Global optimisation and
search space pruning in
spacecraft trajectory
design
Victor Becerra
Cybernetics
The University of Reading, UK
Semi-plenary talk
IEEE Colloquium on Optimisation for Control, Sheffield, UK, 24 April 2006
Introduction
Introduction
 Basics of Space Mission Design
 Multiple Gravity Assist (MGA) Trajectories
 Optimal control and MGA mission design
 Search space pruning
 Examples
 Conclusions

Introduction: the Cassini Huygens
mission

The Cassini spacecraft is the first to explore the
Saturn system of rings and moons from orbit.

Cassini entered orbit on 30 June 2004.

•. European Space Agency's Huygens probe
The
explored Titan's atmosphere in January 2005.

The instruments on both spacecraft are providing
scientists with valuable data and views of this
region of the solar system
Source of image: http://saturn.jpl.nasa.gov
Introduction
The mission sequence was EVVEJS with orbit insertion in Saturn
Source of image: http://saturn.jpl.nasa.gov
Introduction
Source of video: http://saturn.jpl.nasa.gov
Basics of mission design


A central aspect of the design of missions such
as Cassini Huygens is the optimisation of the
trajectory.
It is important to calculate trajectories from Earth
to other planets/asteroids/comets that are fuel
and, ideally, time efficient
Basics of mission design

Objective is to maximise the mass of the probe
that may be used for scientific payload



Desired velocity for leaving Earth’s gravitational field
determines maximum mass of probe
Amount of thrust required by probe determines the
proportion of the probe that must be fuel
Gravity assist trajectories allow significant
reductions in both launch velocity and thrust
Basics of space mission design
Spacecraft are provided with sets of
propulsive devices so they can maintain
stability, execute manoeuvres, and make
minor adjustments in trajectory.
 The propulsive action is often impulsive,
but there are now low thrust engines which
provide continuous thrust over extended
periods of time.

Gravity assist manoeuvres



In a gravity-assist manoeuvre, angular
momentum is transferred from the orbiting
planet to a spacecraft approaching from behind
the planet in its progress about the sun.
These manoeuvres can be powered (impulsive
thrust is applied) or unpowered.
This gives extra velocity to the spacecraft and
yields fuel and time savings in a mission.
Gravity assist manoeuvres
Other manoeuvres
Deep space manoeuvres (impulsive)
 Low thrust arcs
 Orbit insertion (braking)

Optimal control and mission design

1.
2.
3.
4.
The trajectory design problem has all the
ingredients to generate optimal control
problems:
Nonlinear dynamics (orbital mechanics)
An objective function
Control action (thrust)
Inequality constraints
Dynamics




Interplanetary travel requires the understanding of the
“restricted N-body problem”.
If the spacecraft is sufficiently close to a celestial body, it
is possible to approximate the dynamics by neglecting
the influence of other celestial bodies, and to analyse the
dynamics as a “restricted two-body problem”.
The region inside which this approximation is valid is
known as the sphere of influence of the celestial body.
If the spacecraft is not inside the sphere of influence of a
planet or moon of the solar system, it is considered to be
under the influence of the sun only.
Dynamics




Because the sphere of influence of the sun is much
larger than that of the planets, when studying MGA
trajectories we may consider one main attracting body
(the sun) and then join the various trajectories using
what is known as the “patched conics” approach.
Hence the problem may be reduced to a sequence of
“restricted two-body problems”
In the unforced case the restricted two body problem
admits solutions that are known to be conics (elliptic,
parabolic or hyperbolic orbits).
The forced case is becoming relevant with the new “lowthrust engines” and other recent propulsion concepts.
Dynamics – Lambert problem




The problem of travelling
between two points with a preassigned time-of-flight along a
ballistic trajectory is called
“Lambert Problem”
Solution gives the spacecraft
velocity vector at the beginning
and at the end of the arc.
Under certain assumptions the
solution is unique.
Numerical integration is avoided.
r

r




r3

r (t0 )  rA , r (t1 )  rB
Gravity assist calculations



A gravity assist model is
used to calculate the
impulsive thrust required
at periapsis during a
swingby
This impulse is often
required to keep a safe
distance from the planet.
The angle a and the
periapsis radius rp are
related.
The patched conics approach
Simplifying Assumptions




Preliminary mission design
 Its goal is to allow exploration of different mission options, rather
than calculate an very accurate trajectory
Several simplifying assumptions are used
 Spacecraft mass is negligible compared with celestial bodies
 Sun/planets are point masses
 Spacecraft transfers between planets are perfectly elliptical
 Instantaneous hyperbolic transfers occur at planets
We will concentrate on MGA trajectories with powered
gravity assists, but without deep space manoeuvres
or low thrust arcs.
These assumptions yield a constrained continuous
optimisation problem with one dimension per planet
involved
Simplified Search Space

Decision vector, x = [t0, T1, T2, … ,TN+1]
 t0





is the launch date from first planet
T1 is transfer time to from first to second planet
T2 is transfer time to second to third planet
And so on…
N is the number of planets where a gravity assist
manoeuvre is performed.
Each element of x can be bounded, so we are looking
for x within a hyper-rectangle:
xI  I 0  I 1 
I
N 1
Ephemeris
Given an arrival (or departure) time at a
planet, say t1, planetary ephemeris are
used to provide the desired position of the
spacecraft at t1.
 There are publicly available ephemeris
routines and solar system object
databases which can be used to
determine the position of celestial bodies
as a function of time.

