Lunar Landing Trajectory Optimization with SOCS
Contents
1. 1 descent interface altitude 10 000 meters 100 200 300 400 simulation time seconds Lunar Landing Trajectory Optimization Maximize final spacecraft mass initial circular orbit altitude 100 km descent interface flight path angle 1 descent interface altitude 10 000 meters 100 200 300 400 simulation time seconds page 8 The final two plots illustrate the behavior of the thrust and thrust steering angle during the descent Lunar Landing Trajectory Optimization Maximize final spacecraft mass 6000 initial circular orbit altitude 100 km descent interface flight path angle 3 10 a descent interface altitude 10 000 meters 5000 4000 3000 thrust newtons 2000 1000 0 0 100 200 300 400 simulation time seconds Lunar Landing Trajectory Optimization 80 Maximize final spacecraft mass initial circular orbit altitude 100 km descent interface flight path angle 1 descent interface altitude 10 000 meters thrust angle degrees 20 0 100 200 300 400 simulation time seconds page 9 Problem setup for SOCS This section provides additional details about the SOCS software implementation It briefly explains such things as path constraints and the performance index options 1 Performance index For the maximize final spacecraft mass optimization the performance index is simply J m where m is the spacecraft mass at t2. algorithm control options and parameters for the SOCS software KKEKKKKKKKKKKKKKKKKK KKK KKK KKK algorithm control parameters KKKKKKKKKKKKKKKKKKKKKKKKKKKEK This input defines the relative error in the objective function relative error in the objective function performance index 1 0d 5 The next input defines the relative error in the solution of the differential equations relative error in the solution of the differential equations 1 0d 7 The next input is an integer that defines the maximum number of mesh refinement iterations maximum number of mesh refinement iterations 20 The next input is an integer that defines the maximum number of function evaluations page 4 maximum number of function evaluations 100000 The next input is an integer that defines the maximum number of algorithm iterations maximum number of algorithm iterations 10000 The level of output from the SOCS NLP algorithm is controlled with the following integer input KKK KKK KKK KK KK KKK KKK KKK KKK KK sparse NLP iteration output KEK KKK KKK KKKKKKKKKK KKK KK KKK 1 none 2 terse 3 standard 4 interpretiv 5 diagnostic 2 The level of output from the SOCS optimal control algorithm is controlled with the following integer input Please note that option 4 will create lots of information KKK KK KKK KKK KKKKK KKK KKK optimal control output KKEKKKKKKKKKKK KKK KKK KKK 1 none 2 terse 3 standard 4 interp
3. equations of motion relative to a non rotating spherical Moon with the propulsive thrust aligned opposite to the direction of motion are as follows altitude dh h Vsin di 4 speed pa _ Tcosa po dt m ad flight path angle d V T si dy au sing gcosy dt r mV V page 11 propellant flow rate dm T m dt gl a where h altitude V speed flight path angle a thrust angle g gravity uf r gravitational constant of the moon r selenocentric radius of the spacecraft T propulsive thrust m spacecraft mass g Earth surface gravity I specific impulse The thrust angle is defined with respect to the velocity of the vehicle It is similar to the angle of attack for vehicles flying within an atmosphere It is measured positive above the velocity and negative below References and Bibliography 1 Direct Trajectory Optimization Using Nonlinear Programming and Collocation C R Hargraves and S W Paris AIAA Journal of Guidance Control and Dynamics Vol 10 No 4 July August 1987 pp 338 342 2 Optimal Finite Thrust Spacecraft Trajectories Using Direct Transcription and Nonlinear Programming Paul J Enright Ph D Thesis University of Illinois at Urbana Champaign 1991 3 Improved Collocation Methods with Application to Direct Trajectory Optimization Albert L Herman Ph D Thesis University of Illinois at Urbana Champaign 1995 4 S
4. specifies the type of collocation method to use during the solution process For most simulations the trapezoidal method is recommended KIKKIKKAKKKKKAKAAKAKKAKKKKKAKAKKA discretization collocation method KIKIKKAIKKKKAKAAKAKKAKAAKAKAAKAK KA 1 trapezoidal 2 separated Hermite Simpson page 3 3 compressed Hermite Simpson 4 Runge Kutta 4 stage The next integer defines the number of initial grid points to use in the collocation modeling of the descent trajectory number of initial guess grid points to use 25 The software also creates a comma separated variable csv ascii data file that contains the optimal control solution and other flight parameters The name of this output file is specified in the next line of information Please consult Appendix B for information about the contents of this file name of comma delimited solution data file lunlandl csv This next input specifies the type of solution data file to create KKK KKK KKK KKK KEK KKK KKK KKK KK KKK KKK KKK KK KKK KKK type of comma delimited solution data file KKK KKK KKK KKK KKK KK KKK KKKKKKKKKKKKK KKK KK KKK 1 SocS defined nodes 2 user defined nodes 3 user defined step siz For options 2 or 3 this input defines either the number of data points or the time step size of the data output in the solution file number of user defined nodes or print step size in solution data file 10 0 The next series of program inputs are
