On Attitude Control of Microsatellite Using Shape Variable Elements 形状可変機能を用いた超小型衛星の姿勢制御について

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Ethelbert Gibson
5 years ago
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1 The 4th Workshop on JAXA: Astrodynamics and Flight Mechanics, Sagamihara, July 015. On Attitude Control of Microsatellite Using Shape Variable Elements By Kyosuke Tawara 1) and Saburo Matunaga ) 1) Department of Mechanical and Aerospace Engineering, Tokyo Institute of Technology, Tokyo, Japan ) The Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Kanagawa, Japan We propose a shape variable attitude control system to improve the attitude control performance of micro-satellites. Driving deployable solar array paddles by motor is regarded as a variable shape system in the paper. Then micro-satellite attitude can be controlled with the variation of motor angle and its inertial mass characteristics. Due to small mass of micro-satellites, the variation can results in large-angle attitude maneuvering. In this paper, the results of the investigation with a concrete mission scenario for microsatellite is reported. 形状可変機能を用いた超小型衛星の姿勢制御について超小型衛星の姿勢制御性能向上を目的とした形状可変システムを提案する. 具体的には, 展開式の太陽電池パドルをモータ駆動することにより, システムの形状を可変とし, その慣性質量特性の変化を利用して姿勢制御を行う. 超小型衛星は質量が小さいため, この方式によって大アングルマヌーバを行うことができる可能性がある. 本稿においては, 具体的なミッション例に関して有効性を検討した結果について報告する. Key Words: Attitude Control, Microsatellite, Variable Shape Function, Large-Angle Maneuvering Nomenclature 1 Introduction ω i : ω i/j : angular velocity vector of body i relative to an inertial frame angular velocity vector of body i relative to body j J i : inertia dyadic about body i s mass center m i : mass of body i p ci : p c : position vector from inertial origin to body i s mass center position vector from inertial origin to mass center of a multibody satellite system Ω : rotational speed of a motor λ i : maneuver angle of body i b j : ρ p ρ r x unit base vector of the frame fixed to body 0 (B frame) liner density of a solar array paddle liner density of a rod components of a vector x In recent years, a frequency of launching microsatellites has increased. Furthermore, microsatellites which perform challenging mission has proposed. For example, a microsatellite TSUBAME, developed by Laboratory for Space Systems (LSS) at ISAS/JAXA and Tokyo Institute of Technology, aimed to perform advanced missions for a 50 kg class microsatellite 1). One of the missions is construction and demonstration of a high performance microsatellite bus. In the future, microsatellites with much more performances are expected to be increasingly important. In this paper, a new attitude control method for microsatellites is proposed focusing on a variable shape function. Hereinafter, a variable shape attitude control is referred to as VSAC. This study is focused to assess the usefulness of VSAC for microsatellite and this is a fundamental study towards establishment of the new attitude control method for a microsatellite. To start with, here we describe a concrete concept of VSAC. Then, we formulate attitude variation caused by VSAC and show a driving law for VSAC. Finally, we describe the conclusion.

Attitude Control Using Variable Shape Function A microsatellite which tries to perform advanced missions often drives a part of its body such as deployment of solar array paddles to supply enough

