Utafiti uliotengenezwa na Kenta Endo, Manami Kanamaru na Keita Tanaka unapendekeza mkakati wa hatua mbili wa usimamizi wa momentum kwa Variable-Speed Control Moment Gyroscope (VSCMG) ili kupunguza trade-off kuu kati ya reorientation ya haraka na targeting sahihi katika nanosatelaiti za daraja la 3U. Mfumo hutumia wheel angular momentum ya juu katika sehemu ya kwanza ya large-angle maneuver ili kuzalisha gyroscopic torque kubwa, kisha wheel momentum hupunguzwa mapema pale attitude error inaposhuka chini ya 5°. Kwa njia hii, satellite inapokaribia target attitude, gyroscopic torque inayoweza kusababisha oscillation kubwa katika body ya 3U yenye low inertia hupunguzwa. Katika closed-loop simulation iliyoungwa mkono na gimbal model iliyotambuliwa kwa majaribio na wheel dynamics zilizopimwa kwenye bench, maneuver ya 90° ilikamilika katika sekunde 25,30, average reorientation rate ilikuwa 3,55°/s na final attitude accuracy ilikuwa 0,42°. Kikomo muhimu zaidi ni kwamba spacecraft-level results hizi zilipatikana katika simulation, si kwenye satellite halisi.

Katika comparison ya utafiti, classic three-reaction-wheel system iliconverge katika sekunde 116 kwenye maneuver ileile ya 90°, ikiwa na maximum body rate ya 0,78°/s na accuracy ya 0,08°. Two-step CMG solution inatoa reorientation rate takriban mara 4,5 juu kuliko reference hii. Katika traditional single-momentum-level CMG simulation kulikuwa na final error ya takriban 5,2°, huku two-step system ikifikia 0,42°. Improvement ya accuracy ya takriban mara 12 katika general comparison ya waandishi iko kati ya hali hizi mbili zenye tuning tofauti; katika separate fixed-feedback-gain controlled comparison ya makala, momentum switch pekee ilionyeshwa kupunguza accuracy error kutoka takriban 0,72° hadi 0,36°, yaani kuboresha takriban mara mbili.

Engineering approach ya utafiti inategemea kusimamia physical momentum state ya actuator badala ya kutengeneza continuous control law ngumu zaidi. Katika large-error region, high momentum level \(h_1=1.41\times10^{-3}\,\mathrm{N\,m\,s}\) hutumiwa kwa kila CMG, huku karibu na target ikitumika nusu yake, \(h_2=7.05\times10^{-4}\,\mathrm{N\,m\,s}\). Kwa hivyo high maneuverability inadumishwa katika sehemu kubwa ya maneuver, lakini torque authority hupunguzwa kabla ya kuingia precise-pointing phase.

Matokeo yanaonyesha pia kwamba method haitoi automatic high-accuracy guarantee katika operating conditions zote. Katika reduced-model Monte Carlo sweep yenye conditions 60 ambapo inertia ilibadilishwa ±10%, control gain ±20% na actuator noise ±50%, maneuvers zote ziliconverge lakini final accuracy ilibadilika kati ya 0,51° na 1,57°. Hivyo, hasa katika high actuator noise, target ya 1° inaweza kuzidiwa.

Kwa nini speed na accuracy ni vigumu kupata kwa wakati mmoja katika 3U CubeSat?

Reaction wheels zinaweza kutoa high pointing accuracy katika small satellites; lakini torque yake ni ndogo, hivyo large-angle maneuvers hufanyika polepole. Control Moment Gyroscope (CMG) inaweza kuzalisha torque kubwa zaidi kwa kuelekeza angular momentum ya wheel inayozunguka kwa high speed kuzunguka gimbal axis. Gharama yake, hasa katika low-inertia nanosatellites, ni kwamba high stored momentum inaweza kusababisha excessive torque, overshoot na cross-axis oscillation satellite inapokaribia target.

