DESIGN OF ATTITUDE STABILITY SYSTEM FOR PROLATE DUAL-SPIN SATELLITE IN ITS INCLINED ELLIPTICAL ORBIT  J. Muliadi   *   , S.D. Jenie   †   and A Budiyono   ‡  Abstract— In general, most of communication satellites were designed to be operated in geostationary orbit. And many of them were designed in prolate dual-spin configuration. As a prolate dual-spin vehicle, they have to be stabilized against their internal energy dissipation effect. Several countries that located in southern hemisphere, has shown interest to use communication satellite. Because of those countries’ southern latitude, an idea emerged to incline the communication satellite (due to its prolate dual- spin configuration) in elliptical orbit. This work is focused on designing Attitude Stability System for prolate dual-spin satellite in the effect of perturbed field of gravity due to the inclination of its elliptical orbit. DANDE (De-spin Active Nutation Damping Electronics) provides primary stabilization method for the satellite in its orbit. Classical Control Approach is used for the iteration of DANDE parameters. The control performance is evaluated based on time response analysis. Keywords— dual spin, prolate configuration, classical control approach, attitude control, DANDE Abstrak—   Secara   umum   kebanyakan   satelit   komunikasi   dirancang   untuk beroperasi pada orbit geostasioner. Dan kebanyakan dari satelit tersebut dirancang pada konfigurasi dual-spin prolate. Sebagai wahana dual-spin prolate, satelit harus distabilkan dari pengaruh disipasi energi internal. Beberapa negara yang terletak pada belahan   selatan   bumi   telah   menunjukkan   minatnya   untuk   menggunakan   satelit komunikasi. Karena letak negara yang berada di latituda selatan, sebuah ide muncul untuk menginklinasi satelit pada orbit eliptik. Makalah ini secara khusus membahas perancangan sistem stabilitas sikap untuk satelit dual-spin prolate pada pengaruh medan gravitasi terganggu akibat inklinasi dari orbit elipsik wahana. DANDE (De-spin Active Nutation Damping Electronics) memberikan metoda penstabilan utama pada satelit pada orbitnya. Pendekatan kendali klasik digunakan pada iterasi parameter DANDE. Unjuk kerja kendali dievaluasi berdasarkan analisis respons waktu. Keywords— dual spin, konfigurasi prolate, pendekatan kendali klasik, kendali sikap, DANDE  *   Department of Aeronautics and Astronautics, ITB, mie_pn@yahoo.com †   Department of Aeronautics and Astronautics, ITB, sdjenie@ae.itb.ac.id ‡   Department of Aeronautics and Astronautics, ITB, the author to whom all correspondence to be addressed, agus.budiyono@ae.itb.ac.id
1.   INTRODUCTION  For a stability criterion, most of early dual-spin vehicles were designed in an oblate configuration. However, Kaplan [8] states that launch vehicle   shroud   constraints   limited rotor diameters.   In   addition,   the   major   axis stability   condition   effectively   limited spinning spacecraft sizes. Since U.S. Air Force   successfully   by-passed   this limitation by operating TACSAT, many of  prolate   spinners were launch to orbit. The first communication dual-spin satellite in  prolate   configuration is INTELSAT IV. As a case of study, this work used Palapa   B2R   physical   data   to   design feedback   control   parameters   for   the attitude   stability   system.   Near   the satellite’s End of Life (EOL) time, several governments of Africans and Polynesians countries have shown interest to buy and re-use   Palapa   B2R.   