CHINESE

JOURNAL OF

AERONAUTICS

Chinese Society of ¿Aeronautics and Astronautics 'Beihan/i 'University

Accepted Manuscript

Full Length Article

Spacecraft attitude maneuver control using two parallel mounted 3-DOF spherical actuators

Li Guidan, Li Haike, Li Bin

PII: DOI:

Reference:

S1000-9361(16)30223-0 http://dx.doi.org/10.1016/j.cja.2016.12.004 CJA 737

To appear in:

Chinese Journal of Aeronautics

Received Date: Revised Date: Accepted Date:

14 January 2016 4 May 2016 31 October 2016

Please cite this article as: L. Guidan, L. Haike, L. Bin, Spacecraft attitude maneuver control using two parallel mounted 3-DOF spherical actuators, Chinese Journal of Aeronautics (2016), doi: http://dx.doi.org/10.1016/j.cja. 2016.12.004

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Chinese Journal of Aeronautics 28 (2016) xx-xx

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Chinese Journal of Aeronautics

journal homepage: www.elsevier.com/locate/cja

JOURNAL OF

AERONAUTICS

Final Accepted Version

Spacecraft attitude maneuver control using two parallel mounted 3-DOF spherical actuators

Li Guidana'*, Li Haikeb, Li Bina

aSchool of Electrical Engineering and Automation, Tianjin University,

Tianji, I3003t

State Grid Tianjin Power Dongli Power Supply Branch, Tianjin Received 14 January 2016; revised 4 May 2016; accepted 31 October 2016

'2, China 00, China

Abstract

A parallel configuration using two 3-degree-of-freedom (3-DOF) spherical electromagnetic momentum exchange actuators is investigated for large angle spacecraft attitude maneuvers. First, the full dynamic equations of motion for the spacecraft system are derived by the Newton-Euler method. To facilitate computation, virtual gimbal coordinate frames are established. Second, a nonlinear control law in terms of quaternions is developed via backstepping method. The proposed control law compensates the coupling torques arising from the spacecraft rotation, and is robust against the external disturbances. Then, the singularity problem is analyzed. To avoid singularities, a modified weighed Moore-Pseudo inverse velocity steering law based on null motion is proposed. The weighted matrices are carefully designed to switch the actuators and redistribute the control torques. The null motion is used to reorient the rotor away from the tilt angle saturation state. Finally, numerical simulations of rest-to-rest maneuvers are performed to validate the effectiveness of the proposed method.

Keywords: Attitude maneuver; Spherical actuator; Parallel configuration; Backstepping control; Singularity; Null motion

1. Introduction

Control moment gyros (CMGs) are widely used in

spacecraft attitude control, which is attributed to the

advantages of high torque capacity and no propel-

lants.1"3 Especially, the single gimbal CMG (SGCMG) features the torque-amplification-capability. How-

ever, complex gimbal structures, large servo parts and commonly required cluster configurations limit their applications to small spacecraft. In contrast, multi-degree-of-freedom (multi-DOF) spherical electromagnetic momentum exchange actuator (SEMEA) has great advantages of reducing attitude control

system (ACS) mass, volume and power requirements

because of their higher structural integration.4 Furthermore, its largest asset is that a single device is capable of generating three-axis control torques because the variable-speed rotor can be tilted in any direction, which shows great prospect in 3-axis spacecraft attitude control.5' 6

Over the past decades, a variety of structural forms of spherical actuators have been proposed, which commonly have a spherical rotor or a spherical stator. Downer et al.7 proposed a magnetic rotor suspension system including a magnetic annulus rotor and a spherical stator. An armature is used to induce rotation of the rotor and the spin axis can be gimbaled by selectively exciting the control coils on the stator. A

* Corresponding author. Tel.: +86 22 27890983. E-mail address: lgdtju@tju.edu.cn.

