Practical problem solving on fast trajectory optimization

AstosSolutions

Practical Problem Solvingon Fast Trajectory

Optimization

Senior Lecture on Trajectory Optimization3rd Astrodynamics Workshop, Oct. 2 2006, ESTEC

Astos Solutions [email protected]

www.astos.de

(c) Astos Solutions GmbH 2

AstosSolutionsIntension

• What can be done with optimization?• What means PRACTICAL?

– What dominates the optimization work?• CPU time• Operator time

• What means FAST?– Using state of the art technology and hardware– CPU-time is defined by computational accuracy

and model complexity


AstosSolutionsOptimization in Retrospect

• 1996: – Straight forward optimization

– 500 optimizable parameter– CPU critical

• 2006: – up to 150,000 parameter and more– Trajectory optimization and vehicle design

optimization in parallel– Low Thrust problems– Not CPU critical

„Optimization with more than several dozen of parameters makes no sense.“

„Don‘t waste time on the initial guess.“


AstosSolutionsContent

• Overview: applications of trajectory optimisation• Requirements for fast and practical optimisation

software• Existing Software Solutions• Possible Improvements• Outlook


AstosSolutions

Ascent

EntryDestruction

Reentry

Constellations

Rendezvous

Orbit Transfer

Formations

Interplanetary

Aero-AssistedManeuvers

Station Keeping

Typical AerospaceOptimisationApplications

Libration PointMissions


AstosSolutionsReentry Applications

• Reentry– Entry Manoeuvre– Entry trajectory

– Minimum possible loads– Reference trajectory for

entry guidance

– Determination of entry and landing window

– Cross-/Downrange computations


AstosSolutionsFlight-Path - Planning

• Special trajectory for ATD flight experiments


AstosSolutionsAscent & Branched Trajectories

Performance Indices• maximize payload• minimize fuel consumption• minimize structural mass

Boundary Conditions:• initial conditions (launch pad)• target orbit• return of rocket stages• staging conditions• visibility from ground stations• splash down of stages• ...Path Constraints:• max. dynamic pressure• max. heat-flux

• bending moment (qα)• max. acceleration• constraints on flight path• ...


AstosSolutionsSafety Analysis

• Entry destruction analysis of upper stages (ASTOS-EDA)• Trajectory modifications to ensure safe impact points in case of an failure• Ballistic coefficients analysis

• Abort trajectory scenarios• Collision avoidance

during low-thrust flight

main main trajectorytrajectory

stage stage breakbreak--upup

demisedemise

EDA Impact

Impact with Drag

Impact without Drag


AstosSolutionsVehicle Design

• trajectory and vehicle parameter optimization– structural masses of stages– tanks– engine parameters at chemical equilibrium– Considering constraints (loads, safety)– Shape optimization

• performance assessment of upper stage modifications• Examples of design studies

– Mars ascent vehicle (MAV) – Heavy Lift Launch Vehicle (HLLV)– VEGA: upper stage with low thrust


AstosSolutionsSystem Concept Validation

• Design reviews• Nominal vs. non-nominal performance

• Sensitivity analysis• Adjustment of mission parameters

• Investigation of alternate stages of a launcher– different engine performance vs payload– Different tank design

– LOX vs. Kerosene


AstosSolutionsLow-Thrust Orbit Transfer Mission

GTO-GEO transfer• Optimization of

– Minimum transfer time– Minimum fuel consumption

– Minimum degradation

– Pareto optimal solutions

– Consideration of– Disturbances– Eclipses– Battery power– Phasing with target longitude – Slew rates


AstosSolutionsLow-Thrust Orbit Transfer Mission


AstosSolutions

Key points of anoptimization model

• Point mass• No moments• Attitude control

– can be considered as commanded control

– Only two controls(side slip angle = 0)

– Optimised attitude controls allows to integrate the flight-path, but does not ensure, that this trajectory is flyable or useful for 6-dofsimulation

• 6-dof attitude control with inverse dynamics provides – 3 attitude controls– Required control torque ⇒ Additional constraints

• No geometry unless used for computation of– Forces– Volume of tanks– Diameter of nozzles and

stage


AstosSolutionsSoftware Requirements

• How to handle all these different applications?

• A specialized tool for each application

• Difficult maintenance

• Duplication of code• Learning time

• One tool which is – Flexible/Modular

• Model definition• Optimisation

methods

– Complex like the problems

– User Guidance System

• manageable by non-expert users

– Continuously maintained


AstosSolutionsA Single Tool Solution

Common properties• EoM of one body• Central body• Cost functions related to

– Time– Mass– Other typical

astrodynamic values• Constraints

– Position– Velocity – Acceleration– Forces

• One tool for atmospheric flight– Launcher– Reentry

• Possible extensions– Orbit transfer

• Additional perturbations

• Various solvers– Gradient methods– Global optimization


AstosSolutionsASTOS®

AeroSpace Trajectory Optimization Software• Completely data configurable (frequent changes in model data)

• Easy, intuitive Graphical User Interface

• Various optimization techniques

• Easy generation of Initial Guess

• Automatic scaling techniques

• Handles flat minima

• Large convergence radius

• Robust w.r.t. “bad models”

• Handles linear data interpolation

• Data visualization


AstosSolutionsProduct Interfaces

GESOPGraphical Environment

for Simulation and OPtimization

ASTOSAeroSpace TrajectoryOptimization Software

GUI

Command Line

Win32 DLL‘s, so-libsAda, C, F77, ..