Because   of   those countries’   location   in   the   southern latitudes, an idea emerged to incline the satellite’s orbit. When Palapa B2R is operating in its orbit, DANDE (De-spin Active Nutation Damping Electronics (Ref. [2] pp. 62-68, [1]   pp.   127-129,   [4],   [5]))   provides primary   stabilization   method   for   the vehicle,   while   ANC   (Active   Nutation Control (Ref. [1])) supplies the back-up mode. With the use of DANDE as the stabilization   mode,   the   current   paper elaborates   the   tuning   of   the   feedback control parameters using classical control approach.  2.   REFERENCE COORDINATE SYSTEM  The reference coordinate systems used in describing the satellite are the body, stability and inertial axes. Ref [9] provides the definition and illustration of those axes in detail.   Body Axes with their origin at the satellite’s c.g. while   Error! Reference source not found.   showed the axes in the space, as explained in Ref [9].  Fig. 1 Stability Reference Coordinate System In   particular,   the   Stability   Reference Coordinate   System   (Stability   Axes)   is defined as a set of local horizon axes for the satellite. It is a target axes for the satellite’s Body Axes to point its antennae to   the   Earth.   The   stability   axes   is presented in Fig. 1.  3.   EULER ANGLES (ORIENTATION ANGLES)  The   orientation   of   satellite   in space with   respect   to   a   certain   reference coordinate   system   is   described   by   the Euler angles. Fig. describes the attitude of the satellite with respect to the Inertial Axes.  Fig. 2 Orientation of Body Axes in Inertial Axes
The more complete explanation of the use of   Euler   angles   in   describing   satellite orientation is explained in Ref [9].  4.   STATE SPACE MODEL of PROLATE DUAL SPIN SATELLITE in INCLINED ELLIPTICAL ORBIT  In order to stabilize its attitude and pointing direction, Palapa B2R uses its rotor   spinning.   Control   moments   were produced by the angular acceleration and deceleration   of   the   rotor’s   spin.   The satellite’s   motion   in   the   yaw   mode   is coupled with its roll mode. In addition, the satellite’s motion in the   pitch   mode is coupled with its yaw mode. With those couples,   the   satellite’s   attitude   can   be controlled by the angular acceleration and deceleration of the rotor that spun in pitch mode. In Ref [9], the authors derive State Space Model for Dual-Spin Satellite in Stability Axes as follows,  [   ]   [   ]         • +                       • =                        n e r q p r q p  δ δ ψ θ φ ψ θ φ   B A  S S S S S S         Eq. 1  where the [ A ] and [ B ] matrices are:  [   ]                     − =  0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 A 0 A A A 0 A 0 A A A 0 0 A A 0 0 35 33 32 31 25 23 22 21 14 13  n n  δ δ  A  [   ]                     =  0 0 1 0 0 0 0 B 0 B 0 0 31 21  B  By defining   ∆ I   as follows, ) (   2 YZ T Y Z Y   I I I I I I   − + = ∆  the [ A ] matrix elements in 1 st   line are: ) ( ) ( 1 A   0 R S T X 13   Ω ⋅ ⋅ + − =   I I I  X T X 14   ) ( 1 A   G I I   ⋅ + − =  the [ A ] matrix elements in 2 nd   line are: ) ( A   0 R S YZ 21   Ω ⋅ ⋅ ∆ − =   I I I            + ⋅           + ⋅ ⋅ ∆ + =   1 A S Y . . V T Z 22   I I c R K N I I c d I          ⋅           + ⋅ ⋅ ∆ + =  S YZ . . V T Z 23 A   I I c R K N I I c d I  Z YZ Y T Z 25 ) ( A   G I G I I I I  ⋅ ∆ + ⋅ ∆ + − =  the [ A ] matrix elements in 3 rd   line are: ) ( A   0 R S Y 31   Ω ⋅ ⋅ ∆ =   I I I            + ⋅           + ⋅ ⋅ ∆ − =   1 A S Y . . V YZ 32   I I c R K N I c d I          ⋅           + ⋅ ⋅ ∆ − =  S . . V 2 YZ 33 1 A   I c R K N I c d I  Z Y Y YZ 35 A   G I G I I I  ⋅ ∆ − − ⋅ ∆ =  the [ B ] matrix elements in 1 st   column are:          ⋅ ∆ + =  . . T Z 21 B  c d I   R N I I          ⋅ ∆ − =  . . YZ 31 B  c d I   R N I  and  R V n  θ θ   − = − =    ~  R V R V n n n  ) ( ) ( ~ ~ ) (  θ  0 0 θ  0 0  − = − = − = θ θ δ     where   n   = orbital angular velocity, shown in Fig. 1.