Chinese Journal of Aeronautics

similar ball joint type magnetic bearing for tilting body can be found in Ref.8. Note that if the armature is moved outside the stator, it will allow a larger tilting range, increasing the amount of angular momentum exchangeable between the actuator and the spacecraft. Based on this idea for structural improvements, Chetelat et al.9 put forward a reaction sphere actuator with an 8-pole permanent magnet spherical rotor and a 20-pole electromagnet stator. The rotor can be electronically accelerated in any direction, and it is by magnetic levitation that the rotor is held in position. Instead of a multipole magnet, Chabot et al.10,11 proposed a design using a spherical dipole magnet as the rotor,which is inexpensive and readily available. Similarly, in Ref.12, we proposed a new type of spherical momentum exchange device based on a permanent magnet spherical motors (PMSM)13 and the detailed design consideration is presented in Ref.14. Compared with multi-axis magnetic momentum wheels,15 its spherical-profile and dihedral-shell PMs can maintain the uniformity of the air-gap magnetic flux density when the rotor is in motion, and can help acquire a larger tilting range. From the perspective view, a single spherical actuator can be an alternative to conventional CMG clusters. However, its rotor tilt range limited and the singularity occurs when the rotor tilt angle is saturated. Therefore, the control law and steering logic need to be concerned with the singularity. To overcome this drawback, an n-step incremental rotation strategy16 was introduced in Ref.12. In fact, the steering strategy belongs to an open-loop scheme, which is sensitive to the unexpected external disturbances, spacecraft parameters and initial attitude errors. In general, more practicable singularity avoidance schemes and robust feedback control laws are desired.

In this paper, we focus on the attitude maneuver control using spherical actuators. In CMG systems, cluster configuration17 and path planning18 are effective singularity avoidance strategies. When the system falls into the singularity state, null motion can be used to reconfigure the CMGs to preferred gimbal angles. Referring to this method, a parallel configuration for SEMEAs is investigated to avoid the tilt angle saturation singularity and simultaneously to provide redundancy. The dynamic equations of motion are derived by the Newton-Euler approach. Noting that the control system has a cascaded structure, we adopt a backstepping control law.19,20 When the tilt angle saturation singularity is encountered, a modified weighed pseudo inverse steering law based on null motion is applied, and the weighted matrices are carefully designed. To validate the effectiveness of the proposed method, numerical simulations of rest-to-rest maneuvers are carried out.

2. Introduction to SEMEA

The prototype and schematic of the SEMEA are presented in Fig. 1. The SEMEA is mainly composed of an electromagnetic stator and a PM rotor. The universal mechanical shaft in the PMSM is cut off only for momentum exchange purpose. Its variable-speed rotating rotor can be tilted in any direction, thus realizing three-dimensional momentum exchange with the spacecraft platform.

The actuator works on the electromagnetic torque T, whose characteristics are determined by stator currents I, and the arrangements of stator windings and rotor permanent magnets (Fig. 1(b)). The relationship can be expressed as T =KTI where KT is the defined static torque characteristic matrix. Thus, the rotation and tilt of the rotor can be controlled by the stator currents I. Related control laws and electrifying strategies can be found in Refs.21,22. For simplicity, ideal rotor trajectory tracking is assumed in this paper. Note that the mechanical structure and the air-gap magnetic field distribution limit the rotor tilting range (the maximum of the rotor tilt angle 11=15°).

■ PM poles (b) Schematic of SEMEA

Fig. 1 Illustration of SEMEA.

3. Analytical model of spacecraft with two SEMEAs

In this section, the Newton-Euler method is employed to derive the complete dynamic equations of motion for a spacecraft with two parallel mounted SEMEAs. The attitude kinematics is described in terms of quaternions.

