Fuel-free Attitude Control of Bias-momentum Solar Sail
By Yuya MIMASU1), Go ONO1), Yuichi TSUDA1)
1)The Institute of Space and Astronautical Science, JAXA, Sagamihara, Japan (Received 1st Dec, 2016)
Solar sail is the fuel-free thrust system. Its trajectory can be changed without fuel by using the solar radiation pressure (SRP) induced by the sail. In general, however, the direction and the magnitude of the photon acceleration are controlled by the attitude of the sail, and the attitude control fuel is required. That is why, it is necessarily to reduce the fuel consumption of the attitude control in order to realize a real fuel-free solar sail. This research is one solution for this issue. The SRP is affected not only to the orbit, but also to the attitude of the spacecraft. This effect can be utilized to control the attitude of spacecraft. There is an appropriate example on Hayabusa2 case during coasting phase. The attitude dynamics model of the Hayabusa2 under the SRP had been studied. Especially, the attitude motion in the one-wheel bias-momentum control mode has been modeled in detail, and calibrated by using the actual flight data of Hayabusa2. The one-wheel bias-momentum control mode is the control mode in which the spacecraft is controlled by using only Z-axis reaction wheel. In this mode, the angular momentum vector precesses due to the SRP. According to the established model and flight result of Hayabusa2, the precession trajectory is changed by the body-phase angle with respect to the Sun direction. By utilizing this feature, the direction of angular momentum vector can be controlled by changing phase angle with respect to the Sun angle. The phase angle can be changed only by using one reaction wheel of Z-axis and usually unloading is not needed because just change the attitude orientation around Z-axis. It means that the spacecraft attitude can be controlled without fuel.
Key Words: Bias-momentum, Precession, Angular Momentum Control, Orbit Control
Nomenclature However, the attitude control of the solar sail needs fuel in general. Although there have been several study about fuel-
S0 : solar constant free attitude control system of solar sail, usually it needs the c : light speed extra mechanism or technology. In this paper, we propose that
RS/C : solar distance of spacecraft the fuel-free attitude control method for bias-momentum solar
RE : representative solar distance of the Earth sail without any new technology or additional mechanism. This method applies the solar radiation pressure torque to Cspe : specular coefficient control the spacecraft attitude, and firstly verified on the Cdif : diffusive coefficient cruise phase of the Hayabusa2 mission. Therefore, we Cabs : absorption coefficient introduce the attitude control method of Hayabusa2 probe at B : Lambertian coefficient f first. : thermal emissivity
s : Sun direction vector 2. Overview of Hayabusa2 n : normal vector of the effective area : right ascension in the inertial frame The main mission of the probe is to sample pieces of : declination in the inertial frame asteroid, and bring it back to the Earth in order to conduct : rotation angle around Z-axis of the body more advanced analysis on the ground. Hayabusa2 is planned I : moment of inertia tensor to arrive at the target asteroid in 2018, and return to the Earth : angular rate of the body-fixed frame in 2020 1, 2). with respect to the inertial frame During the cruise phase, Hayabusa2 controls its attitude by h : inertial angular momentum of the only one reaction wheel to bias the momentum around Z-axis reaction wheels of the body. This is to save the operating life of reaction st C1 : 1 integration constant wheels for other axes, because we experienced that two nd C2 : 2 integration constant reaction wheels of three equipped on Hayabusa were broken Subscripts after the touchdown mission. s : Sun direction 3. One-wheel attitude control mode 1. Introduction In the one wheel control mode, the angular momentum It is well known that solar sail is the fuel free trust system. direction is slowly moved in the inertial space (generally
