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Separation Analysis with OpenModelica

1. A 2 Translation The translation from coordinate system b to coordinate system a is defined by g qtv FER A 1 where v 0 v and v is the vector from the origin of coordinate system a to the origin of coordinate system b expressed in coordinate system a 87 88 Quaternions A 3 Rotation The rotation from coordinate system b to coordinate system a R of a position x is defined by x Rb 2 a x q A 2 A 4 Euler rotation expressed in quaternions The initial orientation of the spacecraft is defined by Euler angles 0 y with the rotations from the reference frame performed in the order 4 0 y The quaternion to represent this rotation is given as w 2 sin 1 2 sin 1 2 cos w 2 sin cos w 2 cos _ sin Y 2 cos aa cos w 2 sin cos w 2 cos 0 2 cos 6 2 cos 6 2 cos 0 2 sin 9 2 sin 9 2 cos 9 2 sin 9 2 0 2 0 2 0 2 0 2 sin 2 sin 2 sin p 2 cos 2 an A 3 SEHE en sin SS Re N Appendix B User s manual This appendix describes how to use the program developed in this master s thesis OMSep both with and without using the Monte Carlo function B 1 Files The files described in this section are needed to be able to run the program OMSep The mo files are OpenModelica files and consists of classes and functions The py files are Python script files these are described in Chapter 7 and the Inp
2. 8 2 3 Error testing When something does not work it can be hard to find where the error is because of the poor error messages as described in the previous section When building a model in OpenModelica an equation system creates where the equations with common variables depend on each other Therefore it is hard to test only parts of the model This means that it is not always advantageous that the Modelica language is acausal and that the data flow can take place in both directions The acausality makes it hard to take a part of the model give it input signals and see which output signals it result in since there are no input and output signals when using the Modelica language The variables can exist in many equations at the same time and if one variable is set to a value as an input signal it is hard to know which of the equations to remove An equation has to be removed because the number of variables has to be equal to the number of equations and if one variable is set to a value it is an equation When one equation is removed it often causes problems in the other equations which depend on this equation This result in that more variables have to be set to values and more equations have to be removed to be able to simulate the model If the model can be simulated after removing these equations it is hard to know which equation equations was were wrong Compared to Simulink this is a disadvantage In Simulink parts of the model can easily be
3. The input data to the Monte Carlo function are written in the file MC Inp For a description of how to use the input file see Appendix B When using the Monte Carlo function the separation system model will be simulated a user defined number of times The variables which exist in the file MC Inp will vary from simulation to simulation and these variables are uniform distributed random numbers The maximum allowed variation in the parameters are set in the file 60 OMSep implementation MC Inp and the mean values of the variables are the values which have been set in the input files Sep Inp and Spring Inp Two Python scripts are used to perform this readMCinput py and mc py readMCinput py reads the input data from the input file MC Inp and saves them as variables readMCinput py can be seen on the next page it illustrates the syntax of the script files The script mc py produces uniform random numbers for the variables in the file MC Inp and writes the output data from the simulations to file The output data from OMSep when using the Monte Carlo function consists of the values that the variables have at the end time of each simulation i e when the separation has fulfilled and the satellite and the launch vehicle have lost their contact The script mc py writes the output data to the Matlab file MCOutput m This file can be run in Matlab to analyse the result 7 4 Monte Carlo implementation Write OpenModelica mos script
4. Bil Theory 2 342222 ER Ran 35 5 1 1 Newton s laws of motion 2 222 222m nennen 35 931 22 LORGUES a 2 a as A Ao ty A teat Se Sele ee ee 36 5 2 Coordinate systems 2 2 2 2 onen 37 5 3 Velocity and position sosse nommen 40 A KA et cae a San ct ds Are Sead ee Hla xa xe war age na RE TPS es eh 41 5 5 Angular rate and orientation quaternion 41 5 6 Forces and torques acting on the system 42 5 6 1 Spring local frame 2 Como 43 5 6 2 Spring position and force 43 5 6 3 Components which are not included in the model 48 Monte Carlo methods 49 6 1 The use of Monte Carlo 2 2 o nn 49 6 1 1 Simulating physical models 49 6 1 2 Mathematical problems 50 6 1 3 Applications oo o a v ad a Je s kd r de de erfara sl a a a r 50 6 2 Basis components 0 00000 ee ee 50 6 3 Anexample e 50 6 4 Monte Carlo and random numbers 51 6 4 1 Pseudo random numbers ooo mess e 51 6 4 2 Probability distributions 222 2 222m 52 6 5 Monte Carlo history 22 2 a aa non 53 OMSep implementation 55 GI Input data u a ae de D ran E la eis eo de 55 7 2 Model implementation e 56 2A Bunctions 2 x iii 2 Kr Ya a a a 57 14222 Olas SES u qe Se candy a Al AEA Bers SAT NET NET NGE Wome ae Ses ee ct 57 7 2 3 Simulation of the model 59 1 3 O tp t datano ne A OPAL Se
5. Iscwse 5 24 Wse Ase 5 25 and the following equation for the launch vehicle Monet Xlv Ly Wily X iwi 5 26 wm Aly 5 27 The torque will cause an angular rotation of the satellite launch vehicle this rotation is defined by quaternions Q 90 91 92 93 see Appendix A for a de scription of quaternions The quaternions describe the relation between the SC LV frame and the reference frame and they are used to transform between the frames The following algorithm is used to calculate the orientation quaternion Q and this algorithm can be used both to calculate the satellite orientation quaternion Qsc and to calculate the launch vehicle orientation quaternion Qw do 0 w 4 wg do q lio 0 wg w2 q 5 x 5 28 Q da 2 ww w Q wi q2 d3 w3 w2 4u wo q3 5 6 Forces and torques acting on the system The forces and torques acting on the bodies are provided by springs and umbilical connectors see Section 3 2 3 respective 3 2 4 There are two kinds of springs fixed 5 6 Forces and torques acting on the system 43 springs and free springs and the forces and torques that they produce are wanted by the system The umbilical connectors do not have the purpose to provide the system with forces and torques they can instead be seen as disturbances to the system But they affect the system in the same way as the springs and are therefore modeled as the springs These components are described later on in
6. load modules import sys os re codecs Open Input file for reading dSCm fileObject1 readline split 0 1 dSCcog fileObject1 readline split 0 3 dSCmoi fileObject1 readline split 0 3 dSCpoi fileObjecti1 readline split 0 3 dLVm fileObjecti readline split 0 1 dLVcog fileObjecti readline split 0 3 dLVmoi fileObjecti readline split 0 3 dLVpoi fileObjecti readline split 0 3 seednumber fileObject1 readline split 0 1 Close Input file fileObject1 close Chapter 8 OpenModelica evaluation An evaluation between OpenModelica see Section 4 1 and other modeling pro grams such as Matlab Simulink C C and JAVA has been done In this chap ter some advantages and disadvantages with OpenModelica which were discovered while building and simulating the separation system model are discussed 8 1 Advantages with OpenModelica In this section some advantages with OpenModelica compared to other modeling programs are discussed 8 1 1 Readable code The modeling language is acausal see Chapter 4 This means that the equations can be written in their original form the way they are usually found in textbooks and there are no need to manipulate them This makes the code easier to write because no time has to be put on reformulating equations It also makes the code readable and simple to understand The aim
7. part of the work is to evaluate OpenModelica and compare it with other modeling programs such as Matlab Simulink C C and JAVA to see advantages and disadvantages with OpenModelica Nyckelord Keywords Modeling Modelica OpenModelica Separation system Monte Carlo Abstract When launching a satellite a separation system is used to keep the satellite attached to a launch vehicle during ascent and to separate it from the launch vehicle while in space In separation analysis the separation is studied by simulations to see if requirements on the system can be fulfilled The purpose of this master s thesis is to investigate if separation analysis can be done using the modeling program OpenModelica and to evaluate OpenModelica and compare it to other modeling programs OpenModelica is free software implementing the Modelica language which is an object oriented language for modeling and simulation of complex physical systems Modelica uses equation based modeling this means that the physical behaviour of a model is described by differential algebraic and discrete equations and no particular variable needs to be solved manually The work is divided into two parts The main part is to implement a mathe matical model of a separation system in OpenModelica simulate it and study the behaviour of the system A Monte Carlo method which randomly generates values for uncertain model parameters is used when simulating the model The othe
8. rcos 5 31 where l is the spring length The spring force Fs is modeled as m Fixed U spring zZ Figure 5 6 The orientation of a fixed spring is defined by the tilt angle a around the Z axis Fs Foros cosa 5 32 F Fs sina 5 33 The magnitude of the spring force F is defined by three parameters initial force cut off force and stroke the spring extension length as defined in Figure 5 7 5 6 Forces and torques acting on the system 45 Force Initial force Cut off force gt Length Stroke spring extension length Figure 5 7 The magnitude of the spring force Fs is defined by three parameters initial force cut off force and stroke 46 Model description Free springs The orientation of a free spring is defined by the Euler angles 9 and w as described in Figure 5 8 The spring position S is modeled as A A Free spring bi Free spring Y fe Z Z Figure 5 8 The orientation of a free spring is defined by the Euler angles w left and 9 right Sp h lcoso 5 35 Sp rsind Isin 0 sin Y 5 36 Sp rcoso lsin cosy 5 37 where is the spring length The spring force Fs is modeled as Fs Fa sin 0 cosy 5 38 F Fa sin 0 siny 5 39 The magnitude of the spring force Fs is the same as for fixed springs i e it is defined by the three parameters initial force cut off force and stroke as described in Figure 5 7 5
9. 1 where the left hand side of the statement is assigned a value calculated from the expression on the right hand side an equation may have expressions on both its right and left hand sides for example 2x y 7z w 4 2 The Modelica language is acausal i e the equations have no pre defined causality This means that the user do not have to define which variables are inputs and which are outputs in contrast to for example Simulink where the input output causality is fixed In Modelica the simulation engine must manipulate the equa tions symbolically to determine their order of execution to solve the equation system 18 27 28 Modelica 4 1 OpenModelica OpenModelica 20 is a free software implementation of the Modelica language and it is a project at Link pings universitet The goal is to create a complete Modelica modeling compilation and simulation environment based on free soft ware distributed in source code form or binary form OpenModelica is intended for research teaching and industrial usage It is also used to experiment with new language features and language design for the ongoing development of the Modelica language 20 OpenModelica is written in a language called RML Relational Meta Lan guage This language is based on natural semantics which is a popular formalism for describing the semantics for compilers By using the RML language this for malism is combined with efficient compilation into optimize
10. 2 Model implementation 57 7 2 1 Functions CrossProduct and mm2m are the only functions in Figure 7 2 all the others are classes CrossProduct calculates the cross product of two 3x1 vectors and mm2m translates from millimeter to meter It is also one function Constants D2R from the Modelica Standard Library see Section 4 3 used in the model This func tion translates from degrees to radians The functions are used to facilitate the calculations in the classes 7 2 2 Classes All the sub models of the model are described using classes because in Modelica everything is described using classes and it is the only way to build abstractions The model classes are described in this section Evaluate There are a main class Evaluate where all the calculations which uses more than one of the model classes are done It uses all the other classes to calculate the relative separation velocity between the launch vehicle and the satellite and the satellite angular rate This class brings all the sub models together to form a whole model of a separation system QuatRot The class QuatRot calculates quaternion rotation and the class Evaluate uses this class for rotating between the reference frame and the other frames For a description of the frames see Section 5 2 and for a description of quaternions see Appendix A Body A satellite and a launch vehicle have many variables in common such as mass moment of inertia velocity and pos
11. 6 Forces and torques acting on the system 47 Umbilical connector The umbilical connector position U is modeled as Up h 5 41 Up rsing 5 42 Up rcosd 5 43 The umbilical connector force Fu is modeled as Fe Pura 5 44 The magnitude of the umbilical connector force Fu spring extension phase and a pin retraction phase Fu tot tot consists of two parts a is defined by five param eters initial force cut off force min force strokel spring extension phase and stroke2 pin retraction phase as defined in Figure 5 9 Force Initial _ force Cut off force Friction gt Strokel spring extension length Stroke2 spring retraction phase Figure 5 9 The magnitude of the umbilical connector force Fu is defined by five parameters initial force cut off force min force strokel and stroke2 48 Model description 5 6 3 Components which are not included in the model This model only describes the forces caused by springs and umbilical connectors As described in Section 3 the separation system comprises some more components such as separation nuts or a clamp band with bolt cutters or with a CBOD The springs used in the model can be extended they can for example get parame ters which describe the friction in the springs and parameters which describe the deflection in the springs Further work which can be done with the model is to implement these components to read more a