3.1. Dynamics equations of motion

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To simplify the development, we first consider a rigid spacecraft with only one SEMEA. Afterwards, we extend the result to the complete system. As shown in Fig. 2(a), the stator housing and the spacecraft body are treated as one platform. A reference frame B with basis (b1, b2, b3) is fixed with the platform. The center of mass of the overall system O is taken as the origin of coordinates. The spacecraft platform is free to translate and rotate with respect to the inertial frame N, with i1, i2 and i3 the unit vectors. The origin of coordinates is at the rotor's center of mass OR and rR is the distance vector from O to OR.

Fig. 2 Models for derivation of equations of motion.

To facilitate computation, a rotor's outer virtual gimbal frame G with orthogonal unit vectors (g1, g2, g3) and inner virtual gimbal frame H with orthogonal unit vectors (h1, h2, h3) are established to define the orientation of the rotor in the spacecraft platform (Fig. 2(b)). The unit vectors g2, h1 and h3 are parallel to the outer virtual gimbal axis, inner virtual gimbal axis and spin axis, respectively. The frame B transforms to the frames G and H by Euler angle rotations through the outer virtual gimbal angel and inner virtual gimbal angel, respectively. The spherical rotor rotates around the spin axis at speed rate Q. When the initial gimbal angles are zero, the unit vectors (h1, g2, h3) coincide with the unit vectors (b1, b2, b3) of frame B. The unit vector g2 stays fixed relative to the frame B, and any unit vector gi or hi can be obtained by the following direction cosine matrices:

cosa 0 -sina 0 1 0 sina 0 cosa 10 0 " 0 cos ß sin ß 0 - sin ß cos ß

nner vir

where a and 3 are outer and inner virtual gimbal angles, respectively. In the vector expressions, the subscripts indicate the relative motion. The absolute angular momentum HR of the rotor with respect to its center of mass OR is given by

\Hr - IROJr

°rh + mhg + mg

where «R stands for the absolute angular velocity

of the rotor; arh is the relative angular velocity of

or with respect to frame H, ahg the relative

r velocity of frame H with respect to frame G, the relative angular velocity of frame G with

respect to frame B, and a the absolute angular velocity of the spacecraft platform; IR is the rotor inertia matrix. Let Ih be the moment of inertia of the rotor about its spin axis. Assume that the spherical rotor is completely symmetrical, and then in any frame IR is a constant diagonal matrix

Ir = diag (Ih, Ih, Ih ) (3)

According to the definition of the rotor virtual gim-bal coordinate frames,

colh =¿211,. i0hg =ßgl, iogh =dg2 Therefore

Hr = Ir (nh3 + ßgx + àg2 + CO)

Since there is no angular momentum for the virtual gimbals, the total angular momentum of the overall system with respect to its center of mass O is given by

H - Hb + HR + MR ( rR2 E - VR rRT ) m

where HB is the absolute angular momentum of the spacecraft platform, and mR the rotor mass; E is

Let Ib be

unit matrix; rR is the modulus of rR the inertia matrix of the platform with respect to O and then H is rewritten as

Chinese Journal of Aeronautics

H = /„ m + H

4 = 4 + ™r(>R E " *R >R)

Let L represent the external disturbance torques experienced by the system. According to Euler's equation, the initial time derivative of H is given by

IB(b + (oxIB(o + HR =Le

Where HR is the time derivative of HR and it is obtained as

hr =/rK +«»<(e>gb +cohg +wrh) + wgb

X (Mhg + Mrh)+Mhg X Mrh] + ®R X 4®R

Substitute Eqs. (4) and (5) into Eq. (9), and the dynamic equation of motion of the system is obtained as follows:

Iscb = - 10 x (IBio + HR )-IR

ß äß

where IS = IB + IR is a constant matrix. The right of Eq. (11) represents the output torques produced by the actuator. The first term represents the torque caused by the motion of the spacecraft body, the second term represents the torque caused by the rotor accelerations, and the third term represents the torque caused by the rotor tilt rate or rotation acceleration.

