Abstract:
The present invention provides a variable reluctance actuator system and method that can be adapted for simultaneous rotation and translation of a moving element by applying a normal-direction magnetic flux on the moving element. In a beneficial example arrangement, the moving element includes a swing arm that carries a cutting tool at a set radius from an axis of rotation so as to produce a rotary fast tool servo that provides a tool motion in a direction substantially parallel to the surface-normal of a workpiece at the point of contact between the cutting tool and workpiece. An actuator rotates a swing arm such that a cutting tool moves toward and away from a mounted rotating workpiece in a controlled manner in order to machine the workpiece. Position sensors provide rotation and displacement information for a swing arm to a control system. A control system commands and coordinates motion of the fast tool servo with the motion of a spindle, rotating table, cross-feed slide, and in feed slide of a precision lathe.

Description:
RELATED APPLICATION  
       [0001]     This application claims priority from U.S. Provisional Patent Application No. 60/701,176, entitled “HYBRID ROTARY AND LINEAR FLUX-BIASED VARIABLE RELUCTANCE ACTUATOR BASED ROTARY FAST TOOL SERVO SYSTEMS AND METHODS,” filed on Jul. 20, 2005, and is incorporated by reference in its entirety. 
     
    
       [0002]     The United States Government has rights in this invention pursuant to Contract No. W-7405-ENG-48 between the United States Department of Energy and the University of California for the operation of Lawrence Livermore National Laboratory. 
     
    
     BACKGROUND OF THE INVENTION  
       [0003]     1. Field of the Invention  
         [0004]     The present invention relates to a servo system and method, more particularly, the present invention relates to hybrid linear/rotary fast servo systems and methods to enable, for example, fabrication of complex three-dimensional surface features on a variety of components, and steering a light beam in an optical system.  
         [0005]     2. Description of Related Art  
         [0006]     A fast tool servo is a well-known device that can be added to a new or existing machine tool to provide an additional axis of motion between the cutting tool and a workpiece. A fast tool servo most notably distinguishes itself by its ability to move the tool at a much higher bandwidth that is at a high speed of controlled, repetitive motion, on its axis relative to the other machine tool axes, with accuracy equal to or better than that of the other tool axes. Fast tool servos fall into two broad categories: rotary and linear. A rotary fast tool servo produces relative motion between the cutting tool and a workpiece by rotation of a swing arm that carries the tool at a fixed radius from the axis of rotation. A linear fast tool servo produces relative motion between the cutting tool and a workpiece by producing a linear translation of the tool. A steering mirror is a well-known device that can be added to an optical system to allow deflecting an electromagnetic beam.  
         [0007]     Background information on a rotary fast tool servo system is described in U.S. Patent Application Publication No. 2005/0166726 A1, entitled “Rotary Fast Tool Servo System and Methods,” to Montesanti et al., published Aug. 4, 2005, including the following: “The present invention is directed to a rotary fast tool servo system that improves the accuracy and speed to enable and meet manufacturing goals for, for example, fabricating three-dimensional surface features . . . . In a preferred embodiment, the rotary fast tool servo system includes a cutting element mounted to a rotating arm that is driven by an actuator. The arm is mounted to the fast tool servo base by flexures on at least one side of the cutting element. Each flexure preferably includes orthogonally positioned flexure elements that extend from the rotating arm to the base. The rotating arm can be oriented vertically, horizontally, or in any other desired orientation.” 
         [0008]     Background information on a fast tool servo system having linear or rotational movement about a single axis is described in U.S. Patent Application Publication No. US2005/0056125 A1, entitled “Flux-biased electromagnetic fast tool servo systems and methods,” to David L. Trumper, published Mar. 17, 2005, including the following: “The movement of the tool servo can be constrained in translation (linear FTS) or in rotation (rotary FTS) by any of the bearing technologies used in precision motion control systems. These include flexures, rolling element bearings, air bearings, hydrostatic bearings, or magnetic bearings.” 
         [0009]     Further background information on a fast tool servo system having linear or rotational movement about a single axis is described in U.S. Patent Application Publication No. US2005/0223858 A1, entitled “Variable reluctance Fast Positioning System and Methods,” to Lu et al., published Oct. 18, 2005, published Mar. 17, 2005, including the following: “A linear fast motor 682 as described herein is used to position article 686 along a longitudinal (X) axis. Alternatively, a rotary fast motor 684 is used to rotate article 686 around axis through angle 0.” 
         [0010]     Background information on a magnetic bearing system arranged for tilt and tip movement, in addition linear movement about a defined plane, is described in U.S. Pat. No. 4,634,191, entitled “Radial and Torsionally Controlled Magnetic Bearing,” to Studer, patented Jan. 6, 1987, including the following: “A magnetic bearing including a circular stator member having a plurality of circumferential pole faces and a suspended annular ring member with corresponding number of inward facing circumferential pole faces separated by respective air gaps. A source of DC magnetic flux circulates flux between the circumferential pole faces of the stator and the ring to provide axial stability along a central longitudinal axis. Flux coil means are included on the stator member for providing variable flux density along predetermined radial paths to provide active radial stabilization. Additionally, flux coil means are included on the stator to actively modulate the magnitude of the magnetic forces as well as their direction of differential flux control involving the DC magnetic flux to produce torquing moments about a pair of mutually orthogonal axes which are perpendicular to the central axis.  
         [0011]     A need exists for new and/or improved flux-biased variable reluctance actuators that can provide simultaneous rotary motion or rotary and linear motion. Such a system and method of the present invention is directed to such a need.  
       SUMMARY OF THE INVENTION  
       [0012]     Accordingly, the present invention provides systems with improved accuracy and speed to enable and meet manufacturing goals of, for example, fabrication of three-dimensional surface features. The embodiments of the present invention provide a high level of bandwidth and precision control to form short spatial wavelength features of, for example, 50 micron long features with 5 micron peak to valley dimensions at 10 kHz or more.  
         [0013]     The rotary and/or hybrid rotary/linear fast tool servo system of the present invention includes a cutting element mounted to a reciprocating arm that is driven by an actuator so as to operate on workpiece mounted on, for example, a spindle of a precision lathe which can rotate the workpiece during such operation. As another arrangement the rotary and/or hybrid rotary/linear fast tool servo system of the present invention can be arranged with an operatively coupled or manufactured optical surface of the rotor so as to direct electromagnetic radiation (e.g., laser optical wavelengths) to a desired target for desired applications, such as, but not limited to, treating, cutting, milling, inspecting and or communication.  
         [0014]     A beneficial arrangement of the present invention includes a hybrid linear/rotary positioning apparatus having a stator; a rotor moveably coupled with the stator and configured with at least one broad surface parallel to a desired XY plane; and a means to induce a steering flux and a bias flux in the stator so as to rotate the rotor relative to a rotational axis and translate linearly in an orthogonal direction with respect to the XY plane.  
         [0015]     Another aspect of the present invention provides a one degree of freedom actuator that includes a stator configured with at least one central pole and at least two outer poles; a rotor moveably coupled with the stator and configured with at least one central rotor pole, at least two outer rotor poles, and configured with at least one broad surface parallel to an XY plane; and a means to induce a steering flux and a bias flux in the stator so as to rotate the rotor relative to a rotational axis.  
         [0016]     A final aspect of the present invention provides a two degree of freedom actuator that includes a stator configured with at least one central pole, at least two left/right outer stator poles, and at least two top/bottom outer stator poles; a rotor moveably coupled with the stator and configured with at least one central pole, at least two left/right outer rotor poles, at least two top/bottom outer rotor poles, and configured with at least one broad surface parallel to an XY plane; and a means to induce a steering flux and a bias flux in said stator so as to rotate said rotor relative to a first and a second rotational axis.  
         [0017]     Accordingly, such methods and apparatus of the present invention provide accurate high speed beneficial rotary and hybrid linear/rotary servo arrangements so as to enable fabrication of three-dimensional surface features on manufactured components and to enable directing electromagnetic radiation in an optical system. Such arrangements as disclosed herein, improves the bearing system for a moving element by reducing the number of mechanical components supporting the moving element and/or by providing dynamic stiffness and electronic damping to the moving element by a control system. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0018]     The accompanying drawings, which are incorporated into and constitute a part of the specification, illustrate specific embodiments of the invention and, together with the general description of the invention given above, and the detailed description of the specific embodiments, serve to explain the principles of the invention.  
         [0019]      FIG. 1  shows a rotor as described in U.S. Patent Application Publication No. 2005/0166726 A1, entitled “Rotary Fast Tool Servo System and Methods,” to Montesanti et al.  
         [0020]      FIG. 2 , also described in U.S. Patent Application Publication No. 2005/0166726 A1, entitled “Rotary Fast Tool Servo System and Methods,” to Montesanti et al., shows a beneficial method for fixing the outer ends of the flexures shown in  FIG. 1  with a clamping block.  
         [0021]      FIG. 3  shows an example of a rotor with lower flexures configured with radial slots cut into them to create a plurality of ligaments to reduce the axial stiffness that the lower flexures provide to the rotor.  
         [0022]      FIG. 4  shows an example of a rotor with lower flexures configured with an elliptical cross-section to reduce the axial stiffness that the lower flexures provide to the rotor.  
         [0023]      FIG. 5  shows an example of a swing arm shown with a first sensor target and a second sensor target.  
         [0024]      FIG. 6  shows an example of a magnetic circuit for a normal-stress variable reluctance actuator referred to as the “pinched rotor” design.  
