A MAGNETICALLY SUSPENDED, SPHERICAL PERMANENT MAGNETICDIPOLE ACTUATOR

Tyler T. Hamer1, Minkyun Noh1, Lei Zhou1, Joshua Chabot2, and David L. Trumper11Department of Mechanical EngineeringMassachusetts Institute of Technology

Cambridge, MA, USA2MIT Lincoln Laboratory

Lexington, MA, USA

INTRODUCTIONThe attitude control system (ACS) of a spacecraftcontains a minimum of three reaction wheels to ro-tate the spacecraft in 3 degrees of freedom (DoF),but typically contains additional reaction wheelsfor both redundancy and improved pointing accu-racy. Each wheel rotates the spacecraft aboutits axis by imparting an equal-and-opposite torquewhen the spacecraft accelerates the wheel. Sincespace, weight, and power (SWaP) are a premiumon a spacecraft, reaction spheres, which impart anequal-and-opposite torque about an arbitrary axiswhen the spacecraft accelerates the sphere aboutthat axis, have been proposed to reduce the ACSdown to a single device.

While NASA first proposed reaction spheres overa half century ago, limitations with previous de-signs have kept the technology from commercial-ization [1]. These designs can be generalized intotwo categories: asynchronous, induction-type ac-tuators and synchronous actuators similar to DCand hysteresis motors. The induction-type designsare difficult to model and suffer from eddy currentlosses in the rotor while the synchronous designsoften have rotors constructed from multiple mag-nets which presents fabrication, strength, and bal-ance issues. To incorporate the best of both worlds,the mechanical simplicity of an induction motor withthe efficiency and simple model of a brushless DCmotor, a spherical permanent magnetic dipole ro-tor actuated by a stator of surrounding coils hasbeen considered [2, 3]. This paper presents themodeling, design, and vertical suspension of a pro-totype permanent magnetic dipole reaction spheredepicted in Figure 1.

OPERATING PRINCIPLETorque and force on the dipole rotor arise from theinteraction of the rotor’s external magnetic field withthe current passing through each coil. Since therotor is a magnetic dipole, the torque c𝑇 r and forcec𝐹 r on the rotor by a single coil can be found byapproximating the coil as a magnetic dipole as well.

(A)

S

NOuter Screw

Middle Screw

Inner Screw

Adjustment

Screw Thermistor

Dowel Pin

Coil

Hall Effect Sensor

Optical Sensor

Disk Spring

Spring

Set Screw

Delrin Half-Shell

Dipole Rotor

(B)

FIGURE 1. Permanent magnetic dipole reactionsphere (a) bench-level prototype with (b) label crosssection.

The torque and force between these two magneticdipoles are

c𝑇 r = − (𝑚c ×𝐵r) (1a)c𝐹 r = − (𝑚c ·𝐵r) , (1b)

where 𝐵r is the external magnetic flux density ofthe rotor and and 𝑚c is the approximated magneticdipole moment of the coil.

The external magnetic flux density of the rotor is themagnetic flux density of a dipole 𝐵dipole:

𝐵dipole(𝑟) =𝜇0

4𝜋

[3𝑟 (𝑚dipole · 𝑟)

|𝑟|5−

𝑚dipole

|𝑟|3

], (2)

which in spherical coordinates is

𝐵dipole(𝑟) =𝜇0

4𝜋

|𝑚dipole||𝑟|3

(2 cos 𝜃𝑟 + sin 𝜃𝜃

), (3)

where 𝜇0 is the permeability of free space and𝑚dipole is the magnetic dipole moment. 𝑟 is the po-sition of the point of interest with respect to the cen-ter of the magnetic dipole, which in this case will bethe position of the center of the coil’s approximateddipole with respect to the rotor’s center r𝑟c.

