Special optimization problem of control mode of spacecraft motion

Transcription

1 Internatonal Robotcs & Automaton Journal Research Artcle Open Access Spe optmzaton problem o control mode o spacecrat moton Abstract In ths paper, the solvng the orgnal specc problem o optmal control s descrbed. Desgnng optmal algorthm o controllng a spacecrat moton ncreases ecency o onboard control system o a spacecrat and orgnates more econom perormance o spacecrat durng lght on orbt. Analytc soluton o the proposed problem s presented or case when a requred tme nterval o control s nown, and rotaton energy unctonal should be mnmzed. The quaternon method and maxmum prncple are used. For dynamly symmetrc spacecrat, complete soluton o spacecrat reorentaton problem n closed orm s gven. I the controllng torque s lmted, analyt ormulas were wrtten or duraton o acceleraton and brang. It s shown that chosen crteron o optmalty provdes a turn o a spacecrat wth mnmal rotaton energy at xed tme. Example and mathemat smulaton results or spacecrat moton under optmal control are gven. Volume 4 Issue 6-8 Levs MV Research Insttute o Space Systems, Russa Correspondence: Levs MV, Research Insttute o Space Systems, Korolev, Thonravova street, 7, Tel (499) , Russa, Emal Receved: November, 8 Publshed: December 8, 8 Keywords: spacecrat atttude, quaternon, optmal control, maxmum prncple Introducton The problem o brngng a spacecrat nto target orentaton poston s solved. The soluton method and the ormalzed descrpton o spacecrat s rotatonal moton nematcs are based on the method o quaternons. Spatal reorentaton s a transerrng the xed axes o spacecrat s hull rom one nown atttude nto another gven angular poston n nte tme T. Angular orentaton o the bodyxed coordnate system s dened relatve to the chosen reerence bass I. An oten encountered case, when reerence system s nertal coordnate system, s consdered. Numerous wors have been dedcated to studyng the controlled rotatons o rgd body and optmal control problem or spacecrat reorentaton n varous statements. 5 In partcular, the soluton was constructed or spe case when the spacecrat rotates around a nte rotaton vector. The rotaton maneuvers around the prncpal central axe o spacecrat s nerta ellpsod were studed n detal. 4 Notce, many publcatons descrbe solutons when rgd body rotates around the Euler axs, 5 although optmzaton prncples and control algorthms der ncludng use o quaternons, uzzy logc and the method o nverse dynamc problem. 4,5 At the same tme, a turn n the least turn-angle plane s not optmal n many pract cases, no matter how exactly t s executed. Also, tme-optmal maneuvers s nterestng and top, thereore they are popular. 4, 6 Many authors have noted that an analyt soluton to the optmal turn problem n a closed orm, t were ound, would be o great pract nterest, snce t allows the nshed laws o programmed control and modcaton o the optmal traectory o spacecrat moton to be appled onboard the spacecrat. 7,8 Certan solutons are nown or an axally symmetrc spacecrat. 4 However, an analyt soluton to the three-dmensonal turn problem o the spacecrat wth arbtrary mass dstrbuton wth arbtrary boundary condtons or spacecrat s angular poston has not been ound; only certan partcular cases are nown when the problem o a turn has been solved (e.g.,.,7, Thereore, n the general case, we are orced to rely only on the approxmate numer soluton to the problem. In some cases (e.g., 6 ), authors aced dcultes, related to the n homogenety o the optmal control uncton, n an attempt to nd numer solutons to the optmzaton boundary-value problems or spacecrat s optmal turn problems (t s typ or tme-optmal problems). The numer soluton o optmal turn problem or dynamly symmetrc spacecrat s consdered n detal, where authors solved the maxmum-prncple boundary-value problem by replacng the varables and reducng t to the boundaryvalue problem o a turn o a spherly symmetrc body. Also, spe control regme o spacecrat rotaton was studed n the problem o optmal reorentaton. 4 Atttude control o the spacecrats wth nertal actuators (or gyrodns) has specc eatures. 5 8 For control system o the spacecrats, controlled by nertal actuators, the patented method 9 can be used. In ths paper, ndex o optmalty characterzes energy consumpton or spacecrat reorentaton. Issues o cost-ecency also reman relevant or the tme beng or spacecrat moton control. Its mnmzaton remans a very mportant problem n practce o spacecrat lght. Fndng and studyng the optmal atttude control problem or spacecrat reorentaton rom one spatal poston nto another (wth respect to rotaton energy) s topc and subect o ths research. The man results o research wor are the ollowng: the carred out research has reached an overall obectve snce optmal control program o spacecrat reorentaton wth mnmal rotaton energy wthn gven tme nterval was ound; t was demonstrated that two-mpulse control when spacecrat rotates by nerta between acceleraton and brang s optmum; or optmal soluton, estmatons o the relatve growth n the unctonal o qualty due to the lmted controllng moment were done. Angular moton equatons and the control problem statement Angular moton o the spacecrat as rgd body s descrbed by dynamc Euler equatons Submt Manuscrpt Int Rob Auto J. 8;4(6): Levs. Ths s an open access artcle dstrbuted under the terms o the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and buld upon your wor non-commerly.

