eNLP: Application-Centric NLP-Based Optimization in the Aerospace Market Sven Erb
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Sven Erb. eNLP: Application-Centric NLP-Based Optimization in the Aerospace Market. SADCO A2CO, Mar 2011, Paris, France. inria-00585604
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Dr. Sven Erb TEC-ECM, ESA ESTEC, The Netherlands
02/03/2011
ESA UNCLASSIFIED – For Official Use Overview
1. Introduction 2. TEC-ECM Role & Responsibilities 3. Optimal Control Application Examples 4. Description of Optimization Architecture to Solve OCP 5. Low-Thrust GTO-GEO Transfer Optimization 6. GTO-GEO Practical Consideration 7. New European NLP Solver eNLP 8. Conclusions
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 2
ESA UNCLASSIFIED – For Official Use Decyphering the Title
eNLP: Application-Centric NLP-Based Optimization in the Aerospace Market
Optimization is an analysis method to assess the performance of a set of parameters in a model with respect to a precisely defined objective, subject to a set of constraints (occasionally the set has dimension zero).
The work context for a trajectory and performance engineers is application-centric. we have a very Application-Centric. The key concern is to understand the model and best reflect all relevant aspects of the real life application.
Aerospace Market: Reflects the fact that we are mostly interested in Aerospace applications and gives indication of the commercial aspects of our activities.
The primary means of optimization that is used at TEC-ECM is Nonlinear Programming, hence, NLP-Based.
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 3
ESA UNCLASSIFIED – For Official Use Control Systems Division Organisation
Control Systems Division TEC-EC Alain BENOIT
Secretary: Monique DANIEL
Control Systems and Dynamics & Mathematical Guidance, Navigation, and Sensors Section Analysis Unit Control Section TEC-ECC TEC-ECM TEC-ECN Roger JANSSON Guillermo ORTEGA acting Guillermo ORTEGA
TEC-ECM belongs to the Directorate for Technology and Quality Mangement D/TEC of the European Space Agency ESA and is located on the ESTEC campus in The Netherlands.
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 4
ESA UNCLASSIFIED – For Official Use TEC-EC Organisation & Interfaces
Earth TEC-ECC TEC-ECN Observation Human Control Systems & Sensors Section Navigation Guidance & Control Section Spaceflight Definition and Implementation of Control Systems Definition and Implementation of GNC Systems for: Telecom for: - Planetary Exploration Vehicles - Earth Observation Satellites - Launch Vehicles Launchers - Navigation and Telecom Satellites - Formation Flying Vehicles Galileo - Astronomy Observatories
Science and Science and Robotic Robotic Exploration Exploration Technology R/D for: Technology R/D for: - EO, Telecom, Science AOCS and associated - Entry, Descent, Landing Systems FDIR - Re-entry vehicles Technology - High accuracy pointing - RendezVous and Docking Programmes - Image Navigation & Registration - Formation Flying Technology TRP, GSTP, - Antenna Pointing systems - Optical-based navigation Programmes ARTES, EOEP TRP, GSTP, Transverse Divisional R/D for: Transverse Divisional R/D for: ETP, FLPP - Attitude sensors, inertial actuators - Advanced control & estimation Control Hardware Lab (sensors) Control HW Lab (closed loop testing)
Trajectory and orbital mechanics, associated astrodynamics tools Trajectory optimisation, Launcher/satellite performance, Coordination Specialised system analysis TEC-ECM Astrodynamics and Math Analysis Unit Trajectory optimization, astrodynamics tools and techniques, specific mathematical models
ESOC Technology Programme Directorates Programmes
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 5
ESA UNCLASSIFIED – For Official Use TEC-ECM - Mission Arcs
Ascent
Entry
Descent and Landing Atmospheric Flight Aerocapture, Aerobraking
Aerogravity assist
Flight under Parachute, parafoil
Rendezvous and Docking Station-Keeping / Collocation Non-Atmospheric Flight Formation Flying Low Thrust Transfers
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 6
6 Selected Applications
Launch • Maximum P/L performance, • Optimal control under constraints: lift-off conditions, Vehicle pad clearance, azimuth, heat flux, dynamic pressure, axial acceleration, ground station visibility, separation conditions, stage drop zones, • Minimum GLOW, design optimization, engine sizing, • Look-up tables with aero data, etc. Reentry • Maximum safety for vehicle, population, infrastructure, Vehicle • Optimal control under constraints: Heat-flux, dynamic pressure, deceleration, visibility, impact/drop/landing point, • Optimization of TPS, L/D, design, • Completely different regime compared to Launcher. Satellite • Minimize transfer time, • Optimize the thruster control: thrust direction and transfer / thrust on-off, station- • Nonlinear environment with challenging ratio of thrust level vs. perturbations, keeping • Constraints: “the standard”, thruster tech characteristics, sun orientation, GEO crossing, • Minimize fuel-consumption for restricted final time
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 7
ESA UNCLASSIFIED – For Official Use Summing it Up
1. Trajectory optimization problems with Optimal Control properties! 2. Context somewhat similar, 3. Particular applications differ quite substantially with respect to: a. type of suitable equations of motion, b. applicable set of constraints, c. type of objective. 4. Problem formulations are generally nonlinear, 5. Apart from the optimizable control, there is a set of design parameters that is also optimizable.
