On Spacecraft Magnetic Attitude Control

On Spacecraft Magnetic Attitude Control M. A. A. Desouky,∗ Kaushik Prabhu,† and Ossama Abdelkhalik‡ Michigan Technological University, Houghton, MI, 49931, USA

When using magnetorquers in attitude control system, satellites usually utilize magnetometers to measure the external magnetic field in the satellite environment, to compute the control command to the magnetorquers. This paper presents a magnetic control algorithm that eliminates the need for magnetometers in spacecraft attitude control with magnetorquers. During attitude maneuvers, the measured system response to torque commands can be used to estimate the magnetic field. A Kalman filter is used to estimate the magnetic field as well as the spacecraft angular velocities. The measurements are the spacecraft position and angular velocities. A model for the earth magnetic field is assumed available onboard the spacecraft. Model errors as well as measurements errors are simulated in this work. Simulation results show good attitude control performance.

Nomenclature PN ED PECEF PREF R ϕ λ Ix , Iy , Iz I ωx , ωy , ωz Tx , Ty , Tz β Tmag M b Mx , My , Mz Bx , By , Bz TC h˙ wy Td N S imag Xpredicted Xn−1 A C

Position in a NED, North East Down, system Corresponding ECEF, Earth-Centered Earth-Fixed, position The reference ECEF position (where the local tangent plane originates) Rotation Matrix for transformation from the ECEF to NED frame of reference Latitude corresponding to PREF Longitude corresponding to PREF Spacecraft moment of inertia in X,Y, and Z directions Spacecraft tensor matrix Spacecraft angular velocity in X,Y, and Z directions in body frame Applied torque on spacecraft in X,Y, and Z directions Quaternion vector Spacecraft applied torque vector from Magnetorquer Magnetic dipoles moment vector for the magnetorquers Magnetic field vector in body frame Magnetic dipoles for the magnetorquers in X,Y, and Z directions Magnetic field components in body reference frame in X,Y, and Z directions Spacecraft applied torque vector Reaction wheel angular moment Disturbance torque Number of turns in magnetorquer coil Magnetorquer coil cross section area Magnetorquer coil current Propagated state vector Previous state vector State transition matrix Control matrix

∗ PhD student, ME-EM department, 1400 Townsend Dr., 815 R. L. Smith ME-EM Building, Houghton, MI 49931, [email protected]. † Graduate student, ME-EM department, 1400 Townsend Dr., 815 R. L. Smith ME-EM Building, Houghton, MI 49931, [email protected]. ‡ Associate Professor, ME-EM department, 1400 Townsend Dr., 815 R. L. Smith ME-EM Building, Houghton, MI 49931, [email protected], AIAA senior member.

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Un Ppredicted Pn−1 Q Zn H R X Xn Pn kp kd eEuler e˙ Euler

Control vector Propagated estimate of the average error for exact part of the state Previous average error for exact part of the state Estimated process error covariance Measurement vector Observation matrix for translating the state to the measurement domain Estimated measurement error covariance State vector Newest estimate of the current true sate Newest estimate of the average error for exact part of the state Controller proportional gains vector Controller derivative gains vector Euler angles error Derivative error of Euler anglers

I.

