Patent Publication Number: US-9853528-B2

Title: Spherical induction motor

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of PCT Application No. PCT/US2012/050326, filed on Aug. 10, 2012, entitled “Spherical Induction Motor” which claims the benefit of U.S. Provisional Patent Application No. 61/574,980 filed on Aug. 12, 2011, the entire contents of which are incorporated herein by reference. 
     This application is related to U.S. Pat. No. 7,847,504 filed on Oct. 10, 2007, the entire contents of which are incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     This document relates generally to motors and more particularly to spherical induction motors. 
     BACKGROUND 
     Spherical motion is associated with many important applications. For example, robotic wrists and shoulder joints and positioning mechanisms for antennas, sensors, detectors, and cameras can all utilize a spherical rotor to facilitate spherical motion and improve performance. Despite these important applications, no motor has previously been designed that can rotate a spherical rotor continuously through arbitrarily large angles among any combination of three independent axes via an induction principle. 
     SUMMARY 
     In accordance with the teachings herein, systems and methods are provided for an induction motor. An induction motor includes a spherical rotor and a plurality of curved inductors positioned around the spherical rotor. The plurality of curved inductors are configured to rotate the spherical rotor continuously among any combination of three independent axes. 
     As another example, a method of rotating a spherical rotor continuously among any combination of three independent axes includes placing the spherical rotor within a plurality of curved inductors, where each of the curved inductors includes a plurality of windings. A magnitude and frequency of electric currents applied to each of the windings of each of the curved inductors is individually varied to induce a travelling magnetic wave in the spherical rotor to rotate the spherical rotor. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1  depicts two example implementations of a spherical induction motor. 
         FIG. 2  depicts an application of a spherical induction motor in the form of a dynamic balancing mobile robot. 
         FIG. 3  depicts another implementation of a spherical induction motor in the form of a motorized wheel chair. 
         FIG. 4  is a diagram depicting a spherical rotor transmitting forces and torques to a surface on which the rotor sits. 
         FIGS. 5A and 5B  are cross-section diagrams depicting a layered implementation of a spherical rotor and a curved inductor of a spherical induction motor. 
         FIG. 6  depicts an example lamination structure of a curved inductor. 
         FIGS. 7A-C  depict additional configurations for laminations of a curved inductor. 
         FIG. 8  depicts an example curved inductor having wire windings positioned in the slots between the teeth of the inductor. 
         FIG. 9  is a diagram depicting an example implementation of a spherical induction motor. 
         FIG. 10  is a bottom view of a spherical induction motor including a spherical rotor, inductors, bearings, and velocity sensors. 
         FIG. 11  is a bottom view of a spherical induction motor without a spherical rotor that includes inductors, bearings, and velocity sensors. 
         FIG. 12  is a cross-section diagram of a spherical rotor that includes a pattern on the traction or wear layer to improve velocity sensing. 
         FIGS. 13A-C  depict electrical models for implementing control systems for a spherical induction motor. 
         FIGS. 14A-C  depict estimations of the magnetization axis corresponding to the unknown rotor current vector. 
         FIG. 15  is a block diagram depicting a closed-loop feedback scheme controlling each inductor. 
         FIG. 16  is a block diagram depicting an example spherical induction motor control scheme combined with ball rotation sensing and vector drivers that includes a higher-level controller. 
     
    
    
