The Euler equation is used to build the magnetically suspended rotor dynamic model. A magnetically suspended SGCMG consists of the rotor and the gimbal. The magnetic force is produced from the unique relation between the rotor and the bearing. In order to perform the satellite attitude control, the direction of angular momentum is changed through magnetic bearing torque by the gimbal rotation, and then the gyroscope torque is generated, and is transmitted to the satellite by the magnetic bearings.

In this paper, the rotor is supposed to be asymmetric, and the dynamic and static vibration induced by the geometry shape are ignored. Only the force induced from the gimbal rotation of the CMG frame relative to the inertial frame is considered. The rotation relationship between CMG frame and the gimbal frame is given in Figure 2.

As shown in Figure 2, δ̇, θ̇ and ϕ̇ are the angular velocity of CMG frame relative to gimbal frame. δ is the rotation angle from CMG frame to gimbal frame. δ̇ is the gimbal rate about X_cmg axis. Figure 3 denotes the view from A-side about the Y_cmg-direction in Figure 2. It shows the relationship between the gimbal frame and the installed magnetic bearing frame. f_X and f_Z denote the radial magnetic bearing force about the X_f-axis and Z_f-axis. f_Y is the axial magnetic bearing force. α and β are the rotation angle of the rotor about the X_f -axis and Z_f -axis, respectively. α̇ and β̇ are the angular velocity about the X_f -axis and Z_f -axis, respectively. Ω is the rotor speed relative to the rotor frame, and it is about Y_f -axis. i_AX and i_AZ are the magnetic bearing control current about X_f -axis and Z_f -axis of the A-side.

Firstly, only one magnetically suspended SGCMG is analyzed. M^r is the torque acting on the rotor. By using the Euler equation, the magnetically suspended rotor dynamic model in the rotor frame can be obtained: (1) M r = H ˙ r + ω i r r × H r

In Equation (1), ω ir r denotes the absolute angular velocity in rotor frame. It includes ω rf f, ω icmg cmg, and ω^g. ω rf f is the angular velocity of the rotor. It is the magnetic bearing frame relative to the rotor frame, and is shown in the magnetic bearing frame. ω icmg cmg is the angle velocity of CMG frame relative to the inertial frame. ω^g is the gimbal angular velocity: (2) ω ir r = C f r ( ω rf f + C g f C cmg g ω ig cmg )

The absolute angle velocity Ω ir r of the rotor includes the spin rate Ω^r and the rate ω ir r of the rotor frame: (3) Ω ir r = Ω r + ω ir r = Ω r + C f r ( ω rf f + C g f C cmg g ω ig cmg )

Because α and β are very small, then cosα ≈ 1, cosβ ≈ 1, sinα ≈ α, sinβ ≈ β, αβ ≈ 0. Then: (4) C f r = [ cos β sin β 0 − sin β cos β 0 0 0 1 ] [ 1 0 0 0 cos α sin α 0 − sin α cos α ] ≈ [ 1 β 0 − β 1 α 0 − α 1 ] , ω r f f = [ α ˙ 0 β ˙ ] C g f = [ cos 45 ° 0 − sin 45 ° 0 1 0 sin 45 ° 0 cos 45 ° ]

By rotating δ about X_cmg axis from the CMG frame to the gimbal frame, then: (5) C cmg g = [ 1 0 0 0 cos δ sin δ 0 − sin δ cos δ ] ,     ω g = [ δ ˙ 0 0 ] ,               ω ig cmg = ω icmg cmg + ω g = [ φ ˙ θ ˙ ϕ ˙ ] + [ δ ˙ 0 0 ] = [ δ ˙ ^ θ ˙ ϕ ˙ ]where δ ˙ ^, θ̇ and ϕ̇ show the absolute angular velocity of the CMG frame, and the detailed expression will be given in Section 2.2. By substituting the coordinate transformation matrix and the relative angular velocity, then: (6) Ω ir r = [ 2 2 ( δ ˙ ^ − ϕ ˙ cos δ + θ ˙ sin δ ) + α ˙ + β ( θ ˙ cos δ + ϕ ˙ sin δ ) Ω − 2 2 β ( δ ˙ ^ − ϕ ˙ cos δ + θ ˙ sin δ ) − α ˙ β + ( θ ˙ cos δ + ϕ ˙ sin δ ) − 2 2 α ( δ ˙ ^ + ϕ ˙ cos δ − θ ˙ sin δ ) + α β ˙ 2 2 ( δ ˙ ^ + ϕ ˙ cos δ − θ ˙ sin δ ) + β ˙ − α ( θ ˙ cos δ + ϕ ˙ sin δ ) ]