         [0025]      FIG. 7  illustrates a pair of graphs generated by an analysis of the magnetic forces and torques acting on the rotor in  FIG. 6 , and illustrates the two-direction independence of torque and force.  
         [0026]      FIG. 8  shows an example of a magnetic circuit for a normal-stress variable reluctance actuator referred to as the “sandwiched rotor” design.  
         [0027]      FIG. 9  illustrates a pair of graphs generated by an analysis of the magnetic forces and torques acting on the rotor of  FIG. 8 , and illustrates the one-direction dependence of force on torque.  
         [0028]      FIG. 10  shows an example of a magnetic circuit for a normal-stress variable reluctance actuator referred to as the “reversed roles sandwiched rotor” design.  
         [0029]      FIG. 11  illustrates a pair of graphs generated by an analysis of the magnetic forces and torques acting on the rotor in  FIG. 10 , and illustrates the two-direction independence of torque and force.  
         [0030]      FIG. 12  provides a comparison of the pros and cons of the magnetic circuits shown in  FIGS. 6, 8 , and  10 .  
         [0031]      FIG. 13  shows an example of a variation of the magnetic circuit of  FIG. 8 , with a permanent magnet used to provide the bias flux.  
         [0032]      FIG. 14  shows a modification of the magnetic circuit of  FIG. 8 , with the back-iron air gaps removed to illustrate what happens when a high reluctance path is not included in the high magnetically-permeable back-iron.  
         [0033]      FIG. 15  shows the top view of a first reduced-flexure rotor assembly that can be employed as the rotor in the magnetic circuits shown in  FIG. 6, 8 ,  10 , or  13 .  
         [0034]      FIG. 16  shows the top view of a second reduced-flexure rotor assembly that can be employed as the rotor in the magnetic circuits shown in  FIG. 6, 8 ,  10 , or  13 .  
         [0035]      FIG. 17  shows the top view of a third reduced-flexure rotor assembly that can be employed as the rotor in the magnetic circuits shown in  FIG. 6, 8 ,  10 , or  13 .  
         [0036]      FIG. 18  shows a simple, potentially low-cost, rotary actuator that uses a rubber sheet permanent magnet to produce a bias flux, separate the steering flux path from the bias flux paths, and to form a bearing between the rotor and the stator.  
         [0037]      FIG. 19  shows the system of  FIG. 18  with the permanent magnet material moved out of the rubber sheet permanent magnet, resulting in a separate rubber sheet bearing and permanent magnet.  
         [0038]      FIG. 20  shows the system of  FIG. 19  with the permanent magnet replaced by a combination of a permanent magnet and two low magnetic-permeability spacers.  
         [0039]      FIG. 21  shows the rotary actuator of  FIG. 20  with the rotor removed to better illustrate the other components.  
         [0040]      FIG. 22  shows a possible arrangement of two displacement measuring sensors for measuring the displacement of the rotor relative to the stator.  
         [0041]      FIG. 23  shows the rotary actuator of  FIG. 20  with the addition of the two displacement sensors.  
         [0042]      FIG. 24  shows the addition of a separate optical element to the front surface of the rotor.  
         [0043]      FIG. 25  shows the rotary actuator of  FIG. 20  with the addition of a cutting tool.  
         [0044]      FIG. 26  shows the tool moved to a location near an edge of the rotor.  
         [0045]      FIG. 27  shows the system of  FIG. 25  with the tip of the tool located in front of the rotation axis of the rotor.  
         [0046]      FIG. 28  shows the tip of the tool coincident with the rotation axis of the rotor.  
         [0047]      FIG. 29  shows a simple, potentially low-cost, rotary actuator having two rotary degrees of freedom.  
         [0048]      FIG. 30  shows the second stator, its two outer stator poles, and its central stator pole.  
         [0049]      FIG. 31  shows the first stator positioned with the second stator.  
         [0050]      FIG. 32  shows a permanent magnet positioned against the first stator and the central stator pole of the second stator.  
         [0051]      FIG. 33  shows the common bias flux pole piece that is positioned against the permanent magnet. Two axes of rotation intersect at the center of a spherical seat in the common bias flux pole piece.  
         [0052]      FIG. 34  shows the addition of a spherical bearing.  
         [0053]      FIG. 35  shows the spherical bearing accepting the rotor central pole.  
         [0054]      FIG. 36  shows the addition of the steering flux coils to the two stators.  
         [0055]      FIG. 37  is a first cross-sectional view of the system shown in  FIG. 29 .  
         [0056]      FIG. 38  is a second cross-sectional view of the system shown in  FIG. 29 .  
         [0057]      FIG. 39  shows a possible arrangement of four displacement measuring sensors for measuring the displacement of the rotor relative to the stators.  
         [0058]      FIG. 40  shows the rotary actuator of  FIG. 29  with the addition of the four displacement measuring sensors.  
         [0059]      FIG. 41  shows a cross-sectional view of the system of  FIG. 40  to better illustrate two of the four displacement measuring sensors.  
         [0060]      FIG. 42  shows the addition of a separate optical element to the front surface of the rotor.  
         [0061]      FIG. 43  shows the rotary actuator of  FIG. 40  with the addition of a cutting tool.  
         [0062]      FIG. 44  shows the tool moved to a location near an edge of the rotor.  
         [0063]      FIG. 45  shows the tip of the tool located in front of the intersections of the two rotation axes of the rotor.  
         [0064]      FIG. 46  shows the tip of the tool coincident with the intersections of the two rotation axes of the rotor. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0065]     Referring now to the drawings, specific embodiments of the invention are shown. The detailed description of the specific embodiments, together with the general description of the invention, serves to explain the principles of the invention.  
         [0066]     Unless otherwise indicated, numbers expressing quantities of ingredients, constituents, reaction conditions and so forth used in the specification and claims are to be understood as being modified by the term “about.” Accordingly, unless indicated to the contrary, the numerical parameters set forth in the specification and attached claims are approximations that may vary depending upon the desired properties sought to be obtained by the subject matter presented herein. At the very least, and not as an attempt to limit the application of the doctrine of equivalents to the scope of the claims, each numerical parameter should at least be construed in light of the number of reported significant digits and by applying ordinary rounding techniques. Notwithstanding that the numerical ranges and parameters setting forth the broad scope of the subject matter presented herein are approximations, the numerical values set forth in the specific examples are reported as precisely as possible. Any numerical value, however, inherently contain certain errors necessarily resulting from the standard deviation found in their respective testing measurements.  
         [0000]     Specific Description  
         [0067]      FIG. 1  shows a rotor, generally designated by the reference numeral  1 , as described in U.S. Patent Application Publication No. 2005/0166726 A1, entitled “Rotary Fast Tool Servo System and Methods,” to Montesanti et al., and which is herein incorporated by reference in its entirety. Such a rotor  1  includes, but is not limited to, a swing arm  2 , a counter weight  4 , a seal  6 , such as an o-ring, one or more rotor cores  10 , and of note for purposes of the discussion of  FIG. 1 , upper flexures  8  and lower flexures  12  having a high in-plane stiffness compared to a low out-of-plane bending stiffness. Such a design creates a radial and axial constraint at both ends of rotor  1 , which is therefore over-constrained in the axial direction (as shown by the dashed line and denoted by the letter A).  
         [0068]      FIG. 2 , as also detailed in incorporated by reference U.S. Patent Application Publication No. 2005/0166726 A1, shows a beneficial method for fixing the outer ends of the flexures (e.g., lower flexures  12 , as shown in  FIG. 2 , with a clamping block (e.g., reference numeral  14 ) and upper flexures  8  with a clamping block. If the axial length of rotor  1 , as detailed in  FIG. 1 , changes relative to the axial spacing of the bottom portion  17  and top portion  18  of the stator housing  16  and  18 , as shown in  FIG. 2 , that fix the outer ends of flexures,  8  and  12 , via, for example, clamping block  14 , then an unwanted stress can develop in the flexures (e.g., upper flexure  8  or lower flexure  12 ). Thermal growth of the rotor  1 , as shown in  FIG. 2  but more detailed in  FIG. 1 , that is not matched by thermal growth of the stator housing  16  and  18  is one mechanism that can cause such a difference in axial length. Magnetic losses in rotor core  10  (such as in a configured laminate stack core, as shown generally in  FIG. 1 ) generates heat and therefore can cause a change in the axial length of rotor  1 , as shown in  FIG. 1 , during operation.  
         [0069]     Similar magnetic losses in the stator cores, not shown, generate heat in stator housing  16  and  18  that can cause a change in the axial spacing of the outer ends of the flexures during operation. Stresses in the flexures due to the differential axial growth can be kept within a tolerable level by: (1) prescribing strict operating conditions so that the differential thermal growth between the rotor and stator housings is kept within an acceptable level; (2) the use of a cooling fluid to maintain an acceptable differential temperature between the rotor and stator housings; (3) modify the in-plane stiffness of either the lower or upper flexures so that the rotor is less over-constrained in the axial direction.  
         [0070]     Solutions (1) and (2) can be employed with the example embodiment shown in  FIG. 2  and detailed in incorporated by reference U.S. Patent Application Publication No. 2005/0166726 A1. Solution (3) can be realized by the alternate example embodiments shown in  FIG. 3  and  FIG. 4  discussed below. The upper flexures  8  and lower flexures  12 , as shown in  FIG. 1  and  FIG. 2  allows using the same mounting hardware for both (e.g. upper hub  35  and lower hub  36 , as shown in  FIG. 3 , and outer clamping blocks  14 , as shown in  FIG. 2 ). Moreover, the stiffness of the radial and torsional constraints provided to the rotor  1 , as shown in  FIG. 1 , by the lower flexures  12  is the same as the stiffness of the radial and torsional constraints provided by the upper flexures  8 . In specifying a particular embodiment, a trade-off can be made between the advantages of having identical mounting hardware and the same radial and torsional constraint stiffness at both ends of the rotor against the disadvantages of having an over-constrained rotor.  
