Force and Torque Analytical Models of a Reaction Sphere Actuator Based on Spherical Harmonic Rotation and Decomposition

Abstract

Abstract-This paper presents an analytical model for the force and torque developed by a reaction sphere actuator for satellite attitude control. The reaction sphere is an innovative momentum exchange device consisting of a magnetic bearings spherical rotor that can be electronically accelerated in any direction making all the three axes of stabilized spacecrafts controllable by a unique device. The spherical actuator is composed of an 8-pole permanent magnet spherical rotor and of a 20-coil stator. Force and torque analytical models are derived by solving the Laplace equation and applying the Lorentz force law. The novelty consists in exploiting powerful properties of spherical harmonic functions under rotation to derive closed-form linear expressions of forces and torques for all possible orientations of the rotor. Specifically, the orientation of the rotor is parametrized using seven decomposition coefficients that can be determined noniteratively and in a linear fashion by measuring the radial component of the magnetic flux density from at least seven different locations. Therefore, force and torque models for all possible orientations of the rotor are expressed in closed form as linear combination of mutually orthogonal force and torque characteristic matrices, which are computed offline. The proposed analytical models are experimentally validated using a developed laboratory prototype.Index Terms-Force and torque model, reaction sphere, satellite attitude control, spherical actuator.

Spherical Magnetic Dipole Actuator for Spacecraft Attitude Control

“…To Figure 1.5 [29,30]. Analytical force and torque models for this actuator are developed and confirmed through finite-element modeling [31] and experimental investigations [32][33][34]. Also, their studies examine optimal stator sensor placement [31], rotor design optimization [35], back-EMF modeling [30], and eddy current losses [36,37].…”

Closed-loop magnetic bearing and angular velocity control of a reaction sphere actuator

“…A first RS laboratory prototype was manufactured to validate the force and torque analytical models [28]. In this prototype, the eight PM poles of the spherical rotor were discretized using a mosaic of 728 cylindrical magnets to approximate the desired fundamental spherical harmonic.…”

“…2) are applied to the rotor to obtain a spherical surface suitable for this approach. Then, nine single-axis Hall sensors (CY-P15A) are employed to measure the radial component of the magnetic flux density at different locations equidistant from the rotor, which allows computing in closed-form force and torque models for any possible orientation of the rotor [28,31] and estimating the rotor angular velocity [32]. Force and torque models as well as the angular velocity of the rotor are necessary for real-time closed-loop operation of the actuator.…”

“…As one can observe, spherical harmonic coefficients of a given degree n are expressed as a linear combination of spherical harmonic coefficients of the immobile rotor with the same degree n. Spherical harmonic decomposition coefficients c l n;imm of the immobile rotor can be computed from FEM simulations or measured values [31]. Finally, methods to compute coefficients c m n ða; b; cÞ for real-time control operation, which do not require expression (6), are proposed in [28,31]. The analytical expression of the magnetic flux density with a non-magnetic stator can be easily obtained from Eq.…”

“…By exploiting the superposition principle, the torque developed by the RS can be expressed in matrix form as [28,31] T…”