12. Appendix B An example of an input file for the satellite and the launch vehicle to the pro gram SepSin is shown below This input data file was used to produce the results in Chapter 9 Separation of 800S Lisa PDR 2006 06 01 Hasse A F 0 0 0 0 0 0 Firm rocket T F gravity m s2 0 2 0 00003 tMax dtPrint s 0 0002 0 00002 0 2 dti dt2 dt3 s 0 0 0 0 0 0 P L orient phi theta psi deg 3 Main rotation axis 1 X 2 Y 3 Z 435 0 240 0 0 0 mSat m3stage mAdapter kg 0 0 0 01 0 0 Cg3Sat x y z m 0 000 0 0 0 0 Cg3stage x y z m not inc adapter 0 0 0 0 0 0 CgAdapter x y z m 180 0 180 0 180 0 Ixx Iyy Izz Satellite kgm2 0 0 0 0 0 0 Pxy Pxz Pyz kgm2 105 0 105 0 105 0 Ixx Iyy Izz 4th stage kgm2 0 0 0 0 0 0 Pxy Pxz Pyz kgm2 0 0 0 0 0 0 Ixx Iyy Izz Adapter kgm2 0 0 0 0 0 0 Pxy Pxz Pyz kgm2 0 0 0 0 0 0 Initial rotation wx wy wz deg s 0 5 0 0 0 0 Test measurement rMark hMark m psiMark deg 0 012 0 05 11 12 F Measurement output in local frame T F Spring Inp Spring data file 95 96 SepSim input data An example of an input file for the springs to the program SepSim is shown below This input data file was used to produce the results in Chapter 9 8085 LISA PDR 2006 06 01 HA Modified Ident F 4 15 0 365 0 0 90 0 180 0 270 0 0 0 100 193 22 43 20 0 67 5 0 0 1 99999000 0 99999000 0 99999000 0 9999900
13. GAAS 59 7 4 Monte Carlo implementation 0 59 OpenModelica evaluation 63 8 1 Advantages with OpenModelica 2 2 22 63 8 1 1 Readable code 0 e a 63 8 1 2 No pre determined data flow direction 63 8 1 3 Reusable code o nenn 64 8 1 4 Object oriented soo mee er rr rr rss ses a 64 8 1 5 Suited for multi domain modeling 64 8 1 6 Modelica libraries some o o es 65 8 2 Disadvantages with OpenModelica 65 8 2 1 Documentation aa vag een 65 8 2 2 Error Messages 02 0 a oh ta a gh f r te dee 65 8 23 Error testi is 4 4 4 goa aan ara aes 66 Contents xi 8 2 4 Input and output data 2 2 onen 66 8 2 5 Simulation time 67 9 OMSep evaluation 69 9 1 Comparing the models 2 Cum nn 69 9 2 Monte Carlo results 2 CC 0 0 0 eee nnn 74 9 3 Further works a 2 2 nen 2 KE BEARS 79 10 Conclusions 81 Bibliography 83 A Quaternions 87 A T Basic definitions s La aaa at es EEE A E 87 A 2 Translation 54 dett c oe MAB 87 ACS Rotationts 604 204 de as a aed db A 88 A 4 Euler rotation expressed in quaternions 88 B User s manual 89 Belk Piles ce ae ae a 2 A REA IIS en a 89 B 2 Run 0MSep ar era ae anni een 90 C SepSim input data 95 Index 99 Chapter 1 Introduction The purpose of this master s thesis is to investigate if separation analysis can be don
14. asia mos ra ga ra ne ee ee A 1 2 Problem description 0 0 0000000004 133 PULP SE 2 5 a u a 2 2 Go A Ae ee Se 1 4 Disposition of the thesis o 2 Launching satellites 2 1 Satellite 1 Kran aaa nahe 2 1 1 Satellite orbits o sea 2 1 2 Launched satellites sosse on nn 2 22 Adapter et E Abe k r VN Mal ee oe De dl ee te RL A 2 3 Launch vehicle a aii y sve nee oa ar en nun a en 2 3 1 Expandable and reusable 23 2 Mass and stages un sun san eee ad 2 3 3 Nation and space agency 2 nn nn 2 4 Flight sequence during launch o o 3 Satellite separation oil Requirements 4 m a an area Rd re en ALA 3 2 Hardware 40 Wassers ee EE EERE EEE Des 3 2 1 Connection device o e e e 3 2 2 Release mechanisms e 3 2 3 Separation springs 2 2 2 mn nn 3 2 4 Umbilical connectors some 22 on nn 3 3 Separation analysis 2 00 eee eee eee 3 3 1 Release and ejection 22 22 on nn 3 3 2 Collision analysis cos s oo onen 4 Modelica 4 1 OpenModelica 2 2 2 2 nn nn 4 2 Modeling with Modelica 2 2 o e e 42T Models sar amna te daa te ra ee a a A 4 2 2 Formulation of equations 4 3 Modelica libraries 2 22 222m non nn 4 3 1 Add a new Modelica library o ix N NN Hm Ow oro a Contents Model description 35
15. be used only once and their compo nents are not recovered after the launch These launch vehicles usually separate from their payload and break up during atmospheric reentry Reusable launch vehicles on the other hand are designed to be recovered intact and used again for subsequent launches Most launch vehicles for launching satellites are expandable the only example of a reusable launch vehicle in operations is Nasa s Space Shut tle which can be seen in Figure 2 7 This is the spacecraft currently used by the United States government for its human spaceflight missions It carries astronauts and payload such as satellites or space station parts into low earth orbit Usually five to seven astronauts ride in the spacecraft The weight and height of the Space Shuttle can be seen in Table 2 2 2 3 2 Mass and stages Launch vehicles are often characterized by the amount of mass they can lift into orbit or the number of stages they employ A stage is mounted on top of or attached next to another stage The result is effectively two or more rockets stacked on top of or attached next to each other Taken together these are called a launch vehicle see Figure 2 1 Each stage contains its own engines and fuel By jettisoning stages when they run out of fuel the mass of the remaining rocket is decreased This staging allows the thrust of the remaining stages to more easily accelerate the rocket to its final speed and altitude Two stage rockets are qui
16. f r elektronisk version http www control isy liu se http www ep liu se exjobb isy 2007 4061 Titel Title F rfattare Author Separationsanalys med OpenModelica Separation Analysis with OpenModelica Malin K lldahl Sammanfattning Abstract When launching a satellite a separation system is used to keep the satellite at tached to a launch vehicle during ascent and to separate it from the launch vehicle while in space In separation analysis the separation is studied by simulations to see if requirements on the system can be fulfilled The purpose of this mas ter s thesis is to investigate if separation analysis can be done using the modeling program OpenModelica and to evaluate OpenModelica and compare it to other modeling programs OpenModelica is free software implementing the Modelica language which is an object oriented language for modeling and simulation of complex physical systems Modelica uses equation based modeling this means that the physical behaviour of a model is described by differential algebraic and discrete equations and no particular variable needs to be solved manually The work is divided into two parts The main part is to implement a mathe matical model of a separation system in OpenModelica simulate it and study the behaviour of the system A Monte Carlo method which randomly generates values for uncertain model parameters is used when simulating the model The other
17. files are input files to the program Body mo base class for the classes LaunchVehicle and SpaceCraft Body comprises all the variables and computations that these classes have in com mon crossproduct mo calculates the cross product of two 3x1 vectors Evaluate mo main class where all the calculations which need more than one class are performed FixedSpring mo includes the variables and calculations for a fixed spring FreeSpring mo includes the variables and calculations for a free spring LaunchVehicle mo includes the variables and calculations for a launch ve hicle mm2m mo translates from millimeter to meter Quaternions mo performs quaternion rotation between two frames SpaceCraft mo includes the variables and calculations for a satellite SpringModel mo base class for the classes FixedSpring FreeSpring and Um bilicalConnector SpringModel comprises all the variables and computations that these classes have in common 89 90 User s manual UmbilicalConnector mo includes the variables and calculations for an um bilical connector compileModel py sim py readSepinput py writeOMScript py Sep Inp input file for satellite and launch vehicle parameters Spring Inp input file for spring and umbilical connector parameters To be able to run Monte Carlo simulations the following files are needed too mc py readMCinput py MC Inp input file for the Monte Carlo function The program
18. in Chapter 9 Because of this long simulation times it is hard to run Monte Carlo simulations when using OpenModelica Chapter 9 OMSep evaluation To see if the separation system model described in Chapter 5 gives correct results it has been compared to a model of a separation system which is now used at Saab Space A program which is called SepSim Separation Simulation Software 26 is used at Saab Space to give their model input variables simulate it and produce output variables The corresponding program for the model developed in this master s thesis has got the name OMSep In this chapter OMSep is evaluated and compared to SepSim both when using Monte Carlo simulations see Chapter 6 and when not using it Further work which can be done with OMSep are also discussed in this chapter 9 1 Comparing the models The most important variables in the models are the relative separation velocity between the satellite and the launch vehicle and the satellite angular rate These variables are important to be sure that the satellite will be ejected from the launch vehicle in a controlled way and to be sure that the requirements on the system will be fulfilled see Section 3 1 In this section the relative separation velocity which can be seen in Figure 9 1 and the satellite angular rate which can be seen in Figures 9 3 9 4 and 9 5 from the two models are compared The dotted lines are the results from OMSep and the solid lines are the resul
19. is required to launch a satellite into this type of orbit than into any other orbit Satellites that point toward deep space and provide scientific information generally operate in this type of orbit An example of a LEO is the sun synchronous polar orbit it passes almost directly over the North and South poles A slow drift of the orbit position is coordinated with Earth s movement around the sun in such a way that the satellite always crosses the equator at the same local time on Earth Because the satellite goes over all latitudes its instruments can gather information on almost the entire surface of Earth These satellites can be used to study how natural cycles and human activities affects the climate on Earth 2 2 Adapter 9 Figure 2 5 This launch vehicle Soyuz R 7A was used to launch the world s first artificial satellite Sputnik 1 on October 4 in 1957 5 2 1 2 Launched satellites The Soviet Union launched the first artificial satellite Sputnik 1 on October 4 in 1957 The launch vehicle used to launch this satellite Soyuz R 7A can be seen in Figure 2 5 Since then the United States and about 40 other countries have developed launched and operated satellites Today about 3 000 useful satellites are orbiting Earth Table 2 2 describes some examples of launch vehicles and which nation and space agency they belong to 16 2 2 Adapter An adapter is a physical structure used to connect a satellite to a launch vehicl
20. r att simulera modellen Den andra delen av arbetet r att j mf ra OpenModelica med andra modelleringsprogram som t ex Matlab Simulink C C och JAVA f r att hitta f r och nackdelar med OpenModelica vi Acknowledgments Many people have helped make this thesis what it has become and I would like to express my gratitude to all of you here First of all I would like to thank Albert Thuswaldner my supervisor at Saab Space for all the help with the work the report and the presentation Thank you for always having time to help me with my problems and for keeping me up when I was tired of writing this report and just wanted to put it in a shredder I would also like to thank my supervisor at ISY Johanna Wall n for the help with the report and for reading it over and over again to find things that could be better Further I would like to thank my examiner Anders Helmersson for taking time to read the report I would also like to thank Magnus Larsson my opponent for reading my report and for many interesting questions Thanks to all of you at Saab Space in Link ping first of all for making it pos sible for me to do this master s thesis but also for being so nice and for welcoming me with embrace I have really enjoyed this time Last but not least I would like to thank my sister Therese for always being by my side and for listening to me when I was complaining on the report vil Contents 1 Introduction 1 1 Background 244
21. spacecraft and launcher tolerances T Random position of push rod within upper beraing T F F Place springs to minimize disturbations not in function 4 0 Number of springs set 1 amp 2 47 7 47 7 Spr nom des energy 1 2 Nm 100 0 100 0 Spring efficiency set 1 amp 2 0 0 0 0 Set energy tolerance set 1 amp 2 7 1 4 3 0 0 6 2 4 Spring ind tol measured eff set 1 100 0 0 0 Sep dist Spring ind tol measured eff set 2 0 0 Umb start numb of force inputs 0 0 0 0 10 75 N 61 pin Umbilical force tolerances 1 N 0 0 0 0 Umbilical force tolerances 2 N 0 0 0 0 0 0 Spr pos dhSat drSat dtSat mm 0 0 0 0 0 0 dhRoc mm dAngR dAngT deg 100 0 Spr house length mm 0 1 0 28 15 0 0 0 p r gap mm dFri dtFri ms dfFri 0 0 0 0 0 0 2 Initial rates wx wy wz deg s 32 3 Release pert model distr type 0 0 Delaymodel DelayDistr 91223 5 0 10 0 S C Mol envelope type 35 0 1 0 1 S C dMass kg kMass 5 0 5 0 45 0 1 0 1 dCog 1 3 mm kCoG 30 0 30 0 10 0 1 0 1 dMoI 1 3 kgm2 kMol 5 0 5 0 5 0 1 0 L dPoIl 1 3 kgm2 kPol 20 0 1 0 1 L V dMass kg kMass 50 0 10 0 10 0 1 0 a dCog 1 3 mm kCoG 10 0 15 0 15 0 1 0 dMoI 1 3 kgm2 kMol 20 0 20 0 20 0 1 0 Sigl dPoI 1 3 kgm2 kPol 1 345678 Random seed number 98 SepSim input data Index Adapter 9 Bolt cutter 19 CBOD 19 Clamp band 18 Collision analysis 26 Comparison of OM
22. springs are used to eject the satellite from the launch vehicle a separa tion spring can be seen in Figure 3 6 The velocity depends on how many springs 20 Satellite separation Figure 3 2 A clamp band is used to secure a satellite to a launch vehicle during ascent the CBOD is a release mechanism it is used to release the clamp band at separation and the catchers catch the clamp band when it has released 24 3 2 Hardware 21 Left thread Flywheel Pin puller Figure 3 3 A CBOD clamp band opening device is a release mechanism which is used to reduce the clamp band opening shock at separation 24 22 Satellite separation Clamp band Figure 3 4 This separation system which uses a clamp band and a CBOD is developed at Saab Space 3 2 Hardware 23 Figure 3 5 The function of the separation nuts is to release the mating bolt at command and they are pyrotechnically operated 24 24 Satellite separation Figure 3 6 Separation springs are used to eject the satellite from the launch vehicle 24 that are used and how much energy they employ The angular rate of the satellite depends on how the springs are placed There are two types of separation symmetric separation where the separation springs do not contribute to the satellite rotation after separation and asymmetric separation where the separation springs gives contribution to the satellite rotation after separation The symm
