Practical Problem Solving On Fast Trajectory Optimization

Astos Solutions

Practical Problem Solving on Fast Trajectory Optimization Senior Lecture on Trajectory Optimization 3rd Astrodynamics Workshop, Oct. 2 2006, ESTEC Astos Solutions GmbH [email protected] www.astos.de

Astos Solutions

Intension

• 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

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Optimization in Retrospect

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

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

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Astos Solutions

Content

• Overview: applications of trajectory optimisation • Requirements for fast and practical optimisation software • Existing Software Solutions • Possible Improvements • Outlook

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Interplanetary

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Aero-Assisted Maneuvers Ascent Formations Rendezvous

Reentry

Constellations Entry Destruction

Typical Aerospace Optimisation Applications

Station Keeping

Orbit Transfer

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Reentry Applications

Astos Solutions

• Reentry – Entry Manoeuvre – Entry trajectory – Minimum possible loads – Reference trajectory for entry guidance – Determination of entry and landing window – Cross-/Downrange computations

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Flight-Path - Planning

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• Special trajectory for ATD flight experiments

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Ascent & Branched Trajectories

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

Astos Solutions

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 • ...

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Astos Solutions

Safety 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 trajectory

EDA Impact Impact with Drag Impact without Drag

stage break-up

demise

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Astos Solutions

Vehicle 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

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System Concept Validation • • • • •

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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

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Low-Thrust Orbit Transfer Mission

Astos Solutions

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

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Low-Thrust Orbit Transfer Mission

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Astos Solutions

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Key points of an optimization 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-dof simulation

Astos Solutions • 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

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Software Requirements • How to handle all these different applications? • A specialized tool for each application • Difficult maintenance • Duplication of code • Learning time

Astos Solutions

• One tool which is – Flexible/Modular • Model definition • Optimisation methods – Complex like the problems – User Guidance System • manageable by nonexpert users – Continuously maintained

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A 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

Astos Solutions

• One tool for atmospheric flight – Launcher – Reentry • Possible extensions – Orbit transfer • Additional perturbations • Various solvers – Gradient methods – Global optimization

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Astos Solutions

ASTOS ® 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

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Product Interfaces 3rd party models

User interface

HTG/EDA

GUI

NASA/CEA NASA/GRAM99 JPL/Ephemeris

ASTOS

Command Line

AeroSpace Trajectory Optimization Software

Model interface Win32 DLL‘s, so-libs

Pre-/Post processing Mathworks Matlab

Ada, C, F77, ..

GESOP Graphical Environment for Simulation and OPtimization

MS Excel Data Import/Export AGI / STK

Astos Solutions

Celestia

Mathworks Simulink Intec / Simpack

OrbiterSim (c) Astos Solutions GmbH

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ASTOS User Interfaces

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Astos Solutions

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Astos Solutions

Optimisation Workflow

User Action

Mission & Model Definition

Software

Vehicle & Mission Requirements

User & Software

Initial Guess Generation Using Control Laws or existing solutions

Control and State Discretization

Change of Mission Requirements

Specification of Constraints and Cost function considering quality of initial guess Optimization Refinement of Constraints, (c) AstosCost Solutions andGmbH Discretization

Yes Converged Result?

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Initial 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 – ...

Astos Solutions

Examples • Launcher Start Sequence – Vertical Take-Off – Pitch Over – Constant Pitch – Gravity Turn – Bi-linear tangent law

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Astos Solutions

Model 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 (c) Astos Solutions GmbH

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User coded models

Astos Solutions

• Provides functions for User programming interface definition and computation for of – Propulsions – Controls (thrust vector) – Aerodynamics – Design Parameter – Vehicle Components – Constraints • provides functions for • Geometry computation of • Engine => max Isp – Forces – Cost functions – Masses – Auxiliary States • as function of • Can be linked to ASTOS as – user defined variables – DLL – ASTOS state vector: tBurn, tcurrent, h, p, ρ, Ma, α, β, – so-lib q, mtotal, a, dynamic viscosity (c) Astos Solutions GmbH

