Profle Optmzaton of Satellte Antenna for Angular Jerk Mnmzaton Jangwon Lee, Hyosung Ahn, Kwanghee Ko 3 and Semyung Wang 4 Gwangu Insttute of Scence and Technology, Gwangu, Korea, 500-7 and Daekwan Km 5, Sun Cho 6, Okchul Jung 7 and Daewon Chung 8 Korea Aerospace Research Insttute (KARI), Daeon, Korea, 305-333 Ths paper presents the optmzaton of the satellte antenna profle (SAP) to mnmze the angular erk whch s the tme dervatve of the angular acceleraton. The method of movng asymptotes (MMA), whch s a gradent-based optmzaton algorthm, s employed to solve the optmzaton problem. The sequental angle of the SAP s defned as the desgn varable for the optmzaton. The off-pontng margn angle, whch s the mum allowable range of the satellte antenna rotaton for the communcaton wth the ground staton, s used for the sde lmts of the desgn varable. Three constrant sets wth one obectve functon are formulated as the optmzaton problem. The obectve functon s the total sum of the squared angular erk. The frst set of constrants s the angular velocty and acceleraton, the second s the angular erk alone, and the thrd s the angular velocty, acceleraton, and erk. In numercal examples, two real SAPs are used for mplementng the proposed optmzaton algorthm. The optmzaton results show the effectveness of the proposed algorthm wth great reducton of the angular erk. The obectve functon and the computaton tme for the three sets of the optmzaton problems are compared and dscussed. I. Introducton OREA mult-purpose satellte 3 (KOMPSAT-3), orbtng 685 km above the earth, s maneuvered to take very Khgh-resoluton ground mages usng a multspectral camera (MSC). The accuracy of the MSC s 0.7 m of panchromatc and 3. m of multspectral mages. An mage taken by the MSC s transmtted to the ground staton usng an X-band antenna. The X-band antenna can rotates n two drectons of azmuth and elevaton angle n order to pont toward the ground staton antenna as trackng the satellte antenna profle (SAP). The off-pontng margn boundary s the mum range of the antenna pontng error for communcaton. The SAP and the off-pontng margn boundary consst of sequental angles wth a unform samplng tme. Therefore, the X-band antenna must reach each sequental angle wthn the off-pontng margn angle on the unform samplng tme for communcaton. Reference presented a detaled procedure of schedulng the SAP and employed a renforcement learnng approach to satsfy the off-pontng margn boundary and the mechancal constrants of the antenna rotaton velocty and acceleraton. Durng the rotaton of the X-band antenna, undesrable vbraton can arse and affect the satellte body. Vbraton of the satellte body causes degradaton of the hgh-resoluton mage qualty and moton control ablty. The angular erk, whch s the thrd dervatve of rotaton angle wth respect to tme, nduces transent vbraton and Ph. D. Student, School of Mechatroncs. Professor, School of Mechatroncs. 3 Professor, School of Mechatroncs. 4 Professor, School of Mechatroncs, AIAA Senor Member. 5 Researcher, Satellte Control System Department. 6 Researcher, Satellte Msson Operatons Department. 7 Researcher, Satellte Msson Operatons Department. 8 Researcher, Satellte Msson Operatons Department. Copyrght 0 by the, Inc. All rghts reserved.