Fig. 1 Rotation of Local Horizon Axes (Stability Axes),   n  5.   CONTROL STRATEGY  In   elliptic   and   inclined   orbit,   the satellite attitude dynamics acts as a time- varying system. The value of A 14 , A   25 , A 35 and   δ n   will be time-varying for elliptical or inclined orbit. However, the elements of [ A ] are constant value only for circular orbit   at   equatorial   plane.   Therefore,   to apply   Classical   Control   Approach,   the iteration for control parameter was held in equatorial and circular orbit.  5.1.   Control Strategy for Longitudinal Motion  Fig. 2 Response of   θ S   without Gravity Gradient Moment Effect  Fig. 3 Response of   θ S   in Gravity Gradient Moment Effect  F ig. 2 is   θ S   response in equatorial orbit without Gravity Gradient Moment. Fig. 3 is   θ S   response   in   equatorial   orbit   in Gravity Gradient Moment Effect. The comparison from   θ S   response in elliptic   orbit   at
Fig. 2   and   Fig.   3   showed   that,   “Gravity Gradient   Moment   tends   to   stabilize longitudinal motion of the prolate dual- spin satellite.” Therefore,   to   simplify   the   iteration process of longitudinal control parameter, the   authors   omit   the   Gravity   Gradient Moment terms in [ A ] matrix elements. The valueA   14 , A 25 , and A   35   are set to zero.                      × × × − × − = − − − −  0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 10 1.1912 - 10 3636 . 3 4.0326 - 0 0 0 10 4402 . 3 10 7138 . 9 49773 . 0 0 0 0 3.7113 0 0 6 5 5 4  A  Eq. 2  The value of and [ B ]’s elements are:                      × × − = − −  0 0 1 0 0 0 0 10 7735 . 1 0 10 1218 . 5 0 0 5 4  B  Eq. 3  For   q - feedback , the damping values are too small for harmonics oscillation modes. The   author   uses   θ S -feedback   for longitudinal controlling shown in Fig. 4.  Blok Umpan Balik  θ S Dinamika Gerak Satelit   Dual-Spin  di TAK Kestabilan ( Open-Loop )  p q r e  δ  S  φ S  θ S  ψ  p  q r  S  φ  S  θ  S  ψ ref e  δ θ K  + -  S  θ  Longitudinal Closed-Loop  Fig. 4   θ S -feedback   Diagram  For   θ S -feedback   shown in Fig. 4, the root locus diagram is  Fig. 5   Root-Locus   for   θ S -feedback   with   K θ   < 0  To   increase   the   damping   ratio   for harmonics oscillation modes, the authors design a compensator. The authors start with a   pole   placement at   1 1  + s  (or   s   pc =-1), then place a   zero   to push the harmonics oscillation   poles   more to the left. With a   zero   placement at   s +0.2, the locus is:  Fig. 6   R-Locus,   θ S ,   compensated, with   s +0.2/ s +1  With a   zero   placement at   s +0.46, the locus is:
Fig. 7   R-Locus,   θ S ,   compensated, with   s+0.46/s+1  With a   zero   placement at   s +0.4819, the locus is:  Fig. 8   R-Locus,   θ S ,   compensated, with  s +0.4819/ s +1  With a   zero   placement at   s +0.5210, the locus is:  Fig. 9   R-Locus,   θ S ,   compensated, with  s +0.5210/ s +1  By detail, the compensator was chosen as follows 1 498 . 0  + +  s  s  Eq. 4  with   Root   Locus   Diagram   (negative feedback for Fig. 4),  Fig. 10   R-Locus,   θ S ,   compensated, with  s +0.498/ s +1  Fig. 11 Zoom for harmonic oscillation locus (upper)