MathworksSimulink

Intec / Simpack

NASA/CEA

NASA/GRAM99

JPL/Ephemeris

HTG/EDA3rd party models User interface

Model interface

Pre-/Post processingMathworks

Matlab

MS Excel

AGI / STK Celestia OrbiterSim

DataImport/Export


AstosSolutionsASTOS User Interfaces


AstosSolutionsOptimisation Workflow

Vehicle &

Mission R

equirements

Initial Guess GenerationUsing Control Laws or existing solutions

Specification of Constraintsand Cost function considering

quality of initial guess

Mission & Model Definition

Control and State Discretization

Optimization

Refinement of Constraints, Cost and Discretization

Change of Mission Requirements

Converged Result?

Yes

User Action

Software

User & Software


AstosSolutionsInitial Guess with Control Laws

Obtain Initial Guess from• Existing state/control history• Global optimisation• Control Laws for

Attitude Controls– Constant or Linear Law– Profile as Function

of Time or Machnumber– Vertical Take Off– Gravity Turn– Required Velocity– Target Orbit– Bi-Linear Tangent Law– Dynamic Pressure Controllers

for ascent and decent– Constant Turnrate– ...

Examples• Launcher Start Sequence

– Vertical Take-Off– Pitch Over– Constant Pitch– Gravity Turn– Bi-linear tangent law


AstosSolutionsModel Library

• Heart of application• Object oriented design to ensure

– Flexibility, how to transcribe a subcomponent by a coded model object.

– Maintainability• capsulated code• easy to extend

• Fully data driven approach increases reliability– no coding of developer/user to change the

problem• Becomes expandable due to user programming

interface


AstosSolutionsUser coded models

User programming interface for– Propulsions– Aerodynamics– Vehicle Components

• provides functions for computation of– Forces– Masses

• as function of– user defined variables – ASTOS state vector:

tBurn, tcurrent, h, p, ρ, Ma, α, β, q, mtotal, a, dynamic viscosity

• Provides functions for definition and computation of– Controls (thrust vector)– Design Parameter– Constraints

• Geometry• Engine => max Isp

– Cost functions– Auxiliary States

• Can be linked to ASTOS as– DLL– so-lib


AstosSolutionsOptimizers of ASTOS

Collocation Methods

• TROPIC / SNOPT– 3rd party solver SNOPT– 5000 parameters

• SOCS– automatic mesh refinement– sparse solver 150,000

parameters

Multiple Shooting Methods

• PROMIS / SLLSQP– integrated solver SLLSQP– 500 parameters

• PROMIS / SNOPT– 3rd party solver SNOPT– 5000 parameters

• CAMTOS / SNOPT– hybrid optimizer

(colloc. & shoot.)– indirect methods– 5000 parameters

Genetic Algorithm– incl. local search

refinement


AstosSolutions

Transcription Methods and NLP Solvers

Situation

• Transcription methods like collocation and multiple shooting have achieved a technical sophisticated level.

• Sparse NLP solver can solve large problems in acceptable time.

• The CPU time is comparable with operator time.


AstosSolutionsAnalysis Methods

• NLP- Solver output– Constraint violation– Merit function– DoF– Step Size– …

• Review Iteration Monitor– Graphical – History for each iteration

of• NLP status • Optimizable

parameters and constraints

• Additional Optimiser Output– Gradient check

• Additional Optimiser Functions– Automatic Mesh

Refinement• But at the end the operator

has to – analyse the complex

output– bring it in relation to the

real problem– Know how to influence

the behaviour of the optimiser


AstosSolutions

TSTO Saenger ascent from Istres with branched lower stage return

128 iterat.


AstosSolutionsWhat are the real user wishes?

Important Requirements of an Engineer

• Accurate?• Fast?• Robust –

is most important!

How can robustness be improved

• Reduction of operator time– Start from

• bad initial guess• infeasible point

– Robust w.r.t. “bad models”

– Support in case of problems

• Reduction of complex know how:“Current point cannot be improved”


AstosSolutionsASTOS - Daily Work

Are these requirements applicable to the daily

optimisation work?