The magnetic dipole moment of the rotor 𝑚r is

𝑚r = –𝑉r𝐵rem

𝜇0𝑛r, –𝑉r =

4

3𝜋𝑅3

r , (4)

where –𝑉r, 𝑅r, and 𝐵rem, are the rotor’s volume, ra-dius, and remanence respectively, and 𝑛r is a unitvector along the rotor’s axis of magnetization. Theapproximated magnetic dipole moment of the coil is

𝑚c = 𝑁𝑖c𝐴ave𝑛c, (5)

where 𝑁 is the coil’s number of turns, 𝑖c is the cur-rent through the coil, 𝐴ave is the average area en-closed by each turn, and 𝑛c is a unit vector alongthe coil’s axis.

Substituting Equations (2), (4) and (5) into Equa-tion (1) for when the rotor’s center lies along thecoil’s axis (r𝑟c/ |r𝑟c| = 𝑛c), the torque and force onthe rotor by a single coil are

c𝑇 r = 𝐾T𝑖c(𝑛c × 𝑛r) (6a)

c𝐹 r =1

2𝐾F𝑖c[3(𝑛r · 𝑛c)𝑛𝑐 − 𝑛r], (6b)

where 𝐾T and 𝐾F are respectively the reactionsphere’s torque and force constants for a single coilwhile the rotor’s center lies along that coil’s axis:

𝐾T =𝜇0 |𝑚r| |𝑚c|4𝜋 |r𝑟c|3 𝑖c

=1

3𝑁𝐴ave𝐵rem

(𝑅r

|r𝑟c|

)3

(7a)

𝐾F =3𝜇0 |𝑚r| |𝑚c|

2𝜋 |r𝑟c|4 𝑖c=

2𝑁𝐴ave𝐵rem

𝑅r

(𝑅r

|r𝑟c|

)4

.(7b)

The overall torque ctot𝑇 r and force ctot𝐹 r on the rotorare the superposition of the individual torques andforces from each coil. For a rotor centered within aradial array of identical surrounding coils (thus therotor’s center lies along each coil’s axis), these are

ctot𝑇 r =

𝑏∑𝑎=1

c𝑏𝑇 r = 𝐾T

𝑏∑𝑎=1

𝑖c𝑏(𝑛c𝑏 × 𝑛r) (8a)

ctot𝐹 r =

𝑏∑𝑎=1

c𝑏𝐹 r =1

2𝐾F

𝑏∑𝑎=1

𝑖c𝑏 [3(𝑛r · 𝑛c𝑏)𝑛c𝑏 − 𝑛r] .

(8b)

Cube Octahedron

Dodecahedron Icosahedron

FIGURE 2. Distributions of coils centered on facesof platonic solids. Green lines represent singulardipole orientations (3 DoF suspension of dipole ro-tor is lost) for cube and octahedron configurations.

In order to maintain symmetry, radial arrays corre-sponding to the coils centered on the faces of pla-tonic solids are considered. Applying Equation (8)over these distributions, a minimum of 12 equidis-tant coils is needed for full 3 DoF magnetic suspen-sion of the dipole rotor independent of the rotor’sorientation as visualized in Figure 2.

BENCH-LEVEL PROTOTYPE DESIGNFigure 1 depicts the reaction sphere bench levelprototype with a labeled cross section. Twelveequidistantly distributed coils levitate and rotate thedipole rotor while 12 optical sensors and 12 Hall ef-fect sensor, coaxially placed within each coil, mea-sure the rotor’s position and the orientation of therotor’s dipole respectively. These position and ro-tation measurements are fed back to a real-timecontroller which drives 12 transconductance poweramplifiers, each which drive a coil. The design ofthe actuation, sensing, mechanical structure, powerelectronics, and controller are described below.

ActuationThe 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 𝑀r =0.22 kg and remanence of 𝐵rem = 1.31 T. The statorconsists of two mating, symmetric Delrin half-shells,which together form a dodecahedron with a 40.1mm diameter spherical pocket that the rotor is con-tained in. Twelve equidistantly distributed cylindricalcoils, one centered on each face of the dodecahe-dron stator at a diameter of 42.1 mm, levitate androtate the rotor. Thus when the rotor is centeredwithin the spherical pocket, the prototype has a 1mm mechanical air gap and 2 mm magnetic air gapand the rotor can translate up to ±1 mm in 3 DoF.

TABLE 1. Actuation Key Parameters.