Fig. 12 Zoom for coalescent/breakaway point  Compensated  θ S   -Feedback Block Attitude Dynamics  Dual-Spin  Satellite in Stability Axis  (Open-Loop)  p  q r e  δ  S  φ  S  θ  S  ψ  p q r  S  φ  S  θ  S  ψ  ref e  δ   + -  S  θ  1 498 . 0 29800   + + ⋅ −   s  s  Longitudinal Closed-Loop  Fig. 13 Longitudinal Controlling Diagram by   θ S - feedback  The   iteration   result   for   longitudinal controlling is   θ S -feedback , shown in Fig. 13, with gain 1 498 . 0 29800   + + ⋅ −   s  s  Eq. 5 5.2.   Control Strategy for Lateral Motion  Fig. 14 Response of   φ S   without Gravity Gradient Moment Effect  Fig. 15 Response of   φ S   in Gravity Gradient Moment Effect  Fig. 14   is   φ S   response in equatorial orbit without Gravity Gradient Moment.  Fig. 15 is   φ S   response in equatorial orbit in Gravity Gradient Moment Effect. The comparison from   θ S   response in elliptic orbit at   Fig. 14   and  Fig. 15 showed that, “Gravity Gradient Moment   tends   to   de-stabilize   lateral motion of the prolate dual-spin satellite.” Therefore,   the   iteration   processes   of lateral control parameter should included the Gravity Gradient Moment terms in [ A ] matrix elements. The value A 14 , A 25 , and A 35   are non-zero,                      × × × × × − × − × = − − − − − − −  0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 10 1.3422 - 0 10 1.1912 - 10 3636 . 3 4.0326 - 0 10 7.2937 0 10 4402 . 3 10 7138 . 9 49773 . 0 0 0 10 6.1872 - 3.7113 0 0 7 6 5 7 5 4 7  A  Eq. 6  and the value of and [ B ]’s elements as in Eq. 3. For   p-feedback   as lateral controlling (shown in Fig. 16),
Blok Umpan Balik   p  Dinamika Gerak Satelit   Dual-Spin  di TAK Kestabilan ( Open-Loop )  p  q r e  δ  S  φ S  θ S  ψ  p q r  S  φ  S  θ  S  ψ ref e  δ  p K  + -  Lateral Closed-Loop  p  Fig. 16   p-feedback   Diagram  the Root Locus Diagram is  Fig. 17   Root-Locus   for   p-feedback   with   K p   > 0  To push the divergence pole   across imaginary   axis,   negative   p-feedback  applied results the Root Locus Diagram: Pushing the divergence pole to the left side imaginary axis will destabilize the harmonic oscillation modes. Once again, the author design a compensator for   p -  feedback .  Fig. 18   Root-Locus   for   p-feedback   with   K p   < 0  By trial ‘n error with   sisotool   in  MATLAB ®   , the compensation function is,  (   ) (   ) (   ) 63 . 2 9 . 25 1 . 4  + ⋅ + + ⋅   s s s K   p  Eq. 7  and the Root Locus Diagram becomes