affects the stablzaton of the satellte. Therefore, the SAP must be optmzed to mnmze the angular erk so that the transent vbraton s reduced and satellte control accuracy s ncreased. The two broad categores of drect or search methods and ndrect or optmalty crtera methods are most wdely used for profle optmzaton. Based on the two key drawbacks of both methods, namely, the lack of a global search capablty and the requrement of a sutable ntal guess, a hybrd method combnng a gradent-based method and a global search method, such as a genetc algorthm (GA), has been studed. 3-6 Although the GA s not computatonally compettve aganst gradent-based methods, t s useful n fndng the approxmate ntal guess of the optmal soluton due to global search compatblty. Optmzaton of the profle wth restrcton of erk has been reported n several papers. The polyhedron search algorthm s appled to the optmzaton of path plannng usng the cubc splne functons to mnmze the travelng tme of robot manpulators wth ont velocty, acceleraton, and erk constrants. 7-9 Snce optmzaton of the cubc splne usng the polyhedron search algorthm yelds a local optmal soluton, algorthms for fndng a global optmal soluton are presented. Methods based on nterval analyss (IA) and a GA to mnmze the total travellng tme subect to constrants on ont veloctes, acceleraton, and erk were proposed, 0, and a hybrd optmzaton method combnng the GA and sequental quadratc programmng (SQP) was proposed. Unlke the use of erk for a constrant, the ntegral of the squared erk s appled for the obectve functon, 3 and the mum value of the erk s mnmzed usng the mn approach. 4 In prevous studes, there has been no publcaton coverng the optmzaton of the SAP to mnmze or reduce the angular erk. Ths paper ntroduces the mnmzaton of the angular erk of the SAP usng the gradent-based optmzaton method. Wthout the use of a global search algorthm to fnd the ntal guess, the ntal value of the desgn varable s defned as the SAP whch s generated by a geometrcal relatonshp between the satellte and targeted ground staton. The sequental angle of the SAP s defned as the desgn varable. The off-pontng margn angle s used for the sde lmt of the desgn varable. The desgn varable s updated wthn the sde lmt durng optmzaton. The angular velocty, acceleraton, and erk are determned by the fnte dfference method (FDM). The obectve functon s the total sum of the squared angular erk, and three sets of constrants are formed. The frst set of constrants s the angular velocty and acceleraton, the second s the angular erk alone, and the thrd s the angular velocty, acceleraton, and erk. The method of movng asymptotes (MMA), 5,6 whch was ntroduced by Svanberg n 987, s used to solve the optmzaton problems. The MMA s a gradent-based optmzaton algorthm whch has been used n many felds. 7 Gradent-based optmzaton schemes requre desgn senstvty analyss (DSA), whch s performed by the dervatves of the obectve functon and the constrants wth respect to the desgn varable. 8 Snce the obectve functon and the constrants are explctly expressed by the desgn varable, the dervatves are easly computed n ths paper. The proposed method s appled to the numercal examples of two real SAPs, and then the optmzaton results from three optmzaton problems are dscussed. The remander of ths paper s organzed nto the followng sectons. Secton descrbes the SAP and the offpontng margn boundary. In Secton 3., the optmzaton problems n terms of the desgn varable are formulated. The desgn senstvty s derved n Secton 3.. In Secton 4, numercal steps of the optmzaton are explaned. Secton 4. and 4. show the optmzaton of two numercal examples and compare the optmzaton results of three optmzaton problems. Fnally, conclusons are gven n Secton 5. II. Satellte Antenna Profle The X-band antenna rotates to both angles of the azmuth and the elevaton to drect to the ground staton antenna for data communcaton. Fg. schematcally shows the azmuth over elevaton system of the X-band antenna. The rotaton ranges of the azmuth and elevaton angle are from 0 degree to 360 degree and from 4.8 degree to 45 degree, respectvely. The ntal azmuth and elevaton angles are calculated from the relatonshp of the satellte atttude accordng to the earth-centered fxed (ECF) frame, the orentaton of antenna n the body frame, satellte poston accordng to ECF, the poston of targeted ground staton accordng to ECF, and the poston of proected nadr pont. The central axs of the satellte and the ground staton antennas are matched at the ntal angles. The mum allowable rotaton angle for the satellte antenna to communcate wth the ground staton s called the off-pontng margn angle. The azmuth and the elevaton angles can have any value wthn the off-pontng margn angle for communcaton. The SAP and the off-pontng margn angle are descrbed n Fg.. Fg. 3 shows the SAP of the azmuth and elevaton angle. Fgure. Dagrammatc representaton of the rotaton of the X-band antenna. Copyrght 0 by the, Inc. All rghts reserved.