Symbol Parameter Value𝑀r rotor mass 0.22 kg𝐵rem rotor remanence 1.31 T𝑅r rotor radius 19.05 mm|r𝑟c| rotor-coil center distance 26.6 mm𝑁 coil turns 348𝐴ave turn average enclosed area 201 mm2

𝐾T single-coil torque constant 11.2 mN·m/A𝐾F single-coil force constant 2.52 N/A

Each cylindrical coil is wound from 𝑁 = 348 turnsof 0.3 mm diameter (AWG 28) magnet wire mak-ing a footprint of 20 mm outer-diameter × 12 mminner-diameter × 10 mm long. The average areaenclosed by a turn of each coil is 𝐴ave = 201 mm2.Additionally, each coil has a series self-inductanceof 𝐿s = 1.6 mH and a series resistance of 𝑅s = 4.1Ω as measured at 100 Hz by an Agilent 4284A LCRmeter. When the rotor is centered, FEMM simula-tion determined an effective distance between therotor’s center and the center of each coil’s effectivedipole is |r𝑟c| = 26.6 mm. Substituting these val-ues into Equation (7), the torque constant and forceconstant of each coil are estimated to be 𝐾T = 11.2mN·m/A and 𝐾F = 2.52 N/A respectively, requiringa predicted 0.84 A for 1 DoF suspension by a singlecoil directly above the rotor. The key parameters foractuation are summarized in Table 1.

SensingThe stator contains 12 equidistantly distributedsensing modules, one coaxially placed within eachof the coils, for feedback. Each sensing mod-ule consists of a reflective optical sensor (FairchildQRE1113GR) for measuring the translational po-sition of the rotor and a Hall effect sensor (Alle-gro A1308KUA-1-T) for measuring the orientationof the rotor’s dipole (rotational position). Withineach sensing module, each sensor is mounted onits own printed circuit board (PCB) spaced 15 mmapart. This allows the optical sensor to be posi-tioned about 1 mm from the rotor when closest forhighest sensitivity and the Hall effect sensor to bepositioned at least 15 mm from the rotor when clos-est to avoid saturation. When the rotor is centered,the prototype has an optical sensing air gap of 2mm and a Hall effect sensing air gap of 17 mm.

The sensitivity of the optical sensor and the Hall ef-fect sensor with respect to the rotor’s position andorientation of its dipole are provided in Figure 3.To measure these sensitivities, a sensor moduleis mounted to an aluminum spacer above a steelXY linear stage (Newport 461-XY-M) and a steelZ linear stage (by Oriel Corporation) with the rotor

FIGURE 3. Measured sensor module (optical sen-sor and Hall effect sensor) voltage as functions ofthe rotor’s translation and rotation of its dipole.

mounted on an aluminum spacer above a steel ro-tary stage (Melles Griot 98 mm dia. stage) acrossfrom it. The aluminum spacers keep the rotor atleast 50 mm from the stages to ensure the stages’steel does not distort the rotor’s dipole magneticfield. The Y-and Z-axes orient the sensor moduleradially with the rotor. Then the rotary stage setsthe dipole’s orientation and the X-axis moves thesensor module radially.

With the 0∘ orientation denoting the rotor’s northpole facing the sensor module, the rotary stage in-dexes the rotor in 45∘ increments. For each ori-entation of the rotor’s dipole, the X-axis steps thesensor module from 0.65 mm to 20.5 mm from therotor’s surface in 20 𝜇m increments. At each posi-tion, the voltage of both sensors are acquired witha sampling rate of 10 kHz using a National Instru-ments (NI) PXI-6289 M-Series analog input card of750 kHz bandwidth, 18 bit resolution, and an inputrange of ±10 V for the optical sensor and ±5 V forthe Hall effect sensor.

The optical sensor’s voltage did not vary with thedipole’s orientation, but varied as a cubic polyno-mial with its distance from the rotor. Linearizedabout the optical sensor’s distance from a centeredrotor (2 mm), the optical sensor has a sensitivity of𝐻opt = 2.75 V/mm. The Hall effect sensor’s voltagevaried sinusoidally with the dipole’s orientation andas a cubic inverse with its distance from the rotorjust like the radial component of the rotor’s mag-netic flux density in Equation (3). Thus the Hall ef-fect sensor’s voltage varied linearly with the radialcomponent of the rotor’s magnetic flux density witha sensitivity of 𝐻Hall = 12.8 V/T.