         [0071]     Referring to  FIG. 3 , a lower flexure  37  can be configured with radial slots  38  cut into them to create a plurality of ligaments  39  to reduce the axial stiffness that the lower flexures provide to the rotor, now shown generally designated as reference numeral  31 , compared to the higher axial stiffness that the upper flexures  32  provide to rotor  31 . Rotor  31  is axially fixed by the upper flexures  32  and has a relaxed axial constraint from the lower flexures  37 . The size of the slots  38  and number of ligaments  39  can be varied to adjust the stiffness characteristics of the lower flexures  37 . The upper flexures  39  and lower flexures  37  both provide a radial constraint to the rotor, and together guide rotation of rotor  31  around an axis of rotation (denoted by the letter A). In this embodiment a differential axial length change of rotor  31  relative to the bottom portion  17  and top portion  18  of stator housing  16 , as shown in  FIG. 2 , is accommodated by the reduced stiffness of the lower flexures  37 , as shown in  FIG. 3 , and the stresses in all of the flexures are lower than in the case with identical upper and lower flexures, e.g., flexures  8  and  12  as shown in  FIG. 1 .  
         [0072]     It is to be appreciated that the stiffness of the radial and torsional constraints provided to the rotor  31  of  FIG. 3  by the lower flexures  37  is not the same as the stiffness of the radial and torsional constraints provided by the upper flexures  32 , which can lead to additional vibration modes of the rotor compared to the case of identical upper and lower flexures  8  and  12  as shown in  FIG. 1 . Increasing the overall height of the lower flexure blades enables setting the total cross-sectional area of ligaments  39  to be equal to the cross-sectional area of upper flexures  32 . In such an arrangement, the stiffness of the radial and torsional constraints provided to the rotor is substantially the same for the upper and lower sets of flexures. In specifying a particular embodiment, a trade-off can be made between the benefits of a less over-constrained rotor and identical mounting hardware against the deleterious effects of possible additional vibration modes.  
         [0073]      FIG. 4  shows an example rotor arrangement, generally designated by reference numeral  41 , wherein configured lower flexures  48  have an elliptical cross-section to reduce the axial stiffness that the lower flexures provide to the rotor  41  compared to the higher axial stiffness that the upper flexures  42  provide to the rotor. Rotor  41  can be axially fixed by upper flexures  42  and has a relaxed axial constraint from the lower flexures  48 . Upper flexures  42  and lower flexures  48  both provide a radial constraint to rotor  41 , and together guide rotation of rotor  41  around an axis of rotation (again shown denoted by the letter A).  
         [0074]     In such an example embodiment, a differential axial length change of rotor  41  relative to bottom portion  17  and top portion  18  of the stator housings  16  and  18 , as shown in  FIG. 2 , is accommodated by the reduced stiffness of lower flexures  48 , and the stresses in all of the flexures are lower than in the case with identical upper and lower flexures, e.g., flexures  8  and  12  as shown in  FIG. 1 . The cross-section dimensions “a” and “b” of lower flexures  48  can be chosen so that the stiffness of the radial and/or torsional constraints provided to rotor  41  by lower flexures  48  is the same as the stiffness of the radial and torsional constraints provided by the upper flexures  42 .  
         [0075]     In an alternate example embodiment, lower flexures  48  can be configured with rectangular cross-sections. The lower flexures  48 , whether having an elliptical cross-section or a rectangular cross-section, require different mounting hardware for fixing them to the rotor (e.g., a lower hub  47 ) and to the stator (not shown) than the mounting hardware for fixing upper flexures  42 . Moreover, the present invention can be arranged to match the stiffness of the radial and/or torsional constraints provided to the rotor  41  by upper flexures  42  and lower flexures  48  so as to avoid/reduce the additional vibration modes of the rotor that arise in the example embodiment depicted in  FIG. 3 . In specifying a particular embodiment, a trade-off can be made between the benefits of a less over-constrained rotor against the deleterious effects of non-identical mounting hardware.  
         [0076]     Referring to  FIG. 5 , a swing arm, generally designated by reference numeral  50 , is shown with a first sensor target  55  that is communicatively coupled to a first displacement sensor (not shown), and a second sensor target  56  that is communicatively coupled to a second displacement sensor (not shown). First sensor target  55  can be substantially displaced from an axis of rotation (again denoted by the letter A) and second sensor target  56  can be substantially aligned with axis of rotation A. From such an arrangement first sensor target  55  and its displacement sensor (not shown) provides information on the displacement of a tool  90  due to rotation and translation of swing arm  50 . Second sensor target  56  and its displacement sensor (not shown) provides information on the displacement of the axis of rotation A. The information from the first sensor that is communicatively coupled to first sensor target  55  can be used by a control system (not shown) to control a rotation of the swing arm  50 . The information from the second sensor that is communicatively coupled to second sensor target  56  can be used by a control system to control a displacement of the axis of rotation A. The information from the first sensor that is communicatively coupled to first sensor target  55  and the information from the second sensor that is communicatively coupled to second sensor target  56  can be used by a control system of the present invention to separate the displacement of tool  90  caused by a rotation of the swing arm  50  from the displacement of tool  90  caused by a displacement of the axis of rotation A.  
         [0077]     It is well known to one of ordinary skill in the art that adding mechanical damping to a system can improve its controllability. Accordingly, viscous squeeze-film damping can be added to the rotor example embodiments  31  and  41  as respectively shown in  FIGS. 3 and 4 , by constraining a layer of viscous fluid in the air gaps (not shown) between the rotor (e.g., a rotor laminate stack)  30  and  40 , as respectively shown in  FIG. 3  and  FIG. 4 , and the stator pole faces (not shown).  
         [0078]     In another beneficial embodiment, when swing arm  50 , as shown in  FIG. 5 , is integrated with the rotor  31  or  41  of  FIGS. 3 and 4 , viscous squeeze-film damping can also be added to swing arm  50  by constraining a layer of viscous fluid in the air gaps (not shown) between one or both of the sensor targets  55  and  56 , as shown in  FIG. 5 , and their respective sensor (not shown).  
         [0079]      FIG. 6  shows a magnetic circuit for a normal-stress variable reluctance actuator referred to as the “pinched rotor” design because the magnetically permeable rotor  100  appears to be being pinched by the two magnetically permeable stator pieces  200  and  300 . The bias flux coil  90  produces a magnetic flux that circulates in a bias flux path. The triple-headed arrows represent the bias flux.  
         [0080]     For illustrative purposes consider that the bias flux path starts at bias flux coil  90 , goes through the magnetically permeable back-iron  61  to the back-iron air gap  82 , across air gap  82 , through both legs of the stator piece  300  to air gaps  64  and  66 , across air gaps  64  and  66 , through rotor  100  to air gaps  60  and  62 , across the air gaps  60  and  62 , through both legs of the stator piece  200 , across the back-iron air gap  80 , and through the back-iron  61  to bias coil  90 .  
         [0081]     In a beneficial embodiment, the nominal length of the rotor-stator air gaps  60 ,  62 ,  64 , and  66  are equal, the nominal cross-sectional area of the air gaps  60 ,  62 ,  64 , and  66  are equal, and the lengths of the back-iron air gaps  80  and  82  are greater than the lengths of the rotor-stator air gaps  60 ,  62 ,  64 , and  66 .  
         [0082]     One of ordinary skill in the art recognizes that the sections of rotor  100  and stator pieces  200  and  300  that are subjected to a time-varying magnetic flux are most often beneficially constructed from a laminated or powdered magnetic material with high permeability at the operating frequency of interest to reduce eddy current and magnetic hysteresis losses in the material.  
         [0083]     To develop a torque on the rotor  100 , as shown in  FIG. 6 , the first steering flux coil  70  is configured to produce a magnetic flux that circulates in a first steering flux path and the second steering flux coil  72  produces a nominally equal magnitude magnetic flux that circulates in a second steering flux path. The double-headed arrows, as shown in  FIG. 6 , represent the torque-producing flux in each stator piece.  
         [0084]     For illustrative purposes consider that the first steering flux path starts at the steering coil  70 , goes through the stator piece  200  to air gap  62 , across air gap  62 , through rotor  100  to air gap  60 , across air gap  60 , and through the stator piece  200  to the steering coil  70 . Note that back-iron air gap  80  provides a high reluctance path to separate the first steering flux path from the bias flux path. Now consider that the second steering flux path starts at the steering coil  72 , goes through the stator piece  300  to the air gap  64 , across air gap  64 , through the rotor  100  to air gap  66 , across air gap  66 , and through stator piece  300  to the steering coil  72 . Again note that back-iron air gap  82  provides a high reluctance path to separate the second steering flux path from the bias flux path.  
         [0085]     In an alternate embodiment, back-iron air gaps  80  and  82  can be replaced by a spacer having a low magnetic permeability to provide a high reluctance path to also separate the first steering flux path and the second steering flux path from the bias flux path.  
