Instituto de Sistemas e Robótica

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1 Institut de Sistemas e Rbótia Pól de Lisba Attitude Cntrl Strategies fr Small Satellites Paul Tabuada, Pedr Alves, Pedr Tavares, Pedr Lima Setembr 1998 RT ISR-Trre Nrte Av. Rvis Pais 196 Lisba CODEX PORTUGAL Wrk arried ut under the prjet "Cntrl and Stabilizatin f Small Satellites", supprted by the PRAXIS XXI Prgramme (ref. PRAXIS/3/3.1/194/95).

2 Ntatin Vetrs and matries A, v matries and vetrs are written in bld type, I v, v, v vetr v reslved in Cntrl CS, Orbit CS r Inertial CS respetively, v, v, v x, y, and z mpnents f vetr v. x y z List f symbls Ω w Ω Ω w q q, q 4 A( q) angular velity f Cntrl CS w.r.t. Wrld CS, angular velity f Cntrl CS w.r.t. Orbit CS, angular velity f Orbit CS w.r.t. Wrld CS, attitude quaternin representing rtatin f Cntrl CS w.r.t. Orbit CS, vetr part and salar part f q, transfrmatin matrix frm Orbital CS t Cntrl CS (diret sine matrix), i, j, k unit vetr alng x-, y-, z-axis f Orbit CS, ω T h I I x, I y, I z N trl N gg N dist E kin E gg E gyr E tt E Lyap J m B ^ rbital rate, perid f rbit, angular mmentum due t satellite revlutin abut the Earth, inertia tensr f the satellite, mments f inertia abut x-, y-, z-prinipal axis, ntrl trque, gravity gradient trque, perturbatins trque, kineti energy, energy due t gravity gradient, energy due t satellite revlutin abut the Earth, ttal energy, Lyapunv energy andidate funtin, st funtin, magneti mment generated by set f ils, magneti field f Earth (gemagneti field), B gemagneti filed predited by the IGRF mdel, B ~ matrix representatin f prdut B, s sliding variable, S sliding manifld, P psitive membership funtin fr fuzzy lgi ntrl, N negative membership funtin fr fuzzy lgi ntrl, Z zer membership funtin fr fuzzy lgi ntrl, S wz parameter used fr ntrlling the spin angular velity, d( ) disretisatin funtin, R q matrix representatin f quaternin prdut, ( ) 1

3 4 1 Ω extensin f the three dimensinal vetr T vetr [ Ω ] T, Ω t a fur dimensin h, g, psitive nstants, Ω derivative f vetr Ω w.r.t. time, Ωˆ predited value fr Ω, ψ, θ, ϕ Rll, Pith an Yaw angles respetively, α angle between the expeted gemagneti field and the z axis f the Orbit CS, β angle between the measured gemagneti field and the z axis f the Cntrl CS, γ angle between the satellite s bm and the lal vertial, Λ psitive definite gain matrix, q tt Energy gap, t time step, µ i rrelatin prdut fr rule i.

4 Index 1 Intrdutin General 5 1. Related wrk Original ntributins f this wrk Struture f the wrk 6 Satellite attitude dynamis and kinematis 8.1 Crdinate systems desriptin 8. Dynamis 9.3 Kinematis 9.4 Kineti energy 1.5 Ptential energy 1 3 Attitude stabilisatin and ntrl PSAT-1 Stabilisatin Desriptin Simulatin results Disussin f results Sliding mde ntrl Desriptin PSAT Restrited atuatrs Simulatin results Disussin f results Energy based ntrl Desriptin PSAT Restrited atuatrs Simulatin results Disussin f results Preditive stabilisatin Mtivatin Desriptin Stability study Glbal stability Ideal atuatrs (Geneti stabilisatin) PSAT Restrited atuatrs (Brute fre stabilisatin) Simulatin results Disussin f results Overall mparisn f results 34 4 Attitude stabilisatin and spin ntrl PSAT Cntrl Desriptin 36 3

5 4.1. Simulatin results Disussin f results Energy based ntrl Desriptin PSAT Restrited atuatrs Simulatin results Disussin f results Fuzzy lgi ntrl Desriptin PSAT Restrited atuatrs Simulatin results Disussin f results Preditive ntrl Desriptin Simulatin results Disussin f results Overall mparisn f results 51 5 Cnlusins Cnlusins Future wrk and diretins 53 6 Bibligraphy 54 Appendix A Attitude simulatr and rbit mdel 55 Simulatr 55 Orbit mdel 55 Intrdutin 55 The prpagatin mdels 55 Appendix B Simulatin s initial nditins 57 Appendix C Cntrllers perfrmane riteria 6 Appendix D PSAT like atuatr restritins 61 4

6 1 Intrdutin 1.1 General Small satellites are nwadays an easy and heap way t gain aess t spae and t all the advantages a satellite an prvide (telemmuniatins, envirnment mnitring, military infrmatin, et). This lass f LEO satellites (Lw Earth Orbit) may be ntrlled by strit interatin with the gemagneti field. A magneti mment prdued by ils plaed n the satellite will prdue a resultant trque by interatin with the gemagneti field, whih may be used fr attitude ntrl purpses. Nevertheless, this simple, lw pwer nsumptin apprah pses several interesting ntrl diffiulties as the gemagneti field viewed by a satellite, hanges alng its rbit. Besides this time dependeny, this prblem s mathematial desriptin is highly nn-linear, and new ntrl strategies are needed t slve the attitude and ntrl demands f suh a satellite. Several ntrl strategies were implemented and simulated in a realisti envirnment (see appendix A) in rder t hse the mst adequate t eah missin phase. Bth ideal and restrited atuatrs were nsidered viewing a pssible appliatin f suh algrithms t PSAT-1 1 and ther suh satellites. This wrk has been arried ut at Intelligent Cntrl Lab. f ISR/IST, as a final year prjet fr the degree f Aerspae Engineering. This wrk was als integrated in the CnSat prjet. 1. Related wrk Several researhers have already begun t explre and slve the ntrl prblems impsed by a LEO small satellite. [Ong] prpses us sme intuitive ntrl laws t takle this prblem, but the atuatin is very restrited and des nt take advantage f the time-varying nature f this prblem. [Steyn] apprahes the ntrl prblem by using a Fuzzy Lgi Cntrller that ahieves better results than a Linear Quadrati Regulatr (LQR) despite nsidering the nstraint f atuating n a single il at eah atuatin time. This apprah suggests that nn-linear ntrl methdlgies shuld be further explred s that a better prblem understanding and pssible slutins may be fund. [Wisniewski] mpares tw nn-linear slutins: sliding mde ntrl and energy based ntrl, ahieving better results than LQRs based n linear peridi thery. Sme f these ideas and algrithms were implemented r inspired sme new ntrllers prpsed in this wrk. Cnepts and therems f matrix algebra, nn-linear thery and spaeraft attitude dynamis and kinematis are used thrughut the text withut demnstratin. Sme referenes are suggested at the end f the wrk and referred were relevant t mplement the expsitin. 1 PSAT-1 is the first Prtuguese Satellite in rbit, develped in a tehnlgy transfer prgram between University f Surrey and a Prtuguese industrial and eduatinal nsrtium lead by INETI. Stabilisatin and Cntrl f Small Satellites, a PRAXIS XXI prgramme lead by ISR (IST) and UBI. 5