Introduction

Magnetorquers are attractive for their reliability, lightweight, and energy efficiency that turns out that they are suitable in practice in low earth orbit (LEO) satellites,1, 2 . They are widely used for satellite attitude problem,3−12 . Magnetorquers operate on the basis of the interaction between current-driven magnetic coils and the magnetic field of the earth,1, 3, 4 to generate torques onboard a satellite. Magnetometers have been usually used for measuring the external magnetic field, whenever the magnetic coils are used for magnetic attitude control. e.g., Astolfi and Lovera,5 attempted attitude tracking using magnetic actuators and magnetometers, without using rate feedback. The attitude problem is approached by using a low-gain proportional-derivative-like control law to prove the global asymptotic stability of the system. Sugimura et al.,6 used only magnetometers for attitude estimation. Wang et al. approach the magnetic attitude control problem in two steps. First, designing an outer loop within the nonlinear periodic framework using back stepping for virtual control. Second, designing an inner loop for detumbling control and attitude acquisition to track virtual signal using Sliding Mode Controller. The attitude parameters are linearized around the origin to prove the closed loop linearity of the system and thereby guaranteeing the local asymptotic stability according to Floquets Theorems. The saturation is also taken into account in the control torque and good results have been established in terms of performance of the controller for an iso-inertial spacecraft in the Lower Earth Orbit (LEO) where the magnetometer measurements are fundamental part of the proposed methods,2 . Lovera and Astolfi presented a solution based on static attitude with rate feedback and dynamic attitude feedback using magnetic actuators and sensors. A global solution is guaranteed for the former in the case of static attitude and rate feedback with the assumption that the quaternion vector balance lies around zero target attitudes. This is accomplished by extending the PD-like Control Law with the introduction of position and velocity error constants and using the Lyapunov Stability criterion with appropriate scaling. Whereas, the latter yields an almost global solution only in the case of iso-inertial spacecraft. This is demonstrated with the assumption that the inertial magnetic field transformation matrix having values less than unity,7 . Silani and Lovera investigated various linear and nonlinear attitude control methods. The former summarizes classical control, optimal periodic control and robust control methods while the latter is explored from the Lyapunov Stability perspective. It offers a new approach towards pursuing the attitude control problem using only magnetic actuators based on prediction of parameters in discrete time intervals. This is accomplished by assuming the knowledge of the magnetic field and computing a feasible projected torque in an optimal sense,1 . Gerhardt and Palo derived their control strategy depending on the previous and current time step magnetic fields, which are used to determine the induced magnetism in the rod,8 . Humphreys et al. present a magnetometer-based filter and smoother for estimating attitude, rate, and boom orientations for a spinning spacecraft that has wire booms. The estimator is initialized with the measured angular rate of each sub-payload at ejection and thereafter relies solely on three-axis magnetometer data,9 . Ivanov et al. solved spacecraft attitude control and determination problem with three magnetorquers and three-axis magnetometer. They presented a control scheme with inertia tensor uncertainty and unknown natural disturbance,10 . Weng et al.,11 and Siahpush,12 provide control schemes for spacecraft magnetic attitude control


using three-axis magnetometer. The literature has many studies employing the magnetometer measurements for regulating the spacecraft attitude. However, each three-axis magnetometer measurement provides two axes of attitude information. Therefore, an Euler model for propagation of the spacecraft dynamics between measurements is devoted to solve the lack of attitude estimation for single frame,9 . This study aims at developing an algorithm that eliminates the need for magnetometers for magnetic attitude control, using only measurements of the vehicle motion such as the satellite angular velocities. The key idea of this algorithm is measurement of the system response to known control commands can be used to estimate the external magnetic field. The control command, in a magnetorquers setup attitude control system, consists of currents to the magnetorquers. The generated magnetic field in the magnetorquers, due to these control currents, interacts with the external magnetic field to produce the torque that drives the satellite. By measuring the response of the satellite (rotational motion) and given the control command, it is possible to estimate the external magnetic field relying on the spacecraft dynamics and control commands for the satellite attitude. The paper is organized as follows: in section II, the proposed attitude control system will be described. Section III, presents the coordinate frames, spacecraft rotational dynamics and kinematics, and actuator dynamics. Section IV is devoted to the magnetic field propagation model, magnetic field computation from the spacecraft motion, angular velocity, and controller commands, and the estimation process by the Kalman filter. Section V is devoted for control law, problem setup, and simulation results. Section VI concludes this study with the expected future work.

II.

Proposed attitude control system

Figure 1: Conceptual diagram of the proposed ACS Figure 1 depicts the conceptual diagram of the proposed attitude control system using magnetorquers without magnetometers. The onboard computer contains three main blocks. The first block is the magnetic field estimation block, where the spacecraft external magnetic field is computed from the controller commands and the spacecraft angular motion measurements using the spacecraft dynamics to estimate the torque experienced by the spacecraft. Another set of computed earth magnetic field values are obtained from the World Magnetic Model (WMM) using the orbit propagator with the spacecraft position measurements. The Kalman filter (KF) estimates the magnetic field from the first calculated magnetic field method, which is considered as the measurements inside KF and the second computed magnetic field, which is propagated using orbit propagator. The second block inside the onboard computer is the angular velocity estimation block. The estimated


earth magnetic field from the first block will be employed to compute the propagated angular velocity. Whereas the Gyroscope measurement will be considered as the measurement of the angular velocity inside the angular velocity Kalman filter. The estimated angular velocity from the Kalman filter is used to compute the current attitude, which is compared with the attitude command to generate a correction command where it is executed in the third block to issue this command to the attitude actuators.