FIGURE 4. Histograms of senor module noise.

Using the same setup, each sensor’s noise level ismeasured over a period of 10 sec for the 0∘, 90∘,and 180∘ orientations of the rotor’s dipole at dis-tances from the rotor corresponding to the rotor’s-1 mm, 0 mm, and +1 mm positions. Figure 4shows histograms for these 9 stationary configu-rations. Both sensor’s noise levels are Gaussiandistributed and independent of the dipole’s orienta-tion and distance from the rotor. The optical sen-sor’s voltage noise has a standard deviation of 0.67mV, 9 times larger than the quantization error, cor-responding to a position noise standard deviationof 0.24 𝜇m. The Hall effect sensor’s voltage noisehas a standard deviation of 0.61 mV, 16 times largerthan quantization error, corresponding to a mag-netic flux density noise standard deviation of 48 𝜇T.

Mechanical DesignThe stator consists of two mating, symmetric Del-rin half-shells. The bottom half-shell is held up by 5standoffs running to 80-20 pillars mounted to an op-tical breadboard. These half-shells are made fromDelrin as eddy currents formed from the rotor rotat-ing in metal half-shells would shield the Hall effectsensors. Further, the surface of spherical pocketcan then be used as a touch down bearing. To en-sure the Delrin does not overheat, four embeddedthermistors monitor the stator’s temperature.

The rotor can be easily placed within or removedfrom the stator by separating the two half-shells. Toensure the half-shells mate in a repeatable fashion,the mating surfaces use two dowel pins, one in aslot and the other in a hole. The position of the ro-tor within the stator’s spherical pocket can be man-ually adjusted externally using 3 adjustment screwswithin the stator. This allows the rotor’s rotation tobe tested separately from its suspension if needed.

To coaxially place each sensing module within itsaccompanying coil, but allow for radial adjustmentif needed, a series of three hollow screws is used.The inner screw provides clearance for the sens-

FIGURE 5. Circuit diagram for power amplifier witha DC gain of 0.2 A/V and 5 kHz bandwidth [4].

ing module’s wires and with a spring, preloads thesensing module against a retaining ring in the mid-dle screw. The outer screw with a plastic diskspring, preloads the coil into a radial pocket cen-tered on one of the stator’s 12 outer faces. Themiddle screw translates along the coil’s inner boreand outer screw, positioning the sensing module ra-dially while keeping the module in coaxial alignmentwith the coil. Set screws in the stator and two outerscrews later lock each hollow screw into place.

Power ElectronicsEach coil is driven by its own transconductance lin-ear power amplifier, whose circuit is provided in Fig-ure 5. These amplifiers were previously used todrive a bearingless hysteresis motor [4] and lateradjusted for use with these coils. A sense resis-tor of 𝑅s = 0.1 Ω feeds back the current throughthe coil 𝑖c to a control stage containing of 3 high-voltage op amps (Texas Instruments OPA 445 orOPA 552) to provide PI compensation. The con-trol stage drives a power stage containing a linearpower amplifier (Apex Microtechnology PA12) witha gain in the feedback path below 0.25 for stability.The entire circuit drives current through the coil witha DC transconductance gain of 𝐺p = 0.2 A/V and abandwidth of 5 kHz.

ControllerSignals from the 12 sensing modules are acquiredusing two NI PXI-6289 M-Series analog input cards.These are fed back to a control algorithm imple-mented in LabVIEW on a NI PXIe-8135 embeddedcontroller and used to drive the 12 power amplifiers,and in turn their corresponding coils, via an NI PXIe-6739 dedicated analog output card. For initial test-ing, a control algorithm running at 2 kHz levitatesthe rotor using a single coil directly above the ro-tor via feedback from the optical sensor in that coil’scoaxial sensing module. The coil above the rotoris used as the other DoF besides rotation aboutthe dipole are passively stable. Figure 6 shows theblock diagram for this control system.