5. 000000000000 1 00000000000003 90 0000000000000 555 640683701348 444 359316298652 1729 04078952476 degrees meters meters second degrees kilograms seconds meters meters second degrees kilograms kilograms meters second The following are trajectory plots created from the user defined summary file The first two plots illustrate the behavior of the altitude and velocity during the descent page 6 altitude meters velocity meters second 10000 8000 6000 4000 2000 2000 1600 1200 800 400 Lunar Landing Trajectory Optimization Maximize final spacecraft mass initial circular orbit altitude 100 km descent interface flight path angle 1 descent interface altitude 10 000 meters 100 200 300 400 simulation time seconds Lunar Landing Trajectory Optimization Maximize final spacecraft mass initial circular orbit altitude 100 km descent interface flight path angle 1 descent interface altitude 10 000 meters 100 200 300 400 simulation time seconds page 7 This next plot summarizes the flight path angle and spacecraft mass as a function of time since the descent interface flight path angle degrees spacecraft mass kilograms 100 1000 900 800 700 600 500 Lunar Landing Trajectory Optimization Maximize final spacecraft mass initial circular orbit altitude 100 km descent interface flight path angle
6. Program lunland Lunar Landing Trajectory Optimization with SOCS This document is the user s manual for a Fortran computer program called 1unland that uses the Sparse Optimal Control Software SOCS object code library developed by Boeing Phantom Works www boeing com phantom socs to solve the finite burn lunar landing trajectory optimization problem The trajectory is modeled as a single phase with user defined initial and final boundary conditions This computer program attempts to maximize the final spacecraft mass or minimize the total flight time The type of optimization is selected by the user The important features of this scientific simulation are as follows e Automated deorbit delta v algorithm e User defined flight path angle and altitude at the descent interface e 2 DOF flight path equations of motion relative to a spherical non rotating Moon e Thrust magnitude and thrust direction control variables SOCS is a direct transcription method that can be used to solve a variety of trajectory optimization problems using the following combination of numerical methods e collocation and implicit integration e adaptive mesh refinement e sparse nonlinear programming Additional information about the mathematical techniques and numerical methods used in SOCS can be found in the book Practical Methods for Optimal Control Using Nonlinear Programming by John T Betts SIAM 2001 The lunland software consists of Fortran routines tha
7. e information contained in the CSV data file produced by the lunland software The comma separated variable disk file is created by the odeprt subroutine and contains the following information time seconds simulation time since descent interface in minutes altitude meters altitude relative to a spherical Moon in meters velocity mps Moon relative velocity in meters per second fpa degrees Moon relative flight path angle in degrees mass kilograms spacecraft mass in kilograms thrust newtons propulsive thrust in newtons thrust angle deg thrust angle in degrees deltav mps accumulated delta v in meters per second downrange meters downrange distance in meters thrust to weight ratio of thrust to weight in Earth g s Notes 1 The accumulated delta v is determined from a cubic spline integration of the thrust acceleration at all collocation nodes 2 The downrange distance is determined from a cubic spline integration of the range rate at all collocation nodes The range rate equation is given by T Vcosy r h where r is the radius of the Moon A is the altitude V is the velocity and y is the flight path angle page 15 APPENDIX C Fortran Functions and Subroutines This appendix is a brief summary of the major Fortran functions and subroutines included in the lunland computer program lunland f atan3 for csint for cdeorbit for eci2orb for odeinp for odeprt for oderhs fo
8. he final time For the minimize flight time optimization the performance index is simply where t f is the final time The value of the maxmin indicator in SOCS tells the software whether the user is minimizing or maximizing the performance index 2 Path constraints This section summarizes how the software bounds the trajectory characteristics and spacecraft mass during the optimization altitude where h is the user defined initial altitude velocity O l lt vsv where v is the user defined initial velocity flight path angle 90 lt y lt 90 spacecraft mass 0 05m lt m lt 1 05m where m is the user defined initial spacecraft mass page 10 Technical Discussion Deorbit delta v The scalar magnitude of the deorbit maneuver that satisfies the user defined altitude and flight path angle at the descent interface is determined from the following expression where ala Mp Pg r radius ratio har e eq V TA local circular velocity at reentry altitude r e eq y flight path angle at descent interface h altitude of initial circular lunar orbit altitude at descent interface ar II equatorial radius of the moon gravitational constant of the moon This algorithm is described in Deboost from Circular Orbits A H Milstead The Journal of the Astronautical Sciences Vol XIII No 4 pp 170 171 Jul Aug 1966 Equations of motion The first order flight path