7 1.3 Original ntributins f this wrk Sme new slutins fr the attitude stabilisatin prblem are presented in this wrk. A summarised desriptin fllws: A new attitude stabilisatin algrithm is presented fr bth ideal and restrited atuatrs at setins and A stability study fr this new algrithm is presented at setin The same algrithm is prved t slve the attitude stabilisatin and spin ntrl prblem as desribed at setin Struture f the wrk This final year prjet reprt begins with a summarised mathematial desriptin f the satellite's attitude dynamis and kinematis. The fllwing hapters are devted t eah f the simulated ntrl algrithms. These are gruped arding t tw different ntrl bjetives: attitude stabilisatin and attitude stabilisatin with spin ntrl. In the first grup, the atual PSAT-1 ntrller, a sliding mde ntrller, an energy based ntrller and a preditive regulatr are studied. In the send grup the existing spin ntrller fr PSAT-1, an energy based ntrller, a fuzzy lgi ntrller and finally a preditive ntrller are studied. In the setins nerning eah f the ntrllers studied, the algrithm is first desribed fr ideal atuatrs and an extensin is presented fr PSAT like restrited atuatrs. Finally sme nlusins are presented as well as tpis fr further researh n this field. The hapters are rganised as fllws: Chapter, Satellite attitude dynamis and kinematis This hapter prvides definitins f rdinate systems used thrughut the reprt. A summarised desriptin f the satellite mtin, based n quaternins, is given. Chapter 3, Attitude stabilisatin and ntrl In this hapter several attitude stabilisatin and ntrl algrithms are presented. The fllwing different apprahes were studied: PSAT-1 Stabilisatin In this setin the atual regulatr fr PSAT is desribed, and simulatin results are presented fr mparisn with the ther studied ntrl algrithms. Sliding mde ntrl A sliding mde ntrl apprah is studied fr bth ideal and restrited (PSAT like) atuatrs. It is shwn that this type f ntrl is nt suitable fr the restrited atuatrs. Energy based ntrl An energy based ntrl is studied. Mdifiatins are intrdued fr restrited atuatrs. A ntrller fr revery f an inverted bm situatin is presented fr bth ideal and restrited atuatrs. 6

8 Preditive stabilisatin In this setin a preditive regulatr is studied. The essene f the preditive stabilisatin is the minimisatin f a kineti energy like st funtin. Overall mparisn f results The abve mentined algrithms are mpared using different perfrmane measures. Chapter 4, Attitude stabilisatin and spin ntrl In this hapter several attitude stabilisatin and spin ntrl algrithms are presented. The fllwing different apprahes were studied: PSAT-1 Cntrl In this setin the atual ntrller fr PSAT is desribed, and simulatin results are presented fr mparisn with the ther studied ntrl algrithms. Energy based ntrl An energy based ntrl is prpsed fr stabilisatin f the satellite and simultaneusly ahieving spin ntrl. Results fr bth ideal and restrited atuatrs are presented. Fuzzy lgi ntrl Spin ntrl and attitude stabilisatin are ahieved thrugh fuzzy lgi ntrl. Rules and membership funtins based n physial insight f the prblem are presented. Preditive ntrl Mdifiatins n the hapter 3 preditive regulatr algrithm are prpsed fr attitude stabilisatin and spin ntrl f a satellite. Overall mparisn f results The abve mentined algrithms are mpared using different perfrmane measures. Chapter 5, Cnlusins This hapter ntains the nluding remarks and remmendatins fr future wrk and diretins. Appendix A, Attitude simulatr and rbit mdel A brief desriptin f the used simulatr and rbit mdel is presented. Appendix B, Simulatin initial nditins The initial nditins used fr the simulatins are presented and explained. Appendix C, Cntrllers perfrmane riteria Several riteria used fr mparisn f the algrithms are defined. Appendix D, PSAT like atuatrs restritins The main PSAT atuatin limitatins and restritins are presented. 7

9 Satellite attitude dynamis and kinematis.1 Crdinate systems desriptin Befre presenting the any mathematial desriptin, the rdinate systems (CS) used thrughut ut this wrk are defined: Cntrl CS: This CS is a right rthgnal CS inident with the mment f inertia diretins, and with the rigin plaed at the entre f mass. The x axis is the axis f the maximum mment f inertia and Z the minimum. Bdy CS: This CS is a right rthgnal CS with its rigin at the entre f gravity. The z axis is parallel t the bm diretin and pint tward the bm tip. The x axis is perpendiular t the shrtest edge f the bttm satellite bdy, and pints away frm the bm anister. The y axis is perpendiular t the lngest edge f the bttm satellite bdy. It is the referene CS fr attitude measurements and the magnetrquers. Orbital CS: This CS is a right rthgnal CS fixed at the entre f mass f the satellite. The z axis pints at zenith (is aligned with the Earth entre and pints away frm Earth), the x axis pints in the rbit plane nrmal diretin and its sense inides with the sense f the rbital angular velity vetr. The Orbit CS is the referene fr the attitude ntrl system. Inertial CS: This CS is an inertial right rthgnal CS with its rigin at the Earth's entre f mass. The z axis is parallel t the Earth rtatin axis and pints tward the Nrth Ple. The x axis is parallel t the line nneting the entre f the Earth with Vernal Equinx and pints twards Vernal Equinx (Vernal Equinx is the pint where elipti rsses the Earth equatr ging frm Suth t Nrth n the first day f spring). In this wrk the satellite is nsidered t be hmgeneus and axisymmetri s the Bdy CS and the Cntrl CS are assumed t be the same, but that des nt need t be the ase. Nte that these CSs definitins are the same used by [Wisniewski] with the exeptin that the Wrld CS is nw alled Inertial CS. Figure.1 a) Definitin f the Cntrl CS in the Orbit CS. b) Definitin f the Bdy CS. The Bdy CS refers t gemetry f the satellite main bdy, its axes are perpendiular t the satellite s faets. 8