FIGURE 6. Block diagram for vertical suspension of the dipole rotor by the coil and optical sensor above it.

The rotor’s weight passes through the inverse ofthe single-coil motor constant for a centered rotor togenerate the single-coil steady state bias current, apredicted 𝑖c-ss = 0.84A. The bias current then passesthrough the inverse of the power amplifier’s DC gainto generate the steady state bias control effort 𝑢ss tocompensate for the rotor’s weight. This control ef-fort is added to the small displacement control effort𝑢𝛿 responsible for compensating for rotor’s smallvertical deviations from the center.

The small displacement control effort is generatedby a lead compensator 𝐶(𝑠) from the difference be-tween the rotor’s desired vertical position 𝑧ref (0 mmcorresponding to a centered rotor) and the rotor’svertical position measured by the optical sensor𝑧meas. To design the lead compensator, a linearizeddynamic model 𝑃 (𝑠) for the rotor’s vertical positionfrom the center 𝑧 with respect to small changes inthe current through the coil 𝑖c-𝛿 is built. It is repre-sented by the following transfer function:

𝑃 (𝑠) =𝑧

𝑖c-𝛿=

𝐾F

𝑘

(𝑘/𝑀r

𝑠2 − 𝑘/𝑀r

), (9)

where 𝑘 is the negative stiffness of the centered ro-tor’s vertical suspension given by

𝑘 = 4𝐾F𝑖c-ss/ |r𝑟c| . (10)

Substituting in the actuation key values, the rotor’ssuspension instability has a predicted break fre-quency of

√𝑘/𝑀r = 6.1 Hz.

Using this model, the DC gain of the transconduc-tance amplifier, and the sensitivity of the opticalsensor, the following lead compensation is used tocrossover at a predicted 𝜔c = 60 Hz with a predictedphase margin of 𝜑M = 25∘:

𝐶(𝑠) = 𝐾p𝛼𝜏d𝑠 + 1

𝜏d𝑠 + 1, (11)

where 𝐾p = 14.8, 𝛼 = 3, and 𝜏d = 0.0015 sec.

INITIAL EXPERIMENTATION & RESULTSUsing the controller above, the rotor is success-fully levitated in the center of the stator’s spherical

TABLE 2. Vertical Suspension Performance.

Symbol Parameter Predicted Measured𝑖c-ss bias current 0.84 A 0.86 A𝐾F force constant 2.52 N/A 2.45 N/A√𝑘/𝑀r instability freq. 6.1 Hz 6.7 Hz𝜔c crossover freq. 60 Hz 50-64 Hz𝜑M phase margin 25∘ 10-22∘

pocket by the coil above it. The single-coil steadystate bias current is measured to be 𝑖c-ss = 0.86Acorresponding to a single-coil motor force constantof 𝐾F = 2.45 N/A. Both of these values are within3% of their predicted values. The performance ofthe rotor’s vertical suspension is evaluated in boththe frequency and time domain. Table 2 summa-rizes the overall performance.

Frequency ResponseTo evaluate the rotor’s vertical suspension in thefrequency domain, the rotor’s desired vertical po-sition is varied sinusoidally with an amplitude of 50𝜇m over 170 frequencies from 1 Hz to 1 kHz. Ateach of these frequencies, the frequency responseof the rotor’s linearized dynamic model 𝑃 (𝑠), theloop return ratio 𝑅𝑅(𝑠) = 𝐶(𝑠)𝐺p𝑃 (𝑠)𝐻opt, and theclosed loop 𝑅𝑅(𝑠)/[1 + 𝑅𝑅(𝑠)] is measured. Eachof these measured frequency responses along withtheir corresponding predicted frequency responsesare displayed as Bode plots in Figure 7.

