Analysis and Experiment of 5-DOF Decoupled Spherical Vernier-Gimballing Magnetically Suspended Flywheel (VGMSFW)

Due to the capacity of outputting both the high precision torque and the instantaneous large torque, the vernier-gimballing magnetically suspended flywheel (VGMSFW) is regarded as the key actuator for spacecraft. In this paper, a 5-DOF active VGMSFW is presented. The 3-DOF translation and 2-DOF deflection motions of the rotor are respectively realized by the spherical magnetic resistance magnetic bearings and the Lorentz magnetic bearing. The mathematical model of the deflection torque is established, and the decoupling between the 2-DOF deflections is demonstrated by the numerical analysis method. Compared with the conventional cylindrical magnetic bearings-rotor system, the spherical system is proven to eliminate the coupling between the rotor translation and deflection. In addition, a set of spherical magnetic resistance magnetic bearings with six-channel decoupling magnetic circuit are adopted to achieve the 3-DOF translation decoupling. The rotor dynamic model is derived, and the control system is established. The decoupling experiments and the torque experiments of the prototype are carried out. The results show that the decoupling among 5-DOF motions is realized and the instantaneous large torque can be obtained, which indicates that the requirements of the spacecraft can be highly satisfied by the spherical VGMSFW.


I. INTRODUCTION
The ball bearing flywheels and the control moment gyroscopes are usually used for three-axis attitude stability control of spacecraft [1]- [3]. Flywheels are suitable for high-precision attitude pointing and high-stability attitude control because of the high precision output torque [4], [5]. However, the rapid maneuvering requirement for the spacecraft is hard to meet. By means of control moment gyroscopes with the advantage of large output torque, it is easy to achieve the fast attitude maneuver for the large-scale spacecraft [6]- [8], but difficult to guarantee the control precision [9], [10]. The limitations above can be remedied when The associate editor coordinating the review of this manuscript and approving it for publication was Halil Ersin Soken . the gimballing flywheels integrating advantages of the flywheels with high precision torque and the control moment gyroscopes with large torque is used [11], [12]. The attitude control precision of spacecraft with gimballing flywheel is insufficient on account of the mechanical friction and vibration of the ball bearing [13], [14]. Owing to the advantages of no stiction-friction effect, virtually zero wear, long service life, high control precision, micro vibration and so on [15], [16], the VGMSFWs [17]- [30] are the attractive inertial actuator for the remote sensing satellites and the space telescopes. Two working modes including high precision and agile maneuver can be used for the VGMSFWs. In the former mode, the high precision attitude control is realized by controlling the rotor rotating speed. In the latter mode, the instantaneous large control torque is generated by tilting VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ the high-speed rotor. Simultaneously, the rapid stability of the spacecraft attitude is realized when the rotor vibration is suppressed with the help of the magnetic suspension technique. The magnetic bearing (MB) can be categorized as the magnetic resistance MB and the Lorentz MB. The former with the advantages of high rigidity and low power consumption are widely adopted in the early VGMSFWs. C. Murakami et al. [17] presented a 3-DOF VGMSFW with an axial magnetic resistance MB for three-axis control. The axial