23. x 35 5 Scientific ENVISAT 7 8211 10 0 x 26 0 Table 2 1 Examples of satellites and their missions The dimensions are satellite height x wingspan The numbers after the names of the satellites are the references to the sources where more information about these satellites can be found 8 Launching satellites Figure 2 4 A navigation satellite like this Global Positioning System GPS satellite sends signals that operators of aircraft ships land vehicles and people on foot can use to determine their location 16 orbit LEO Most orbits of these three types are circular See 9 16 for more information about satellite orbits these are the sources for this section GEO satellites lie above the equator at an altitude of about 36 000 kilometers Communications satellites which relays radio television and other signals between points in space and on Earth are put into these high altitude orbits The aim with the GEO is that it has a constant distance to the surface of Earth MEO is the region of space around the Earth above LEO and below GEO Radio signals sent from a satellite at medium altitude can be received over a large area of the surface of Earth These orbits are stable and they have wide coverage which makes them ideal for navigation satellites such as GPS satellites see Figure 2 4 A LEO is just above Earth s atmosphere where there is still some air that cause drag on the satellite and reduce its speed Less energy
24. 0 0 oooooorR 0 0 0 o0o0Oo0Oo0roowoooo 0 oOoOooooo0 Free springs T fixed springs F number of springs Release shock duration ms no frict CP height mm Spring radius mm Phi spring deg Spring tilt angle deg Spring assembly length mm Initial force N Cut off force N Stroke mm Pushrod diameter mm Gap Length between bearings neg gt const Length above upper bearing Bearing friction coefficient Pushrod satellite friction coeff Pushrod satellite friction fact Spring bracket stiffness X N mm gt 0 Spring bracket stiffness Y N mm Push off pad stiffness X N mm Push off pad stiffness Y N mm Definition of tolerances T true f false Force coefficient P L dx height mm P L dz radius mm P L dy tangential deg Launcher dx height mm Launcher dz radius mm Launcher dy tangential deg 97 An example of a Monte Carlo input file to the program SepSim This input data file was used to produce the results in Chapter 9 0 Start value for the simulation counter normally 0 5000 End value als F simulation with one tolerance for each simulation 0 Case number for selection of one simulation normally O 1 selection of plot files instead of log files only one sim 1 Selection of Monte Carlo performance output 1 2 or 11 F Simulations with spring tolerances T Siumlations with
25. Institutionen f r systemteknik Department of Electrical Engineering Examensarbete Separation Analysis with OpenModelica Examensarbete utf rt i Reglerteknik vid Tekniska h gskolan i Link ping av Malin K lldahl LITH ISY EX 07 4061 SE Link ping 2007 UU ely a Nas un Link pings universitet TEKNISKA H GSKOLAN Department of Electrical Engineering Link pings tekniska h gskola Link pings universitet Link pings universitet SE 581 83 Link ping Sweden 581 83 Link ping Separation Analysis with OpenModelica Examensarbete utf rt i Reglerteknik vid Tekniska h gskolan i Link ping av Malin K lldahl LITH ISY EX 07 4061 SE Handledare Albert Thuswaldner Saab Space Johanna Wall n ISY Link pings universitet Examinator Anders Helmersson ISY Link pings universitet Link ping 5 December 2007 GS UN P oo The Avdelning Institution Datum U KA Division Department Date E a u a Ay Division of Automatic Control a ar x a of Electrical Engineering 2007 12 05 eN y ink pings universitet ER goc SE 581 83 Link ping Sweden Spr k Rapporttyp ISBN Language Report category IM Svenska Swedish Licentiatavhandling ISRN amp Engelska English X Examensarbete LITH ISY EX 07 4061 SE C uppsats Serietitel och serienummer ISSN D uppsats Title of series numbering Ovrig rapport URL
26. OpenModelica 1 4 3 also has to be installed on the computer to be able to run OMSep B 2 Run OMSep The following have to be done to run OMSep without using the Monte Carlo func tion Give input data for the satellite and the launch vehicle in the file Sep Inp and give input data for the separation springs and the umbilical connectors in the file Spring Inp An example of how the input files look like can be seen later on in this appendix Run compileModel py this script compiles the model Run sim py this script simulates the model and produces the output data Output data for the relative separation velocity and the satellite angular velocity can be found in the file SepOutput m To run OMSep with the Monte Carlo function the following have to be done Give input data for the satellite and the launch vehicle in the file Sep Inp give input data for the separation springs and the umbilical connectors in the file Spring Inp and give input data to the Monte Carlo function in the file MC Inp An example of how the Monte Carlo input file look like can bee seen later on in this appendix Run compileModel py this script compiles the model B 2 Run OMSep 91 e Run mc py this script performs the Monte Carlo function simulates the model and produces output data e Output data for the relative separation velocity and the satellite angular velocity can be found in the file MCOutput m The variables used in the input and o
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28. Sep and SepSim 69 Conclusions 81 Dymola 28 Fixed spring 44 Flight sequence during launch 14 Free spring 46 Further work 79 Implementation of Input data 55 Monte Carlo 59 Output data 59 The separation system model 56 Launch vehicle Expandable and reusable 11 Mass and stages 11 Nation and space agency 13 Table of 13 Launch vehicle velocity 40 MathModelica 28 Modelica Building models 28 Classes 29 Description of 27 Equations 29 Functions 29 Libraries 31 99 Monte Carlo Basis components 50 Applications 49 Description of 49 Example of 50 History 53 Monte Carlo results 74 Newton s laws of motion 35 OMSep classes 57 OMSep evaluation 69 OMSep functions 57 OMSep implementation 55 OpenModelica 28 Advantages 63 Disadvantages 65 Evaluation 63 Probability distributions 52 Pseudo random numbers 51 Quaternion Basic definitions 87 Rotation 88 Translation 87 Satellite orbits 5 Satellite separation Description of 17 Hardware 18 Requirements 17 Satellite velocity 40 Satellites Description of 5 Table of 7 Separation analysis 26 Separation nut 19 100 Index Separation spring 19 Separation system model Coordinate systems 37 Description of 35 Spring local frame 43 Space Shuttle 11 Sputnik 1 9 Torque 36 41 Umbilical connector 24 47 User s manual 89 Index 101 yM j LINK PING UNIVERSITY 5 ELECTRONIC P
29. a body and the force point of support is not the center of gravity of the body the force will cause a torque to the body For a rigid body this torque is calculated as the cross product 5 2 Coordinate systems 37 M rxF 5 5 where M is the torque r is the position of the force point of support relative the center of gravity of the body and F is the force acting on the body 21 If a torque is acting on a rigid body this will cause an angular rotation of the body The relation between the torque and the angular rotation of the body is described as SM lo 5 6 where M is the torque I is the moment of inertia and a is the angular acceleration 5 2 Coordinate systems The coordinate systems used in the separation system model are described in this section Three right handed Cartesian coordinate systems see Figure 5 2 are defined 26 e The launch vehicle frame LV frame with the X y axis parallel to the LV symmetry axis in the flight direction and the Yzy and ZLy axis initially in the separation plane e The not moving reference frame which initially coincides with the Lv frame e The satellite frame SC frame with the origin in the separation plane the Xsc axis parallel to the SC symmetry axis in the satellite flight direction and the Ysc and Zsc axis initially in the separation plane An example of how the coordinate systems used in the model can look like after the satellite and the launch vehicle have separated a bi
30. a libraries In Modelica related classes in particular areas are grouped into packages to make them easier to find Modelica Standard Library 17 is a standardized prede fined package It provides constants types and model classes of components from various application areas which are grouped into sub packages of the Modelica Standard Library package The currently available libraries in Modelica Standard Library can be seen in Table 4 1 The libraries in Modelica Standard Library can be used in OpenModelica accept from Mechanics MultiBody library and Media li brary which are not implemented in OpenModelica yet The libraries can be used freely for both commercial and noncommercial purposes Additional libraries are available in application areas such as thermodynamics hydraulics power systems data communication etc The full documentation as well as the source code of these libraries appear at the Modelica web site 18 The most of these additional libraries are implemented in OpenModelica There are also some libraries for com mercial usage such as libraries for vehicle dynamics hydraulic components and air conditioning systems 17 Because OpenModelica has open source code everyone can develop new libraries both for personal usage and to share with others This means that the Modelica libraries are growing all the time 4 3 1 Add a new Modelica library To build a model in OpenModelica the easiest way is to first find out if some o
31. alculate the velocity and the position of the satellite and the launch vehicle Ifthe separation is asymmetric see Section 3 2 3 the satellite and the launch vehicle will get angular rates when they separate from each other Newtons second law of motion gives the following equations Fue ma wx mv 5 7 v a 5 8 Dp v 5 9 where the term w x mv in equation 5 7 describes the satellite and launch ve hicle rotation in the SC frame respective the LV frame relative to the reference frame The acceleration a ar dy az the velocity v vz Uy vz the an gular rate w Wy wy wz the position p pz py pz and the total force Pin Enets Pnet P nen are defined in the sc Lv frame Fret is the net force acting on the body i e the vector sum of all the forces acting on the body Only the springs and the umbilical connectors which are described later on in this chapter will cause forces on the bodies therefore Fnet Fs Fs Fs Fua Fuco Fach 5 10 where F Fs Fs are the forces caused by the n springs used in the model and Fuci Fuca Fue are the forces caused by the m umbilical connectors used in the model Newton s third law of motion see Section 5 1 1 can be used to calculate the velocity and the position of the satellite and of the launch vehicle because the satellite and the launch vehicle are modeled as two rigid bodies This law tells that if a force F acts on the satellite then a fo
32. and when the satellite separates it causes a mechanical shock to the satellite Sometimes there are such requirements on the system that this shock has to be low To meet this requirement a clamp band opening device CBOD see Section 3 2 2 has been developed An other requirement is that housekeeping data has to be transmitted to the ground during ascent This data is information about the satellite and its health and safety The design of the system has to be made in such a way that it meets all the requirements on the system The design used at Saab Space which is described 17 18 Satellite separation Umbilical Connector Satellite a Satellite Before separation After separation Separation alang Separation spring Figure 3 1 A cross section of a satellite separation as can be seen the separation system releases and ejects the satellite 15 in Section 3 2 is one way to accomplish this For more information about the requirements on a separation system see 24 3 2 Hardware When designing a separation system it is desirable to have an simple design be cause the most important is to make the separation system reliable i e the satel lite has to separate every time A separation system can be designed in various ways to meet the requirements described in Section 3 1 The hardware Saab Space uses in their designs of separation systems are described in this section and more information about the hardwa
33. ations e The physical or mathematical system must be described by a set of proba bility distribution functions e A random number generator must be available e An implementation for sampling from the specified probability distribution functions with use of the random numbers e The outcomes must be accumulated into the quantities of interest 6 3 An example A trivial example of the Monte Carlo method is to estimate the area of a circle It is trivial because there is a well known formula which is easy to use but this example has most of the elements of more complex applications 1 Draw a square 2 Draw a circle in the square such that the diameter of the circle and the side length of the square are the same 6 4 Monte Carlo and random numbers 51 BER RD Figure 6 1 This figure illustrates an example of estimating the area of a circle with use of a Monte Carlo method 3 Randomly cover the surface of the square with dots see Figure 6 1 4 Count all the dots then count the ones inside the circle the area of the circle is then estimated as dotsinside Acircle estimated Asquare x 6 1 dotsall By increasing the number of dots the accuracy of the estimation will in crease When using Monte Carlo methods the accuracy of the estimation increases as number of experiments 6 4 Monte Carlo and random numbers Monte Carlo simulation methods do not generally require truly random numbers to be us
34. aunching a satellite see Chapter 2 it is important that the satellite is secured to the launch vehicle during ascent and that the satellite separates while in space A connection device is needed to secure the satellite to the launch vehicle and a release mechanism is needed to be able to separate the satellite When the satellite has been released it has to be ejected from the launch vehicle and be given the provided kinetic energy The system used to perform all this are called a separation system In Figure 3 1 a satellite separation can be seen and a separation system can be seen in Figure 3 4 24 The requirements on a separation system and the hardware Saab Space uses in their separation system are described in this chapter To see if the requirements are fulfilled separation analysis is done this is also discussed here 3 1 Requirements A separation system has some requirements that have to be fulfilled which depends on the mission The most important requirement is that the separation occur in a controlled way There are also requirements on the separation velocity and the angular rate to make sure that the satellite will be placed into orbit Other requirements on the system are that it has to be able to attach the satellite to a launch vehicle during ascent and that the satellite can be released and ejected with the provided energy to be able to be placed into orbit while in space A satellite separation is an abrupt course of events
