Satellite Dynamics Toolbox - Principle, User Guide and tutorials
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1. Figure 2 7 Inverse dynamics model of the assembly hub appendage with a revo lute joint and a local mechanism or controller model K s expressed in frame R e to take into account that an appendage A can itself be the support the parent of another appendage A at point P Thus one can model any open kinematic chain of bodies that can be described through a connection tree To do this one has to build the global model starting from the tree leaves and using the previous approach where the body A is considered as a base hub or the parent of appendage A However because of assumption H1 each intermediate body must be rigid Flexible bodies are allowed only at the end of the kinematic chain The intrinsic formulae they do not depend on a projection frame that enable transport 2 3 or addition 2 4 of direct dynamics models allow to build the global model and to use it recursively The SDT toolbox was developed on this principle For example one can write the global direct dynamics model of the system shown on Figure 2 8 as follows ME Ta D Tip DA TA p Mi S rmi n TPB ThB Mg s Tp B One can also use this generalization to model revolute joints between elements as detailed in the previous section Thus it allows for the modelling of a gyroscopic actuator CMG fitted on a hub using three bodies as shown on Figure 2 9 e the hub B e the fork A2. 1 From 2 9 and 2 11 the direct dynamics model of the appendage at point A is Al 0 0 0 M amp As T 43 ve ge 3x3 Al 03x3 03x3 ino z Transporting this model to point P we get M s T1 M4 s rAp that leads to equation 2 7 when taking the first of the following remarks into account ISAE DMIA ADIS Page 22 58 od IS dE nstitut Sup rieur de l A ronautique et de l Espace 2 4 INVERSE DYNAMICS MODEL 23 Remarks independent of the projection frame used r 03x3 03x3 T 03x3 03x3 Arp e T ES ucc sd 03x3 inQ z 03x3 21 0 Zy e It clearly appears in 2 8 that the model of an embedded angular momentum appendage is a second order 6 x 6 model containing 2 integrators These 2 integrators were used to model the hub s rotation vector W from the angular acceleration W equation 2 11 that is the last 3 inputs of direct dynamics model e When computing a minimal realization of the inverse dynamics model M2 P s augmented with the 6 integrators needed to compute the speeds from the ac celerations these previous integrators disappear e When considering a hub fitted with N embedded angular momentum ap pendages A i 1 Na for instance reaction wheels the dynamics model Na of the assembly Mi AP which is a 2N order model should reduce to a second order model because the matrix h is rank 2 for all vectors h and 2 MIT 1 1 LEH DO
3. 8 NOMENCLATURE Figure 1 Example of a tree open chain structure Chapter 2 explains the modelling principles used in the Satellite Dynamics Toolbox Chapter 3 presents the MATLAB implementation of the toolbox and more partic ularly e the setup of the data files describing the structure to be modeled e structure examples Spacecraft1 m is a data file describing a spacecraft composed of a main body and two cantilevered flexible appendages see section 3 4 Spacecraft2 m is the same as Spacecraft1 m but the first appendage is connected to the main body through a revolute joint and tilted with a 10 degrees angle see section 3 5 Spacecraft5 m is the same as Spacecraft1 m but includes 3 additionnal appendages corresponding to angular momentums along to X Y and Z axes taken into account to represent spinned RWAs see section 3 6 FOUR CMGs m is a data file describing a platform fitted with four iden tical CMGs see section 3 7 The CMG is described by the data file dataCMG m This data correspond roughly to the experimental CMG platform TETRAGYRE developed by ONERA DCSD and presented at http www onera fr dcsd gyrodynes SpaceRoboticArm m is a data file describing a platform fitted with a 6 d o f rigid space robotic manipulator see section 3 8 the 6 links of the manipulator are described in the files Segment1 m Segment2 m Segment6 m All these files and a demo script file demoSTD m
4. 03e 02 8 03e 00i 5 02e 03 8 03e 00 4 03e 02 8 03e 001 5 02e 03 8 03e 00 5 06e 02 9 02e 00i 5 61e 03 9 02e 00 5 06e 02 9 02e 00i 5 61e 03 9 02e 00 9 62e 02 1 70e 01i 5 67e 03 1 70e 01 9 62e 02 1 70e 01i 5 67e 03 1 70e 01 1 09e 01 1 80e 01i 6 02e 03 1 80e 01 1 09e 01 1 80e 01i 6 02e 03 1 80e 01 1 11e 01 2 09e 01i 5 29e 03 2 09e 01 1 11e 01 2 09e 01i 5 29e 03 2 09e 01 ISAE DMIA ADIS Page 39 58 E IS dB Institut Sup rieur de l A ronautique et de l Espace A0 3 IMPLEMENTATION WITH MATLAB Frequencies expressed in rad seconds gt gt gt in addition to flexible modes one can see a very low frequency gt gt around 0 1 rd s mode corresponding to the gyroscopic mode gt gt figure gt gt sigma Mi gt a very low fequency resonance gt gt Angular model gt gt Mi Mi 4 6 4 6 gt gt Integrations between angular acceleration and rates gt gt int tf 1 1 0 gt gt Mi_int minreal Mix int eye 3 2 states removed gt gt gt 2 states removed extra integrators to model gyroscopic couplings gt gt are removed as wanted The frequency domain response is shown on Figure 3 3 Singular Values dB Singular Values Frequency rad s Figure 3 3 Frequency domain response Singular Values of the inverse dynamics model 3 7 Example 4 FOUR_CMGs m This file describes the example of a platform fitted with a cluster of
5. 15 208 9 27 79 y9 30 01 385 8 225 1 11 91 y10 385 8 7804 3983 145 3 yii 225 1 3983 1 088e 04 265 9 y12 11 91 145 3 265 9 2464 Static gain gt gt 4 This model is Static since all links are rigid 3 9 Example 6 Spacecraftiu m This version of the toolbox SDT Version 1 3 also allows to take parametric varia tions of parameters into account and more particularly the main structural param eters such as the mass the inertias the geometry position of the centre of mass and position of the anchoring points the angular frequencies dampings and modal par ticipation factors To sum up all the parameters defined by the fields of type double of the structures MB and SA i see table 3 1 and 3 2 except the angular parameters SA i TM SA i pivotdir and SA i angle can be defined with parametric un certainties The varying parameters can be defined by the function ureal from the RoBusT CONTROL TOOLBOX RCT Then the output variables of the SDT are uncertain matrices or state space representations and are fully compatible with the analysis tools of the RCT The file Spacecraftiu m is based on example 2 and adds parametric variations on dynamics and geometric parameters of the hub and of the appendage The following Page 48 58 Eu ac Nr Institut Sup rieur de l A ronautique et de l Espace ISAE DMIA ADIS 3 9 EXAMPLE 6 SPACECRAFTIU M 49 MATLAB session shows how to perform a sensitivity analysis of the frequency res
6. DA 59 ap Bley 2 7 where D comes from 2 5 where T Kg m is the appendage s inertia about its spin axis spinning top defined by unit vector Z and where is the spinning top s speed rad s Complementary assumptions When the appendage is an embedded angular momentum we will further suppose that H4 Q and Z the angular speed and the spin axis of the spinning top are constants in frame Ra H5 the spinning top is balanced H6 non restrictive the spin axis is along the z axis of the appendage body frame Ra Use Zo Zal ISAE DMIA ADIS Page 21 58 D ISdE Institut Sup rieur de l A ronautique et de l Espace 22 2 SATELLITE DYNAMICS TOOLBOX PRINCIPLES Under assumptions H5 and H6 one can write 0 0 Ly D alr ec ON O Ra where 1 is the spinning top s principal inertia about Z and 1 its radial inertia When expressed in frame R equation 2 7 becomes 03x3 03x3 A A NE m ls 03x3 0 I s 0 MP ln is als 03x3 Ale raphe 03x3 IyQ 1 s 0 0 0 0 0 2 8 Proof of the model 2 7 At point A the appendage s centre of mass one can write Faja m d 2 9 dH dH S In D A HA he D A H from H1 2 10 where HA Ag QZ is the total angular momentum of the appendage dt o H4 01021 Mo a OE Oz Tena Dot du ao O AI V A 1 22 YS