     DETAILED DESCRIPTION 
     A spherical induction motor, as described herein, provides an ability to rotate a spherical rotor continuously through arbitrarily large angles among any combination of three independent axes.  FIG. 1  depicts two example implementations of a spherical induction motor. The implementations each include a spherical rotor  102 . The spherical rotors  102  are surrounded by a plurality of curved inductors  104 ,  106 ,  108 ,  110 . The curved inductors  104 ,  106 ,  108 ,  110  are positioned around their respective spherical rotor  102 . The curved inductors  104 ,  106 ,  108 ,  110  are configured to rotate the spherical rotor  102  continuously among any combination of three independent axes. For example, the plurality of inductors can be positioned in a non-degenerate arrangement so as to provide a set of torque axes, enabling rotation of the spherical rotor among any combination of three independent axes. In the left example, the curved inductors  104  are positioned askew relative to one another along or parallel to a great circle of the spherical rotor  102 . In the right example, certain of the curved inductors (e.g., curved inductors  108 ,  110 ) are positioned orthogonally to one another. 
     As noted above, spherical motion has a wide array of applications.  FIG. 2  depicts one such application in the form of a dynamic balancing mobile robot (a ballbot). A ballbot  200  is a robot that is tall enough to interact with people at a reasonable height and is slender enough to maneuver about an area without colliding with things or people. The ballbot  200  balances on a single spherical rotor that is controlled via a drive unit  204  in the form of a spherical induction motor. The drive unit  204  imparts inductive forces on the spherical rotor  202  to cause controlled rotation of the spherical rotor  202  to maintain the balance and stability of the ballbot, even when the ballbot is pushed by an outside force. 
       FIG. 3  depicts another application of a spherical induction motor in the form of a motorized wheel chair. In this application, a spherical rotor  302  is controlled via a spherical induction motor  304  according to commands received from a joystick  306 . The spherical induction motor  304  may monitor the current rotational velocity of the spherical rotor  302  and the current directional command from the joystick  306  to determine an appropriate inductive force to apply to the spherical rotor  302  to maintain appropriate motion. 
       FIG. 4  is a diagram depicting a spherical rotor transmitting forces and torques to a surface on which the rotor sits. As the spherical rotor  402  sits on the surface  404 , the spherical rotor  402  can be rotated among any of the three depicted independent axes  406 . Such rotations impart a force and/or torque to the floor  404 , enabling translation and/or rotation motion of the spherical rotor  402 , and any apparatus connected to or carried by the spherical rotor  402 , with respect to the floor surface  404 . 
       FIG. 5A  is a cross-section diagram depicting a layered implementation of a spherical rotor and a curved inductor of a spherical induction motor. The outer (stator) layer  502  includes the plurality of curved inductors that rotate the spherical rotor  504  among any combination of three independent axes. In one example, the stator layer  502  has at least three uni-directional excitation sources (e.g., curved inductors) responsible for driving the spherical rotor  504  in three degrees of freedom. The inductors have a non-degenerate arrangement that also leaves a sufficiently large area of exposed rotor for contacting a surface (e.g., a floor). To first order, one can consider the inductors as sheet currents on the inner surface of the stator. The spherical rotor  504 , of radius r e , has a first, inner layer  506  of high magnetic permeability (such as Iron, Vanadium Permendur, 1010 alloy) of thickness f and a second, high electrical conductivity (such as Copper or Silver) layer  508  of thickness d. A traction or wear layer  510  covers the high conductivity layer  508 . The composition of the traction or wear layer  510  may be selected to provide a sufficiently high level of friction with the contacting surface (e.g., the floor of  FIG. 4 ) to avoid slipping. The traction or wear layer  510  may also protect the surface of the high electrical conductivity layer  508 . 
     The spherical rotor  504  may be constructed in a variety of ways. For example, the inner permeable rotor layer  506  can be initially made as two hemispheres, which can be formed using a variety of processes such as machining, spin forming, deep drawing, and hyroforming. The two hemispheres can then be joined by processes such as brazing, welding, and adhesive fastening. The outer conductive rotor layer  508  can be formed by electrodeposition directly onto the surface of the inner permeable rotor  506  or can be formed as two hemispheres using a variety of processes such as machining, spin forming, deep drawing, and hyroforming. The hemispheres thus formed of conductive material can be applied to the inner permeable rotor and fastened by adhesive bonding, soldering, and brazing. 
     The hemispheres may be annealed before joining to maximize certain properties, such as magnetic properties. Additionally, as shown in  FIG. 5B , the high conductivity layer  508  may include interspersions of a high magnetic permeability material  514  that extends through the high conductivity layer  508  and is in substantial magnetic contact with the inner layer  506 . A traction or wear layer  510  may be applied to the outer conductive layer  504 . The traction or wear layer  510  may be formed from a soft or hard material such as urethane or epoxy and may itself be interspersed with conductive particles, meshes, or strands. 
     The spherical rotor  504  and stator  502  are separated by an effective air gap  512  of thickness e which is maintained in a variety of ways (e.g., a mechanical bearing, an air bearing, a gas bearing, a ball bearing, a magnetic bearing, or other bearing system). Referring again to  FIG. 5A , fields are measured with components parallel and transverse to the radial R, with rotor rotational velocity ω rot . 
     In one example, each of the inductors is wound with three sets of windings which are excited sinusoidally, differing 120 degrees in phase from one another causing a travelling magnetic wave to be induced in the permeable layer  506  of the rotor  504 . The inductor core with p pole pairs has a number of slots s. For example, in one embodiment, there are s=3p+3 slots, p+1 for each phase (three), p poles inductor. The changing magnetic fields, in turn, generate reactive currents in the conductive layer  508  producing torque. Under the assumptions that fields are concentrated in the air gap  512 , currents in the inductors can be treated as sheet currents at the interface between the stator inductor  502  and the air gap  512 , negligible fringing fields, and linear and infinitely permeable inductor material, analytical modeling may be used to derive the rotor torque from the magnetic diffusion equation 
                         ∇   2     ⁢     A   -&gt;       μσ     =         ∂     A   -&gt;         ∂   t       -       ω   rot     ⁢     r   e     ×     ∇     ×     A   -&gt;                     (   1   )               
where {right arrow over (A)} is the magnetic vector potential, μ is the rotor Iron  506  permeability, σ is the rotor conductive  508  conductivity, ω rot  is the rotor angular velocity, and r e  is the rotor  504  radius. In the two-dimensional model of  FIG. 5A , the vector potential {right arrow over (A)} reduces to the radial component
 
