| dc.description.abstract | This thesis presents decoupled, closed-loop, suspension and rotation of a spherical dipole permanent magnet. The spherical magnet is used as the rotor in a bench-level prototype reaction sphere actuator for spacecraft attitude (orientation) control. Reaction spheres rotate spacecraft either with an equal-and-opposite torque about their rotation axis when accelerated about that axis, similar to reaction wheels, or with a more ecient gyroscopic torque used to reorient their rotation axis, similar to control moment gyroscopes (CMGs). This thesis focuses on the modeling, design, fabrication, control, and operation of a prototype for generating reaction torque. However, modifications to the prototype for generating gyroscopic torque are discussed. Compared to conventional reaction wheels and CMGs, reaction spheres can potentially generate higher torque and store more angular momentum for lower SWaP (Size, Weight, and Power), lower pointing jitter, and over longer mission durations due to the elimination of mechanical bearings and gimbal structures. Thus, such a reaction sphere can potentially enable attitude control performance traditionally found in large, expensive, monolithic spacecraft in fleets of small, low-cost spacecraft as the demand for small, low-cost spacecraft only continues to increase.
In the reaction sphere prototype, 12 equidistant stator coils, each centered on one face of a dodecahedron, levitate and rotate the enclosed dipole rotor using translation and rotation feedback from 12 equidistant sensing modules, each coaxially placed within one of the coils. The dipole rotor is a 38.1 mm (1.5 in) diameter, permanently axially magnetized, NdFeB (Grade 42) sphere (K&J Magnetics SX8) with a mass of 0.22 kg and a remanence of 1.22 T. The rotor can translate up to 1.0 mm from center and rotate about the 2 axes orthogonal to its magnetization axis as the coils cannot generate torque about the rotor’s magnetization axis. Each stator coil, a 12.0 mm ID x 20.0 mm OD x 10.0 mm long hollow cylinder wound from 348 turns of AWG 28 magnet wire, has a nominal center-to-center distance of 26.1 mm from a centered rotor. Each sensing module’s optical sensor (Fairchild QRE1113GR) measures the rotor’s translation along its axis with a noise-limited standard deviation of 0.24 mm. Taken as a full set, these optical sensors estimate the rotor’s translation along the inertial (X-, Y-, and Z-) axes with a standard deviation of 60 nm. Each sensing module’s Hall eect sensor (Allegro A1308KUA-1-T) measures the rotor’s radial magnetic flux density with a noise-limited standard deviation of 47 mT. Taken as a full set, these Hall eect sensors estimate the direction of the rotor’s magnetization axis with a standard deviation of 160 mrad (9 mdegree), from which the rotor’s angular velocity is estimated with a standard deviation of 6.1 rpm (0.64 rad/s).
Given the rotor’s external magnetic flux density is that of a magnetic dipole, we approximate the coil’s external magnetic flux density as that of a magnetic dipole as well. With this approximation, we can approximate a single coil’s force and torque on the rotor by the interaction of 2 magnetic dipoles. Superimposing the force and torque from each coil, the overall force and torque on the dipole rotor resembles the governing equations for voice coils and brushless DC motors respectively. Thus, the overall force and torque equations are in forms familiar to mechanical and electrical engineers to design with. Using the overall force and torque equations, we (1) determine a minimum of 12 equidistant coils are needed for 3 DoF (degree of freedom) suspension independent of the rotor’s orientation and (2) size the prototype such that a single coil can levitate the rotor against gravity by itself. Additionally, we develop commutation laws which: (1) decouple the rotor’s DoFs, reducing the control of this multi-input multi-output (MIMO) system to the control of several parallel single-input single-output (SISO) systems, and (2) reorient the rotor such that the rotor’s magnetization axis is orthogonal to the axis about which desired torque is commanded. Thus, despite the rotor’s inherent underactuation, we can rotate the rotor about an arbitrary axis in steady state. Using classic loop shaping techniques, we design a SISO controller for each of the 3 translational DoFs and another SISO controller for each of the 3 rotational DoFs. The suspension controller is a filtered proportional-integral-derivative (PID) controller with a loop crossover of 60 Hz and phase margin of 30°. The rotation controller is a filtered proportional (P) controller with a loop crossover of 20 Hz and phase margin of 60°. We implement the commutation laws and controllers in real-time on a National Instruments PXIe-8135 controller using the LabVIEW programming language with a loop sampling rate of 4 kHz.
Using this control scheme, we successfully demonstrate control of the rotor’s position and rotation. Specifically, the rotor’s position can independently step ±500 mm along each inertial axis with a measured crossover of 60 Hz and a measured phase margin of 30° as designed for. A single coil directly above the rotor requires 0.86 A to suspend the rotor against gravity, corresponding to a measured force constant of 2.45 N/A, which is within 4 % of the predicted force constant of 2.55 N/A. Additionally, the rotor’s angular velocity can independently step ±900 rpm (±94.3 rad/s) about each inertial axis with a measured crossover of 20 Hz and a measured phase margin of 60° as designed for. A sequentially applied 7.5 A coil current corresponding to torque about an arbitrary axis accelerates the rotor to a maximum angular velocity of 13 600 rpm (1420 rad/s) about that axis in 250 ms, corresponding to a measured torque constant of 21.0 mN·m/A, which is with 5 % of the predicted torque constant of 22.1 mN·m/A. | |