7. ISdE Institut Sup rieur de l A ronautique et de l Espace 3 2 USER DEFINED DATA FILE NOM FICH M 31 e antisym m the syntax M antisym V computes the skew symmetric matrix M eV associated to the vector V V 3 components projected in the R a frame R see nomenclature 3 2 User defined data file nom fich m From a reference frame Rp the user defined data file nom fich m describes the geometry the dynamics parameters of the main body hub and of the different appendages 4 Running this file through gt gt nom fich must creates 3 variables e MB structure describing the main body B The different required fields of this structure are detailed in table 3 1 e nappend the number of appendages N integer e SA vector of N cells Each cell SA i i 1 nappend describes the appendage A The different required fields of the structure SA i are detailed in table 2 2 3 2 1 Remarks One can also define appendage SA i in an independent data file nom fich i m see table 3 2 line 6 that creates the same variables MB nappend and SA as nom fich m It allows the user e to use the recursivity of main m in order to model chains of rigid bodies see example data files SpaceRoboticArm m and files Segment1 m to Segment6 m provided with the toolbox They model a robotic manipulator on a space platform e to use only one data file when the platform is fitted with several identical appendages We can meet this
8. Page 43 58 a ISdE Institut Sup rieur de l A ronautique et de l Espace 44 3 IMPLEMENTATION WITH MATLAB 7 11e 15 6 32e 01i 1 12e 16 6 32e 01 7 11e 15 6 32e 01i 1 12e 16 6 32e 01 8 29e 17 8 59e 01i 9 65e 19 8 59e 01 8 29e 17 8 59e 01i 9 65e 19 8 59e 01 Frequencies expressed in rad seconds gt gt gt angular frequecies of gyroscopic modes have changed gt gt First integrators angular rates on outputs on nominal model Mi gt gt int tf 1 1 0 gt gt Gv minreal int eye 7 Mi 3 states removed gt gt Precession rate servo loop inner loop gt gt bf2 feedback Gv diag 1 1 1 0 06 0 06 0 06 0 06 eye 4 4 7 4 7 gt gt bf2 minreal bf2 3 states removed gt gt damp bf2 3 states removed OK Eigenvalue Damping Frequency 3 00e 01 3 81e 01i 6 18e 01 4 85e 01 3 00e 01 3 81e 01i 6 18e 01 4 85e 01 3 00e 01 3 81e 01i 6 18e 01 4 85e 01 3 00e 01 3 81e 01i 6 18e 01 4 85e 01 5 99e 01 1 00e 00 5 99e 01 3 00e 01 7 85e 01i 3 57e 01 8 40e 01 3 00e 01 7 85e 01i 3 57e 01 8 40e 01 Frequencies expressed in rad seconds gt gt gt gyroscopic modes are correctly damped gt gt gt gt os Platform attitude servo loop 23 The jacobian is defined as the DCgain of the transfer between the 4 gt gt precession rates reference inputs and the 3 platform angular rates Jacob dcgain bf2 1 3 4 7 Jacob 0 0120 0 0120 0 0
9. along the embedded angular momentum L Spinning top s principal inertia I Spinning top s radial inertia If there is a revolute joint between the appendage and the hub we add the following definitions T Unit vector along the revolute joint axis T r Yr Ss e 6 Revolute joint s angular acceleration Cm Revolute joint s torque General definitions Xir X model vector or tensor expressed in frame R ax R U Derivative of vector X with respect to frame R UAG Cross product of vector d with vector A v w v Uv Scalar product of vector with vector v w v ar W e VRi S LAPLACE s variable Ij Identity matrix n x n zo Zero matrix n x m AT Transpose of A diag w Diagonal matrix N x N diag w i 4 w i 1 N Pista i44 rm Sub system of P s from inputs to m to outputs to j ISAE DMIA ADIS Page 6 58 T ISde Institut Sup rieur de l A ronautique et de l Espace Introduction This document presents the principles and the MATLAB implementation of the Satellite Dynamics Toolbox SDT version 1 3 available for download following this link http personnel isae fr daniel alazard matlab packages satellite dynamics toolbox html This toolbox allows the user to compute the linear dynamics model of a satellite fitted with one or more flexible appendage solar generators antennas possi bly taking into account the joints and actuators
10. are included in the SDT package ISAE DMIA ADIS Page 8 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace NOMENCLATURE 9 Remark this new version of the toolbox can also manage uncertain parameters to perform sensitivity analyses In this case the outputs of the toolbox main func tions are uncertain elements matrices or models compatible with the MATLAB Robust Control Toolbox see file Spacecraftiu m as an example section 3 9 The development of this version of the toolbox was done with MATLAB R2013a ISAE DMIA ADIS Page 9 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace EMPTY PAGE 11 Chapter 1 Rigid body dynamics based on Newton Euler equations This chapter is a reminder on rigid body dynamics based on NEWTON EULER equa tions see also 6 The interest of NEWTON EULER equations is to consider in the same model the 6 degrees of fredom 3 translations and 3 rotations of the rigid body This chapter will also present the linearity assmuptions which will be adopted in the following chapters 1 1 Newton Euler equations at the center of mass Let us consider a body B with a center of mass B see Figure 1 1 Fa ext B Figure 1 1 A rigid body X ISAE DMIA ADIS Page 11 58 LT ISa institut Sup rieur de l A ronautique et de l Espace 12 1 RIGID BODY DYNAMICS BASED ON NEWTON EULER EQUATIONS The NEWTON EULER equations reads Pi
11. in 06 X Y Z m SA 1 TM cos 80 pi 180 0 sin 80 pi 180 T A6A7 0 1 0 sin 80 pi 180 O cos 80 pi 180 SA 1 pivot 0 4 cantilevered joint SA 1 filename Spacecraft1 ISAE DMIA ADIS Page 50 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace 3 10 EXERCISE 51 Figure 3 9 A manipulator arm fitted on a spatial vehicle holding a debris spacecraft Secondly the data file Spacecraft1 m must be completed by the following instruc tions to take into account the new reference point P instead of the previous one O gt DEBRIS case taking into account P7_07 MB cg MB cg 0 3 0 1 2 SA 1 P SA 1 P 0 3 0 1 2 SA 2 P SA 2 P 0 3 0 1 2 Then the model of the assembly chaser robotic arm debris can be computed using gt gt clear all close all gt gt global config gt gt mod main SpaceRoboticArm ISAE DMIA ADIS Page 51 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace EMPTY PAGE 53 References 1 M Rognant Ch Cumer Configurations 6 actionneurs gyroscopiques Etude bib liographique ONERA RT 1 22821 DCSD Janvier 2016 2 D Alazard Ch Cumer and K Tantawi Linear dynamic modeling of spacecraft with various flexible appendages and on board angular momentums 7th International ESA Conference on Guidance Navigation amp Control Systems 01 05 Jun 2008 Tralee Ireland 3 G Nicola
12. l Espace 36 3 IMPLEMENTATION WITH MATLAB gt gt Inverse Dynamic Model gt gt Mi mod InverseTotalModel gt gt damp Mi The 5 6 1 flexible modes Eigenvalue Damping Frequency 4 03e 02 8 03e 00i 5 02e 03 8 03e 00 4 03e 02 8 03e 00i 5 02e 03 8 03e 00 5 07e 02 9 03e 00i 5 61e 03 9 03e 00 5 07e 02 9 03e 00i 5 61e 03 9 03e 00 9 64e 02 1 70e 01i 5 67e 03 1 70e 01 9 64e 02 1 70e 01i 5 67e 03 1 70e 01 1 09e 01 1 80e 01i 6 02e 03 1 80e 01 1 09e 01 1 80e 01i 6 02e 03 1 80e 01 1 11e 01 2 09e 01i 5 29e 03 2 09e 01 1 11e 01 2 09e 01i 5 29e 03 2 09e 01 Frequencies expressed in rad seconds gt gt figure gt gt sigma Mi Frequency domain response The frequency response is presented on Figure 3 1 Singular Values Singular Values dB _o 24 3 10 10 10 10 Frequency rad s Figure 3 1 Frequency response Singular Values of the inverse dynamics model ISAE DMIA ADIS Page 36 58 Toad ISdE Institut Sup rieur de l A ronautique et de l Espace 3 5 EXAMPLE 2 SPACECRAFT2 M 3T 3 5 Example 2 Spacecraft2 m This file describes the same platform as example 1 but now the link between the solar array and the hub is a revolute joint along Z Za The following MATLAB session see also the script file demoSTD m shows how to use this file with the SDT and perform complementary analyses more particularly study the response of the revolute joint