Kwa 3U satellite iliyotumika katika utafiti, inertia matrix ilichukuliwa kuwa:

\[ I = \operatorname{diag}(0.034,\;0.034,\;0.0067) \quad [\mathrm{kg\,m^2}] \]

Principal moments of inertia kwenye X na Y axes ni 0,034 kg·m², wakati value kwenye Z axis ni 0,0067 kg·m² tu. Z-axis inertia ni takriban mara tano ndogo kuliko axes nyingine mbili. Kutokana na asymmetry hii, hata cross-axis CMG torque ndogo inaweza kuzalisha yaw oscillation inayoonekana.

CMG hardware iliyotengenezwa imepangwaje?

Katika utafiti, CMG nne zimewekwa katika pyramid arrangement yenye inclination ya 54,73° ili kusambaza torque kwa balance kwenye body axes tatu. Physical CMG module ina dimensions za 90 × 90 × 105 mm na total mass ya gramu 584. Kila prototype wheel ina diameter ya 13,4 mm na imetengenezwa kwa brass.

Angular momentum ya kila wheel inaelezwa kwa:

\[ h_i = I_w\Omega_i \]

Hapa \(I_w\) ni moment of inertia ya wheel kuzunguka spin axis, na \(\Omega_i\) ni angular velocity ya wheel ya i. Katika nominal condition, wheels zote nne ziliendeshwa kwa same momentum magnitude.

Kuna scale distinction muhimu hapa. Moment of inertia ya physical prototype wheel ni takriban \(1.35\times10^{-7}\,\mathrm{kg\,m^2}\), na katika takriban 10.000 rpm huzalisha momentum ya \(1.41\times10^{-4}\,\mathrm{N\,m\,s}\). Phase 1 momentum inayotumika katika spacecraft simulation ni mara kumi zaidi, \(1.41\times10^{-3}\,\mathrm{N\,m\,s}\). Waandishi walitumia prototype si kama direct demonstration ya flight-scale momentum generation, bali kuthibitisha feasibility ya required speed transitions na control envelope.

Sifa ya kutofautisha strategy hii ni kubadilisha amount ya physical angular momentum iliyohifadhiwa kwenye actuator badala ya kubadilisha control gains continuously. Kwa hiyo method si sawa na classic gain scheduling, momentum dumping au reference manager inayoshape commanded torque pekee.

Kwa nini CMG torque inapungua momentum inapopunguzwa?

Torque inayozalishwa na four-CMG pyramid group imeandikwa katika utafiti kama:

\[ T_C=C_j\dot{\delta} \]

\(T_C\) ni CMG torque inayowekwa kwenye satellite body, \(C_j\) ni Jacobian matrix inayotegemea gimbal geometry na wheel momentum, na \(\dot{\delta}\) ni angular-rate vector ya gimbals nne.

Jacobian matrix nzima huzidishwa na wheel momentum magnitude \(h\). Hivyo, kwa gimbal motion ileile, kupunguza \(h\) hupunguza moja kwa moja available torque magnitude. Hii ndiyo physical logic ya msingi ya method: high \(h\) hutoa fast maneuver; lower \(h\) karibu na target hupunguza over-excitation ya low-inertia body.

Kwa nini switching threshold ya 5° ilichaguliwa?

Lengo la waandishi lilikuwa kuhakikisha wheel-speed reduction inakamilika kabla ya kuingia precise-pointing region, bila kukatisha high-momentum phase mapema mno. Kwa hiyo 5° error threshold ilitumika kama nominal value.

Katika reduced model, threshold ilifanyiwa sweep tofauti kwa 2°, 5°, 8° na 12°. Katika sweep hii, peak body rate ilibaki 3,55°/s na final accuracy takriban 0,30° katika hali zote nne; value iliyobadilika wazi ilikuwa momentum transition start time. Kwa threshold ya 2°, transition ilianza katika sekunde 28,4, na kwa 12° ilianza katika sekunde 23,9.

Hata hivyo, threshold-sensitivity results hizi hazikutolewa kutoka same model kama main simulation yenye full gimbal dynamics. Waandishi wanaeleza wazi katika Section 3.5 kwamba huu ni reduced model uliotumika kwa fast parameter sweep. Kwa mfano, minimum singularity indicator ni \(m_{\min}=1.09\) katika reduced model, lakini takriban 0,07 katika full model. Kwa hiyo threshold table inapaswa kutafsiriwa kama indicator ya relative effect ya threshold change, si absolute performance prediction.

