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Preliminary Investigations on Ground Experiments Of PRELIMINARY INVESTIGATIONS ON GROUND EXPERIMENTS OF VARIABLE SHAPE ATTITUDE CONTROL FOR MICRO SATELLITES *Yusuke Shintani1, Tsubasa Tsunemitsu 1, Kei Watanabe1, Yohei Iwasaki1, Kyosuke Tawara 1, Hiroki Nakanishi 1, 1 Saburo Matunaga 1 Tokyo Institute of Technology, Isikawadai 1st building, 2-12-1 I1-63 Ookayama, Meguro-ku, Tokyo, 152-8552, Japan, E-mail: [email protected] ABSTRACT In this paper, we describe numerical simulations and ground experiments of variable shape attitude control proposed by the authors to achieve both the agile maneuvering and the high pointing stability. In the numerical simulations, the optimal control theory is applied for the attitude control in the shortest time. In the ground experiments, an attitude control performance evaluation experiment is conducted in a Figure 1: Concept of Variable Shape Attitude Control two-dimensional microgravity environment using an air levitation device. Also the vibration measurement In the VSAC, the maneuver angle of the satellite is carried out and the influence on the attitude of the main body can be determined by driving the amount satellite on orbit is discussed. of the paddle angles, and it is also possible to control the attitude angular velocity of the body by the 1 INTRODUCTION driving speed of the paddle. Moreover, the attitude is Micro satellites occupy important positions in the settled at the same time as the driving paddle is field of space development as they can demonstrate stopped, and the disturbance of the rotating body advanced missions on orbit. The launch number of during the stationary state is small if there is no micro satellites are now increasing, and it is flexible part in the satellite. As a result, we can expected that the missions will be further advanced. expect compatibility between the agile maneuvering As the missions become more sophisticated, and the high pointing stability. However, the agility demands on bus systems become much severer. For and pointing stability might be mutually affected example, Matunaga Lab at Tokyo Tech is from not only the adopted control algorithm but also considering an astronomy satellite called "Hibari"[1]. the adopted drive mechanism. Thus, it is necessary Its missions are “position determination and optical to simulate the behavior on orbit by ground observation of gravitational wave sources”, and experiments using the actual drive mechanism and accordingly, “compatibility between agile to evaluate the influence on agility and pointing maneuvering and high pointing stability” is required. stability. For micro satellites, power, volume and mass are In this paper, first, we explain about results of severely restricted and it is difficult to mount all numerical simulations using the VSAC to achieve necessary components. both the agile maneuvering and the high pointing Thus, the authors proposed a Variable Shape stability. Next, we introduce a set-up of two- Attitude Control (VSAC) as shown in Fig.1 [2-6]. dimensional ground experiment using an air The attitude of a satellite body is changed by the levitation device. Finally, we describe results of reaction to rotate the solar paddles. Although some ground experiments for the VSAC to take the studies on attitude variation by rotating robot arms vibration-induced effect into the consideration. and solar paddles were conducted [7-8], few studies 2 NUMERICAL SIMULATION have been applied to a practical satellite attitude control by arm drive, and almost no demonstration 2.1 Formulation was shown on orbit in the practical mission. In this section, the fundamental principles of the VSAC are briefly described as applied to the attitude 푛−1 control of a rigid spacecraft. The objective here is to = −푰−1 ∑{풓 × 푚 (풓̇ + 흎 × 풓 ) present a simple mathematical model of a satellite 푘 푘 푘 0 푘 system equipped the VSAC. We formulate a 푘=0 relationship between the angular velocity of each +푱푘흎푘/0} body and the rotational speed of motors [3]. The 푇 nomenclature is described in the Appendix. Because 풓푘 = 풇푘(푞1, 푞2, … , 푞푛−1) = [푓푘1 푓푘2 푓푘3] , 풓̇ 푘 is obtained as First, we consider a multibody system consisted of 푛 bodies, and the main body has a number 0 and 휕풇푘 풓̇ 푘 = 흃̇ (6) appendages has numbers 푖 (푖 = 1, 2, … , 푛 − 1) . 