In the experiment, the single axis air bearing table is utilized for simulating the rotational motion of a certain physical axis of the satellite, and according to the feature of large angle and rapid maneuver, the pitch axis (Y) is chosen as the physical axis. The dynamic model of the single axis air bearing table is governed by the following differential equation: (1) J ω ˙ = u + T dwhere ω denotes the air bearing table rotation angle rate, J is the moment of inertia of the air bearing table, u is the control torque input, and T_d is the disturbance torque.

Assuming that the “3-1-2” Euler angle rotation is adopted, the relationship between the attitude quaternion and Euler angle is described as follows: (2) q = [ q 0 q 1 q 2 q 3 ] = [ cos φ 2 cos θ 2 cos ψ 2 − sin φ 2 sin θ 2 sin ψ 2 sin φ 2 cos θ 2 cos ψ 2 − cos φ 2 sin θ 2 sin ψ 2 cos φ 2 sin θ 2 cos ψ 2 + sin φ 2 cos θ 2 sin ψ 2 sin φ 2 sin θ 2 cos ψ 2 + cos φ 2 cos θ 2 sin ψ 2 ]where q₀ denotes the scalar part of the quaternion, q̄ = [q₁ q₂ q₃]^T is the vector part of the quaternion. φ, θ and ψ denote the roll, pitch and yaw angle, respectively. Considering the case of φ = 0 and ψ = 0 for the single axis air bearing table, then the attitude of the air bearing table is rewritten in the form of quaternion as follows: (3) q = [ cos θ 2 0 sin θ 2 0 ] T

If the Euler angle θ_c defines the command air bearing table rotation angle, the corresponding command attitude quaternion is given as: (4) q c = [ q c 0 q ¯ c T ] T = [ cos θ c 2 0 sin θ c 2 0 ] T

Then the error quaternion, which represents the error between the current attitude quaternion and the command attitude quaternion, is defined as: (5) q e v = [ q c 0 q ¯ c T − q ¯ c − [ q ¯ c × ] + q c 0 I 3 × 3 ] q

Substituting (3) and (4) into (5) gives the error quaternion: (6) q e v = [ q e 0 q e 1 q e 2 q e 3 ] = [ cos ( θ − θ c 2 ) 0 sin ( θ − θ c 2 ) 0 ]and then the error quaternion kinematics differential equation is expressed as: (7) q ˙ e v = 1 2 [ − q e 2 ⋅ ω 0 q e 0 ⋅ ω 0 ] = [ − sin ( θ − θ c 2 ) ⋅ ω 0 cos ( θ − θ c 2 ) ⋅ ω 0 ]

The MSDGCMG consists of inner and outer gimbal servo systems as well as a high-rate rotor system. The high-rate rotor is mounted in the stator housing and provides a constant angular momentum. With a specific rotation of inner and outer gimbal, the vector direction of the rotor angular momentum is changed to produce the desired gyroscope torques to meet the requirements of the attitude control. The main parameters of the MSDGCMG used in this paper are:

Angular momentum: ≥15 Nms.

Furthermore, the circuit box of MSDGCMG is used to control the inner and outer gimbal servo systems as well as the high-rate rotor system. Assume the inner gimbal axis, outer gimbal axis and rotor axis are perpendicular to each other at the initial time, then the MSDGCMG coordinate system is defined as in Figure 1, outer gimbal axis as X axis, inner gimbal axis as Z axis, and Y axis in agreement with the Right-handed Rule. When the rotor angular momentum revolves about the positive half of the Z axis, the inner gimbal angle changes α. When the rotor angular momentum revolves about the positive half of the X axis, the outer gimbal angle changes by −β.

In this paper, MSDGCMG is put on the single axis air bearing table. At the initial time, the Y axis of the MSDGCMG coordinate system is parallel to the rotation (pitch) axis of the air bearing table, whereas the inner gimbal axis (Z axis) and the outer gimbal axis (X axis) are perpendicular to the pitch axis. Consequently, the installation matrix of MSDGCMG is a 3 × 3 identity matrix. According to the definition of the MSDGCMG coordinate system and the installation matrix, the CMG angular momentum vector is represented in the matrix form as: (14) h g = h 0 [ − sin α cos α cos β − cos α sin β ]where h₀ is the angular momentum magnitude of the individual MSDGCMG, α and β are the inner and outer gimbal angle, respectively.