Fig. 19   Root-Locus   for   p-feedback compensated  with   K p   > 0  Fig. 20   Root-Locus   for   p-feedback compensated  with several critical gains  Fig. 21   Root-Locus   for   p-feedback compensated  gain margin  The   iteration   result   for   lateral controlling is   p-feedback , shown in Fig. 22, with gain  (   ) (   ) (   ) 63 . 2 9 . 25 1 . 4 10 5 . 1   6  + ⋅ + + ⋅ ×   s s s  Eq. 8  Compensated  p -Feedback Block Attitude Dynamics  Dual-Spin  Satellite in Stability Axis ( Open-Loop )  p  q r e  δ  S  φ S  θ S  ψ  p  q r  S  φ S  θ  S  ψ ref e  δ   + -  Lateral Closed-Loop  p (   ) (   ) (   ) 63 . 2 9 . 25 1 . 4 10 5 . 1   6  + ⋅ + + ⋅ ×   s s s  Fig. 22 Lateral Controlling Diagram by   p- feedback  5.3.   Control Strategy for Directional Motion  Because of the coupled between lateral and   directional   motion,   the   Gravity Gradient   Moment   Effects   are   also included in the iteration for Directional Control Parameters. With [ A ] matrix elements showed in Eq. 6 and the value of and [ B ]’s elements as   in   Eq.   3,   for   the   r-feedback   as directional controlling (shown in Fig. 23),  Blok Umpan Balik   r  Dinamika Gerak Satelit   Dual-Spin  di TAK Kestabilan ( Open-Loop )  p  q r e  δ  S  φ  S  θ  S  ψ  p q r  S  φ S  θ  S  ψ  ref e  δ r K  + -  Directional Closed-Loop  r  Fig. 23   r-feedback   Diagram  the Root Locus Diagram is
Fig. 24   Root-Locus   for   r-feedback   with   K r   > 0  Fig. 25 Zoom for   Root-Locus   for   r-feedback   with several critical gains  Fig. 26 Zoom for   Root-Locus   for   r-feedback   with breakaway gain  To   increase   the   damping   value   for harmonic mode, the authors use directly the Open Loop Root Locus and choose the gain   as   constant.   The   directional controlling is   r-feedback , shown in Fig. 27, with gain 000 300 = r K  Eq. 9  r -Feedback Block Attitude Dynamics  Dual-Spin  Satellite in Stability Axis ( Open-Loop )  p  q r e  δ  S  φ  S  θ  S  ψ  p q r  S  φ  S  θ  S  ψ  ref e  δ   + -  Control- Loop  5 10 3 ×   r  Fig. 27 Directional Controlling Diagram by   r- feedback  6.   INTERPRETATIONS of CLOSED LOOP SIMULATION RESULTS  To   evaluate   the   performance   of   the Attitude   Control   System,   numerical simulations conducted with Simulink from Matlab 6.5. The effect of the orbit’s eccentricity,   e , were induced with the variation of orbital  drift rate   in Stability Axes,   δ n   together with variation of satellite’s distance from the center of the Earth,   R . Inclination of orbit will effect the simulation due to the variation of   R ZI   , the Z-Axis component of satellite’s position vector in Inertial Axes. Attitude controlling was done with the deviation of armature voltage in the de- spin motor,   δ e , from its stationer value. The   reference   input   is   the   command signal,   δ e ref   . This   paper   only   reports   simulation results on Euler Angles since the most important   thing   in   Satellite’s   Attitude Stability is to keep pointing error under pointing   budget   and   to   reduce   any nutation angle. For clarity in presenting diagrams and graphics,   the   concept   of   simulation programming   and   the   Euler   angles graphics are presented in Appendices. The concept   of   simulation   programming diagrams describe simulation algorithm of the controlled dual-spin satellite in each mode.