From here on, we extend the result to the case of the spacecraft with two SEMEAs. Then the dynamics equation of motion of the overall system can be obtained from Eqs. (9) and (11) in the following form:

Iscb = - cox (IB(o + HR) - A^A- AJj + Le

4 = 1B + 4l + ^R2

HR = HR1 + HR2

t] = [ àx, ß, , ä2, ß2, Ù, J

^g = 4 [ g21, Sil, g31, g22' g12' g32 ]

Ah = 4 [~^h2i, Qhi, V

2h12, ^32 ]

3.2. Attitude kinematics

'21, 11, "31, " ¿L"22 , ^"12,

with the subscripts 1 and 2 indicating the two SEMEAs.

In this paper, the quaternion q = \qx, q2, q3, q4 ] is

used to describe the attitude of the spacecraft and the desired attitude is adopted as the inertial frame, i.e.,

the command quaternion qc = \0, 0, 0, 1]T . In this

case, the kinematic differential equation in terms of error quaternion is expressed as follows:23

ase, the ki rror quater.

Àev =

M +2 Ve4M

<7e4 ="

where qev =\qel, qe2, qe3 ] and qeA are the vector and scalar parts of the error quaternion q , respectively, w = \®1, a2, co3 ]T ; and ^q*v J is the slew-symmetric matrix defined by

T 0 -qe3 qe2 [ qexv ]= qe3 0 - qe1 (15)

_-qe2 qe1 0 .

4. Nonlinear backstepping control law design

Note that the attitude control system described by Eqs. (12) and (14) has a cascade structure, and the effective backstepping method can be used to develop the feedback law. The control block diagram is presented in Fig. 3.

Fig. 3 Backstepping control block diagram.

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It is assumed that the current state of the system a, P, Q, m and q can be measured in real time. We first consider the subsystem described by Eq. (14). To bring the spacecraft to the desired final attitude, the tracking law = [®fl ,®f2 ,®f3 ]T can be considered as pseudo control input. The error state variables e and e are defined as

e1 = tfev

(16) (17)

Let Va be the following Lyapunov candidate function:

K =1 [ eT e + (1 - qe4)2 ] (18)

The time derivative of Va is obtained as 1 T

K = -(I,A =-</cvW

To make Va < 0, we select the linear tracking function19 as follows:

% =-kqei i =1,2,3

ques are bounded by

sup||\Le\\x <Im (25)

Substituting Eqs. (20) and (24) into Eq. (23) gives

V = el (Le-1M sign{e2))-Pele2 ~Y,kd, < (supllLe IIL - lM ) IIe2 11 - e2 - S kiqh (26)

Accordingly, the backstepping control law guarantees the asymptotically stability of the c losed-loop system according to the Lyapunov theory.

5. Singularity avoidance steering law design

As shown in Eq. (13), Ag does not contain Q and thus is much smaller compared to A , and it is usually dropped. The steering law constraint given in Eq. (24) is then simplified as

where k are positive constants.

After a is determined, the real command input should be determined to guarantee the pseudo-control input to be achieved. We define the following Lyapunov candidate function for the overall system:

V =1 [ +(1 - q e4)2 ]+1

e2 Ise2

2L 1 1 * " J 2 The time derivative of V can be written as follows:

V = -qe4+e2l,S<?2

Substituting Eqs.(12), (16) and (20) into Eq. (22) gives

x(HR +IBa>) + Le -/sß>f]

^-A^A-AJj-m

To guarantee V < 0, the feedback control law is selected as

= ~e\ (HR + 7Bto) - fscbf + Pe2 + ^Msign(^2)

Each column vector of A represents the output control torques produced by the rotor tilt motion or spin acceleration, corresponding to CMG mode and RW mode, respectively. Note that the inner virtual gimbal angle P never equals 90° (¿m=15°) and rank (A ) = 3. This is to say, within the tilting range, the output torque of a single SEMEA spans the entire space. However, when the rotor tilts to the bound, there exists a direction in which an output torque cannot be generated. It is perpendicular to the spin axis and points outwards. This direction is called the "tilt angle saturation singularity direction". When the required torque lies in this direction, a single SEMEA cannot avoid the singularity because there is no null space to reorient the rotor. In contrast, for a parallel configuration with two SEMEAs, a modified null motion strategy can be resorted to in order to avoid this singularity.