III. A.

Spacecraft dynamics and kinematics

Definition of geometric frames

For spacecraft attitude control system, the following reference frames are used:

Figure 2: NED, ECEF, and ECI frames 1) Orbital reference frame. The origin of this frame is the satellite center of mass. The X-axis points to the Earths center, the Y-axis is perpendicular to the X-axis in the orbital plane in the velocity direction and the Z-axis is normal to the satellite orbit plane and completes the right- handed orthogonal triad 2) Satellite body frame. The origin of this frame is in the satellite center of mass and it is attached to the spacecraft 3) Earth-centered inertial frame (ECI). The origin of this frame is in the center of the Earth, Fig.2 A detailed description of these frames in,3, 4, 13 . 4) North East Down, NED, frame. The origin is the satellite center of gravity, the down axis is the direction to the earth center and the east is the local east direction,14 Fig.2 5) Earth-Centered Earth-Fixed, ECEF, frame. The origin is the center of mass of the earth. Its axes are aligned with the international reference pole (IRP) and international reference meridian (IRM) that are fixed with respect to the surface of the earth,15 Fig.2 NED and ECEF coordinates are related to each other according to the following relation: PN ED = RT (PECEF − PREF ) Where



−sinϕcosλ  R = −sinϕsinλ cosϕ

−sinλ cosλ 0

 −cosϕcosλ  −cosϕsinλ −cosλ

(1)

(2)

The NED and ECEF frames are used only for completing the transformation of the earth magnetic field vector from NED to body frame. 4 of 11 American Institute of Aeronautics and Astronautics

B.

Spacecraft rotational dynamics and kinematics

The satellite body frame is chosen to be the principal frame where the moment of inertia matrix I, is diagonal:   Ix 0 0   (3) I =  0 Iy 0  0 0 Iz The attitude dynamics of the rigid spacecraft is expressed using Euler’s equations as follows13 : Ix ω˙x = − (Iz − Iy ) ωy ωz + Tx

(4)

Iy ω˙y = − (Ix − Iz ) ωx ωz + Ty

(5)

Iz ω˙z = − (Iy − Ix ) ωy ωx + Tz 3

(6) 3

Where ω ∈ R is the vector of the spacecraft angular velocity. T ∈ R is the applied torque vector on spacecraft. The spacecraft attitude will be expressed by four Euler parameters, the quaternions, as they do not suffer from the singularity problem that is found in the three Euler angles representation; moreover, they are less computationally expensive than the direction cosine matrix,13 . Hence, the kinematics is represented as follows:      0 −ωx −ωy −ωz β0 β˙0   ˙ 1 0 ωz −ωy  β1  ωx β1  (7)    ˙ =  β2  2 ωy −ωz 0 ωx  β2  ωz ωy −ωx 0 β3 β˙3 Where β ∈ R4 is the quaternions, β0 represent the quaternion scaler value and β1−3 are the quaternions vector components. C.

Spacecraft Attitude control actuator

There are different choices of attitude control actuators for small spacecraft in LEO. However, here magnetorquers used to generate the external torque. The magnetic attitude control torques are generated by a set of three magnetic coils according to the following law. Tmag = M × b However, Eq. (8) can be rewritten in matrix form as follows:  0 Bz  Tmag = B (b) M = −Bz 0 By −Bx

(8)

  −By Mx   Bx  My  0 Mz

(9)

It is clear from the above equation that skew-symmetric matrix B(b) is of rank two and the kernel of B(b) is specified by the vector b itself. In addition, due to the cross product in Eq. (8) it is impossible to generate torque along the earth magnetic vector b. This singularity problem is solved in the literature by several methods such as: 1. Solving the attitude control problem with 2-D actuation from magnetotorqers,2, 10−11 . 2. Using magnetic rods along with a momentum wheel,1 . 3. Replacing one of the magnetorquers with one reaction wheel,4 . The last method is adopted in this paper. In such a case, we adapt Equ.(10) to be the following equation    0 0 −By Mx    TC = B (b) M = −Bz 1 −Bx  h˙ wy  (10) By 0 0 Mz 5 of 11 American Institute of Aeronautics and Astronautics

Therefore, Eqs.(4)-(6) can be written in matrix form as follows: I ω˙ = −ω × Iω + TC + Td

(11)

Where the disturbance torque on the body of the spacecraft occur naturally and have different sources, such as gravity gradient, aerodynamics, solar radiation and residual magnetic dipoles.