Fgure. SAP: ntal angle, and off-pontng margn angle. Fgure 3. Desred angle and off-pontng margn angle: azmuth angle w.r.t tme, and elevaton angle w.r.t tme. III. Formulaton of Optmzaton Problem The formulaton of the optmzaton problem, whch nfluences the fnal goal of the optmzaton, s sometmes trcky. The followng three steps are requred to formulate the optmzaton problem. 9 The frst step s the dentfcaton of the desgn varable. The second s the dentfcaton of the obectve functon n terms of the desgn varable. The thrd s the dentfcaton of the constrants n terms of the desgn varable. The dentfcaton of the desgn varable s often the most dffcult part of the entre optmzaton procedure. In ths paper, the sequental angle of the SAP, q, q, q3, L, q n -, q n, s defned as the desgn varable. Here, n s the number of desgn varables. The angular velocty q &, the angular acceleraton q &&, and the angular erk q &&& are the frst, second, and thrd dervatves of the angle wth respect to tme, whch are calculated as follows: & q -q q3 -q q -q - q =, & q =,, & n n L qn- = Dt Dt Dt & q - & q & q3 - & q & q - - & && q - q =, && q =,, && n n L qn- = Dt Dt Dt && q - && q && q - && q && q - - && &&& q - q =, &&& q =,, &&& n n L qn-3 = Dt Dt Dt 3 3 () () (3) where Dt s the samplng tme. 3 Copyrght 0 by the, Inc. All rghts reserved.
A. Optmzaton Problem The angular erk s used for the functon of quantfyng the smoothness of the profle. 0 Snce ths paper ams to mnmze the angular erk to reduce the vbraton of the satellte antenna, the total sum of the squared angular erk s consdered for the obectve functon. The hgh angular erk s crtcal to satellte vbraton, whereas the low one s trval. The large angular erk becomes domnant n the obectve functon through takng the square as follows: Mnmze 0 ( ) n-3 f q = å &&& q (4) = Three sets of constrants are appled for the optmzaton problem. The frst set of constrants ncludes both the angular velocty and the angular acceleraton. The second s the angular erk. The thrd s the angular velocty, the angular acceleraton, and the angular erk. The angular velocty and acceleraton are respectvely restrcted by & q of 6 degree/s and && q of degree/s due to dsturbance by the angular momentum of the antenna and due to the lmted power of the antenna pontng system (APS). The constrant of the mum angular erk &&& q s defned as 0 % of the mum erk of the ntal SAP. Three sets of constrants are gven as follows: Set : Set : Set 3: where subect to - & q & q & q for =, K, n - - && q && q && q for =, K, n - subect to - &&& q &&& q &&& q for =, K, - 3 (6) n subect to - & q & q & q for - && q && q && q for =, K, n - =, K, n - -&&& q &&& q &&& q for =, K, n - 3 The sde lmt of the desgn varable s obtaned as follows: Dq s the off-pontng margn angle of th desgn varable. Dq Dq q - q q + for =,, K, n (8) B. Desgn Senstvty Analyss The DSA s requred to determne the drecton of desrable desgn change to mprove the performance measures of the system n the gradent-based optmzaton method. Desgn senstvty s calculated by the dfferentaton of the obectve functon and the constrant functon wth respect to the desgn varable. The desgn senstvty of the obectve functon n Eq. (4) s calculated as (5) (7) æ ö é ù df ç ê ú = è ø = ë û for =,, K, n dq dq dq n-3 n-3 d &&& åq d å( q+ 3-3q + + 3q + -q ) 0 = = 6 Dt (9) The desgn senstvtes of the angular velocty, acceleraton, and erk n Eqs. (5)-(7) are calculated as ( q q ) d & q d - + = for =,, K, n dq Dt dq (0) 4 Copyrght 0 by the, Inc. All rghts reserved.