2 Copyrght: Spe optmzaton problem o control mode o spacecrat moton 8 Levs 44 + ( ) ωω =, J ω + ( J J ) ωω = M, ω ( ) J ω J J M J + J J ωω = M () where J are the spacecrat s prncpal central moments o nerta, M are proectons o the man moment M o the orces onto the prncpal central axes o the spacecrat s nerta ellpsod, ω are proectons o the spacecrat s absolute angular velocty vector ω onto the axes o the body-xed bass E ormed by the prncpal central axes o spacecrat s nerta ellpsod (=, ). To descrbe the spacecrat s spatal moton, the mathemat apparatus o quaternons (Euler Rodrgues parameters) s used. Moton o the body-xed bass E relatve to the reerence bass I wll be gven by a quaternon Λ. The angular postons o the ntal and nal spacecrat atttude wth respect to the reerence bass I are gven by quaternon s Λ and Λ, respectvely. Wthout loss o generalty, t s assumed that bass I s nertal. In ths case, the ollowng nematc equatons hold: λ = λω λ ω λω, λ = λω + λω λω () n λ = λω + λω λω, λ = λω + λω λ ω Where λ are components o quaternon Λ (=, ). For smplcty, the quaternon Λ specyng the current spacecrat orentaton s assumed the normalzed quaternon ( Λ = ). In space lght, an mportant characterstc eature o spacecrat moton control s that the dsturbng moments caused by the vehcle s nteracton wth external elds and envronmental resstance are small. The spacecrat moton control relatve to ts center o mass s done by changng the moment o orces M (external or nternal, spacecrat orentaton control s done wth nertal actuators,.e., powered gyroscopes). The boundary condtons are wrtten as: Λ () = Λ () n Λ ( T ) = Λ (4) where T s the tme o endng the reorentaton process, and the quaternon s Λ and Λ determnng the orentaton o spacecrat n xed axes at the ntal and nal tme nstants have arbtrary predened values satsyng the condton Λ = Λ =. For optmzaton o reorentaton control, quadratc crteron o qualty s used. 5 The coecents o the unctonal o quadratc crteron o qualty was chosen n that way, at whch ecency o the control s estmated by the ntegral value T ω ω ω n G = ( J + J + J ) dt (5) The reorentaton optmal control problem s ormulated as ollows: spacecrat must transer rom poston () nto poston (4) accordng to equatons () wth mnmal value o the unctonal (5). The equatons () are necessary or determnng the moments M durng optmal rotaton, ater solvng ths control problem ()-(5). The ndex (5) s the ntegral o the netc rotaton energy on the gven tme nterval. The tme T when the spacecrat reorentaton maneuver should end s xed. The assumed optmalty crteron allows or estmaton o an energetly advantageous angular moton traectory along whch the spacecrat wll turn rom ts ntal poston Λ nto the requred nal angular poston Λ, and or ndng the correspondng control mode. Also, the chosen crteron o optmalty provdes turn o a spacecrat n wth mnmal rotaton energy at xed tme. Solvng the optmal control problem The proposed problem o optmal control s the class nematc rotaton problem. To nd the moton law ω (t) optmal wth respect to crteron (5), t s assumed that the tme o endng the reorentaton maneuver T s gven. The value o the optmalty unctonal (5) does not explctly depend on the moment M o controllng (expresson (5) does not contan M ). It s consdered, angular velocty proectons ω are control varables, and compute moments M by substtutng the ound optmal unctons ω (t) nto equatons () (=, ). The orm o unctonal (5) s spe case o typ (relatvely standard) orm o the optmzed unctonal 5 ( tae nto account the act that the spacecrat angular velocty vector ω s the control varables). The restrcton or phase varables λ s nsgncant because t s always satsed (or any possble spacecrat moton around ts center o mass). Varables λ (beng the components o quaternon Λ) have a characterstc eature: due to equatons (), the norm Λ o the quaternon Λ s constant, Λ =const. For solvng the posed problem, the Pontryagn s maxmum prncple s used. 6 Snce the mnmzed unctonal (5) does not depend on poston coordnates λ the unversal varables r proposed earler can be used (=, ), and to ntroduce the conugate varables correspondng to the phase varables s not oblgatory. The Hamltonan H o problem ()-(5) loos le the ollowng: H = ω r + ω r + ω r J ω J ω J ω (6) Where the unctons r are r = ( λψ + λψ λψ λψ )/ r = ( λψ + λψ λψ λψ )/ r = ( λψ + λψ λψ λψ )/ ψ are the conugate varables correspondng to components o quaternon λ (=, ). The rst terms (t contans the unversal varables r ) comprse the nemat part o the Hamltonan H whch s responsble or the geometrc propertes o optmal moton. The other terms (t contans the spacecrat s moments o nerta) corresponds to the chosen optmalty crteron. The uncton H does not tae nto account the phase constrant Λ =, snce Λ () =. For the unversal varables r, the ollowng system o equatons are true : r = ω r ω r, r = ω r ω r, r = ω r ω r (7) The change n vector r ormed by the unversal varables r s gven by the soluton o the ollowng equaton r = ω r (t s vector orm o the equatons (7) varables r assume proectons o vector r on the axes o the body-xed bass E. The symbol denotes the vector product o two vectors. As t s now, the Ctaton: Levs MV. Spe optmzaton problem o control mode o spacecrat moton. Int Rob Auto J. 8;4(6):4 4. DOI:.546/rat

3 Copyrght: Spe optmzaton problem o control mode o spacecrat moton 8 Levs 45 vector r turns out to be motonless relatve to the nertal bass I, and r =const. Thus, the problem o ndng an optmal control reduces to solvng the system o equatons o spacecrat s angular moton () and equatons (7) under the condton that the control tsel s chosen by maxmzng the Hamltonan. The system o equatons (7) whch determnes the behavor o vector r relatve to xed axes replaces the conugate system o equatons. The optmal vector uncton r(t) s related wth atttude quaternon Λ (t) by the ormula c, where c r() E E n n r = Λ Λ = Λ Λ = const The drecton o vector c E depends on the ntal and nal spacecrat postons. In order, or the spacecrat to have the requred orentaton at the rght end Λ ( T ) = Λ, the vector c E (or the value o vector r at the ntal moment) by the correspondng solutons o system () should be determned. The system o derental equatons (7), together wth the maxmalty condton o the Hamltonan H, s necessary condtons o optmalty. Constrant equatons are gven by the system o equatons () whch descrbes the spacecrat s moton relatve to ts center o mass. The maxmum condtons o uncton H determne sought soluton ω (t). Boundary poston condtons Λ() and Λ(T) determne solutons Λ (t) and r(t). The boundary problem o the maxmum prncple s to nd the value o the vector r() or whch the