Kwa nini singularity problem ni muhimu?

Katika CMG systems, baadhi ya gimbal geometries zinaweza kusababisha kinematic singularities ambapo required torque haiwezi kuzalishwa katika direction fulani. Utafiti unafuatilia hali hii kwa manipulability indicator:

\[ m= \sqrt{\det(A_gA_g^T)} \]

\(A_g=C_j/h\) ni dimensionless geometric Jacobian. \(m=0\) inaonyesha singular configuration. Utafiti unatumia approximate \(m_0=0.1\) warning threshold inayotumiwa sana katika literature.

Katika main 90° maneuver, mfumo ulikaribia threshold region hii mara mbili na takriban \(m_{\min}=0.07\) ilionekana. Waandishi walisema kutegemea pseudo-inverse steering pekee si robust katika region hii na wakatumia Generalized Singularity-Robust (GSR) inverse.

GSR steering law ni:

\[ \dot{\delta} = C_j^T \left( C_jC_j^T+\lambda E \right)^{-1} T_r \]

Hapa \(T_r\) ni reference torque, na \(\lambda E\) ni regularization term inayoregularize matrix inversion karibu na singularity. Operability ikiwa juu, regularization effect huwa ndogo sana; mfumo unapokaribia singularity, effect huongezeka na kuzuia uncontrolled growth ya gimbal-rate commands.

Je, VSCMG inaweza pia kutumika kama reaction wheel?

Ndiyo. Utafiti pia ulisimulate RW mode ambapo gimbals nne zilifungwa katika angles maalum na torque ikazalishwa tu kwa acceleration na deceleration ya VSCMG wheels. Mode hii haikutathminiwa kwa high speed bali kwa precise attitude holding.

Katika 15° roll maneuver, satellite ilifikia 0,1° error band katika sekunde 351, na worst-case steady-state error katika sekunde 10 za mwisho ilikuwa 0,013°. Kwa hiyo same VSCMG cluster inaweza kuhudumia mission phases tofauti kati ya high-torque CMG mode na RW mode iliyo polepole sana lakini accurate zaidi.

Kwa nini kupunguza wheel speed kwa nusu hupunguza micro-vibration?

Prototype wheels zilitathminiwa pia kwa imbalance-induced micro-vibration. Synchronous radial force inayotokana na residual mass eccentricity ilimodeliwa kama:

\[ F(\Omega)=U\Omega^2, \qquad U=m_we \]

\(m_w\) ni wheel mass, \(e\) residual eccentricity na \(\Omega\) wheel angular velocity. Kwa kuwa force huongezeka na square ya speed, wheel speed ikipunguzwa kutoka 10.000 rpm hadi 5000 rpm, synchronous vibration component hii hupungua hadi takriban robo.

Chanzo kinakadiria synchronous radial force ya takriban 16–99 mN katika 10.000 rpm na takriban 4–25 mN katika 5000 rpm kwa eccentricity range ya 2,5–15 µm. Hizi si payload jitter values zilizopimwa orbit, bali actuator-level estimates zinazotokana na manufacturing tolerances.

Mbinu na Matokeo ya Utafiti

Spacecraft dynamics ilimodeliwaje?

Rotational dynamics ya satellite ilimodeliwa kwa Euler equation:

\[ I\dot{\omega}_s+ \omega_s\times(I\omega_s) = T_C+T_d \]

Hapa \(I\) ni inertia matrix ya satellite, \(\omega_s\) ni body angular velocity, \(T_C\) ni CMG control torque na \(T_d\) ni external disturbance torque.

Katika main closed-loop model, external disturbance ilichukuliwa kama:

\[ T_d=T_a+T_g \]

na aerodynamic drag pamoja na gravity-gradient effects katika altitude ya 560 km zilijumuishwa. Atmospheric density iliyotumika ilikuwa \(2.108\times10^{-13}\,\mathrm{kg/m^3}\), cross-sectional area 0,03 m² na drag coefficient 2,2.