Problem formulation
For a mission with N gravity assist manoeuvres, find:
x  [t0 , T1 , T2 ,
to minimise:
, TN 1 ]  I
N 1
f (x)   Vi ( x)
subject to:
i 0
V0 (x)  V0max
Vi (x)  Vi max , i  1,
rp ,i (x)  rpmin
i  1,
,i ,
VN 1 (x)  VNmax
1
Launcher thrust constraint
,N
,N
Thrust constraint at each GA
Periapsis radius constraint at each GA
Braking manoeuvre constraint
Objective Function
The objective function f(x) seeks to
minimise the total thrust / maximise
payload.
 Thrust is measured as instantaneous
changes of velocity provided by the
engine.
 The initial thrust is provided by a
launcher which then separates from
the probe.

Evaluating the objective function
x=[t0, T1,...TN+1]
Ephemeris
routine
(N+2)
r={r0, r1,...rN+1}
Lambert
solver
(N+1)
Gravity
{v1in , v1out , vinN , vout
N }
assist solver
{a1,...a N }
(N)
vN+1
Eccentricity, e
Radius of
periapse, rp
v0
Braking
manoeuvre
{v1,…,vN}
{rp1,...rpN}
vN+1
Objective function and constraints
f(x)
Constraint
violations
Local minima
The number of local
minima grows with
the number of
stages of an MGA
mission.
Local minima located with SQP in the EJS transfer
problem
The presence of a
large number of
local minima calls
for the use of global
optimisation
techniques
Pruning the search space


Previous work has shown that the vast
majority of this search space I corresponds
to infeasible solutions
How can we identify these infeasible
regions and prune them from the search
space?
Overview of Gravity Assist Space
Pruning (GASP) Algorithm







Deterministic algorithm
Relies on efficient grid sampling of the search space
Exploits domain knowledge to effectively constrain
space
User defined constraints on launch energy, gravity
assist thrusts, swingby periapsis radii, and braking
thurst
Provides intuitive visualisation of high dimensional
MGA search spaces
Allows simple identification of solution families
Produces a set of reduced box bounds (between 6
and 9 orders of magnitude smaller than original space)
Example: Earth-Mars transfer
Consider an Earth-Mars transfer
 -1200<t0<600 MJD2000, 25<T1<525 days
 Grid sampled at resolution of 10 days

Effect of launch velocity constraint
V0max  5 km/s
V
max
0
 5 km/s
V0max  10 km/s
Note: Arrival time = Departure time + transfer time (t0+T1)
Adding a Braking Constraint

Adding a braking manoeuvre constraint at Mars of 5
km/s yields only 4% of the search space valid.
Optimising launch windows



Reduced box bounds automatically calculated for each launch window
Each launch window has been optimised separately using Differential Evolution
Different solution families can be examined separately and then the most
appropriate chosen
GASP algorithm

For single interplanetary transfer
Initial velocity constraint
 Braking manoeuvre constraint

Allows simple identification of prospective
departure/arrival windows.
 Significantly reduces the search space.
 How can these ideas be applied to
multiple gravity assists?

Two and more phases…
Infeasible
Arrival Times
Therefore, it must be infeasible to depart
from the next planet on these dates…
Complete GASP Algorithm

Perform sampling as sequence of 2D spaces






Earth departure/Mars arrival
Mars departure/Venus arrival
Apply initial velocity constraint to first phase
Forward constraining through all phases
Apply braking manoeuvre constraint
Backward constraining through all phases

Invalidate arrival dates based on departure dates
Forward constraining


Infeasible arrival date constrains departure from
the planet on that date
Horizontal axis constrains vertical in the next
phase
Phase (k + 1)
Phase k
2000 MJD2K
Invalidate
corresponding
departure date
Invalid
arrival date
1000 MJD2K
Earth departure
1000
2000
Mars departure
Scaling of GASP algorithm
GASP algorithm scales polynomially – this
is due to the characterisation of the search
space as a sequence of connected 2D
search spaces
 This is true both in memory requirements
and computational expense
 Copes well with the curse of dimensionality

Example: EVVEJS Trajectory
Lower bound: 250000 fold
reduction in size of search space
EVVEJS With New Bounds
EVVEJS Optimised
Differential Evolution was applied to the
reduced bounds
 Best solution found was 5225.7m/s
 Launch velocity: 3737m/s
 Probe velocity: 1488 m/s
 A direct transfer requires launch velocity of
approx 10000m/s

Conclusions






Introduced the multiple gravity assist problem
Showed relations to optimal control and gave a
formulation for a specific MGA problem with no deep
space manoeuvres or low thurst arcs.
Have described the Gravity Assist Space Pruning
algorithm (GASP)
Computationally efficient deterministic method for
pruning infeasible solutions with polynomial time and
space complexity
Allows effective visualisation of high dimensional search
space
Identifies launch windows which can be optimised
separately
Acknowledgements



The work presented here comes from a project
commissioned by the European Space Agency
under contract No. 18138, Project Ariadna 4101.
Special thanks to Darren Myatt, Slawek Nasuto
(Reading), Mark Bishop (Goldsmiths), and Dario
Izzo (ESA).
The final report of this project can be
downloaded from:
http://www.esa.int/gsp/ACT/doc/ACT-RPT-03-4101-ARIADNAGlobalOptimisationReading.pdf