Detumbling of a rigid spacecraft via torque wheel assisted gyroscopic motion

Acta Astronautica 93 (2014) 1–12

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Detumbling of a rigid spacecraft via torque wheel assisted gyroscopic motion Yiing-Yuh Lin n, Chin-Tzuo Wang Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 70101, Taiwan, ROC

a r t i c l e i n f o

abstract

Article history: Received 21 May 2012 Received in revised form 4 June 2013 Accepted 18 June 2013 Available online 26 June 2013

A time and energy efficient two-part method for detumbling a rigid spacecraft using an onboard torque wheel and a set of three-axis magnetic torquer is presented in this paper. Part-1 of the method manipulates the speed of the wheel, whose spin axis is parallel to a designated body axis of a tumbling spacecraft, and induces a desired gyroscopic-like ! motion to align the designated axis with its total angular momentum, H T . The procedure ! in effect detumbles the spacecraft to rotate about the designated axis and distributes H T , which is conserved during this control period, between the body and the wheel. After the alignment is achieved, Part-2 control, activated with a specified momentum transfer parameter, η, can either quickly stop the body rotation by transferring its angular momentum to the wheel or offload most of the momentum into space, using the wheel and the magnetic torquer. Convergence criteria and control laws for both parts are derived from the Lyapunov stability analysis and the method of feedback linearization. The wheel performs as a momentum storing and transferring device regulating the angular momentum between the wheel and the body. It can also provide gyroscopic stiffness to stabilize the system while the magnetic torquer is offloading the momentum. Simulation results from the included cases indicate that significantly fast detumbling of the spacecraft can be achieved with Part-1 of the proposed method. The results also show that, under ! the same condition, either by transferring almost all H T to the wheel or dumping it, the two-part method, with a chosen η and final residual momentum condition, requires much less time and energy needed than the B-dot method does. Moreover, the stability nature of the two-part method is heuristically substantiated as the wheel torques and the dipole moment were constrained in the simulation. & 2013 IAA. Published by Elsevier Ltd. All rights reserved.

Keywords: Detumbling spacecraft Gyroscopic-like motion Lyapunov stability Feedback linearization Momentum transfer B-dot method

1. Introduction When a spacecraft is deployed from the last stage of a launcher into orbit, it is released with a spin, which usually results in tumbling [1]. Also, it has become a trend in recent years to bundle several smaller satellites and eject them from a launcher in a single trip [2]. Notably, the Vegas, a new launcher of Arianespace for light-weight lifting purpose, had its successful test and maiden flight on

n

Corresponding author. Tel.: +886 62757575/63673. E-mail address: [email protected] (Y.-Y. Lin).

13 February 2012 [3]. At the critical moment after separation, the solar panels are not in place and the on-board electric power from the battery is limited but the spacecraft needs to use the magnetic torquer to perform despin control and sometimes subsequent attitude acquisition [4,5]. Safety considerations may require enough spin rate to make the course steady when the satellites are exiting from the launcher to prevent any collision with the launcher or among themselves. The need for an effective method to detumble these satellites is evident [6,7]. The simple B-dot control method [8–10] has been commonly used on spacecraft either in the detumbling mode following deployment or in the safety mode after a

0094-5765/$ - see front matter & 2013 IAA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actaastro.2013.06.021

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sudden malfunction of the attitude control during in-orbit operation. Although it is a stable algorithm, the B-dot method has the problems of excessive control period and ineffective power consumption as its control torque depends on the interactions between magnetic dipole moments generated by the spacecraft and Earth's magnetic field. Also, because of the uncontrollable magnitude and direction of the geomagnetic field, which is very weak, the magnetic torque induced is small and the desired control function is difficult to achieve. Although many papers discussed the strategies on attitude stabilization and control, relatively little research has been reported in the study of detumbling schemes due to limited types of actuators one can apply other than magnetic torquers. The advent of new technologies in the torque wheel and the attitude sensor renders the possibility of developing a fast, power-saving method for the detumbling application. Nowadays, for efficient and accurate attitude control, most of the spacecrafts, large or small, carry a torque wheel [11–14]. It would be convenient and effective to detumble the spacecraft with the wheel and also help reduce the weight and size of the battery, the consumption of electric power, and the time of initial attitude acquisition because of the much larger torque available from the wheel. The approach is not without example as applying single rotor actuation with momentum transfer has been investigated in the flat spin recovery problem, another form of detumbling, for the dual-spin satellite [15–18]. In this paper, a two-part detumbling method using a torque wheel and a 3-axis magnetic torquer as actuating devices is presented. Part-1 of the method aligns a designated body axis with the total angular momentum, ! H T , bringing the tumbling spacecraft into single axis rotation as shown in Fig. 1, and then Part-2 despins the platform by transferring the momentum from it to the wheel or dumping most of the momentum, depending on the mission requirements or wheel characteristics. The