Characteristic disturbance torques zilizokadiriwa katika chanzo ni takriban \(4.9\times10^{-8}\,\mathrm{N\,m}\) kwa maximum gravity gradient, takriban \(8.0\times10^{-9}\,\mathrm{N\,m}\) kwa aerodynamic drag, takriban \(4.4\times10^{-9}\,\mathrm{N\,m}\) kwa solar radiation pressure, na kati ya \(2.5\times10^{-8}\) na \(2.3\times10^{-7}\,\mathrm{N\,m}\) kwa residual magnetic dipole.

Solar radiation pressure na residual magnetic torque hazikujumuishwa katika main simulation. Waandishi wanakokotoa kwamba katika fast maneuver ya takriban sekunde 25, angular impulse inayotokana na hizi ni ndogo sana kuliko body angular momentum inayobadilishwa na CMG. Hata hivyo, utafiti unasema wazi kwamba residual magnetic dipole inaweza kuwa disturbance muhimu katika time scales ndefu zaidi.

Quaternion feedback controller

Command torque huzalishwa kwa quaternion error feedback:

\[ T_e=-K_{qp}q_v-K_{qd}\omega_s \]

\(q_v\) ni vector part ya error quaternion kati ya target na current attitude; \(K_{qp}\) ni proportional attitude gain; \(K_{qd}\) ni angular-rate damping gain.

Lyapunov candidate iliyotumika kwa nominal stability analysis ni:

\[ V= \frac{1}{2} \omega_s^TI\omega_s+ 2K_{qp}(1-q_4) \]

na derivative yake katika ideal conditions ni:

\[ \dot V= -K_{qd}\|\omega_s\|^2 \le 0 \]

Matokeo haya yanaunga mkono asymptotic stability ya nominal quaternion feedback loop. Hata hivyo, chanzo kinasema wazi kwamba huu si formal stability proof kwa full two-step switched VSCMG system; gimbal saturation, switching, actuator delay na model uncertainty hazijajumuishwa katika proof kamili.

Gimbal model ilitolewaje kwa majaribio?

Gimbal motor, driver na encoder zilichukuliwa pamoja kama black-box system. Frequency sweep ya 0,1–100 Hz yenye amplitude ya 6 V iliwekwa kwa sampling rate ya 400 Hz na second-order transfer function ikapatikana:

\[ G(s)= \frac{108600} {s^2+210.9s+1460} \]

Model ilitoa %93,2 time-domain fit katika hold-out validation segment. Hata hivyo, utafiti pia unaandika kama limitation kwamba repeated tests katika amplitudes tofauti au independent step-input validation hazikufanywa.

PID, LQR na Integral-type Optimal Servo (IOS) zililinganishwa kwa gimbal speed control. Katika 0,5 Hz sinusoidal tracking, IOS ilionyesha phase lag ya takriban sekunde 0,02; PID takriban 0,31 s na LQR takriban 0,14 s. Katika step disturbance, IOS ilifikia asymptotically zero steady-state error, huku steady-state deviation ya −16,9°/s ikiripotiwa kwa PID na 260°/s kwa LQR. Kwa hiyo IOS ilichaguliwa katika closed-loop simulations.

Wheel dynamics si synthetic; inategemea transition iliyopimwa

Wheel speed controller ilikalibishwa kwenye prototype. Measured deceleration time series kutoka 10.000 rpm hadi 5000 rpm iliingizwa moja kwa moja kama dynamic input katika closed-loop spacecraft simulation. Measured transition inaweza kuwakilishwa na first-order approximate model yenye time constant \(\tau\approx0.6\) s, fit ya %94,7 na RMSE ya takriban 70 rpm; %95 settling time ni takriban sekunde 1,8.

Matokeo kuu katika maneuver ya 90°

Traditional CMG condition bila two-step control iliconverge katika takriban sekunde 27,61 lakini ikatoa 5,17° undershoot kwenye roll axis na oscillation ya takriban 3,3° kwenye yaw axis.

