUTOMATED PATCH POINT PLACEMENT FOR SPACECRAFT TRAJECTORY TARGETING Galen Harden, Amanda Haapalay, Kathleen C. Howell z, and Belinda Marchand x

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Official Full-Text Publication: Automated Patch Point Placement for Spacecraft Trajectory Targeting on ResearchGate, the professional network for scientists.

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AAS 14-272 AUTOMATED PATCH POINT PLACEMENT FOR SPACECRAFT TRAJECTORY TARGETING Galen Harden∗, Amanda Haapala†, Kathleen C. Howell‡, and Belinda ...

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Automated patch point placement for autonomous spacecraft trajectory targeting. Galen K Harden, Purdue University. Abstract. The lofty goal of autonomous ...

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Autonomous targeting and guidance, specifically that which relies on iterative gradient-based processes, is sensitive to the quality of the startup guess ...

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Automated patch point placement for autonomous spacecraft trajectory targeting . By Galen K Harden. Abstract. The lofty goal of autonomous, onboard ...

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Automated patch point placement for autonomous spacecraft trajectory targeting by Harden, Galen K., M.S.A.A., PURDUE UNIVERSITY, 2013, 157 pages; 1549361

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Automated patch point placement for autonomous spacecraft trajectory targeting. Author . Harden, Galen K.

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Belinda Marchand 2011 ... Targeting and Guidance ... A., Howell, K.C., and Marchand, B., “Automated Patch Point Placement for Spacecraft Trajectory ...

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View Amanda Haapala ... Automated Patch Point Placement ... Representations of Higher-Dimensional Poincaré Maps with Applications to Spacecraft Trajectory ...

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