body-fixed wheel with spinning axis along a designated body axis provides a control torque that induces the needed gyroscopic-like motion for the alignment in Part1. In Part-2, the wheel also acts like a momentum sink, absorbing the platform angular momentum and despinning the spacecraft, and the magnetic torquer is mainly for dumping the momentum. 2. Attitude dynamics To conveniently derive the proposed detumbling method, the attitude dynamics of a rigid spacecraft is formulated in terms of angular momentum [15–17]. Consider a rigid spacecraft as shown in Fig. 1. The center of mass of the spacecraft including the wheel is chosen as the origin of the set of body-fixed coordinates ði; j; kÞ, which are also the principle axes of the spacecraft system. Total angular ! momentum of the spacecraft, H T , is defined as ! ! ! ð1Þ H T ¼ H P þ h w; ! ω is the angular momentum of the spacewhere H P ¼ I P ! ! craft platform and h w ¼ hw j is the wheel angular momentum whose spin axis is parallel to the body j-axis. Also, I P ¼ dia½I 1 I 2 I 3  is the principal moment of inertia matrix and ! ω ¼ ω1 i þ ω2 j þ ω3 k is the platform angular rate that can be measured from a three-axis gyro set onboard. The wheel angular momentum is hw ¼ I w ðω2 þ ΩÞ, where Iw is the wheel axial moment of inertia about its spin axis and Ω is the wheel speed relative to the spacecraft platform. The transverse moment of inertia of the wheel along i- and kaxes is included in the IP. Expressing in terms of momentum components, Eq. (1) becomes ! H T ¼ H 1P i þ ðH 2P þ hw Þj þ H 3P k;

ð2Þ ! ¼ þ ðH 2P þ hw Þ þ where H T ¼ j H T j. and Consider that the environmental disturbance torques in ! space are very small and can be ignored. As such, H T is H 2T

H 21P

2

H 23P ,

Fig. 1. Detumbling of the spacecraft from (a) initial state to (b) final state with Θ ¼ 0.

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3

Substituting Eqs. (4)–(6) into Eq. (12) leads to V_1 ¼ −Δ31 H 1P H 3P H T :

ð12Þ

As HT is a positive constant, Eq. (12) shows that keeping the condition V_1 o 0 requires the inequality Δ31 H 1P H3P 4 0 to be valid, which in effect increases the term H 2P þ hw as indicated in Eq. (5) until it reaches the maximum value of H T ¼ H 2P þ hw with H 1P ¼ 0, H 3P ¼ 0, and V 1 ¼ 0. Observing the precession angle Φ on the k−i plane in Fig. 2, one finds that two cases are possible to establish the condition V_ 1 o 0: (a) if Δ31 4 0, it requires 01 o Φo 901 or 1801 o Φo 2701 to obtain H 1P H 3P 4 0; (b) if Δ31 o0, it requires 901 oΦ o1801 or 2701 oΦ o 3601 to obtain H1P H 3P o0.

Fig. 2. Definition of nutation angle Θ and precession angle Φ.

conserved and its time derivative gives _ ! ! d! HT ¼ HT þ! ω  H T ¼ 0: dt

ð3Þ

Expressing Eq. (3) in component form becomes [15–17] H3P H_ 1P ¼ Δ23 H 2P H 3P þ hw ; I3

ð4Þ

H_ 2P ¼ Δ31 H 3P H 1P −h_ w ;

ð5Þ

H 1P H_ 3P ¼ Δ12 H 1P H 2P − hw ; I1

ð6Þ

where I 2 −I 3 I 3 −I 1 I 1 −I 2 ; Δ31 ¼ and Δ12 ¼ : I2 I3 I3 I1 I1 I2 ! Also, H T can be expressed in terms of body-fixed coordinates shown from the geometric sketch in Fig. 2 and represented by the following relations:

Δ23 ¼

It is not applicable for the case when Δ31 ¼ 0. However, the condition is not likely to happen unless it is a perfect axisymmetrical satellite. The two cases will induce an interesting dynamic phenomenon, the gyroscopic-like motion that increases the value of H 2P þ hw and, at the same time, reduces both H1P and H3P. Also, the nutation angle Θ shall go to zero and ! the j-axis shall gradually align with the vector H T automatically without direct control effort. To satisfy the stability condition, a wheel control law is formulated utilizing the method of feedback linearization. Differentiating Eqs. (8)–(10) and then substituting the results into Eqs. (4)–(6) yield _ ¼ −Δ31 H T sin Θ cos Φ sin Φ; Θ

ð13Þ

_ ¼ ðΔ23 þ Δ31 sin 2 ΦÞH T cos Θ þ hw : Φ I2

ð14Þ

H 1P ¼ H T sin Θ sin Φ;

ð7Þ

H 2P þ hw ¼ H T cos Θ;

ð8Þ

H 3P ¼ H T sin Θ cos Φ;

ð9Þ

_ in Eq. (14) gives The time derivative of Φ
€ ¼ Δ31 H T sin 2Φ 1 H T ðΔ23 þ Δ31 sin 2 ΦÞð1 þ cos 2 ΘÞ Φ 2  h_w hw þ cos Θ þ : ð15Þ I2 I2

where Φ is the precession angle and Θ is the nutation angle.

The wheel torque is formulated with the method of feedback linearization [18,19]:

3. Control law design and analysis

_ þ ΓÞ h_w ¼ −I 2 ðαΦ

The tumbling problem can be studied more easily in terms of angular momentum as the attitude of the spacecraft is not considered. The dynamic model and kinematic relations derived in the previous section can greatly simplify the stability analysis and the design of the twopart detumbling control in the following. 3.1. Part-1 alignment from Lyapunov stability Consider the Lyapunov function of the form [19] V 1 ¼ 12 ðH T −H 2P −hw Þ2 þ 12ðH 21P þ H 23P Þ;

ð10Þ

where V 1 ≥0. Taking the time derivative of V1 results in V_1 ¼ −ðH_ 2P þ h_ w ÞðH T −H 2P −hw Þ þ H_ 1P H 1P þ H_ 3P H 3P :

ð11Þ

¼ Δ31 H T sin 2Φ½ 12 H T ðΔ23

ð16Þ 2

where Γ þ Δ31 sin ΦÞð1 þ cos 2 ΘÞ þðhw =I 2 Þ cos Θ, and α is a positive damping coefficient to be determined. Substituting Eq. (16) into Eq. (15) results in a simplified form € þ αΦ _ ¼ 0; Φ

ð17Þ

where the wheel torque provides a simple damping effect to the precession motion. The control law in Eq. (16) manipulates the wheel speed and slows down the spacecraft rotation about the ! j-axis as shown in Eq. (17) so that the projection of H T , or the Φ angle, will fall in the proper quadrant of the k−i plane keeping the condition V_ 1 o 0 valid. Also, the wheel torque, h_w , is only activated in the quadrant where V_ o 0 and is otherwise set to zero to ensure that Φ angle will linger in the proper quadrants and slip through others.

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When the alignment control objective is achieved in Part-1, the components H1P and H3P go to zero and the momentums are transferred to H2P and hw. However, it may take very long time to reach Θ ¼ 0 as the term Δ31 H 1P H 3P in Eq. (5) becomes smaller and smaller. 3.2. Part-2 transfer and dumping of angular momentum The momentum components in Part-2 are defined as H 1P ¼ ϵ1 and H 3P ¼ ϵ3 with jϵ1 j 51 and jϵ3 j 51 to represent their relatively small values and to distinguish the formulations from those in Part-1. Eqs. (4)–(6) are rewritten as ϵ_ 1 ¼ Δ23 H 2P ϵ3 þ

ϵ3 hw þ T 1 I3

H_ 2P ¼ Δ31 ϵ3 ϵ1 −h_ w þ T 2 ϵ_ 3 ¼ Δ12 ϵ1 H 2P −

ϵ1 hw þ T 3 I1

ð18Þ ð19Þ ð20Þ

! where, in terms of body coordinates, T ¼ ½T 1 ; T 2 ; T 3 T is the magnetic torquer generated from the interactions between ! the magnetic dipole moment, m ¼ ½m1 ; m2 ; m3 T , and the ! Earth's magnetic field, B ¼ ½B1 ; B2 ; B3 T , by the relation ! ! ! T ¼m B. A second Lyapunov function is defined for the stability analysis in Part-2 as V2 ¼