translation and radial deflections of the rotor are controlled by two pairs of independent coils, and the radial translation stability are passively realized by permanent magnets. Seddon and Pechev [18] proposed another axial passive suspension scheme of VGMSFW for 4-DOF attitude control. Its radial translations and deflections are realized by a radial magnetic resistance MB. In order to remedy the limitations of passive suspension schemes in [17], [18] with the low precision torque, Horiuchi et al. [19] adopted the electromagnetic magnetic resistance MBs to achieve 5-DOF active suspension of the flywheel rotor. The suspension consumption is high as the bias magnetic flux and the control magnetic flux are both generated by the coil currents. Wen and Fang [20] and J. Tang et al. [21] respectively proposed another VGMSFW with the permanent magnet biased magnetic resistance MBs. The bias magnetic flux of the electromagnetic magnetic resistance MBs in [19] is replaced by that of the permanent magnet. Based on it, Peng et al. [22] introduced a synchronous vibration control method with a two-stage notch filter to suppress the vibration of the rotor. To improve the inertia-mass ratio of the flywheel rotor, Xie et al. [23] proposed an outer rotor VGMSFW with conical configuration. The load-bearing MBs were used for simulating the space weightless environment during ground testing. Saito et al. [24] developed another similar VGMSFW, which was carried on the ''SERVIS-2'' satellite launched in 2010. The control precision in [17]- [24] are relatively low due to the poor linearity of the magnetic resistance MB, and the complexity of the control systems is increased.
The Lorentz MBs with good linearity and no displacement stiffness, are more suitable for the high-precision and stable suspension of the flywheel rotor. Gerlach et al. [25] introduced the scheme of the VGMSFW manufactured by the Rockwell Collins Deutschland GmbH (formerly Teldix). The radial/axial translations and deflections are all controlled by Lorentz MBs. Liu et al. [26] put forward another similar scheme, and the attitude control and the attitude sensitivity of VGMSFW were analyzed systematically. When the coil current is constant, the electromagnetic force generated by the Lorentz MB is less than that produced by the magnetic resistance MB.
Combining the magnetic resistance MBs with large bearing capacity and the Lorentz MBs with high control precision, Li et al. [27] presented a hybrid VGMSFW. Similar to the schemes in [17], [18], the control precision is limited on account of the passive axial suspension force. Xiang and Tang et al. [28], Tang et al. [29] proposed a conical hybrid VGMSFW to achieve 5-DOF active control. The decoupling between the deflection force and the translation force is realized when the forces passes through the rotor centroid. Based on it, Liu et al. [30] introduced another hybrid VGMSFW with conical spherical hybrid magnetic resistance MBs.
In the schemes above, when the rotor is tilted, the interference torque is induced due to the change of the magnetic air gap with cylindrical shell, thin-wall and conical shell. To eliminate the coupling among 5-DOF motions, a novel spherical VGMSFW developed to meet the ever-increasing precision and maneuverability requirements of the spacecrafts is presented in this paper. The spherical magnetic resistance MBs-rotor system is adopted for decoupling between the rotor translation and deflection. The mathematical models of the rotor deflection and translation are established. The electromagnetic forces and the electromagnetic torques are analyzed. The decoupling experiment and torque experiment of the prototype are carried out.