In Equation (1), H^r is the rotor angular momentum in the rotor frame: (7) H r = I ˜ r Ω ir r = [ I rx 0 0 0 I ry 0 0 0 I rz ] Ω ir rwhere Ĩ_r denotes the wheel moment of inertia in the spin direction. I_rx and I_ry are the radial inertia of the magnetically suspended rotor in the x and y direction, respectively. I_rz is the axial inertia of the rotor. The angular momentum variation rate relative to the rotor frame can be obtained from the time derivatives: (8) H ˙ r = [ I r x ( 2 2 ( δ ¨ ^ − ϕ ¨ cos δ + ϕ ˙ δ ˙ ^ sin δ + θ ¨ sin δ + θ ˙ δ ˙ ^ cos δ ) + β ˙ θ ˙ cos δ + β θ ¨ cos δ − β θ ˙ δ ˙ ^ sin δ + β ˙ ϕ ˙ sin δ + β ϕ ¨ sin δ + β ϕ ˙ δ ˙ ^ cos δ + α ¨ ) I r y ( 2 2 ( − β ˙ δ ˙ ^ − β δ ¨ ^ + β ˙ ϕ ˙ cos δ + β ϕ ¨ cos δ − β ϕ ˙ δ ˙ ^ sin δ − β ˙ θ ˙ sin δ − β θ ¨ sin δ − β θ ˙ δ ˙ ^ cos δ + α ˙ δ ˙ ^ + α δ ¨ ^ + α ˙ ϕ ˙ cos δ + α ϕ ¨ cos δ − α ϕ ˙ δ ˙ ^ sin δ − α ˙ θ ˙ sin δ − α θ ¨ sin δ − α θ ˙ δ ˙ ^ cos δ ) + Ω ˙ + θ ¨ cos δ − θ ˙ δ ˙ ^ sin δ + ϕ ¨ sin δ + ϕ ˙ δ ˙ ^ cos δ − α ¨ β + α β ¨ ) I r z ( 2 2 ( δ ¨ ^ + ϕ ¨ cos δ − ϕ ˙ δ ˙ ^ sin δ − θ ¨ sin δ − θ ˙ δ ˙ ^ cos δ ) − α ˙ θ ˙ cos δ − α θ ¨ cos δ + α θ ˙ δ ˙ ^ sin δ − α ˙ ϕ ˙ sin δ − α ϕ ¨ sin δ − α ϕ ˙ δ ˙ ^ cos δ + β ¨ ) ]

By substituting Equations (4)–(8) into Equation (1), the rotor torque in rotor frame can be obtained: (9) M r = [ − I ry Ω ( 2 2 ( δ ˙ ^ + ϕ ˙ cos δ − θ ˙ sin δ ) + β ˙ ) + ( I rz − I ry ) ( θ ˙ cos δ + ϕ ˙ sin δ ) ( 2 2 ( δ ˙ ^ + ϕ ˙ cos δ − θ ˙ sin δ ) + β ˙ ) + I rx ( 2 2 ( δ ¨ ^ − ϕ ¨ cos δ + ϕ ˙ δ ˙ ^ sin δ + θ ¨ sin δ + θ ˙ δ ˙ ^ cos δ ) + β ˙ θ ˙ cos δ + β ˙ ϕ ˙ sin δ + α ¨ ) ( I rx − I rz ) ( 2 2 ( δ ˙ ^ + ϕ ˙ cos δ − θ ˙ sin δ ) + β ˙ ) ( 2 2 ( δ ˙ ^ − ϕ ˙ cos δ + θ ˙ sin δ ) + α ˙ ) + I ry ( 2 2 ( − β ˙ δ ˙ ^ + β ˙ ϕ ˙ cos δ − β ˙ θ ˙ sin δ + α ˙ δ ˙ ^ + α ˙ ϕ ˙ cos δ − α ˙ θ ˙ sin δ ) + Ω ˙ + θ ¨ cos δ − θ ˙ δ ˙ ^ sin δ + ϕ ¨ sin δ + ϕ ˙ δ ˙ ^ cos δ − α ¨ β + α β ¨ ) I ry Ω ( 2 2 ( δ ˙ ^ − ϕ ˙ cos δ + θ ˙ sin δ ) + α ˙ ) + ( I ry − I rx ) ( θ ˙ cos δ + ϕ ˙ sin δ ) ( 2 2 ( δ ˙ ^ − ϕ ˙ cos δ + θ ˙ sin δ ) + α ˙ ) + I rz ( 2 2 ( δ ¨ ^ + ϕ ¨ cos δ − ϕ ˙ δ ˙ ^ sin δ − θ ¨ sin δ − θ ˙ δ ˙ ^ cos δ ) − α ˙ θ ˙ cos δ − α ˙ ϕ ˙ sin δ + β ¨ ) ]