1 2 1 2 η 2 1 ϵ þ H þ h þ ϵ2 ; 2 1 2 2P 2 w 2 3

ð21Þ

where η is a non-negative momentum transfer coefficient used to regulate the momentum transfer between the spacecraft platform and the wheel. Taking the time derivative of V2 results in V_ 2 ¼ ϵ1 ϵ_ 1 þ H 2P H_ 2P þ ηhw h_ w þ ϵ3 ϵ_ 3 ;

ð22Þ

and substituting Eqs. (18)–(20) into Eq. (22) gives V_ 2 ¼ −Δ31 ϵ1 ϵ3 hw þ ϵ1 T 1 þ H 2P T 2 þ ϵ3 T 3 −ðH2P −ηhw Þh_ w : ð23Þ Also from the method of feedback linearization, the wheel torque is formulated as Δ31 ϵ1 ϵ3 hw h_ w ¼ − þ ðH 2P −ηhw Þ: ðH 2P −ηhw Þ

ð24Þ

Substituting together the wheel torque and the magnetic torque equations into Eq. (23) finds ! ! ! V_2 ¼ m  ð B  H P Þ−ðH 2P −ηhw Þ2 ; ð25Þ ! where H P ¼ ϵ1 i þ H2P j þ ϵ3 k. ! ! The condition V_2 ≤0 is satisfied with m ¼ −km ð B  ! H P Þ. The control gain km is determined by km ¼ M=jr m j, where M 4 0 is the bound for the dipole moment and jr m j is the largest value chosen among jr 1 j, jr 2 j, and jr 3 j, given ! ! ½r 1 ; r 2 ; r 3 T ¼ B  H P . The formulation defines a positive ! time varying gain, keeping each component of m on or between the upper and lower bounds to save the power. The function of Part-2 is to transfer H2P either quickly to the wheel or gradually dump it and stop the spacecraft

platform rotation, which can be determined from the choice of η and the final angular speed. 4. Simulation results and discussions To examine the functions of the proposed method, two simulation examples are included. Example 1. A rigid spacecraft of mass 1032 kg in a circular orbit of 650 km height and 981 inclination is considered, with moment of inertia matrix I P ¼ dia½86:2; 85:1; 114 2 kg m [18] and wheel axial moment of inertia I w ¼ 2 0:5 kg m . The initial tumbling conditions for the simulation are specified in the following: ! HT ¼20.12 N m s, H T ¼ ½14:27 10:06 −9:99T N m s, hw ¼0 N m s, j! ω j ¼ 0:22 rad=s (12.61/s), Θ ¼ 601, and Φ ¼ 1251. The bounds on the wheel and the dipole moment of each axis are 0.05 Nm and 15 A m2 , respectively. Total work done by the wheel, WE, is calculated with t ¼ tf ðΩ þ ω Þh_ w Δt and the energy the equation W ¼ ∑ E

t¼0

2

consumed by the dipole moment, ME, is computed with t¼t

M E ¼ ∑t ¼ 0f ð∑3i ¼ 1 jmi jÞΔt. Also, the Earth's magnetic field is modeled by the international geomagnetic reference field (IGRF) [1]. The control is switched from Part-1 to Part-2 when the condition, jΘj ≤0:011, is reached to expedite the simulation process. Three cases are included for comparison, which are Case A: Two-Part method with η ¼ 0, Case B: Two-Part method with η ¼ 0:1, and Case C: B-dot method. The formulation of the B-dot method is listed in the appendix. Final condition specified to end the simulation is when the platform angular rate equals or is smaller than 0.1331/s, which is about twice the orbit rotation speed. Further, for all cases, the duty cycle of the magnetic torquer in Part-2 is set at 80% in every 5-s control cycle and the hysteresis in the torquer coils is modeled by a first-order transfer function, GðsÞ ¼ 0:5=ðs þ 0:5Þ, representing its time lag effect. The data of overall performance and final states from the simulations are summarized in Table 1. Because the results from Case A are about the same as those in Case B from 0 to 0.26 h, only the simulation trajectories in Case B are displayed in Figs. 3–14. Several observations from Table 1 and the figures are discussed in the following: (a) For Case A, the data shown in Table 1 indicate that it takes only an average 2:95  10−3 W of wheel power over a 0.16 h control period to induce and maintain the gyroscopic-like motion to achieve the alignment in Part-1. With the setting η ¼ 0, the control in Part-2, using 1.09 W of average wheel power for 0.1 h, trans! fers almost all H T from the platform to the wheel as it shows hw ¼19.8 N m s. The time and energy required in Case A are much less than those in the other two cases to detumble the spacecraft, because it stores almost all the momentum in the wheel. (b) The data in Table 1 show that, for Case B, the results in Part-1 are the same as those in Case A. But for η ¼ 0:1, the control in Part-2 dumps 90% of the momentum and