         [0086]     In a beneficial embodiment, the bias flux and steering fluxes are designed to add in air gaps  60  and  64 , and subtract in air gaps  62  and  66 , producing a net torque on rotor  100  from the magnetic forces acting on it, resulting in the counterclockwise rotation (shown denoted by θ and an accompanying directional arrow) of the rotor shown in  FIG. 6 . Reversing the directions of the two steering fluxes reverses the direction of the torque acting on rotor  100  and hence its angular rotation θ. Moreover, each of the steering flux coils  70  and  72  can be made up of one or more coils acting together or independently for producing a particular desired steering flux.  
         [0087]     In another example beneficial embodiment, the steering fluxes (denoted by double arrows) are controlled by a closed-loop feedback control system that uses a measurement of the angular rotation θ for controlling rotation and for providing dynamic rotation stiffness to the rotor. As another arrangement, the steering fluxes are controlled by a closed-loop feedback control system that uses a measurement of the rotation θ for providing electronic damping to the rotational motion of the rotor.  
         [0088]     In a beneficial embodiment, nominally unequal magnitude steering fluxes in the first and second steering flux paths can create a force on rotor  100  in  FIG. 6 , resulting in a translation along a linear direction (as denoted by the letter Z and a directional arrow). This can be accomplished by increasing the current in one of steering coils  70  or  72 , or by decreasing the current in one of such steering coils, or by increasing the current in one of the steering coils and decreasing the current in the other steering coil. The single-head arrows represent the force-producing flux in each stator piece. In an example embodiment, the steering fluxes are controlled by a closed-loop feedback control system that uses a measurement of the translation Z for controlling that translation and providing dynamic translation stiffness to the rotor. In another arrangement, the steering fluxes are controlled by a closed-loop feedback control system that uses a measurement of the translation Z for providing electronic damping to the translation motion of the rotor.  
         [0089]     According to description above for the magnetic circuit in  FIG. 6 , the torque and force produced on the rotor are independent of each other: changing one nominally does not affect the other.  FIG. 7  shows a pair of graphs generated by an analysis of the magnetic forces and torques acting on the rotor  100  in  FIG. 6 , and illustrates the two-direction independence of torque and force.  
         [0090]     The magnetic circuit shown in  FIG. 6  is a variation on the magnetic circuit for a normal-stress variable reluctance actuator that is taught in U.S. Patent Application Publication No. 2004/0035266 A1, entitled, “Rotary fast tool servo system and methods,” by Richard C. Montesanti and David L. Trumper, and of which, is herein incorporated by reference in its entirety. The present invention improves upon the device described in US 2004/0035266 A1 by providing a high reluctance path for separating the steering flux and the bias flux when a coil is used to provide the bias flux.  
         [0091]     It is well known to one of ordinary skill in the art that magnetic flux preferentially follows a low reluctance path instead of a high reluctance path. It is also well known that the reluctance of a path depends on the magnetic permeability of the material that the path is made of, and that the permeability for a magnetic material depends on the frequency of oscillation of the magnetic field. The high reluctance paths created by the back-iron air gaps  80  and  82 , as shown in  FIG. 6 , substantially confine the steering flux to the steering flux paths, and the bias flux to the bias flux path, at the intended frequencies and ranges of motion for the actuator. Such high reluctance paths created by the back-iron air gaps  80  and  82  substantially separate the steering flux paths and the bias flux path, ensuring that the actuator generates a desired rotary motion for a particular set of currents in the steering flux coils, and ensures that the actuator generates a desired linear motion for a particular set of currents in the steering flux coils.  
         [0092]     Moreover, the high reluctance paths created by the back-iron air gaps  80  and  82  magnetically decouples the two stator pieces  200  and  300 , making it easier to independently control the steering magnetic flux in each stator piece. Independent control of the steering magnetic flux in each stator piece allows the creation of a controllable force on the rotor. In an example embodiment, the actuator is intended to operate at 10,000 cycles per second or greater. In another embodiment, the air gaps between the rotor and stator have a nominal length of about 50 micrometers, and the rotor makes contact with the stator during operation, closing at least one of the air gaps.  
         [0093]     It is to be appreciated that in contrast to the similar but different arrangement of  FIG. 6  of the present invention that is taught in US 2004/0035266 A1, the embodiment shown in  FIG. 6  of this application allows easy adjustment of the bias flux; by changing the amount of electrical current flowing through the bias coil  90 .  
         [0094]      FIG. 8  shows a magnetic circuit for a normal-stress variable reluctance actuator referred to as the “sandwiched rotor” design because the magnetically permeable rotor  100  appears to be sandwiched between the two magnetically permeable stator pieces  200  and  300 . The bias flux coil  90  produces a magnetic flux that circulates in a bias flux path. The triple-headed arrows represent the bias flux.  
         [0095]     For illustrative purposes consider that the bias flux path starts at the bias coil  90 , goes through the magnetically permeable back-iron  61  to the back-iron air gap  82 , across air gap  82 , through both legs of the stator piece  300  to air gaps  64  and  66 , across air gaps  64  and  66 , through rotor  100  to air gaps  60  and  62 , across the air gaps  60  and  62 , through both legs of the stator piece  200 , across the back-iron air gap  80 , and through the back-iron  61  to the bias coil  90 .  
         [0096]     As a beneficial arrangement, the nominal length of the rotor-stator air gaps  60 ,  62 ,  64 , and  66  are equal, the nominal cross-sectional area of the air gaps  60 ,  62 ,  64 , and  66  are equal, and the lengths of the back-iron air gaps  80  and  82  are greater than the lengths of the rotor-stator air gaps  60 ,  62 ,  64 , and  66 .  
         [0097]     One of ordinary skill in the art recognizes that the sections of rotor  100  and stator pieces  200  and  300  that are subjected to a time-varying magnetic flux are most often beneficially constructed from a laminated or powdered magnetic material with high permeability at the operating frequency of interest to reduce eddy current and magnetic hysteresis losses in the material.  
         [0098]     To develop a torque on rotor  100 , as shown in  FIG. 8 , the first steering flux coil  70  produces a magnetic flux that circulates in a first steering flux path and the second steering flux coil  72  produces a nominally equal magnitude magnetic flux that circulates in a second steering flux path. The double-headed arrows represent the torque-producing flux in each stator piece.  
         [0099]     For illustrative purposes consider that the first steering flux path starts at the steering coil  70 , goes through the stator piece  200  to the air gap  62 , across air gap  62 , through the rotor  100  to air gap  60 , across air gap  60 , and through the stator piece  200  to the steering coil  70 . Note that the back-iron air gap  80  provides a high reluctance path to separate the first steering flux path from the bias flux path.  
         [0100]     For illustrative purposes consider that the second steering flux path starts at the steering coil  72 , goes through the stator piece  300  to the air gap  64 , across air gap  64 , through the rotor  100  to air gap  66 , across air gap  66 , and through the stator piece  300  to the steering coil  72 . Note that the back-iron air gap  82  provides a high reluctance path to separate the second steering flux path from the bias flux path. In an alternate embodiment, back-iron air gaps  80  and  82  can be replaced by a spacer having a low magnetic permeability to provide a high reluctance path to separate the first steering flux path from the bias flux path.  
         [0101]     In a beneficial arrangement, the bias flux and steering fluxes are designed to add in air gaps  60  and  64 , and subtract in air gaps  62  and  66 , producing a net torque on the rotor  100  from the magnetic forces acting on it, resulting in the counterclockwise rotation θ of the rotor shown in  FIG. 8 . Accordingly, by reversing the directions of the two steering fluxes reverses the direction of the torque acting on the rotor  100  and hence its rotation θ. Moreover, each of the steering flux coils  70  and  72  can be made up of more than one coil acting together or independently for producing a particular desired steering flux.  
         [0102]     As another arrangement, the steering fluxes can be controlled by a closed-loop feedback control system that uses a measurement of the rotation θ for controlling the rotation and for providing dynamic rotation stiffness to the rotor. Moreover, the steering fluxes can be controlled by a closed-loop feedback control system that uses a measurement of the rotation θ for providing electronic damping to the rotational motion of the rotor.  
         [0103]     It is to be appreciated that nominally unequal magnitude steering fluxes in the first and second steering flux paths can create a force on the rotor  100  in  FIG. 8 , resulting in the translation Z. This can be accomplished by increasing the current in one of the steering coils  70  or  72 , or by decreasing the current in one of the steering coils, or by increasing the current in one of the steering coils and decreasing the current in the other steering coil. The single-head arrows represent the force-producing flux in each stator piece. As one example embodiment, the steering fluxes can be controlled by a closed-loop feedback control system that uses a measurement of the translation Z for control for controlling that translation and providing dynamic translation stiffness to the rotor. As another arrangement, the steering fluxes can be controlled by a closed-loop feedback control system that uses a measurement of the translation Z for providing electronic damping to the translation motion of the rotor.  
         [0104]     According to the example embodiment described above for the magnetic circuit in  FIG. 8 , the torque and force produced on the rotor are not independent of each other: changing the force nominally does not affect the torque, but changing the torque does affect the force.  
         [0105]      FIG. 9  shows a pair of graphs generated by an analysis of the magnetic forces and torques acting on the rotor  100  of  FIG. 8 , and illustrates the one-direction dependence of force on torque. A force can be produced by the magnetic circuit in  FIG. 8  as long as a torque is being produced by it. In practice, an external torque (not shown) needs to be maintained on the rotor  100  so that a torque producing flux can be present to act as an operating point for creating a force.  
         [0106]     It is to be noted that  FIG. 8  of the present invention shows an arrangement that is similar to a circuit disclosed in incorporated by reference  
         [0107]     U.S. Patent Application Publication No. 2005/0166726 A1. The present invention improves on such a disclosed circuit by configuring back-iron air gaps  80  and  82 , as shown in  FIG. 8 , to provide a high reluctance path to separate the steering flux paths from the bias flux path when a bias coil  90  and back-iron  61  are used instead of at least one permanent magnet to provide the bias flux.  