soluton o the system o derental equatons (),(7) together wth smultaneous maxmzaton, at every current moment o tme, o the Hamltonan H satses reorentaton condtons (), (4) (the restrcton on phase varables λ s not taen nto account snce the equalty Λ (t) = always holds due to equatons ()). To nd the control uncton ω (t) (the optmal control problem) and the optmal vector r, the condtons o maxmum or Hamltonan H must be ormalzed. From ormula (6) ollows, Н s a concave quadratc uncton o the angular velocty ω, and hence ts maxmal value s acheved at the pont o the lo extremum. Applyng the necessary condtons H / ω =, the equatons r J ω = s obtaned. The uncton Н s maxmal the relatons ω = r /J (8) are satsed. The problem o constructng an optmal control has been reduced to solvng the system o equatons o spacecrat s angular moton () and equatons (7) under the condton that control uncton ω s ormed rom requrement (8). The ormulated control problem () (5) s solved completely. Due to the act that r =const= r(), or smplcty, to the normalzed vector p=r/ r s used, p =. For the vector p, t holds that p = ω p, or p = ω p ω p, p = ω p ω p, p = ω p ω p (9) In what ollows, the components p o the vector p wll be used. Note that r = r() p. The necessary condton o optmalty can be wrtten as Jω = bp. () where b> s a sar value. Let-hand sdes o equatons () are proectons o the spacecrat s angular momentum vector onto the body-xed bass axes E. Expressons () lead to the concluson that spacecrat rotaton durng optmal moton s done wth a constant drecton o angular momentum relatve to the nertal coordnate system. The value o b equals the modulus o the spacecrat s angular momentum L. The trple p, p, p represents drectonal cosnes o the vector L relatve to the body-xed bass axes E. Equatons () clearly show that n the geometrc representaton, the vector p s smply the unt vector o the spacecrat s angular momentum vector L n the spacecrat s xed system o coordnates. Thus, the optmal (n the sense o mnmzng the energy ntegral) spacecrat reorentaton s perormed along the traectory o ree moton. Equatons (9), together wth equaltes (), orm a closed system o equatons whch determne unque propertes o optmal moton. Orgnal soluton s determned by close ths system o equatons (9), () by equatons () wth condtons (), (4) or soluton Λ (t). Here and n what ollows, t s assumed that the traectory o ree moton s a amly (totalty or multtude) o angular postons (values o Λ ) that a rgd body occupes durng ts rotaton by nerta. In the geometrc nterpretaton, the traectory o ree moton s a trace o the representng pont Λ (t), where Λ (t) s a soluton o the system o derental equatons (), () when M =M =M = and ω. Moreover, t s nown that the end o the angular velocty vector ω (t) durng the ree moton o rgd body moves along a centrod. 7,8 Let us nd the proporton between the netc energy E and angular momentum L durng optmal slew maneuver. The netc energy E and the value b are related by expresson E = b ( p J + p J + p J )/ Thereore, the proporton E / L = ( p J + p J + p J )/. It s shown that ths proporton E / L s constant on the entre moton nterval t T. The Hamltonan H s ndependent o tme n explct orm,.e. H / t =. Thereore, H=const nsde the entre nterval o control t T. 7 From ormulas (6), the equalty ω r + ω r + ω r J ω J ω J ω = const s obtaned. Ater substtuton the values (8) n ths equalty, the condtons and const ω culated by equatons r J r J r J const + + =, p J + p J + p J = are satsed because r =const. Thereore, E crteron (5). / L = const. It s ey property o optmal moton or Thus, the problem o constructng the optmal control ω (t) has been manly reduced to ndng such a value o vector p() that as a result o the spacecrat s moton, accordng to equatons (), (9), and (), the equalty (4) wll satsy. It s vrtually mpossble to nd a general soluton o ths system o equatons. The problem s to nd boundary condtons on p() and p(t) whch are related by expresson Λ p( T ) Λ = Λ p() Λ () n n The property L =const ollows rom relatons (8) whch relect the Hamltonan H maxmzng property. It s demonstrated above, proporton E / L s constant, so E =const durng optmal rotaton. An exact value o the netc energy E and modulus o angular momentum are determned by the end tme o the reorentaton maneuver T (by the condton that as a result o solvng the system (), (9), (), the boundary condton (4) s satsed at the rght end). The ound Ctaton: Levs MV. Spe optmzaton problem o control mode o spacecrat moton. Int Rob Auto J. 8;4(6):4 4. DOI:.546/rat

4 Copyrght: Spe optmzaton problem o control mode o spacecrat moton 8 Levs 46 modulus L wll correspond to the mnmal possble spacecrat s netc rotaton energy E that provdes or reachng the nal poston Λ n allotted tme T. Note, the vectors ω and p are related as ω = E p + + p J p J p J J () Let us now nd the controllng moments necessary to support the spacecrat s optmal rotaton mode n tme nterval <t<t durng the reorentaton. Let us substtute the unctons ω (t) computed by expressons (8) nto dynamc equatons () wth tang nto account the act r =const. As result, all component o moment M are M =. In the soluton optmal wth respect to crteron (5), the spacecrat s reorentaton s done wth zero controllng moment M=. Ths ollows rom the analyss o equatons (8) that show a relaton between the angular momentum L and the vector r o unversal varables. The act that L=r/ and L =const, eepng n mnd the mmoblty o vector r n the nertal bass I, mples that the spacecrat s angular momentum vector s constant relatve to nertal coordnate system durng the reorentaton. The netc energy E s constant also. Thus, t s proven the ollowng concluson: spacecrat s reorentaton occurs wth the mnmal value o the netc energy ntegral and only the spacecrat durng ths reorentaton rotates as a ree moton (the man moment o orces s zero),.e., when the spacecrat s rotaton (as a rgd body) s an Euler Ponsot moton. 