35. avity and the moment of inertia for the satellite and for the launch vehicle are allowed to variate between the simulations The input data in the files Sep Inp and Spring Inp describes which values these variables can vary about and they also give input data to the rest of the model variables The input data to the satellite the launch vehicle and the springs used in this section are the same as used in the previous section In this example the variables that can variate are uniform distributed see Section 6 4 2 for a description of uniform distribution In this section histograms of the results when the Monte Carlo function was used are compared In Figure 9 7 the relative separation velocity between the satellite and the launch vehicle v can be seen The satellite angular rates wy and wy can be seen in Figure 9 8 and 9 9 and the satellite total transverse angular rate wr w2 w2 can be seen in Figure 9 10 In all these figures the result from SepSim is to the left and the result from OMSep is to the right The variable values in the histograms are the values that the variables have at the stop time for the simulations i e when the separation is completed When simulating the model with the Monte Carlo function random numbers are used this means that the results will not be exactly the same for the models but as can be seen in the histograms the results from OMSep and SepSim are similar to each other Another explanation for th
36. bout further work see Section 9 3 Chapter 6 Monte Carlo methods Monte Carlo methods are widely used for simulating the behavior of various phys ical and mathematical systems They randomly generate values for unknown pa rameters to simulate models of the systems The methods are distinguished from other simulation methods by being stochastic i e they are using random num bers or in practice pseudo random numbers These randomly generated values are often uniform or normal distributed Monte Carlo methods are based on the fact that the average value of a number of independent observations gets closer and closer to the mean value as the number of observations increases This means that the result will be normal distributed when using a Monte Carlo method To read more about Monte Carlo methods see 3 2 which are the sources in this chapter 6 1 The use of Monte Carlo Two common situations where Monte Carlo are used are when simulating physical models and to find solutions to mathematical problems These and some more applications are discussed in this section 6 1 1 Simulating physical models When simulating physical models it is important to do many simulations with some small differences in the input data for the uncertain variables if there are any to get as reliable results as possible This is to make sure that the model always will fulfil all its requirements Monte Carlo simulation is a method to perform this it randoml
37. d elica 18 is an object oriented language for modeling and simulation of complex physical systems It has a JAVA and Matlab like syntax Modelica uses equation based modeling this means that the physical behaviour of a model is described by differential algebraic and discrete equations and no particular variable needs to be solved manually 2 Introduction 1 2 Problem description When launching a satellite a separation system is used to keep the satellite attached to a launch vehicle during ascent and to separate it from the launch vehicle while in space In separation analysis the separation is studied by simulations to see if requirements on separation velocity and satellite angular velocity can be fulfilled To be able to simulate separations a mathematical model of the separation system is needed This model will be implemented in OpenModelica during the work with this master s thesis A Monte Carlo method will be used to get as reliable results as possible When using a Monte Carlo method the model will be simulated many times with some small differences in the input data this method randomly generates values for uncertain parameters to simulate a model This is done to make sure that the system always will meet all its requirements To see if OpenModelica is suited for modeling this kind of systems it will be evaluated and compared with other modeling programs such as Matlab Simulink JAVA and C C 1 3 Purpose The purpose
38. d C code 1 The OpenModelica environment consist of a compiler that translates Modelica code into flat Modelica which basically is the set of equations algorithms and variables needed to simulate the compiled Modelica model The environment also includes a shell i e an interactive command and expression interpreter similar to a Matlab prompt where models can be entered computations can be performed and functions can be called In this environment it is also possible to execute Modelica scripts i e Modelica functions or expressions executed interactively or a set of algorithm statements defined in a text file 1 4 2 Modeling with Modelica Commercial software products such as MathModelica and Dymola have been de veloped for modeling with Modelica It is also an open source project the Open Modelica Project 20 These programs have open model libraries which means that the users are free to create their own libraries or modify the already made libraries This is to better match the users unique modeling and simulation needs All pro grams that use the Modelica language can use the Modelica Standard Library described in Section 4 3 Both MathModelica and Dymola have an interactive graphical environment They also have some additional libraries in the various application areas such as biochemical magnetic and vehicle dynamics libraries To see which additional libraries exist in MathModelica see 13 and to see which additional l
39. distributions The randomly generated numbers used in Monte Carlo methods follows a proba bility distribution The two most common used distributions uniform distribution and normal distribution are described in this section Uniform distribution Uniform distribution is a probability distribution such that each number on the support of the distribution are equally probable The support is defined by two parameters a and b which are the minimum and maximum values of the support The distribution is often abbreviated U a b The probability density function for the uniform distribution U a b is 1 Ha ia fora lt a lt b 0 else 6 2 and it can be also be seen in Figure 6 2 6 5 Monte Carlo history 53 iawn Hawa PAE noch gt yess AR u m m m Figure 6 3 Probability density functions for some normal distributions with the mean 4 and the variance o Normal distribution A normal distributed also called the Gaussian distributed variable is defined by two parameters mean average value 4 and variance variability o The dis tribution is often abbreviated N u 7 The normal distribution is the most widely used family of distributions in statistics Many statistical tests are based on the assumption of normality and if having enough numbers from some distribution the numbers will be asymptotically normal distributed Probability density functions for some normal distributions N w o can b
40. e 13 2 3 3 Nation and space agency Launch vehicles are also characterized by nation or space agency responsible for the launch and the company that manufactures and launches the vehicle Some examples of launch vehicles their nations space agencies weight height and how many stages they employ can be seen in Table 2 2 Launch Vehicle Nation Company Weight Height m Stages nations agency 10 kg Space Shuttle 16 USA NASA 2 029 58 12 2 Soyuz 2 5 Russia RSA 305 46 1 2 or 3 Ariane 5 7 Europe ESA TTT 59 2 HIIA H2A 12 Japan JAXA 285 53 2 GSLV 11 India ISRO 402 49 3 Long March 4B 4 China CALT 254 44 1 3 Shavit 2 10 Israel ISA 0 25 3 76 2 Dnepr 28 Ukraine Yuzhmash 211 34 3 3 VSL 23 Brazil AEB 14 8 0 2 Table 2 2 Examples of launch vehicles from various nations and space agencies The numbers after the names of the launch vehicles are the references to the sources where more information about these launch vehicles can be found 14 Launching satellites 2 4 Flight sequence during launch Figure 2 8 describes the flight sequence when launching a satellite in this case for the launch vehicle LM 3C 22 which can be seen in Figure 2 1 The first step is the lift off of the launch vehicle Then the boosters and a bit later the first stage will be separated from the rest of the launch vehicle The boosters are used to assist with
41. e some adapters that have been developed at Saab Space can be seen in Figure 2 6 An adapter has a bolted launcher interface at the bottom and a satellite interface at the top The satellites can as described in Section 2 1 have various of sizes depending on their missions The adapter makes it possible to use the same kind of launch vehicle for all kinds of satellites The adapter also includes a separation system described in Chapter 3 which is used to separate the satellite while in space 24 2 3 Launch vehicle A launch vehicle is a rocket used to carry a payload from the surface of Earth into outer space Usually the payload is a satellite which is placed into orbit A launch vehicle can be seen in Figure 2 1 There are various types of launch vehicles and there are various ways to characterize them for example if they are expandable or reusable by the amount of mass they can lift into orbit or the number of stages 10 Launching satellites Figure 2 6 Saab Space Modular Payload adapter family is shown These adapters are used to connect a satellite to a launch vehicle The adapters are of various sizes to match various sizes of satellites 24 2 3 Launch vehicle 11 they employ by nation or space agency responsible for the launch this is described in this section To read more about launch vehicles see 9 which is the source for this section 2 3 1 Expandable and reusable Expandable launch vehicles are designed to
42. e differences in the results is that the SepSim model is more complex than the OMSep model as described in Section 9 1 The only histograms that differs are the ones for wy see Figure 9 8 the values on the x axis differs a bit but as described in Section 9 1 these values are small remark the scale on the x axis which means that the difference is small The output variables will be normally distributed when using Monte Carlo simulations see Chapter 6 for a description of this In the histograms it can be seen that all variables apart from wr are approximately normal distributed But only the output variables and linear combinations of the output variables will be normal distributed when using Monte Carlo simulations if some variables are nor mal distributed then all linear combinations of them are normal distributed too Because of wr yw w2 9 1 includes a square root and quadrats it is not a linear combination of any of the output variables Therefor wr will not be normal distributed and it will not get any of the standard distributions this can be seen in Figure 9 10 9 2 Monte Carlo results 75 400 400 350 350 300 300 250 250 200 200 150 150 100 100 50 50 0 0 0 46 0 48 0 5 0 52 0 46 0 48 0 5 0 52 Ye m s SepSim Ye m s OMSep Figure 9 7 Comparison of the relative separation velocity between the satellite and the launch vehicle vez the left from SepSim and the right from OMSep The number of cases can b
43. e graphical interface of the Modelica language and they makes icons for the libraries extends Icon Library and extends Icon Library2 should not be deleted if a graphical editor is used If these libraries are not removed from the code it will be errors when try ing to simulate the packages because if no graphical editor is used the icon libraries will have no function Use the command loadModel Modelica Use the command loadFile for each package mo files to use Simulate the packages These packages are exactly as all other mo files i e they have to be simulated before they can be used otherwise the functions do not exist in the OpenModelica environment The functions in the packages can now be used In the packages every function has a description of how it shall be used Chapter 5 Model description The separation system model which has been developed in this master s thesis describes two rigid bodies which separates from each other The two bodies are a satellite and a launch vehicle which are both modeled with six degrees of freedom often denoted as 6DoF 6DoF refers to motion in three dimensional space combined with rotation about three perpendicular axes In this Chapter the equations used in the model and the theory behind them are described To read more about the model equations used in this Chapter see 26 5 1 Theory The theory behind the equations used in the separation system model i e the theory of how forc
44. e seen in Figure 6 3 6 5 Monte Carlo history Monte Carlo methods were originally practiced under more generic names such as statistical sampling The name Monte Carlo is a reference to the famous casino in Monaco because the methods use of randomness and the repetitive nature of the process are analogous to the activities at a casino Chapter 7 OMSep implementation The program developed in this master s thesis OMSep OpenModelica Separation consists of a model of a separation system a function which gives input data to the model a function which simulates the model and a function which saves the output data An overview of OMSep can be seen in Figure 7 1 and the implementation of the program is described in this chapter A Monte Carlo method can also be used when simulating the model the implementation of the function to perform this is also described in this chapter Output data Input data Separation system model Figure 7 1 An overview of OMSep the program developed in this master s thesis It consists of a model of a separation system a function which gives input data to the model a function which simulates the model and a function which saves the output data 7 1 Input data The input data to OMSep are written in the files Sep Inp and Spring Inp In Sep Inp the satellite and launch vehicle input data are written and the spring input data are written in Spring Inp Examples of input files and a description of
45. e seen on the y axis As can be seen the results are quite similar to each other and Vrel gets approximately normal distributed for both of the models There are some small differences in the results they depend on that the two models differs a bit and that some of the input variables are randomly generated 76 OMSep evaluation 1000 1800 900 1600 800 1400 700 1200 600 1000 500 800 400 600 300 aii 400 100 200 0 D 1 05 0 05 1 4 2 0 2 4 w rad s SepSim 40 wy rad s OMSep 40 Figure 9 8 Comparison of the satellite angular rate around the X axis wz the left from SepSim and the right from OMSep The number of cases can be seen on the y axis As can be seen the results for Vs gets approximately normal distributed for both of the models The results differs a bit but the values are small remark the scale of the x axis which means that the difference is quite small 9 2 Monte Carlo results TT 350 350 300 300 250 250 200 200 150 150 100 100 50 50 0 0 0 04 0 02 0 0 02 0 04 0 04 0 02 0 0 02 0 04 w rad s SepSim w rad s OMSep Figure 9 9 Comparison of the satellite angular rate around the Y axis w the left from SepSim and the right from OMSep The number of cases can be seen on the y axis As can be seen the results are quite similar to each other and vy gets approximately normal distributed for both of the models There are some small differences in the results they depend on that the tw