         [0108]      FIG. 10  shows a magnetic circuit for a normal-stress variable reluctance actuator referred to as the “reversed roles sandwiched rotor” design because the magnetically permeable rotor  100  appears to be sandwiched between the two magnetically permeable stator pieces  200  and  300  and the roles of the coils and the nature of the flux paths for producing a torque on the rotor of  FIG. 10  are reversed from the roles of the coils and flux paths as shown and as discussed above for the configuration of  FIG. 8 .  
         [0109]     For  FIG. 10 , the first bias flux coil  70  produces a magnetic flux that circulates in a first bias flux path, and the second bias flux coil  72  produces a nominally equal magnitude bias flux that circulates in a second bias flux path. The triple-headed arrows represent the bias flux.  
         [0110]     For illustrative purposes consider that the first bias flux path starts at bias coil  70 , goes through the stator piece  200  to the air gap  62 , across air gap  62 , through the rotor  100  to air gap  60 , across air gap  60 , and through the stator piece  200  to bias coil  70 . Note that the back-iron air gap  80  provides a high reluctance path to separate the first bias flux path from the torque flux path in the back-iron  61 .  
         [0111]     For illustrative purposes consider that the second bias flux path starts at bias coil  72 , goes through the stator piece  300  to the air gap  64 , across air gap  64 , through the rotor  100  to air gap  66 , across air gap  66 , and through the stator piece  300  to the steering coil  72 . Note that the back-iron air gap  82  provides a high reluctance path to separate the second bias flux path from the torque flux path in the back-iron  61 . In an alternate embodiment the back-iron air gaps  80  and  82  can be replaced by a spacer having a low magnetic permeability to provide a high reluctance path to separate the first steering flux path from the bias flux path.  
         [0112]     To develop a torque on rotor  100 , as shown in  FIG. 10  with the stated example embodiment, the torque flux coil  90  in  FIG. 10  produces a magnetic flux that circulates in a torque flux path. The double-headed arrows represent the torque-producing flux.  
         [0113]     For illustrative purposes consider that the torque flux path starts at the torque coil  90 , goes through the magnetically permeable back-iron  61  to the back-iron air gap  82 , across air gap  82 , through both legs of the stator piece  300  to air gaps  64  and  66 , across air gaps  64  and  66 , through the rotor  100  to air gaps  60  and  62 , across the air gaps  60  and  62 , through both legs of the stator piece  200 , across the back-iron air gap  80 , and through the back-iron  61  to the torque coil  90 .  
         [0114]     In a beneficial embodiment, the nominal length of the rotor-stator air gaps  60 ,  62 ,  64 , and  66  are equal, the nominal cross-sectional area of the air gaps  60 ,  62 ,  64 , and  66  are equal, and the lengths of the back-iron air gaps  80  and  82  are greater than the lengths of the rotor-stator air gaps  60 ,  62 ,  64 , and  66 .  
         [0115]     One of ordinary skill in the art recognizes that the sections of rotor  100  and stator pieces  200  and  300  that are subjected to a time-varying magnetic flux are most often beneficially constructed from a laminated or powdered magnetic material with high permeability at the operating frequency of interest to reduce eddy current and magnetic hysteresis losses in the material.  
         [0116]     In an example embodiment, the bias flux and torque fluxes are designed to add in air gaps  60  and  64 , and subtract in air gaps  62  and  66 , producing a net torque on the rotor  100  from the magnetic forces acting on it, resulting in the counterclockwise rotation θ of the rotor shown in  FIG. 10 . Reversing the direction of the torque flux reverses the direction of the torque acting on the rotor  100  and hence its rotation θ. Moreover, each of the bias flux coils  70  and  72  can be made up of more than one coil acting together or independently for producing a particular desired bias flux.  
         [0117]     In a beneficial example embodiment, the torque flux produced by torque coil  90  is controlled by a closed-loop feedback control system that uses a measurement of the rotation θ for controlling the rotation and providing dynamic rotation stiffness to the rotor. In another example embodiment, the torque flux is controlled by a closed-loop feedback control system that uses a measurement of the rotation θ for providing electronic damping to the rotational motion of the rotor.  
         [0118]     It is to be appreciated with respect to the configuration shown in  FIG. 10  that nominally unequal magnitude bias fluxes in the first and second bias flux paths can create a force on the rotor  100 , resulting in a translation Z. Such a method can be accomplished by increasing the current in one of the bias flux coils  70  or  72 , or by decreasing the current in one of the bias flux coils, or by increasing the current in one of the bias flux coils and decreasing the current in the other bias flux coil. The single-head arrows represent the force-producing flux in each stator piece.  
         [0119]     In an example arrangement, the bias fluxes are controlled by a closed-loop feedback control system that uses a measurement of translation Z for controlling the translation and for providing dynamic translation stiffness to the rotor. In another arrangement, the bias fluxes are controlled by a control system that uses a measurement of translation Z for closed-loop feedback control for providing electronic damping to the translation motion of the rotor.  
         [0120]     According to description above for the magnetic circuit of  FIG. 10 , the torque and force produced on the rotor are independent of each other: changing one nominally does not affect the other.  FIG. 11  shows a pair of graphs generated by an analysis of the magnetic forces and torques acting on the rotor  100  in  FIG. 10 , and illustrates the two-direction independence of torque and force.  
         [0121]      FIG. 12  provides a comparison of the pros and cons of the magnetic circuits shown in  FIGS. 6, 8 , and  10 ; suggesting reasons for preferring one over the other for a particular application.  
         [0122]      FIG. 13  shows a variation of the magnetic circuit of  FIG. 8 , with a permanent magnet  81  used to provide the bias flux and create a high reluctance path in the back-iron  61  and  62  (all other reference numerals are the same as shown in  FIG. 8 ). It is to be noted that one of ordinary skill in the art understands that certain high strength permanent magnets have a low magnetic permeability approaching that of free space, and therefore the permanent magnet  81  replaces the function of the air gaps  80  and  82 , as shown in  FIG. 8 , which are not present in  FIG. 12 .  
         [0123]      FIG. 14  shows another modification of the magnetic circuit of  FIG. 8 , with the back-iron air gaps  80  and  82  removed to illustrate what happens when a high reluctance path is not included in the high magnetically-permeable back-iron  61 . When an alternating current is applied to coil  70 A, an alternating magnetic flux can be produced primarily in two paths. The first path comprises the first stator piece  200 , the air gap  62 , the rotor  100 , and the air gap  60  back to the stator piece  200 . The second path comprises the first stator piece  200 , the back-iron  61 , the two legs of the second stator piece  300 , across air gaps  64  and  66 , through rotor  100 , and across air gap  60  back to the first stator piece  200 . One of ordinary skill in the art readily recognizes that if the stator pieces  200  and  300  and the back-iron  61  are made of a high magnetically-permeable material, then current  70 A can produce a significant magnetic flux in the first and second paths, which can couple with coils  70 B,  70 C, and  70 D and induce an alternating current in them.  
         [0124]     A comparison of  FIG. 8 ,  FIG. 10 , and  FIG. 14  illustrates how the high reluctance paths provided by the back-iron air gaps  80  and  82 , e.g., as shown in  FIG. 8 , decrease the coupling between alternating electrical currents flowing through coils  70  and  72 . Referring to  FIG. 8 , when an alternating current is applied to coil  70 , an alternating magnetic flux can be produced primarily in a path comprising the first stator piece  200 , the air gap  62 , the rotor  100 , and the air gap  60  back to the stator piece  200 . A second flux path exists in this case, comprising the first stator piece  200 , across the air gaps  62  and  64  via the rotor  100 , through the second stator piece  300 , across the air gaps  66  and  60  via the rotor  100 , and back to the first stator piece  200 . One of ordinary skill in the art recognizes that the flux induced by current introduced into coil  70  in the second path can be substantially less than the flux induced by current introduced in the first path because of the higher reluctance of the second path due to a nominal doubling of the air gaps in the second path, and therefore the alternating current induced in coil  72  is less than the current applied to coil  70 . A similar argument exists for describing what happens when an alternating current is applied to coil  70 . A third flux path exists in this case, comprising the first stator piece  200 , across the back-iron air gap  80 , through the back-iron  61 , across the back-iron air gap  82 , through the two legs of the second stator piece  300 , across the air gaps  64  and  66 , through the rotor  100 , and across air gap  60  back to the first stator piece  200 . Those of ordinary skill in the art readily recognizes that the flux induced by current in the third path induces an alternating current in coil  72  when the induced flux in the two legs of stator piece  300  is not balanced, as would be the case when the lengths of rotor-stator air gaps  64  and  66  are not equal, or when coil  72  includes two separate coils with one on each leg of stator piece  300 . Moreover, one of ordinary skill in the art also recognizes that the flux induced by current in the third path can be made significantly less than the flux in the first path by setting the lengths of the back-iron air gaps  80  and  82  to be significantly longer than the lengths of the rotor-stator air gaps  60 ,  62 ,  64 , and  66 . Therefore, the back-iron air gaps  80  and  82  create a high reluctance path in the back-iron  61  that magnetically decouples the stator pieces  200  and  300  when a coil is used to provide the bias flux. Those practiced in the art also recognize that the difficulty of independently controlling the currents in coils  70  and  72  decreases when the magnetic coupling between the stator pieces  200  and  300  decreases.  