8 I allow a step le change o the angular velocty vector ω, then the proposed optmal control problem (the nematc reorentaton problem) can be consdered solved: Equatons (), (9), and () completely dene the necessary moton ω (t). The tas o the onboard control system or realzaton o optmal control s to norm the spacecrat o the ntal moton condtons, namely the culated angular velocty at tme moment t= and reducng the netc energy to zero at tme moment t=t, when Λ () t = Λ (ater the spacecrat reaches ts nal poston Λ ). From the moment o reachng the necessary ntal angular velocty ω and untl the reorentaton s nshed, when the spacecrat wll be n the neghborhood o the requred poston Λ, there s no torque M actng on the spacecrat s body; the spacecrat perorms uncontrolled rotaton (M=),.e. ree moton. Creatng the ntal angular velocty and dampng the nal rotaton happens n an mpulse (as ast as the spacecrat s actuators wll allow). Between the mpulsve mpartng o angular momentum and the mpulsve suppressng o angular momentum J ω + J ω + J ω = const, J ω + J ω + J ω = const () Topty o the consdered problem s due to the act that by mnmzng ntegral (5) the energy spent to perorm spacecrat reorentaton rom poston Λ nto poston Λ n tme T s n mnmzed. The energy E obeys the ollowng relaton: T t T ω ω ω T max E ( J + J + J ) dt / Consequently, mnmum o quantty T t T ω ω ω max E = ( J + J + J ) dt / T max E s t T The mnmal value o energy, requred or spacecrat rotaton satsyng gven boundary condtons (), (4), s reached n the only one case when J ω + J ω + J ω = const and the ntegral (5) s mnmum. Both last condtons satsy or moton under the equatons (7), (8) whch are the necessary condtons o optmalty (see () also). As a result, when the unctonal (5) s mnmzed, we actually arrve at the mnmum rotary energy max E and, as corollary; the t T mnmum energy o the control orces s acheved. The spacecrat rotaton regme accordng to condtons (7), (8) (or (9), ()) gves the absolute mnmum o the controllng orces wor snce n tme nterval <t<t the spacecrat rotates wthout control (wth moment M=). Fndng the optmal moton o a spacecrat n the partcular cases Constructng the optmal reorentaton regme wth mnmal value o ntegral (5) s non-trval tas. In the optmal reorentaton problem (n constructng the optmal programmed moton ω (t)), t s cru to nd the ntal vector p() and the correspondng angular velocty ω (+) (the angular velocty ω (+) s culated by ormulas ()). The vector p() depends on reorentaton parameters Λ and the spacecrat characterstcs J, J, J. For arbtrary values J J J, t s hard to nd the soluton o the consdered problem o spacecrat s three-dmensonal reorentaton or arbtrary values Λ and Λ. The dculty s to nd boundary vectors p() and p(t) whch are related by (). Analyt soluton o the system o equatons (), (9), and () exsts or dynamly spher and dynamly symmetrc bodes only. For a spherly symmetrc spacecrat (when J =J =J ), the soluton p(t), ω (t) have elementary orm: p(t)=const and ω (t)=const, or n detal ν arccos ν p = ν ν+ ν+ ν, and ω = T ν + ν + ν Where ν, ν, ν, ν are components o the reorentaton quaternon Λ = Λ Λ. t n For a dynamly symmetrc spacecrat (or example, when J =J ), the optmal control problem can be solved completely also. For ths dstrbuton o spacecrat s mass, the ollowng derental equatons: = ( ) ωω, J ω = ( J J ) J ω J J ωω are satsed under condton ω = const. Last system o derental equatons descrbes the oscllator (wth the parameter ω = const ), or whch n ω and ω are harmonc unctons o tme. Thereore, р =const=р and harmonc oscllatons o the unctons р and р are observed. In ths spe case, the optmal moton s the smultaneous rotaton o the spacecrat as a rgd body around ts axal axs OX and around spacecrat s angular momentum L whch s constant n the nertal space and whch consttutes a certan constant angle ϑ wth the spacecrat s axal axs. Angular veloctes wth respect to OX and p axes have a constant rato (as s shown above, vectors L and p are parallel). The soluton o system (), (9), (), necessary or solvng the control problem, s regular precesson. For the regular precesson case t Ctaton: Levs MV. Spe optmzaton problem o control mode o spacecrat moton. Int Rob Auto J. 8;4(6):4 4. DOI:.546/rat

5 Copyrght: Spe optmzaton problem o control mode o spacecrat moton 8 Levs 47 poβ / eα / Λ =Λ e e n where p =p(); e s the unt vector o the spacecrat s axal axs; α s the spacecrat s rotaton angle around ts axal axs; β s the spacecrat s rotaton angles around the vector p, e s quaternon exponent. It s assumed that α π, β π. For a dynamly symmetrc spacecrat wth moments o nerta J J = J, the soluton p(t) s wrtten as ollows: p = p = const = cos ϑ, p = p cos κ + p sn κ, p = p sn κ + p cos κ (4) where t J J κ = ω () t dt. J In ths case, dependences (4), together wth equaltes (), orm a soluton o the system o equatons (), (9) under condton (). At the same tme, the vector p also generates a cone around the axal axs OX n the body-xed coordnate system. The specc value o р s determned exclusvely by the requrement that, accordng to equatons (),(9), (), boundary condtons () and (4) be satsed. In ths type o control, the spacecrat s angular momentum preserves a constant drecton n the nertal reerence bass I, whle the axally symmetrc body moves along a conc traectory. To move the spacecrat rom poston Λ, t rotates smultaneously around the Λ nto poston n vector c E, whch s constant relatve to the nertal bass I, by the angle β, and around ts own longtudnal axs by the angle α. Usng the mathemat ormalsm o quaternons to descrbe rotatons o rgd body about the center o mass, relatons relectng a dependence between the values p, α, and β are wrtten. The dependence o parameters p, α, and β on the boundary angular postons Λ s gven by the ollowng system o equatons: β α β α cos cos p sn sn = ν, p β α β α sn cos + p sn sn = ν, β α β α cos sn + p sn cos = ν β α β α p sn cos + p sn cos = ν, α = Λ J J p J β n and For a gven reorentaton tme T, angular rotaton speeds around the OX and p axes are equal to α = α / T,and β = β / T. The magntude o angular momentum durng optmal rotaton s L =J β /T. The programmed values o controllng unctons ω (proectons o the angular velocty vector ω ) have the ollowng orm: ω = α + β p, ω β cos( α σ ) = p t + where σ = arctg ( p / p ). ω β sn( α σ ), = p t + Notce, optmal values p, α, and β correspondng to soluton o last system o ve transcendent equatons and whch correspond to ree moton rom poston Λ nto poston Λ can be determned n wth usng the devce. 