II. STRUCTURE AND PRINCIPLE OF THE SPHERICAL VGMSFW
As shown in Figure 1, the VGMSFW is mainly composed of a sphere rotor, three spherical magnetic resistance MBs, a Lorentz MB, a motor and twelve eddy current displacement sensors. The sphere rotor driven by the brushless DC motor is utilized to output the gyroscopic moment by changing the rotor speed and the rotating shaft of the rotor at high speed. The stable suspension and the translational control in radial and axial directions are respectively realized by a radial spherical magnetic resistance MB and a pair of axial spherical magnetic resistance MBs. The 2-DOF deflections of the rotor are achieved by the Lorentz MB. To sense the real-time position of the rotor, the four radial eddy current displacement sensors are adopted for measuring the radial displacements, and the eight axial eddy current displacement sensors are used for detecting the axial displacements and the deflection angles. The displacement signals measured by the sensors are transported to the control system for adjusting the rotor attitude. In addition, the vacuum environment is kept by the three gyro houses. A pair of touchdown bearings is utilized to protect the rotor when the MBs are off or invalid.

III. THEORETICAL ANALYSIS OF 5-DOF DECOUPLING A. DECOUPLING BETWEEN 2-DOF DEFLECTIONS
In this VGMSFW, the structure of the Lorentz MB is shown in Figure 2a. The permanent magnet rings and the magnetic flux rings are fixed in the rotor edge slot. Four sets of the coils wound around the stator frame are placed in the air gap between the permanent magnet rings. The magnetic flux flowing across the four evenly distributed coils through two magnetic flux rings is produced by the four permanent magnet rings. The closed magnetic circuits plotted in the blue solid lines are got as the adjacent permanent magnet rings with opposite magnetization directions. The ampere forces for 2-DOF rotor deflection are generated by loading the control currents in coils.
There are two working modes of Lorentz MBs including the active vibration suppression and the torque output. In former mode, the high-frequency and small-amplitude torques are got to compensate the interference torque acting on the suspension rotor. In latter mode, the instantaneous large torque can be generated by actively tilting the high-speed rotor.
In Figure 2b, the ampere force generated by the coil current will be changed with the angle β when the rotor is tilted around X axis with deflection angle β. The angle β can be expressed as, where λ is the longitude angle of coils. The ampere force microelements df up and df down respectively produced by the upper and lower coils can be analytically got based on the Ampere's rule, which can be written as, where n is the number of the coil turns, B is the air gap magnetic flux density, i l is the control current in the Lorentz MB coils, and r is the radius of the coils. Therefore, the deflection torque microelement dT up and dT down can be obtained as, where, r up and r down are the force arms of the upper and lower coils, which can be given by, where X 0 , Y 0 and Z 0 are the unit vectors in the X, Y, Z directions, and a is half the axial height of the coils. According to (3) and (4), the torque microelement dT generated by a set of coils can be integrated as, When the two control currents with the same value i ly and the opposite directions are loaded in the two sets of coils in the Y direction, a pair of ampere forces with the same value and opposite directions are generated to drive the rotor deflection around X axis. The deflection torque T x around X axis is written as, where λ 0 is the half of the center angle of single coil. Similarly, when the two control currents with the same value i lx and opposite directions are applied to the two sets of coils in the X direction, the deflection torque T y around Y axis is presented as, According to (7) and (8), the control currents in ±X direction are only utilized for generating the deflection torques around Y axis, and the control currents in ±Y direction are only used for producing the deflection torques around X axis. It indicates that the 2-DOF deflections of the Lorentz  MB are decoupled whether the rotor is tilted. Based on the design parameters listed in Table 1, the relationship of the deflection torques versus the rotor deflection angle and the control current is shown in Figure 3. As shown in Figure 3a and 3b, when the rotor is tilted around the X axis within ±2 • , the change of the deflection torque T y is obviously less than that of the deflection torque T x under the same excitation current. Similarly, when the rotor is tilted around the Y axis within ±2 • , the deflection torque T y will be obviously changed compared with the deflection torque T x with constant value. In Figure 3a, the variation range of the deflection torque T x is less than 2 mNm, which is far less than itself about 1.47 Nm. Thus, the deflection torque is approximately linear with the control current, and the current stiffness is about 1.5 Nm/A. Based on the analysis above, the high-precision deflections of the rotor are realized by precisely controlling the currents in the coils.