9. ilograms 1000 0d0 maximum thrust newtons 5000 0d0 minimum thrust newtons 1000 0d0 page 2 specific impulse seconds 300 0d0 lower bound for thrust angle degrees 90 0d0 upper bound for thrust angle degrees 90 0d0 The following series of data items allow the user to define guesses for the final time flight conditions and spacecraft mass To fix one or more conditions the user should input identical lower and upper bounds Please note the units and valid data range for each item Also note that the final speed must be greater than zero KKK KKK KKK KKK KKK KK KKK KKK KKK KKK KKK KKK KKK final time mass and flight conditions KEK KKK KKK KKK KKK KKK KKKKKAKKKKK KKK KKK KKKK initial guess for final time seconds 50 0 initial guess for final spacecraft mass kilograms 900 0d0 initial guess for final altitude meters 10 0d0 lower bound for final altitude meters 10 0d0 upper bound for final altitude meters 10 0d0 initial guess for final speed meters second 0d0 lower bound for final speed meters second upper bound for final speed meters second initial guess for final flight path angle degrees 90 0d0 lower bound for final flight path angle degrees 90 0d0 upper bound for final flight path angle degrees 90 0d0 The next series of program inputs are algorithm control options and parameters for the SOCS software The first input is an integer that
10. kkkkkkkkkkkkk please input the name of the simulation definition file The source code that reads the name of an input file included on the command line is c if present use command line argument 1 for input file call getarg 1 inputfname istatus The source code that creates the file name input prompt is as follows c clear screen isys system cls if istatus eq 1 then kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk input filename not on command line request name of simulation definition input file kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk ann a print page 13 print print x kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkxk print x program lunland bi print i x print 7 lunar landing trajectory XI print ka optimization with SOCS ki print x print x October 31 2005 Ki print x KEKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKEKE print print print amp please input the name of the simulation definition file read inputfname end if If your compiler does not accept input from a command line you will have to modify this source code for your particular Fortran compiler You may also choose to eliminate the code that accepts a command line input file Please note also that your compiler may have a different command to clear the screen page 14 APPENDIX B Contents of the Simulation Summary CSV File This appendix is a brief summary of th
11. r orb2eci for readfpn for readint for readtext for utility for uvector for vcross for vdot for vecmag for xmod for SOCS main executive program four quadrant inverse tangent function cubic spline integration of tabular data subroutine subroutine that computes deorbit deltav convert eci state vector to classical orbital elements subroutine SOCS simulation input subroutine SOCS print subroutine creates comma separated variable file SOCS subroutine that evaluates the equations of motion and any algebraic equations convert orbital elements to eci state vector subroutine read and echo floating point number from an input file subroutine read and echo an integer from an input file subroutine read and echo text from an input file subroutine number and text manipulation functions and subroutines unit vector subroutine vector cross product subroutine vector dot product subroutine vector scalar magnitude function modulo 2 pi function page 16 APPENDIX D Example Fortran Subroutine This appendix contains the source code for a single Fortran 77 routine and illustrates typical programming conventions used in the lunland software This subroutine is the differential algebraic equations routine required by the SOCS software subroutine oderhs iphase t y ny p np f nf iferr C computes the right hand sides of the c differential algebraic dae equations c dynamic variables y 1 altit
12. retiv The level of output from the SOCS differential equations algorithm is controlled with the following integer input Please note that option 5 will create lots of information KKK KKK KKK KK KKK KKK KKK KKK KK KKK differential equation output KKK KKK KKK KKKKKKKKKK KKK KKK KKK 1 none 2 terse 3 standard 4 interpretiv 5 diagnostic The level of output can be further controlled by the user with this final text input This program option sets the value of the SOCOUT character variable described in the SOCS user s manual To ignore this special output control input the simple character string no KKKKKKKKKKK KKK KKK KK user defined output input no to ignore KKKKKKKKKKKKKKKKKKK a0b0c0d0e0f0g0h01031k010m0n000p0qg0r0 page 5 SOCS solution and graphics The following is the SOCS trajectory summary for this example program lunland input data file gt lunlandl in lt maximize final mass gt initial circular orbit altitude impulsive deorbit delta v 100 000000000000 23 1907400535548 1 00000000000000 kilometers meters second flight path angle at interface conditions at descent interfac altitude speed flight path angle spacecraft mass final conditions time altitude speed flight path angle spacecraft mass propellant mass deltav 10000 0000000000 1693 20179797398 1 00000000000000 1000 00000000000 355 040890866099 10 0