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Astos Solutions

Optimizers of ASTOS

Multiple Shooting Methods

Collocation Methods •

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

•

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

• PROMIS / SLLSQP – integrated solver SLLSQP – 500 parameters • PROMIS / SNOPT • CAMTOS / SNOPT – 3rd party solver SNOPT – hybrid optimizer – 5000 parameters (colloc. & shoot.) – indirect methods – 5000 parameters

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Genetic Algorithm – incl. local search refinement 25

Transcription Methods and NLP Solvers

Astos Solutions

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.

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Astos Solutions

Analysis 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

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Astos Solutions

TSTO Saenger ascent from Istres with branched lower stage return 128 iterat.

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What are the real user wishes? Important Requirements of an Engineer • Accurate? • Fast? • Robust – is most important!

Astos Solutions

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”

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ASTOS - Daily Work

Astos Solutions

Are these requirements applicable to the daily optimisation work?

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Example: 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

Astos Solutions

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

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Concurrent Design

Astos Solutions

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

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Reduced view of trajectory optimization Prop. Sys. Definition

Astos Solutions

Trajectory

Thrust dm/dt

Propell. Type Tank System

3-DoF view

mass Propellant loading

constraint cost

θ,ψ,α,µ

L/D force

Shape design

TPS

Control

Trimming

Aerodynamics (c) Astos Solutions GmbH

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Work methodology

Trajectory: not just a result, connecting part

Mission Level

Astos Solutions

Reduced Level Trajectory

Ma-regime Accelerations Loads Altitudes Overall masses ...

Propulsion

M&S

ATD

GNC

Costs Constraints Objectives ...

Propulsion

M&S

Cycle Program

CAD

ATD

Navier Stokes

Subsystem Level

GNC

Costs

...

Specialist Level (c) Astos Solutions GmbH

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Astos Solutions

Mass 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 kg 7000

L O X /L H 2

6000 5000 4000 3000 2000 1000 kN

0 0

1000

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2000

3000

4000

5000

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Astos Solutions

Trajectory optimization C*(r, p0)

Ce (p0=100bar; r=5)

2500 p0=10

2300

p0=50 p0=100 p0=150

2100 1900

p0=200 p0=300

1700

r

1500 0

5

10

15

20

4750 4700 4650 4600 4550 4500 4450

Optimizable Parameters Ae/At 0

100

200

300

CEA software

400

c = f ( p0 , At , r ) c = f ( p , A , r , Ae ) e 0 t *

Thrust & Massflow

At

t ⋅p ⋅A m& b = r 0* t c

m& = 1.01⋅ m& b

ASTOS

L O X /L H 2

Ae = expansion ratio At

At = throat area

 A  T =  m& b ⋅ ce ⋅η Isp − p a ⋅ At ⋅ e  ⋅ (# engines ) At   kg 7000

p0 = chamber pressure

r = mixture ratio

# engines

t r = throttle factor

6000 5000

Engine Mass: Correlation from existing engines

4000 3000 2000 1000

Aerodynamic Area = f ( At ,

Ae , # engines) At

kN

0 0

1000

2000

3000

4000

5000

Propulsion System Mass

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Exhaust and Characteristic Velocity At Chemical Equilibrium Blue plane = Upper Isp (m/s) limit

Astos Solutions

C* LH2

2450 2400 2350 2300

200

2250 2200 7

150 100 6

50 5 mixture ra tio

Only the surface below the plane provides realistic values with today’s technology (c) Astos Solutions GmbH

4

3

0 c hamber pre s sure

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Practical Aspects of Engine Design

Astos Solutions

• 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

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Summary of Efficient Trajectory Optimisation

Astos Solutions

• 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. (c) Astos Solutions GmbH

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Future of Optimisation

Astos Solutions

• “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 In 10 years every engineer will use optimization software similar to Matlab today • 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 (c) Astos Solutions GmbH

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