2 Attitude Control Using Variable Shape Function A microsatellite which tries to perform advanced missions often drives a part of its body such as deployment of solar array paddles to supply enough stable power for mission requirement. We call such function as variable shape function. Further, appending appropriate actuators and motors to the deployment hinge parts, the satellite easily get a function for driving them. When a satellite drives an appendage, its main body attitude varies due to antitorque caused by driving the appendage (for example, solar array paddle, robot arm and so on). A method utilizing this attitude variation positively for attitude control is VSAC (variable shape attitude control). Figure 1 shows a concept of VSAC. comparative ease. Further, constraint on the microsatellite mass is frequently flexible compared to other constraint (e. g. length and volume). Therefore, we expect that microsatellites may perform agile large-angle maneuver using VSAC. In development of a large scale satellite, it will be demanded to appropriate a certain amount of budget in improvement of actuators performance before adding Variable Shape Function. However, particularly in development of a microsatellite, where compromises have frequently to be made in reliability, it is attractive that to be improved in attitude control performance by only adding a driving mechanism to a portion that originally needed to deploy. 3 Attitude Variation of Satellite Figure 1 Image of a variable shape attitude control, VSAC In this section, we formulate a relationship between angular velocity of each body and rotational speed of a motor. First, let us consider a multibody system consisted of n bodies. Satellite has a number 0 and appendages has numbers i (i=1,,, n-1). Then, total angular momentum h c about mass center of the system is written as follows and h c is conserved when external torque equals zero 4). Attitude variation due to driving an appendage is studied in some previous studies. However, the variation is treated as perturbation ) in the studies whereas VSAC proactively utilize the variation to control an attitude of a satellite. Although there are some studies investigating attitude control by robot arm motion 3), these are not employed for practical attitude control for any spacecraft. VSAC is to be effective in practical use particularly in microsatellite. Table 1 shows a relationship between mass of solar array paddles and total mass in various satellites. According to this table, the ratio of paddles mass to total mass is usually 5-10%. The larger the moment of inertia of the appendage is, the more variation of attitude with VSAC is extensive. Due to smallness of microsatellites mass, we are able to increase the percentage of mass of an appendage to mass of the microsatellite with Table 1 Relationship between paddles mass and total mass in various satellites Spacecraft Launch [year] Total mass [kg] Paddles mass [kg] Paddles mass per unit area [kg/m ] Percentage of paddles mass [%] ETS-III MOS-I ADEOS-II INDEX TSUBAME Hayabusa h c = {(p ck p c ) m k p ck + J ck ω k } (1) Further, considering the definition of mass center, following statement is valid. (p ck p k ) m k p c = 0 () By summing Eq. (1) and Eq. (), total angular momentum h c is rewritten as follows. h c = {(p ck p c ) m k (p ck p c) +J ck ω k } (3) Let us consider body 0 drives body j (j = m+1, m+,, n-1) at angular velocity ω j/0 and initial angular momentum h c, initial angular velocity of each body and external torque equal zero. Note that the norm of angular velocity ω j/0 equals to rotational speed of a motor Ω. When we assume undriven body q (q = 1,,...,m) to be fixed to satellite (body 0), each angular velocity ω q/0 satisfies following equations.

ω 0/q = 0 (4) ω q = ω 0 (5) Then, the angular velocity of the satellite is calculated by following equation. from θ 1 =90 deg to 0 deg and body is fixed to the satellite.

4 m 0.4 m 0.67 kg/m 50 kg In the next place, we construct a two dimensional satellite model to reduce argument as shown in Figure.

3 ω 0/q = 0 (4) ω q = ω 0 (5) Then, the angular velocity of the satellite is calculated by following equation. from θ 1 =90 deg to 0 deg and body is fixed to the satellite. Constants used in the calculation is set as shown in Table. Table Values of the constants 0 = (p ck p c ) m k (p ck p c) m + J ck ω 0 + J ck (ω 0 + ω k/0 ) k=m+1 (6) Constants d ζ ρ r m 0 Value 0.4 m 0.4 m 0.67 kg/m 50 kg In the next place, we construct a two dimensional satellite model to reduce argument as shown in Figure. Adding to Table, we set the initial conditions of the simulation as shown in Table 3. Table 3 Vaues of the initial conditions I. C. Value θ 1 90 deg 90 deg θ Figure A Two-dimensional model of the satellite Note that degrees of freedom between the rod and the solar array paddle are ignored in this model. Assumed each body doesn t have possibility of outplane-motion, we can define a maneuver angle λ i as integration of each body s angular velocity shown in following. Because 30 deg attitude maneuvering is a typical large angle maneuvering, the parameters should be chosen such that attitude control using VSAC has sufficient capacity to conduct large angle the microsatellite mass is frequently flexible in development of a microsatellite. Further, it is difficult to deploy large rods during orbit. Then, we chose the parameters as following. t λ i = ω i dt 0 (7) Note that ω i is the third component of the frame fixed to body 0 (B frame). Using Eq. (7), we estimate the attitude variation of each body. 4 Attitude Maneuver Using VSAC In this section, two kinds of numerical examples are conducted and the results are shown. First, we assess the magnitude of λ 0 under various parameter combinations and study desirable parameter maximize λ 0 under the design constraint. Second, we conduct attitude control simulation with a concrete mission scenario. 4.1 Magnitude of Attitude Variation Parameters in the model as shown in Figure are the rod length ξ and the density of the paddle ρ p. Here, we swept the parameters and calculate λ 0 in various condition. Note that only body 1 is driven Figure 3 conditions Attitude variation plot under various 3

4. Attitude Control Simulation Missions which require large angle maneuver include an observation of Gamma-ray burst (GRB). GRB observation sequence is shown in Figure 4.

Each parameter of the simulation is set as Table and Table 3. Additionally, the density of paddles ρ p and rod length ξ are chosen as 1.5 kg/m and 0.6 m. Control input is ω 0.