13. nie 6 3 0 BExunple3 Spacecraftb m gt 4 44 4 4 Dia ba ER XEE X ER ag ample dr POUR CMOS E look Louis od Pod mad ist ISAE DMIA ADIS Page 3 58 od 4 CONTENTS 38 Example D SpacstoboticArm m e ca 46 9o hbase niai 45 39 Example 6 Spacecraftiu m 4 4 ocu RR xS 48 DID EXE eee Re ee Sels ess Ed xu dat 50 References 53 A Inline help 55 Ad help oi function main d es s xo dex ee Be Es 55 A 2 help of function translate dynamic model m 56 A 3 help of function rotate dynamic model m 56 A 4 help of function kinematic model m 56 A5 help of function antisym m e s sossa 999 d og os 57 ISAE DMIA ADIS Page 4 58 a S dE Institut Sup rieur de l A ronautique et de l Espace Nomenclature Ra O Zo Yo Z t Ra P Zas Ya Za HQ x ty B v amp e 8 gt w GS S x ssl EL eL eL B A P oL DSO x23 Soe C Main body hub B reference frame O is a reference point on the main body xj Ye are unit vectors Appendage A reference frame P is the appendage s anchor ing point on the hub Za Ya Z are unit vectors Hub s B centre of mass Appendage s A centre of mass Overall assembly s centre of mass B A Direction cosine matrix of the rotation from frame R to frame Ra that contains the coordinates of vectors Za Ya Za x pressed in frame R4 Inertial acceleration
14. of the spacecraft the inverse of the previous one MODEL liste IO0s Description and ordering of the 6 NBPIVOTS inputs outputs used in the dynamic model Reference Alazard D Cumer C and Tantawi K Linear dynamic modeling of spacecraft with various flexible appendages and on board angular momentums ISAE DMIA ADIS Page 55 58 T ISdE Institut Sup rieur de l A ronautique et de l Espace 56 A INLINE HELP In Proceedings of the 7th International ESA Conference on Guidance Navigation amp Control Systems Tralee Ireland 1 5 June 2008 A 2 help of function translate dynamic model m TDM translate dynamic model vec A vec B DM translate the direct dynamic model from point A to point B in the same reference frame Inputs vec 3x1 coordinate vector of point A vec B 3x1 coordinate vector of point B DM dynamic model at point A Output TDM dynamic model at point B It is assumed that the first 6x6 block of DM represent the dynamic model between the 6 dof acceleration vector and the 6 dof external force vector A 3 help of function rotate dynamic model m DM OUT rotate dynamic model DM IN ANGLE AXIS computes the dynamic model after a rotation of ANGLE deg around the axis AXIS DN IN dynamic model of a body in a frame R ANGLE rotation angle deg AXIS 3 components in the frame R of the unitary vector along the rotation axis DN OUT dynamic model of the ro
15. 000 0 0000 0 0000 0 0000 0 0120 0 0120 0 0035 0 0035 0 0035 0 0035 gt gt Second integrators platform angular positions and rates on outputs ISAE DMIA ADIS Page 44 58 Eu ac 150 Institut Sup rieur de l A ronautique et de l Espace 3 8 EXAMPLE 5 SPACEROBOTICARM M 45 gt gt sint ss 0 1 1 0 0 1 gt gt Gpv minreal append sint sint sint 1 1 1 1 bf2 gt gt CMGs guidance gt gt Gpv Gpv eye 3 zeros 3 zeros 4 3 pinv Jacob gt gt Attitude servo loop bandwitdh outer loop gt gt w 3 gt gt bf feedback Gpv diag 1 1 1 w w w eye 3 4 5 6 1 3 5 damp bf Eigenvalue Damping Frequency 3 08e 00 1 00e 00 3 08e 00 3 25e 00 1 00e 00 3 25e 00 3 25e 00 1 00e 00 3 25e 00 2 84e 01 3 69e 01i 6 09e 01 4 66e 01 2 84e 01 3 69e 01i 6 09e 01 4 66e 01 2 84e 01 3 69e 01i 6 09e 01 4 66e 01 2 84e 01 3 69e 01i 6 09e 01 4 66e 01 5 99e 01 1 00e 00 5 99e 01 2 84e 01 7 80e 01i 3 43e 01 8 30e 01 2 84e 01 7 80e 01i 3 43e 01 8 30e 01 Frequencies expressed in rad seconds gt gt gt all the modes are correctly damped gt gt figure gt gt step bf 1 3 5 4 5 61 2 The step response is shown on Figure 3 6 3 8 Example 5 SpaceRoboticArm m This file describes a spatial vehicle fitted with a manipulator arm with 6 degrees of freedom and a high aspect ratio as shown on Figure 3 7 We suppose that the segments are all rigid The obtained model ha
16. 0e 03 Main inertia in 0 0086 1 0e 03 4 0907 1 0e 03 0 0114 1 0e 03 0 X1 Y1 Z1 0 0265 1 0e 03 0 0114 1 0e 03 0 4606 1 0e 03 Kg m 2 I MB IO MB m antisym MB cg 2 Huygens theorem MB Ixx MB I 1 1 Main Inertia in MB Iyy MB I 2 2 CG1 X1 Y1 Z1 kgm 2 MB Izz MB I 3 3 MB Ixy MB I 1 2 Cross Inertia in MB Ixz MB I 1 3 CG1 X1 Y1 Z1 kgm 2 MB Iyz MB I 2 3 ISAE DMIA ADIS Page 32 58 LE Nur Institut Sup rieur de l A ronautique et de l Espace 3 3 FAQ 33 In this example the HUYGENS thereom is applied to the main body MB but the same procedure can be applied to any appendage SA i 3 3 FAQ e The field liste IOs returned by the function main is wrong the function main m is called recursively and uses persistent variables In a nominal use the persistent variables are managed inside the function main m but if the function returned on an error message due to wrong data in nom fich m for instance it is then required to clear persistent variables for the next call by the command clear main e The fields DynamicModel or InverseTotalModel returned by the function main contains some NaN the mass or inertia seen from one of the d o fs is null There is a mistake in the file nom fich m e The ss model returned by the function main in the field DynamicModel or InverseTotalModel contains unstable dynamics 1 the mass or inertia of one or more bodies is not definite positive The
17. 2 nappend 2 flex ible appendages SA 1 and SA 2 cantilevered in 2 different points of the platform e a solar array SA 1 Name Solar Array described by 3 flexible modes e an antenna SA 2 Name Antenna described by 3 flexible modes The frequencies w i 1 2 3 of the 3 modes are the same for both appendages Because of the symmetry of the two modal participation factors along the z axis of each appendage one flexible mode is uncontrollable It is reduced when computing a minimal realization of the model The following MATLAB session see also the script file demoSTD m shows how to use this file with the SDT and perform complementary analyses gt gt mod main Spacecraft1 2 states removed gt gt 1 flexible mode removed due to symmetry of flexible appendage gt gt mod TotalCg the position of the total centre of mass G ans 0 1909 0 3909 0 gt gt mod liste IOs list of channels ans Trans X Platform Trans Y Platform Trans Z Platform Rot X Platform Rot Y Platform Rot Z Platform gt gt Md mod DynamicModel direct dynamic model gt gt Translation of the direct dynamic model from the global centre of mass G gt gt to the hub s centre of mass B gt gt Spacecrafti to get all the data in the workspace gt gt MD_B translate_dynamic_model mod TotalCg MB cg Md ISAE DMIA ADIS Page 35 58 D ISdE Institut Sup rieur de l A ronautique et de