10 . Dynamis The dynamis f a rigid satellite may be desribed as [Wertz], [Chbtv], [Thmsn], [Wiesel]: Ω i = Ωi I Ωi + N trl + N gg N dist. (..1) I + The ntrl trque is btained as: N trl = m B (..) and the magneti mment as [Alns and Finn]: m = n i A (..3) il il il hene, given a lal gemagneti field vetr the ntrl trque an be hanged by regulating the value f i il. The gravity gradient may be expressed as [Wertz]: gg ( k I k ) N = 3ω. (..4) The disturbane trque is due t aerdynami drag, slar pressure, eentriity f the rbit, and several ther effets..3 Kinematis The kinematis is expressed in Euler symmetri parameters als knwn as quaternins, thrugh the integratin f the angular velities: 1 q = R( 4 1Ω q (.3.1) ) where the R matrix represents the quaternin prdut [Wertz], [Chbtv]. The kinematis an als be expressed by tw different equatins, ne fr the vetr part f the quaternin and anther fr the salar part: 1 q = Ω 1 q 4 = q Ω 4 1 q Ω q (.3.) The use f quaternins in the kinematis desriptin is justified beause this redundant representatin (4-dimensinal vetr in a 3-dimensinal spae) allws fr a singularity free mdel. Nevertheless Euler angles will be used when simulatin results are analysed, ne they prvide a better physial understanding f the satellite attitude. The transfrmatin matrix A (diret sine) an be parameterised by quaternins r by a 13 series f rtatins using Euler angles [Chbtv]. Frm this relatins it is pssible t btain the relatins between quaternins and Euler angles: a a 3 33 whih an be slved as: a1 = tgψ, sin ϕ = a31, = tgϕ (.3.3) a 11 9

11 1 ψ = artg q1 q θ = arsin ϕ = artg q ( q q q q ) ( ( q1q3 + qq4 ) ( q q + q q ) 1 q 4 1 q + q q + q (.3.4) different expressins an be btained by nsidering ther relatins between quaternins and Euler angles..4 Kineti energy The kineti energy nsidered here is nly a part f the ttal kineti energy. The ttal kineti energy has a ntributin frm the satellite revlutin abut the Earth, and a ntributin frm the mvement f the Cntrl CS w.r.t. (with respet t) the Orbit CS. The first ntributin is nstant sine ω is apprximately nstant (the rbit's eentriity is very small, e=.1 fr PSAT-1 rbit). The send ntributin is the nly ne nsidered here and is given by: Ekin 1 T = ΩI Ω (.4.1).5 Ptential energy The ptential energy has a ntributin frm the gravity gradient and anther frm the revlutin abut the Earth, s: E = E + E (.5.1) pt gg gyr The gravity gradient ptential energy is given by: E gg T ( k I I ) 3 = ω k zz (.5.) and the ptential energy assiated with the revlutin abut the Earth is given as: E 1 = ω gyr I xx T ( i I i ) (.5.3) 1

12 3 Attitude stabilisatin and ntrl The stabilisatin algrithms desribed in this setin aim damp r even eliminate the libratin mvement (align the Orbital CS z axis with the Cntrl CS z axis) and t eliminate any existing spin velity. Attitude ntrl algrithms align the Cntrl CS with a desired referene CS in this ase the Orbital CS. Fr the simulatin and mparisn f the different algrithms it is assumed that the attitude an be determined withut any errr and that it desn't exist any perturbatin influening the mtin f the satellite (ex. slar pressure, aerdynami drag, et.). 3.1 PSAT-1 Stabilisatin Desriptin The PSAT attitude stabilisatin ntrller, as desribed in [Ong], is a simple ntrl law based n the angle (α) between the expeted gemagneti field (based n the IGRF mdel) and the z axis f the Orbital CS, α = artan ^ x B + ^ z B ^ y B (3.1.1) and the angle (β) between the measured gemagneti field and the z axis f the Cntrl CS: x y ( B ) + ( B ) β = artan z B (3.1.) The algrithm nly uses the z il thrugh the fllwing law: m z dβ dα = k dt dt (3.1.3) and the magnetrquers are fired when the satellite latitude is 3.4º r 3.4º whih rrespnds t fur firings per rbit. Further details n the algrithm are available in [Ong]'s wrk. Althugh this is the algrithm desribed in the literature, better results were attained with the fllwing mdified algrithm: m z dβ dα z = sign max( m ) (3.1.4) dt dt where max( z m ) represents the maximum prduible magneti mment with the z il. Nte that this is the same ntrl law as (3.1.3), but the magneti mment amplitude is nw nstant and f maximum amplitude prduing a faster deay n the satellite ttal energy. 11

13 3.1. Simulatin results The fllwing simulatin results were btained with ntrl law in the simulatin test desribed in appendix B. γ settling time (rbits) γ fr 3 rbits (º) γ fr 5 rbits (º) γ fr 15 rbits (º) Energy (J) Test 1 >15 rbits Test >15 rbits Test 3 >15 rbits Test 4 >15 rbits Test 5 >15 rbits Test 6 >15 rbits Test 7 >15 rbits Test 8 >15 rbits Test 9 >15 rbits Test 1 >15 rbits Mean >15 rbits Std. Dev. >15 rbits Best ase >15 rbits Table Simulatin results fr the mdified PSAT ntrller. Nte that in rder t prdue the average results test 1 and 5 were nt taken int aunt beause the satellite mved t an inverted bm nfiguratin. Gamma (º ) Figure Gamma evlutin fr test 5. m z (Am ) Figure 3.1. Cntrl magneti mment fr test 5. 1