 {right arrow over (A)}=A ( r )exp[ j (ω sup   t−p/r   e θ re ]  (2)
 
where ω swp  is the inductor excitation frequency, p is the number of inductor pole pairs, θr e  is the circumferential distance along the rotor  504 , and {tilde over (r)} is a unit vector in the radial direction. The flux density {right arrow over (B)} and field strength {right arrow over (H)} can then be determined from
 
 {right arrow over (B)}=∇×{right arrow over (A)}   (3)
 
and the constitutive relationship
 
{right arrow over (B)}=μ{right arrow over (H)}  (4)
 
     Boundary conditions may be applied to each layer shown in  FIG. 5A . The field intensity {right arrow over (H)} equals zero in the stator layer  502  and also at R equals zero. Between adjacent layers i and i+1,
 
 {circumflex over (r)}× ( {right arrow over (H)}   i   −{right arrow over (H)}   i+1 )= {right arrow over (j)}   (5)
 
and
 
 {circumflex over (r)} ·( {right arrow over (B)}   i   −{right arrow over (B)}   i+1 )=0  (6)
 
where {right arrow over (j)} is the surface current on the stator  502 .
 
     The torque dτ per unit surface dS can be found by evaluating the Maxwell stress tensor on the rotor  504  surface:
 
 dτ=r   e μ 0   H   ω   H   R   dS   (7)
 
where H ω  and H R  are the azimuthal and radial magnetic field components in the air gap and μ 0  is the permeability of free space. The total torque is obtained by integration over the surface of the inductor, and has the form
 
τ= p·S ·( N·I   1 ) 2   ·r   e   ·g (μ, σ, ω rot   , γ, e, d )  (8)
 
where S is the inductor surface area, N·I I  is the ampere-turns of impressed excitation, γ is the slip ratio 1−(ω rot /ω sup ) between the rotor speed and magnetic field speed, and g is a complicated nonlinear function. Note the squared dependence on stator current I I  and the cubic dependence on the radius r e  due to the fact that S can scale as r e   2 .
 