9 For a non-symmetrc spacecrat (when J J J ), the system (), (9),() can be solved by numer methods only (e.g., usng the method o successve approxmatons or teratons methods wth consecutve approach to true soluton). To nd the vector p, t s necessary the solvng the boundary problem Λ () = Λ, n Λ ( T ) = Λ, tang nto account the equatons (), () mposed upon the moton, n whch M =. As a result, the value o the angular velocty vector at the ntal tme moment ω, or whch the spacecrat s moved wth ts ree rotaton wth respect to the center o mass (M=) rom the state Λ ( T ) Λ () = Λ, ω ()= ω to the state n = Λ, wll be ound ( ω (T) s arbtrary here). In partcular, the method o solvng the boundary problem and determnng the vector p was descrbed n detal n artcle. The value o vector p relates to ω as ð = J ω ( Jω ) + ( Jω ) + ( Jω ) The nown algorthms presented n patent and system 4 can be used or ndng culated values ω and p also. These algorthms,,4 are relable and provde asymptotc approachng or sought value p. Other culaton schemes can be useul only n some specc cases. Desgnng the optmal program o spacecrat reorentaton under restrctons on the controllng moment In many pract tass, reorentaton s made n stuaton when ntal state satses condton ω ()= and nal angular velocty must be absent ω (T)= (these cases occur very requently, espely atttude control s done relatve to nertal coordnate system). It s obvous, n nstants o tme t= and t=t, angular velocty culated accordng to the ormula (), correspondng to nomnal program o optmal rotaton maneuver, s not equal to zero. Thereore, segments o acceleraton and brang at the begnnng and the endng o slew maneuver are nevtable. For the optmal moton, spacecrat reorentaton rom one angular poston Λ to another poston Λ s done by mpulsve mpartng the necessary angular velocty (the nomnal value o the angular momentum vector) to the spacecrat, rotaton o the spacecrat wth constant netc energy and modulus o angular momentum, and short-term (mpulse) reducton o the rotaton energy to zero. Very mportant characterstc or reorentaton maneuver s ntegral T S = L() t dt (5) The value o characterstc S s determned only by the rotaton condtons Λ, Λ, and the spacecrat s prncpal central moments o n nerta J, J, J. I tme o reachng the culated angular velocty whch s equal ω = E JC p nom n Ctaton: Levs MV. Spe optmzaton problem o control mode o spacecrat moton. Int Rob Auto J. 8;4(6):4 4. DOI:.546/rat

6 Copyrght: Spe optmzaton problem o control mode o spacecrat moton 8 Levs 48 and duraton o suppressng the angular velocty to zero are nntesmal, then modulus o angular momentum durng uncontrolled moton (between acceleraton and brang) s L =S/T, where ntegral (5) s culated as pr ω ω ω S = t J + J + J Where t pr s the predcted tme o achevng the condton Λ ( t ) = Λ durng ree rotaton rom the poston pr Λ () = Λ wth ntal angular velocty ω ()= ω (accordng to the equatons (), () n whch all values M =). Note, the value S and vector p, whch satsy optmal moton, are computed together. Denote C = p J + p J + p J For a spherly symmetrc spacecrat and or a dynamly symmetrc spacecrat, ey characterstcs and constants o control law are determned straghtorwardly, wthout ntegraton o moton equatons (), (). For a spherly symmetrc spacecrat, the ntegral S s culated as S = J arccosν Accordngly, optmal modulus o angular momentum durng uncontrolled rotaton s L = J arccos ν / T opt and netc energy s E = J arccos ν / T For an axally symmetrc spacecrat (when J =J ), the ntegral S s equal S = J β Optmal modulus o angular momentum durng uncontrolled rotaton s L = J β / T opt n and netc energy s E = J β (cos ϑ / J + sn ϑ / J ) / ( T ) I the moment M s lmted, some non-zero tme τ s requred or mpartng the requred angular momentum to the spacecrat and or suppressng the exstng angular momentum to zero. A restrcton on the magntude o easble controllng moment leads to the appearance o restrcted ntervals when spacecrat ncreases and decreases ts angular velocty. These ntervals lead to an ncrease n the angular momentum necessary on the stage o ree rotaton n order to comply wth the gven tme T. The netc energy E ncreases (accordng to the constant proporton E / L ). The ntegral (5) ncreases also (as compared to the mpulse, step-le control). Thereore, mnmzaton o tmes o acceleraton and deceleraton s necessary. On the acceleraton stage, the control problem s to transer the spacecrat rom the state ω ()= to the state ω = ω n mnmal tme. On the deceleraton stage, the control problem s to transer the spacecrat rom the state ω to the state ω (T)= n mnmal tme (n order that nstant o begnnng o brang s maxmal not ar rom gven tme T). In many cases o atttude control, the eld o avalable moment M s descrbed by the nequalty M J + M J + M J u (6) where u > s some postve value specyng power o actuators o spacecrat atttude system. In uture, or ndng the optmal rotaton rom poston Λ n nto poston Λ, let us assume that the moment M satsy the condton (6). Laws o optmal control or astest acceleraton and brang are nown when restrcton on the controllng moment M has the orm (6). For spn-up at mnmal tme, optmal moment M s culated by the ormula ω ω ω ω M = uj J + J + J (7) Optmal moment M and angular momentum L are parallel durng acceleraton at mnmal tme. Derentaton o let and rght parts o equaltes (7) gves the ollowng derental equatons (angular acceleratons ω are taen rom dynamc equatons ()): Rewrte last equatons n vector orm. M = ω M M = ω M ω M, M = ω M ω M, M = ω M ω M (8) The obtaned derental equaton, or the controllng moment M, means ts mmoblty relatve to nertal coordnate system. As consequence, M =const durng acceleraton stage. For astest brang, optmal moment M s ω ω ω ω M = uj J + J + J (9) Ater derentaton o equaltes (9) we obtan derental equatons (8) rom whch the property M =const appears or the entre brang stage. Thus, equalty M =const s satsed or optmal rotaton durng acceleraton and brang phases. It s very mportant property o optmal moton and optmal control. Both at mpartng the culated angular momentum, and at dampng o rotaton, the moment M has constant magntude (drecton o vector M s not changed relatve to nertal bass I);.e. wthn acceleraton and brang segments, optmal moment M s the xed vector relatve to nertal coordnate system. For zero boundary condtons ω()=ω(т )=, slew maneuver ncludes two phases durng whch magntude o the moment М s maxmal possble: acceleraton and brang, and phase o uncontrolled moton at whch equatons () are satsed. A detaled analyss o the man reorentaton stages: speedup, brang, and uncontrolled spacecrat rotaton wth constant netc energy and angular momentum (relatve to nertal coordnate system), shows that all three stages have a common property, namely that spacecrat rotates along the traectory o ree moton. It s characterstc or the Ctaton: Levs MV. Spe optmzaton problem o control mode o spacecrat moton. Int Rob Auto J. 8;4(6):4 4. DOI:.546/rat