IV.

Earth magnetic field model

The National Oceanographic and Atmospheric Administrations and National Geophysical Data Center (NOAA/NGDC) provide the World magnetic Model (WMM) which is updated and released every 5 years. The WMM consists of a 12 order spherical-harmonic main (i.e., core-generated) field model comprised of 168 spherical-harmonic Gauss coefficients. The Secular Variation (SV) is also taken into account since the earths liquid-iron outer core that contributes to the majority of the earths magnetic field intensity used in the computation called core field, and it changes distinctively from year to year. This SV is calculated by a linear SV model in the WMM. However, due to non-linear variations, the WMM has to be updated every 5 years. The Epoch year considered for the purpose of this study is Epoch 2015. Where, the magnetic field components are calculated in geodetic coordinates. Therefore, the above coordinate systems are utilized for computing the earth magnetic field parameters in the spacecraft body frame. More details in,16 . A.

Earth magnetic field propagation model

In practical applications, spacecraft mostly have navigation equipment such as GPS or GLONASS, which are utilized during the different phases in the spacecraft lifetime. An earth magnetic field propagation model employs the GPS readings, latitude, longitude, and altitude, or the ground station spacecraft position update inside the orbit propagator model. The orbit propagator provides an estimate about the spacecraft position in the orbit. This estimate is fed to the WMM to provide the spacecraft with an estimation of the external earth magnetic field parameters in the orbital reference frame. B.

Earth magnetic field computation

The estimated Earth magnetic field parameters rely on the measured angular velocities, from gyroscope, and the controller command. The controller will provide the magnetic rods with current and the dipole moment is computed using to Eq. (12),17 . M = N Simag (12) The spacecraft external torque can be estimated from the measurements of the spacecraft angular velocities using the spacecraft dynamics equations of motion Eqs. (4)-(6). Equation (10) has a singularity problem when computing the earth magnetic field vector. Hence, one of the components of the earth magnetic field vector is assumed at each time step using its value at the previous time step. This assumed component is altered between Bx and Bz . C.

Earth magnetic field refinement using Kalman filter Xpredicted = AXn−1 + CUn Ppredicted = APn−1 AT + Q y = Zn − Xpredicted SS = HPpredicted H T + R K = Ppredicted H T SS −1 Xn = Xpredicted + Ky Pn = (1 − KH) Ppredicted


(13)

The Kalman filter is used to refine the earth magnetic field parameters. The model used in the Kalman filter is described in section IV-B, and the pseudo measurement is described in section IV-A. The refinement process uses Eqs. (13). Where: Zn is the pseudo measurement earth magnetic field parameters, and Xpredicted is the propagated earth magnetic field parameters. The Xn is the estimated earth magnetic field parameters and is used to predict the angular velocity inside the control matrix in the angular velocity KF.

V.

Simulations

The convention used here assumes that the roll angular motion is about the X-axis (along the velocity vector), the pitch angular motion is about the Y-axis (along the negative orbit normal) and the yaw angular motion is about the Z-axis (in the Nadir direction). The selected Euler rotation sequence is 3 − 2 − 1 (i.e., Z-Y-X). The classical PD controller is implemented in this study. Measurements are the Euler angles rates .There are many other control strategies that can be employed,1−12 . Equation. (14) represents the controller output, which is the driving current to the magnetorquers. imag = kp eEuler + kd e˙ Euler

(14)