B. DECOUPLING BETWEEN TRANSLATION AND DEFLECTION
According to the Maxwell electromagnetic suction equation [21], the electromagnetic force f generated by the magnetic resistance MB stator is given by, where is the total magnetic flux generated by the magnetic resistance MB, µ 0 is the vacuum permeability, A is the area of magnetic pole, N is the number of the coil turns, i is the control current in coils, and g is the air gap between the magnetic pole and the rotor. The mathematical models of air gap are established for analyzing the effect of the rotor motions on electromagnetic forces. The air gap model of the conventional cylindrical magnetic resistance MBs-rotor system is shown in Figure 4. where e x , e y and e z are respectively the rotor offset components along the X, Y, Z directions, θ is the latitude angle of the point P, r c is the radius of the cylindrical rotor and z is the half of the rotor rim height. The radial air gap g cr can be calculated as, (11) shown at the bottom of next page, where R c is the radius of the cylindrical magnetic resistance MB stator. Similarly, the axial air gap g ca is written as, where R a is the half the distance between the stator magnetic poles of the two axial magnetic resistance MBs, M is the arbitrary point on the end of the rotor shaft, r is the distance in XOY between the point O and the point M on the end of the rotor shaft, and z 0 is the half the axial height of cylindrical rotor. When the rotor is tilted only around the X axis, the deflection interference torques T cr and T ca generated by the radial and axial magnetic resistance MBs are given by, sin θ (r c sin θ sin β +z cos β) − cos θ (r c sin θ sin β +z cos β) cos θ (r c sin θ cos β − z sin β − r c sin θ) where f cr and f ca are the electromagnetic forces generated by the cylindrical radial and axial magnetic resistance MBs, which can be expressed as, where N cr and N ca are the number of the coil turns of the cylindrical radial and axial MBs, and i cr and i ca are the coil control currents in cylindrical radial and axial MBs. The rotor is tilted around the X axis with angle 1.5 • . The design parameters of the VGMSFW are substituted into (13). The interference torques T cr and T ca along +X direction are about 0.06 Nm and 0.05 Nm, respectively. Both of them are larger than 3% of the deflection torque T x about 1.47 Nm. Similarly, the rotor deflection interference torques around the Y axis can be obtained. Based on the analysis above, the interference torques generated by the magnetic resistance MBs are the main factor affecting the deflection control precision under the rotor deflection.
To remedy the limitation of cylindrical magnetic resistance MB with the large interference torque, the spherical magnetic resistance MB is presented. The MBs-rotor system is built where r sr is the radial radius of the sphere rotor, θ and ϕ are the latitude and longitude angle of the point Q. The radial air gap g sr is obtained as, (16) shown at the bottom of next page, where R sr is the radius of the spherical radial magnetic resistance MB stator. Similarly, the axial air gap g sa can be expressed as, (17) shown at the bottom of next page, where R sa is the radius of the spherical axial magnetic resistance MB stator, and r sa is the axial radius of the sphere rotor. It can be seen in (16) and (17) that there is irrelevance between the air gap and the deflection angle. So no interference torques occur when the rotor is tilted, and the rotor decoupling between the translation and the deflection is realized.

C. DECOUPLING AMONG 3-DOF TRANSLATIONS
The radial/axial spherical magnetic resistance MBs are utilized to support the rotor, and the MBs-rotor system is shown in Figure 6. Two axial spherical magnetic resistance MBs are located at the upper and lower ends of the spherical rotor. The four stator cores with two magnetic poles are placed circumferentially in the ±X and ±Y directions. The eddy current loss in radial direction is far larger than that in axial direction, as the obvious change of the radial air gap g cr = R c − |proj PlaneXOY (OP)| = R c − (e x +r c cos θ) 2 + e y +r c sin θ cos β −(e z +z) sin β 2 (11) VOLUME 8, 2020 magnetic flux density when the rotor rotates. To decrease the radial eddy current loss, the homopolar radial MB is adopted. As shown in Figure 6, the four upper magnetic poles with the same polarities are opposite to the four lower magnetic poles. The rotor is offset along +X direction with e x . According to (16) and (17), the radial and axial air gaps g sr and g sa are given by, x + r 2 sr + 2r sr e x sin θ cos ϕ g sa = R sa − e 2 x + r 2 sa + 2r sa e x sin θ cos ϕ Based on (9) and (18), the resultant force f x in X direction generated by radial magnetic resistance MB can be got as, where ϕ 0 is the half of the center angle of a single magnetic pole in XOY, θ 1 and θ 2 are the latitude angles of the single magnetic pole upper and lower edges, N r is the number of the coil turns, i rb is the bias current for keeping the rotor stably suspending in radial direction, i rx is the coil control current in X direction, and σ r is the electromagnetic flux leakage coefficient of the radial MB. As shown in (19), the resultant force f x is only related to the offset e x and the control current i rx . Similarly, the resultant force f y in Y direction only related to the offset e y and the coil control current in Y direction i ry is obtained, when the rotor is offset along +Y direction with e y . If the rotor is offset along +Z direction with e z , the radial and axial air gap are written as,    g sr = R sr − e 2 z + r 2 sr + 2r sr e z cos θ g sa = R sa − e 2 z + r 2 sa + 2r sa e z cos θ (20) The resultant force f z in Z direction generated by two axial magnetic resistance MBs can be given by, where θ 3 is the half of the latitude angle of the inner magnetic pole, θ 4 and θ 5 are the half of the latitude angles of the inner and outer edge of the outer magnetic pole, N a is the number of the axial MB coil turns, i ab is the bias current for keeping the rotor stably suspending in axial direction, i az is the coil control current of the upper and lower axial MBs, and σ a is electromagnetic flux leakage coefficient of the axial MBs.
The resultant force f z is only determined by the offset e z and the control current i az . Based on the analysis above, the decoupling among the 3-DOF translations is realized whether the rotor is offset. The design parameters are listed in Table 2. The forcedisplacement and force-current characteristics of the magnetic resistance MBs are plotted in Figure 7. It can be seen in Figure 7a that the force-current stiffness and forcedisplacement stiffness of radial MB are 387.1N/A and −553.1N/mm. As shown in Figure 7b, the force-current stiffness and force-displacement stiffness of axial MBs are 580.5N/A and −829.3N/mm, respectively.