6.1.   Simulation in Longitudinal Controlled Mode  The   results   of   attitude   dynamics simulation (Fig. 28) show good damping characteristics   when   responding   to   the doublet input.  6.1.1.   Effect of Eccentricity in Inclined Orbit ( i   = 30°)  Graphical interpretation for Effect of Eccentricity in Inclined Orbit for   θ S   (Fig. 29 and Fig. 30): Longitudinal controlling reduced   θ S   deviation amplitude –1.5° into damped oscillation with amplitude 0.01° in   s 30   .   The   subsidence   mode   of   θ S deviation becomes oscillation mode with damping time   s 60 ±   . For 10 orbit period (   s 2256.7 7   ) the long periodic oscillation mode of   θ S   were oscillating regularly from 6 10 2   − × −   to   deg 10 6   6 − × +   .  6.1.2.   Effect of Inclination in Elliptic Orbit ( e   = 0.2)  Graphical interpretation for Effect of Inclination in Elliptic Orbit for   θ S   (Fig. 31 and Fig. 32): Inclination increment will give   random   effect   to   θ S   in   order   to deg 10   12 −   for   s 500   . For 10 orbit period (   s 2256.7 7   ), the increment of inclination has no effect on the response of   θ S .  6.2.   Simulation in Lateral Mode  The   results   of   attitude   dynamics simulation (Fig. 33) show good damping characteristics   when   responding   the doublet input.  6.2.1.   Effect of Eccentricity in Inclined Orbit ( i   = 30°)  Graphical interpretation for Effect of Eccentricity in Inclined Orbit for   φ S   (Fig. 34   and   Fig.   35):   Lateral   controlling reduced roll librations mode of   φ S   with amplitude   deg 10 1   3 − ×   becomes damped oscillation   mode   with   maximum amplitude   deg 10 1   4 − ×   with   damping time   s 5 2   . The increment of eccentricity induced long period oscillation modes of  φ S   with smaller amplitude from its   open- loop   values   ( open-loop :   3 10 4   − × −   to deg 10 6   3 − × +   ).   This   mode   oscillates from   3 10 5 . 1   − × −   to   deg 10 5 . 1   3 − × +   ..  6.2.2.   Effect of Inclination in Elliptic Orbit ( e   = 0.2)  Graphical interpretation for Effect of Inclination in Elliptic Orbit for   φ S   (Fig. 36 and Fig. 37): In   s 500   of simulation, the inclination increment from 0° to 30° will amplify   the   φ S   deviation   amplitude   to deg 10 5 . 1   8 − ×   .   For   simulation   time   in order of orbital periode, the increasing of inclination from 0° to 30° will amplify the amplitudes   of   long   period   oscillation mode of   φ S   for   deg 10 2   8 − × ±   .  6.3.   Simulation in Directional Mode  The   results   of   attitude   dynamics simulation (Fig. 38) show good damping characteristics   when   responding   the doublet input.  6.3.1.   Effect of Eccentricity in Inclined Orbit ( i   = 30°)  Graphical interpretation for Effect of Eccentricity in Inclined Orbit for   ψ S   (Fig. 39 and Fig. 40): Directional controlling reduced the yaw librations mode of   ψ S   and changing it into non-oscillatory mode. The  ψ S   response followed the doublet input in command   signal   δ e ref .   The   orbit eccentricity can be seen since   s 200   and deviates   ψ S   to   deg 10 4   7 − × +   at   s 500   . The orbit eccentricity also develops the undamped   mode   of   long   period oscillation.   The   long   period   oscillation mode of   ψ S   oscillates from   3 10 5 . 5   − × −   to deg 10 6 . 1   3 − × +   in   order   of   orbital period. This deviation still under Palapa B2R pointing error, N-S: 0.047° and E-W: 0.047° Ref. [5].