14 x Figure Disussin f results Ω evlutin fr the best ase. Figure Satellite energy evlutin fr the best ase. The results btained with the mdified PSAT ntrller shw that this ntrller has a very pr perfrmane. After 15 rbits the γ angle redued nly t 3º in the best ase. In test 1 and 5 the algrithm was nt apable f maintaining the satellite in a bm up nfiguratin as an be seen in fig The angular velity dissipatin is very slw (fig. 3.1.) and the algrithm is nt apable f dissipating the z axis angular velity as explained in [Ong]. The ttal energy f the satellite als deays very slwly (fig ) and the perfrmane f this algrithm an nly be evaluated in days instead f rbits [Ong]. The nly advantages are the lw mputatinal needs and the lw pwer nsumptin ( J) as the magnetrquers are nly fired 4 times per rbit. 3. Sliding mde ntrl 3..1 Desriptin The sliding mde ntrl algrithm implemented is very similar t the ne prpsed by [Wisniewski] s there will nt be presented any dedutins r stability nsideratins as they an be nsulted in the supra-ited referene. Cnsider the fllwing sliding variable: s Ω + Λ q q (3..1) where Λ q is a psitive definite gain matrix. The sliding manifld is defined as the subspae f the state spae where the sliding variable is zer: { q, : s = } S Ω (3..) [Wisniewski] shwed that when n the sliding manifld the slutin rbit will T nverge t the referene Ω = [ ], q = [ ] T, therefre it is nw 13

15 neessary t find a ntrl law that will make the slutin rbit nverge t the sliding manifld. The desired trque is defined as: N N des eq = = N Ω eq i λ I 1 IΛ q s s Ω i ω ( k I k ) + ωi( i Ω ) ( Ω q Ω q) 3 (3..3) 4 where the equivalent trque mpensates the system dynamis and the term s will make the slutin nverge t the sliding surfae in an expnential way. λ s [Wisniewski] als shwed that nly the mpnent f N des that is parallel t the sliding variable vetr is respnsible fr dereasing the distane t the sliding manifld, s the fllwing ntrl law in prpsed: where prl N des B m = (3..4) B N prl des N des s = s. (3..5) s Equatin (3..4) mputes the magneti mment frm the parallel mpnent f the desired trque, whih is btained by prjeting the desired trque n the sliding variable vetr (equatin 3..5). As an be seen in the wrk f [Wisniewski] this ntrl law is prved t be lally asympttially stable. The ntrl law presented at equatin (3..3), (3..4) and (3..5) was implemented nsidering nly ne restritin: the magneti mment prduible has a lwer and upper limit equal t the PSAT limits. 3.. PSAT Restrited atuatrs Due t the nature f PSAT atuatrs restritins (single-il-atuatin) it is nt pssible t use sliding mde ntrl with PSAT atuatrs as will be shwn next. The ntrl trque generated by the satellite ils is btained by equatin.3. and an als be expressed in matrix frm as: N trl = ~ m B = mz m y m m x z m y mx B x By B z (3..6) By inspetin f the ~ m matrix we an see that if m has nly ne mpnent ~ m different frm zer the matrix will have a line f zers. This means that the ntrl trque will be zer in ne dimensin; there will always be a dimensin were it will nt be pssible t mpensate the system dynamis, nr make the slutin rbit nverge t the sliding manifld. This restritin bemes even mre severe if we regnise that the dynamial mdel f the satellite (..1) is a upled system, 14

16 therefre the perturbatin impsed by the atuatin nstrains will prpagate t the ther dimensins and the system will beme unntrllable. Frm the abve arguments it is visible that this kind f ntrl is nt suited fr systems with the referred atuatin nstraints Simulatin results As was shwn at setin 3..3 PSAT an nt be ntrlled by this kind f ntrl, therefre we will nly present simulatin results fr ideal atuatrs. The simulatin results presented here d nt rrespnd t the simulatin test desribed in appendix B beause initial nditins are utside stability margins f the Sliding mde ntrller. Therefre the initial spin velity had t be dereased frm rad s t.3 rad s, but even with this redutin there were ases were the algrithm diverged as an be seen in table The simulatins used the fllwing values fr the gains: Λ q. =., λ s.1 that were fund empirially. =.1 (3..7) γ settling time (rbits) γ fr 3 rbits (º) γ fr 5 rbits (º) Pinting auray (º) Energy (J) Test e Test e Test e Test 4 Diverge Diverge Diverge Diverge Diverge Test e Test e Test e Test 8 Diverge Diverge Diverge Diverge Diverge Test 9 Diverge Diverge Diverge Diverge Diverge Test 1 Diverge Diverge Diverge Diverge Diverge Wrst ase e Table 3..1 Simulatin results fr Sliding mde ntrl with initial spin f.3 1 rad s 15

17 The algrithm diverged almst in half f the simulatins s average results are nt shwn as their nfidene degree wuld be very lw. 5 Rll (º ) Pith (º ) Gamma (º ) Yaw (º ) Figure 3..1 Euler angles (13) fr the wrst ase. Figure γ angle evlutin fr the wrst ase. m x (Am ) x 1-4 m y (Am ) Energy (J) 1.5 m z (Am ) Figure 3..3 Magneti mment evlutin fr the wrst ase Figure 3..4 Satellite energy evlutin fr the wrst ase Disussin f results Althugh this algrithm has a strng drawbak, its lal stability prperties, it als has an imprtant prperty: its pinting auray. Pinting auraies f.1e-3º (table 3..1) shwn in almst every simulatin and 3 axis stabilisatin (fig 3..1) are harateristis that may be neessary fr missin phases where high pinting auray is an essential fatr. Hwever in real nditins the pinting auray will be degraded due t the errrs in the attitude and angular velities prdued by the attitude determinatin system and due t the errrs indued by the magnetrquers 16

