Large Scale Optimal Control Problems - [email protected]

Large Scale Optimal Control

Problems

Sven Schäff, Astos Solutions

SADCO Second Industrial Workshop

Stuttgart, February 2nd 2012

Outline

Large scale aerospace optimization problems

Objectives for mission analysis

Brief low-thrust model description

Examples

© 2012 Astos Solutions GmbH SADCO Second Industrial Workshop, Stuttgart, February 2nd 2012 2

Low-Thrust Aerospace Applications


Orbit Transfers

Lunar Transfers

Interplanetary

Transfers

Multi-Revolution

Long Transfers

Multiple Targets

Constraints

Large Scale Problems

Challenge

• 10,000s of optimizable parameters, typically up to 200,000

• 10,000s of constraints

• Constraints: boundary conditions, path constraints

• Cost terms: Mayer costs, Lagrange costs

Complex and challenging optimal control problems with

huge number of optimizable parameters

Solution

• Transforming the optimal control problem into a discrete

NLP using direct method with collocation

GESOP with SOCS


Objectives for Mission Analysis

Determination of initial mission specifications is governed by

Varying levels of sophistication

Perturbations (Earth oblateness effects, perturbational bodies, ...)

Radiation belt modelling, power degradation modelling

Varying propulsion and system configurations

Propulsion components, thruster characteristics

System driven restrictions, e.g. solar cells orientation, recharge cycle

Trade-off aspects

Restrictions on orbit geometry, e.g. sub-synchronous transfers

Changing objectives, e.g. power output, payload, fuel, trip time

Mission constraints

Power management

Geometrical path constraints, e.g. sub-syncronous transfer

Visibility and navigational constraints

Target orbit definition


Objectives for Mission Analysis

Requirements for mission analysis optimization software

• Utilization of low-thrust transfers (continuous thrust)

• Calculate time or fuel optimal transfer trajectories

• Computation of optimal control history

• Allow quick modification/in-/exclusion of boundary and path constraints and cost components

• Time economic and reliable computation of transfer trajectories

• Robust with respect to changing dynamics

• Provide optimal results that can easily be compared

• Relieve user from tuning of optimizer setting


Low-Thrust Tool

Low-Thrust Tool for

Orbit transfers

Moon transfers

Interplanetary transfers

Model

Perturbations (oblateness, 3rd bodies, solar radiation pressure, ...)

Environment (radiation, eclipses, ...)

Operational aspects (slew rates, GEO ring, visibility, ...)


Low-Thrust Tool


Minimum fuel transfer Minimum time transfer Composite objective

Phasing Intermediate orbits

Starting orbit Mission epoch

Perturbations

Solar eclipses

Radiation damage

Constrained slew rates

Managerial

constraints

Optimal Trajectory

Geo constraints

Visibility constraints

Initial Guess Generation

Initial guess generation is based on a straight forward simulation

• Construction using standard control laws

• Generic control history is sufficient to allow steady optimization

• Enhanced performance with more sophisticated initial control

histories

• Use of earlier trajectories of lower-level computations is possible

Benefits:

• Non need to compute abstract adjoint variables

• Non need to newly generate model equations (see indirect/hybrid

methods)

• Pure utilization of physical relations

• Preparation of optimization algorithm is not required (0 minutes)


Don‘t waste time on the initial guess (< 5 minutes)

Scenario

Orbital Life Extension Vehicle

Spiralling from a transfer orbit (GTO) to the geostationary orbit (GEO)

using low-thrust solar electric propulsion

Docking with client spacecraft (telecommunication satellite in GEO)

and taking over attitude and orbit control


Low-Thrust Orbit Transfer


GTO-GEO Transfer Optimization of

• Minimum transfer time

• Minimum fuel consumption

• Minimum degradation

• Pareto optimal solutions

Consideration of

• Initial orbit (e.g. orientation wrt Sun)

• Disturbances

• Slew rates

• Eclipses (bang-bang control)

• Power management

• Phasing with target longitude

Phasing to Target Longitude

Without phasing

With phasing


0 100-1

-0.5

0

0.5

1

Time [days]

Radial Component

0 100-1

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1

Time [days]

Tangential Component

0 100-1

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1

Time [days]

Normal Component

0 100-1

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0

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1

Time [days]

Radial Component

0 100-1

-0.5

0

0.5

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Time [days]

Tangential Component

0 100-1

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Time [days]

Normal Component

Orbit Transfer - Results


Time Optimal Transfer

• Permanent and constant thrust

• Engine shut-down in eclipses

• Super-synchronous transfer

Optimal Solution

Transfer duration 179.0 days

Fuel consumption 165.4 kg

Eclipses 186

GEO ring crossings 7

Collision Risk in GEO Ring - Challenge

Would you like to mitigate the risks of multi-revolution low-

thrust transfers to the geostationary ring?

Definition of GEO box:

± 2 km radial

± 75 km normal


GEO box Terminal position

Collision Risk in GEO Ring - Solution


• Crossings of the GEO ring pose a serious threat with assets

• Solutions

− Sub-synchronous transfer works well, but expensive in time and fuel

− Super-synchronous transfer prevent crossings of the GEO ring

Mars Sample Return


Launch

© ESA

Earth Escape

Earth-Mars (Min Time) Earth-Mars (Min Fuel)

Mars Arrival (Payload Jettison)

© ESA

Mars Stay

© NASA

Rendezvous & Docking

© NASA Mars Escape

Mars-Earth (Min Time) Mars-Earth (Min Fuel)

Earth Re-entry © ESA

Mars Sample Return

Three multi-revolution phases

• Earth escape (up to 1 year)

• Mars capture (> 1,000 revolutions)

• Mars escape (> 1,000 revolutions)

Two interplanetary phases

• Outbound trip

• Inbound trajectory

What‘s next ...

Mission setup in one problem?


Asteroid Double Rendezvous

• Spacecraft visiting two near-Earth asteroids

• Solar electric propulsion

• Thrust level depends on available power


Restrictions in maximal

usable thrustlevel

Thrust peaks used for e.g.

inclination change

Asteroid Double Rendezvous


Usage of low-thrust

model with SOCS

• Modeling of whole mission

in one problem under

consideration of all

mission constraints

Stepwise refinement of

the trajectory under

consideration of

• operational constraints

(e.g. station visibility)

• navigational constraints

(e.g. target visibility)

Jupiter Moons Sample Return

• Sample return mission from one of the Jovian moons

• NEP driven spacecraft

• Hohmann-like low-thrust transfers

• Fuel optimal transfers

• Mission duration: ~10 years


Comet Sample Return


• Sample return mission from comet nucleus

• Solar electric propulsion

• Fuel optimal transfers

• Constrained comet arrival: during perihelion passage

Conclusions

The NLP setup in GESOP/SOCS for large scale optimal control

problems with low-thrust transfers has proven to

• be easily extendable/adaptable to new mission requirements

• be reliable/stable with respect to the sophistication of the dynamics

• produce solutions that are superior to control law applications

For mission analysis the NLP setup offers

• optimality w.r.t. time/fuel/safety, depending on mission focus

• the flexibility to consider a broad set of constraints and effects

• reduced need for trajectory updating


Documents

Large Scale Optimal Control Problems - [email protected]