46. e small remark the scale of the y axis 72 OMSep evaluation x 107 wy rads m Sepoim 0 005 0 1 0 15 time Figure 9 4 Comparison of the satellite angular rate around the Y axis wy The solid line is the result from SepSim and the dotted line is the result from OmSep The difference depends on that the models are not exactly the same the SepSim model is more complex than the OMSep model But the difference is quite small remark the scale of the y axis 10 w rad s m epom EBENE OMSep 0 05 01 0 15 time Figure 9 5 Comparison of the satellite angular rate around the Z axis wz The solid line is the result from SepSim and the dotted line is the result from OmSep as can be seen the results are almost the same 9 1 Comparing the models 73 u 0 05 0 1 0 15 time s Figure 9 6 The difference of w for the two models is as can be seen quite small It depends on that the models are not exactly the same the SepSim model is more complex than the OMSep model 74 OMSep evaluation 9 2 Monte Carlo results A Monte Carlo method see Chapter 6 and Section 7 4 have been used with 5000 simulations each for the OMSep model and the SepSim model to get the results described in this section The input data files to OMSep can be seen in Appendix B and the input data files to SepSim can be seen in Appendix C The input data file to the Monte Carlo function describes how much the mass the center of gr
47. e using the program OpenModelica The work is made at Saab Space in Link ping and it is divided into two parts The main part of the work is to implement a mathematical model of a separation system in OpenModelica simulate it and study the behaviour of the system The other part of the work is to evaluate OpenModelica and compare it with other modeling programs such as Matlab Simulink C C and JAVA to see advantages and disadvantages of the programs 1 1 Background Saab Space develops and manufactures equipment for space The main office is located in G teborg and they have a division in Link ping with two main products separation systems for launch vehicles and control systems for sounding rockets A separation system supports a satellite during ascent releases the satellite upon command and ejects the satellite from the launch vehicle A sounding rocket is an instrument carrying rocket designed to take measurements and perform scientific experiments in space Saab Space wants to investigate if the program OpenModelica 20 described below which uses the language Modelica can be used for modeling and simulation of their separation systems Modelica will maybe be a standard language for modeling and therefore Saab Space wants to evaluate OpenModelica to see if this is a program they can use OpenModelica is free software with the goal to create a complete Modelica modeling compilation and simulation environment based on free software Mo
48. eful Many of the most useful techniques use pseudo random sequences described later on in this section which makes it easy to test and re run sim ulations The only quality usually necessary to make good simulations is that the pseudo random sequence appear randomly enough What this means depends on the application but typically the sequence should pass some statistical tests Testing that the numbers are uniformly distributed normal distributed or follow another desired distribution is the simplest and most common ones 6 4 1 Pseudo random numbers A pseudo random number is a number that appears random but is not truley ran dom Pseudo random sequences typically have statistical randomness while being generated by an entirely deterministic causal process Such a process is easier to produce than a genuine random one and it can be used again and again to pro duce exactly the same numbers This is useful when testing software There is 52 Monte Carlo methods je tT a a um auf a b Figure 6 2 The probability density function for the uniform distribution U a b no known method to produce true randomness because if any factor determines the outcome it would mean that the produced number is not random at all The random number generation functions provided in all software packages are there fore pseudo random as for example the function randn in Matlab which produces normally distributed random numbers 6 4 2 Probability
49. es see Section 5 1 2 The torque is modeled in the same way for the springs and the umbilical connectors which gives the following cross product M Sp x F 5 19 for each of the springs and the following cross product M U x Fa 5 20 for each of the umbilical connectors where Ms Ms Ms Ms is the torque caused by the spring Mu Mu Mus Mu is the torque caused by the umbilical connector Sp is the spring position Up is the umbilical connector position and F F is the force acting on the body caused by the spring umbilical connector 5 5 Angular rate and orientation quaternion If a torque see Section 5 1 2 is acting on the system the satellite and the launch vehicle will get angular rates the following equation is used to calculate the angular 42 Model description rate and this equation can be used both for the satellite and for the launch vehicle Mne Ia wx Iw 5 21 vo a 5 22 where the term w x Iw describes the satellite and launch vehicle rotation in the sc frame respective the LV frame relative to the reference frame The the vector sum of all the torques acting on the satellite launch vehicle Mnet the angular acceleration a and the angular rate w are defined in the sc Lv frame The moments of inertia J are defined by moment of inertia tensors Lox k Ipz I Iy In u 5 23 l yg Iyz Lzz This gives the following equation for the satellite Mnet Asc Ise Wee X
50. es and torques act on rigid bodies is described in this section 5 1 1 Newton s laws of motion When having rigid bodies Newton s second and third laws of motion are used to calculate the forces acting on the bodies and the movements of the bodies These laws are described in this section Newton s second law of motion Newton s second law of motion tells that the net force on a particle is proportional to the time rate of change of its linear momentum linear momentum is the prod uct of mass and velocity d mv Sirs 7 5 1 where F is the force acting on the particle m is the mass of the particle and v is 35 36 Model description Figure 5 1 An illustration of Newton s third law of motions the velocity of the particle This law is often stated as SF ma 5 2 where a is the acceleration of the particle 21 Newton s third law of motion Newton s third law of motion tells that whenever a particle A exerts a force on another particle B B simultaneously exerts a force on A with the same magnitude in the opposite direction This together with Newton s second law of motion gives the following equations for a system of two rigid bodies A and B F mada 5 3 where ma and mp are the masses of particle A respective particle B and a and ap are the accelerations of particle A respective particle B An illustration of this can be seen in Figure 5 1 21 5 1 2 Torque If a force is applied to
51. esk help toolbox aeroblks index html access helpdesk help toolbox aeroblks quaternionrotation html Accessed 19 September 2007 NASA URL http visual gsfc nasa gov images separation_web jpg Accessed 26 October 2007 John E Oberright Satellite Artificial URL http www nasa gov worldbook artificial_satellites_ worldbook html Accessed 14 September 2007 Martin Otter and Hilding Elmqvist Modelica Language Libraries Tools Workshop and EU project RealSim URL http www modelica org June 2001 Martin Otter and Hilding Elmqvist Modelica A Unified Object Oriented Language for Physical Systems Modeling Language Specification URL http www modelica org 5 September 2007 Gunther s Space Page URL http space skyrocket de index frame htm http space skyrocket de doc sdat astra 1k htm Accessed 16 October 2007 Programming Environment Laboratory PELAB Department of Computer and Information Science IDA at Link ping Univer sity Openmodelica Users Guide URL http www ida liu se pelab modelica OpenModelica html June 2007 Andrew Pytel and Jaan Kiusalaas Engineering Mechanics Statistics Brooks Cole Publishing Company Pacific Grove CA 2 edition 2004 ISBN 91 44 02443 6 China Academy of Launch Vehicle Technology Lm 3c User s manual 5 June 2003 AEB Sonda 3 URL http www astronautix com lvs sonda3 htm Ac cessed 16 October 2007 Albert Thuswaldner Saab Space Risk analysis Functio
52. etric separation is to prefer if no satellite rotation caused by the separation is wanted But if a certain angular rate on the satellite is wanted the asymmetric separation can be used to achieve this The springs sizes and placements decides how the separation will occur and the requirements on the system decides which type of separation will be used 3 2 4 Umbilical connectors During ascent it is required that housekeeping data can be transmitted to the ground to make sure that everything is working properly This is performed by using umbilical connectors an umbilical connector can be seen in Figure 3 7 The umbilical connectors are not needed to be able separate the satellite but they give a disturbance to the separation and how they affect the separation are described in Chapter 5 Therefore they are a part of the separation system 3 2 Hardware 25 Figure 3 7 Umbilical connectors are used to transmit housekeeping data to the ground This data is information about the satellite and its health and safety 24 26 Satellite separation 3 3 Separation analysis In separation analysis the separation is simulated to see if requirements on sepa ration velocity and satellite angular rate are fulfilled This is done to determine if the satellite will be placed into the wanted orbit Separation springs are used to achieve this velocity and this angular rate The amount of energy for the springs the numbers of springs and how the sprin
53. f the functions needed for the modeling already exist in the available libraries If some of the libraries can be used then add these libraries and use the functions To add a new Modelica library and to be able to use the functions in the library the following has to be done 32 Modelica Table 4 1 A description of the libraries currently included in the Modelica Standard Modelica Library Description Modelica Blocks Continuous discrete and logical input output blocks Modelica Constants Common constants from mathematics physics etc Modelica Electrical Common electrical component models Analog Digital etc Modelica Icons Graphical layout of icon definitions Modelica Math Definitions of common mathematical functions Modelica Mechanics Mechanical components Rotational Translational and MultiBody Modelica Media Media models for liquids and gases Modelica SIUnits Type definitions with SI standard names and units Modelica StateGraph Hierarchical state machines Modelica Thermal Thermal phenomena heat flow etc Modelica Utilities Utility functions especially for scripting Library 17 4 3 Modelica libraries 33 Open the source code for the packages mo files in the library and delete extends Icon Library and extends Icon Library2 everywhere in the code Icon Library and Icon Library2 are used for th
54. formulating equations This is why OpenModelica is well suited for modeling of mathematical and physical systems OpenModelica is free software and there are quite few people using the program A disadvantage is that this makes it hard to find documentation and it is also hard to find help when problems occur Another disadvantage is the error messages they are unfortunately very poor they often only tells that something is wrong in the code and not where the error had occurred and what is wrong But the largest disadvantage with OpenModelica is that the simulation times are very long The fact that it takes about 50 hours to run 5000 simulations of the OMSep model makes it very hard to use a Monte Carlo method when simulating the model To summaries OpenModelica is well suited for modeling this kind of systems and it is simple to use as long as no problems occur but when problems arise it is very hard to find help OpenModelica is a young program and it will maybe 81 82 Conclusions be simpler to use it in a couple of years because if more people starts to use OpenModelica the documentation and the error messages will probably be better Bibliography 10 11 12 13 Peter Aronsson Peter Fritzson Levon Saldamli Peter Bunus and Kaj Nys tr m Meta Programming and Function Overloading in OpenModelica Pro ceedings of Modelica 2003 conference Link ping Sweden 3 4 November 2003 James O Berger Statist
55. function does not work it is hard to know if the function is not implemented in OpenModelica or if the function was used in an incorrect way Because of this it can take long time to determine if the function can be used in OpenModelica or not 8 2 2 Error messages Another disadvantage with OpenModelica is that there are often very poor infor mation in the error messages apart from syntax errors If it is a syntax error OpenModelica tells exactly where the error had occurred and what is wrong Oth erwise the error messages often do not tell any more than that it is something wrong somewhere in the code Two common error messages is Too few equations under determined system and Too many equations overdetermined system Both these messages means that the number of equations is not equal to the number of variables but they do 66 OpenModelica evaluation not give any information of where the error had occurred The only way to find out is to search through all the sub models to find the missing or extra equations This can take quite a long time if the model is a bit complex and comprises a lot of sub models The worst and unfortunately one of the most common error messages is Vari able getErrorString not found in scope lt global scope gt The only information this error message gives is that the model could not be simulated because something is wrong Good luck and patience are needed to find out why it does not work