Two-step momentum control ilipotumika, target ileile ya 90° roll ilitekelezwa katika sekunde 25,30, average body reorientation rate ilikuwa 3,55°/s na final attitude error ilikuwa 0,42°.

Katika comparison hii, two-step CMG inatoa reorientation rate takriban mara 4,5 juu kuliko reaction-wheel reference, lakini final accuracy ya 0,08° ya reaction wheel ni bora zaidi. Kwa hiyo utafiti hauungi mkono hitimisho kwamba “CMG ni bora kuliko reaction wheel kwa kila kipengele”; faida ya method inayopendekezwa ni kutoa operating point yenye balance bora kati ya speed na sub-degree accuracy.

Je, improvement ya accuracy ya mara 12 inatokana na momentum switch pekee?

Hapana; makala yenyewe imedhibiti point hii kando. Katika main comparison, quaternion feedback gains ni tofauti kati ya simulations za two-step na no-two-step. Kwa hiyo si sahihi kuhusisha mabadiliko yote ya takriban 5,2° → 0,42° na momentum switch pekee.

Katika fixed-gain controlled comparison ya Figure 13, final accuracy iliboreshwa kutoka takriban 0,72° hadi 0,36° wakati momentum switch iliwashwa kwenye P–D controller ileile. Kwa hiyo, control gains zinapowekwa fixed, contribution ya momentum switch yenyewe ni improvement ya accuracy ya takriban mara mbili.

Katika comparison ileile, kuongeza derivative gain mara mbili pekee au kuongeza gyroscopic feedforward hakukuzalisha final accuracy iliyotolewa na two-step momentum management. Waandishi wanatafsiri matokeo haya kuwa component kuu ya improvement ni kupunguza actuator momentum kimwili.

Nini hutokea bila GSR singularity avoidance?

Katika singularity test ya utafiti, GSR ilipokosekana mfumo ulifikia singular configuration katika takriban sekunde 6,04 na haukuweza kufikia target attitude kwa sababu haukuweza kuzalisha required torque direction.

GSR inverse ilipowashwa, satellite ilifikia target katika sekunde 26,44. Operability indicator ilishuka hadi \(m=0.0519\) katika sekunde 6,97, lakini kutokana na regularization mfumo ulipita near-singularity region na kurudi kwenye gimbal configuration iliyo better conditioned.

Je, mfumo unaendelea kufanya kazi CMG moja ikishindwa?

Utafiti pia ulisimulate three-CMG condition ambapo CMG4 ilifeli kabisa na corresponding column ikaondolewa kutoka Jacobian. Katika scenario hii, roll error component iliconverge katika takriban sekunde 85,06 na yaw component katika takriban sekunde 89,54. Steady-state attitude accuracy ilibaki ndani ya 0,95°.

Hata hivyo, wakati wa failure yaw error ilifikia 39,40° katika sekunde ya 23. Large transient deviation hii inaonyesha kwamba active CMGs tatu haziwezi kila wakati kuzalisha optimal torque direction inayotoa shortest attitude path kwenda target. Kwa hiyo matokeo hayamaanishi “failure ya CMG moja haiathiri performance”; yanaonyesha tu kwamba mfumo unaweza bado kufikia target licha ya severe performance degradation.

Convergence ya takriban sekunde 85 ni polepole takriban mara 3,4 kuliko sekunde 25,3 katika four-CMG two-step condition. Hata hivyo, bado ni haraka kuliko reaction-wheel reference ya sekunde 116 ya utafiti.

Switching threshold sensitivity

Threshold sweep iliyotumika katika reduced model ilitoa matokeo yafuatayo:

Sweep hii inaonyesha kwamba maneuver rate si sensitive sana kwa threshold ndani ya range ya 2°–12°; threshold kubwa zaidi hasa huanzisha momentum reduction mapema zaidi. Hata hivyo, table hii si ya full gimbal model na absolute accuracy value ya 0,30° haipaswi kuchanganywa na main full-model result ya 0,42°.

Monte Carlo analysis inaonyesha method ni robust kwa kiwango gani?