         [0125]      FIG. 15  shows the top view of a rotor assembly  5  that can be employed as the rotor  100  in the magnetic circuits shown in  FIG. 6, 8 ,  10 , or  13 . Similar in construction to the rotor  1  in  FIGS. 1 and 2 , rotor assembly  5  in  FIG. 15  includes of a rotor core  10  coupled to at least an upper flexure hub  100  that in turn is connected to at least two flexure blades  110  and  120  that are fixed at their outer ends to a base  150 . One of ordinary skill in the art recognizes that as drawn, the two flexure blades in  FIG. 15  substantially constrain only two of the three translation degrees of freedom of the rotor: one in the direction  222 , and a second in a direction in and out of the plane of the page and denoted by the centerline  200 . Those of ordinary skill in the art further recognize that rotor core  10  is substantially free to rotate in the direction  210  about the centerline  200 , and that the centerline  200  is not constrained in the translation direction  220  by the flexure blades  110  and  120 . In a beneficial embodiment, rotor assembly  5  is employed as the rotor discussed above in any of the magnetic circuits shown in  FIG. 6, 8 ,  10 , or  13 , which are all capable of producing a torque on the rotor via the two forces  300  to cause the rotation  210 , and are all capable of producing a force to provide the rotor a dynamic stiffness and electronic damping to control its displacement in the translation direction  220  when the two forces  400  on the rotor are controlled by a control system that uses the translational displacement  220  for closed-loop feedback control.  
         [0126]     Recall from the earlier discussion that the magnetic circuit in  FIG. 8  requires that an external torque (not shown) be maintained on rotor  100  so that a torque producing flux is present to act as an operating point for creating a force. Turning back to  FIG. 15 , the helper spring elements  130  and  140  are fixed at their outer ends to a base  150  and are connected to the at least upper flexure hub  100 . Spring elements  130  and  140  can be used to augment the dynamic stiffness provided to rotor core  10  in the translation direction  220 . In one extreme case the spring elements  130  and  140  are not present and all of the stiffness in the translational direction  220  is provided by an actuator operating in accordance with  FIG. 6, 8 ,  10 , or  13 . In another extreme case the spring elements  130  and  140  are replaced with flexure blades identical to  110  and  120 , and the rotor assembly becomes the same as rotor assembly  31  and  41 , as respectively shown in  FIGS. 3 and 4 .  
         [0127]      FIG. 16  shows a first variation of the rotor assembly  5  shown in  FIG. 15 . In  FIG. 16  the flexure blades  110  and  120  are oriented to substantially constrain the three translation degrees of freedom of a rotor core  10 . Here, rotor core  10  is substantially free to rotate in the direction  210  about a centerline  200 , and that centerline  200  can wander in the translation directions  220  and  222  when the rotor is rotated in the direction  210 . In an example embodiment, the rotor assembly  5  is employed as the rotor  100  in any of the magnetic circuits shown in  FIG. 6, 8 ,  10  or  13 , which are all capable of producing a torque on the rotor via the two forces  300  to cause the rotation  210 , and are all capable of producing a force to provide the rotor a dynamic stiffness and electronic damping to control its displacement in the translation direction  220  when the two forces  400  on the rotor are controlled by a control system that uses the translational displacement  220  for closed-loop feedback control.  
         [0128]     In the case of the rotor assembly  7  shown in  FIG. 16 , motion of rotor core  10  in the translation direction  222  is not controlled by an actuator operating in accordance with  FIG. 6, 8 ,  10  or  13 . Spring elements  130  and  140  can be used to augment the dynamic stiffness provided to rotor core  10  in the translation direction  220 , and are discussed in the description for  FIG. 15 .  
         [0129]      FIG. 17  shows a second variation of the rotor assembly  5  shown in  FIG. 15 . In  FIG. 17  flexure blades  110  and  120  are oriented to substantially constrain the three translation degrees of freedom of rotor core  10 . One of ordinary skill in the art recognizes that rotor core  10  is substantially free to rotate in the direction  210  about the centerline  200 , and that centerline  200  can wander in the translation directions  220  and  222  when the rotor is rotated in the direction  210 . In an example beneficial embodiment, rotor assembly  9  is employed as the rotor in any of the magnetic circuits shown in  FIG. 6, 8 ,  10  or  13 , which are all capable of producing a torque on the rotor via the two forces  300  to cause the rotation  210 , and are all capable of producing a force to provide the rotor a dynamic stiffness and electronic damping to control its displacement in the translation direction  220  when the two forces  400  on the rotor are controlled by a control system that uses the translational displacement  220  for closed-loop feedback control. In the case of the rotor assembly  9 , as shown in  FIG. 17 , motion of rotor core  10  in the translation direction  222  is not controlled by an actuator operating in accordance with  FIG. 6, 8 ,  10  or  13 . Spring elements  130  and  140  can be used to augment the dynamic stiffness provided to rotor core  10  in the translation direction  220 , and are discussed in the description for  FIG. 15 .  
         [0130]     Taken together,  FIGS. 15, 16 , and  17  represent three possible arrangements of at least two flexure blades  110  and  120  and optional at least two helper springs  130  and  140  for providing a bearing for a rotor core  10 . Those of ordinary skill in the art recognize that a continuum exists for the possible angles between the flexure blades  110  and  120  and for the possible angles between the flexure blades and rotor core  10 .  
         [0131]      FIG. 18  shows a simple, potentially low-cost, rotary actuator that uses a rubber sheet permanent magnet  700  to produce a bias flux, to separate the steering flux path  685  from the bias flux paths  701  and  702 , and to form a bearing between the rotor  500  and the stator  600 . The arrow shown on the rubber sheet permanent magnet  700  indicates the north-pole direction. Note the large potential work zone at the front face of the rotor  500 . If the actuator is used in a fast tool servo, then the cutting tool can be arranged to engage a large workpiece. Alternatively, if the actuator is used to rotate an optical element, that optical element can be made integral with the exposed front face of the rotor or mounted directly to it. Potential optical elements include a reflective element, e.g., a mirror configured with a flat, concave, convex, or complex surface, a refractive element, or a diffractive element.  
         [0132]     A rubber sheet permanent magnet is a sheet of elastic material that has permanent magnet properties. For example, a rubber sheet magnet can be up to and/or greater than about a 0.5 mm thick layer of rubber impregnated with particles of ceramic permanent magnet that are oriented in a common direction. For a beneficial embodiment, rubber sheet permanent magnet  700 , stator central pole  630 , and rotor central pole  540  have mating curved surfaces that have a common center on the axis of rotation  550  of rotor  500 . In another beneficial embodiment, rubber sheet permanent magnet  700  can be made up of multiple layers of rubber sheet permanent magnets and high magnetic permeability material, and the particular composition and thickness of the layers is chosen to achieve desired mechanical and magnetic properties of rubber sheet permanent magnet  700 . One of ordinary skill in the art can recognize that if the axis of rotation  550  is chosen so that it passes through the combined center of mass of the moving elements  500  and  540 , then the moving elements may not develop a linear acceleration and attendant reaction force.  
         [0133]     The use of one or more rubber sheet permanent magnets provide a source of magnetic flux that flows through two elements, to separate a steering flux path from a bias flux path, and to form a bearing between those two elements. A significant aspect of a rubber sheet bearing as utilized herein is that it can tolerate the compressive stress that results from the attraction of a rotor and stator in certain magnetic circuit topologies. This allows considering magnetic topologies such as the one shown in  FIG. 18 , which can be difficult to realize if flexure blades were used (as shown in  FIGS. 3 and 4 ) because of their intolerance to compressive stresses.  
         [0134]     For a beneficial and desired embodiment of the device shown in  FIG. 18 , the bias flux paths  701  and  702  start at the rubber sheet permanent magnet  700 , enter the magnetically permeable stator  600  at the stator central pole  630 , split and circulate through the stator to the left stator pole  610  and right stator pole  620 , cross the air gaps  510  and  520  to enter the magnetically permeable rotor  500 , travel through the rotor to the rotor central pole  540 , and return to the rubber sheet permanent magnet  700 . In an example embodiment, the lengths of the air gaps  510  and  520  are equal, and the cross-sectional area of the air gaps  510  and  520  are equal. Those of ordinary skill in the art recognizes that the sections of the rotor  500  and stator  600  that are subjected to a time-varying magnetic flux are most often beneficially constructed from a laminated or powdered magnetic material with high permeability at the operating frequency of interest to reduce eddy current and magnetic hysteresis losses in the material.  
         [0135]     To develop a torque on the rotor  500  in  FIG. 18 , the first steering flux coil  680  and the second steering flux coil  690  are arranged to produce magnetic flux that circulates in a common steering flux path  685 . The steering flux path starts at the first steering coil  680  in the left stator pole  610 , goes through the stator  600  to the right stator pole  620 , through the second steering flux coil  690  to the air gap  520 , across the air gap  520  to the rotor  500 , through the rotor  500  to the air gap  510 , across the air gap  510  to the left stator pole  610 , and returns to the first steering coil  680 . Note that the rubber sheet permanent magnet  700  extends across substantially the entire width of the stator central pole  630  and the rotor central pole  540 , and is significantly thicker than the nominal length of the left rotor-stator air gap  510  and right rotor-stator air gap  520 . Therefore, the rubber sheet permanent magnet provides a high reluctance path that substantially prevents the steering flux from flowing across the space between the central poles of the stator and rotor, separating the steering flux path  685  from the bias flux paths  701  and  702 . In a beneficial embodiment just described, the bias flux and steering fluxes are configured to add in air gap  520  and subtract in air gap  510 , producing a net torque on the rotor  500  from the magnetic forces acting on it, resulting in the counterclockwise rotation  530  of the rotor around the axis of rotation  550  shown in  FIG. 131 . Note that the flux addition and subtraction in the outer air gaps  510  and  520  does not appreciably change the magnitude of the net magnetic force acting on the rotor, but it does cause a lateral shift of that force. One of ordinary skill in the art recognizes that reversing the direction of the steering flux  685  can reverse the direction of the torque acting on the rotor  500  and hence its rotation  530 , and that each of the steering flux coils  680  and  690  can be made up of more than one coil acting together or independently for producing a particular desired steering flux. In a beneficial embodiment, the steering flux is controlled by a control system that uses a measurement of the rotation  530  for closed-loop feedback control for controlling that rotation and providing dynamic rotation stiffness to the rotor. In another example beneficial embodiment, the steering flux is controlled by a control system that uses a measurement of the rotation  530  for closed-loop feedback control for providing electronic damping to the rotational motion of the rotor.  