Therefore, the MSDGCMG output torque vector ḣ_g is derived as: (15) h ˙ g = h 0 ⋅ C ⋅ δ ˙where δ̇ = [α̇ β̇]^T is the gimbal angle rate, C is the Jacobian matrix: (16) C = [ − cos α 0 − sin α cos β − cos α sin β sin α sin β − cos α cos β ]

It is clearly seen from the above equation that, the Jacobian matrix is a nonlinear function of the gimbal angles α and β.

For the attitude control of the single axis air bearing table, particular attention should be paid to the MSDGCMG output torque along the pitch axis. Therefore A is written by extracting the second row of Jacobian matrix C: (17) A = [ − sin α cos β − cos α sin β ]

The command gimbal rate δ̇, referred to as the pseudoinverse steering law, is then obtained as: (18) δ ˙ = − A + u / h 0 where A + = A T ( A A T ) − 1where u denotes the command control torque derived by the attitude control law in the preceding section.

For the reason that the high-rate magnetically suspended rotor of the MSDGCMG is actually active-control elastic supporting with certain stiffness and damping, the gimbal movement will disturb the magnetically suspended rotor. For simplicity, the phenomena caused by the gimbal movement are called the moving-gimbal effects. On the other hand, the response bandwidth of the CMG gimbal is constrained by the output torque ability of gimbal motor, thus the gimbal servo systems are unable to respond to the large gimbal angle acceleration. Consequently, the limit of the gimbal angle velocity and the acceleration should be given full consideration, to make sure that the high-speed rotor would work steadily around the operating point.

Furthermore, the singularity problem is inherent in the CMG. When the MSDGCMG is trapped in the singular state, the gimbal of the MSDGCMG will vibrate seriously so as to influence the stability of the MSDGCMG. Therefore, the maximum gimbal angle acceleration δ̈_max is determined on the basis of the singularity measurement: (19) δ ¨ max = { δ ¨ max 1 i f 0 < D < d δ ¨ max 2 i f D ≥ dwhere D = det(AA^T) is the singularity measurement, d is the piecewise point.

In the experiment, the maximum limit of the gimbal angle acceleration is firstly taken into consideration. The inner and outer gimbal angle acceleration δ̈_k at moment k is calculated as: (20) δ ¨ k = ( δ ˙ k − δ ˙ k − 1 ) / T swhere δ̇_k and δ̇_k−1 are the command gimbal angle rates at moment k and k − 1, respectively. T_s denotes the control cycle. Then the gimbal angle acceleration is constrained as follows: (21) δ ∼ ¨ k = { δ ¨ k i f ‖ δ ¨ k ‖ ∞ < δ ¨ max δ ¨ k δ ¨ max ‖ δ ¨ k ‖ ∞ i f ‖ δ ¨ k ‖ ∞ ≥ δ ¨ maxwhere ‖•‖_∞ is the infinite norm of the vector, and then the command gimbal rate at the moment k is modified as: (22) δ ˙ k = δ ˙ k − 1 + δ ∼ ¨ k ⋅ T s

Finally, the gimbal angle velocity is calculated as follows: (23) δ ∼ ˙ k = { δ ˙ k i f ‖ δ ˙ k ‖ ∞ < δ ˙ max δ ˙ k δ ˙ max ‖ δ ˙ k ‖ ∞ i f ‖ δ ˙ k ‖ ∞ ≥ δ ˙ maxwhere δ̇_max denotes the allowed maximum gimbal angle rate. Then the command gimbal angle rate _k is derived from Equation (23) taking account of the limit of the gimbal angle rate and acceleration.

Note that the maximum limit of the gimbal angle rate and the acceleration defined as above has the same direction as that of themselves before maximum limit. That is, the MSDGCMG output torque is kept the same direction as the command control torque, but the magnitude of the actual torque is reduced.