18. 4 identical gyroscopic actuators in a pyramidal configuration The geometries weights and inertias of the different bodies are very close to those of the experimental platform TETRAGYRE developed by ONERA DCSD see http www onera fr dcsd gyrodynes ISAE DMIA ADIS Page 40 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace 3 7 EXAMPLE 4 FOUR CMGs M 41 and Figure 3 4 The default configuration is obtained when the spinning tops axes are horizontal and when the sum of the angular momentums is zero Figure 3 4 The bench TETRAGYRE The main data file is FOUR_CMGs m It describes the platform hub B and the nom inal configuration of the 4 CMGs One difference with the example files detailed above is that we add a global variable angles 4 x 1 that represents the current configuration of the precession axes It also enables the computation of the dy namics model in any configuration without modifying the data file This variable is initialized with the nominal angular configuration angles 0 0 0 0 The file dataCMG m describes one CMG see Figure 2 9 and is called 4 times by the file FOUR_CMGs m The CMG is modelled as a fork A for the main part of the appendage and a spinning top W as the appendage s appendage The following MATLAB session see also the script file demoSTD m shows how to use this file with the SDT performs complementary analyses and proposes a 3 axes attitude control law This cont
19. E ER 2e le P Zu SCH where h gt KIS is the total angular momentum of the global assem bly containing the N spinning tops 2 4 Inverse dynamics model Figure 2 4 shows the block diagram of the inverse dynamics model MP s expressed in frame Ry It also shows that the appendage s direct dynamics model is a feedback for the hub s inverse dynamics model D3 M s DS Is M SIDS The usefulness of this representation instead of the direct inversion turns up when conducting sensitivity or robustness analyses with respect to parametric variations of the appendage s model Indeed there is no need to invert the direct dynamics model of the appendage M2 s Moreover one can assure that each characteristic parameter of the model M s m IA wi amp lip Q appear a minimum number of occurences ISAE DMIA ADIS Page 23 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace 24 2 SATELLITE DYNAMICS TOOLBOX PRINCIPLES Me s Fea Tp Taar Ra B Tha 03x3 Tha 03x3 T 03x3 Tha 03x3 Tha Tbe Ro PB Twa Taan Ry Ta d LU Ry Ro Figure 2 4 Inverse dynamics model block diagram of the appendage M s expressed in frame Rp 2 5 Revolute joint between the appendage and the hub Let us consider Figure 2 5 in which the joint between the hub B a
20. I PB Indeed y T a BP F ext lever arm x force The time derivation of 1 3 in the inertial frame gives p dpt PBAU AD DA P AD 1 7 B 0 H1 or equivalently p _ n CPR s _ care 1 8 wo 033 Ts uj 03x1 TPB Using 1 2 1 6 and 1 8 one can derive the NEWTON EULER equations at point P Pao mh BRB ap mB RP E m BP B m B p T 1 m BP NE T TEpDBTBP 1 9 1 3 Linearized Newton Euler equations Assuming that is small linearity assumption H2 quadratic terms in PODES can be neglected Then equations 1 2 and 1 9 becomes Tas ms Osa da PI B d ML NC DB D is the direct dynamics model of the body B about the centre of mass B ISAE DMIA ADIS Page 13 58 D ISdE nstitut Sup rieur de l A ronautique et de l Espace 14 1 RIGID BODY DYNAMICS BASED ON NEWTON EULER EQUATIONS 7 D Tgp DB TBP 1 10 s m mP Is m CBP Tanp mE CBP 18 m BP D D3 rep is the direct dynamics model of the body B about the point P Equations 1 10 is very convenient to transport the direct dynamics model from one point to another considering a third point R one can also write D Ton D TBR Thr Tp DE TPBTBR 1 11 DR TprDPTPR 1 12 Without loss of generality one can write the linear dynamics model of a mechanical system at its center of mass B and
21. S Page 49 58 T S dE Institut Sup rieur de l A ronautique et de l Espace 50 3 IMPLEMENTATION WITH MATLAB 3 10 Exercise Statement Using the data file SpaceRoboticArm m see section 3 8 describing a chaser space craft with a robotic arm and the data file Spacecraft1 m see section 3 4 describing a spacecraft with two symmetrical flexible appendages the objective of this exercise is two build the data file to analyse the dynamic behavior of the chaser holding the spacecraft considered as a debris see Figure 3 9 The interface point P between the arm end effector and the debris is characterized by e its coordinate vector in the frame R attached to the end effector Segment6 m EE 204 0 0 0 55 Rag e its coordinates vector in the frame Ra attached to the debris spacecraft PE Spacecraft1 m nra 0 3 0 13 R ar The direction cosine matrix between frames Ras and Ra see nomenclature is cos 0 sind M ais 0 1 0 with 0 80 deg sin 0 cos Solution Firstly the data file Segment6 m must be completed by the following instructions to take into account that Segment6 the end effector hold an appendage described in the data file Spacecrafti m nappend 1 HL aa APPENDAGE 1 SA 1 DEBRIS SPACECRAFT SA 1 Name DEBRIS Name of next appendage the debris SA 1 P 0 0 0 55 4 position of connection point P7
22. T 1500 ONERA Institut Sup rieur de l A ronautique et de l Espace THE FRENCH AEROSPACE LAB Satellite Dynamics Toolbox Principle User Guide and tutorials Daniel Alazard Christelle Cumer June 2014 EMPTY PAGE Contents Nomenclature Rigid body dynamics based on Newton Euler equations 11 NEWTON EULER equations at the center of mass 1 2 NEWTON EULER equations at any reference point 1 3 Linearized NEWTON EULER equations Satellite Dynamics Toolbox principles 2 ABPO enera RR de EO ee Ce GRR duo 22 Direct dynamics model poh se das mu das de OO eue 2 3 Appendage dynamics model MA s 2 3 1 Case of a rigid appendage 2 3 2 Case of a flexible appendage 2 3 3 Case of an embedded angular momentum 24 Inverse dynamics model 4 ca sero wao t pu due ue da 2 5 Revolute joint between the appendage and the hub 2D Ganeralisation es en she dus Bi dan Dent Des du Implementation with Matlab 3 1 Satellite Dynamics Toolbox functions 3 2 User defined data file nomfich m K MMS o esce xo qox a Xo X CR OX ede ooh ROS 3 2 2 A first tutorial on inertia tensor specified at any reference cog PP k oe ee SS HS Ee Be EDS Mo FAU s siu so Bea DE OED amp eR DMA Ow A ores 24 Example 1 Spasseraftlam 4 242249 YS ES XA 3 5 Example 2 Spacecraft2 m 2429 93 ee VRE
23. U T U A B __ nB T ba U3x3 A ba U3x3 d DE Ed gm TX DG ot real where Ty is the direction cosine matrix of the rotation from frame R to frame Ra see nomenclature Figure 2 2 represents this model s block diagram and matching physical signals Mg s Tp F gja pi m Tiji Ra Tha 03x3 Tha 03x3 03x3 Tha 03x3 Tia TPe r res Rp E W p DS Figure 2 2 Direct dynamics model block diagram M7 P s expressed in frame Ry The following operation allows for the transport of the direct dynamics model to point G the global assembly s centre of mass Mg s T5 G Mg P s TBc 2 3 Appendage dynamics model M s There are 3 different cases to consider in order to write the appendage s dynamics model MA s at anchoring point P between A and B see 2 ISAE DMIA ADIS Page 19 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace 20 2 SATELLITE DYNAMICS TOOLBOX PRINCIPLES 2 3 1 Case of a rigid appendage mI 0 Mets D e ra 2 5 3x3 A This is a static model 2 3 2 Case of a flexible appendage N g M4 s D4 gt PA 2 6 p s P iP Pd D S w2 i 1 in which De comes from 2 5 and the various parameters w amp et l p see nomen clature can be provided by finite element software used to model the appendage see 2 or 3 for more explanations Other repr
24. case when modelling a platform fitted with a cluster of gyroscopic actuators see example data file presented in section 3 7 ISAE DMIA ADIS Page 31 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace 32 IMPLEMENTATION WITH MATLAB 3 Variable name Notation Type Unit MB Name char MB cg OB double 3 x 1 m MB m m i double Kg MB IXX Am 1 1 double Kgm MB Iyy 15 Ra 2 2 double Kgm MB Izz 1 r 3 3 double Kgm MB Ixy 15 Rp 1 2 double Kgm MB 1xz 5 3 1 3 double Kgm MB Iyz 1 r 2 3 double Kgm nappend Na integer Table 3 1 Description of the fields of the structure MB 3 2 2 A first tutorial on inertia tensor specified at any ref erence point In the user defined data file nom fich m the tensor of inertia of each body fields MB I and SA i I must be defined w r t to its centre of mass In some applications data are provided at a reference point O different from the centre of mass D then HUYGENS thereom 1 15 can be used in the script file nom fich m by using the following MATLAB sequence based on the function antisym MB Name Platform cg 0 0063 0 0200 1 7744 MB MB MB MB m 905 6290 Name of the body position of gravity center in 0 X Y 2 m mass kg I02 3 8991 1 0e 03 0 0086 1 0e 03 0 0265 1