From Equation (9), it can be seen that, for every magnetically suspended SGCMG, I ry Ω δ ˙ ^ is the gyro coupled torque produced by the gimbal rotation relative to the inertial frame. I rr δ ¨ ^ is the inertial coupled torque produced by the gimbal. I_ryΩϕ̇cosδ and I_ryΩθ̇sinδ are the gyro coupled torque produced by the satellite rotation. I_rrϕ̈cosδ and I_rrθ̈sinδ are the inertial coupled torque produced by the satellite rotation. I_ryΩβ̇ and I_ryΩα̇ are the gyro coupled torque produced by the rotor tilt relative to the magnetic bearing. I_rrα̈ and I_rrβ̈ are the inertial coupled torque produced by the rotor accelerating tilt relative to the magnetic bearing.

Equation (9) includes the gimbal angular velocity and the satellite angle acceleration. It shows that there is strong coupling between the rotor, the gimbal and the satellite. Moreover, there are many trigonometric functions, for example, sinδ and cosδ, which shows strongly nonlinear characteristics.

In this paper, the pyramid configuration using four magnetically suspended SGCMGs is adopted (Figure 4). The frame of the n-th CMG can be obtained by rotating the satellite body frame γ_n about the Z_b axis, and then rotating σ_n about the X_b axis. γ = [γ₁  γ₂  γ₃  γ₄]^T = [90°  180°  270°  0]^T, σ = [σ₁  σ₂  σ₃  σ₄]^T = [53.13°  53.13°  53.13°  53.13°  ]^T.

By rotating the rotor frame 45° in the negative direction of the Y_g axis, rotating −δ about the X_g axis, and then transforming according to the installation matrix of the pyramid configuration, the attitude transformation matrix C r b from rotor frame to satellite body frame can be obtained. Because the magnetic bearing gap is small, the rotation displacement can be ignored relative to the rotation of the gimbal and the satellite, then C r f ≈ E: (10) C r b = C cmg b C g cmg C f g C r f ≈ C cmg b C g cmg C f gwhere (11) C cmg b = [ sin σ sin γ cos γ − cos σ sin γ − sin σ cos γ sin γ cos σ cos γ cos σ 0 sin σ ] ,     C g cmg = ( C cmg g ) − 1 ,     C f g = [ cos 45 ° 0 sin 45 ° 0 1 0 − sin 45 ° 0 cos 45 ° ]

The transformation matrix from gimbal frame to body frame is: (12) C g b = C cmg b C g cmg = [ sin σ sin γ cos γ cos δ − cos σ sin γ sin δ − cos γ sin δ − cos σ sin γ cos δ − sin σ cos γ sin γ cos δ + cos σ cos γ sin δ − sin γ sin δ − cos α cos γ cos δ cos σ sin σ sin δ sin σ cos δ ]

Equation (10) can be written as: (13) C r b = [ 2 2 sin δ cos γ + 2 2 sin σ sin γ + 2 2 cos δ cos σ sin γ cos δ cos γ − sin δ cos σ sin γ − 2 2 sin δ cos γ + 2 2 sin σ sin γ − 2 2 cos δ cos σ sin γ 2 2 sin δ sin γ − 2 2 sin σ cos γ − 2 2 cos δ cos σ cos γ cos δ sin γ + sin δ cos σ cos γ − 2 2 sin δ sin γ − 2 2 sin σ cos γ + 2 2 cos δ cos σ cos γ 2 2 cos σ − 2 2 cos δ sin σ sin δ sin σ 2 2 cos σ + 2 2 cos δ sin σ ]

Then, the absolute angular velocity of the CMG frame can be obtained: (14) C b cmg ω ib b + ω g = [ sin σ sin γ − sin σ cos γ cos σ cos γ sin γ 0 − cos σ sin γ cos σ cos γ sin σ ] [ ω ibx b ω iby b ω ibz b ] + [ δ ˙ 0 0 ]

The torque generated by the rotor in body frame can be obtained after the sum of four CMGs: (15) M b = ∑ n = 1 4 C r , n b M n r

By Equation (15), M^b, the torque of the magnetically suspended SGCMG cluster acting on the three inertial principal axis of the satellite can be obtained.