6.3.2.   Effect of   Inclination   in Elliptic Orbit ( e   = 0.2)  Graphical interpretation for Effect of Inclination in Elliptic Orbit for   ψ S   (Fig. 41 and Fig. 42): For   s 500   simulation the increment of inclination from 0° to 30° will amplify the amplitude of   ψ S   deviation to   deg 10 5 . 1   10 − ×   . For simulation time in the order of orbital period the increment of inclination from 0° to 30° will effects the amplitude of long period oscillation mode   of   ψ S   from   5 10 75 . 0   − × −   to deg 10 5 . 0   5 − × +   .  7.   CONCLUDING REMARKS  In   Closed-Loop   simulation,   the feedback parameter successfully improved the   stability   of   satellites   attitude   from impulsive   perturbation.   Because   of controlling, pitch librations on   θ S   can be damped <60s, roll librations on   φ S   can be damped <30s, and yaw librations on   ψ S can be damped <30s. Longitudinal   controlling   with   θ S - feedback   also   successfully   reduces   the effect of eccentricity. In elliptic orbit, the amplitude of   θ S   oscillation reduced from o 20 −   to   o 90 +   becomes 0° (±100%). Lateral   controlling   with   p -feedback also reduces the effect of eccentricity. In elliptic   orbit,   the   amplitude   of   φ S oscillation reduced from   deg 10 4   3 − × −  to   deg 10 6   3 − × +   becomes deg 10 5 . 1   3 − × −   to   deg 10 5 . 1   3 − × +   . Directional controlling with   r -feedback also   cannot   reduce   the   effect   of eccentricity,   but   the   deviation   can   be tolerated. In elliptic orbit, the amplitude of  ψ S   is oscillating from   deg 10 6   3 − × −   to deg 10 2   3 − × +   .   (Pointing   error   N-S: 0.047° & E-W: 0.047°, Ref. [7]). The increment of inclination,   i , has no effect on longitudinal controlled motion in the   dual-spin   satellite.   However, increasing the inclination of orbital plane,  i ,   can   amplify   the   amplitude   of   long period oscillation modes in   φ S -controlled and   ψ S -controlled.  REFERENCES  [1]   Agrawal, Brijn N., 1986,   Design of Geosynchronous   Spacecraft , Prentice-Hall   Inc.,   Englewood Cliffs (New Jersey). [2]   Bryson, Arthur Earl, 1994,   Control of   Spacecraft   and   Aircraft , Princeton   University   Press, Princeton (New Jersey). [3]   Cornelisse,   J.   W.   with   H.   F.   R. Schöyer and K.F. Wakker, 1979,  Rocket Propulsion and Spaceflight Dynamics , Pitman Publishing Ltd., London. [4]   Hughes   Aircraft   Company,   1982,  Palapa-B   System   Summary ,  Hughes Aircraft Company. [5]   Hughes   Aircraft   Company,   1990,  Palapa-B2R System Summary HS- 376E-22868 ,   Hughes   Aircraft Company. [6]   Jenie,   Said   D.,   2002,  Astrodinamika 1 (Catatan Kuliah PN-350) ,   Departemen   Teknik Penerbangan   Institut   Teknologi Bandung, Bandung. [7]   Jenie,   Said   D.,   2003,   Kendali Terbang (Diktat Kuliah PN-4041) , Departemen   Teknik   Penerbangan Institut   Teknologi   Bandung, Bandung. [8]   Kaplan, Marshal H. , 1976,   Modern Spacecraft   Dynamics   &   Control , John Wiley & Sons, New York. [9]   Muliadi, Jemie, Said D. Jenie and Agus   Budiyono,   2005,   Modeling and Simulation of Prolate Dual- Spin   Satellite   Dynamics   in   an Inclined   Elliptical   Orbit:   Case Study   of   Palapa   B2R   Satellite , Scientific   Paper   for   Siptekgan, Jakarta. [10]   Roy,   Archie   E.,   1982,   Orbital Motion , Adam Hilger Ltd., Bristol (England)