d&&& q dq d && q dq d ( q - q + q ) for,, K, n + + = = Dt dq d ( q - 3q + 3q -q ) for,, K, n + 3 + + = = 3 Dt dq () () IV. Numercal Implementaton The MMA s used to solve the optmzaton problem of mnmzng the angular erk. The MMA uses a specal type of convex approxmaton. For each step of the teratve process, a strctly convex approxmatng subproblem s generated and solved. The generaton of these subproblems s controlled by the so-called movng asymptotes, whch both stablze and speed up the convergence of the general process. 6 The gradent-based optmzaton of the SAP usng MMA s carred out as follows: Step 0. Choose an ntal desgn varable. Step. Evaluate the obectve functon and constrants n Eqs. (4)-(7). Step. Calculate the gradents of the obectve functon and constrants n Eqs. (9)-(). Step 3. Optmze usng MMA. Step 4. Update the desgn varable. Step 5. Check f the convergence test s satsfed. If not, go to step. In step 0, the desred angle of the SAP s used for the ntal desgn varable. An absolute tolerance of the mum change of the desgn varables s typcally used for the convergence test. In ths paper, the teraton number s used for the convergence crtera to compare the tme and the performance of usng three sets n Table. The defned teraton number for termnatng the optmzaton loop s 50. As the numercal mplementaton for our proposed optmzaton method, we employ two examples of actual flght data. In sectons 4. and 4., the optmzaton results accordng to three sets of optmzaton problems are compared to the ntal values of the SAP. The computaton tme for the optmzaton, whch needs to be reduced, s an mportant factor n evaluatng the optmzaton performance and n schedulng the satellte operaton. The computaton tmes of the three sets are compared n addton to the obectve functon and constrants. The computer used to calculate the numercal examples was an Intel Core Quad CPU @.5 GHz wth GB RAM, and the optmzaton program was developed by Matlab. A. Example : General Case The number of angles for the optmzaton s 80, and the samplng tme s s. The mum angular veloctes of the ntal azmuth and elevaton profle are 7.4685 degree/s and.73 degree/s, respectvely. The mum angular acceleratons of the ntal azmuth and elevaton profle are.456 degree/s and 0.0 degree/s, respectvely. 0 % of the mum angular erk of the ntal azmuth and elevaton profle are respectvely 0.084 degree/s 3 and 0.067 degree/s 3, whch are appled for the constrant of the angular erk n Eqs. (6) and (7). The azmuth angular velocty volates the mechancal constrant of 6 degree/s. Tables and summarze the optmzaton results usng three sets of optmzaton problems. All constrants of the three sets are satsfed. The optmzaton results of set are consderably worse than the results of the others, especally n the reducton of the obectve functon. As seen n the comparson of sets and 3 n Tables and, set s more effectve than set 3. Although the optmzaton results of both sets are almost the same, the computaton tme of set s sgnfcantly less than that of set 3 because of the number of constrants. Table. Optmzaton results of azmuth profle Comparson crtera Set Set Set 3 Reducton rato of obectve functon (%).895 9.547 9.944 Maxmum velocty (ntal value: 7.4685) (degree/s) 6.0000 5.564 5.46 Maxmum acceleraton (ntal value:.456) (degree/s ).573 0.7778 0.73 Maxmum erk (ntal value:.0848) (degree/s 3 ) 0.7675 0.07 0.009 Total tme (s) 7 03 3 5 Copyrght 0 by the, Inc. All rghts reserved.
Table. Optmzaton results of elevaton profle Comparson crtera Set Set Set 3 Reducton rato of obectve functon (%) 0.4689 95.8858 96.053 Maxmum velocty (ntal value:.73) (degree/s).507 0.747 0.703 Maxmum acceleraton (ntal value: 0.0) (degree/s ) 0.900 0.086 0.0845 Maxmum erk (ntal value: 0.67) (degree/s 3 ) 0.444 0.04 0.0 Total tme (s) 67 6 64 Fg. 4 shows the convergence hstores of optmzng the azmuth and elevaton profles usng set. Fgure 4. Optmzaton hstory of set : azmuth angle, elevaton angle. Fg. 5 shows the optmzed results usng set compared to the ntal values of the SAP. (c) (d) 6 Copyrght 0 by the, Inc. All rghts reserved.