13. t delete any of these annotation lines or increase or decrease the number of lines reserved for each comment However you may change them to reflect your own explanation The annotation line also includes the correct units and when appropriate the valid range of the input ASCII text input is not case sensitive but must be spelled correctly The first six lines of any input file are reserved for user comments These lines are ignored by the software However the input file must begin with six and only six initial text lines KEK KKK KKK KKK KKK KKK KKK KKKKKKKKKK KKK KKK KKK xx lunar landing trajectory optimization xx optimal thrust level and steering xx lunlandl in xx October 31 2005 KKK KKK KKK KKK KEK KKK KKK KK KKK KKK KKK KKK KKK KKK The first program input is an integer that defines the type of trajectory optimization to perform type of trajectory optimization KEK K KKK KK KKK KKKKKKKKK KKK KKK KKK K 1 maximize final mass 2 minimize flight time 1 The following series of data items are reserved for user defined initial conditions This information includes initial flight conditions propulsive characteristics and lower and upper bounds for the thrust angle Please note the units and valid data range for each item altitude of initial circular lunar orbit kilometers 100 0 altitude at descent interfac meters 10000 0d0 flight path angle at descent interface degrees 1 0d0 initial spacecraft mass k
14. t perform the following tasks e main program that sets algorithm control parameters and calls the SOCS transcription optimal control subroutine e define problem definition and perform initialization related to scaling lower and upper bounds initial conditions etc e evaluate the right hand side differential equations e define and compute any point and path constraints e display the optimal solution results The SOCS software will use this information to automatically transcribe the user s problem and perform the optimization using a sparse nonlinear programming method The software allows the user to select the type of collocation method and other important algorithm control parameters With the appropriate substitution of fundamental constants this simulation can also be used to model landings on other airless celestial bodies page 1 Typical input file The lunland software is data driven by a user created text file The following is a typical input file used by this computer program In the following discussion the actual input file contents are in courier font and all explanations are in times italic font This data file defines a typical descent analysis starting from a 100 kilometer circular lunar orbit and ending at an altitude of 10 meters and a speed of 1 meter per second This simulation maximizes the final spacecraft mass Each data item within an input file is preceded by one or more lines of annotation text Do no
15. ude meters c y 2 velocity meters second c y 3 flight path angle radians C y 4 mass kilograms c control variables c y 5 thrust newtons c y 6 thrust angle radians c KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK implicit double precision a h o z include socscoml inc integer iphase ny np nf iferr dimension y ny p np f f nf set function error flag iferr 0 c extract current flight conditions altitude y 1 velocity y 2 fpa y 3 xmass y 4 Cc extract current thrust newtons thrust y 5 Cc extract current angle of attack radians alpha y 6 page 17 current spacecraft radius meters rscm rm altitude acceleration of gravity at altitude m s 2 agrav 1 0d9 xmu rscm 2 altitude derivative meters f 1 velocity sin fpa velocity derivative meters second f 2 agrav sin fpa thrust cos alpha xmass flight path angle derivative radians 3 velocity rscm agrav velocity cos fpa amp thrust sin alpha xmass velocity propellant flow rate kg sec f 4 thrust vexhaust return end page 18
16. urvey of Numerical Methods for Trajectory Optimization John T Betts AIAA Journal of Guidance Control and Dynamics Vol 21 No 2 March April 1998 pp 193 207 5 Parametric Study of Lunar Landing Techniques Using a Predetermined Thrust Orientation Sam H Harlin Jr NASA CR 61075 June 1965 6 Powered Descent Guidance Methods for the Moon and Mars R Sostaric and J Rea AIAA 2005 6287 AIAA Guidance Navigation and Control Conference 15 18 August 2005 page 12 APPENDIX A Compiling and Running the Software This appendix describes how to compile and run the lunland computer program This software was created using version 6 3 3 of SOCS and Compaq Visual Fortran A DOS Windows version of lunland using Compaq Visual Fortran version 6 6C can be created with the following command df arch host lunland f for c 1socs1socs633 1ib advapi32 1lib This command assumes the SOCS library is located in the subdirectory c socs An input file created by the user can be run from the command line or a simple batch file with a statement similar to the following lunland lunlandl in If the software is executed without an input file on the command line the computer program will display the following information screen and file name prompt kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk x program lunland a ki lunar landing trajectory xi x optimization with SOCS Kk k ki x October 31 2005 x kkkkkkkkkkkkkkkkkkkkkkkk
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