The measured frequency response of the rotor’s dy-namics has a break frequency for its instability of√𝑘/𝑀r = 6.7 Hz. Additionally, its magnitude un-

expectedly drops at 50 Hz until 250 Hz where it re-covers after passing through a notch-and-peak pair.This pair implies a pair of complex-zeros and-polesso it is hypothesized that this magnitude drop-then-recover occurs from the stator’s mass coupling toand then decoupling from the rotor’s mass. Thismagnitude drop has two effects: (1) It causes thereturn ratio to crossover through unity magnitude ata measured 𝜔c = 50 Hz with a measured phase mar-gin of 𝜑M = 10∘, both lower than expected. (2) Sincethe drop is near crossover, it distorts the closed loopmagnitude’s resonance peak.

10-7

10-4

10-1

Mag.

[abs]

100 101 102 103

Frequency [Hz]

-405

-315

-225

-135

Phase [

deg]

data

model

(A)

10-2

100

102

Mag.

[abs]

100 101 102 103

Frequency [Hz]

-405

-315

-225

-135

Phase [

deg]

data

( )

model

10-2

10-1

100

101

Mag.

[abs]

100 101 102 103

Frequency [Hz]

-360

-270

-180

-90

0

Phase [

deg]

data

model

( )

FIGURE 7. Frequency response of (a) the rotor dy-namics, (b) return ratio, and (c) closed loop duringvertical suspension of the rotor by the coil above it.

Time ResponseTo evaluate the rotor’s vertical suspension in thetime domain, the rotor’s desired position is stepped100 𝜇m from -50 𝜇m to 50 𝜇m. Figure 8 shows therotor’s response to this step. The rotor steps 104𝜇m, peaking in 8.0 ms with 49% overshoot and set-tling in 51.7 ms. These correspond to a measured𝜔c = 64 Hz with a measured phase margin of 𝜑M =22∘, similar to the frequency domain evaluation.

CONCLUSIONReducing space, weight, and power (SWAP) is es-sential in the design of spacecraft. One possibleway to reduce SWAP is to replace the three reactionwheels of a spacecraft’s ACS with a single reactionsphere. This paper presents a spherical perma-nent magnetic dipole rotor driven by 12 surround-ing equidistant coils for consideration as a reactionsphere. Such a design has a mechanically simplerotor like induction motors with the model simplic-ity of hysteresis-and brushless DC-motors. In fact,it was shown that the torque and force on the rotorcan be modeled as the superposition of the torqueand force generated by 12 sets of two dipoles.

FIGURE 8. Step response of sphere’s height for a100 𝜇m input step.

After presenting the reaction sphere’s actuation, thesensing, mechanical structure, power electronics,and controller were presented. The design uses12 sets of optical sensors and Hall effects sensors,placed coaxially with in each coil, to provide positionand rotation feedback respectively. This design wasphysically constructed. Using a lead compensator,the coil above the rotor successfully levitated therotor by itself with position feedback from the coil’scoaxial optical sensor. The rotor required 0.86 A tolevitate corresponding to a force constant per coil of2.45 N/A while the rotor is centered. The frequencyresponse and the step response of the rotor weremeasured to evaluate the suspension. The suspen-sion was found to crossover just below 60 Hz, wellabove the 6.7 Hz instability of the rotor’s dynamics,with a phase margin around 16∘. Next steps involvetesting 1 DoF rotation and then 3 DoF-suspensionand-rotation separately, then together.

ACKNOWLEDGMENTSThis work is performed in collaboration with andsupported by MIT Lincoln Laboratory.

REFERENCES[1] Zhu L, Guo J, Gill E. Review of re-

action spheres for spacecraft attitude con-trol. Progress in Aerospace Sciences. 2017May;91:67–86.

[2] Wang J, Jewell GW, Howe D. Analysis, designand control of a novel spherical permanent-magnet actuator. IEE Proceedings - ElectricPower Applications. 1998 January;145(1):61–71.

[3] Chabot J. A Spherical Magnetic Dipole Actu-ator for Spacecraft Attitude Control [Master’sThesis]. Aerospace Engineering Sciences,University of Colorado. Boulder, CO; 2015.

[4] Noh M, Gruber W, Trumper DL. Hystere-sis Bearingless Slice Motors With Homopo-lar Flux-Biasing. IEEE/ASME Transactions onMechatronics. 2017;22(5):2308–2318.