       FIG. 6  depicts an example base structure of a curved inductor. The curved inductor  602  is formed from a plurality of laminations (layers) joined face to face. The curved inductor  602  includes a number of spaces (slots)  604  within which wire windings are placed. Currents are applied to the windings and are varied in magnitude and frequency to apply a travelling magnetic wave to a spherical rotor, as described in further detail below. The slots  604  are defined by a number of teeth  606  that support the windings and provide structure to the curved inductor  602  as well as creating magnetic circuits and strengthening magnetic field intensity. The teeth  606  may be formed of a variety of materials such as Silicon Iron or Vanadium Permendur. The individual laminations may be of uniform shape. Alternatively, the individual laminations may be varied in shape to better conform to the shape of the spherical rotor and better transmit the travelling magnetic wave to the spherical rotor. For example, the laminations depicted in  FIG. 6  are of consistent width and are positioned parallel to one another. However, the heights of the teeth of the laminations are skewed relative to one another, such that the teeth on the outside laminations  608  are taller than the teeth on the inside laminations  610 , giving the curved inductor a substantially spherical shape that can be juxtaposed with the spherical rotor. The curved inductor may also be formed from a single piece of material. Curved inductors can be tapered by grinding or other machining to provide a substantially spherical surface. Insulating material (e.g., thin plastic adhesive sheets) may be applied to the outside of an assembled set of laminations to provide insulation and protection of windings. 
       FIG. 7  depicts additional configurations for laminations of a curved inductor. The inductor lamination scheme at  702  is similar to the scheme depicted in  FIG. 6 , where the laminations are of consistent width and positioned parallel to one another at skewed heights on the rotor side relative to one another. The inductor lamination scheme at  704  is a radial stacking scheme where the laminations are of consistent width and positioned in a radial fashion relative to the spherical rotor. The inductor lamination scheme at  706  is a tapered scheme where the laminations are of tapered width (e.g., the top  708  of the laminations is thicker than the bottom  710  of the laminations). The torque force and power loss of a curved inductor tends to increase with the square of the driving current, where torque peaks at a certain optimal frequency. Torque may increase with tooth width and may decrease with tooth height beyond a particular threshold. Torque tends to decrease with air gap length. 
       FIG. 8  depicts an example curved inductor having wire windings positioned in the slots between the teeth of the inductor. In one example, the curved inductor  802  is interspersed with three sets of winding loops  804 ,  806 ,  808 . The winding loops  804 ,  806 ,  808  are repeated along the length of the curved inductor. Electric currents are applied to the three winding loops  804 ,  806 ,  808  to control the travelling magnetic wave that is applied to the spherical rotor. Such currents may be applied independently in one example. In another example, the magnitude and frequency applied to the winding loops  804 ,  806 ,  808  may be varied, where the currents applied to the winding loops  804 ,  806 , and  808  are applied 120 degrees out of phase with one another to produce the travelling magnetic wave. 
       FIG. 9  is a diagram depicting an example implementation of a spherical induction motor. A spherical induction motor includes a spherical rotor  902  positioned within a plurality of curved inductors  904  that rotate the spherical rotor  902  continuously among any combination of three independent axes. A plurality of bearings  906  (e.g., mechanical bearings, air bearings, gas bearings, ball bearings, magnetic bearings) provide a gap between the spherical rotor  902  and the spherical induction motor, enabling the spherical rotor  902  to rotate. The spherical induction motor may further include one or more velocity sensors  908  that are configured to measure a velocity of rotation of the spherical rotor  902 . The velocity sensors  908  may transmit a signal representative of the measured velocity to a processor. The processor considers the signals from the velocity sensors, any commanded movement of the spherical induction motor, and/or other inputs in commanding the variation of the magnitudes and frequencies of the currents transmitted to the curved inductors to control rotation of the spherical rotor.  FIG. 10  is a bottom view of a spherical induction motor including a spherical rotor  1002 , inductors  1004 , bearings  1006 , and velocity sensors  1008 .  FIG. 11  is a bottom view of a spherical induction motor without a spherical rotor that includes inductors  1104 , bearings  1106 , and velocity sensors  1108 . 