56. gs are placed decides the separation ve locity and angular rate Which springs to use and where to place them to meet the requirements on satellite velocity and angular rate can be analysed to find a design that can be used in the real system 25 The separation analysis is done to verify that the chosen design fulfil all the requirements on the system and that it can be used see Section 3 1 for a description of the system requirements To make the analysis reliable it has to comprise all uncertainties that can affect the system variables To do this in an efficient way Monte Carlo simulations see Chapter 6 are used 25 3 3 1 Release and ejection A separation analysis covers two phases release and ejection The release phase is defined to be the phase when the satellite and the launch vehicle have contact through the separation plane The duration of this phase normally is in the mag nitude of a millisecond In this phase it is studied how the release mechanism affects the separation 25 The ejection phase comprises the time from the contact in the separation plane has ceased until all of the separation springs have lost contact with the satellite surface The duration of this phase normally is in the magnitude of a few tenths of a second In this phase it is studied how the springs and umbilical connectors affect the separation 25 3 3 2 Collision analysis Sometimes also collision analysis is performed This is to make sure that t
57. he satellite will not collide with the launch vehicle when they separate In this analysis the points on the satellite and the points on the launch vehicle which are most possible to collide are studied with simulations to see if the satellite and the launch vehicle will collide 25 Chapter 4 Modelica This chapter gives an introduction to Modelica 18 which is an object oriented language for modeling of complex physical systems The aim is to construct a standard language for describing physical models Modelica is suited for multi domain modeling for example modeling of mechatronic systems within automotive and robotics applications Such systems are composed of mechanical electrical and hydraulic subsystems as well as control systems Modelica uses equation based modeling the physical behaviour of a model is described by differential algebraic and discrete equations This means that the equations can be written as they are with no need to manipulate them 18 There are two important differences between Modelica and other object oriented programming languages such as C C or JAVA Modelica is a modeling lan guage rather than a true programming language and the primary content of the classes is a set of equations and not statements or blocks with assignments as in other object oriented programming languages The equations do not describe as signment but equality In contrast to a typical assignment statement such as 2 3y 5 4
58. here are fixed free uc Number of springs uc Stroke fixed mm Stroke free mm Stroke uc mm Spring length fixed mm Spring length free mm Spring length uc mm Spring hight fixed mm Spring hight free mm Spring hight uc mm Spring radius fixed mm Spring radius free mm Spring radius ucmm Spring phi fixed deg Spring phi free deg Spring phi uc deg Fmax fixed N Fmax free N Fmax uc N Fmin fixed N Fmin free N Fmin uc N Spring tilt angle deg Spring orientation psi deg Spring orientation theta deg Fmin2 N uc Stroke2 mm uc B 2 Run OMSep 93 An example of how the file MC Inp can look like is shown below The input data for the Monte Carlo simulations are set in this file How much the parameters are allowed to vary between the simulations are set here This input data file was used to produce the results in Chapter 9 5 35 0 5 0 5 0 45 30 0 30 0 10 5 0 5 0 5 20 0 50 0 10 0 10 10 0 15 0 15 20 0 20 0 20 345678 L V Number of simulations S C dMass kg kMass dCog 1 3 mm kCoG dMoI 1 3 kgm2 kMoI dPol 1 3 kgm2 kPoI dMass kg kMass dCog 1 3 mm kCoG dMoI 1 3 kgm2 kMoI dPoI 1 3 kgm2 kPoI Random seed number 94 User s manual Appendix C SepSim input data In this appendix the input data files to the program SepSim is shown the corre sponding input files to the program oMSep can be seen in
59. here the input output causality is fixed An ex ample is the following equation F ma 8 4 which can be used in two ways in Modelica as follows F ma 8 5 a F m 8 6 In other object oriented languages with fixed input output causality the assign ments have to be used instead of the equation which makes it harder to reuse the code Because of this it is simple to write a structured code with use of inheritances when using the Modelica language 8 1 4 Object oriented Modelica is an object oriented language this means that it is simple to reuse and structure the code due to classes and inheritances It is also simple to add additional components to the model which means that it is no problem to extend the model if some more functions or sub models are needed These are large advantages compared to for example building models in Matlab Simulink which is not object oriented In Matlab Simulink the models can not inherit from each other and the code will therefore be less structured 8 1 5 Suited for multi domain modeling The Modelica language is also suited for multi domain modeling It offers the possibility to create hybrid models i e models that consist of both continuous 8 2 Disadvantages with OpenModelica 65 and discrete components that interact with each other Therefore there are no need to transform between the time domain and the frequency domain This means that the language is well suited for modeling of all ki
60. how to use them can be seen in Appendix B A Python script readSepinput py is used to read the input data from the input files This script only reads the input data and saves them as variables 55 56 OMSep implementation 7 2 Model implementation The separation system model see Chapter 5 is implemented in OpenModelica see Section 4 1 and it comprises a model of a satellite a model of a launch vehicle and a model of a separation system These sub models are build as classes and there are some functions to facilitate the evaluations see Section 4 2 1 To get the model structured the classes that have common properties share a base class which they inherit from The system is described below and a system overview with the structure of the model classes and functions can be seen in Figure 7 2 The model equations are described in Chapter 5 QuatRot CrossProduct SpringModel LaunchYehicle SpaceCraft Umbilical FixedSpring FreeSpring Figure 7 2 An overview of the model classes and functions Evaluate is the main class where the calculations are done and CrossProduct mm2m and QuatRot facilitates the eval uations Evaluate have objects of the classes LaunchVehicle SpaceCraft FixedSpring FreeSpring and UmbilicalConnector LaunchVehicle and SpaceCraft have a base class Body which they inherit from FixedSpring FreeSpring and UmbilicalConnector have a base class SpringModel which they inherit from 7
61. ible in the model In some equations initial values have to be set to get the right solution For example the differential equation Tr ax 4 3 where a is a constant will result in the solution x 0 if no initial value is set to x If the wanted solution instead is z e 4 4 then the initial value of x has to be set to 1 If statements When using if statements it is important that the number of equations in the if part is equal to the number of equations in the else part Because if this is not true the number of equations will not be equal to the number of variables An example of this is the two if statements below which will produce the same result The first is is written with equations as in Modelica class if example Real a start 0 Real b equation if time gt 0 1 then a 1 b 2 else a 0 b 1 end if end if example The second is written with assignments as in other programming languages such as C C or JAVA 4 3 Modelica libraries 31 class if example Real a start 0 Real b algorithm if time gt 0 1 then acre s b 2 else b 1 end if end if example Real a start 0 gives a the initial value 0 in both examples In the second if statement a will have the initial value 0 as long as the if statement is false But in the first if statement an equation is needed to perform this because otherwise the number of equation will not be the same as the number of variables 4 3 Modelic
62. ibraries exist in Dymola see 6 4 2 1 Models Modelica models consists of several smaller sub models which can be joined to gether to form larger and more advanced models The models are built as classes and functions can be used to facilitate the evaluations In Modelica everything is described using classes it is the only way to build abstractions and it enables structured modeling 18 4 2 Modeling with Modelica 29 Classes A class contains variable declarations an algorithm section with assignments and an equation section describing the behaviour of the class Similar classes have a base class with the common properties which they inherit from The syntax of a class can be seen below where name is the name of the class 18 class name variable declarations algorithm algorithm section equation equation section end name Functions A number of mathematical functions like abs sqrt mod etc are predefined in the Modelica language whereas others such as sin cos exp etc are available in the Modelica standard mathematical library Modelica Math See Section 4 3 for a description of the Modelica libraries User defined functions can also be useful when modeling 18 Modelica functions are mathematical functions with no memory they always return the same results given the same arguments The syntax of a function definition where name is the name of the function can be seen below and it is quite close to
63. ical decision theory and Bayesian analysis Springer cop Berlin 2 edition 1985 ISBN 3540960988 Gunnar Blom Jan Enger Gunnar Englund Jan Grandell and Jan Holst Sannolikhetsteori och statistikteori med till mpningar Studentlitteratur Lund 5 edition 2005 ISBN 91 44 02442 8 CALT Launch vehicle URL http www cnsa gov cn Accessed 16 Octo ber 2007 The Soyuz Company Document ID ST GTD SUM 01 Soyuz user s manual Issue 3 April 2001 Dynasim URL http www dymola com Accessed 4 October 2007 ESA ENVISAT URL http envisat esa int Accessed 16 October 2007 United States Air Force Global positioning system URL http www af mil factsheets factsheet asp id 119 Accessed 16 October 2007 Peter Fortescue and John Stark Spacecraft Systems Engineering Wiley Chichester 2 edition 1995 ISBN 0 471 95220 6 ISA Shavit URL http www most gov il English Units Science Israel SpacetAgency default htm Accessed 16 October 2007 ISRO GSLV URL http www isro org gslv htm Accessed 16 October 2007 JAXA Japanese Aerospace Exploarating Space About JAXA URL http www jaxa jp index_e html Accessed 16 October 2007 MathCore URL http www mathcore com products mathmodelica Ac cessed 4 October 2007 83 84 Bibliography 14 15 16 17 18 19 20 21 22 23 24 25 26 27 The MathWorks URL http www mathworks com access helpd
64. ion for anyone to read to download to print out single copies for your own use and to use it unchanged for any non commercial research and educational purpose Sub sequent transfers of copyright cannot revoke this permission All other uses of the document are conditional on the consent of the copyright owner The pub lisher has taken technical and administrative measures to assure authenticity security and accessibility According to intellectual property law the author has the right to be men tioned when his her work is accessed as described above and to be protected against infringement For additional information about the Link ping University Electronic Press and its procedures for publication and for assurance of document integrity please refer to its WWW home page http www ep liu se Malin K lldahl
65. ition and therefore they have a base class Body with the common properties which they inherit from SpringModel Fixed springs free springs and umbilical connectors also have many variables in common such as forces and orientation and they have some calculations in com mon such as computation of torques and forces Therefore they have the base class SpringModel which they inherit from The class FixedSpring can be seen on the next page it illustrates the syntax of the classes 58 OMSep implementation class FixedSpring Includes variables and calculations for a fixed spring extends SpringModel Real alpha NoS Spring tilt angle rad equation if Spring 0 then Spring O if there are no fixed springs Ftot 0 Sposx 0 Sposy 0 Sposz 0 Fi 0 0 0 else for i in 1 NoS loop Position of the spring expressed in the reference frame Sposx i springh i springllil cos alphali Sposy i SpringRadius i cos SpringPhi i springlli l sin alphali Sposz i SpringRadius i sin SpringPhi i Force caused by the spring expressed in the reference frame Ftotli if distance lt stroke i then Fmax i Fmax i Fmin i stroke i distance else 0 Fi 1 i Ftot i cos alpha i Fi 2 i Ftot i sin alpha i Fi 3 i 0 end for end if end FixedSpring 7 3 Output data 59 7 2 3 Simulation of the model To simulate the model a Python script wri
66. nal Analysis and FMECA Functional Categories and Definition Characteristics Critical Points list Document ID P MPA NOT 00009 SE 29 January 2007 Hasse Andersson Saab Space Kapitel Analys Realease amp Separation Analysis Document ID D LR NOT 5392 SE 12 February 2002 Hasse Andersson Saab Space Separation Simulation Software Description and Users Manual Document ID D LR NOT 05441 SE 29 May 2006 James R Wertz Spacecraft Attitude Determination and Control D Reidel Publishing Company Dordrecht 4 edition 1985 ISBN 90 277 0959 9 Bibliography 85 28 Yuzhmash Rockets URL http www russianspaceweb com dnepr html Accessed 16 October 2007 Appendix A Quaternions In this appendix it is described how translation and rotation from a coordinate system a to a coordinate system b is performed with quaternions See 14 27 for more information about quaternions these are the sources used in this appendix A l Basic definitions A quaternion q Q R4 qq 1 is a four tuple of real numbers It is denoted by q 90 91 92 93 Alternatively it is denoted by q qo q where go is called the scalar part and q the vector part of the quaternion Some of the operators used for quaternions are the following Addition p q po q P q Multiplication p q poqo Pq Poq qop p X q Conjugation a qo 9 A Cartesian vector x R is represented in quaternions as 0 x R4
67. nds of physical and mathematical systems 8 1 6 Modelica libraries There are functions in the Modelica libraries see Table 4 1 which can be used to facilitate the modeling This simplifies the modeling because it is much simpler to add an existing function from a library and use it than create an own function Because OpenModelica has open source code anyone can develop new libraries and share with others If some functions are needed in a particular area when building a model maybe they are already created and exist in a library Then it is just to add the library and use the functions This means that the more people using Modelica the more libraries will be created and the simpler it will be to build models in Modelica 8 2 Disadvantages with OpenModelica In this section some disadvantages with OpenModelica compared to other modeling programs are discussed 8 2 1 Documentation OpenModelica is a free software and there are quite few people using the program Therefore it is hard to find documentation and it is also hard to find help when problems occur Because of this it often takes long time to solve the problems that have occurred An example is the Modelica libraries see Section 4 3 Most of them can be used in OpenModelica but not all of them and it is hard to find information about which libraries are implemented in OpenModelica and which are not The simplest way to find out is to try the functions in OpenModelica But if a
