Parameter Estimation of Solar Radiation Pressure Torque of IKAROS

Transcription

1 Parameter Estimation of Solar Radiation Pressure orque of IKAROS Yuya Mimasu 1), Ryu Funase ), akanao Saiki ), Yuichi suda ), and Jun ichiro Kawaguchi ) 1) Department of Aeronautics and Astronautics, Kyushu University, Fukuoka, Japan ) JAXA Space Exploration Center, Japan Aerospace Exploration Agency, Sagamihara, Japan Abstract Japan Aerospace Exploration Agency (JAXA) have operated the small demonstration solar sail spacecraft IKAROS (Interplanetary Kite-craft Accelerated by Radiation Of the Sun), which was launched in May, 1. he main objective of this spacecraft is to deploy the m class sail membrane, and demonstrate to accelerate the spacecraft by the solar radiation pressure induced by that sail. In order to model the photon acceleration induced by this sail membrane, we approaches by two sequences. he first one is to model based on the several parameters obtained from the on-ground experiments and the shape model calculated by the numerical method. he other one is to estimate the parameters of SRP model by analyzing the dynamics in orbit with the on-orbit attitude data. In this paper, we present the latter approach to obtain the parameters to construct the photon torque model mainly focusing on the analysis of the attitude dynamics. IKAROS における太陽輻射圧トルクパラメータの推定 三桝裕也 1), 船瀬龍 ), 佐伯孝尚 ), 津田雄一 ), 川口淳一郎 ) 1) 九州大学大学院工学府航空宇宙工学専攻宇宙航空研究開発機構月 惑星探査プログラムグループ ) 摘要宇宙航空開発機構 (JAXA) では現在, 小型ソーラー電力セイル IKAROS (Interplanetary Kite-craft Accelerated by Radiation Of the Sun) を 1 年 5 月に打ち上げ後, 定常運用を行っている. 本探査機の主目的は,m 級の膜を宇宙空間で展開し, それにより発生する太陽輻射圧を用いた加速実証である.IKAROS における太陽輻射圧による光子加速度のモデル化では, 地上での試験等をベースに, 膜の物性値や数値計算による形状モデル等から加速度を求める手法と, 軌道上でのダイナミクスを解析することによりパラメータを推定する手法の つのアプローチを行うことになる. 本論文においては, 軌道上でのデータを用いた姿勢ダイナミクスの解析から光子トルクモデルに必要なパラメータを推定する手法に関する研究を紹介する. 1. Introduction 1.1 Background he propulsion system using the Solar Radiation Pressure (SRP) force is called as the solar sailing system. In this system, the propellant is not needed to obtain the thrust force. his feature has motivated people to research this system and it has been supposed to widely spread the deep space exploration 1). Although a lot of research for the solar sail spacecraft had been done all over the world, there had been no actual flight solar sailing spacecraft in the history. evertheless, the Japan Aerospace Exploration Agency (JAXA) succeeded the deployment of the membrane in the space with the S experiment in 4 ). Also JAXA launched the world s first demonstration spacecraft for the solar power sail craft named IKAROS (Interplanetary Kite-craft Accelerated by Radiation Of the Sun). his spacecraft has demonstrated already several technologies mainly; the photon propulsion, thin film solar power generation system and the deployment of the sail membrane. he IKAROS spacecraft deploys and spans -meter class membrane by taking the advantage of the spin centrifugal force. he spacecraft weighs approximately 31 kg, was launched together with the agency s Venus Climate Orbiter, PLAE-C in May 1, 1. Both spacecrafts are boosted by the H-IIA vehicle directly onto the transfer orbit bound for the Venus. he sail spacecraft controls its attitude orientation to demonstrate managing photon acceleration in accordance with Fig.1 Solar sail spacecrafts JAXA launched and plan to launch the guidance strategy. 1

2 1. Objective and Motivation his spacecraft aims at the technical demonstration to generate the acceleration induced by the Solar Radiation Pressure (SRP). One of the most significant tasks to achieve this solar sail mission, is to estimate the thrust force and the torque induced by the photon, namely to establish the precise SRP model in the view point of a navigation. he estimation and modeling of the SRP force for this spacecraft from the orbit dynamics is already studied 3). In addition to this force model, IKAROS project team initialized to establish also the SRP torque model. his study may contribute to make the acceleration model more precisely or compensate it by estimating the common parameters between the force and torque such as the parameters to express the shape of the sail membrane.. Overview of IKAROS he solar sail demonstration spacecraft IKAROS