For Ah is never rank deficient, naturally, the standard Moore-Penrose inverse can be used to obtain a minimum norm solution for i;, and then the

resulting simplified velocity steering law is given by

where Lr and Lo stand for the required torque and the output control torque, respectively; p and lM are positive constants. The external disturbance tor-

Note that it is not applicable in practice if the solution tends to exceed the restricted range but the rotor has reached up to the bound. It is eagerly anticipated that the tilt angle will decrease automatically at the next time. However, the ideal case is in-

e = w—w

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frequent and the tilt angle saturation singularity is more likely to happen. To avoid the singularity, when one of the actuators falls into the saturation state, the other one should switch to provide effective control torques, meanwhile, the saturated rotor should be reoriented to a preferred position away from tilt angle saturation. A modified weighted pseudo inverse steering logic based on null motion can be implemented as

N = E6 - W2 AT ( aw aT )-1 Ah

where E6 represents a 6x6 identity matrix; W1 and W are weighted matrices used to switch the actuators and redistribute the control torques. They are defined to be

W = diag ( wm , W2V1, Ws^, w4^2, ^6^2 ) [W2 = diag ( w1, w2, w3, w4, w5, w6 )

where w{ are positive scalars which control how heavily the SEMEAs are to perform in reaction wheel mode or CMG mode.24 For simplicity, here wi are all set to 1. The parameters / and / are switch weights. Let S1 and S2 represent the two rotors' tilt angle respectively, then / and / are functions of Sx and S2. They are defined to be

|M = sign(^mi -3)> Ml = 1 -M If M- = 1 [/M = sign(^m2 M = 1 -M2 If M = 0

where Smi represents the maximum of the two rotors' tilt angle, and u- the value of / at the last moment. And Nd is the SEMEA null motion. Let constant vector nf be the desired rotor position, and the vector d is selected as

d = keW3(«f - n)

where ke is a positive gain to be appropriately chosen, and W3 a diagonal matrix associated with rotor 's reorienting movement given by

W3 = diag (U2,U2,U2,U1,U1,U1) (33)

As shown in Eq. (31), / and u2 are either 0 or 1. If / is 1, this means that the resulting steering law will be performed with rotor 1 to be reoriented to the desired position and at the same time rotor 2 providing effective control torques onto the spacecraft. As we can see, AhNd = 0 , i.e., the SEMEA null motion produces no torques onto the

spacecraft. The stability of null motion has been demonstrated in Ref.25.

6. Numerical simulations

According to the dynamics model, nonlinear control laws and steering laws have been discussed, and numerical simulations of rest-to-rest maneuvers are performed for two main objectives: (1) to confirm the asymptotically stability of the backstepping feedback law; (2) to demonstrate the effectiveness of the proposed singularity avoidance steering logic for the parallel configuration. The external disturbance torques are selected as

3.5cos(0.01t ) +1 3.5cos(0.01t )-1 4sin(0.01t) +1

c10"3 N • m

Two cases with different tilting ranges are considered in our simulations. The detailed simulation parameters are listed in Table 1. In Case 1, the maximum tilt angles are just as normal (¿m1=^m2=15°). During the maneuvers, it may not encounter the tilt angle saturation. In order to demonstrate the working principle of the steering law clearly, a singularity case is needed. Therefore, in Case 2, the maximum tilt angle of rotor 1 is modified (¿m1=8°, ¿m2=15°) to ensure that the singularity will happen. The simulation results are presented in Figs. 4-6 Note that the two cases share the same responses of the spacecraft platform, but different SEMEA responses. The rotor tilt angle response illustrates the working process of the steering law clearly.