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keeps 10%, or hw ¼2 N m s, of it in the wheel, which is also indicated at the end of the hw trajectory in Fig. 9, maintaining the spacecraft attitude with some gyroscopic stiffness. Also, the data in the table show that the method in Case B requires only 62% of time and 37% of energy needed by the B-dot method in Case C. (c) Due to the field distribution of geomagnetic lines, the smallest final body rate by the B-dot method is 2:32  10−3 rad=s at that orbital height and inclination, which is twice the orbit angular rate rotating about the largest moment of inertia axis of the spacecraft [1], while the proposed method can reduce the rate by specifying smaller final residual momentum as the ! torque generated is opposite to H P instead of ! ω. (d) Shown in Figs. 3–6 are the plots of Part-1 alignment from 0 to 0.16 h and the beginning of Part-2 from 0.16 Table 1 Summary of simulations for Example 1. Case Settling time (h)

A

Final momentum (N m s)

10−3 (rad/s)

W E1 ¼ 1:93 W E2 ¼ 398 M E ¼ 5196 T E ¼ 5596

H1P ¼ −0:0046 H2P ¼ 0:198 H3P ¼ −0:001 hw ¼ 19:8

2.32 (0:1331/s)

H1P ¼ 0:0046 H2P ¼ 0:19 H3P ¼ −0:0024 hw ¼ 1:99

2.32 (0:1331/s)

Ts ¼ 15.3

W E1 ¼ 1:93 W E2 ¼ 848 M E ¼ 832 044 T E ¼ 832 894

2.32 (0:1331/s)

Ts ¼ 24.7

H1P ¼ −0:0631 H2P ¼ −0:0519 H3P ¼ −0:237 T E ¼ 1 900 684 hw ¼ 0

P1 ¼ 0.16 P2 ¼ 0.1 Ts ¼ 0.26

B

Final rate j! ωj

Control energy (W s)

P1 ¼ 0.16 P2 ¼ 15.1

C

Note: P1: control time in Part-1; P2: control time in Part-2; WE1: wheel energy in Part-1; WE2: wheel energy in Part-2; ME: magnetic energy in Part-2; T s ¼ P 1 þ P 2 ; T E ¼ W E1 þ W E2 þ M E .

5

to 0.3 h in Case B, indicating that wheel torque starts to induce gyroscopic-like motion from about 0.02 h in Fig. 3 by keeping the precession angle Φ to stay at 2401 and pulling the nutation angle Θ from 1501 to zero in Fig. 4. The momentum components H1P and H3P quickly go to zero at about 0.15 h, transferring most of the momentum to H2P in Figs. 5 and 6 and a very small quantity to hw in Fig. 3. (e) For the positive value η ¼ 0:1 in Case B, the momentum H2P quickly transfers to the wheel first from 0.16 to 0.25 h as shown in Figs. 3 and 5 and then slowly dumps from the wheel to the platform and offload qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi from the body as hw, H2P, and ∑3i ¼ 1 H 2iP continuously decrease in Figs. 9, 11 and 12, respectively. The dumping process starts from 0.16 h at the beginning of Part2 control as shown in Figs. 7 and 8. (f) It is also indicated in Fig. 10 that the spacecraft attitude is kept after the control in Part-2 is terminated at 15.3 h as the nutation angle Θ grows to about 0:951 but returns to a little less than 0:21, demonstrating the restoring effect from the weakening gyroscopic stiffness even from the loss of momentum in the platform and the wheel. (g) One can also observe from Figs. 13 and 14 that the magnetic dipoles m1 and m3 are mostly on the bound generating negative torque component T2 to offload the momentum component, H2P. The results from Example 1 depict the advantages of the two-part method and the effectiveness of the gyroscopic-like motion applied to the detumbling of the spacecraft. Further investigations on the proposed method for its performance with heavier spacecrafts were done with the following example: Example 2. Consider to detumble two model spacecrafts of the masses two and three times larger than the one in

Fig. 3. Case B wheel angular momentum and torque in Part-1.