investigated in JAXA has the square sail membrane. IKAROS is a spinning spacecraft and has the simple cylindrical body. It carries a drum around which a membrane is wound to be re-wound via a special mechanics aboard. he sail of IKAROS consist of the 7.5 [mm] Polyimide evaporated 8 [nm] aluminum, and there are several devices on the sail; i.e., the flexible solar array, Reflectivity Control Devices (RCD), dust counter, and so on. he attitude is estimated by the measurement Sun angle, Earth angle and the spin rate calculated by the Sun pulse timing of the Sun sensor. he attitude control is nominally performed by the Reaction Control System (RCS) by using the gas-liquid equilibrium thrusters 4). he main specifications of the IKAROS spacecraft related to the attitude estimation are summarized in able 1. able 1. Main specifications of IKAROS Description Values Moment of Inertia (Ix, Iy, Iz) = (434, 434, 868) [kgm ] Error of Sun Angle.15 [deg] (1) Error of Earth Angle.1 [deg] (1) Error of Spin Rate 1.e-4 [rpm] (1) 3. Estimation Strategy he photon acceleration model should be calibrated by the on-orbit data to use for the navigation of the solar sail. he acceleration of the spacecraft in orbit is mainly given by the Orbit Determination (OD) process 3) with estimating the position and the velocity. In addition to this force model, the IKAROS project aims to establish the SRP torque model of IKAROS as well. he estimated torque parameters have several parameters to express the shape of the sail membrane. herefore, this study may give some knowledge to modeling the precise shape of the IKAROS membrane dynamics Attitude Determination of IKAROS he Attitude Determination (AD) of IKAROS is not required to be performed on board, because the attitude is controlled only by means of the Sun pulse timing. herefore, the AD is implemented on ground with the telemetry data of the Sun sensor down-linked from the spacecraft, and the Observation minus Calculation (O-C) data of the range rate. We can obtain the Earth angle by extracting the amplitude of the spin modulation of the range rate. hen, we can estimate the spacecraft s attitude from consideration of the geometry of the obtained two angles. he attitude direction of the spacecraft must be on the arc of both the Sun angle s and the Earth angle e on the surface of the unit sphere as illustrated in Fig.. cos A S, cos A E (1) - () s e Attitude Direction Sun Direction E S s Earth Direction Attitude Unit Sphere e Fig. Attitude solution using two arc-length where A denotes the attitude direction of the spacecraft, S denotes the Sun direction, and E denotes the Earth direction from the spacecraft. As shown in Fig., two attitude solutions are obtained, and the ambiguity of these two solutions are removed by a priori attitude in this study. By using these two angles described in Eqs. (1) and (), the measurement equation can be obtained. At first, we define the state vector x and measurement vector y as follow; x (,, ), s, cos e, obs ) y (cos (3) - (4) where is the out-of-orbit-plane angle with respect to the Sun direction, is the in-orbit-plane angle with respect to the Sun direction. he measurements at a point of time t i can be related to the states as follow; y i h(,, ) i εi [ H] i xi εi [ H] i[ ( t, ti )] x ε (5) i

3 1 h(,, ) i with orb orb orb A [ H] i E1 E E3 (6) xi xi 1 i where, A denotes the attitude direction vector and this can be defined as A = (-coscos, -cossin, sin) in the orbit reference frame (the components in the matrix is also defined in the orbit frame), [(t, t i )] denotes the State ransition Matrix (SM) 5), denotes the measurement error. he [H] i finally becomes 3 x 3 matrix. As described in Eq. (5), the SM which is constructed by based on the dynamics equation 6) is needed to propagate the state vector from epoch to the present time. herefore, we need consider the dynamics model, i.e., the kinematic equation of motion and Euler equation. In order to derive these equations for this study, we introduce the spin-free coordinate system here as blue axes shown in Fig. 3. he origin is defined as the center of mass of the spacecraft. he z-axis of this coordinate system corresponds to the spin axis direction. he x-axis of this coordinate system is rotated angle