25. connected through a revolute joint precession axis Ta Za with B at point P e the spinning top A W linked with A at point P G and with its angular momentum along Z according to the assumptions H6 ISAE DMIA ADIS Page 27 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace 28 2 SATELLITE DYNAMICS TOOLBOX PRINCIPLES Figure 2 8 Example of an open kinematic chain Figure 2 9 Illustration of a CMG ISAE DMIA ADIS Page 28 58 Vd ISdE Institut Sup rieur de l A ronautique et de l Espace 29 Chapter 3 Implementation with Matlab The MATLAB package SDT_V1 3 zip to implement the Satellite Dynamics Tool box can be downloaded from http personnel isae fr daniel alazard matlab packages satellite dynamics toolbox html This toolbox requires a user defined data file describing the structure of the space craft The variables to be declared by such a script file are described in section 3 2 The MATLAB package contains also several examples of such a script file detailled in sections 3 4 to 3 9 and the associated demo file demoSTD m It is advised to the user to run step by step this demo file and to use one of the data file examples as a guide to create its own data file The next section describes the various functions included in the toolbox in relation with the general notations used in the document and summarized in the nomenclature The inline help of these functions a
26. en introducing the 3 integrators be tween the 3 angular accelerations and speeds of the hub All the integrators added into the dynamics model to take into account gyroscopic couplings dis appeared in the reduction as expected ISAE DMIA ADIS Page 38 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace 3 6 EXAMPLE 3 SPACECRAFT5 M 39 gt gt clear all close all gt gt mod main Spacecraft5 2 states removed 2 states removed 2 states removed gt gt 1 flexible mode removed due to symmetry of flexible appendages gt gt 2 double integrators removed gt gt Md mod DynamicModel direct dynamic model gt gt damp Md only 2 integrators to represent the 3 angular momentums Eigenvalue Damping Frequency 1 36e 15 1 00e 00 1 36e 15 5 68e 15 1 00e 00 5 68e 15 4 00e 02 8 00e 00i 5 00e 03 8 00e 00 4 00e 02 8 00e 00i 5 00e 03 8 00e 00 4 00e 02 8 00e 00i1 5 00e 03 8 00e 00 4 00e 02 8 00e 00i 5 00e 03 8 00e 00 7 50e 02 1 50e 01i 5 00e 03 1 50e 01 7 50e 02 1 50e 01i 5 00e 03 1 50e 01 7 50e 02 1 50e 01i 5 00e 03 1 50e 01 7 50e 02 1 50e 01i 5 00e 03 1 50e 01 1 00e 01 2 00e 01i 5 00e 03 2 00e 01 1 00e 01 2 00e 01i 5 00e 03 2 00e 01 Frequencies expressed in rad seconds gt gt Mi mod InverseTotalModel gt gt damp Mi Eigenvalue Damping Frequency 2 14e 11 9 21e 02i 2 32e 10 9 21e 02 2 14e 11 9 21e 02i 2 32e 10 9 21e 02 4
27. esentations can be used for M s e the hybrid cantilevered model Ta i p pal T Tiar w Loi dp 7j diag 2 w n diag w n Lp EN e the second order generic model E A E B 06x6 Osx N E B 06x6 Dox Lp Iw 7 Onxe diag 2 ur 7 Ovxe diag w7 Pu Taat e the state space representation with a feedforward matrix D a P where d and Fert rl Los which is the residual mass of the appendage A rigidly cantilevered to the hub B at point RUR HE BAIE diag w2 diag 2 w ie mi P Hou Taie LE diag w LE diag 2 u 7 gt c pp d pi DA Po e or the block diagram on Figure 2 3 This diagram easily allows for the intro duction of parametric uncertainties on w Ei Lp and D because they appear with a minimum number of occurences ISAE DMIA ADIS Page 20 58 nsti 150 itut Sup rieur de l A ronautique et de l Espace 2 3 APPENDAGE DYNAMICS MODEL M s 21 MP 8 n 9 1 l J i 1 diag w diag w 4 zn or 1 i 2 7 i Y i Lp a diag 2 amp 4 Tu Lr i k i i n pale Ps 2 E T Figure 2 3 Direct dynamics model block diagram of the appendage M s expressed in frame Ra 2 3 3 Case of an embedded angular momentum 0 0 MAG
28. g E mPIs m E B 03x3 B dA 1 1 ext B DB where e Fag total force acting on the body T ad p total torque acting about the centre of mass e m mass of the body 5 Inertia tensor of the body about the centre of mass T g acceleration of the centre of mass e W angular velocity of the body Eq 1 1 is intrinsic and can be projected in any frame Using the skew matrix associated with vector W see nomenclature Eq 1 1 can be written pe m l oa 1 2 Newton Euler equations at any reference point Let us consider another point P on the body B Since the body is rigid assumption H1 the velocity of P can be expressed as a function of the velocity of B and the angular velocity W S ABUPY uA 1 3 Using the skew matrix PB associated with vector BP see nomenclature one can write Vp UV PB 1 4 Considering the dual velovity vectors 6 components at point B and P one can SLE BIB e TPB ISAE DMIA ADIS Page 12 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace 1 3 LINEARIZED NEWTON EULER EQUATIONS 13 Tpp is called the kinematic model or jacobian between points P and B Property jg Tap Considering the dual force vectors 6 components at point B and P one can also write Ta I3 03x3 F eat T ioe 1 6 ext B P I3 ext P T
29. iagram of Figure 2 6 represents this operation It also shows the con nection of the first six inputs and outputs between P 4 S and the hub s direct model D in order to get the assembly model P2 P s expressed in frame Ry ISAE DMIA ADIS Page 25 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace 26 2 SATELLITE DYNAMICS TOOLBOX PRINCIPLES Taking into account a revolute joint between the hub and an appendage with an embedded angular momentum then allows for the modelling of CMGs Control Moment Gyros the axis of the joint being the precession axis of the CMG Pele Figure 2 6 Direct dynamics model 7 x 7 block diagram of the assembly hub appendage Pare s with revolute joint expressed in frame Rp For control laws synthesis one can use the following inverse model d 1 F ext p Rp P Rp s Tas Rp 0 G 2 15 that allows the user to introduce between the seventh input to the seventh output a local model of the joint mechanism or controller K s according to Figure 2 7 2 6 Generalisation One can generalise the previous approach e to take Na appendages A i 1 Na linked with the hub at points P into account ISAE DMIA ADIS Page 26 58 T ISdE Institut Sup rieur de l A ronautique et de l Espace 2 6 GENERALISATION 27 m
30. k 5 Link 4 Joint Link 4 Link 3 Joint Link 3 Link 2 Joint Link 2 Link 1 Joint Link 1 Main_body gt gt mod TotalCg Position of the global centre of mass ans 0 0270 0 3741 1 4893 gt gt mod DynamicModel ans d ul u2 u3 u4 yl 1400 6 311e 30 6 311e 30 7 272e 31 y2 6 311e 30 1400 3 864e 46 4 547e 13 y3 6 311e 30 3 864e 46 1400 3 411e 13 y4 T 272e 31 4 547e 13 3 411e 13 1 361 e 04 y5 0 3 088e 30 2 132e 14 57 02 y6 3 411e 13 2 132e 14 2 361e 30 266 3 y7 0 0 0 0 3375 y8 8 472e 16 32 74 18 9 202 1 y9 4 216e 15 25 2 43 65 241 7 y10 1 002e 13 598 8 1037 4377 y11 5 978e 14 1500 523 7 1 07e 04 y12 523 7 37 8 0 266 3 u5 u6 u7 u8 yl O 3 411e 13 O 8 472e 16 y2 3 088e 30 2 132e 14 0 32 74 y3 2 132e 14 2 361e 30 0 18 9 ISAE DMIA ADIS Page 47 58 E IS dB Institut Sup rieur de l A ronautique et de l Espace 48 3 IMPLEMENTATION WITH MATLAB y4 57 02 266 3 0 3375 202 1 y5 1 141e 04 1269 4 133e 17 15 53 y6 1269 2934 5 646e 17 26 91 y7 4 133e 17 5 646e 17 0 3375 2 067e 17 y8 15 53 26 91 2 067e 17 32 09 y9 19 45 11 23 0 3375 2 562e 15 y10 223 7 129 2 0 3375 2 562e 15 yii 57 02 225 4 0 3375 208 9 y12 1269 2267 5 646e 17 27 79 u9 u10 uil ui2 yl 4 216e 15 1 002e 13 5 978e 14 523 7 y2 25 2 598 8 1500 37 8 y3 43 65 1037 523 7 0 y4 241 7 4377 1 07e 04 266 3 y5 19 45 223 7 57 02 1269 y6 11 23 129 2 225 4 2267 y7 0 3375 0 3375 0 3375 5 646e 17 y8 2 562e 15 2 562e