IV. EXPERIMENT AND ANALYSIS A. SPHERE ROTOR DYNAMICS MODEL AND CONTROL SYSTEM
When the rotor is suspended stably, the forces and torques acting on the sphere rotor are shown in Figure 8. f x , f y and f z are the translation electromagnetic forces generated by the radial and axial magnetic resistance MBs. T x and T y are the deflection torques around X and Y axes. According g sr = R sr − |OQ| = R sr − e 2 x + e 2 y + e 2 z + r 2 sr + 2r sr e x sin θ cos ϕ + e y sin θ sin ϕ + e z cos θ x + e 2 y + e 2 z + r 2 sa + 2r sa e x sin θ cos ϕ + e y sin θ sin ϕ + e z cos θ  to the Newton's second law and the gyro-kinetic equation, the dynamic model of the rotor system can be integrated as,   where m is the rotor mass, J x , J y and J z are the rotary inertial momentum around X, Y and Z axes, x, y, z are the rotor displacements along X, Y and Z directions, β and γ are the rotor deflection angles around X and Y axes, k ir and k ia are the force-current stiffnesses of radial and axial magnetic resistance MBs, k er and k ea are the force-displacement stiffnesses of radial and axial magnetic resistance MBs, k il is the torque-current stiffness of the Lorentz MB, and ω is the rotor rotating speed. Equation (22) can be rewritten in the form Step signal applied to radial X-axis translation channel. (c) Step signal applied to radial Y-axis translation channel. (d) Step signal applied to deflection channel around X axis.
(e) Sine signal applied to deflection channel around Y axis.
of matrix, where, the mass matrix M= diag (m, m, m, J x , J y ), the displacement matrix q= [x, y, z, β, γ ] T , the coupling matrix  model, the control system is designed, which is shown in Figure 9. Where, F= diag (f x , f y , f z , T x , T y ) is the force matrix, (s)= (Ms 2 +K c s) −1 is transfer function of the control system, K w and K s are respectively the amplification coefficient and the sensor sensitivity.

B. EXPERIMENTAL SETUP
The prototype of the VGMSFW is manufactured. The three rotor spherical surfaces with the sphericity of 3 µm and the spherical surface roughness of 0.1 µm are machined by the mesh grinding method. The coincidence between the sphere center and the rotor centroid is realized by adjusting the thickness of the aligning ring. The on-line dynamic balance test is carried out, and the unbalanced mass of the sphere rotor is compensated by the counterweight screws in the upper and lower counterweight surfaces. The experimental platform of VGMSFW is shown in Figure 10, and the control parameters are listed in Table 3. The control current is supplied to the MBs through the power amplifier. Based on the CAN bus, the real-time control and monitoring of the system are realized by the telemetry computer.