around y-axis of the inertial frame from the x-axis of the inertial frame. he y-axis is defined to be the right hand frame. In addition to this spin-free coordinate system, the Sun-pointing frame is introduced as the reference frame. In this frame, the z-axis corresponds to the Sun direction, the y-axis is the normal to the orbit plane, and the x-axis is defined to be the right hand coordinate system. he components of the kinematics equation defined between the Sun-pointing reference frame and the spin-free body fixed frame can be obtained as follow; d dt x y / cos s where denotes the elevation angle of the spin axis direction from the orbit plane, denotes the azimuth angle in the orbit plane from the Sun direction, and s denotes the longitudinal angular rate of the Sun direction in the orbit plane, x and y denote the angular rate components around the x and y axis of the spin-free body fixed frame. he Euler equation is defined between the Sun-pointing frame and the spin-free body fixed frame. he initial formulation can be expressed as follow; dh S / C ωsf H S / C (8) dt where H S/C denotes the spacecraft s angular momentum vector in the spin-free body fixed frame, SF denotes the angular rate vector defined between the Sun-pointing reference frame and the spin-free body fixed frame, denotes the perturbation torque. he angular momentum vector and angular rate vector can be expanded as follow; S / C (I x I y I S) H, Fig.3 Definition of coordinate systems SF ( x y ) (7) ω (9) where I denotes the moment of inertia around the transverse direction toward the spin axis, I S denotes the momentum inertia around the spin axis direction, denotes the spin rate around the spin axis. Here we note that the angular momentum vector is the absolute vector, whereas the angular rate is defined as the relative vector between the spin-free body fixed frame and the Sun-pointing reference frame. By using these components of each vector, the components of the Euler equation can be obtained as follow: x I S y / I x / I d y I Sx / I y / I (1) dt z / I S where x, y and z denote the components of the perturbation torque. In the case of the deep space mission, the dominant torque exerting to the spacecraft is the solar radiation pressure. herefore we only model the solar radiation pressure torque in the next section for the torque components in Euler equation. 3.. Solar Radiation Pressure orque Model Before the deployment of the sail, we assume that the SRP torque is induced only by the offset of the center of pressure from the center of mass in the direction of the spin axis. However, after the deployment of the sail, the observed attitude motion cannot be explained by this assumed torque. herefore, we re-model the SRP torque. he main reason why the attitude motion does not obey the predicted model is due to the asymmetry sail shape. his asymmetry is mainly caused by 3

4 the distortion and deflection from the flat plate model. If we consider that we use the SRP torque model in the guidance of IKAROS, it is easy to use when the torque model is formulated with the attitude angles ( and ) rather than Sun angle. herefore, at first we model the SRP torque with the angles and that express the attitude (spin axis) direction in the inertial frame (see Fig. 3). We start to model the sail shape from the small element of the membrane as shown in Fig. 4. he normal vector of the small element in the spin-free body fixed frame is described as follow; n SF ( ) (sin cos cos sin sin, sin sin cos sin cos, cos cos ) (11) Fig. 4 Geometry of small elements of membrane where, denotes the deflection angle, denotes the distortion angle, denotes the phase angle around the z-axis (= the spin axis). ote the small element of the membrane is fixed in the spinning body frame and the general small element is rotating around the spin axis in the spin-free body fixed frame, thus the position in the spin-free body fixed frame must be defined with the phase angle around the spin axis. he lever arm vector L in the spin-free body fixed frame is described as follow; body L ( ) ( R cos, R sin, h) (1) where R is the radial length of the small element from the center of mass, h denotes the offset from the center of mass to the spin axis direction. In order to