     It may be desirable to measure the three-dimensional rotational velocity of the rotor for at least three reasons: i) to determine the optimal current and drive frequency at any instant in time (as described herein below); ii) to provide virtual damping for rotor control in the presence of external torques; and iii) to provide an odometry reference for travel along a surface such as the floor. Because, in some implementations, there are no shafts attached to the spherical rotor, it may not be possible to use encoders for sensing motion. Thus, an “axis independent” method for measuring rotation may be used to measure such motion. Such a method can employ optical, capacitive, inductive, or other principles. An optical velocity sensor may be implemented in a variety of forms. In one implementation, an optical velocity sensor uses optical mouse sensors (e.g., 2 or more) that measure surface velocity in two orthogonal directions by tracking patterns of small optical features such as texture or scratches. By using more than 2 velocity sensors, the signal-to-noise ratio of the velocity measurement can be improved by averaging multiple outputs in generating three degree of freedom angular rates and rotation angles. An optical velocity sensor can use light emitting diodes as light sources and are good at detecting slower motion at low resolutions. Higher cost velocity sensors (e.g., using laser or gaming mouse components) utilize laser diodes as light sources, resulting in higher precision at higher speeds (e.g., 5 μm resolution and 3.8 m/s, with processing rates of 12,000 frames/s. 
       FIG. 12  is a cross-section diagram of a spherical rotor that includes a pattern on the traction or wear layer to improve velocity sensing. A layer of high magnetic permeability  1202  is positioned within a layer of high electrical conductivity  1204 . A traction or wear layer  1206  is positioned outside of the layer of high electrical conductivity  1204  on the spherical rotor. The traction or wear layer  1206  includes an incorporated pattern  1208  (e.g., a pattern of contrasting colors, shadings, lines) that enhances sensing by a velocity sensor. For example, the pattern may be a visual pattern that enhances detection of velocity by an optical scanner. As another example, the pattern may be mechanically embossed on the traction or wear layer  1206  to facilitate inductive or capacitive sensing. 
     Control systems for implementing a spherical induction motor may be realized using a vector control scheme. High-speed processors and MOSFET and IGBT power devices aid in making such schemes practical. A variety of approaches can be used in implementing a control system for a spherical induction motor. For example,  FIG. 13  depicts at  1302  a transformer model, where an impressed voltage e 1  generates an electromotive force e 2  through primary and secondary inductances L 1  and L 2  whose mutual inductance is M. The load R accounts for the output work and resistive (eddy current) losses. As shown at  1304 , three-phase windings R, S, and T, with reverse windings R bar , S bar , and T bar , generate a rotating field in the rotor, with respect to fixed orthogonal axes α and β, which lags behind the impressed field because of magnetic diffusion.  FIG. 13  depicts at  1306  multiple sets of coils and their relationship with the rotating field vector whose components are expressed in the α−β frame fixed in the respective inductor. A Clarke transformation can be used to project the three phase drive quantities (voltages or currents) onto the α and β axes. 
     The frame of the rotating field may be denoted by the direct (d) and quadrature (q) orthogonal axes in both the inductor and rotor as shown in the diagram of  FIG. 14 .  FIG. 14  depicts estimations of the magnetization axis corresponding to the unknown rotor current vector. The inductor current i d =i R  and i q =(1/√{square root over (3)})(i s −i T ). Because i R +i S +i T =0, this reduces to:
 