7 Copyrght: Spe optmzaton problem o control mode o spacecrat moton 8 Levs 49 traectory o ree moton that the spacecrat s angular momentum remans constant n nertal coordnate system. Tang nto account that durng the entre reorentaton (on the entre tme nterval [, T]) the torque M s parallel to the angular momentum vector L (.e., ether ponts n the same drecton as L, or n the opposte drecton, or equals zero), can conclude that M L=, and, thereore, there are no reasons or a rotaton o the angular momentum L n nertal coordnate system. For spacecrat reorentaton wth lmted control, ey property o optmal moton remans vald, the proporton E / L between the netc energy E and angular momentum L s constant on the entre nterval o tme t T, ndependently o duraton o acceleraton and brang (ndependently o presence or absence o acceleraton and brang stages). Angular momentum L and rotaton energy E are contnuous unctons o tme. Thereore, the proporton ρ =E / L s contnuous uncton. For acceleraton (or brang), the equaltes (7) (or (9)) and M =const are satsed. Thereore, the proporton ρ s constant wthn acceleraton and brang. Between acceleraton and brang, the equatons () are satsed; as result, ρ =const durng ree rotaton. Hence, the proporton ρ =E / L =const wthn tme nterval [, T] because o a contnuty o uncton ρ. As consequence, modulus o moment M s dent or acceleraton and brang, and t s equal to same magntude m = u ρ = u / C snce M = u L E Angular momentum L does not change the drecton relatve to nertal coordnate system durng acceleraton, brang and at rotaton between acceleraton and brang (when spacecrat rotates by nerta), so, a drecton o angular momentum n nertal coordnate system nvarably on the entre nterval o tme [, T]. Hence, n the presence o restrcton (6), the equatons (9) and () are satsed at the entre nterval o rotaton rom t= to t=т. At acceleraton and brang М =const=m, where m > s maxmal admssble magntude o the moment М n drecton o angular momentum L. Snce on entre nterval o control J ω // L =p, the value m s dent durng both phases o acceleraton and brang, and t s equal u /C, where C s the earler ntroduced constant whch s unambguously speced by the vector p and moments o nerta J, J, J. The condton M m s satsed wthn the entre nterval o control, where m =u /C. Optmal moment M s parallel to motonless lne relatve to nertal coordnate system. Drecton o ths motonless lne s determned by drecton o the vector p (snce the drectons o angular momentum L and the vector p concde). Hence, M=m(t) p. The sar uncton m(t) s speced as m(t)=(m L +M L +M L )/ L or m(t)=m p +M p +M p. Control uncton m(t) s three-postonal relay. The uncton m(t) can be wrtten as: m(t)=m L <L opt and t<t/; m(t)=-m t T τ ; m(t)= τ t < T τ. Here, L opt s the modulus o angular momentum durng ree rotaton; τ s duraton o acceleraton (brang); L = m τ. Note the spacecrat s angular opt momentum satses the nequalty L L opt or any tme t. For optmal control, spacecrat acceleraton contnues untl ts angular momentum equals the target level L = L Λ Λ p() Λ Λ tag opt n n The begnnng o brang wll be determned rom the act that as angular velocty ω reduces to zero, the angular momentum value L (t) changes lnearly. Durng brang, the modulus o the controllng moment s constant, and the tme moment rom whch brang wll be started s speced by the ollowng condton: 4arcsn + K + K δ δ ω ω = ( J ) ( J ) m ( J ) ( J ) ω + ω ω + ω Where m s the maxmal controllng moment magntude that can be provded by the actuators o spacecrat s atttude control system; δ are the components o the dscrepancy quaternon Λ () t Λ ; К = L(t) s the current magntude o the spacecrat s angular momentum. The sad condton or ndng the start moment o brang phase allows the onboard control system to orm the angular velocty reducton sgnal based on the normaton on current spacecrat orentaton and measurements o ts angular velocty. Use o ths condton ncreases the precson o transerrng the spacecrat nto nal poston Λ. The case when the regon o easble controllng moments s gven by nequalty (6) was consdered. The problem o optmal control o spacecrat orentaton, when restrcton has the orm M m, s solved analogously. I control s lmted by condton M m, then, n order to gan (suppress) the angular momentum L as ast as possble, t s necessary (and sucent) that the moment M s parallel to the angular momentum L; speed o change o the angular momentum modulus (as a measure o moton ntensty) does not depend on angular velocty ω. Optmal value o the moments M s M = ± mjω / L (sgn + corresponds to an acceleraton, sgn - corresponds to a brang). Optmal moment M s collnear to the vector p whch s unt vector o angular momentum L. Dependence o the controllng moment M on the vector p and all relatons obtaned or control uncton m(t) are true or optmal rotaton under restrcton M m. The system o equatons (9), () s satsed also. Thus, we have arrved to the ollowng concluson: spacecrat reorentaton wth a mnmal energy ntegral s perormed along the traectory o ree moton, on whch the drecton o the spacecrat s angular moment remans constant n nertal coordnate system on the entre tme nterval rom t= to t=t. For cases when τ << T the constructon acceleraton o rotaton, uncontrolled rotaton, dampng o rotaton s optmum or optmal control problem ()-(5) ω ()= ω (T)= n slew maneuver. The assumed crteron o optmalty provdes spacecrat moton wth mnmal netc energy o rotaton durng reorentaton maneuver. I at tme moments t= and t=t the angular velocty ω changes abruptly, steple, then the energy ntegral G and the reorentaton tme T would be nversely proportonal to each other, and the level o netc energy E o the spacecrat rotaton would relate to the reorentaton tme T as T E =const=c S. Let us estmate the relatve growth n unctonal G due to the nonzero tme t taes to gan and suppress the angular momentum. As s now, value o the ntegral (5) does not depend on the character o varaton o the angular momentum modulus angular moton satses the system o equatons (9), (). For spacecrat moton along the traectory o ree moton, t holds that ( T ( t + t ) / ) L = S ac br opt Ctaton: Levs MV. Spe optmzaton problem o control mode o spacecrat moton. Int Rob Auto J. 8;4(6):4 4. DOI:.546/rat