consider the torque in the inertial frame, we transform these vectors with the transformation matrix as follow; cos sin sin sin cos [ ine / SF (, )] cos sin (13) sin cos sin cos cos where denotes the longitudinal angle and as shown in Fig. 5. he vectors n and L can be transformed from the spin-free body fixed frame to the inertial frame by using this transformation matrix. On the other hand, the Sun direction vector in the inertial frame can be defined as follow by using the Sun longitudinal angle s. ine (sin s,, cos s ) s (14) By using the inner and cross products of these vectors, the SRP torque elements due to the specular reflectivity, diffusive reflectivity and absorption in the inertial frame can be obtained as follow; 1 spe, ine PACspe ine ine ine ine d ( n s ) ( L n ) (15) 1 dif, ine PACdif ine ine ine ine d ( n s )( L n ) (16) 3 1 abs, ine PA Cabs Cdif ine ine ine ine d ( )( n s )( L s ) (17) where P denotes the solar radiation pressure, A denotes the effective area, C spe denotes the specular reflectivity, C dif denotes the diffusive reflectivity, C abs denotes the absorptivity. ote that the SRP torque in the spin-free body fixed frame should be averaged along to the spin rotation, thus the each torque element is integrated by the phase angle, and divided by. ow we obtain the torque model in the inertial frame with the attitude angles and. However, if we would use this torque model in the Euler equation defined in the spin-free fixed frame, we must transform these torque elements again to the spin-free body fixed frame. he transformed torque elements can be described with the transformation matrix defined in Eq. (13). Each torque element is the function of the and. ow, by using the relationship = + s, and assuming that the attitude angles and can be approximated in second order, and also shape angles and can be approximated in first order, the torque component can be expanded as follow 7) ; k1 k SF [ ine / SF ] ine P k1 k (18) k3 k4 1 1 h with k1 { Cspe Cdif ( Cabs Cdif )(1 )} ii Ai Ri (19) 3 R i1 1 1 k { Cspe Cdif ( Cabs Cdif )} ii Ai Ri () 3 i1 4

5 k 1 3 ( Cspe Cdif ) ii Ai Ri, k i i i 3 4 Cspe ( Cabs Cdif i A R )} i1 i1 { (1) () where subscript i denotes the index number of the small element. As we see in Eq. (18), it is seen that the z component exist due to the distortion angle, and it has the sensitivity with respect to square of the attitude angle. Each parameter has the different sensitivity with respect to the attitude angles or ; k 1 multiplied by, k multiplied by, k 3 does not multiplied by any angle, and k 4 multiplied by square of.by using this torque model, the torque parameters can be estimated in the attitude determination process. Although we can choose arbitrarily the estimated parameter from the elements in the parameters k 1, k, k 3 and k 4, the observability is not satisfied in the case when the estimated parameters are too much compared to the amount of measurements. herefore, we estimate k 1, k, k 3 and k 4 directly including the uncertainty for the composition elements in Eqs.(19) (). 4. Actual Flight Results 4.1 Estimation Periods As the one of estimation results during these periods, we show the results in the following period: July, 1 - July 7, 1 (5 days) Although the period length is middle in a number of coasting periods of IKAROS, there is spacecraft operation everyday in this period. In addition, the spacecraft was close to the Earth in these early days, so the telemetry data of the Sun angle is more than the later periods. In order to verify the adequacy of the attitude estimation, we implement the overlap analysis. his analysis is to compare the estimation result of the attitude during the overlapping period. For this nd period, we split up to the following periods with overlapping the middle period. 