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 8
ESA UNCLASSIFIED – For Official Use Discretize then Optimize
Describe the Optimal Control Problem with its true physical characteristics Best for the application expert Discretize the OCP Mostly automatable
Transcribe the OCP into a simple parameter optimization problem Flexible / Adaptable
Solve the problem using an NLP algorithm ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 9
9 Discretize the States, Controls, Constraints
Shooting Method Collocation Method
Monitor the ODE tolerance, Refine the grid,
ESA PresentationGIGO: |Garbage Dr. Sven Erb In| 02/03/2011 – Garbage | Slide 10Out.
10 Nonlinear Programming Problem
Nonlinear Program: n Minimize the cost function f (x), x subject to equality constraints h(x) 0, hme and inequality constraints. g(x) 0, g (mme )
Lagrange-Function: L(x,, ) f (x) T g(x) T h(x)
Optimal solution: (y* )T (x* )T (* )T ( * )T which satisfies the Karush-Kuhn-Tucker conditions g(x* ) 0 h(x* ) 0 *g(x* ) 0 * 0 * * * * T * T * Lx (x , , ) f x (x ) g x (x ) hx (x ) 0 ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 11 11 General Architecture
Launch Case Reentry Case Transfer / Station-Keeping
NLP GESOP Solver ASTOS
Application Solution Expert to OCP
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 12
ESA UNCLASSIFIED – For Official Use EP Transfer as Part of the Business Case
OLEV – Orbital Life Extension Vehicle: 1. Target is to provide commercially viable life extension services for fuel depleted, operational GEO telecom satellites, 2. OLEV docks with client satellite and takes over station-keeping tasks from the client.
ConeXpress2: 1. ConeXpress will be a smaller-than-small platform for GEO applications
CX2 (Courtesy Dutchspace)
Launch costs are lowered by flying as additional payload to GTO, Transfer from GTO to GEO, repositioning and station-keeping is performed with low- thrust electric propulsion.
Docked OLEV ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 13 (Courtesy Kayser- Threde) GTO-GEO Control Law Strategies
z 1. Combinations of simple control laws, ih ir ith 2. Superposed or consecutive, 3. With coast arcs/thrust arcs, r y x Strategy suggested by J.E. Pollard for GTO-GEO: vernal 1. Phase: Primarily increase semi major axis a to final value equinox
ur 0
uth cos
uh sin 2. Phase: Decrease eccentricity and inclination to zero; a = const. : Ecc. anom. arc switch. cond. cos 1 e2 sinE : Out - of - plane angle u r 1 ecosE E : Eccentric anomaly (i i )(3 cos sin) cos cosE e tan 2 1 e : Eccentricity e 1 e 1 uth 2cos sin ln 2 1 e e i : Inclination 1 ecos E 2 1 e2 1 e1 1 : Argument of perigee uh sin
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 14 Spacecraft and Mission Model Description for GTO-GEO Transfer
Wet mass BOL: 1335.7 kg
Thrust data: Constant thrust 2 * 0.0891 N = 0.1782 N Fuel flow rate 2 * 5.4733 E-6 = 10.9466 E-6 kg/s
Launch Orbit: Inclination 6 deg Perigee height 250 km Apogee height 35,943 km Argument of perigee 178 deg
Disturbances: Earth gravity field including J2 yes Gravitation of Sun and Moon yes Solar radiation fixed Sun radiation, 1388 W/m2 Drag no Gravity gradient no Magnetic no
Target Orbit: Eccentricity 0.0 Semi major axis 41,810 km (GEO radius – 1 day of spiral thrust increase: 354.2 km) Inclination 0.0 deg
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 15 Nonlinear Programming Based Optimization
1. NLP can solve long EP transfer trajectory optimization problems, 2. Order of 100,000 parameters and constraints, 3. First discretize, then optimize, 4. Complex model with EP patterns, perturbing accelerations, eclipses can be considered, Result is an optimal thrust vector history with huge degree of parameterization.
