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.
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Lyapunov Exponents
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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
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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
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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
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Note that the LLE contour for a given
arc will change according to the
normalizing factor selected
Patch Point Placement
on an LLE Surface
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Patch Point Placement
on an LLE Surface
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Patch Point Placement
on an LLE Surface
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Patch Point Placement
on an LLE Surface
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Patch Point Placement
on an LLE Surface
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Patch Point Placement
on an LLE Surface
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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.
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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.
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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.
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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)
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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.
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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.
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Two-Stage Corrector: Performance Comparison Across
Patch State Selection Strategies
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Motivating Example #2: Orion trans-Earth Trajectory
2x2BP Initial Guess in Earth-Moon 3BP
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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
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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.
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