1 st Period: July, 1 - July 5, 1 (4 passes) nd Period: July 4, 1 - July 7, 1 (4 passes) 4. Estimation Results As the estimation result, we show the estimated attitude in this study (by Precise Attitude Determination System (PADS)) by comparing with the (Quasi-) Real ime Attitude Determination (RADS) results, and also residuals of each measurement; i.e., the Observed - Calculation (O-C) data of the Sun angle, Earth angle and spin rate. Figures 5 and 6 show the attitude determination results by the RADS and PADS. As shown in these figures, the random noise of the estimated attitude by PADS is filtered out. able 3 shows the estimated torque parameters in this period. Out-orbit-plane Sun Angle (phi) [deg RADS PADS In-orbit-plane Sun angle (psi) [deg] Fig. 5 Residual of Sun angle during 1 st period Out-orbit-plane Sun Angle (phi) [deg RADS PADS In-orbit-plane Sun angle (psi) [deg] Fig. 6 Residual of Sun angle during nd period able 3 Estimated parameters Parameters k 1 k k 3 k 4 Value Figures 7 ~ 9 show the results of the overlap analysis. Figure 7 shows the estimated in-orbit-plane Sun angle in each period. Figure 8 shows the estimated out-orbit-plane Sun angle in each period. From these two figures, the difference of each angle is shown in Fig. 9. In-orbit-plane Sun Angle (psi) [deg] Psi_7-5 Psi_ ime from Epoch [day] Fig. 7 Estimated in-orbit-plane Sun angle during periods Out-orbit-plane Sun Angle (phi) [deg] Phi_7-5 Phi_ ime from Epoch [day] Fig. 8 Estimated out-orbit-plane Sun angle during periods 5

6 From Figure 9, the maximum difference of is.5 [deg], and that of is.3 [deg], thus the total maximum difference of estimated attitude is.58 [deg]. As the cause of this difference, two reasons can be considered, i.e., the inconsistency of the SRP torque model and/or the low sensitivity of estimates to the measurement (measurement noise is too high). For the IKAROS case, the former reason is closer because of the uncertainty of the sail shape. In order to confirm this hypothesis, we introduce the results of the measurement residuals. Figures 1 and 11 show the Fig. 9 Difference of estimated attitude during overlapping period residuals of the Sun angle, Figs. 1 and 13 show the residuals of the Earth angle, and Figs. 14 and 15 show the residuals of the spin rate. Especially from the residuals of the Sun angle and spin rate measurements, it can be seen that the estimation is not performed adequately in the nd period. his results indicates the torque model somehow cannot follow the actual attitude motion. O-C of Sun Angle [deg] O-C of Earth Angle [deg] O-C of Spin Rate [rpm] ime from Epoch [day] Fig. 1 Residual of Sun angle during 1 st period ime from Epoch [day] Fig. 1 Residual of Earth angle during 1 st period ime from Epoch [day] Fig. 14 Residual of spin rate during 1 st period Difference [deg] O-C of Sun Angle [deg] O-C of Earth Angle [deg] O-C of Spin Rate [deg] ime from Epoch [day] Fig. 11 Residual of Sun angle during nd period ime from Epoch [day] Fig. 13 Residual of Earth angle during nd period ime from Epoch [day] Fig. 15 Residual of spin rate during nd period 5. Conclusions In this study, we present the original torque model for the attitude estimation by based on the actual flight results. By using this torque model, the attitude estimation is dramatically well performed during relatively long estimation period. In addition, we show the adequacy of the attitude determination by analyzing the overlapping periods. From results of this overlap analysis, it can be found that there is still un-modeled torque effect. As the future work, we make clear the cause of this degradation of the estimation adequacy. References 1) Colon, R. McInnes, Solar Sailing, echnology, Dynamics and Mission Applications, Springer-Praxis, ) suda, Y., Mori, O., akeuchi, S. and Kawaguchi, J. Flight Result and Analysis of Solar Sail Deployment Experiment using S-31 Sounding Rocket, Space echnol., 6 (6), pp ) Yamaguchi., et al. rajectory Analysis of Small Solar Sail Demonstration Spacecraft IKAROS Considering the Uncertainty of Solar Radiation Pressure, 9-d-59, ISS7, sukuba, 9. 4) Yamamoto,., Mori, O. and Kawaguchi, J., ew hruster System for Small Satellite: Gas-Liquid Equilibrium hruster, ransactions of the Japan Society for Aeronautical and Space Sciences, Space echnology Japan, Vol. 7 (9), ists6, pp.d_9-d_33, 9. 5) Montenbruck, O., and Gill, E., Satellite Orbits, Springer, Corrected 3rd Printing, 5, pp ) Wertz, J. R. (Editor), Spacecraft Attitude Determination and Control, Kluwer Academic Publishers, etherlands, 1978, pp ) 津田雄一, 佐伯孝尚, 三桝裕也, 船瀬龍, 山口智宏, 中宮賢樹, 白澤洋次, 池田人, IKAROS における姿勢軌道ダイナミク ス同定, 第 54 回宇宙科学技術連合講演会, 静岡,1 年,11 月 Difference_Psi Difference_Phi ime from Epoch [day] 6