(e) (f) (g) (h) Fgure 5. Intal and optmzed results usng set : azmuth angle, elevaton angle, (c) azmuth angular velocty, (d) elevaton angular velocty, (e) azmuth angular acceleraton, (f) elevaton angular acceleraton, (g) azmuth angular erk, and (h) elevaton angular erk. B. Example : The Worst Case The mum angular velocty, acceleraton, and erk of the elevaton profle, whch are respectvely.8753 degree/s,.9 degree/s, and 0.694 degree/s 3, are not crtcal. However, example s one of the worst SAPs, because the mum angular velocty and acceleraton of the azmuth profle, whch are respectvely 9.804 degree/s and.0 degree/s, sgnfcantly volate the mechancal restrctons of 6 degree/s and degree/s. The mum azmuth angular erk s.787 degree/s 3, whch s consdered to affect the vbraton. The number of angles s 5. Tables 3 and 4 summarze the optmzaton results. The optmzaton of set gves the best results n terms of the hghest reducton rato of the obectve functon and the least computaton tme. All constrants of optmzng the elevaton profle are satsfed, but the azmuth angular velocty and acceleraton volate the mechancal restrctons. The volatons of the constrants are unsolvable n ths example. In other words, the optmzaton results of the azmuth angular velocty and acceleraton are mum possble reductons wthn the off-pontng margn boundary. Table 3. Optmzaton results of azmuth profle Comparson crtera Set Set Set 3 Reducton rato of obectve functon (%) 9.93 97.8009 95.436 Maxmum velocty (ntal value: 9.804) (degree/s) 7.6686 7.89 7.93 Maxmum acceleraton (ntal value:.0) (degree/s ) 3.73.607 3.808 Maxmum erk (ntal value:.787) (degree/s 3 ).0669.0740.647 Total tme (s) 67 84 973 7 Copyrght 0 by the, Inc. All rghts reserved.
Table 4. Optmzaton results of elevaton profle Comparson crtera Set Set Set 3 Reducton rato of obectve functon (%) 78.465 99.4740 99.5078 Maxmum velocty (ntal value:.8753) (degree/s).9768.448.4 Maxmum acceleraton (ntal value:.9) (degree/s ) 0.8865 0.93 0.86 Maxmum erk (ntal value: 0.694) (degree/s 3 ) 0.4575 0.048 0.038 Total tme (s) 73 45 68 Fg. 6 shows the ntal and optmzed results of the azmuth angle, angular velocty, and angular acceleraton n the volaton secton. In Fg. 6, the profle of the optmzed angle makes the gentlest slope whle satsfyng the off-pontng margn boundary. (c) (d) Fgure 6. Volaton secton of the azmuth profle. Fg. 7 shows the convergence hstores of optmzng the azmuth and elevaton profles usng set. Fgure 7. Optmzaton hstory of set : azmuth angle, elevaton angle. Fg. 8 shows the optmzed results usng set compared to the ntal values of the SAP. 8 Copyrght 0 by the, Inc. All rghts reserved.
(c) (e) (d) (f) (g) (h) Fgure 8. Intal and optmzed results usng set : azmuth angle, elevaton angle, (c) azmuth angular velocty, (d) elevaton angular velocty, (e) azmuth angular acceleraton, (f) elevaton angular acceleraton, (g) azmuth angular erk, and (h) elevaton angular erk. 9 Copyrght 0 by the, Inc. All rghts reserved.