68. o models differs a bit and that some of the input variables are randomly generated 78 OMSep evaluation 350 350 300 300 250 250 200 200 150 150 100 100 50 50 0 01 0 02 0 03 0 01 0 02 0 03 wy rad s SepSim wy rad s OMSep Figure 9 10 Comparison of the satellite total transverse angular rate wr w2 w2 the left from SepSim and the right from OMSep The number of cases can be seen on the y axis As can be seen the results are quite similar to each other wr is not normal distributed which depends on that wr is not an output variable from the models and it is not a linear combination of any of the output variables 9 3 Further work 79 9 3 Further work As described in Section 9 1 the SepSim model are more complex than the OMSep model which means that further work can be done with the OMSep model Some functions which exist in the SepSim model and not exist in the OMSep model see Chapter 5 are described in this section 26 e A spring deflection model which calculates the transverse force due to de flection of the push rod e A friction model which is applied on the force between the satellite push off pad and the push rod top e Modeling of play in the bushings A bushing is designed to reduce friction or restrain motion of mechanical parts e A model of the stiffness in the adapter spring brackets and payload push off pads e Modeling the effects of perturbation from clamp band release e Im
69. of this master s thesis is to e Make a mathematical model of a separation system and implement it in OpenModelica e Simulate the model to see if separation analysis can be performed e Use a Monte Carlo method when simulating the model e Evaluate OpenModelica and compare it with other modeling programs to see advantages and disadvantages 1 4 Disposition of the thesis The structure of the thesis is described in this section Chapter 2 Launching satellites includes facts about satellites adapters and launch vehicles These are the components needed for launching a satellite Chapter 3 Satellite separation describes the separation system Chapter 4 Modelica gives an introduction to the modeling language Modelica and to the program OpenModelica Chapter 5 Model description describes the mathematical model of the separation system and includes the equations 1 4 Disposition of the thesis 3 Chapter 6 Monte Carlo describes Monte Carlo methods Chapter 7 OMSep implementation presents the implementation of OMSep the program developed in this master s thesis Chapter 8 OpenModelica evaluation discusses advantages and disadvantages with OpenModelica compared to other modeling programs such as Matlab Simulink JAVA C C Chapter 9 OMSep evaluation compares the results from OMSep with the results from SepSim a corresponding program used at Saab Space now Chapter 10 Conclusions discusses the result of this master
70. plement the uncertainties for the springs in the Monte Carlo function Chapter 10 Conclusions One part of this master s thesis was to investigate if separation analysis can be done using the program OpenModelica 20 A program named OMSep has been developed and it consists of a mathematical model of a separation system and a function which makes it possible to use a Monte Carlo method when simulating the model see Chapter 5 and Chapter 6 respectively OMSep has been compared to a corresponding program SepSim which is now used at Saab Space to see if it gives correct results As can be seen in Chapter 9 the results are almost the same for the two programs both when using the Monte Carlo function and when not using it but there are some small differences One explanation for this is that the separation system model used in SepSim is more complex and includes more variables than the OMSep model This means that the OMSep model can be used for separation analysis and that it can be extended with the functions that only exists in the SepSim model now see Section 9 3 to give better results The other part of this master s thesis was to evaluate OpenModelica and com pare it to other modeling programs to see advantages and disadvantages The largest advantage with the language is that the equations can be written in their original form and there are no need to manipulate them This makes the code simple to write because no time has to be put on re
71. r part of the work is to evaluate OpenModelica and compare it with other modeling programs such as Matlab Simulink C C and JAVA to see advantages and disadvantages with OpenModelica Sammanfattning F r att skjuta upp en satellit anv nds en b rraket ett separationssystem ser till att satelliten h lls fast till b rraketen under uppskjutningen och att satelliten sep areras n r den har kommit upp i rymden F r att studera separationen med simu leringar och f r att se om krav p systemet uppfylls anv nds separationsanalys Syftet med det h r examensarbetet r att unders ka om det g r att anv nda mod elleringsprogrammet OpenModelica f r att g ra separationsanalys och att j mf ra OpenModelica med andra modelleringsprogram Programmet OpenModelica r fritt och det anv nder sig av spr ket Modeli ca Modelica r ett objektorienterat spr k som r utvecklat f r att anv ndas vid modellering av komplexa fysikaliska system Det h r spr ket anv nder sig av ek vationsbaserad modellering vilket betyder att ett systems beteende beskrivs av differentialekvationer och algebraiska och diskreta ekvationer och att ingen vari abel beh ver l sas ut f r hand Arbetet r uppdelat i tv delar Huvuddelen best r av att implementera en matematisk modell av ett separationssystem i OpenModelica simulera den och studera systemets beteende En Monte Carlo metod som slumpvis genererar v r den f r os kra modellvariabler anv nds f
72. rce F will act on the launch vehicle which gives the following equations for the satellite Fret MsgcAse Wse X MscUsc 5 11 Veo dsc 5 12 Pse Use 5 13 and the following equations for the launch vehicle Fret MyAy Ww X MiyUly 5 14 Vy Aw 5 15 Dv Vw 5 16 5 4 Torque 41 To calculate the relative separation velocity between the satellite and the launch vehicle vret the velocities vs and vw has to be transformed to the reference frame Then vre is calculated as Vrel VUscO Vwo 5 17 where vsco is the satellite velocity transformed to the reference frame and vwo is the launch vehicle velocity transformed to the reference frame The distance d between the satellite and the launch vehicle is calculated in the same way the positions ps and pi has to be transformed to the reference frame and then d is calculated as d Psco Pwo 5 18 where Psco is the satellite position transformed to the reference frame and pwo is the launch vehicle position transformed to the reference frame 5 4 Torque The forces acting on the bodies are provided by springs and umbilical connec tors which are described later on in this chapter If the placement of the springs and or the umbilical connectors have any asymmetry such as the amount of en ergy of the springs umbilical connectors differ or the center of gravity of the body is not placed in origo of the reference frame they will cause torques on the bodi
73. re described here can be found in 24 A design of a separation system which uses a clamp band and a CBOD can be seen in Figure 3 4 these components are described later on in this section 3 2 1 Connection device To secure the satellite to the launch vehicle a clamp band or separation nuts can be used these components are described in this section The design of the satellite decides which of these components to use Clamp band The clamp band see Figure 3 2 is used to secure the satellite to the launch vehicle during ascent During separation the clamp band releases and makes it possible for the satellite to separate from the launch vehicle Bolt cutters or a CBOD is used to release the clamp band see Section 3 2 2 When the clamp band has released 3 2 Hardware 19 there are catchers which are used to prevent the release of the clamp band from affecting the separation Separation nuts When using separation nuts the satellite is secured to the launch vehicle with bolts A separation nut can be seen in Figure 3 5 3 2 2 Release mechanisms Which release mechanism to use depends on how the satellite is secured to the launch vehicle There are two ways to release the clamp band by using bolt cutters or by using a CBOD clamp band opening device These release mechanisms are described in this section When using separation nuts the release of the satellite occurs in a third way which is also described in this section Bolt c
74. s thesis Appendix A Quaternions includes facts about quaternions Appendix B User s manual describes how to use the program OMSep Appendix C SepSim input data describes the input files to the program SepSim Chapter 2 Launching satellites When launching a satellite apart from the satellite a launch vehicle and an adapter are needed These components are described below and can be seen in Figure 2 1 2 1 Satellite A satellite is a smaller object that rotates around a larger object Satellites that have been placed into orbit by human are sometimes called artificial satellites to distinguish them from natural satellites such as the moon Most satellites orbit the Earth The satellites are designed for the missions they shall perform such as weather satellites communications satellites and navigation satellites Weather satellites see Figure 2 2 observe atmospheric conditions over a large area to help study weather patterns and forecasting the weather A communications satellite see Figure 2 3 relays radio television and other signals between points in space and on Earth Navigation satellites see Figure 2 4 send signals that operators of aircrafts ships land vehicles and people on foot can use to determine their location 16 Satellites are also used to study the universe Such satellites have orbited the moon the sun asteroids and the planets Venus Mars and Jupiter These satel lites mainly gather information abou
75. t from each other can be seen in Figure 5 3 The Sc orientation is defined by rotations with the Euler angles 0 y with the rotations from the reference frame performed in order 4 0 y is rotation about the Y axis 0 is rotation about the new X axis and is rotation about the new Z axis For example the application 90 45 90 is illustrated in Figure 5 4 26 38 Model description Figure 5 2 Right handed Cartesian coordinate system this is how the three coordinate systems used in the model initially will look like 26 ASC 2sc Ysc x ziy xly Ylw Figure 5 3 An example of how the coordinate systems used in the model can look like after the satellite and the launch vehicle have separated a bit from each other 5 2 Coordinate systems 39 Figure 5 4 Rotation from the reference frame to the SC frame with the Euler angles 0 w performed in order 4 0 amp vis rotation about the Y axis 0 is rotation about the new X axis and is rotation about the new Z axis The application 90 45 90 is illustrated in this Figure 26 40 Model description 5 3 Velocity and position This section describes the equations used in the separation system model to cal culate the satellite and launch vehicle velocity and position The satellite and the launch vehicle can be described as two rigid bodies There fore Newton s second and third laws of motion see 5 1 1 is used to c
76. t the bodies they orbit to help scientists to investigate these bodies Some examples of satellites and their weights dimensions and orbits can be seen in Table 2 1 16 2 1 1 Satellite orbits Depending on the mission satellite orbits have a variety of shapes Some are circular while others are highly elliptical Orbits also vary in altitude For example some circular orbits are just above the atmosphere at an altitude of about 250 kilometers while others are about 36 000 kilometers above Earth Many types of orbits exist but most artificial satellites that orbit Earth travel in one of three types geostationary earth orbit GEO medium earth orbit MEO and low earth 5 6 Launching satellites Satellite Booster Figure 2 1 An example of a launch vehicle Long March 3C an adapter and a satellite The stages token together with the boosters are called a launch vehicle A launch vehicle and an adapter are used when launching a satellite 22 2 1 Satellite 7 Figure 2 2 This is a weather satellite it observes atmospheric conditions over a large area to help scientists study and forecast the weather 16 Figure 2 3 A communications satellite such as Astra 1K shown here relays radio television and other signals between points in space and on Earth 16 Mission Example Weight kg Dimensions m Communication Astra 1K 19 5250 6 6 x 37 0 Navigation Navstar GPS 8 1705 2 4
77. te common but rockets with as many as five separate stages have been successfully launched The main reason for multi stage launch vehicles is that once the fuel is burnt the space and structure which contained the fuel and the motors themselves are useless and only add weight to the vehicle which slows down its future acceleration By dropping the stages which are no longer useful the rocket gets lighter The thrust of the future stages is able to provide more acceleration than if the earlier stages were still attached or than a single large rocket would be capable of When a stage drops off the rest of the rocket is still traveling near to the speed that the whole assembly reached at burn out time This means that it needs less total fuel to reach a given velocity and or altitude A further advantage is that each stage can use its own type of rocket motor with each stage motor tuned for the conditions in which it will operate On the downside staging requires the vehicle to lift motors which are not being used until later and makes the entire rocket more complex and harder to build But the savings are so great that every rocket currently used to deliver a payload into orbit uses staging 12 Launching satellites Figure 2 7 Nasa s space shuttle is the only example of a reusable launch vehicle in operations it is the spacecraft currently used by the United States government for its human spaceflight missions 16 2 3 Launch vehicl