1 called precession) due to the solar radiation torque. This Solar Array attitude motion caused by the balance of the total angular momentum and solar radiation pressure is known to trace the Center of Mass Sun direction automatically with ellipsoidal and spiral motion Sun Direction = s around Sun direction. Based on the knowledge in the past, the Angular Momentum Direction = L attitude dynamics model for Hayabusa2 mission had been Torque Direction 3) developed before the launch . According to the newly Center of ˆ Pressure T (L s) developed attitude dynamics model of Hayabusa2, the precession trajectory is almost the ellipsoid around the attitude Fig. 2 SRP torque direction equilibrium point, and this equilibrium point is determined mainly by the phase angle around Z-axis of the body. In Hayabusa and IKAROS mission4,5), this attitude motion In the actual operation of Hayabusa2, the spacecraft already was actually observed in the flight operation, and we have experience the one wheel control mode, and the attitude accumulated the experience and knowledge of the attitude motion in this mode is almost corresponds to the expected dynamics under the solar radiation pressure. Based on the motion based on the dynamics model developed before the knowledge in the past, the attitude dynamics model for launch. The precession trajectory is ellipsoid around the Hayabusa2 mission had been developed before the launch3). equilibrium point, and the attitude dynamics model is verified The detail about the dynamics is introduced in the section 5. by the actual flight data. In this one wheel operation, the In general three axis control operation, Hayabusa2 should Sun-aspect angle is restricted within a certain limit angle in follow the Sun direction in order to keep the Sun aspect angle terms of the thermal condition of the spacecraft. Because the within a certain restriction determined from the thermal precession radius is determined by the initial attitude and the condition. It takes fuel to keep Sun aspect angle because the equilibrium point, the Sun-aspect angle almost exceed the Sun direction automatically moves about 1 degree/day due to limit angle due to the precession without change of the the orbit motion. However, by using the attitude motion due to equilibrium point. At this operation, we execute the attitude the SRP, the angular momentum vector can trace the Sun maneuver around Z-axis to change the equilibrium point in direction automatically and fuel free to keep the Sun-aspect order to reduce the Sun-aspect angle and succeeded. After that, angle. The attitude motion in the inertial frame and we execute the maneuver again to change the equilibrium Sun-pointing frame is illustrated in Fig. 3. As shown in Fig.3, point to close point in order to make the small precession the angular momentum makes circle trajectory below the Sun trajectory. direction around the equilibrium point in the Sun-pointing frame. 4. Sun-direction-tracking mode ◆ Inertial Frame ◆ Sun-pointing Frame
During the cruise phase, Hayabusa2 controls its attitude Orbit Plane Equilibrium only by one reaction wheel to bias the angular momentum Direction around Z-axis of the body. There are two main reasons: ・ To save the operating life of reaction wheels for other axes Precession ・ To save the fuel consumption. Angular Momentum Direction First reason is from the redundancy concept learned from Fig. 3 Sun tracking motion in inertial frame (left) and Sun-pointing Hayabusa experience. The second reason is related to utilize frame (right) the Solar Radiation Pressure (SRP). In this one wheel control mode, the angular momentum direction is slowly moved in the In the actual operation, we should consider about the transition inertial space (generally called precession) due to the SRP of the control mode. The 3-axis attitude of Hayabusa2 is torque. This attitude motion caused by the balance of the total nominally controlled by three RW’s as bias-momentum. Thus, the angular momentum and SRP is known to trace the Sun momentum of the X and Y axis should coast down before direction automatically under the appropriate condition spacecraft transits to the OWC mode. If the momentums are between SRP torque and angular momentum. The schematic coasted down without control, however, the reaction torque of the Sun tracking motion is illustrated in Fg.1 and the affects the spacecraft attitude as the disturbance and the attitude geometry of the angular momentum vector and the SRP torque starts tumbling. In order to avoid this, the attitude control mode is direction is shown in Fig.2. firstly transit to the 3-axis control mode by the thrusters called y Sun-Pointing Frame R3AX (RCS three-axis) control mode. In this R3AX mode, the S/C Fixed Frame y thrusters are ignited when the attitude or the angular rate of the Orbit spacecraft are over the limits of the state (few degrees for the Revolution x Direction attitude and few 0.1 degree/sec for the angular rate). Therefore,
x the attitude is kept by thrusters when the RW’s are coasted down, z Sun H and after that the control mode transits to the OWC mode. Indeed, 0 z there are few degrees residual angle error and few 0.1 degree/sec residual angular rate, so the initial orientation of the angular SRP torque momentum vector of RW-Z is affected by these residual states Fig. 1 Sun tracking motion just after the transition to the OWC mode.