31. latform gt gt Md mod DynamicModel direct dynamic model gt gt Model restricted to 3 axes angular motion CMGs channels gt gt Md Md 4 10 4 10 ISAE DMIA ADIS Page 42 58 T ISdE Institut Sup rieur de l A ronautique et de l Espace 3 7 EXAMPLE 4 FOUR CMGs M 43 gt gt One can check that the total angular momentum is null gt gt M3x3 minreal Md 1 3 1 3 6 states removed M3x3 d ui u2 u3 yl 5 315 O 0 005015 y2 0 5 315 0 005015 y3 0 005015 0 005015 10 61 Static gain gt gt gt The 3x3 model is static no more gyroscopic couplings between platform gt gt and inertial frame gt gt Mi inv Md Inverse dynamic model gt gt damp Mi Eigenvalue Damping Frequency 3 55e 15 4 85e 01i 7 33e 17 4 85e 01 3 55e 15 4 85e 01i 7 33e 17 4 85e 01 5 33e 15 4 85e 01i 1 10e 16 4 85e 01 5 33e 15 4 85e 01i 1 10e 16 4 85e 01 7 25e 17 8 40e 01i 8 63e 19 8 40e 01 T 2be 17 8 40e 01i 8 63e 19 8 40e 01 Frequencies expressed in rad seconds gt gt gt the pulsations of the 3 gyroscopic modes gt gt Dynamic model for a new angular configuration gt gt global angles gt gt angles 10 20 5 25 gt gt mod bis main FOUR CMGs Angular configuration 10 20 5 25 2 states removed damp mod bis InverseTotalModel Eigenvalue Damping Frequency 1 24e 16 3 13e 01i 3 95e 18 3 13e 01 1 24e 16 3 13e 01i 3 95e 18 3 13e 01 ISAE DMIA ADIS
32. nd the appendage A at point P is a revolute joint along Fr axis We use the following definitions e is the angular acceleration inside the revolute joint do Wag F or ays Yra Ra ds e Cm is the torque if present along P Z applied by an actuator located inside the revolute joint The objective is to compute the augmented direct model P2 P s 7 x 7 of the assembly A 4 B such that F d ext lt 2 MR POP s w C ISAE DMIA ADIS Page 24 58 Vd ISdE Institut Sup rieur de l A ronautique et de l Espace 2 5 REVOLUTE JOINT BETWEEN THE APPENDAGE AND THE HUB 25 E Figure 2 5 Assembly of the base B and the appendage A linked with a revolute joint along Z Because of the revolute joint the projection of the torque Ta A P exerted by the base on the appendage at point P along T axis is either null in case of a free revolute joint or equal to Cm in case of an actuated joint Gps Toss 2 12 Expressing the direct dynamics model M s of the appendage at point P in frame R enables us to write that fua B A P dp 243 paw 219 MAG From 2 12 and 2 13 one can write the augmented direct model 7 x 7 of the appendage PS p S at point P and expressed in frame Ra 0 0 gt B A Ia A 0 RH Taar Ra E 0 0 0 Tra Yra Zra MP s fe Tra aa Ra Cm Yr 0 Zra Oh PA 2 2 14 The block d
33. nels are the transfers between the torques Cm and accelerations local to each revolute joint linking two bodies if activated e translate dynamic model m the syntax MB translate dynamic model A B MA translates or transports the direct dynamics model MA projected in a frame Ri from point A vector of the 3 components of the point A in the frame R to point B vector of the 3 components of the point B in the frame R Inputs and output arguments are expressed in the same frame R Mo Go rue ies tala re 1 where N is the number of revolute joints in the structures e rotate dynamic model m thesyntax MB rotate dynamic model MA ANG AXIS computes the direct or the inverse dynamics model input argument MA pro jected in a frame R after a rotation of ANG deg around the axis AXIS unit vector of the 3 components of the rotation axis r in the frame R Inputs and output arguments are expressed in the same frame R This function computes the direction cosine matrix 7 associated to the rotation and then T 03x3 O3xn TT Osa O3x Ny Malr S Osxn T Osxn Ma g s Osxw I Osxn On x3 On x3 In On x3 On x3 In where N is the number of revolute joints in the structures e kinematic_model m the syntax TAU kinematic_model A B computes the kinematic model TAU betwen points A and B vectors of the 3 components of points A and B in a frame R mE AB TAB n Ri 03x 3 I3 ISAE DMIA ADIS Page 30 58 od
34. ponse of the inverse dynamics model see Figure 3 8 gt gt clear all close all gt gt mod main Spacecraftilu gt gt mod TotalCg the position of the global centre of mass ans Uncertain matrix with 3 rows and 1 columns The uncertainty consists of the following blocks MB_m Uncertain real nominal 100 variability 10 10 1 occurrences SA1_CGx Uncertain real nominal 1 variability 10 10 1 occurrences SA1_Px Uncertain real nominal 0 range 0 01 0 01 1 occurrences SA1 Py Uncertain real nominal 1 variability 10 10 1 occurrences 10 10 1 occurrences SA2_CGy Uncertain real nominal 1 variability 10 10 1 occurrences SA1 m Uncertain real nominal 50 variability SA2_Py Uncertain real nominal 1 variability 10 10 1 occurrences SA2 Pz Uncertain real nominal 0 range 0 01 0 01 1 occurrences SA2_m Uncertain real nominal 20 variability 10 10 1 occurrences Type ans NominalValue to see the nominal value get ans to see all properties and ans Uncertainty to interact with the uncertain elements gt gt Mi mod InverseTotalModel inverse dynamic model gt gt figure gt gt sigma Mi gt gt sensitivity of the frequency domain response to parametric variations Singular Values Figure 3 8 Frequency responses Singular Values of the inverse dynamics model for several parameter configurations ISAE DMIA ADI
35. r t to its centre of mass by the user In some applications data are provided at a reference point O different from the centre of mass B then equation 1 15 can then be used and can be easily implemented in the SDT see section 3 2 2 as a tutorial ISAE DMIA ADIS Page 15 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace EMPTY PAGE 17 Chapter 2 Satellite Dynamics Toolbox principles We consider a satellite as a main body B the hub and one appendage cantilevered to the hub at point P as shown on Figure 2 1 The appendage is considered as a dynamic sub structure either because of its flexibility or because of an embedded angular momentum reaction wheels or CMG The objective is two fold first com puting the direct dynamics model M s i e the 6 by 6 transfer between the d dual vector of the translational and rotational accelerations e of the hub seen Ww from the hub centre of mass B and the dual vector of the external forces and torques 7 applied to the hub at point B ext B fe M s E ext B Second computing the inverse dynamics model MA s 1 Considering the result of the previous chapter equation 1 10 one can then com pute the direct and inverse dynamics models about any arbitrary reference point a R that is the models between dual vectors of accelerations _ and forces W Pas seen from the reference point R e
36. re given in appendix A 3 1 Satellite Dynamics Toolbox functions Once the user defined data script file is created 5 functions can be used main translate dynamic model rotate dynamic model kinematic model and antisym Let us called nom fich m the user defined data script file e main m using the syntax mod main nom_fich the function outputs a structure variable mod with 5 fields mod TotalMass the total mass of the satellite Dm mi m in kg mod TotalCg the coordinates of the global centre of mass expressed in the reference frame Fy oc in m b mod DynamicModel the direct dynamics model expressed at the global ISAE DMIA ADIS Page 29 58 D ISdE Institut Sup rieur de l A ronautique et de l Espace 30 3 IMPLEMENTATION WITH MATLAB f Y Ai B centre of mass G in the reference frame Ry u le s in SI b units meaning that the accelerations in input of the model are in m s and in rad s and that the forces and torques outputted by the model are in N and Nm mod InverseTotalModel the above inverse model inputs are forces outputs are accelerations mod liste IOs a string of chars that describes the different channels of the square model mod DynamicModel and built from the names of the bodies written in nom fich m The first 6 channels are always relative to the 3 translations and the 3 rotations of the hub base The optional 7th and beyond chan