The satellite attitude dynamic equation including magnetically suspended SGCMGs can be obtained by the law of angular momentum conservation: (16) J ω ˙ ib b + ω ib b × ( J ω ib b ) + M b = u dwhere J is moment of inertia of the satellite. ω ib b is the angular velocity vector relative to the inertia frame, and it is shown in the body frame. u_d is the environmental disturbance torque, such as aerodynamic drag gravity gradient, solar radiation pressure and Earth magnetic torque. ω ib b × ( J ω ib b ) represents the gyroscope torque generated by the satellite rotation.

Equation (16) can be simplified to J ω ˙ ib b = u c + u d, where u_c is the command control torque acting on the satellite. The command torque for magnetically suspended SGCMGs is: (17) u g = − u c − ω ib b × H g − ω ib b × ( J ω ib b )where H_g is the total angular momentum of the actuator relative to the satellite body frame. The gimbal angle of four magnetically suspended SGCMGs is δ = [δ₁  δ₂  δ₃  δ₄]^T, where δ_n represents the gimbal angle of the n -th CMG. By substituting γ into Equation (12), the total momentum of four magnetically suspended SGCMGs is: (18) H g = h 0 [ − cos σ 1 sin δ 1 − cos δ 2 + cos σ 3 sin δ 3 + cos δ 4 cos δ 1 − cos σ 2 sin δ 2 − cos δ 3 + cos σ 4 sin δ 4 sin σ 1 sin δ 1 + sin σ 2 sin δ 2 + sin σ 3 sin δ 3 + sin σ 4 sin δ 4 ]where h₀ = I_ryΩ denotes the rotor angular momentum. Then the variation rate of CMG angular momentum in the satellite body frame can be obtained: (19) u g = H ˙ g = C ( δ ) δ ˙where C(δ) is: (20) C ( δ ) = [ − cos σ 1 cos δ 1 sin δ 2 cos σ 3 cos δ 3 − sin δ 4 − sin δ 1 − cos σ 2 cos δ 2 sin δ 3 cos σ 4 cos δ 4 sin σ 1 cos δ 1 sin σ 2 cos δ 2 sin σ 3 cos δ 3 sin σ 4 cos δ 4 ]

The command gimbal rate is calculated by the robust pseudo-inverse steering law with null motion [15]: (21) δ ˙ = δ ˙ C + δ ˙ Nwhere δ̇_C denotes the command value calculated by robust pseudo-inverse steering law, δ̇_N denotes the command value calculated by null motion.

The attitude kinematics are described with the quaternion of the satellite body frame relative to the orbit frame, and then the attitude kinematics equation is: (22) q ˙ = 1 2 q ⊗ ω obwhere q = [q₀  q₁  q₂  q₃]^T represents the attitude quaternion, ⊗ means quaternion multiply. The satellite body rate relative to the orbit is ω_ob = ω_b − Aω_o. A is the attitude rotation matrix relative to the orbit frame. ω_o = [0  −ω₀  0]^T is the orbit velocity.

In this section, aiming at the dynamic characteristics of a single gimbal magnetically suspended CMG, the simulation is performed without the attitude control loop. The gimbal acceleration is 120°/s², the maximal gimbal rate is 10°/s, and the simulation period is 3 s. The gimbal rate limit is δ̇_max = 10°/s. Figures 6 and 7 are the simulation results of with and without feedforward compensation. From the local zoom in of Figures 6(a) and 7(a), it can been seen that, the vibration displacement range is decreased to some degree after using the feedforward compensation, and then a faster response and the higher precision of the output torque can be obtained. From the comparison of Figure 6(b) with Figure 7(b), it can be seen that the overshooting of control current is reduced by using feedforward compensation, and the power consumption is decreased. Because the magnetic bearing control frequency is highly relative to the satellite motion, the disturbance torque induced from the satellite angle velocity can be compensated by the feedforward loop.

Figures 6(c) and 7(c) give the extent of tilt of the rotor relative to the magnetic bearing. It can be seen that after using feedforward compensation, the tilt extent is reduced, but the relative tilt angle velocity is increased, which can be seen from Figures 6(d) and 7(d). Figures 6(e,f) and 7(e,f) correspond to the displacement of the two sides of the rotor. It can be seen that the rotor displacement in the maneuvering process is reduced apparently by the feedforward compensation.

The attitude control period is 0.5 s. The maximal gimbal rate is δ̇_max = 10°/s. The simulation results of the three-axis attitude control of the satellite with feedforward compensation are given in Figures 8–13. Figure 8 is the satellite attitude angle. Figure 9 is the satellite angle velocity. Figure 10 gives the gimbal rates of four magnetically suspended SGCMGs. Figure 11 gives the gimbal angles of four magnetically suspended SGCMGs. Figure 12 is the satellite angle acceleration.