78. teOMScript py is used It assigns the input data to the model variables and loads all the needed OpenModelica files i e all the classes and functions that the model consists of This script also simulates the model and saves the output variables 7 3 Output data The most important variables in the model are the relative separation velocity between the satellite and the launch vehicle and the satellite angular rate therefore the output data consists of these variables The output data from OMSep consists of the values that the variables have in each point of time during the simulation and it also consists of a time vector A Python script sim py writes the output data to the Matlab file SepOutput m This file can be run in Matlab and plots of the results can be performed to analyse the result 7 4 Monte Carlo implementation OMSep also includes the possibility to use a Monte Carlo method see Chapter 6 for a description of Monte Carlo methods when simulating the model This section describes how this function is implemented and an overview of the program which includes the Monte Carlo function can be seen in Figure 7 3 Separation system model Figure 7 3 An overview of OMSep when using a Monte Carlo method It consists of a model of a separation system a function which gives input data to the model a function which performs a Monte Carlo method a function which simulates the model and a function which saves the output data
79. tested by giving them input signals and see if the output signals are correct then it is easy to find in which sub model the error had occurred 8 2 4 Input and output data The input data often exist in a text file or another type of file outside OpenModel ica Then it is preferable to be able to export the input data to the OpenModelica model in an automatic way instead of having to write them directly in the model Sometimes there are need of exporting the output data to another program for example Matlab to compare the result with the result from another program There is no function implemented in OpenModelica which can be used to read input data from file and write output data from the simulations to file There are neither any documentation which can easily be found about this The problem can be solved by using scripts as described in chapter 7 but this is quite complicated It had been much simpler if the reading of input data and writing of output data could have been done inside OpenModelica 8 2 Disadvantages with OpenModelica 67 8 2 5 Simulation time It takes long time to simulate models in OpenModelica even if the model is not very complex This is a problem when trying to use Monte Carlo simulations see Chapter 6 because then the model has to be simulated many times If for example the OMSep model see Chapter 5 is simulated 5000 times it takes about 50 hours This is the number of simulations used to perform the results
80. the differences are small One problem when comparing the models is that the SepSim model is more complex and includes more functions than the OMSep model such as more complex models for the springs and a model for how the clamp band release affects the system See Section 9 3 for a description of the functions which only exists in the SepSim model The input files to SepSim can be seen in Appendix C When simulating the models the variables that only exist in SepSim have been given values in such a way that they will affect the result as little as possible This was done to make the models more similar to each other and simpler to compare The models are not equivalent and because of this they do not give exactly the same results as can be seen in the plots 9 1 Comparing the models 71 4 x 10 10 Sep Sin Vi UhMSep O 0 05 0 1 0 15 time s Figure 9 2 The difference of vre for the two models is as can be seen quite small It depends on that the models are not exactly the same the SepSim model is more complex than the OMSep model 410 Wy rad s sepsim Perrier OMSep O 0 05 0 1 0 15 time s Figure 9 3 Comparison of the satellite angular rate around the X axis wz The solid line is the result from SepSim and the dotted line is the result from OmSep The difference depends on that the models are not exactly the same the SepSim model is more complex than the OMSep model But the difference is quit
81. the lift off and these are sometimes referred to as stage 0 The boosters and the stages separate when they have run out of fuel because then they are no longer useful and the rocket gets lighter to read more about how this works see Section 2 3 2 The fairing it is used to protect the satellite while in Earth s atmosphere this is no longer needed when outside the atmosphere and the launch vehicle will therefor jettison the fairing then After this has happened the third stage will be separated The last thing that will happen is that the satellite will separate from the launch vehicle To perform this separation a separation system described in Chapter 3 will give the satellite the needed velocity and angular rate to be ejected from the launch vehicle The following are a description of the steps in the flight sequence which can be seen in Figure 2 8 the numbers in the list refers to the numbers in the Figure 22 1 Lift off 2 Pitch over Booster separation First second stage separation Fairing jettison Second third stage separation Third stage first powered phase Third stage coast phase o 0 N 9 0 A WwW Third stage second powered phase 10 Attitude Adjustment 11 Satellite launch vehicle separation 2 4 Flight sequence during launch 15 Figure 2 8 Flight sequence when launching a satellite A list of the steps in the figure can be seen in Section 2 4 22 Chapter 3 Satellite separation When l
82. the syntax of a class definition The body of a function is an algorithm section that contains algorithmic code to be executed when the function is called Input and output parameters are also defined in the function 18 function name input declarations output declarations algorithm algorithm section end name 4 2 2 Formulation of equations To help Modelica solve an equation system and to give the right solution it is important how the equations are formulated But the most important is that the number of equations is equal to the number of variables Some aspects about formulating the equations are described in this section 18 Division by zero and square roots During a simulation a lot of problems that stop the solving process can occur One problem is division by zero Formulations involving square roots could be another problem because the square root function is returning complex numbers 30 Modelica for negative values This is a problem if no complex numbers are wanted in the model It is important to reformulate the equations in a way that these problems will not occur Tnitial values When Modelica solves an equation system each variable has the default initial value zero To increase the probability to find a solution and to do it in a more efficient way it is possible to set a more appropriate initial value To find the solution fast it is important to have a good initial value of as many variables as poss
83. this section 5 6 1 Spring local frame A spring local frame Xsp Ysp Zsp is defined to describe the position of the springs and the umbilical connectors The spring top connection point to the satellite is defined by the polar coordinates r and and the distance above the separation plane h expressed in the reference frame Xo Yo Zo The origin is defined to be the spring attachment point to the launch vehicle The spring local frame and the reference frame can be seen in Figure 5 5 h is assumed to be zero Figure 5 5 The reference frame Xo Yo Zo for the separation system and a spring local frame for the spring Xsp Ysp Zsp which is defined by the polar coordinates r and with origin in the spring attachment point to the launch vehicle 5 6 2 Spring position and force The input data to the model describes the position of the spring top connection point to the satellite the position of the spring local frame origin expressed in the reference frame Sp x y z is needed to be able to determine the torque caused by the spring How to calculate S and the force F caused by the springs 44 Model description depends on if it is a fixed spring a free spring or an umbilical connector this is described in this section Fixed springs When using fixed springs the spring can tilt an angle a as described in Figure 5 6 The spring position Sp is modeled as Sp h lcosa 5 29 Sp rsind lsina 5 30 Sp
84. ts from the SepSim The input data used to perform these plots can be seen in Appendix B and in Appendix C When simulating with these parameters the model describes a satellite separation where four symmetrically placed fixed springs see Section 3 2 3 and Section 5 6 for a description of springs are used to eject the satellite from the launch vehicle The satellite gets an angular rate due to that the satellite center of gravity and the launch vehicle center of gravity are not placed in origo of the reference frame see Section 5 2 for a description of the reference frame The relative separation velocity ve gives almost the same result for the two models as can be seen in Figure 9 1 the difference of vre can be seen in Figure 9 2 The angular rate around the Z axis wz also gives almost the same result for the two models as can be seen in Figure 9 5 the difference of w can be seen in 69 70 OMSep evaluation Ya ms 5ep Sim cana OMSep 0 0 05 0 1 0 15 time s Figure 9 1 Comparison of the relative separation velocity Ure between the satellite and the launch vehicle when using the input data files shown in Appendices B and C The solid line is the result from SepSim and the dotted line is the result from OMSep as can be seen the results are almost the same Figure 9 6 The angular velocities wy and wy differs a bit as can be seen in Figures 9 3 and 9 4 but their values are very small in the plots which means that
85. utput files are described in Chapter 5 and examples of the input data files can be seen later on in this appendix The output files can be run i Matlab and plots and histograms of the variables can be done to analyse the result An example of how the file Sep Inp can look like is shown below The input data for the satellite and the launch vehicle are set in this file This input data file was used to produce the results in Chapter 9 Input data Spacecraft and launch vehicle 435 0 Spacecraft mass 240 0 Launch Vehicle mass 0 0 0 0 0 0 Spacecraft initial orientation deg 0 0 0 01 0 0 Spacecraft Cog m 0 0 0 0 0 0 Launch Vehicle Cog m 180 0 180 0 180 0 Ixx Iyy Izz kgm2 Spacecraft 0 0 0 0 0 0 Pxy Pxz Pyz kgm2 105 0 105 0 105 0 Ixx Iyy Izz kgm2 Launch Vehicle 0 0 0 0 0 0 Pxy Pxz Pyz kgm2 92 User s manual An example of how the file Spring Inp can look like is shown below The input data for the springs and the umbilical connectors are set in this file This input data file was used to produce the results in Chapter 9 When using this input file only fixed springs will be used and the parameters for the springs are set here Input data Springs 1 0 0 4 0 0 43 8 43 8 43 8 43 8 100 0 100 0 100 0 100 0 0 0 0 0 0 0 0 0 365 0 365 0 365 0 365 0 0 0 90 0 180 0 270 0 193 0 193 0 193 0 193 0 22 0 22 0 22 0 22 0 0 0 0 0 0 0 0 0 1 if t
86. utters When using bolt cutters the clamp band consists of two band halves named straps for attaching the satellite to the adapter The straps are joined together by two strap joints that include connecting bolts At separation the connecting bolts are severed by two pyrotechnically operated bolt cutters CBOD The purpose of the CBOD mechanism is to release the satellite in a controlled way and compared with using bolt cutters it reduces the clamp band opening shock at separation The tension in the clamp band is reacted by the CBOD when closed This device will operated by an electrical signal from the launch vehicle release the tension in the clamp band and facilitate the proper release of the clamp band If a system requirement is that the clamp band opening shock has to be low this release mechanism is used The CBOD consists of a flywheel which is constructed with a left and a right thread and a pin puller A screw on the left thread and a screw on the right thread puts the flywheel and the clamp band together and tenses the clamp band The pin puller is the part that releases the flywheel and thus starting the release of the clamp band A CBOD with descriptions of these parts can be seen in Figure 3 3 A separation system which uses a CBOD can be seen in Figure 3 4 Separation nuts The function of the separation nuts is to release the mating bolt at command and they are pyrotechnically operated 3 2 3 Separation springs Separation
87. with the Modelica language is that the equations can be written as they are without any manipulations 8 1 2 No pre determined data flow direction Because the Modelica language is acausal and equation based there is no need to specify which variables are inputs and which are outputs No particular variable has to be solved manually and the data flow can take place in both directions This is a large advantage compared to for example Simulink where the blocks have a pre determined data flow from inputs to outputs To solve for example this equation system F mxp M pxF 8 2 63 64 OpenModelica evaluation when the mass m and the torque M are input variables and the position p is the output variable the equations can be written as they are in Modelica But if a language which is not acausal is used then the second equation M pxF 8 3 has to be reformulated in a way that p is on the left side and M and F are on the right side of the equation This makes the code harder to both write and understand This example clearly shows how advantageous it can be to use an acausal language compared to using a language with a pre determined data flow 8 1 3 Reusable code The Modelica language is object oriented and therefore the code can be reused The acausality of the language also makes the code more reusable than code from other object oriented languages such as C C or JAVA because these languages contain assignment statements w
88. y generates values for uncertain variables to simulate a model In OMSep the program developed in this master s thesis a Monte Carlo method is used This method randomly generates uniform distributed values for the uncer tain parameters and simulates the model with these values The implementation of this Monte Carlo function is described in Section 7 4 49 50 Monte Carlo methods 6 1 2 Mathematical problems Monte Carlo algorithms are also often used to find solutions to mathematical problems that are too complicated to solve analytically by for example integral calculus or other numerical methods For many types of problems its efficiency relative to other numerical methods increases as the dimension of the problem increases The most common application of the Monte Carlo method is to use it to do integration 6 1 3 Applications Monte Carlo methods have many application areas such as the following e Studying systems with a large number of coupled degrees of freedom such as liquids disordered materials strongly coupled solids and cellular structures e Evaluation of definite integrals e Calculation of risk in business e Computational physics physical chemistry and related applied fields 6 2 Basis components As can be seen in Section 6 1 there are many types of Monte Carlo methods and depending on application areas the methods can differ a bit The following components comprise the basis of most Monte Carlo applic

Separation Analysis with OpenModelica

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