C. DECOUPLING EXPERIMENT
To verify the performance of the spherical VGMSFW, the decoupling experiments of the prototype are implemented. When the rotor is stably suspended, the step or sine current signal is applied to the arbitrary channel of the VGMSFW system, and the displacement fluctuation curves of the other channels are plotted in Figure 11.
As shown in Figure 11a, the sphere rotor is offset along +Z direction with 50 µm due to the step signal applied to the axial channel. The radial displacement amplitude is 20 µm. The deflection displacement amplitude is 30 µm, and its corresponding deflection angle is 0.036 • . Both the radial and deflection displacements are almost unchanged. It can be seen in Figure 11b and Figure 11c that, when the step signals are respectively loaded into the X and Y translation channels, the sphere rotor is respectively offset along +X and +Y direction with ±50 µm, the displacements in the other channels are constant. The radial and axial rotor amplitudes are respectively less than 20 µm and 12 µm, and  the deflection displacement amplitudes are within 30 µm with corresponding deflection angle of 0.036 • . It indicates that there is no interference between the 3-DOF translations decoupled mutually and 2-DOF deflections. Similarly, the step and sine signals are respectively applied to the deflection channels around X and Y axes. There is no obvious displacement variation in other four channels, which can be seen in Figure 11d and Figure 11e. The interference of 2-DOF deflections to 3-DOF translations can be ignored, and the decoupling between the 2-DOF deflections is realized. Therefore, the 5-DOF motions of the spherical VGMSFW can be considered as decoupling mutually. That is accordance with the theoretical analysis in part III, which provides the basis for the high-precision and stable control of the rotor.

D. TORQUE EXPERIMENT
As the decoupling among the 5-DOF motions of the sphere rotor, the high-precision gyroscopic moment M gyro can be obtained by tilting high speed sphere rotor, which can be expressed as, where ω and are the rated speed and the procession angular velocity of the sphere rotor. The rotor rated speed is measured by the Hall switch sensors in the motor stator. The speed measurement waveform with the period of 1.25 ms is plotted in Figure 12, and the measurement frequency f m about  801.9 Hz is got. Since the six pairs of the motor magnetic poles, the rated speed is about 8019 r/min corresponding to the speed accuracy of 2.4 r/min. The sine signal with amplitude of 350 µm and frequency of 4 Hz is loaded into the Lorentz MB coils in Y direction. The axial displacement responses measured by the four axial sensors in ±Y directions are shown in Figure 13. The difference method is used for calculating the axial displacement. The procession angular velocity is got by taking the derivate of the terminal displacement curve, which is shown in Figure 14. The measurement and reference curves of the procession angular velocity are plotted in the blue and red solid line. The maximum measurement/reference angular velocities are respectively about 6.28 • /s and 6 • /s. Thus, the procession angular velocity accuracy of the sphere rotor is about 0.047 • /s, which is better than that about 0.32 • /s of the VGMSFW with conical MB in [28].
Based on (24), the curves of the gyroscopic moment generated by the sphere rotor are plotted in Figure 15. When the procession angular velocity of the rated speed rotor is at the maximum, the maximum actual and reference gyroscopic moments are about respectively 1.54 Nm and 1.47 Nm, and its corresponding accuracy is about 0.048 Nm. Therefore, the high-precision agile maneuver of the spacecraft can be achieved by the spherical VGMSFW.

V. CONCLUSION
In this paper, a 5-DOF decoupled spherical VGMSFW is developed for spacecrafts. Its structure and principle are introduced. The electromagnetic force models of the Lorentz MB and magnetic resistance MBs are built, and the decoupling among 5-DOF motions is demonstrated by the numerical analysis method. The spherical VGMSFW control system is established, and the decoupling and torque experiments are carried out based on the prototype. The decoupling among the 5-DOF motions is verified by the decoupling experiment. The torque experiment results show that the maximum gyroscopic moment about 1.54 Nm and its corresponding moment accuracy about 0.048 Nm are obtained when the sphere rotor is actively tilted. Both the two experiments indicate that the high-precision control and agile maneuver requirements for spacecraft can be effectively fulfilled by the novel spherical VGMSFW.