휕흃 Then, the total angular momentum 푯c about the 휕풇푘 where is the Jacobian matrix of 풇푘 respect to 흃 mass center of the system is written below and 푯c is 휕흃 푇 conserved when the resultant external torque equals and 흃 = [휉1 휉2 ⋯ 휉푛−1] . Additionally, the zero. angular velocity of the body 푘 are decomposed by 푛 휉푘̇ at hinge ℎ푘 as 푁푑풑 푯 = ∑{(풑 − 풑 ) × 푚 푐푘 c 푐푘 푐 푘 푑푡 (1) 흎 = ∑ 흎 = ∑ 휉̇ 풆 = 훀 흃̇ 푘=0 푘/0 푗/푗 푗 푗 푘/0 (7) 푗∈푘̂ 푗∈푘̂ +푱푘흎푘} By considering Eqs. 5-7, 흎 is function of 흃. Thus, The time derivative of the vector (ex: 풂 = 0 we can determine the angular velocity of the satellite [푎 푎 푎 ]푇) with respect to 퐵 frame and 푁 frame 1 2 3 푘 main body (body 0) caused by driving the is represented as shown in Eq. 2. appendages using Eq. 4 ,the position, the velocity, 푁 퐵 푑풂 푘푑풂 and the angular velocity of each bodies 풑푘, 풑̇ 푘, 흎푘, = + 흎푘/푛 × 풂 respectively. In addition, the quaternion kinematic 푑푡 푑푡 (2) differential equation is introduced for the main body ≡ 풂̇ + 흎푘/푛 × 풂 attitude. 푇 푞̇ 0 휔 −휔 휔 푞 where 흎푘/푛 = [휔1 휔2 휔3] is an angular velocity 1 3 2 1 1 1 vector of body 푘 with respect to 푁 frame. The cross 푞̇2 −휔3 0 휔1 휔2 푞2 [ ] = [ ] [푞 ] (8) product of two vectors is presented with the matrix 푞̇3 2 휔2 −휔1 0 휔3 3 푞 notation as follows. 푞̇4 −휔1 −휔2 −휔3 0 4 Additionally, the commanded attitude quaternion 0 −휔3 휔2 푎1 풒 = [푞 푞 푞 푞 ]푇 and the current attitude 흎푘/푛 × 풂 = [ 휔3 0 −휔1] [푎2] 푐 푐1 푐2 푐3 푐4 [ ]푇 −휔2 휔1 0 푎3 (3) quaternion 풒 = 푞1 푞2 푞3 푞4 are related to the 푇 attitude error quaternion 풔 = [푠1 푠2 푠3 푠4] , as ≡ 흎̃ 푘/푛풂 following. Here to simplify the discussion, suppose that 푯푐 = 푠1 푞푐4 푞푐3 −푞푐2 −푞푐1 푞1 0 at the initial time 푡 = 0 . We derive the main 푠 −푞 푞 푞 −푞 푞 푖 [ 2] = [ 푐3 푐4 푐1 푐2] [ 2] (9) body’s angular velocity from Eq. 1 as following. 푠3 푞푐2 −푞푐1 푞푐4 −푞푐3 푞3 푠4 푞푐1 푞푐2 푞푐3 푞푐4 푞4 푛−1 푁 −1 푑풑푐푘 흎0 = −푰 ∑{(풑푐푘 − 풑푐) × 푚푘 Next, we describe a VSAC steering logic to 푑푡 푘=0 (4) calculate 흃 for the desired 흎0. Eq. (5) is rearranged as follows. +푱푘흎푘/0} 푛−1 푛−1 where 푰 = ∑푘=0 푱푘. (푰 − ∑ 푚푘풓̃푘풓̃푘) 흎0 푁푑풑 Note that ∑푛−1(풑 − 풑 ) × 푚 푐 = 0 by 푘=0 푘=0 푐푘 푐 푘 푑푡 푛−1 푛−1 considering the definition of the mass center. Thus, 휕풇 (10) = − {∑ 푚 풓̃ 푘 + ∑ 푱 휴 } 흃̇ Eq. 4 is rewritten as follows. 푘 푘 휕흃 푘 푘/0 푘=0 푘=0 푛−1 푁푑풓 −1 푘 = 푷흃̇ 흎0 = −푰 ∑{풓푘 × 푚푘 푑푡 (5) 푘=0 By using 푷+ as the Moore-Penrose pseudoinverse + 푱푘흎푘/0} matrix of 푷, Eq. (10) is rearranged as follows. 푛−1 + 흃̇ = 푷 (푰 − ∑ 푚푘풓̃푘풓̃푘) 흎0 (11) 푘=0 If we command the desired angular velocity vector • No external torque of the satellite main body, the rotational speed of the • Maximum paddle rotation speed : 10 deg/s hinge 흃̇ can be computed by Eq. (11). The target attitude maneuver angle is set to 60 deg Next, we describe a simulation model of the satellite with an azimuth angle of 20 deg as shown in Fig. 3. system considered in this paper. The system with variable shape function is as shown in Fig.2, which consists of 4 paddles and a satellite main body. Each body is modeled as shown in Table 1. Figure 3: Target attitude angle 2.2.2 Simulation Result Results of the satellite angular velocity, paddle angles and the attitude error angles obtained by the simulation are shown in Fig. 4. Figure 2: Simulation model Table 1: Models of rigid body (a) satellite angular velocity 2.2 1-axis Rotation In this section, consider the case of the 1-dof in motion where each paddle can rotate only on the x- axis in the configuration of Fig. 2. 2.2.1 Simulation Conditions (b) paddle angle In this paper, we apply an optimal control theory to the attitude maneuver of the satellite with variable shape function, and we consider the shortest time attitude maneuver problem. As a method to solve the optimal control problem, we adopt Direct Collocation with Nonlinear Programming (DCNLP) which is a kind of direct method for numerically obtaining optimum orbital solution (Trajectory Optimization). (c) error angle The following simulation conditions are set. Figure 4: Satellite angular velocity, paddle angle and • No initial angular momentum attitude error angle The graph shows that it takes 33s to maneuver the attitude. In the case of the attitude control introducing the optimal control, when the attitude maneuver angle is 45 degrees or less, it is possible to maneuver the attitude with high rapidness of about 6s to 7s. However, due to the limitation of the driving angle of the paddles, the attitude maneuver angle is restricted, and in the case of the paper, the attitude maneuver time greatly increases when the attitude change angle is larger than 45 deg. (a) satellite angular velocity In order to satisfy the sophisticated demands, the agile maneuvering is required at every attitude change angle.
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