V. Concluson Ths paper proposed the gradent-based optmzaton of the SAP to mnmze the angular erk. The sequental angle of the SAP s defned as the desgn varable, and the off-pontng margn boundary s used for the sde lmt. The angular velocty, acceleraton, and erk are determned from the sequental angle wth respect to the samplng tme. Three optmzaton problems were formulated by one obectve functon and three sets of constrants. The obectve functon was to mnmze the total sum of the squared angular erk. The frst set of constrants was the angular velocty, and acceleraton. The second set was the angular erk. The thrd set was the angular velocty, acceleraton, and erk. Three optmzaton problems were compared n terms of optmzaton results obtaned usng two numercal examples. The second set of constrants s udged to be the best formulaton of the optmzaton problems accordng to the comparson crtera of the computaton tme, the obectve functon, and the constrant satsfacton. Acknowledgments Ths work was supported by Satellte Control System Department and Satellte Msson Operatons Department of Korea Aerospace Research Insttute (KARI). References Ahn, H. S., Jung, O. C., Cho, S. J., Son, J. H., Chung, D. W., and Km, G. S., An optmal satellte antenna traectory usng renforcement learnng, IEEE Transactons on Systems, Man, and Cybernetcs-Part C: Applcatons and Revews, vol. 4, no. 0, 0, pp. 306-3. Betts, John T., Survey of Numercal Methods for Traectory Optmzaton, Journal of Gudance, Control, and Dynamcs, vol., no., 998, pp. 93-07. 3 Stryk, O. yon, and Bulrsch, R., Drect and ndrect methods for traectory optmzaton, Annals of Operatons Research, vol. 37, 99, pp. 357-373. 4 Luo, Y. Z., Tang, G. J., and L, H. Y., Optmzaton of multple-mpulse mnmum-tme rendezvous wth mpulse constrants usng a hybrd genetc algorthm, Aerospace Scence and Technology, vol. 0, no. 6, Sep. 006, pp. 534-540. 5 Yokoyama, N. and Suzuk, S., Modfed genetc algorthm for constraned traectory optmzaton, Journal of Gudance, Control, and Dynamcs, vol. 8, no., Jan. 005, pp. 39 44. 6 Balesdent, M. and Berend, N., A survey of multdscplnary desgn optmzaton methods n launch vehcle desgn, Structural and Multdscplnary Optmzaton, vol., no., 998, pp. 93 07. 7 Ln, C. S., Chang, P. R., and Luh, J. Y. S., Formulaton and optmzaton of cubc polynomal ont traectores for ndustral robots, IEEE Transactons on Automatc Control, vol. AC-8, no., Dec. 983, pp. 066-074. 8 Wang, C. H., and Horng, J. G., Constraned mnmum-tme path plannng for robot manpulators va vrtual knots of the cubc B-splne functons, IEEE Transactons on Automatc Control, vol. 35, no. 5, 990, pp. 573-577. 9 Jamhour, E. and André, P. J., Plannng smooth traectores along parametrc paths, Mathematcs and Computers n Smulaton, vol. 4, 996, pp. 65-66. 0 Pazz A. and Vsol, A., Global mnmum-tme traectory plannng of mechancal manpulators usng nterval analyss, Internatonal Journal of Control, vol. 7, no. 4, 998, pp. 63-65. Tse, K. M. and Wang, C. H., Evolutonary optmzaton of cubc polynomal ont traectores for ndustral Robots, Proceedngs of the IEEE Internatonal Conference on Man, System and Cybernetcs, San Dego, vol. 4, 998, pp. 37-376. Chettb, T., Lehthet, H. E., Haddad, M., and Hanch, S., Mnmum cost traectory plannng for ndustral robots, European Journal of Mechancs A-Solds, vol. 3, no. 4, 004, pp. 703-75. 3 Smon, D. and Isk, C., A trgonometrc traectory generator for robotc arms, Internatonal Journal of Control, vol. 57, no. 3, 993, pp. 505-57. 4 Pazz, A. and Vsol, A., Golbal mnmum-erk traectory plannng of robot manpulators, IEEE Transactons on Industral Electroncs, vol. 47, no., 000, pp. 40-49. 5 Svanberg, K., The method of movng asymptotes - a new method for structural optmzaton, Int.. numer. methods eng., vol. 4, 987, pp. 359-373. 6 Svanberg, K., A class of globally convergent optmzaton methods based on conservatve convex separable approxmatons, SIAM Journal on Optmzaton, 00, vol., no., pp. 555-573. 7 Lee, J. W., Kook, J. H., and Wang, S. M., Mult-doman topology optmzaton of pulsed magnetc feld generator sourced by harmonc current exctaton, IEEE Transactons on Magnetcs, vol. 48, no., Feb. 0, pp. 475-478. 8 Cho, K., and Km, N., Structural Senstvty Analyss and Optmzaton. Sprnger, 004. 9 Arora, J. S., Introducton to optmum desgn, McGraw-Hll Book Co., New York, 989. 0 Hogan, N., Adaptve Control of Mechancal Impedance by Coactvaton of Antagonst Muscles, IEEE Transactons on Automatc Control, vol. 9, 984, pp. 68-690. 0 Copyrght 0 by the, Inc. All rghts reserved.