18 whih are nt able t prdue the magneti mments with the desired preisin. Anther prblemati harateristi f this algrithm is that satellite energy (fig. 3..4) inreases drastially in the first rbit. This inrease in energy is required t fre the slutin rbit t the sliding surfae, ne n the sliding surfae the energy dereases, and the slutin nverges smthly t the referene as an be seen in fig Energy based ntrl Desriptin As the desriptin f the previus ntrl algrithm, the energy apprah t magneti attitude ntrl vered in this setin is based n [Wisniewski], s we will nly present a summarised desriptin f the basi algrithm, and prpse a slightly different energy apprah t use with the PSAT-1 restrited atuatrs. A magneti generated mehanial trque is always perpendiular t the gemagneti field vetr. The nsequene is that the satellite is nly ntrllable in tw diretins at any single pint in time. With the gemagneti field varying alng an rbit this implies, e.g. that in the Earth's plar regins the yaw angle is unntrllable, whereas it an be ntrlled again when the satellite is in the equatrial regins. Sine the ntrl trque is always perpendiular t the gemagneti field vetr, it is desirable that the magneti mment is als perpendiular t the gemagneti field vetr, as nly this mpnent prdues a nn-zer ntrl trque. It is nluded that magneti ntrl mment must inlude infrmatin abut the angular velity f the satellite [Wisniewski], and als abut time prpagatin f the gemagneti field. A andidate fr generatin f the magneti mment is an angular velity feedbak m ( t) = h Ω ( t) B( t) (3.3.1 ) where h is a psitive nstant (the velity feedbak an nly use salar gain in rder t prve asymptti stability [Wisniewski]). There are tw main reasns t suggest this feedbak: l. - It ntributes t the dissipatin f kineti energy f the satellite.. - It prvides fur stable equilibrium pints. The equilibrium pints are suh that the z axis f the Cntrl CS (the axis f the minimal mment f inertia) pints twards the diretin f the z axis f the Orbit CS, and the unit vetr f the x axis f the Cntrl CS (the axis f the largest mment f inertia) is parallel t the x axis f the Orbit CS. One f these equilibrium pints is the desired referene. Using the ttal energy as a Lyapunv andidate funtin it an be prven that ntrl law (3.3.1) is asympttially stable arund fur equilibrium pints (see [Wisniewski] fr the mplete demnstratin). One ther imprtant result in the demnstratin is the energy derivative, that will be needed in the next hapter: E tt = Ω T N trl (3.3.) 17

19 A ntrl law that makes all equilibrium pints but the referene unstable will be presented next. The referene nsidered is: ( Ω, k, i ): (, k, i ) (3.3.3) The three axis attitude stabilisatin an be amplished when sme attitude infrmatin is added int the velity ntrl law. m ( t) = h Ω ( t) B( t) ε q( t) B( t) (3.3.4) where h and ε are psitive nstants. It is prven in [Wisniewski] that the ntrl law (3.3.4), with the prper values fr h and, is asympttially stable abut the referene (3.3.3). If the ttal energy, the sum f E gg, E gyr and E kin is abve the energy level E x gg + E z gyr (the maximum ptential energy required fr the bm axis t rss the hrizntal plane), then the kineti energy has nnzer bias, and the satellite will tumble, i.e., the bm axis will evlve between upright and upside-dwn attitude. Whereas if the ttal energy is belw E y gg (the minimum ptential energy neessary t rss the hrizntal plane), and the initial attitude is suh that the bm axis is abve the lal hrizn, then it mves abve the hrizn frever. The time prpagatin f the slutin trajetry fr the energy level between E x gg + E z gyr and E y gg, where the energy gap tt is tt = E x gg + E x gyr E y gg = ω 3 ) ω ( I x I z ( I y I z remains undetermined. A ntrl law taking this unertainty int aunt is: Predure 1 ) (3.3.5) 1. If E tt > E x gg + E z gyr (see 3.3.7) ativate the angular velity ntrller (3.3.1). Else wait until k z hanges the sign frm negative t psitive, then ativate the rate/attitude ntrller (3.3.4) fr z k >. The first stage diminishes the ttal energy using the angular velity feedbak t the level E x gg + E z gyr, then waits until the bm axis rsses the hrizn plane frm upside-dwn t upright t ativate the rate/attitude ntrller. The ntrller needs nly t dissipate a small amunt f energy tt in rder t keep the bm axis abve the hrizn frever. Hene, the slutin nverges asympttially t the referene. The ntrl law in predure 1 is lally stable in the sense that if the bm axis is upside-dwn and the ttal energy is belw E x gg + E z gyr (mre preisely belw E y gg) there are n means t turn the bm axis upright. T get ver this prblem, an alternate predure fr ahieving a glbally stable ntrller is devised. The nept is t apply a destabilising ntrl when the gravity gradient bm is upside-dwn and a stabilising ntrl when it is abve the hrizn. 18

20 Predure 1. If the bm axis is upside-dwn (belw the hrizn) generate the magneti mment arding t (3.3.6) until k > m = g i B (3.3.6) where g is a psitive r negative nstant, a design parameter.. Use predure 1. There are tw reasns t prpse this algrithm: A minimum effrt ntrller is a ntrller whih generates a ntrl trque perpendiular t the lal gemagneti field. The unit vetr i is apprximately perpendiular t B(t) fr all t, sine it is perpendiular t the rbit plane. The resultant ntrl generated arding t (3.3.6) is parallel t i and therefre perpendiular t the lal gemagneti field. The minimum ptential energy neessary t turn the bm axis upright is the rtatin abut the pith axis, whih is at mst E p y y 1 = E ( ) gg + Egyr = ω I y I z + ω ( I x I y ) (3.3.7) 3 The ptential energy neessary t turn the satellite abut the rll axis is at mst z where E p x z = E + E = ω ( I I ) (3.3.8) gg gyr x z E y gyr ( I I ) 1 ω = x = y (3.3.9) Predure is imprved by utilizatin f the angular mmentum due t the satellite revlutin abut the Earth, h. The design parameter g is stritly psitive, thus the angular mmentum h ats in the same diretin as the ntrl trque, and the neessary effrt t turn the satellite upright is dereased. There is a mre elegant frm f a glbally stable ntrller in [Wisniewski], but the ne presented is f simpler implementatin and therefre was the ne used PSAT Restrited atuatrs Due t the restritins assiated with the PSAT atuatrs, the bak-ff time, the pssibility f using nly ne il f the magnetrquer at a time and the nly three different urrents available t prdue the magneti mment, it is prpsed a slightly altered energy ntrl algrithm. The prpsed alteratins are nly in rder t maintain the effetiveness and stability f the algrithm in this restrited atuatrs situatin. Sine the satellite an nly atuate in ne diretin, x, y r z, is hsen the diretin that prdues a mment mre similar t the ne that is prpsed by the ideal atuatr algrithm. This is dne by mparing the magneti trque attained with eah 19