AstosSolutionsExample: Pre-Phase A Study

Launcher Design• Nominal trajectory• Sensitivity analysis

– Engine performance: Isp, thrust

– Structural index• Different payload orbits• Different propellants• Different strategy for

jettisoning the fairing• Different strategy for splash

down of upper stages• Consideration of additional

coast arcs

Requirements• Simple modification of

mission and model definition• restart of optimization based

on old result• Capability to modify phase

structure and used EOM and controls– Because of new mission

requirements– To avoid singularities in

case of changed mission

=> fast over all process


AstosSolutionsConcurrent Design

1. Subsystem mass correlation– No CAD models available, too complex – Fast model, accurate enough within margins

2. Design of Propulsion System (full stage design of launcher)– Thrust and mass flow shall be optimizable– Both values are coupled by chem./physical laws– Complete cycle computation too complex


AstosSolutions

Reduced view of trajectory optimization

3-DoFview

Prop. Sys. Definition

Propell. TypeTank System

ControlPropellantloading

Shape design

TPS Aerodynamics

Trimming

Trajectory

mass

Thrustdm/dt

L/D force

constraintcost

θ,ψ,α,µ


AstosSolutions

Subsystem Level Specialist Level

Work methodology

Propulsion M&S ATD GNC Costs

Mission Level

CycleProgram CAD Navier

Stokes ...

Ma-regimeAccelerations

LoadsAltitudes

Overall masses...

ConstraintsObjectives

...

Propulsion M&S ATD GNC Costs

TrajectoryReduced Level

Trajectory: not just a result, connecting part


AstosSolutionsMass Correlation

• Important criterion: mdry/mpropellant

• Splitting of dry mass into subcomponents which depends on variable quantities:– Tank mass

(propellant mass)– Shell mass (shell area)– Truss mass (over all)– ...– Constant masses

• Definition of analytical relationship

• Definition of correlation factors using linear, quadratic, exponential or logarithmic inter-or extrapolation

L O X /L H 2

0

1 0 0 0

2 0 0 0

3 0 0 0

4 0 0 0

5 0 0 0

6 0 0 0

7 0 0 0

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

k g

k N


AstosSolutions

Trajectory optimization

Propulsion System Mass Aerodynamic Ref Area

),,,( 0t

ete A

ArApfc =),,( 0

* rApfc t=

ASTOS

)(# enginesAA

ApcmTt

etaIspeb ⋅

⋅⋅−⋅⋅= η&

)#,,( enginesAA

AfAreacAerodynamit

et=

L O X /L H 2

0

1 0 0 0

2 0 0 0

3 0 0 0

4 0 0 0

5 0 0 0

6 0 0 0

7 0 0 0

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

k g

k N

Engine Mass:Correlation

from existing engines

Optimizable Parameterspressurechamberp =0

ratioansionexpAA

t

e =

areathroatAt =

engines#

factorthrottletr =

ratiomixturer =

Ce (p0=100bar; r=5)

4450450045504600465047004750

0 100 200 300 400

Ae/At

CEAsoftware

C*(r, p0)

1500

1700

1900

2100

2300

2500

0 5 10 15 20

r

p0=10p0=50p0=100p0=150p0=200p0=300

bmm && ⋅= 01.1

Th

rust

& M

assf

low

*0

cApt

m trb

⋅⋅=&


AstosSolutions

Exhaust and Characteristic Velocity At Chemical Equilibrium

Blue plane = Upper Isp (m/s) limit

Only the surface below the plane provides realistic values with today’s technology

0

50

100

150

200

345

67

2200

2250

2300

2350

2400

2450

c hamber pressure

C* LH2

mixture ra tio


AstosSolutions

Practical Aspects of Engine Design

• System engineer can specify the bounds of parameters and the characteristics of the reduced model

• Engine throttling can be defined depending on the used model.• Mixture Ratio can be considered as

– constant– Optimizable but constant with switching point(s), which are

optimizable– Time variable and optimizable (control)– Model can be easily changed

• Propellant loading of oxidizer and fuel tank is automatically adjusted considering mixture ratio and throttling

• Changing tank masses can be considered using mass correlation


AstosSolutions

Summary of Efficient Trajectory Optimisation

• Interchangeability of data input– Data handling– Exchangeability between software tools of different domains

• Optimization• Subsystem Calculation • GNC design• Visualization

– Version management of data driven model objects

• Improvements of numerical methods– Numerical code more tailored to the requirements of an

engineer– Not pure CPU time is decisive factor but net process time of

operator.


AstosSolutionsFuture of Optimisation

• “Black Box” Optimiser• User is engineer not “mathematician”• He needs to understand physical background of his

problem, but not the difficult background of optimisation methods

• Difficulties during the optimization run will be solved automatically or by intelligent support, where an error is transcribed into the physical meaning of the problem.

=> As important as faster solvers and faster CPUs

In 10 years every engineer will use optimization software similar to Matlab today

Practical problem solving on fast trajectory optimization

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