8 Copyrght: Spe optmzaton problem o control mode o spacecrat moton 8 Levs 4 because the modulus o angular momentum changes accordng to the lnear law durng optmal acceleraton and brang, where L opt s the magntude o angular momentum between acceleraton and brang (.e. when condton L =const s satsed); t and t are the duratons o ganng and suppressng the angular momentum, respectvely. Thereore, L = S / (T ( t + t ) / ), and opt ac br E = C S / (( T ( t + t ) / ) ) The value o unctonal (5) equals ac G = G T( T ( t + t ) / )( T ( t + t ) / ) opt ac br ac br br Where G opt s the value o the energy ntegral (5) or mpulse control (when, t ). The value G opt s G opt =С S /Т t ac Absolute ncrease n ntegral (5) s br ac G = G G. Obvously, G opt s mnmal when t + t =; and mnmal value o unctonal G s ac br G mn =G opt. I t + t >, then a relatve ncrease n the value o G ac br wll be G / G = T( T ( t + t ) / )( T ( t + t ) / ) opt ac br ac br The tme-optmal reachng o the necessary angular momentum s done the controllng moment and the spacecrat s angular momentum have the same drecton. The astest reducton o the angular velocty s done the controllng moment M s drected opposte to the spacecrat s angular momentum. In our case, at the acceleraton and deceleraton stages, the magntude o moment M s constant, tme characterstcs o the optmal moton program can be determned wth sucent precson. Snce magntude o the moment М s dent durng acceleraton and brang M =m =u /C (the constant C s ntroduced earler), speedup tme and brang tme are dent, and nomnal value L opt = m τ, where τ s the tme whch s necessary to gan (suppressng) the angular momentum T 4 τ S = m T The relatve loss wll equal G/ G = TT ( 4 τ / )( T τ) opt The less τ, the less the loss G / G clear to wrte τ τ G / G = opt ( T τ ) T τ opt br wll be. It becomes qute The tme τ changes rom zero to T/. The uncton G / G ncreases everywhere n the range τ <T/. The mnmal value corresponds to the case τ, and the crt pont (maxmum) s the hypothet case τ =T/ as lmt. The excess to a thrd o G opt. G / G opt opt wll amount Numer example and results o mathemat smulaton Let us gve a numer soluton o optmal control problem or spacecrat reorentaton wth mnmal value o the ntegral (5). As an example, let us consder spacecrat reorentaton or 8 degree rom ntal poston Λ that corresponds to the atttude when body axes n concde wth the axes o reerence bass I nto the target poston Λ. Let us assume that ntal and nal angular veloctes are zero, ω ()= ω (T)=. Values o the elements o quaternon Λ that characterzes the target spacecrat atttude are as ollows: λ =, λ =.777, λ =.59, λ =.9 Let us nd the optmal control program or the spacecrat rotaton velocty ω (t) or transerrng the spacecrat rom the state Λ () = Λ, ω ()= to the state Λ ( T ) n = Λ, ω (T)=. The constant u whch characterzes power o actuators s u =. N g /. The spacecrat s mass-nertal characterstcs are as ollows: J = g m, J =8466. g m, J = g m Let us present a numer soluton o the spacecrat reorentaton optmal control problem. Let the reorentaton tme be equal T=4 s. As a result o solvng the nematc reorentaton problem o transtonng the spacecrat rom poston Λ () = Λ nto poston Λ ( T ) = Λ (the optmal reorentaton problem n the mpulse settng), the culated value o vector p ={.48549;.6;.8659} and ntegral S= N ms wll be obtaned Accordng to these culated values, the ntal angular velocty equals ω ={ /s;.59 /s;.477 /s}. Iteratons method guaranteeng successve approach to true value p was used (n most cases, ths method provdes asymptotc approachng). The maxmal value o the controllng moment s m =9. N m. The duratons o speedup and brang s the same and equal to τ = s, whle the angular momentum magntude at the stage o rotaton by nerta equals L opt =85. N m s. Results o the mathemat modelng o the reorentaton process under optmal control are gven on Fgure 4. Fgure shows the character o changng the angular veloctes n the spacecrat-related system o coordnates ω (t), ω (t), ω (t) wth respect to tme. At the stage between spacecrat speedup and brang, the spacecrat rotates wth constant energy equal to E =.7 oules. The value o the energy ntegral, whch characterzes the cost-ecency o the rotaton traectory { Λ (t), ω (t)} ater spacecrat s angular moton rom poston Λ nto poston Λ (the value o unctonal (5)) has n been equal to G=56 J s. Fgure shows the graphs o changes n the components o quaternon Λ (t) that denes current spacecrat orentaton n the process o the rotaton maneuver: λ (t), λ (t), λ (t), λ (t). Fgure shows the dynamcs o components p (t), p (t), p (t) o unt vector p n tme. It s characterstc that that change o proecton p s very small n comparson wth proectons p and p (the angular velocty component ω also changes a lot less on the nomnal rotaton nterval than angular velocty components ω and ω ). Ths conrms the act that the OX axs s longtudnal axs. Unle varables ω, varables p and λ are smooth unctons o tme. Fnally, Fgure 4 shows the behavor o sar uncton m(t). As t s well vsble, change o uncton m(t) has relay character. n Ctaton: Levs MV. Spe optmzaton problem o control mode o spacecrat moton. Int Rob Auto J. 8;4(6):4 4. DOI:.546/rat

9 Copyrght: Spe optmzaton problem o control mode o spacecrat moton 8 Levs 4 Fgure Changng the angular veloctes durng optmal reorentaton maneuver. Fgure Changng the components o orentaton quaternon Λ (t) durng optmal reorentaton. Fgure Changng the components o unt vector p n tme under optmal control. Fgure 4 Changng the sar uncton m(t) or optmal control. Results and dscusson Control algorthm o angular moton s very sgncant element o control system o spacecrat atttude, booster unts and orbtal statons. Desgnng optmal algorthm o controllng a spacecrat moton ncreases ecency o onboard control system o a spacecrat and orgnates more econom perormance o spacecrat durng lght on orbt. In ths paper, the optmal control problem or spacecrat s spatal reorentaton n gven tme s consdered and solved. The optmzaton has been perormed or case when rotaton energy ntegral should be mnmzed. Fndng the optmal mode o spacecrat reorentaton wth a mnmal value o energy expendture s qute top. An analytc soluton o the proposed problem s presented. Formal equatons and computatonal expressons or constructng optmal reorentaton program were obtaned. To solve the ormulated problem, the optmalty condtons n the orm o the maxmum prncple are appled, and use o quaternons sgncantly smples computatonal procedures and reduces the computatonal costs o the control algorthm, whch maes t sutable or onboard mplementaton. Mathemat expressons, whch provde or obtanng nal equatons and relatons descrbng the varaton o control unctons and behavor o a spacecrat durng optmal slew maneuver, s based on unversal varables whch were ntroduced and nvestgated earler. Frst the man characterstc propertes o optmal moton and the type o traectory optmal wth respect to the chosen crteron were determned. The nematc reorentaton problem has been solved completely. Necessary optmalty condtons and determned the optmal control structure were ound; ormalzed relatons to determne the spacecrat s optmal moton s obtaned. Ater that, the spacecrat reorentaton control problem wth regard to the constrants on the controllng moment s studed. For a specc orm o these constrants, the problem o optmal reorentaton was solved. It was demonstrated that optmal soluton s two-mpulse control. I controllng torque s lmted, analyt ormulas were wrtten or duraton acceleraton and brang. It s shown that along the entre reorentaton