37. re is a mistake in the file nom fich m ISAE DMIA ADIS Page 33 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace 34 3 IMPLEMENTATION WITH MATLAB Variable name Notation Type Unit 1 SA i Name char 2 SA i P OP double 3 x 1 m 3 SATI TM E double 3 x 3 A SA i pivot boolean 5 SAti pivotdir Fale Era Yra rl double 3 x 1 6 SA i angle 0 double deg 7T SA i filename char 8 SA il cg PA double 3 x 1 m 9 SA i omega Q double rad s 10 SA i m m double Kg 11 SA i Ixx IA r bD double Kgm 12 SA i Iyy TM 2 2 2 double Kgm 13 SA i Izz IS Ra 3 3 double Kgm 14 SA i Ixy IS ms 1 2 double Kgm 15 SA i Ixz IS g 13 double Kgm 16 SA i Iyz IA r 2 3 double Kgm 17 SA i flex integer 18 SA i L Lle ou Lale double N x 6 Kg VKgm 19 SA i wi wy We wy double 1 x N rad s 20 SA i z i Ea En double 1 x N Table 3 2 Description of the fields of the structure in cell SA i if SA i pivot 1 then there is a revolute joint between the appendage and the hub at point P The direction Ta of the revolute joint axis can be specified in the field 5 SA i pivotdir otherwize the defaut value is SA i pivotdir 0 0 1 In case of a revolute joint the direct dynamics model will be augmented on the ou
38. rol law has the following structure also depicted in Figure 3 5 e a inner control loop feedbacking the precession speeds 6 It is a pure propor tional control and it is the same on the 4 precession axes Cn 008 Gap 0 V4 e d e an outer control loop feedbacking the 3 hub attitudes It is a pure proportional control and it is the same on the hub s 3 rotation axes Tres 3 dref m Oref m 0 Vref wr Rp ISAE DMIA ADIS Page 41 58 Toad ISdE Institut Sup rieur de l A ronautique et de l Espace 42 3 IMPLEMENTATION WITH MATLAB e a guidance law for the cluster of gyroscopic actuators which is the pseudo inverse of the Jacobian Jo J c 5 o 3 x 4 computed from the 4 precession speeds to the hub s 3 rotation speeds when in nominal configuration Fiver Garer Cares GR 7 4 Ra Tuus Ln p X Ai B 2 5 61 Pref Ol ref Cm Lp d s l S s Poy i Bas Cs i m 7 5 JE JJF IR E Figure 3 5 Attitude control using 4 CMGs gt gt clear all close all gt gt mod main FOUR_CMGs Angular configuration 0 0 0 0 2 states removed gt gt mod liste IOs Definition of the various channels ans Trans X Platform Trans Y Platform Trans Z Platform Rot X Platform Rot Y Platform Rot Z Platform Joint CMG 1 Platform Joint CMG 2 Platform Joint CMG 3 Platform Joint CMG 4 P
39. s D Alazard Ch Cumer and C Charbonnel Journal of Dynamic Systems Measurement and Control vol 136 num 2 ISSN 0022 0434 2014 4 Ch Cumer D Alazard It rations design m canique commande T che 1 Mod lisation m canique classique RAv 1 21116 DCSD Avril 2013 On D Alazard Reverse engineering in control design Wiley 2013 6 NEWTON EULER equations http en wikipedia org wiki Newton Euler equations ISAE DMIA ADIS Page 53 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace EMPTY PAGE 99 Appendix A Inline help A l help of function main m MODEL main FILENAME compute the model MODEL of the spacecraft described in the file FILENAME m MODEL is a structure MODEL TotalMass total mass of the spacecraft MODEL TotalCg 3x1 coordinate vector of the global centre of mass CGtot in the reference frame attached to the main body 0 X Y Z MODEL InverseTotalModel 6 NBPIVOTS x 6 NBPIVOTS ss inverse dynamic model between inputs 6 dof external force torque applied on the main body NBPIVOTS torques applied inside the NBPIVOTS pivot joints between main body and appendages and outputs 6 dof linear angular accelerations of main body NBPIVOTS relative angular accelerations of appendages w r t main body around pivots axis This model is written at point CGtot in frame CGtot X Y Z MODEL DynamicModel 6 NBPIVOTS x 6 NBPIVOTS ss direct dynamic model
40. s 12 channels 6 for the degrees of freedom of the hub and 6 for the manipulator The files Segment1 m Segment2 m Segment6 m describe the different segments Each segment is composed of a main body hub and one appendage which is the next segment The global variable config 6x1 can be used to set the angular configuration of the manipulator same as in the previous example The following MATLAB session see also the script file demoSTD m in the STD V1 3 sub directory shows how to use this file with the SDT ISAE DMIA ADIS Page 45 58 S dE Institut Sup rieur de l A ronautique et de l Espace 46 3 IMPLEMENTATION WITH MATLAB Step Response From In 1 From In 2 From In 3 To Out 1 Amplitude To Out 2 To Out 3 Time seconds Figure 3 6 Step response of the CMG platform with attitude control on the 3 axes Figure 3 7 A manipulator arm fitted on a spatial vehicle gt gt clear all close all gt gt global config gt gt mod main SpaceRoboticArm Angular configuration 0 30 120 0 0 0 gt gt mod liste_I0s The 12 channels d o f order ans Trans X Main_body ISAE DMIA ADIS Page 46 58 d ISdE Institut Sup rieur de l A ronautique et de l Espace 3 8 EXAMPLE 5 SPACEROBOTICARM M 47 Trans Y Main body Trans Z Main body Rot X Main body Rot Y Main body Rot Z Main body Joint Link 6 Link 5 Joint Lin
41. s angle to a force impulse applied at the global centre of mass One will notice that thanks to the revolute joint the 6 flexible modes are now both controllable and observable computing a minimal realization of the model will not reduce the number of modes gt gt clear all close all gt gt mod main Spacecraft2 gt gt mod liste_I0s Now the dynamic model is 7x7 Trans X Platform Trans Y Platform Trans Z Platform Rot X Platform Rot Y Platform Rot Z Platform Joint Solar Array Platform gt gt Md mod DynamicModel direct dynamic model gt gt Mi mod InverseTotalModel inverse dynamic model gt gt Mi7i Mi 7 1 Transfer between a force applied on the gt h platform along X axis and the revolute joint s gt gt relative acceleration gt gt ddint tf 1 1 0 0 4 double integrations gt gt impulse response of Mi71 s s gt gt figure gt gt impulse Mi71 ddint 3 response of the revolute joint angle to gt gt h an impulse on the X thruster The impulse response is shown on Figure 3 2 Remark The file Spacecraft3 m describes the same assembly but uses file APPENDAGE1 m to define the solar array File Spacecraft4 m shows the usefulness of the recursivity used in function main m it describes the same assembly as Spacecraft3 m with a third more complex appendage which is the assembly described in Spacecraft3 m ISAE DMIA ADIS Page 37 58 od ISdE Institu
42. solar generator driving mechanism Control Moment Gyro CMG external manipulator More generally it can model an open kinematic chain of bodies sub structures in a space environment gravity efforts are not taken into account Such a chain or structure can be de scribed by a tree see Figure 1 where e each node A corresponds to a body or link or substructure e cach edge corresponds to a joint between two bodies 2 types of joint are considered clamped joint revolute joint around a given axis Classical tree terminology will be used In the example of Figure 1 Apo is the root A3 to A are the leaves 42 is the parent of As and the child of Ap Each body can be either flexible or not with the restriction that the flexible bodies must be the last elements or the leaves of the chain The linear dynamics model is the 6 by 6 transfer between the external forces and torques applied on the main body structure at a reference point and the translational and rotational accelerations at the same reference point When the structure includes joints in this version of the toolbox only revolute joints between the sub structures are considered the model is augmented with the transfer between internal torque and acceleration for each joint Chapter 1 is a reminder on NEWTON EULER equations applied to a single rigid body ISAE DMIA ADIS Page 7 58 od ISdE Institut Sup rieur de l A ronautique et de l Espace
