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Modeling of active magnetic bearing in rotating payload satellite considering shaft motion coupling

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Abstract

This paper establishes the refined dynamic model of the five degree-of-freedom active magnetic bearing used in the earth-oriented satellite system with an unbalanced rotating payload based on the more refined air gap change model for stator motion. The disturbance effects of the thrust active magnetic bearing and the coupling effects of shaft motion are analyzed by the equivalent stiffness and damping model, especially when the satellite system is in the fine pointing phase. Numerical simulation results are provided to verify the accuracy of the proposed simplified stiffness and damping model, the disturbance term effects on active magnetic bearing, and the significance of the refined magnetic bearing dynamic model for improving the payload working ability in the fine pointing phase.

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Abbreviations

o :

Inertial coordinate system

b :

Platform’s body coordinate system whose origin is the centroid of the satellite platform. The z- and x-axes point to the forward and normal directions of the orbit, respectively, and the y-axis constitute the right hand coordinate system with z- and x-axes.

p :

Payload’s body coordinate system whose origin is the center of the rotating payload. The orientation of the co-ordinate axis is the same as that of b.

wi :

Body coordinate system of momentum wheel i (i = x, y, z, t) whose origin is their respective centroid. The orientation of the coordinate axis is the same as that of b.

ak :

Body coordinate system of solar panel k whose origin is located at the connection position between solar panel k and the satellite platform. The orientation of the coordinate axis is the same as that of b.

r 1 :

Vector from the origin of the inertial coordinate system to the origin of platform’s body coordinate system

r 2 :

Vector from the origin of the inertial coordinate system to the origin of payload’s body coordinate system

r 3 :

Vector from the origin of the inertial coordinate system to the actual center of the AMB

r 4 :

Vector from the origin of the inertial coordinate system to the expected center of the AMB

r 5 :

Vector from the expected center of the AMB to the actual center of the AMB

r 6 :

Vector from the actual center of the AMB to the left RAMB action point

r 7 :

Vector from the expected center of the AMB to the left RAMB action point

r 8 :

Vector from the actual center of the AMB to the right RAMB action point

r 9 :

Vector from the expected center of the AMB to the right RAMB action point

r 10 :

Vector from the expected center of the AMB to the equilibrium point of the TAMB at radius r and angle φ

r 11 :

Vector from the actual center of the AMB to the equilibrium point of the TAMB at radius r and angle φ

r 12 :

Vector from the equilibrium point to the action point of the TAMB at radius r and angle φ

r 13 :

Vector from the actual center of the AMB to the TAMB action point at radius r and angle φ

X :

Vector projection under the inertial coordinate system, which is from the origin of the inertial coordinate system to the platform subsystem’s centroid

X p :

Vector projection under the inertial coordinate system, which is from the origin of the inertial coordinate system to the payload subsystem’s centroid

r jb :

Vector projection under the platform’s body coordinate system, which is from the platform subsystem’s centroid to the satellite platform’s centroid

r mp :

Vector projection under the payload’s body coordinate system, which is from the payload subsystem’s centroid to the rotating payload’s centroid

1 1 :

Vector projection under the platform’s body coordinate system, which is from the satellite platform’s centroid to the expected center of AMB

l 2 :

Vector projection under the payload’s body coordinate system, which is from the actual center of the AMB to the origin of the payload’s body coordinate system

r dp :

Vector projection under the payload’s body coordinate system, which is from the origin of the payload’s body coordinate system to the rotating payload’s centroid

A :

Attitude transformation matrix of coordinate system b relative to inertial coordinate system o

A p :

Attitude transformation matrix of coordinate system p relative to inertial coordinate system o

A p b :

Attitude transformation matrix of coordinate system p relative to coordinate system b

L 1x :

Disturbance on kax, \({L_{1x}} = - (4\pi k_{zz}^{\;\;\;\prime } - 2{k_{iz}}{k_{p2}}){\delta _y}{\delta _z}/\varphi \)

L 1y :

Disturbance on kay, \({L_{1y}} = ( - 2{k_{iz}}{k_{p2}} + 4\pi k_{zz}^{\;\;\;\prime }){\delta _x}{\delta _z}/\theta \)

L 2x :

Disturbance on cax, \({L_{2x}} = 2{k_{d2}}{k_{iz}}{\delta _y}{{\dot \delta }_z}/\dot \varphi \)

L 2y :

Disturbance on cay, \({L_{2y}} = - 2{k_{d2}}{k_{iz}}{\delta _x}{{\dot \delta }_z}/\dot \theta \)

L 3x :

Disturbance on kax, \({L_{3x}} = \pi k_{zz}^{\;\;\;\prime }{({R_3} + {R_6})^2}\tan \varphi /(2\varphi )\)

L 3y :

Disturbance on kay, \({L_{3y}} = \pi k_{zz}^{\;\;\;\prime }{({R_3} + {R_6})^2}\tan \theta /(2\theta \cos \varphi )\)

L 4 :

Disturbance on kz, \({L_4} = - 4\pi k_{zz}^{\;\;\;\prime }({\delta _y}\tan \varphi /{\delta _z} - {\delta _x}\tan \theta /({\theta _z}\cos \varphi ))\)

L 5x :

Disturbance on kax, L5x = L1x − δzδy S L4/ϕ

L 5y :

Disturbance on kay, L5y= L1y+δzδxL4/θ

L 6x :

Disturbance on kax, L6x = L5x + L3x

L 6y :

Disturbance on kay, L6y = L5y + L3y

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11772102).

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Authors and Affiliations

  1. School of Astronautics, Harbin Institute of Technology, Harbin, 150001, China

    Yatao Zhao, Xueqin Chen, Feng Wang, Cheng Wei & Yang Zhao

Authors
  1. Yatao Zhao
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  2. Xueqin Chen
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  3. Feng Wang
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  4. Cheng Wei
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  5. Yang Zhao
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Corresponding author

Correspondence to Cheng Wei.

Additional information

Cheng Wei is a Professor of School of Astronautics, Harbin Institute of Technology, Harbin, China. He received his Ph.D. in Aeronautical and Astronautical Science and Technology from Harbin Institute of Technology. His research interest covers dynamics, control and system simulation technology of space flexible multibody system.

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Zhao, Y., Chen, X., Wang, F. et al. Modeling of active magnetic bearing in rotating payload satellite considering shaft motion coupling. J Mech Sci Technol 34, 4423–4437 (2020). https://doi.org/10.1007/s12206-020-1005-7

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  • DOI: https://doi.org/10.1007/s12206-020-1005-7

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