Figure 3: Illustration of the simulation environment A simulation environment is developed to test this concept, Fig. 3 . This study pertains to the case involving a satellite with a reaction wheel along the Y-axis of the body fixed coordinate system and two magnetorquers along the X-axis and Z-axis. Consequently, three-axis attitude control is guaranteed. The spacecraft dynamics in the attitude simulator computed using the Euler Eqs. (4)-(6) for a rigid body. The spacecraft experienced torque which is calculated using the actual external magnetic field, and the dipole moments Eq. (10). Where, the dipole moments computed by controller in Eq. (12) and (14). The Euler error terms referred in Eq. (14), are the inputs to the PD controller and are computed from the four Euler parameter errors, quaternion errors. The WMM is utilized in both the Spacecraft Attitude Simulator and the Estimator block. In the Attitude Simulator, the WMM is used to simulate the external magnetic field for the spacecraft, here random noises are added to account for disturbances such as other magnetic field sources like solar magnetic field. However, in the estimator block, there is another WMM with added random noises with GPS readings as input, used as a part of onboard software. The Estimator block as seen in Fig. 3, see also Fig. 1, has two KF inside it. One of them is for the angular velocity estimation. The estimated values used to estimate the Euler angles which are given as inputs to the PD controller. The output of the spacecraft attitude simulator is the Angular velocities of the spacecraft. These angular velocities are measured using an onboard Gyroscope. Random noises are added to simulate sensor noise. These measurements are the measurement vector inside the angular velocity KF. A linearized model for angular velocity propagation step is used Eq. (15)


ωx(k) − ωx(k−1) = Tx(k−1) tk − tk−1 ωy(k) − ωy(k−1) = Ty(k−1) Iy tk − tk−1 . ωz(k) − ωz(k−1) Iz = Tz(k−1) tk − tk−1 Ix

(15)

This model, however, needs the torque to compute the angular velocity. Therefore, the external magnetic field in the spacecraft environment is needed. A second KF used to estimate the external magnetic field. For the second KF, the measurement vector is the solution of Eq. (10). Where the moment can be calculated from the controller output and the torque can be calculated from the inverse dynamics of the rigid body with knowledge of the angular velocity from the gyroscope. While, the propagated external magnetic field values is calculated using the WMM. The WMM needs the altitude, longitude, and latitude. Therefore, an orbit propagator in the simulation study is employed to calculated theses parameters. In reality, GPS readings will be used instead of orbit propagator. In the simulation conducted in this study, the magnetic field model used in the attitude simulator is different from those in the KF, through added random noise. The KF inside the magnetic field estimation block, Fig. 1, is discussed in section IV. Time domain simulations were performed to evaluate the system capabilities in different configurations. A block diagram of the attitude simulation program is presented in Fig. 4 In Tables 1-3, the spacecraft inertia, initial parameters values, orbital parameters, actuators parameters, and PD controller gains are provided. Table 1: Spacecraft Inertia tensor, Actuators’ parameters and Initial Conditions Moment of inertia, kg/m2 Ix =16.88230 Ix =26 Ix =36.1176

Initial degree 0 0 0

angles,

Initial euler rates, rad/sec 0 0 0

Max. Dipole Moment,Am2 ±50

Table 2: Orbital Parameters Argument of perigee, rad π/2

Right Ascension of ascending node, rad π/4

Initial true animaly, rad π/6

Eccentricity 0

Altitude, km 450

Inclination, rad 87.5 ∗ π/6

Reaction wheel moment of inertia, kg/m2 0.01

Table 3: PD controller gains kp1 kp2 kp3

70 15 110

kd1 kd2 kd3

2800 300 6800

The simulation results shown in Figs. 5-9 prove that the spacecraft is able to follow the attitude command without the need for the magnetometer measurements. The estimated magnetic field from the feedback vehicle angular motion is used to replace measurements of magnetometer. Figure 9 shows a good accuracy of the estimated magnetic field after employing the KF compared to the propagated magnetic field, which relies on the WMM. This estimated magnetic field is the main part in the propagation step of the angular velocity as discussed earlier. The second KF, the angular velocity KF, is used for refining and estimating the angular velocity.


Figure 4: Block Diagram of the Numerical Simulation Prorgam


Figure 5: Attitude angle in each axis

Figure 6: Angular rates in each axis

Figure 7: Torque in each axis

Figure 8: Dipole moment in each axis

Figure 9: Magnetic field, Propagated (B Propagated), Computed (B Computed), and Estimated from KF (B KF), in each axis


VI.

Conclusion

The proposed algorithm eliminates the need for magnetometers for spacecraft magnetic attitude control in LEO, using only measurements of the vehicle motion such as the satellite angular velocities. The key idea is to use the system response to known control commands to estimate the external magnetic field. The simulation results show that by measuring the response of the satellite (rotational motion), and given the control command, it is possible to estimate the external magnetic field and control the satellite attitude accordingly. The simulation shows promising results that prove that the proposed control algorithm is able to fulfill the required attitude command. Future work of this study is to achieve magnetic attitude control with only three magnetometer without use of magnetometer.

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