4 4. Attitude Control Simulation Missions which require large angle maneuver include an observation of Gamma-ray burst (GRB). GRB observation sequence is shown in Figure 4. Figure 4 Mission sequence of GRB observation Here we conducted 30 deg rest to rest attitude control simulations assuming a mission of a satellite was GRB observation. Each parameter of the simulation is set as Table and Table 3. Additionally, the density of paddles ρ p and rod length ξ are chosen as 1.5 kg/m and 0.6 m. Control input is ω 0. The angular momentum of multibody system becomes h c = {R k m k R k + J ck ω k } (8) where R k = p ck p c. To determine the control input, the angular momentum is expressed in B frame as h c = {m k R kr k + J k ω k } (9) where the bolds are component of the vectors. Operator tilde is defined below as 0 x 3 x X = [ x 3 0 x 1 ] (10) x x 1 0 Because R k = f k (θ 1, θ ), R k is obtained as R k = D θ f k θ (11) where D θ f k is the Jacobian matrix of f k respect to θ and θ = [θ 1 θ ] T. The angular velocity ω i/0 is expressed as 0 ω i/0 = C [ 0 ] = Ω i/0 θ (1) θ i Considering eq. (11) and eq. (1), eq. (9) becomes as Iω 0 m i R ir iω 0 i=0 = ( m i R id θ f i + J i Ω i/0 ) θ i=0 i=1 (13) = Pθ where the angular momentum h c is assumed to be zero and I = J i. If we command angular velocity vector, the rotational speed of the hinge θ computed by the pseudoinverse of P become θ = P + (I i=0 m i R ir i)ω 0 (14) where P + = P T (PP T ) 1. Control input ω 0 is chosen as 8 deg/s. The result of the simulation is shown in Figure 5. First, Figure 5 shows angular velocity of satellite and rotational speed of the motors. As can be seen in Figure 5, 30deg rest to rest maneuver is successfully completed within 4s using VSAC. Further, attitude of satellite was settled when stopped driving paddles. As can be seen from eq. (14), the angular velocity of satellites is directly controlled in VSAC. Then, there are no overshoot of λ 0 ideally. This fact can be merit from the viewpoint of attitude control. Second, Figure 5 shows attitude variation of each body. Seen in Figure 5, each final attitude variation of the paddles λ 1, λ is 1deg. Considering this result, the light receiving face of paddles after maneuvering is 10% larger using VSAC than not using (See Figure 6). Attitude variation of paddles with respect to inertial direction becomes smaller because it is sufficient that only a part of system is to be reoriented to desired direction with VSAC whereas all parts of system should be maneuvered with conventional maneuver control methods, such as one using reaction wheels. Figure 5 Time histories of satellite angular velocity and attitude variation Figure 6 Posture relationship between sunlight and solar array paddles

5 For microsatellites, which are always in short power supply due to stuffing many components in small volume, it is very attractive benefit that the area of the light receiving face is not reduced too much after maneuvering. 5. Conclusion We proposed VSAC as a novel attitude control method for microsatellites. In order to show that the method is effective, we conducted two kinds of calculation. First, we showed that microsatellites can achieve large angle maneuvering by VSAC in realistic range of the simulation parameters. Then, we found that VSAC has two advantages at least by conducting the simulation of GRB observation. One arises from the characteristic control method. Control input to a satellite is rotational speed of motor in attitude control using VSAC whereas that is torque in attitude control using reaction wheels. This is to be equivalent to being able to control angular velocity of the satellite directly. The other merit is that there are some condition that we maneuver a satellite into one direction without the loss of power supply. We showed these merits in this paper. In addition to them, it is expected that performance of attitude stability improves with increasing mass of paddles. Showing this merit is a future issue. References 1) M. Matsushita et al., Flight Model Development of the Micro-satellite TSUBAME, 10 th IAA Symposium on Small Satellites for Earth Observation, Berlin, April 015. ) M. Oda, Y. Ohkami, Coordinated Control of Spacecraft Attitude and Space Manipulators, Control Engineering Practice, Vol. 5, Issue 1, 1997, pp ) K. Yamada, Attitude Control of Space Robot by Arm Motion, Journal of Guidance, Control, and Dynamics, Vol. 17, No. 5, 1994, pp ) Attitude Control Research Committee, Handbook of Satellite s Dynamics and Control, Baifukan,

Reactive Fluid Dynamics 1 G-COE 科目複雑システムのデザイン体系第 1 回植田利久慶應義塾大学大学院理工学研究科開放環境科学専攻 2009 年 4 月 14 日. Keio University

Reactive Fluid Dynamics 1 G-COE 科目複雑システムのデザイン体系第 1 回植田利久慶應義塾大学大学院理工学研究科開放環境科学専攻 2009 年 4 月 14 日 Reactive Fluid Dynamics 2 1 目的本 G-COE で対象とする大規模複雑力学系システムを取り扱うにあたって, 非線形力学の基本的な知識と応用展開力は不可欠である. そこで,