43. t Sup rieur de l A ronautique et de l Espace 38 3 IMPLEMENTATION WITH MATLAB Impulse Response T Amplitude L 1 L 5 1 1 5 Time seconds Figure 3 2 Impulse response of the transfer between 0 the angle of the revolute joint and the force applied at the global centre of mass along X 3 6 Example 3 Spacecraft5 m This file describes the same platform as example 1 but now it is fitted with 3 more appendages nappend 5 modelling embedded angular momentums along the 3 directions Zp y in the hub s frame Such devices can model reaction wheels RWA that have a very high speed close to their saturation The following MATLAB session see also the script file demoSTD m shows how to use this file with the SDT and perform complementary analyses One will notice the following facts e The reduction of one uncontrollable mode same as example 1 e The reduction of a double pair of integrators Indeed to model the 3 embed ded angular momentum see the remarks in section 2 3 3 we introduced three pairs of integrators in the direct dynamics model However only 2 integra tors are required to model the gyroscopic couplings due to the total angular momentum i e the sum of the 3 embedded angular momentums e The gyroscopic mode in the inverse dynamics model which is represented by a pair of poles on the imaginary axis at low frequency around 0 1 rd s e The reduction of a pair of integrators wh
44. tated body in the frame R It is assumed that the first 6x6 block of DM_IN represent the dynamic model between the 6 dof acceleration vector and the 6 dof external force vector A 4 help of function kinematic model m JACOB kinematic model vec A vec B calculates the kinematic model JACOB of a body between two points A and B Inputs vec 3x1 coordinate vector of point in a given frame vec B 3x1 coordinate vector of point B in the same frame ISAE DMIA ADIS Page 56 58 T ISdE Institut Sup rieur de l A ronautique et de l Espace A 5 HELP OF FUNCTION ANTISYM M 57 Output JACOB 6x6 kinematic model projected in the same frame eye 3 AB JACOB zeros 3 eye 3 A 5 help of function antisym m MAT antisym VEC computes the antisymmetric matrix MAT associated with a vector VEC if VEC x y z then MAT 0 z y z O x y x 0 ISAE DMIA ADIS Page 57 58 Vd ISdE Institut Sup rieur de l A ronautique et de l Espace EMPTY PAGE
45. then transport it to any other point R where the external force is applied for instance The development of 1 10 is mP s m BB D rp DR Tgp m CBP 18 m B B Ip One can recognize in the bottom right hand term the HUYGENS theorem between the inertia tensor I of B at point B and the inertia tensor I of B at point P 18 m By 1 13 uw or in projection in the frame Ry 0 Zb Yb 2 attached to the body B with X BP y Z Rp y z zy Zz I5 ele m ay 2 yz 1 14 Ro Note that the equation 1 12 works on the 6 x 6 direct dynamics model and is valid for any points P and R on the body B while HUYGENS theorem equation 1 13 works on the 3 x 3 tensor of inertia and is valid only if B coincides with the centre of mass of the body B In the general case BAIR m PR ISAE DMIA ADIS Page 14 58 od L ISdE Institut Sup rieur de l A ronautique et de l Espace 1 3 LINEARIZED NEWTON EULER EQUATIONS 15 Nevertheless HUYGENS theorem equation 1 13 can be used to find I from 15 in the case where the data of the tensor of inertia are provided w r t the body reference frame Ra 8 2 I8 m COB 1 15 Indeed in the MATLAB implementation of the SDT detailled in chapter 3 the tensor of inertia of each body must be defined w
46. tputs by the joint torque Cm and on the inputs by the joint acceleration 6 If SA i pivot 0 then the appendage is cantilevered to the hub B at point P and the fields 5 and 6 are optional or disregarded if SA i pivot 1 then is possible to take into account a tilt of the appendage by SA i angle degrees along the revolute joint axis if the appendage is described by a data file then the fields from line 8 to 20 are optional or disregarded if the field SA i omega exists then the appendage is considered as an onboard angular momentum along Za Then SA i omega and SA i Izz are the angular rate and the main inertia of the spinning top SA i Ixx is its radial inertia fields 12 14 15 and 16 are optional or disregarded and the appendage must be rigid SA i flex 0 if SA i flex 0 then the appendage is supposed rigid and the fields from line 18 to 20 are optional or disregarded If SA i flex 1 then the appendage is flexible and SA i L is the list of the modal participation factors L4 expressed at centre of mass A of the appendage If SA i flex 2 then the appendage is flexible and SA i L is the list of the modal participation factors Lp at anchoring point P We remind that Lp LATAp p Nr Institut Sup rieur de l A ronautique et de l Espace ISAE DMIA ADIS Page 34 58 3 4 EXAMPLE 1 SPACECRAFT1 M 35 3 4 Example 1 Spacecrafti m This file describes a platform MB Name Platform fitted with
47. vector of body B at point B Inertial acceleration vector of body B at point P Angular speed vector of R with respect to the inertial frame External forces vector applied to B External torques vector applied to B at point B Force vector applied by B on A Torque vector applied by B on A at point P Hub s static dynamics model expressed at point B Hub s B mass Inertia tensor 3 x 3 of B at point B Appendage s dynamics model at point P Appendage s mass Inertia tensor 3 x 3 of A at point A TPB Kinematic model between points P and B _ PB TPB 03x3 la ISAE DMIA ADIS Page 5 58 T ISde Institut Sup rieur de l A ronautique et de l Espace 6 NOMENCLATURE a PB Skew symmetric matrix associated with vector P if PB E z Jr 0 z y then PR z 0 r Ri y x 0 Na Number of appendages If there is a flexible appendage we add the following definitions N Number of flexible modes n Modal coordinates vector wi i flexible mode s angular frequency amp i flexible mode s damping ratio l p i flexible mode s vector of modal participations 1 x 6 expressed at point P Lp Matrix N x 6 of the modal participation factors expressed at point P Lp E li P l2 p ur Us p If there is an embedded angular momentum we add the following definitions Q Angular speed rad s of the spinning top Z Unit vector
48. xt R MST s TRR MST s TBR 2 1 1 1 Mg s tre M s Ts 2 2 ISAE DMIA ADIS Page 17 58 D ISdE nstitut Sup rieur de l A ronautique et de l Espace 18 2 SATELLITE DYNAMICS TOOLBOX PRINCIPLES T ext B Figure 2 1 The system hub B appendage A 2 1 Assumptions H1 The hub is rigid dB In H2 Non linear terms in DANS su of second or higher order are disregarded H3 The only external force resp torque applied to the appendage is the force F gja resp torque T gja p applied by the hub B at the appendage s anchorage point P 2 2 Direct dynamics model The direct dynamics model of the assembly M pu s at the hub B centre of mass is the sum 2 4 e of the hub s direct dynamics model at point B D5 PI D HIE 03x 3 which can be projected in frame Ry as follows 03x3 B B u Th Ts 03x3 Dlr E 03x3 Bla e and of the appendage s direct dynamics model at point P MA s moved to point B thanks to the kinematic model rpg see nomenclature and the transport equation 1 10 that leads to M s TM s rpn 2 3 ISAE DMIA ADIS Page 18 58 od ISdE nstitut Sup rieur de l A ronautique et de l Espace 2 3 APPENDAGE DYNAMICS MODEL M s 19 When summing we can write that Mg s D5 TB Mf s ren 2 4 In projection in the hub s frame Ra the assembly s direct dynamics model is the following T Tha
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