21 f the restrited atuatr ils (already quantified t prduible mments), with the magneti trque that wuld be btained with the prpsed magneti mment (ideal atuatr). The diretin that prdues a lser result is the hsen ne. The resizing r quantifiatin f the mment is dne by using the Eulidean distane definitin, where the prpsed mment mpnents (ideal mment) is mpared with the restrited mment and is hsen the restrited mment magnitude that is lser t the prpsed ne. The stability f the algrithm is guaranteed beause the atuatrs are ativated nly when the ttal energy derivative (3.3.) is negative. This leads t think that the best atuatin is the ne that prdues the lwest (negative) energy derivative. An energy like ntrl mpletely thught fr the PSAT s atuatrs and using the previus nept is presented in the next hapter. The destabilising ntrl algrithm in predure that turns the satellite frm an upside-dwn t an upright psitin, desn t wrk when the PSAT atuatrs are used, again beause f the restritin f atuating in just ne il at a time and f the minimum time between atuatins. When used with the alteratins prpsed befre this ntrl law makes the satellite spin instead f induing enugh rll r pith angular velity t turn the satellite upright. By nting that the nly mment that always indues rll r pith trques is the mment prdued by the z ils, is prpsed a ntrl law fr turning the satellite upright frm an inverted psitin that is: m m z x = MAX _ Mz *sgn = m y = z [ E ( N )] MAX _ M (3.3.1) Where MAX_Mz is the maximum magneti mment prduible in the z diretin. The ntrl law makes the satellite atuate nly with the z ils and with the maximum pwer. The sign f the energy derivative gives the rret sign t the atuatin, i.e., if the sign is psitive the mment prdued will inrease the satellite s energy, and this is what is wanted. If the sign is negative, it means that a psitive mment wuld derease the satellite s energy and therefre a negative mment will inrease it Simulatin results The fllwing results were btained using the desribed algrithm, see Appendix B fr further details n the initial nditins fr the simulatin. The results shwn here were attained with a velity gain h=7*1 7, an attitude gain ε=1*1 5, and a destabilising gain g=1*1 7. These values fr the gains were derived thrugh simulatin as they seemed t prdue better results. Nte that the values fr the pinting auray are attained fr 15 rbits, whih means that greater auray uld be btained if a lnger time was used fr the simulatin.

22 γ settling time γ fr 3 rbits γ fr 5 rbits Pinting Energy (J) (rbits) (º) (º) auray (º) Test E Test E Test E Test E Test E Test E Test E Test E Test E Test E Mean E Std. Dev Wrst ase E Table Simulatin results fr the Energy ntrl with ideal atuatrs 18 Euler angles - Ideal atuatrs Rll (º ) Gama (º ) 1 8 Pith (º ) Figure γ evlutin fr the wrst ase (ideal atuatrs) Yaw (º ) Figure 3.3. Euler angles fr the wrst ase (ideal atuatrs) 1

23 γ settling time γ fr 3 rbits γ fr 5 rbits Pinting Energy (J) (rbits) (º) (º) auray (º) Test 1 > Test > Test Test 4 > Test 5 > Test 6 > Test 7 > Test Test Test 1 > Mean N.A Std. Dev. N.A Wrst ase Table 3.3. Simulatin results fr the Energy ntrl with the PSAT atuatrs 7 5 Euler angles - Restrited atuatrs 6 Rll (º ) Gama (º ) Pith (º ) Yaw (º ) Figure γ evlutin fr the wrst ase (PSAT atuatrs) Figure Euler angles fr the wrst ase (PSAT atuatrs)

24 Time fr Ett>Ethres (rbits) Time fr z k > γ settling time (rbits) Energy (J) (rbits) Test Test Test Test Test Test Test Test Test Test Mean Std. Dev Wrst ase Table Simulatin results fr the inverted bm test with ideal atuatrs 18 Euler angles - Ideal atuatrs Rll (º ) Gama (º ) 1 8 Pith (º ) Figure γ evlutin fr the wrst ase (ideal atuatrs) Yaw (º ) Figure Euler angles fr the wrst ase (ideal atuatrs) Magneti mments - Ideal atuatrs m x (Am ) m y (Am ) m z (Am ) Figure Magneti mments fr the wrst ase (ideal atuatrs) 3

25 Time fr Ett>Ethres (rbits) Time fr z k > γ settling time (rbits) Energy (J) (rbits) Test Test N.A Test N.A Test N.A Test Test Test N.A Test N.A Test N.A Test Mean Std. Dev Wrst ase Table Simulatin results fr the inverted bm test with restrited atuatrs 18 Euler angles - Restrited atuatrs Rll (º ) Gama (º ) 1 8 Pith (º ) Figure γ evlutin fr the wrst ase (PSAT atuatrs) Yaw (º ) Figure Euler angles fr the wrst ase (PSAT atuatrs) 5 Magneti mments - Restrited atuatrs m x (Am ) m y (Am ) m z (Am ) Figure Magneti mments fr the wrst ase (PSAT atuatrs) 4