Table 1 Simulation parameters

Parameter Value

Is (kg-m2) [15, 0.22, 0.36; 0.22, 15, 0.36; 0.47,

Ir (kg-m2) 0.36, 20]

[1.6, 0, 0; 0, 1.6, 0; 0, 0, 1.6]

q(t0) [0.42, 0.42, 0.63, 0.5]

qc [0, 0, 0, 1]

Vf [0, 0, 0, 0, 0, 0]

Im (N-m) 0.005

k1, k2 ,k3 1, 1, 1

[«1(t0), a2(t0)] (° ) [0,0]

D»1(t0), M0)] (° ) [0,0]

[Û1(t0), Û2(t0)] (rad/s) [10,10]

fefo), fcfo)] [1,0]

Fig. 4 shows the history responses of the spacecraft. Fig. 4(a) plots the responses of the attitude quaternion, and Fig. 4(b) gives the body angular velocity responses. From the simulation results, it can be seen that the proposed nonlinear control law is asymptotically stable and performs very well. The large angle attitude maneuver is effectively achieved with the

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existence of the external disturbance torques.

ter the singularity. The maximum tilt angle of rotor 1 approximately equals 9.7° (around 2.6 s), and it is within the tilting range. During the maneuver, therefore, only SEMEA 1 works and provides control torques while SEMEA 2 holds the initial states. It is apparent that the tilt saturation may happen if the maneuver mission is changed or the tilting range is decreased, just as in Case 2.

As shown in Fig. 6, in Case 2, at the beginning of the maneuver SEMEA 1 first provides control torques, and around 1.2 s it tilts to the maximum. At this moment, SEMEA 2 switches to produce effective torques onto the spacecraft, meanwhile, null motion drives SEMEA 1 away from the saturation state. Around 5 s, it is brought to the initial position. Thus, the tilt angle saturation singularity is successfully avoided. The drawback of the weighting matrices is that the tilt angular rates of the rotor change extremely sharply at the switch point, which requires the actuator to make a very fast dynamic response in practice.

Fig. 4 History responses of spacecraft As shown in Fig. 5, in Case 1, it does not encoun-

Fig. 5 History responses of SEMEAs for Case 1 (¿m1=^m2=15°)

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Time(s) Time(s)

(c) Rotor spin acceleration (d) Rotor tilt angle

Fig. 6 History responses of SEMEAs for Case 2 (ám1=8°, ám2=15°)

7. Conclusions

(1) The full equations of motion of a rigid spacecraft with two spherical actuators mounted in parallel are derived. Compared with conventional CMG system, the spherical actuator's gimbal-less structure makes the formula more accurate and simple.

(2) A nonlinear control law based on the backstepping control method is developed with the external disturbance overcome. To avoid singularity, a modified version of weighted velocity steering law based on null motion is proposed and the weighted matrices are carefully designed.

(3) The simulation results validate the effectiveness of the proposed control law and the singularity avoidance steering law. Ideal rotor trajectory tracking for the actuator is assumed in the simulation. In practice, fast dynamic response of the SEMEA is crucially required. Device optimization and preferred configuration need to be studied in the future work.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China (No.51677130) and the Independent Innovation Funds of Tianjin University (No.1405).

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Li Guidan received the Ph.D. degree in electrical theory & new technology from Tianjin University in 2009 and currently is an associate professor in Tianjin University. Her main research interests include power electronic technique and its application. E-mail: lgdtju@tju.edu.cn

; Grid Tii

Li Haike is a power dispatcher in State Grid Tianjin Power Dongli Power Supply Branch. He received his master degree in electrical engineering from Tianjin University in 2016. He currently researches on the safe operation of electric power system. E-mail: lihaike@tju.edu.cn

Li Bin received the Ph.D. degree in electrical engineering from Tianjin University in 2006 and now is an associate professor in TJU. His current research interests include electric machine design and control. E-mail: elib@tju.edu.cn