         [0136]     Typical rubber sheet permanent magnets have a low remanent flux of approximately 0.1 Tesla, which drives up the required cross-sectional area of the central stator pole  630  and the central rotor pole  540 . For example, if a total flux of 1.5 Tesla is desired in the outer rotor-stator air gaps  510  and  520 , with 0.75 Tesla provided by a typical rubber sheet permanent magnet having a permanent flux of 0.1 Tesla and 0.75 Tesla provided by the steering flux coils, then the cross-sectional area of the central poles  630  and  540  needs to be 7.5 times larger than the combined cross-sectional area of the outer rotor-stator air gaps  510  and  520 .  
         [0137]     In  FIG. 19 , the permanent magnet material has been moved out of the rubber sheet permanent magnet  700 , as shown in  FIG. 18 , resulting in the rubber sheet bearing  505  and the permanent magnet  750 . The arrow shown on the permanent magnet  750  indicates the north-pole direction. Note that as with the rubber sheet permanent magnet, the permanent magnet  750  extends across substantially the entire width of the stator central pole  631  and the rotor central pole  540 , and is significantly thicker than the nominal length of the left rotor-stator air gap  510  and right rotor-stator air gap  520 . Therefore, the permanent magnet  750  provides a high reluctance path that substantially prevents the steering flux from flowing across the space between the central poles of the stator and rotor, separating the steering flux path  685  from the bias flux paths  701  and  702 . By moving the permanent magnet material out of the rubber sheet permanent magnet, the mechanical and magnetic properties of the bearing  505  and the bias flux source  750  can be more readily tailored to meet the performance goals for the actuator. For instance, this allows: avoiding the possible break down of the rubber in a rubber sheet permanent magnet due to the motion of the particles of ceramic permanent magnet in the rubber during repeated cycles of shear deformation; adjusting the thickness and type of elastomer used in a rubber sheet based on the stiffness requirements for the bearing, independent of the bias flux requirement; and using a stronger permanent magnet to produce a desired bias flux. The rubber bearing  505  can include of a single layer of elastomeric material, or it can be a composite consisting of separate layers of elastomeric material and metal. To better allow optimizing the dimensions of the permanent magnet  750 , it may be desirable to use a combination of the permanent magnet  755  and two low magnetic-permeability spacers  756  and  757 , as shown in  FIG. 20 .  
         [0138]      FIG. 21  shows the rotary actuator of  FIG. 20  with the rotor  500  removed to better illustrate the other components. As one arrangement, the addition of a viscous fluid between the stator poles  610  and  620  and the rotor  500  can be used to provide mechanical damping to the rotor.  
         [0139]      FIG. 22  shows a possible arrangement of two displacement measuring sensors  902  and  904  for measuring the displacement of the rotor  500  relative to the stator  600 . The two sensors measure rotation of the rotor  500  around the axis  550 , and translation of the rotor  500  towards/away from the stator  600 . Candidate displacement sensors include eddy current sensors, capacitance sensors, and laser sensors. Those practiced in the art readily recognizes that the list of candidate sensors is not exhaustive, and that the effects of the time-varying magnetic flux carried by the stator poles  610  and  620  on the performance of the sensors needs to be considered when choosing a particular type of sensor.  
         [0140]      FIG. 23  shows the rotary actuator of  FIG. 20  with the addition of the two displacement sensors  902  and  904 .  
         [0141]      FIG. 24  shows the addition of a separate optical element  910  to the front surface of the rotor  500 . Potential optical elements include a reflective element, e.g., a flat, concave, convex, or complex surface, a refractive element, or a diffractive element. Depending on the intended application and available manufacturing processes, the optical element can be fixedly attached or removeably attached. Alternately, the optical element can be manufactured directly (e.g., via polishing said front face using methods known to one of ordinary skill in the art) onto the front face of the rotor  500 .  
         [0142]      FIG. 25  shows the rotary actuator of  FIG. 20  with the addition of a cutting tool  920 . Rotation of the rotor  500  about the rotation axis  550  causes motion of the tool  920  away/towards the stator  600 . A torsionally flexible element  800  is attached to a base  900  and the rotor  500 . In an example beneficial embodiment, the base  900  is fixed relative to the stator  600 , and the torsionally flexible element  800  is substantially aligned with the axis of rotation  550  of the rotor  500  and constrains translation of the rotor in a direction parallel to the axis  550 . Together, the torsionally flexible element  800  and the bearing  505  form a bearing system that constrains the rotor  500  relative to the stator  600  in all degrees of freedom of motion except for rotation around the axis  550 .  
         [0143]     In  FIG. 26  the tool  920  has been moved to a location near an edge of the rotor  500 . This may be beneficial in some cases when compared to the tool location in  FIG. 25 .  
         [0144]     Those of ordinary skill in the art recognizes that the bearing  505  in  FIGS. 23, 24 , and  25  can also be a gas-film or fluid film bearing. Moreover, it is to be appreciated that the magnetic attractive forces between the stator central pole  631  and the rotor central pole  540 , as shown and detailed in  FIG. 19 , due to the bias flux can provide a preload force on the bearing, and that a torsionally flexible element  800  and its base  900 , as shown in  FIG. 25 , can provide an axial support to the rotor  500 .  
         [0145]      FIG. 27  shows the system of  FIG. 25  with the tip of the tool  920  located in front of the rotation axis  550  of the rotor, and in a plane passing through the rotation axis. This configuration can be desired for an application where an oscillatory back and forth motion of the tool tip is desired.  
         [0146]     In  FIG. 28  the tip of the tool  920  is coincident with the rotation axis  550  of the rotor  500 . This configuration can be desirable for an application where rotation of the tool around its tip is desired. Note that as drawn,  FIG. 28  depicts a system where the combined center of mass of the major rotating elements,  500  and  540  is not coincident with the axis of rotation  550 . If the resultant forces from the linear acceleration of the center of mass is intolerable in a certain application, then balance masses (not shown) can be added to the front face of the rotor.  
         [0147]     Following the single degree of freedom rotary actuator in  FIG. 20 ,  FIG. 29  shows a simple, potentially low-cost, rotary actuator having two rotary degrees of freedom. The rotor  1500  has a first rotation axis  1550  and a second rotation axis  2550 . The first stator  1600  and second stator  2660  act on the rotor to produce rotations  1530  and  2530  around rotation axes  1550  and  2550 , respectively. Note the large potential work zone at the front face of the rotor  1500 . If the actuator is used in a fast tool servo, then the cutting tool can be arranged to engage a large workpiece. Alternatively, if the actuator is used to rotate an optical element, that optical element can be made integral with the exposed front face of the rotor or mounted directly to it. Potential optical elements include a reflective element having, (e.g., a mirror having a flat, concave, convex, or complex surface), a refractive element, or a diffractive element.  
         [0148]      FIG. 30  shows the second stator  2600 , its two outer stator poles  2610  and  2620 , and its central stator pole  2615 .  
         [0149]      FIG. 31  shows the first stator  1600  positioned with the second stator  2600 . The first stator  1600  has two outer stator poles  1610  and  1620 , and an opening to provide a space  1615  between the first stator  1600  and the central stator pole  2615  of the second stator  2600 .  
         [0150]      FIG. 32  shows a permanent magnet  1750  positioned against the first stator  1600  and the central stator pole  2615  of the second stator  2600 . The arrow shown on the permanent magnet  1750  indicates the north-pole direction.  
         [0151]      FIG. 33  shows the common bias flux pole piece  1630  that is positioned against the permanent magnet  1750 . The two axes  1550  and  2550  intersect at the center of the spherical seat  1635  in the common bias flux pole piece  1630 .  
         [0152]      FIG. 34  shows the addition of a spherical bearing  1505  to the spherical seat  1635 . One of ordinary skill in the art recognizes that the bearing  1505  can be an elastomeric layer, a lamination of elastomeric and metal layers, or a gas-film or fluid film bearing. Moreover, it is to be appreciated that the magnetic attractive forces between the common bias flux pole piece  1630  and the rotor central pole  1540  due to the bias flux can provide a preload force on the bearing.  
         [0153]      FIG. 35  shows the spherical bearing  1505  accepting the rotor central pole  1540  and  FIG. 36  shows the addition of the steering flux coils  1680  and  1690  for the first stator  1600 , and the addition of one  2680  of the pair of steering flux coils (the other coil  2690  is hidden in  FIG. 36 ) for the second stator  2600 .  