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5. Attitude dynamics equations of solar sailing mode Substituting Eq. (4) into (9), and solve about and , the analytical solution can be derived as follow:
It is known that the attitude dynamics of the spacecraft in the DH 3 2 2 2 2 t {M (D H) M} N P M NP C e 2hz cos t C tan1 deep space is dominated mainly by the SRP in general. eq 1 2 2 3 2 N 2hz M (D H) M Hayabusa2 is no exception. We start from the general formulation of SRP force is described as follow: (10)
DH 2 t M S R 2hz 0 S / C eq C1e sin t C2 f [ s n (Cabs Cdif )s (s n){Bf C dif C abs 2 C spe s n } n ]dA (1) (11) c R 2hz E The Sun direction vector is described as follow: M, N and P are defined as follows:
2 2 2 2 2 2 sins coss (cos cos sin sin sin ) coss cos s (sin cos cos sin sin ) sins cos sin 3D E G 3H 2DH 6EG 2(H D)(E G)sin 4 {(D H) (E G) }cos 4 M s sin cos (cos sin sin sin cos ) cos cos (sin sin cos sin cos ) sin cos cos s s s s s 2 sin cos sin cos cos cos cos cos sin sin s s s s s (12) (2) N 4(DH EG){2G (D H)sin 2 2 (E G )sin 2 } (13) Assuming the Z-axis of the body-fixed frame is pointing close (D H)cos2 (E G)sin2 to the Sun direction and the equatorial plane, Eq. (2) can be P (14) G E (D H)sin2 (E G)cos2 reduced:
cos sin ( s ) Also the solution for equilibrium point is obtained: s sin cos ( ) hz s (3) eq s [{D H (E G)sin2 (H D)cos2}s 2(DH EG) 1 1 (DI FG)sin (EI FH)cos By using Eq. (3) and the normal vector n, the SRP torque {G E (H D)sin2 (E G)cos2}s ] DH EG formulation can be obtained. Although the normal vector n is (15) derived from the integration of the local body shape, we skip hz eq s [{E G(H D)sin2 (E G)cos2}s the detail explanation here because the formulation is very 2(DH EG) complex in the strict expression3). As the result of Eqs. (1) and (EI FH)sin (DI FG)cos {D H (E G)sin2 (D H)cos2}s ] (3) and the appropriate form of the normal vector n, the SRP DH EG torque can be formulated as follow: (16) According to Eqs. (10) - (14), the precession trajectory can be T D E F cos sin x s ellipsoid around the attitude equilibrium point and there is also T G H I sin cos y s (4) divergent or convergent feature due to the exponential term. Tz J K L 1 1 From Eqs. (15) and (16), the equilibrium point is determined by 6 SRP parameters (D~I), and the phase angle around Z-axis where D~L are the original SRP parameters which are of the body with respect to the Sun direction. In the left figure invented in Ref. 3). Euler equation is described as follow in of Fig. 4, the converged ellipsoidal case of precession the general form: trajectory is plotted based on the analytical solution (Iω h) ω(Iωh) T (5) introduced as Eqs. (10) ~ (16). The trajectory is dependent on In the OWC mode, each vector becomes: the SRP-9 parameters. The SRP parameters are determined from the relative position between center of mass and pressure, I XX x the optical properties of the exposed area, local shape of the I I , ω , h YY y (6) spacecraft, and shadow. It means that the SRP parameters can IZZ z hz be different by spacecraft and the precession trajectory is also different as that result. The 9 parameters for Fig. 4 are
Substituting Eq. (6) into (5), we neglect z and the second determined by the on-ground FEM model of Hayabusa2 based order terms of x and y. In addition, if we assume that the on the parameters of design as the most probable values. time dependency of the precession is enough small compared 10 SUN 10 to the nutation, we can also neglect and : CASE‐1 ×：Equilibrium Point x y CASE‐2 CASE‐3 5 CASE‐4 5 Ty Tx
x y (7) [deg]
h h s z z 0 0 s[deg] 6) δ ‐ δ The kinematics equations for X and Y components can be δ‐δ 太陽相対赤緯 ‐5 derived as follow in the case when the is small: ‐5 TLM_DATA SIMULATION SUN sin cos sin cos d x y x y ‐10 ORBIT PLANE ‐10 cos cos (8) ‐10 ‐50 510 cos sin ‐10 ‐50 510 dt cos sin x y 太陽相対赤経α‐ αs [deg] x y α‐αs[deg] Substituting Eq. (8) into (7), the differential equation for the Fig. 4 Precession trajectory (left) and actual flight result (right) and becomes: 6. Control of precession d 1 cos sin Tx (9) T dt hz sin cos y According to Eqs. (15) and (16), the equilibrium point of the