26 3.3.4 Disussin f results The strng pint f this algrithm is that it is glbally stable. As an be seen frm figure the wrst ase urred due t unfavurable nditins that indued the satellite int an inverted bm nditin, but revered frm it and nverged t the referene very fast. The algrithm is als a fast algrithm (with an average settling time f 3 rbits) and presents a very high pinting auray (that will be very redued after intrdutin f aerdynami perturbatins and f the attitude estimatin algrithm in the simulatr dynamis). Fr the PSAT s atuatrs the results are prer, as was expeted sine it is the same algrithm, but atuating at mst 3% f the time f the ideal ase. If the ther restritins n the atuatrs were taken int aunt the attained results wuld steel be are within the expeted values. Frm lnger simulatins values f º where btained fr the pinting auray with the PSAT s atuatrs. The strngest drawbak t the algrithm due t the restritins are the atuatins in nly ne il at a time, sine the diretin f the magneti mment prdued by the satellite is very imprtant fr the effiieny and stability f the algrithm. Fr the inverted bm situatin it is seen frm table that the algrithm has a very gd perfrmane. In the wrst ase the algrithm ges frm an inverted bm attitude t a 5º attitude errr in less than 7 rbits. W.r.t. the magneti mments presented in figure 3.3.7, the values shwn are the values requested by the algrithm, depending highly n the destabilising gain g that is desirably high, while the real atuating values are limited t the maximum values f PSAT-1. Nte the differene between the time required fr the ttal energy t beme higher than the threshld energy in eq , and the time fr the satellite rever frm an upside-dwn psitin t a definite upright psitin (where k z >). Fr the PSAT atuatrs, the results btained are very gd if we take int aunt the reasns already mentined relative t the PSAT atuatrs. In 4 ut f 1 tests, 5º f pinting auray is ahieved in less than 15 rbits frm a 18º inverted bm situatin. Nte that the transitin t an upright psitin is nt very diret, sine due t the limited atuatins, the satellite in sme ases has an angular velity and energy that makes the satellite just pass thrugh frm a dwn t an upright psitin and then againt an inverted psitin. This is the ase presented in figure and Preditive stabilisatin Mtivatin As was shwn in setin the derivative f the Lyapunv funtin based n the ttal satellite energy is given by eq. 3.3., repeated here fr nveniene: E tt = Ω T N trl (3.3.) The equatin E tt = represents all the ntrl trques that lie n a plane that is perpendiular t Ω, therefre impsing E tt < is the same as impsing that the ntrl trque shuld lie "behind" the plane perpendiular t ntrl trque is btained frm eq..3. als repeated here: Ω. Further mre the 5

27 N trl = m B (..) whih states that the ntrl trque must always be perpendiular t the gemagneti field. In view f this the slutin f this ntrl prblem must satisfy this tw requirements: Ω B T T N N trl trl < = (3.4.1) Frm eq. (3.4.1) it an be seen that althugh the slutin t these nstraints is nt a linear spae, it is hwever an unlimited subset f a plan embedded in a three dimensinal spae, in the general ase, r it desn t exist if Ω is parallel t N trl. The same is t say that the slutins t this ntrl prblem are infinite in the general ase, whih pints t a ntrl algrithm that wuld hse the ptimum magneti mment (r at least the best ne given all the nstrains) at eah atuatin mment t take advantage f the partiular angular velity and gemagneti field. This apprah differs frm the thers already nsidered whih use the same ntrl law fr all situatins when, depending n the urrent angular velity and gemagneti field, an ptimum magneti mment is available frm the set f slutins Desriptin The ntrl algrithm has t hse the best magneti mment arding t the angular velity and gemagneti field nfiguratin at atuatin time. Defining a st funtin based n the kineti energy 3 : J 1 = Ω T Λ Ω Ω (3.4.) where Λ Ω is a psitive definite gain matrix. Mre insight will be given relative t the hie f the st funtin when studding the algrithm stability at setin The dynamial mdel f the satellite is well knwn and understd s it an be used t see the influene f the magneti mment n the angular velity. The angular velity f the Cntrl CS w.r.t. the Inertial CS an be written as: Ω ( q) Ωi = + ω i i = Ω + Ωi = Ω + A Ω (3.4.3) where is used the fat that small satellites are usually launhed int plar rbits with small eentriities (PSAT rbit has an inlinatin f 98º and an eentriity f.1) therefre, the angular velity f the Orbital CS w.r.t. the Inertial CS is apprximately given by: Ω i = [ ω ] T The derivative f eq nw bemes: (3.4.4) Ω i = Ω + ω i Ω (3.4.5) 3 The use f Λ q instead f the inertia matrix was hsen due t the pssibility f defining relative weights fr the angular velities. 6

28 substituting in the dynamis equatin (eq..3.1) and negleting the disturbane trque we get: Ω = I Ω i Ωi + Ω ω i + N gg N trl (3.4.6) I + T use eq t predit the evlutin f the angular velity nditined by sme ntrl trque we disretise it nsidering a small time step t: Ω ( t + t) Ω ( t) 1 1 I ( I Ωi ( t) Ω i ( t) ) + I ( Ω ( t) ω i ( t) ) t I N t + I N t whih may be written as: Ω t + t = Ω f gg ( ) ( ) ( ) ( ) ( t) + t f ( t) + O( t) ) ( t) = I ( I Ωi Ωi ) + I ( Ω ω i ) + I N gg + I N trl and the preditin equatin if given by: ( t + t) = Ω ( t) + t f( t) trl (3.4.8) Ω (3.4.9) where the ^ stands fr preditin. It an be seen frm eq that it is pssible t predit the effet that a determined ntrl trque will prdue n the angular velity 4. Fr this preditin there is nly need t knw the urrent angular velities ( ) Ω and attitude ( q), readily available frm the attitude determinatin system. Using the preditin equatin (3.4.9) and (..) it is pssible t hse frm the available magneti mments the ne that minimises the st funtin (3.4.), ne the gemagneti field value is available frm the magnetmeters Stability study The ptential energy f the satellite is mpsed f tw terms (eq..6.1), where the term that reflets the satellite revlutin abut the Earth (eq..6.3) has a minimum when i [ ] T = ± 1, whih means that the x axis f the ntrl CS is parallel t the x axis f the rbit system. As we are nly aiming at stabilise the satellite (remve the angular velity f the Cntrl CS w.r.t. the Orbital CS) this mpnent f the ptential energy brings n useful infrmatin, s it is nt nsidered. The ptential energy due t gravity gradient has a minimum when the z axis f the Cntrl CS is parallel t the z axis f the Orbital CS. One again there is n need t nsider the infrmatin given by this expressin, as the gravity gradient effet will be refleted n the kineti energy. Cnsider that all kineti energy was dissipated, but the z axis f the Cntrl C.S. is nt parallel t the Orbital C.S.. A trque is being applied t the satellite aused by the gravity gradient effet (eq..3.4), therefre this trque is being mpensated by a ntrl trque, t maintain the kineti energy at zer. The gemagneti field is hanging in diretin and amplitude trugh the rbit, but the ntrl trque annt perfetly math these hanges s a residual trque will appear and will impse an angular velity (kineti energy) different frm zer that will be readily eliminated by the ntrller. Althugh the kineti energy will be zer at these unstable equilibria pints, it will nt stay at zer fr lng. As this shws it is enugh fr ur purpses t 4 Reall that eq. (3.4.8) rrespnds t the Euler methd fr slving numerially first rder differential equatins. 7