         [0154]      FIG. 37  is a first cross-sectional view of the system shown in  FIG. 39 , illustrating the magnetic flux paths through the first stator  1600  and the rotor  1500 . The bias flux paths  1701  and  1702  start at the permanent magnet  1750 , enter the magnetically permeable first stator  1600 , split and circulate through the first stator to the left first stator pole  1610  and right first stator pole  1620 , cross the air gaps  1510  and  1520  to enter the magnetically permeable rotor  1500 , travel through the rotor to the rotor central pole  1540 , pass across the bearing  1505 , through the magnetically permeable common bias flux pole piece  1630 , and return to the permanent magnet  1750 . In an example beneficial embodiment, the lengths of the air gaps  1510  and  1520  are equal, and the cross-sectional area of the air gaps  1510  and  1520  are equal. One of ordinary skill in the art readily recognizes that the sections of the rotor  1500  and first stator  1600  that are subjected to a time-varying magnetic flux are most often constructed from a laminated or powdered magnetic material with high permeability at the operating frequency of interest to reduce eddy current and magnetic hysteresis losses in the material.  
         [0155]     To develop a torque acting in the direction  1530  on the rotor  1500  in  FIG. 159 , the first steering flux coil  1680  and the second steering flux coil  1690  produce magnetic flux that circulates in a common steering flux path  1685 . The steering flux path starts at the first steering coil  1680  in the left first stator pole  1610 , goes through the first stator  1600 , around the second stator central pole  2615  to the right first stator pole  1620 , through the second steering flux coil  1690  to the air gap  1520 , across the air gap  1520  to the rotor  1500 , through the rotor  1500  to the air gap  1510 , across the air gap  1510  to the left first stator pole  1610 , and returns to the first steering coil  1680 . Note that the permanent magnet  1750  provides a high reluctance path that substantially separates the steering flux path  1685  from the bias flux paths  1701  and  1702 . Note that the space  1615  substantially prevents the steering flux  1685  from entering the second stator  2600 . One of ordinary skill in the art recognizes that the space  1615  can be filled with a low magnetically-permeable material that can add mechanical integrity to the assembly. In the embodiment just described, the bias flux and steering fluxes add in air gap  1520  and subtract in air gap  1510 , producing a net torque on the rotor  1500  from the magnetic forces acting on it, resulting in the rotation  1530  of the rotor around the axis of rotation  1550 . Note that the flux addition and subtraction in the outer air gaps  1510  and  1520  does not appreciably change the magnitude of the net magnetic force acting on the rotor, but it does cause a lateral shift of that force. One of ordinary skill in the art recognizes that reversing the direction of the steering flux  1685  reverses the direction of the torque acting on the rotor  1500  and hence its rotation  1530 , and that each of the steering flux coils  1680  and  1690  can be made up of more than one coil acting together or independently for producing a particular desired steering flux. In another example arrangment, the steering flux is controlled by a control system that uses a measurement of the rotation  1530  for closed-loop feedback control for controlling that rotation and providing dynamic rotation stiffness to the rotor. In another example arrangment, the steering flux is controlled by a control system that uses a measurement of the rotation  1530  for closed-loop feedback control for providing electronic damping to the rotational motion of the rotor.  
         [0156]      FIG. 38  is a second cross-sectional view of the system shown in  FIG. 29 , illustrating the magnetic flux paths through the second stator  2600  and the rotor  1500 . The bias flux paths  2701  and  2702  start at the permanent magnet  1750 , enter the magnetically permeable central pole  2615  of the second stator  2600 , split and circulate through the second stator to the left second stator pole  2610  and right second stator pole  2620 , cross the air gaps  2510  and  2520  to enter the magnetically permeable rotor  1500 , travel through the rotor to the rotor central pole  1540 , pass across the bearing  1505 , through the magnetically permeable common bias flux pole piece  1630 , and return to the permanent magnet  1750 . In a beneficial embodiment, lengths of the air gaps  2510  and  2520  are equal, and the cross-sectional area of the air gaps  2510  and  2520  are equal. One of ordinary skill in the art recognizes that the sections of the rotor  1500  and second stator  2600  that are subjected to a time-varying magnetic flux are most often constructed from a laminated or powdered magnetic material with high permeability at the operating frequency of interest to reduce eddy current and magnetic hysteresis losses in the material.  
         [0157]     To develop a torque acting in the direction  2530  on the rotor  1500  in  FIG. 38 , the first steering flux coil  2680  and the second steering flux coil  2690  produce magnetic flux that circulates in a common steering flux path  2685 . The steering flux path starts at the first steering coil  2680  in the left second stator pole  2610 , goes through the second stator  2600 , to the right second stator pole  2620 , through the second steering flux coil  2690  to the air gap  2520 , across the air gap  2520  to the rotor  1500 , through the rotor  1500  to the air gap  2510 , across the air gap  2510  to the left second stator pole  2610 , and returns to the first steering coil  2680 . Note that the permanent magnet  1750  provides a high reluctance path that substantially separates the steering flux path  2685  from the bias flux paths  2701  and  2702 . Note that the space  1615  substantially prevents the steering flux  2685  from entering the first stator  1600 . One of ordinary skill in the art readily recognizes that the space  1615  can be filled with a low magnetically-permeable material that can add mechanical integrity to the assembly. In the embodiment just described, the bias flux and steering fluxes add in air gap  2520  and subtract in air gap  2510 , producing a net torque on the rotor  1500  from the magnetic forces acting on it, resulting in the rotation  2530  of the rotor around the axis of rotation  2550 . Note that the flux addition and subtraction in the outer air gaps  2510  and  2520  does not appreciably change the magnitude of the net magnetic force acting on the rotor, but it does cause a lateral shift of that force. Those of ordinary skill in the art recognizes that reversing the direction of the steering flux  2685  reverses the direction of the torque acting on the rotor  1500  and hence its rotation  2530 , and that each of the steering flux coils  2680  and  2690  can be made up of more than one coil acting together or independently for producing a particular desired steering flux.  
         [0158]     In another beneficial embodiment, the steering flux is controlled by a closed-loop feedback control system that uses a measurement of the rotation  2530  for controlling that rotation and providing dynamic rotation stiffness to the rotor. As another arragnment, the steering flux is controlled by a closed-loop feedback control system that uses a measurement of the rotation  2530  for providing electronic damping to the rotational motion of the rotor.  
         [0159]      FIG. 39  shows a possible arrangement of four displacement measuring sensors  1902 ,  1904 ,  2902 , and  2904  for measuring the displacement of the rotor  1500  relative to the stators  1600  and  2600 . The two sensors  1902  and  1904  measure rotation of the rotor  1500  around the axis  1550 , and translation of the rotor  1500  towards/away from the stator  1600 . The two sensors  2902  and  2904  measure rotation of the rotor  1500  around the axis  2550 , and translation of the rotor  1500  towards/away from the stator  2600 .  
         [0160]     Candidate displacement sensors include eddy current sensors, capacitance sensors, and laser sensors. One of ordinary skill in the art recognizes that the list of candidate sensors is not exhaustive, and that the effects of the time-varying magnetic flux carried by the stator poles  1610 ,  1620 ,  2610 , and  2620  on the performance of the sensors needs to be considered when choosing a particular type of sensor. Note that the addition of a viscous fluid between the stator poles  1610 ,  1620 ,  2610 , and  2620 , and the rotor  1500  can be used to provide mechanical damping to the rotor.  
         [0161]      FIG. 40  shows the rotary actuator of  FIG. 29  with the addition of the four displacement measuring sensors  1902 ,  1904 ,  2902 , and  2904 .  
         [0162]      FIG. 41  shows a cross-sectional view of the system of  FIG. 40  to better illustrate two of the four displacement measuring sensors.  
         [0163]      FIG. 42  shows the addition of a separate optical element  910  to the front surface of the rotor  1500 . Potential optical elements include a mirror (flat, concave, convex, or complex surface), a refractive element, or a diffractive element. Depending on the intended application and available manufacturing processes, the optical element can be fixedly attached or removeably attached. Alternately, the optical element can be manufactured directly onto the front face of the rotor  1500 .  
         [0164]      FIG. 43  shows the rotary actuator of  FIG. 40  with the addition of a cutting tool  1920 . Rotation of the rotor  1500  about the rotation axis  1550  causes motion of the tool  1920  predominantly away/towards the stator  1600 . Rotation of the rotor  1500  about the rotation axis  2550  causes motion of the tool  1920  predominantly up/down relative to the stator  2600 . This combined tool motion may preferable in certain machining applications.  
         [0165]     In  FIG. 44 , the tool  1920  has been moved to a location near an edge of the rotor  1500 . This may be beneficial in some cases when compared to the tool location in  FIG. 43 .  
         [0166]      FIG. 45  shows the system of  FIG. 43  with the tip of the tool  1920  located in front of the intersections of the rotation axes  1550  and  2550  of the rotor. This configuration can be desirable for an application where an oscillatory back and forth motion and/or up and down is desired.  
         [0167]     In  FIG. 46  the tip of the tool  1920  is coincident with the intersections of the rotation axes  1550  and  2550  of the rotor  1500 . This configuration can be desirable for an application where rotation of the tool around its tip is desired. Note that as drawn,  FIG. 46  depicts a system where the combined center of mass of the major rotating elements,  1500  and  1540  is not coincident with the axes of rotation  1550  and  2550 . If the resultant forces from the linear acceleration of the center of mass are intolerable in a certain application, then balance masses (not shown) can be added to the front face of the rotor.  
         [0168]     It should be understood that the invention is not intended to be limited to the particular forms disclosed. Rather, the invention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the following appended claims.