3 precession trajectory is determined by the phase angle around Z-axis. This is because the equilibrium point direction Applying this attitude dynamics to the actual operation, we is almost uniquely determined by the relative position vector keep the +Z-axis direction within the attitude restriction, i.e., between the center of mass and pressure, and its direction in 11 degree limited by the thermal condition. In the case of June the Sun-pointing frame can be changed by the orientation of in Fig. 7, the equilibrium point is controlled to move close to the rotation angle around Z-axis as shown in the left figure in the attitude point. As the result, the precession radius can be Fig. 5. The plot of equilibrium points derived analytically is gradually reduced. This motion is induced only by the SRP shown in the right figure in Fig. 5. torque. The attitude motion is very sensitive even to the small In Hayabusa2 case, both of the center of mass and the center difference of the local shape of the spacecraft because the of photon pressure shift from the geometric center of attitude motion itself is already very small phenomenon. In spacecraft to the other directions. The relative geometry of contrast, the attitude in Fig. 8 is gradually converged in the these two centers produces the bias of the solar radiation different equilibrium point. In both case, the attitude control torque even in the case that the body Z-axis points to the Sun by changing the equilibrium point succeed to reduce the direction. As a result of the bias of the solar radiation torque, precession radius. the equilibrium point is changed due to the phase angle around 10 the body Z-axis illustrated in Fig. 5. SUN June 12 Maneuver Estimated Equilibrium Points (-30deg) Estimated Present Equilibrium Point CASE‐3 X CASE‐2 Orbit Plane 5 June 30 Maneuver Y X (-20deg)
Y Orbit Plane 0
s[deg] Start: June 9
Trajectory of equilibrium point δ‐δ due to the phase angle Y CASE‐1 CASE‐4 ‐5 June 15 Maneuver X Y (+40deg) June 25 Maneuver (+10deg) X ‐10 ‐10 ‐50 510 Fig. 5 Attitude control strategy by changing phase angle with respect to α‐αs[deg] the Sun direction Fig. 7 Flight result of the attitude control in June 2015 From this physical property, we noticed the fact that the precession trajectory can be changed, if we change the Jan. 22 Maneuver equilibrium point by changing the phase angle around (-30deg) Jan. 24 Maneuver Z-axis at certain timing. If we go back to the flight result (+30deg) introduced in section 5 (also shown in right figure of Fig. 6, and simulate to change the equilibrium point at the cross point ] where the precession trajectory and line of equilibrium points g due to the phase angle , it is clearly shown that the s [de δ - precession trajectory is changed due to the change of the δ equilibrium point in the right figure in Fig. 6. This fact indicates that the precession trajectory can be controlled only Jan. 19 Maneuver Feb. 19 Maneuver by switching the equilibrium point direction by changing the (+15deg) (-30deg) phase angle. α-αs [deg] 10 Fig. 8 Flight result of the attitude control in Jan. – March 2016
7. Solar Sailing Hayabusa2 5 The trajectory during this one wheel control mode is Equilibrium Point Change estimated by the orbit determination team every one week in Hayabusa2 project. In this coasting period, the trajectory is 0 s[deg] changed only by the SRP. It means that Hayabusa2 is solar δ‐δ sailing in this period. In order to evaluate the orbit difference TLM_DATA due to the different attitude, trajectories of two periods are ‐5 SIMULATION shown in Fig. 9 and 10. In these figures, the coordinate frame SUN is defined as the Sun-probe fixed frame. The –X direction ORBIT PLANE ‐10 always points to the Sun direction, and +Y direction is always ‐10 ‐50 510 along-track direction of the ballistic trajectory. The relative α‐αs[deg] trajectory is derived by substituting the ballistic trajectory Fig. 6 Simulation result of changing equilibrium point from the actual or simulated trajectory at each time point.
