# Automated Patch Point Placement for Spacecraft Trajectory Targeting

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Information about Automated Patch Point Placement for Spacecraft Trajectory Targeting
Technology

Published on March 21, 2014

Author: BelindaMarchand

Source: slideshare.net

## Description

2014 AAS/AIAA Space Flight Mechanics Meeting

Automated Patch Point Placement for Spacecraft Trajectory Targeting Galen Harden Amanda Haapala Kathleen Howell Belinda Marchand 2014 AAS/AIAA Space Flight Mechanics Meeting Image compliments of www.nasa.gov

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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