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Fig. 4. Case B nutation and precession angles in Part-1.

Fig. 5. Case B components of spacecraft angular momentum in Part-1.

Example 1, while keeping all other conditions, such as orbit, torque wheel, initial tumbling conditions, switching condition, and final terminating condition, the same as in Example 1. Simulation data are summarized in Table 2 below. Because the trajectories of system parameters obtained are similar to those in Figs. 3–14 in Example 1 with slower responses, they are not shown to avoid replications.

Cases A, B, and C represent the spacecrafts detumbled with η ¼ 0, η ¼ 0:1, and B-dot method, respectively. Under ! the same platform density and H T , the spacecraft of mass 2065 kg in Cases A-1, B-1, and C-1 has the platform 2 moment of inertia matrix I P1 ¼ dia½274; 270; 361 kg m with an initial angular speed of 41/s and the spacecraft of mass 3097 kg in Cases A-2, B-2, and C-2 has I P2 ¼ 2 dia½538; 531; 790 kg m with an initial angular speed of

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Fig. 6. Case B total angular momentum of the spacecraft platform in Part-1.

Fig. 7. Case B components of magnetic dipole moment at the beginning of Part-2.

2.031/s. Components of IP1 and IP2 are 25=3 or about 3.18 times and 35=3 or about 6.24 times of IP components in Example 1. Comparing the simulation results in Tables 1 and 2 indicates: (a) Due to slower platform angular velocities and larger platform moments of inertia in Cases A-1 and A-2, the time and the energy required by Part-1 control, respectively, are 2.81 and 7.62 times longer and 14.1 and 23.6 times larger than those in Case A, for inducing and holding the gyroscopic-like motion with the same torque wheel and wheel constraints.

(b) It also shows that the total time, including Part-1 and Part-2, in Cases A-1 and A-2 is 2.08 and 5 times, respectively, of that in Case A but the total energy in Cases A-1 and A-2 is only 0.412 and 0.336 times, respectively, of that in Case A. The longer total time is contributed by Part-1 alignment as mentioned in item (a) but the energy reduction comes from slightly less time used by the energy dominating magnetic torquer in Part 2 as it stabilizes the platform when less H2P is transferred to the wheel in Cases A-1 and A-2. The residual H2P of Cases A, A-1, and A-2, at the final time are 0.198, 0.6, and 1.21 N m s, respectively, while their final platform angular speeds are about the same

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Fig. 8. Case B components of magnetic torque at the beginning of Part-2.

Fig. 9. Case B wheel angular momentum and torque in Part-2.

due to larger and larger moments of inertia. For A cases, HT is kept about the same by observing the final value of H 2P þ hw . (c) The residual H2P in Cases B, B-1 and B-2 are about the same as in Cases A, A-1, and A-2, respectively, having almost the same final platform angular speed in each case. But for B cases, the magnetic torquer in Part-2 will offload both H2P and hw, which slowly transfers

from the wheel to the platform. Comparing to Case B, for spacecrafts with larger moments of inertia as in Cases B-1 and B-2, the process takes less time to reach final platform angular speed, leaving larger amount of momentum on the system as shown from the final value of H 2P þ hw . ! (d) For the same H T , the time and the energy required by the B-dot detumbling method are smaller for larger

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Fig. 10. Case B nutation and precession angles in Part-2.

Fig. 11. Case B components of spacecraft angular momentum in Part-2.

spacecrafts, due to the moments of inertia effect stated in item (b), but still much more than those required by the proposed method. Drawn from the above discussions, under the effect of increased moments of inertia, detumbling larger spacecraft without dumping the angular momentum requires more control time but less total energy, while detumbling the same with angular momentum dumping needs less time and less total energy.