The magnetic bearing force in each channel depends on the magnetic bearing coefficient, the magnetic gap of the two sides of the magnetic bearing, the bias current, the rotor displacement and the control current, etc. When the magnetic bearing design is finished, it is determined by the magnetic bearing coefficient and magnetic bearing gap. By adjusting the control current, the magnetic bearing force is changed to assure the convergence of rotor displacement, but the magnetic bearing force has a nonlinear relationship with the rotor displacement and the control current. The magnetic bearing controller is used to make sure the stability under the working point.

In Figures 14–19, the left column gives the rotor X–Y displacement, the magnitude of rotor displacement and control current without feedforward compensation. The right column gives the results with feedforward compensation.

It can be seen that, after using feedforward compensation, the rotor displacement is reduced to about 30%, which decreases the risk of the rotor instability. Figures 18–19 show that the control current does not vary apparently. Although the rotor displacement and the given control current are reduced, extra control current is needed for feedforward compensation. It is to say, the command torque is invariant. The torque is output by the control current to assure the stable suspending of the rotor, and so the control current did not vary apparently. By using feedforward compensation, the maximal displacement of the rotor is less than 10% of the magnetic bearing protecting gap, which can ensure the requirement of the stability of magnetically suspended rotor. Figures 20–23 give α, β, α̇ and β̇ of the magnetically suspended SGCMG rotor, which describe the motion of the rotor relative to the magnetic bearing.

By using feedforward compensation for the magnetically suspended control system, the response of angle acceleration and angle velocity becomes faster owing to the increase of the control current, which decreases the rotor angle displacement. There is a magnetic saturation problem in the magnetic bearing design, so there exists a magnetic bearing control current saturation limit. When the satellite and the gimbal rate exceed the limit, the additional disturbance torque on the rotor is large, which exceeds the extent of the control current, and then the rotor becomes instable, even resulting in CMG failure. This will influence the stability of the whole system. When the maximal gimbal rate limit is δ̇_max = 12°/s, the results of not using feedforward compensation are shown in Figures 24 and 25.

The terminative condition of the simulation is whether the rotor displacement exceeds the magnetic gap (100 μm) or not. The above figures show that, the simulation finish in less than 1 s. The reason is that magnetically suspended SGCMG3 becomes instable. Because the gimbal angular acceleration command limit is not performed, the overshooting of gimbal servo system results in the real gimbal rate being larger than 12°/s, and then the equivalent additive magnetic bearing torque is large, which results in an increase of the rotor displacement. There exists the magnetic bearing control current saturation limit, it is difficult to provide sufficient magnetic bearing torque, which results in the rotor displacement exceeding the approximately linear zone and reaching the magnetic gap instantaneously, and then the magnetically suspended rotor becomes instable. In the end, the stability of magnetic bearing control system and satellite control system are destroyed.

The curves of gimbal rate and rotor displacement with feedforward compensation are given in Figures 26 and 27.

It can be seen that the rotor displacement is reduced by the feedforward compensation. Although there is overshooting of the gimbal rate, even exceeding 14°/s, the displacement of the rotor is less than 5 μm. This ensures the normal use of the magnetically suspended SGCMG. There are also some research results that are related to the compensation and disturbance attenuation [16–19]. In the future, the advanced algorithm will be employed.

The output torque of magnetically suspended SGCMG includes the inertial coupled torque and the gyro coupled torque, which is produced from the coupling of the satellite, the gimbal and the rotor. Traditionally, the inertial coupled torque is ignored when Equation (19) is used to calculate the output torque according to the Jacobian matrix and the gimbal rate. In this paper, the total torque is produced from the magnetically suspended rotor dynamic equation, as shown in Equation (15). By using this method, a more actual output torque of the actuator can be obtained. This is important for the analysis of the agile satellite attitude control system. Equation (15) also includes the strong coupling relationship between the rotor, gimbal, and satellite. This results in strongly nonlinear in the trigonometric function item that is relative with the gimbal angle. Figure 28 shows the torque calculated from Equation (19). Figure 29 gives the total output torque calculated from Equation (15).

From the simulation results, it can be seen that the trend of torque curves calculated using Equations (15) and (19) are consistent. Figure 29 shows the influence of the rotor imbalance vibration and gimbal system, which is high frequency compared with the satellite attitude control system, so in the simulation, the torque model in Equation (15) should be used, which can emulate the dynamic characteristics of a magnetically suspended SGCMG more really. In the feedforward compensation for a magnetically suspended rotor, the compensation should be performed according to the rotation rate of the CMG frame with respect to the inertial frame, so in a situation of satellite rapid maneuver, the dynamic modeling of the magnetically suspended SGCMG is very important. The simulation results show that it coincides with the theoretical analysis results.