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 16 Growing the Complexity
1. Optimize control in order to minimize transfer time (= fuel, in case of permanent thrust),
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 17
ESA UNCLASSIFIED – For Official Use Growing the Complexity
2. Optimize control in order to perform a time optimal sub-synchronous transfer,
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 18
ESA UNCLASSIFIED – For Official Use Growing the Complexity
3. Optimize control in order to minimize radiation exposure in Van Allen belt, 4. Optimize control, taking account of eclipses and power cycling, 5. Optimal control for a fuel optimal transfer with time limit (thrust on-off),
Transfer Costs & Benefits for Arrival on 01-Mar-2009
200
190
180
170
160
150 Subsynchronous Fuel Consumption [kg] Overshooting 140
1-Jul-2008 1-Jun-2008 11-Jul-200821-Jul-200831-Jul-2008 12-May-200822-May-2008 11-Jun-200821-Jun-2008 10-Aug-200820-Aug-200830-Aug-2008 Launch Date
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 19
ESA UNCLASSIFIED – For Official Use Growing the Complexity
6. Optimize control in order to avoid intermediate crossing through the GEO belt.
Graph shows a time optimal transfer trajectory in a rotating frame; the red box marks GEO +-75km, green + red mark GEO +-150km; every crossing of the blue trajectory through the boxes marks a GEO crossing.
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 20
ESA UNCLASSIFIED – For Official Use Growing the Complexity
7. Optimize control so that it complies with attitude constraints and can be uplinked to the spacecraft.
• Ensure sun-pointing of solar array for power generation, • Limit slew rates of spacecraft to comply with GNC specs., • Optimized manoeuvre plan / attitude history needs to be parameterized and uplinked to spacecraft, • Amount of parameters needs to comply with data rate of communication system and duration of ground contacts.
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 21
ESA UNCLASSIFIED – For Official Use Quaternion History
0.6 q1 q2 1. Source data contains 250 revolutions 0.4 q3 q4 for transfer with 360 nodes each, 0.2 0
2. The NLP solution needs to be treated Quaternion -0.2 further to generate attitude profiles -0.4 for satellite control, -0.6 -0.8 0 200 400 600 800 1000 3. 3-element thrust vector history is No. of Node converted into 4-element quaternion profile,
0.8 q1 4. Requirement on solar array pointing 0.6 q2 q3 0.4 is superimposed in order to secure q4 sufficient power budget, 0.2 0
Quaternion -0.2 Graphs show the quaternion history over -0.4 -0.6
-0.8 three revolutions during: 4.82 4.84 4.86 4.88 4.9 4.92 4.94 4.96 4.98 No. of Node 4 a. Early transfer, x 10
0.2 b. Close to mid-point of transfer, q1 q2 0 q3 c. Towards end of transfer, q4 -0.2
-0.4 Quaternion
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 22 -0.6
-0.8 8.57 8.58 8.59 8.6 8.61 8.62 8.63 8.64 8.65 8.66 No. of Node 4 x 10 Optimal Polynomial Fit
1. Apply least squares method to optimally fit the Chebyshev polynomial to the entire attitude history of the quaternions, 2. Segment the entire history suitably and choose degree of Chebyshev polynomials in order to minimize the required number of coefficients while achieving a maximum error that is below the performance requirement, 3. Empirical study shows that CP of degree 8 is an efficient compromise between increased number of coefficients and increased number of segments to improve the approximation.
Note: Naturally, the true anomaly is the better independent variable.