5. Conclusions Instead of slowing down the tumbling angular velocity as the B-Dot control method does, the proposed Part-1 control detumbles the spacecraft by aligning a designated body axis with the total angular momentum, bringing the spacecraft to spin about that axis. It is achieved with a torque wheel to induce a gyroscopic-like motion in the system and relocate the angular momentum to the designated axis. Then, Part-2 control transfers the angular

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Fig. 12. Case B total angular momentum of the spacecraft platform in Part-2.

Fig. 13. Case B components of magnetic dipole moment in Part-2.

momentum from the spacecraft platform to the wheel first to slow down the platform rotation and then from the wheel back to the platform to be offloaded. The factor η in Part-2 determines the momentum transfer rate as can be comprehended from Eq. (24) and observed in the simulation figures. If η is too large, the wheel will transfer the momentum to the platform in a rather fast rate causing the platform to tumble, and the result of Part-2 will be much like B-dot control. Largely increasing the mass and the size of the spacecraft but keeping the system and simulation parameters the same will also obtain stable and satisfactory

detumbling results. The concept of the method is dynamically interesting and its performance is much efficient in terms of time and energy. It is applicable to large or small spacecraft with proper attitude control and sensing devices, such as torque wheel, three-axis magnetic torquer, and three-axis gyro sensor. Also, the control laws of the method derived from the Lyapunov stability analysis and the method of feedback linearization are simple and easy to formulate. Dumping all the tumbling angular momentum using onboard power may be a waste of time and resource. It likely has no choice in the past; however, with the control

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Fig. 14. Case B components of magnetic torque in Part-2.

Acknowledgments

Table 2 Summary of simulations for Example 2. Case Settling time (h)

A-1

Control energy (W s)

Final momentum Final rate (N m s) j! ω j (1/s)

W E1 ¼ 27:2 W E2 ¼ 361 M E ¼ 1920 T E ¼ 2308

H 1P ¼ −0:00321 H 2P ¼ 0:6 H 3P ¼ 0:01 hw ¼ 19:3

W E1 ¼ 45:5 W E2 ¼ 333 M E ¼ 1502 T E ¼ 1881

H 1P ¼ 0:00236 H 2P ¼ 1:21 H 3P ¼ −0:0268 hw ¼ 18:7

W E1 ¼ 27:2 W E2 ¼ 755 M E ¼ 677 593 T E ¼ 678 375

H 1P ¼ −0:0149 H 2P ¼ 0:63 H 3P ¼ −0:0003 hw ¼ 6:27

Ts ¼ 6.44

W E1 ¼ 45:5 W E2 ¼ 628 M E ¼ 313 268 T E ¼ 313 942

H 1P ¼ 0:0127 H 2P ¼ 1:23 H 3P ¼ 0:00834 hw ¼ 12:4

Ts ¼ 17.6

H 1P ¼ 0:11 H 2P ¼ −0:24 H 3P ¼ −0:76 T E ¼ 1 278 943 hw ¼ 0

Ts ¼ 16.6

H 1P ¼ 0:47 H 2P ¼ 0:12 H 3P ¼ −1:51 T E ¼ 1 197 533 hw ¼ 0

P1 ¼ 0.45 P2 ¼ 0.09 Ts ¼ 0.54

A-2

P1 ¼ 1.22 P2 ¼ 0.08 Ts ¼ 1.30

B-1

P1 ¼ 0.45 P2 ¼ 11.1 Ts ¼ 11.6

B-2

P1 ¼ 1.22 P2 ¼ 5.22

C-1

C-2

0.128

0.130

0.133

0.133

The research presented in this paper was supported by the National Space Program Office, Taiwan, ROC, under the contract 98-NSPO(A)-GE-FA09-02. The authors also thank an anonymous reviewer for the careful reading and the advice in the work.

Appendix A ! The magnetic torque T for B-dot method is formulated _ ! ! ! ! ! as [8] T ¼ m  B : Let m ¼ −kB B where kB is a positive _ ! ! gain, and B ¼ −! ω  B with ! ω being the spacecraft angular rate. Similar to the previous derivation, kB is determined by kB ¼ M=jpm j, where M 4 0 is the bound for the dipole moment and jpm j is the largest component ! ! ω  B in absolute value to keep m among ½p1 ; p2 ; p3 T ¼ ! on or within the upper and lower bounds.

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wheel carrying most of the spacecrafts, the proposed detumbling method can be a much time and energy saving approach.

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