8.   APPENDICES 8.1.   Simulation in Longitudinal Mode  Compensated  θ S -feedback Block  Longitudinal Closed-Loop  q   r   q   p   System Matrix  x x x x x x  p r n  δ  Input Matrix Attitude Dynamics Block in Stability Axis  I Z R  + +  p  q r  S  φ S  θ S  ψ S  φ   S  θ   S  ψ S  φ   S  θ   S  ψ   e  δ  ref e  δ   +-  S  θ  1 498 . 0 10 8 . 29   3  + + ⋅ × −   s  s  Open-Loop  ] [ A  ] [ B x A   • u B   • Bu Ax x  + =   R e  δ n  δ   n  δ   n  δ n  δ n  δ  I Z R I Z R R R  ref e  δ  Fig. 28 Attitude Dynamics Simulation Diagram in Longitudinal Mode  8.1.1.   Effect of Eccentricity in Inclined Orbit ( i   = 30°)  Plot of   θ S   because impulsive input   δ e ref  Fig. 29 Plot of   θ S ; input   δ e ref   ;   i =30deg
Plot of   θ S   because elliptic orbital drift input   δ n  Fig. 30 Plot of   θ S ; input   δ e ref   ;   i =30deg  8.1.2.   Effect of Inclination in Elliptic Orbit ( e   = 0.2)  Plot of   θ S   because impulsive input   δ e ref  Fig. 31 Plot of   θ S ; input   δ e ref   ;   e   = 0.2
Plot of   θ S   because elliptic orbital drift input   δ n  Fig. 32 Plot of   θ S ; input   δ e ref   ;   e   = 0.2; 10 Periode
8.2.   Simulation in Lateral Mode  Lateral Closed-Loop  q   r   q   p   System Matrix  x  p  r  n  δ  Input Matrix Attitude Dynamics Block in Stability Axis  I Z R  + +  p  q r  S  φ S  θ S  ψ S  φ   S  θ S  ψ S  φ   S  θ   S  ψ   e  δ  ref e  δ   +  Open-Loop  ] [ A  ] [ B x A   • u B   • Bu Ax x  + =  R  e  δ p  Compensated  p -feedback Block -  n  δ   n  δ   n  δ  x x x x x  ref e  δ  I Z R I Z R I Z R R R R n  δ n  δ n  δ  (   )  (   ) (   ) 63 . 2 9 . 25 1 . 4 10 5 . 1   6  + ⋅ + + ⋅ ×   s s s  Fig. 33 Attitude Dynamics Simulation Diagram in Lateral Mode  8.2.1.   Effect of Eccentricity in Inclined Orbit ( i   = 30°)  Plot of   φ S   because impulsive input   δ e ref  Fig. 34 Plot of   φ S ; input   δ e ref   ;   i =30deg
Plot of   φ S   because elliptic orbital drift input   δ n  Fig. 35 Plot of   φ S ; input   δ e ref   ;   i =30deg  8.2.2.   Effect of Inclination in Elliptic Orbit ( e   = 0.2)  Plot of   φ S   because impulsive input   δ e ref  Fig. 36 Plot of   φ S ; input   δ e ref   ;   e   = 0.2
Plot of   φ S   because elliptic orbital drift input   δ n  Fig. 37 Plot of   φ S ; input   δ e ref   ;   e   = 0.2; 10 Periode  8.3.   Simulation in Directional Mode  q   r   q   p   System Matrix  x  p r n  δ  Input Matrix Attitude Dynamics Block in Stability Axis  I Z R  + +  p q r  S  φ S  θ S  ψ  S  φ S  θ S  ψ S  φ   S  θ   S  ψ   e  δ  ref e  δ   +  Open-Loop  ] [ A  ] [ B x A   • u B   • Bu Ax x  + =   R  e  δ  - Compensated  r -feedback Block   Control-Loop  5 10 3 ×   r  n  δ n  δ n  δ n  δ  ref e  δ  I Z R I Z R I Z R R R R n  δ  x x x x x  n  δ n  δ  Fig. 38 Attitude Dynamics Simulation Diagram in Directional Mode  8.3.1.   Effect of Eccentricity in Inclined Orbit ( i   = 30°)  Plot of   ψ S   because impulsive input   δ e ref
Fig. 39 Plot of   ψ S ; input   δ e ref   ;   i =30deg  Plot of   ψ S   because elliptic orbital drift input   δ n  Fig. 40 Plot of   ψ S ; input   δ e ref   ;   i =30deg
8.3.2.   Effect of Inclination in Elliptic Orbit ( e   = 0.2)  Plot of   ψ S   because impulsive input   δ e ref  Fig. 41 Plot of   ψ S ; input   δ e ref   ;   e   = 0.2  Plot of   ψ S   because elliptic orbital drift input   δ n  Fig. 42 Plot of   ψ S ; input   δ e ref   ;   e   = 0.2; 10 Periode