Katika reduced closed-loop model, Monte Carlo runs 60 zilifanywa. Spacecraft inertia ilibadilishwa ±10%, feedback gains ±20% na actuator noise ±50%.

Runs zote ziliconverge kwenye target error band na peak body rate ilibaki 3,55 ± 0,00°/s. Mean settling time ilikuwa 32,7 ± 3,8 sekunde. Final accuracy ilikuwa 0,87 ± 0,24° na ilibadilika kati ya 0,51°–1,57°.

Matokeo haya yanaonyesha kwamba maneuver rate ya method ni robust dhidi ya uncertainties zilizochunguzwa, lakini final pointing accuracy inaathiriwa wazi na actuator noise. Kwa kuwa error inaweza kuzidi target ya 1° katika highest-noise conditions, utafiti hautoi guarantee ya sub-degree accuracy katika uncertainty combinations zote.

Matokeo yanayoungwa mkono na utafiti

• Two-level wheel momentum inaweza kuunganisha high maneuverability ya CMG na lower torque requirement karibu na target katika actuator cluster ileile kwenye low-inertia 3U satellite model.
• Katika full closed-loop simulation, 90° roll maneuver ilikamilika katika sekunde 25,30 kwa characteristic rate ya 3,55°/s na accuracy ya 0,42°.
• Maneuver rate iliongezeka takriban mara 4,5 dhidi ya reaction-wheel reference; reaction wheel ilitoa final accuracy ya juu zaidi.
• Fixed-feedback-gain comparison inaonyesha kwamba momentum switch pekee inaweza kuboresha final accuracy kwa takriban mara mbili.
• GSR steering ilifanya kazi kwa robustness zaidi kuliko pseudo-inverse approach katika near-singularity maneuver iliyochunguzwa.
• Katika simulation ya failure ya CMG moja, mfumo ulifikia target lakini maneuver ilipungua kasi kwa takriban mara 3,4 na large transient yaw deviation ikatokea.
• Precise-pointing phase ya 5000 rpm inapunguza imbalance-induced synchronous micro-vibration force kwa takriban mara nne kulingana na modeli ikilinganishwa na phase ya 10.000 rpm.

Matokeo ambayo utafiti hauungi mkono au haujajaribu bado

• 90° maneuver ya sekunde 25,3 na accuracy ya 0,42° haijaonyeshwa experimentally kwenye physical 3U satellite au air-bearing full-system test.
• Physical prototype haizalishi full flight-scale angular momentum inayotumiwa katika simulation; prototype inavalidate speed na gimbal envelope katika lower momentum level.
• Lyapunov analysis si formal stability proof kwa entire two-step switched system.
• Utafiti unaonyesha kwa kina single-axis 90° large-angle main maneuver pekee; arbitrary eigenaxis maneuvers na sequential multi-target missions hazijavalidishwa bado.
• Comprehensive fault detection, isolation and recovery system kwa multiple CMG failures haijajaribiwa.
• COTS motors zinazotumika kwenye prototype si space-qualified; radiation, thermal vacuum, long-life bearing/lubrication na electromagnetic compatibility validation hazijafanywa.
• Micro-vibration values si jitter iliyopimwa kwenye integrated optical payload.
• Kwa kuwa hakuna test iliyofanywa kwenye CubeSat, mission au orbital infrastructure maalum ya Uturuki, direct performance result kwa Uturuki haiwezi kutolewa.

Maelezo ya Chanzo na Mbinu

Aina ya chanzo na publication status: Utafiti huu ni peer-reviewed research article na si preprint. Ulichapishwa katika jarida la Aerospace mwaka 2026 kama volume 13, issue 7, article 582.

Utafiti asilia: Kenta Endo, Manami Kanamaru na Keita Tanaka, “A Two-Step Variable-Speed Control Moment Gyroscope Control Strategy for 3U Nanosatellite Attitude Maneuvers”. DOI: 10.3390/aerospace13070582.

Corresponding author: Keita Tanaka.

Taasisi: Kenta Endo na Keita Tanaka wako Tokyo Denki University Graduate School of Science and Engineering; Manami Kanamaru yuko Shibaura Institute of Technology College of Engineering.

