Automated Patch Point Placement for Spacecraft Trajectory Targeting

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.

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Finite-Time Lyapunov Exponents

and Local Lyapunov Exponents


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:

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Visualization of LLE Contours

as a Function of Normalization Time


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 )

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LLE Contour

Dependence on Horizon Time


Note that the LLE contour for a given

arc will change according to the

normalizing factor selected

Patch Point Placement

on an LLE Surface



on an LLE Surface


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.



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.



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.


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


Evaluation Metric (3)



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.


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.


Two-Stage Corrector: Performance Comparison Across

Patch State Selection Strategies


Motivating Example #2: Orion trans-Earth Trajectory

2x2BP Initial Guess in Earth-Moon 3BP


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

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Motivating Example #2: Orion trans-Earth Trajectory

Converged Solution in Earth-Moon 3BP


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

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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.


Automated Patch Point Placement for Spacecraft Trajectory Targeting

Technology

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