i d −i R   (9)
 
and
 
 i   q =(1/√{square root over (3)}) i   R +(2/√{square root over (3)}) i   S   (10)
 
By using current sensors (e.g., Hall sensors) on only two of the phase currents (e.g., i R  and i S ) one can determine the d and q components of the currents in the inductor frame. A Park transformation can be used to obtain the i d  and i q  currents in the rotating frame of the rotor:
 
 i   d   r   =i   q  sin θ r   +i   d  cos θ r   (11)
 
 i   q   r   =i   q  cos θ r   −i   d  sin θ r   (12)
 
where θ r =θ 0 +ωt, and ω is the angular velocity of the rotating flux.
 
     The torque exerted on the rotor is the product of the d and q currents in the rotating frame:
 
τ= K   T   i   d   r   ·i   q   r   (13)
 
By controlling these two currents at any instant, the torque can be directly controlled. However, i d   r  and i q   r  may not be directly measurable. However, in some instances they can be estimated.  FIG. 14  at  1402  depicts the d−q frame rotating with respect to the stationary α−β frame with the direction of maximum rotor flux indicated by the vector I 1  having components i d   r  and i q   r  along the d and q axes, respectively.
 
     With reference to the transformer model depicted in  FIG. 14  at  1402 , the angular frequency of ω of the rotating flux vector is 
                     ω   ⁡     (   t   )       =         L   2     R     ⁢       i   q   r       I   0                 (   14   )               
where I 0  is the magnitude of the rotating flux vector along the dr-axis. The angle θ can be generated by numerically integrating Equation 14. The quantity L 2 /R can be estimated. Thus, in one implementation, the closed-loop feedback scheme depicted in  FIG. 15  is implemented. Here, measured currents, i R  and i S , from two phases of the pulse width modulation (PWM) drive are transformed by the Clarke block  1502  and combined with rotor angular speed ω rot  measurements from the proposed optical velocity sensor to estimate the rotor flux angle θ. This is equivalent to measuring the slip ratio γ. The Park transform  1504  obtains the rotating frame currents followed by proportional-integral (PI) controllers  1506 ,  1508  that compute the desired currents  d i d   r  and  d i q   r  from the commanded torque τ and flux λ is proportional to I 0 , respectively.  FIGS. 14  at  1404  and  1406  illustrates cases where the estimated flux angle lags behind or moves ahead of the correct angle, respectively. In either case, the closed-loop scheme will tend to converge rapidly to the correct value.
 
     The closed-loop scheme of  FIG. 15  can be implemented for each independent set of inductors surrounding the rotor. The number of controllers implemented may depend on the number and configuration of inductors. Additional higher-level controllers can be implemented in some scenarios (e.g. in a ballbot) to compute the correct instantaneous rotor torque {right arrow over (Γ)}=Σ i   n =1 {right arrow over (τ)} i , where n is the number of independent inductor sets. For balancing in one place, {right arrow over (Γ)} may change rapidly in magnitude and direction to counter environmental disturbances. For moving from place to place, the torque {right arrow over (Γ)} can be used by a higher-level controller to establish a body lean angle proportional to the desired acceleration.  FIG. 16  is a block diagram depicting an example spherical induction motor control scheme combined with ball rotation sensing and vector drivers that includes such a higher-level controller. 
     Examples have been used to describe the invention herein, and the scope of the invention may include other examples. For example, velocity may be sensed according to the following sensing process. Usually, an optical mouse sensor has two sensing axes. There are assumed to be two independent surface velocity sensors in one mouse sensor (whose position is identical but sensing direction are perpendicular to each other). 
     Let the position of the sensor i (i=1 . . . n) be p i  and unit vector of sensing axis s i . When the sphere rotates with angular velocity ω, surface velocity at p i  is
 
ν 1 =ω× p   i  
 
     The sensor can sense the relative speed along s i .
 
ν si   =s   i ·ν i   =s   i ·(ω× p   i )=ω·( p   i   ×s   i )
 
     Using three sensors, numbered 1, 2, and 3, the equation can be written as: 
               (           v     s   ⁢           ⁢   1                 v     s   ⁢           ⁢   2                 v     s   ⁢           ⁢   3             )     =         (             p   1     ×     s   1                   p   2     ×     s   2                   p   3     ×     s   3             )     ⁢     (           ω   x               ω   y               ω   z           )       =     S   ⁢           ⁢   ω             
This equation can be solved when the matrix S is a regular matrix, and the angular velocity can be derived using the surface speed sensor.
 
     
       
         
           
             
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                       ω 
                       x 
                     
                   
                 
                 
                   
                     
                       ω 
                       y 
                     
                   
                 
                 
                   
                     
                       ω 
                       z 
                     
                   
                 
               
               ) 
             
             = 
             
               
                 S 
                 
                   - 
                   1 
                 
               
               ⁡ 
               
                 ( 
                 
                   
                     
                       
                         v 
                         
                           s 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         v 
                         
                           s 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                     
                   
                   
                     
                       
                         v 
                         
                           s 
                           ⁢ 
                           
                               
                           
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                           3 
                         
                       
                     
                   
                 
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     From n sensors, triplets can be chosen of  n C 3 =n!/(3!(n−3)!), while better triplets having a larger |S| are subsequently chosen. The angular velocity can then be calculated, and an average of all triplets provides angular velocity with a better signal to noise ratio (e.g., using a weighted mean). 
     As a further example, torques can be generated using the following equations. Let the position of inductor i be p i  and unit vector of force generating direction (tangent of the rotor) s i . The torque generated by inductor i is expressed by
 
τ i   =p   i ×( f   i   s   i )
 
where f i  is an output command (scalar) for inductor i.
 