Evidence level halisi ya utafiti: Spacecraft-level attitude-control results zinategemea closed-loop numerical simulation. Physical prototype experiments zimewekewa mipaka katika identification ya gimbal motor model, measurement ya wheel speed transition na testing ya feasibility ya required gimbal/wheel speed envelope. Utafiti si system-level experimental attitude-control demonstration.

Hardware: Pyramid module yenye CMG nne ina dimensions za 90 × 90 × 105 mm na mass ya 584 g. Physical prototype wheel ni brass flywheel yenye mass ya 6,06 g na diameter ya 13,4 mm. Wheel moment of inertia ni takriban \(1.35\times10^{-7}\,\mathrm{kg\,m^2}\).

Experimental model identification: Gimbal system ilicharacterize kwa 0,1–100 Hz swept-sine input na second-order dynamic model ikatoa %93,2 fit katika hold-out segment. Wheel speed transition ya 10.000 → 5000 rpm ilipimwa moja kwa moja kwenye bench na kuhamishwa kama data kwenye simulation dynamics.

Main simulation: 90° roll maneuver ilifanywa katika orbital altitude ya 560 km chini ya aerodynamic na gravity-gradient disturbances. Katika two-step strategy, momentum levels ni \(h_1=1.41\times10^{-3}\) na \(h_2=7.05\times10^{-4}\,\mathrm{N\,m\,s}\), na switching threshold ni 5°.

Stability limit: Ingawa abstract ya chanzo ina phrase “Lyapunov-based stability guarantee”, Section 2.2.3 inasema wazi kwamba Lyapunov analysis inatumika tu kwa fixed-momentum nominal loop, ideal torque tracking na zero-disturbance assumptions. Waandishi pia wanasema katika section hiyo kwamba analysis hii si formal stability proof kwa switched two-step system na kwamba common Lyapunov function au average-dwell-time type analysis inapaswa kufanywa baadaye. Verianla haijatoa stability guarantee pana zaidi.

Source-internal numerical-expression inconsistency: Section 2.3 inasema 5° switching threshold inaacha “more than 94° of the 90° maneuver” kwa high-momentum Phase 1. Katika total maneuver ya 90°, sehemu ya zaidi ya 94° haiwezekani kimwili. Intended form ya phrase hii haijakisiwa wala kusahihishwa kimya kimya.

Model-comparison warning: Katika main full model final accuracy ni 0,42° na \(m_{\min}\approx0.07\), wakati threshold sensitivity na Monte Carlo sweeps hutumia reduced model ya haraka zaidi. Kwa mfano, katika model hii threshold-sweep accuracy ni 0,30° na \(m_{\min}=1.09\). Chanzo kinaeleza tofauti hii wazi na kinataka reduced-model results zisomwe kama relative trend/robustness indicators, si absolute performance prediction.

Maelezo kuhusu 12× comparison: Katika comparison ya chanzo ya 5,2° → 0,42°, feedback gains si sawa. Makala ilichunguza confounding effect hii kando na kuripoti kwamba katika fixed-gain experiment, two-step momentum switch ilitoa takriban two-fold accuracy improvement ya 0,72° → 0,36°. Kwa hiyo value ya “mara 12” haijawasilishwa kama isolated effect ya momentum switch.

Ufadhili: Utafiti ulifadhiliwa kwa sehemu na Tokyo Denki University na Tokyo Metropolitan Government University Startup Support Program.

Data availability: Imeripotiwa kwamba data za utafiti zinapatikana kutoka kwa corresponding author kwa request na si public kwa sababu ya ongoing project developments.

Project: Imeelezwa kwamba utafiti ulifanywa kama sehemu ya HATOSAT project.

AI use: Waandishi wameeleza kwamba walitumia AI-assisted tools katika language editing ya makala na wanachukua full responsibility kwa content.

Conflict of interest: Waandishi hawakuripoti conflict of interest.

License: Makala imechapishwa open access chini ya Creative Commons Attribution (CC BY) license.

Chanzo rasmi: https://doi.org/10.3390/aerospace13070582