29 nsider nly a kineti energy like st funtin t minimise 5, beause the nly stable equilibria is k [ ] T = ± 1. Having established that a kineti energy like st funtin is enugh fr stability it is still neessary t shw that this minimisatin methd based n a preditive mdel will wrk. Cnsider a Lyapunv funtin E Lyap as defined in eq. 3.4., the kineti energy based n eq may be expressed as: E Lyap ( t + t) = E Lyap ( t + t) + O ( t) 4 ( ) + O( t) ) (3.4.1) If we nsider that the minimisatin algrithm is wrking rretly, we will have: E ( t + t) E ( t) Lyap < Lyap substituting eq in we get: E Lyap ( t t) E ( t) < O ( t) Lyap 4 ( ) + O( t) ) (3.4.11) + (3.4.1) dividing by t and assuming t as small as wanted, we an write: E lim t Lyap E Lyap ( t + t) ELyap ( t) O ( t) < t < lim t 4 ( ) + O( t) ) t (3.4.13) Therefre glbal unifrm asympttial stability is ensured twards the Ω =, and as previusly shwn als twards k = ± [ 1 ] T r referene [ ] T equivalently t q = q 1 = Glbal stability k = an algrithm similar t the ne prpsed by [Wisniewski] in the energy based ntrl is emplyed. As desribed at setin the energy required t turn the satellite t a bm up nfiguratin is given by eq. (3.3.8), and the ntrl magneti mment shuld inrease the ptential energy when the satellite is n an up side-dwn nfiguratin. T ensure that the atuatin will effetively turn the satellite's bm up nly the z il is emplyed, this way the ntrl trque will indue a libratin mvement. The algrithm emplyed is divided in three steps: Fr the purpse f ahieving glbal stability twards [ 1 ] T If z k < and E ttal < Etreshld hse the magneti mment that maximises the st funtin. If z k < and E ttal > Etreshld wait until the satellite turns up. If z k > hse the ntrl magneti mment that minimises the st funtin. 5 There is anther issue related t minimising the gravity gradient ptential energy, eq..3.4 invlves attitude infrmatin. The preditin f the attitude uld be dne using the same methd as fr the angular velity preditin, but when integrating the angular velity preditin, we wuld als prpagate its errr, whih brings nvergene prblems, and wuld be mputatinally mre demanding. 8

30 3.4.5 Ideal atuatrs (Geneti stabilisatin) Fr ideal atuatrs the minimisatin f the st funtin is dne n a ntinus infinite subset f a plan. Optimum ntrl thery is diffiult t apply in this ase due t the nn-linear time-varying nature f the dynamis equatin. An iterative methd fr the st funtin minimisatin was an alternative, s a Geneti Algrithm (GA) was implemented. The nept f GA's will nt be explained here, as it an be fund n many referenes n that subjet, see fr example [Gldberg]. The implemented GA uses the standard tehniques and a speial peratr, elitism, this means that the best slutin is always preserved and transmitted t the next generatin. Clning has als been used, by whih we insert int the ppulatin the slutin m = [ ] T, beause it has been fund thrugh simulatin that smetimes the algrithm wuld nverge t magneti mments parallel t the gemagneti field after the stabilisatin had been mpleted. The slutin m = [ ] T perfrms same atin (d nthing), but preserves pwer, as it desn't use the magnetrquers fr that purpse PSAT Restrited atuatrs (Brute fre stabilisatin) When nsidering PSAT restrited atuatrs there are nly 19 available magneti mments. Eah il may reeive three different urrents with tw different plarities, whih gives 6 mments per il and 18 mments fr the three ils. The 19 th mment is the d nthing slutin m = [ ] T. With suh a restrited searh spae it is nt neessary t use an iterative minimisatin algrithm, beause all slutins may be evaluated and the best ne (the ne that minimises eq. 3.4.) is hsen Simulatin results These results were btained using the desribed algrithm, see Appendix B fr further details n the initial nditins fr the simulatin. Fr the geneti regulatr a ppulatin f 1 slutins was used and evlved during 1 generatins. Better results uld be ahieved with a larger number f generatins and/r a larger ppulatin f slutins, but thse wuld rise the mputatinal effrt. Mutatin prbability f 3% and rssver prbability f 7% was emplyed, as they seemed t prdue better results. The gain matrix Λ Ω used was the identity matrix. 9

31 γ settling time (rbits) γ fr 3 rbits (º) γ fr 5 rbits (º) Pinting auray (º) Energy (J) Test e Test Test e Test e Test Test e Test e Test Test Test e Mean Std. Dev Wrst ase Table Simulatin results fr Preditive stabilisatin (Ideal Atuatrs). 5 Rll (º ) Pith (º ) Gamma (º ) Yaw (º ) Figure Euler angles (13) evlutin fr wrst ase and ideal atuatrs Figure γ angle evlutin fr wrst ase and ideal atuatrs. 3

32 Fr the brute fre ntrller the matrix. Λ Ω gain matrix used was als the identity γ settling time (rbits) γ fr 3 rbits (º) γ fr 5 rbits (º) Pinting auray (º) Energy (J) Test Test Test Test Test Test Test Test Test Test Mean Std. Dev Wrst ase Table 3.4. Simulatin results fr Preditive stabilisatin (Restrited Atuatrs)..5 q x 1-4 q q3 w z (rad/s) Figure Attitude expressed in quaternins fr Figure wrst ase and restrited atuatrs. z ω evlutin fr wrst ase and restrited atuatrs. 31

33 1 m x (Am ) 7 m y (Am ) m z (Am ) Gamma (º ) Figure Magneti mment fr wrst ase Figure γ angle evlutin fr wrst and restrited atuatrs. ase and restrited atuatrs. Fr the inverted bm test the Λ Ω gain matrix used was ne again the identity matrix and the available set f magneti mments was redued t z max m, max m z when the satellite was n an inverted bm senari. Als { ( ) ( )} the satellite energy threshld used was nt eq. (3.3.8) but E = ( I I ) sine this value imprved the ntrller effiieny. treshld 3ω, xx yy z k > settling time γ settling time (rbits) Pinting auray (º) Energy (J) (rbits) Test Test Test Test Test Test Test Test Test Test Mean Std. Dev Wrst ase Table Simulatin results fr Preditive stabilisatin (Restrited Atuatrs) n an inverted bm senari. 3