     Assuming n inductors, the total generated torque can be calculated according to: 
             Γ   =         ∑     i   =   1     n     ⁢     τ   i       =       ∑     i   =   1     n     ⁢       (       p   i     ×     s   i       )     ⁢     f   i                 
Letting the outer product p i ×s i  be a vector t i =(t ix , t iy , t iz ) T , the above equation can be written in matrix form as:
 
               (           Γ   x               Γ   y               Γ   z           )     =         (           t     1   ⁢   x           …         t   nx               t     1   ⁢   y           …         t   ny               t     1   ⁢   z           …         t   nz           )     ⁢     (           f   1             ⋮             f   n           )       =   Tf           
where T and f are a matrix that consists of t i  and a vector of f i .
 
     Solving this equation, command f i  for each current drive controller can be determined from the desired torque Γ. If n=3 and t r , t 2 , and t 3  are linearly independent, then the equation can be solved: 
     
       
         
           
             
               ( 
               
                 
                   
                     
                       f 
                       1 
                     
                   
                 
                 
                   
                     
                       f 
                       2 
                     
                   
                 
                 
                   
                     
                       f 
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             = 
             
               
                 T 
                 
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                         Γ 
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                         Γ 
                         y 
                       
                     
                   
                   
                     
                       
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     If n&gt;3 and the rank of T is three, f i  can still be defined by one of two methods. First, the pseudo inverse matrix (or similar technique) can be used to solve the equation: 
               (           f   1             ⋮             f   n           )     =       T   +     ⁡     (           Γ   x               Γ   y               Γ   y           )             
where T +  is the pseudo inverse matrix of T. Second, one can choose only three major inductors out of n and use the above equation inverse to decide three f i  (or choose less than n and use the pseudo inverse). This solution can decrease power consumption for magnetizing but may result in lag when switching operating inductors.
 
     As another example, the methods and systems described herein may be implemented on many different types of processing devices by program code comprising program instructions that are executable by the device processing subsystem. The software program instructions may include source code, object code, machine code, or any other stored data that is operable to cause a processing system to perform the methods and operations described herein and may be provided in any suitable language such as C, C++, JAVA, for example, or any other suitable programming language. Other implementations may also be used, however, such as firmware or even appropriately designed hardware configured to carry out the methods and systems described herein. The systems&#39; and methods&#39; data (e.g., associations, mappings, data input, data output, intermediate data results, final data results, etc.) may be stored and implemented in one or more different types of computer-implemented data stores, such as different types of storage devices and programming constructs (e.g., RAM, ROM, Flash memory, flat files, databases, programming data structures, programming variables, IF-THEN (or similar type) statement constructs, etc.). It is noted that data structures describe formats for use in organizing and storing data in databases, programs, memory, or other computer-readable media for use by a computer program. 
     The computer components, software modules, functions, data stores and data structures described herein may be connected directly or indirectly to each other in order to allow the flow of data needed for their operations. It is also noted that a module or processor includes but is not limited to a unit of code that performs a software operation, and can be implemented for example as a subroutine unit of code, or as a software function unit of code, or as an object (as in an object-oriented paradigm), or as an applet, or in a computer script language, or as another type of computer code. The software components and/or functionality may be located on a single computer or distributed across multiple computers depending upon the situation at hand. 
     It should be understood that as used in the description herein and throughout the claims that follow, the meaning of “a,” “an,” and “the” includes plural reference unless the context clearly dictates otherwise. Also, as used in the description herein and throughout the claims that follow, the meaning of “in” includes “in” and “on” unless the context clearly dictates otherwise. Further, as used in the description herein and throughout the claims that follow, the meaning of “each” does not require “each and every” unless the context clearly dictates otherwise. Finally, as used in the description herein and throughout the claims that follow, the meanings of “and” and “or” include both the conjunctive and disjunctive and may be used interchangeably unless the context expressly dictates otherwise; the phrase “exclusive or” may be used to indicate situation where only the disjunctive meaning may apply.