The measurement of the air bearing table moment of inertia is important in the initial stage of the semi-physical simulation. Here the twisting vibration approach [18] is applied for measuring the moment of inertia. This method has the advantages of simple design, easy operation, as well as not needing any knowledge of the air bearing table mass. Only the spring resonance device is needed in our experiment.

During the measurement of the moment of inertia, it should be ensured that the vibration amplitude of the air bearing table is small to satisfy the condition of micro-amplitude. However, the equivalent torsional stiffness is difficult to measure, and the change of torsional stiffness resulting from the reinstallation brings about large error in the moment of inertia. In order to obtain the moment of inertia J precisely, two standard cylinder balance masses are symmetrically placed on the air bearing table, and then the air bearing table moment of inertia is given as: (24) J = f 2 2 f 1 2 − f 2 2 ⋅ Δ Jwhere f₁ denotes the system vibration frequency without the balance mass, f₂ denotes the system vibration frequency when the air bearing table is installed with the balance masses, ΔJ is the increased moment of inertia: (25) Δ J = 2 m 0 L 2 + m 0 r 2where m₀ and r are the mass and the radius of the balance mass, respectively. L represents the distance from the rotating axis of the air bearing table to the centre of the balance mass.

Here the vibration period is calculated by counting the zero-crossing number of the air bearing table angle velocity within a specified time. Multiple measurements are carried out, and an average measurement value is used from ten attempts. The experimental data is shown in Table 1:

In the experiment, the air bearing table is subjected to some external disturbance torques, such as gravity torque resulted from incomplete balancing, friction torque produced from air bearing, the resistance moment caused by communication cable and power supply cable, and so on. Taking account of the influence of the disturbance torques on the control accuracy and attitude stability, the disturbance torque T_d is calculated and the equation is given as follows [19]: (26) T d = J Δ ω Δ twhere Δω denotes the change of angle velocity during time Δt.

Through multiple measurements, the average value of the disturbance torque is measured as 1.09 × 10⁻⁴ Nm.

In the experiment, the parameter segment attitude control law developed in this paper is now integrated to the MSDGCMG steering law. The parameters shown in the preceding sections are properly selected as in Table 2.

Experiments of the following three cases are carried out with different MSDGCMG rotor speed and different command rotation angles of the air bearing table. Case 1, as shown in Figure 4, is to realize a 30° attitude maneuver with 6,000 rpm rotor speed of the MSDGCMG. In this case, the initial air bearing table angle is 0°, and the command angle is set at 30°. Case 2, as shown in Figure 5, is to realize 60° attitude maneuver with 15,000 rpm rotor speed. In this case, the initial angle is 0°, and the command is 60°. Case 3, as shown in Figure 6, is to realize 60° attitude maneuver with 20,000 rpm rotor speed. In this case, the initial angle is 60°, and the command is 0°.

From the experimental results of the three cases, it is indicated that the larger the rotor speed (angular momentum of the MSDGCMG) is, the shorter the time required to realize the same angle maneuver is, meaning that the greater the maneuver ability is. It's easily known from the above figures that, the inner and outer gimbal angle rates follow the command perfectly, and that MSDGCMG output torque almost agrees with the command torque under the moving-base condition. Moreover, the attitude stability with the MSDGCMG as an actuator is superior to 10⁻³°/s, and there exists no transient overshoot during the process of the rapid maneuver and quick stability.

It is clearly seen from the Table 3 that, the MSDGCMG angle momentum determines the maneuver velocity, and that the maximum average maneuver ability is up to 3.57°/s. For the reason of the coupling between the air bearing table, MSDGCMG gimbal and high-rate rotor, the quicker the angle velocity of the air bearing table is, the more serious the influence on the high-rate magnetically suspended rotor becomes. In the experiment, for the reason of the high-precision magnetic bearing control, the MSDGCMG is able to overcome the influence that moving-base effects bring about, even when the maximum angle velocity reaches to 9.49°/s and the speed of high-rate rotor is 20,000 rpm.

In the future, advanced control algorithms [20–22] can be employed to further improve the control performance.