nterval, drecton o spacecrat s angular momentum s constant n the nertal coordnate system, and the spacecrat rotates along the traectory o ree moton. A procedure or mplementng the control mode s descrbed. It s estmated how the duraton o ganng and suppresson o angular momentum nluences energy costs. Expressons or computng temporal characterstcs o the reorentaton maneuver and the condton or ndng the deceleraton start moment based on actual nematc moton parameters udgng by termnal control prncples are presented, whch leads to hgh orentaton precson. Example and results o mathemat modelng or spacecrat moton under optmal control are gven. Thus, ey results are the ollowng: the carred out research has reached an overall obectve snce optmal control program o spacecrat reorentaton wth mnmal rotaton energy wthn gven tme nterval was ound; t was demonstrated that two-mpulse control when spacecrat rotates by nerta between acceleraton and brang s optmum; or optmal soluton, estmatons o the relatve growth n the unctonal o qualty due to the lmted controllng moment were done. Other characterstc propertes o the obtaned optmal moton are determned also. All conclusons are absolutely true snce well-nown mathemat methods were used, and all mathemat ormulas are based on the checed theores. For a dynamc symmetrc spacecrat, a complete soluton o the reorentaton problem n closed orm s presented; optmal values o control law parameters Ctaton: Levs MV. Spe optmzaton problem o control mode o spacecrat moton. Int Rob Auto J. 8;4(6):4 4. DOI:.546/rat

10 Copyrght: Spe optmzaton problem o control mode o spacecrat moton 8 Levs 4 can be ound by the devce. 9 The obtaned results demonstrate that the desgned control method o spacecrat s three-dmensonal reorentaton s easble n practce. Concluson In ths research, new control method o spacecrat atttude s obtaned; the used crteron o optmalty s new and specc. The desgned method o spacecrat s moton control was descrbed n detal. The solved problem s very top. Sgncance and mportance o the executed nvestgatons consst n the act that chosen crteron o optmalty mnmzes netc energy o rotaton wthn the gven nterval o tme. The obtaned method s ders rom all other nown decsons. Man derence conssts n new orm o mnmzed unctonal whch allows to turn spacecrat wth mnmal rotaton energy maneuver tme s gven. Ths useul qualty s advantage o presented control mode because t sgncantly saves the controllng resources and ncreases the possbltes o spacecrat control. Acnowledgments No nan exsts. Conlcts o nterest No conlcts o nterest exst. Reerences. Branets VN, Shmyglevs IP. Use o quaternons n the problems o orentaton o sold bodes Aleseev KB, Malyavn AA, Shadyan AV. Extensve control o spacecrat orentaton based on uzzy logc. Flght. 9; Raushenbah BV, Toar EN. Spacecrat orentaton control.974;6. 4. Ermoshna OV, Krshcheno AP. Synthess o programmed controls o spacecrat orentaton by the method o nverse problem o dynamcs. Journal o Computer and Systems Scences Internatonal. ;9():. 5. Velshchans MA, Krshcheno AP, Tachev SB. Synthess o spacecrat reorentaton algorthms usng the concept o the nverse dynamc problem. Journal o Computer and System Scences Internatonal. ;4(5): L F, Banum PM. Numer approach or solvng rgd spacecrat mnmum tme atttude maneuvers. Journal o Gudance, Control, and Dynamcs. 99;(): Scrvener S, Thompson R. Survey o tme-optmal atttude maneuvers. Journal o Gudance, Control, and Dynamcs. 994; 7():5. 8. Lu S, Sngh T. Fuel/tme optmal control o spacecrat maneuvers. Journal o Gudance. 996;(): Byers R, Vadal S. Quas-closed-orm soluton to the tme-optmal rgd spacecrat reorentaton problem. Journal o Gudance, Control, and Dynamcs. 99;6(): Junns JL, Turner JD. Optmal spacecrat rotatonal maneuvers. Elsever. 986;54.. Levs MV. Pontryagn s maxmum prncple n optmal control problems o orentaton o a spacecrat. Journal o Computer and System Scences Internatonal. 8;47(6): Shen H, Tsotras P. Tme-optmal control o ax-symmetrc rgd spacecrat wth two controls. Journal o Gudance, Control, and Dynamcs. 999;(5): Molodenov AV, Sapunov YaG. A soluton o the optmal turn problem o an axally symmetrc spacecrat wth bounded and pulse control under arbtrary boundary condtons. Journal o Computer and System Scences Internatonal. 7;46():. 4. Molodenov AV, Sapunov YaG. Spe control regme n the problem o optmal turn o an axally symmetrc spacecrat. Journal o Computer and System Scences Internatonal. 7;49(6): Kovtun VS, Mtras VV, Platonov VN, et al. Mathemat support or conductng experment wth atttude control o space astrophys module Gamma. Techn Cybernetcs. 99;: Platonov VN, Kovtun VS. Method or spacecrat control usng reactve actuators durng executon o a programmed turn Levs MV. Features o atttude control o a spacecrat, equpped wth nertal actuators. Mechatroncs, Automaton, Control. 5;6(): Levs MV. Spe aspects n atttude control o a spacecrat, equpped wth nertal actuators. Journal o Computer Scence Applcatons and Inormaton Technology. 7;(4): Levs MV. Method o Controllng a Spacecrat Turn. 997;7.. Levs MV. The use o unversal varables n problems o optmal control concernng spacecrats orentaton. Mechatroncs, Automaton, Control. 4;: Levs MV. On optmal spacecrat dampng. Journal o Computer and System Scences Internatonal. ;5(): Levs MV. Optmal spacecrat termnal atttude control synthess by the quaternon method. Mechancs o Solds. 9;44(): Levs MV. Method o spacecrat turns control and the system or ts realzaton. Inventons Applcatons and Patents Levs MV. Control system o spatal turn o a spacecrat. Inventons Statements and Patents. 994; Krasovs AA. Handboo on the Automatc Control Theory. 987;7. 6. Pontryagn LS, Boltyans VG, Gamreldze RV, et al. The mathemat theory o optmal processes. Gordon and Breach: USA; 987. p Young LG. Lectures on the culus o varatons and optmal control theory. WB Saunders Company: USA; Mareev AP. Theoret mechancs. 44 p. 9. Levs MV. Devce or regular rgd body precesson parameters ormaton. Inventons Applcatons and Patents... Lastman G. A shootng method or solvng two-pont boundary-value problems arsng rom non-sngular bang-bang optmal control problems. Internatonal Journal o Control. 978;7(4): Bertolazz E, Bral F, Da Lo M. Symbolc-numerc ecent soluton o optmal control problems or multbody systems. Journal o Computatonal and Appled Mathematcs. 6;85(): Kumar S, Kanwar V, Sngh S. Moded ecent amles o two and three-step predctor-corrector teratve methods or solvng nonlnear equatons. Journal o Appled Mathematcs. ;():5 58. Ctaton: Levs MV. Spe optmzaton problem o control mode o spacecrat moton. Int Rob Auto J. 8;4(6):4 4. DOI:.546/rat