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 23
23 Chebyshev Polynomials
T (x) 1 1. Sequence of orthogonal polynomials, 0 2. Chebyshev Polynomials (CP) of first kind: T1 (x) x 2 a. Degree 0: T0(x)=1 T2 (x) 2x 1 b. Degree 1: T (x)=x 3 1 T3 (x) 4x 3x c. Degree n: Tn+1(x)=2x Tn(x)- Tn- 4 2 T4 (x) 8x 8x 1 1(x) 5 3 d. x is the normalized independent T5 (x) 16x 20x 5x variable, x [-1, 1], 6 4 2 T6 (x) 32x 48x 18x 1 7 5 3 T7 (x) 64x 112x 56x 7x 8 6 4 2 T8 (x) 128x 256x 160x 32x 1 9 7 5 3 T9 (x) 256x 576x 432x 120x 9x
• Optimum polynomial fit by use of straight forward least squares method, • Ensure continuity and smoothness across segment bounds. ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 24 Example: Results for 1483 Segments: q3 error
1. Error quaternion (residual rotation) as measure for the quality of the
polynomial fit: Qerror(t) = Qcp(t)’ * Qinput(t),
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 25
25 Refinement of Segmentation
1613 segments X= q1error X= q2error X= q3error X= q4error Mean(abs(x)) 0.573e-4 1.89e-4 1.336e-4 0.965e-4 s(x) 0.00041 0.00033 0.00042 0.00064 abs(x) > 0.05 0.0033 % null % 0.0078 % 0.0089 % abs(x) > 0.01 0.13 % 0.04 % 0.17 % 0.04 % abs(x) > 0.001 3.7 % 17.7 % 9.9 % 6.45 %
• Spacecraft autonomy period, communication link and on-board memory size need to be dimensioned to be 200 able to handle transfer precision requirements,
• Figure depicts how many subsequent segments are 150 required to approximate the quaternion profiles for any
upcoming 7 day period, when using the Chebyshev set 100 with 1613 segmentations , • Maximum of 166 segments, 50 • CP degree 8 => 9 parameters, • 6308 4-bit parameters needed (memory requirements), 0 0 200 400 600 800 1000 1200 1400 1600 1800 • Better to treat/smoothen input data, than to improve Total Segments od 7 toDay Period Cover methodology for polynomial fit. No. of Segment
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 26
26 eNLP: Application-Centric NLP- Based Optimization in the Aerospace Market
Dr. Sven Erb TEC-ECM, ESA ESTEC, The Netherlands
02/03/2011
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 27
ESA UNCLASSIFIED – For Official Use Motivation for the Development of a New NLP Solver
• Most existing solvers are sandboxes for academic research, • Maintenance, bug-fixes, enhancement, licensing depend primarily on academic interests and/or passions, • Industrial grade software products are very scarce (coding standards, documentation, support, verification status), • NLP solver performance characteristics are driven by particular interests and not to provide a complete, flexible, state-of-the-art optimization environment, • Most NLP solvers have ONE state-of-the-art solution strategy implemented (with variations), • Development competition is driven by Math Problem Library performance, rather than “real life” engineering problems, • Many solvers do not achieve sufficient optimality performance when it comes to large, sparse problems.
ESA proposed the development of a new NLP solver that meets industrial grade requirements.
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 28
28 The eNLP Team
Dept. Mathematics Univ. of Bremen eNLP
Dept. Mathematics Univ. Birmingham Dept. Mathematics Univ. Coimbra ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 29
29 Some Key Features
1. Modular architecture,eNLP 2. Reverse communication, 3. Hessian 4update Genera lschemes, Architecture
4. Globalization4.1 The schemes, Concept of Modularity 5. Failure output.
Figure 4-1: Modularity
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 30 Benchmark Tests with CUTEr Problem Library
As of Spring 2010 (old): Some very recent numbers: 1. CUTEr tests for eNLP project are run via AMPL (918 test cases available, http://www.princeton.edu/~rvdb), 2. Fully automated process; problems solved with various high-performance benchmark solvers: 3. With 96.1% WORHP-eNLP is the top performer (versatility), 4. Not leading in computation speed yet, but similar to KNITRO, IPOPT, 5. WORHP-eNLP shows superior capabilities for large problems, 6. Further, WORHP-eNLP has been tested and applied to aerspace application cases including interplanetary trajectories.
100.00% 90.00% Latest confirmed record 80.00% (mccormck, 256 GB memory): 70.00%
60.00% > 400,000,000 variables with 50.00% > 800,000,000 constraints 40.00% Solved!
30.00% Problems solved [%] solved Problems 20.00%
10.00%
0.00% ESA PresentationWORHP SNOPT | Dr. filterSQPSven ErbKNITRO | 02/03/2011ipfilter ipopt| Slide LOQO31 MINOS Lancelot Conclusions
1. TEC-ECM is making excellent experience with the Direct Transcription approach for solving Optimal Control Problems in a versatile and application driven environment, 2. TEC-ECM keeps expanding the application range in its domain, 3. Development of new NLP solver in Europe provides performance boost, extension of capabilities and added-value for users. 4. WORHP-eNLP is available as Stand-Alone solver and as NLP solver for ASTOS. 5. The eNLP team setup is geared towards commercial/industrial as well as academic use, 6. WORHP-eNLP has taken the lead when it comes to optimality performance, 7. Current efforts are geared towards pushing WORHP-eNLP higher up in the top tier when it come to computation speed and efficiency (Hessian update, Filter, linear algebra, memory management, etc.). 8. www.worhp.de - “We Optimize Really Huge Problems”
ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 32
32 ESA Presentation | Dr. Sven Erb | 02/03/2011 | Slide 33