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Thus, the origin of this frame is the position of the ballistic trajectory. 8. Conclusion In the period of Fig. 9, the spacecraft is decelerated because the normal direction to the solar array paddle points to the –X The novel attitude control scheme by controlling the direction as shown in Fig. 7. Therefore, the actual orbit equilibrium point of the precession trajectory is presented. determination result is always delayed from the Sun-pointing This scheme is based on the SRP model which was invented attitude case. On the other hand, in the period of Fig. 10, the before launch of Hayabusa23). According to the analytical spacecraft is accelerated due to the attitude orientation shown solution of this SRP model, the equilibrium point is in Fig. 8. Thus, the actual orbit is always preceded from the determined by the phase angle around the Z-axis of the body. Sun-pointing attitude case. The blue and green trajectories are By utilizing this attitude dynamics property, the precession accelerated by the solar radiation pressure, that is why, the trajectory can be controlled by controlling the phase angle trajectory normally shifted to the opposite direction to the Sun within the attitude restriction and succeed to reduce the direction, and the orbit radius is expanded. As the result of precession radius. As the result, the utility of this control expansion of the orbit radius, the trajectories delayed from the scheme is verified in the actual flight operation. This control ballistic trajectory. scheme has possibility to be applied to the future spacecraft Ballistic Trajectory and potentially even to the operational spacecraft which has more than one RW under the SRP. In the coasting operation of Hayabusa2 probe, its trajectory is changed by the SRP following its attitude history under this precession control. Although it is very small change of the orbit, it is verified that this attitude control scheme can be applied to the solar sail,
Orbit Along-track and it indicates possibility to realize the real fuel free solar Direction sailing thrust system.
Sun Direction References 1) Kuninaka H. :Hayabusa2 Project, Deep Space Exploration of Hayabusa-2 Spacecraft, ISTS-2015-k-61, 30th International Fig. 9 Relative trajectory to the ballistic orbit during Aug. 12 - 19 Symposium on Space Technology and Science, Kobe, June 4-10, 2015.
Ballistic Trajectory 2) Tsuda Y., Yoshikawa M., Abe, M., Minamino H., Nakazawa S. :System Design of The Hayabusa 2 – Asteroid Sample Return Mission To 1999 JU3，ActaAstronautica, Vol.90, pp.356-362, doi: 10.1016/j.actaastro.2013.06.028, 2013. 3) Tsuda, Y., Ono, G., Akatsuka, K., Saiki, T., Mimasu, Y., Ogawa, N. and Terui, F.:Generalized Attitude Model for Momentum-biased Solar Sail Spacecraft, AAS Flight Mechanics Conference, AAS15-656, Vail, CO, USA, 2015. 4) Kawaguchi, J. and Shirakawa, K.: A Fuel-Free Sun-Tracking Orbit Along-track Attitude Control Strategy and the Flight Results in Hayabusa Direction (MUSES-C), AAS Flight Mechanics Conference, AAS07-176, Sedona, 2007. 5) Tsuda, Y., Saiki, T., Funase, R., Mimasu, Y.: Generalized Attitude Model for Spinning Solar Sail Spacecraft, AIAA Journal of Sun Direction Guidance, Control and Dynamics, Vol. 36, No. 4, pp. 967-974, 2013. 6) Wertz, J.R. Ed.: Spacecraft Attitude Determination and Control, Kluwer Academic Publishers, Boston, pp.765-766, 1978. Fig. 10 Relative trajectory to the ballistic orbit during Feb. 24 – March 2
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