Abstract:
A stiff anti-backlash gear unit comprising two or more high ratio reducer assemblies provides an energy efficient, low backlash, speed reducer. The gear unit is assembled in such a way as to eliminate backlash without any preloading and may be configured as a conventional, planetary or star gear with co-axial, spaced, angled or multiple output shafts.

Description:
REFERENCE TO EARLIER FILED APPLICATIONS 
       [0001]    This application claims priority to U.S. Provisional Application No. U.S. 61/752,040 filed Jan. 14, 2013 and U.S. Provisional Application No. U.S. 61/752,045 filed Jan. 14, 2013. Each of these patent applications are incorporated herein entirely by reference. 
     
    
     TECHNICAL FIELD 
       [0002]    This disclosure herein relates to a gear device comprising a plurality of gears. More particularly the disclosure herein relates to an apparatus and method providing reducing backlash. 
       BACKGROUND 
       [0003]    Existing methods for reducing backlash in a gear train typically involve pre-loading the internal members of the gear train. One method uses a flexible member to apply a torque between the two halves of a split gear to maintain uninterrupted engagement between corresponding teeth on the two halves of the split gear and opposite sides of the neighboring teeth of the gear being engaged. 
         [0004]    Another method uses a flexible member to apply a torque between two carriers, each of which are rotatably coupled to one half of a split gear. The associated torque maintains uninterrupted engagement between corresponding teeth on the two halves of the split gear and opposite sides of the teeth of the gear being engaged. 
         [0005]    Another method uses a flexible member to apply a radial force to a gear which is slidably coupled to a carrier along a radial axis. The associated force maintains uninterrupted engagement between both sides of the teeth of the gear and the gear being engaged. 
         [0006]    Another method uses a gear with split, flexible teeth that are tangentially compressed when engaged to maintain uninterrupted engagement between the two halves of the split gear teeth and neighboring teeth on the gear being engaged. 
         [0007]    Each pre-loading method introduces a static force which is present regardless of the external load applied to the gear train. This internal force generates friction and reduces energy efficiency under all but the maximum rated loading condition. If the external applied torque exceeds the internal preload torque, the flexible members comply and backlash returns. Consequently, the choice of spring constant trades off energy efficiency with output shaft stiffness, or torque capacity. 
         [0008]    Anti-backlash reducer gears may be used to provide uninterrupted engagement between a sun gear and an orbit gear, thereby resulting in an anti-backlash planetary gear. Planetary gears have a number of advantageous qualities and are found in a variety of configurations. One configuration comprises a sun gear, an orbit gear, and one or more planet pinion gears mounted on a carrier. The orbit gear is the reference, the sun gear is the input, and the carrier is the output. The reducer gears may be conventional or stepped pinion gears. 
         [0009]    Another configuration comprises two sun gears and one or more planet pinion gears mounted on a carrier. One sun gear is the reference, the carrier is the input, and the other sun gear is the output. The reducer gears may parallel or angled with respect to the sun gears. When the reducer gears are parallel, they are stepped pinion gears. When the reducer gears are angled, they may be conventional or stepped pinion gears. 
         [0010]    Another configuration comprises two orbit gears and one or more planet pinion gears mounted on a carrier. One orbit gear is the reference, the carrier is the input, and the other orbit gear is the output. This configuration is commonly referred to as an orbit gear and is capable of higher reduction ratios than a conventional planetary gear. 
         [0011]    Another configuration comprises two sun gears and one stepped ring gear mounted on a carrier. One sun gear is the reference, the carrier is the input, and the other sun gear is the output. This configuration is an alternate version of an orbit gear. 
         [0012]    Another configuration comprises a reference gear, an output gear and a stepped gear mounted on a carrier. The stepped gear is angled with respect to the reference and output gears and follows a nutating path. This configuration is commonly referred to as a nutating gear and is capable of higher reduction ratios than a conventional planetary gear. 
         [0013]    Another configuration is similar in construction to a planetary gear but the carrier is the reference and the orbit is the output. This configuration is commonly referred to as a star gear and is capable of lower reduction ratios than a conventional planetary gear but may be more energy efficient at high speeds since the stationary carrier does not experience mechanical resistance from the internal lubricant. 
         [0014]    The exemplary embodiments disclosed herein each comprise one or more pairs of reducer assemblies to provide a low backlash coupling between a drive and driven gear without any internal pre-load forces. A backlash reduction method biases the two reducer assemblies in each pair in opposite directions to provide a stiff, low backlash, engagement path between the drive gear and driven gear for both directions of rotation. No energy is lost to pre-loading friction and the engagement paths do not comply under heavy loads. No flexible members or adjustment mechanisms are required and the method may be used to compensate for wear. 
       SUMMARY 
       [0015]    Certain exemplary embodiments comprise a first engaging member, a second engaging member, a carrier member, one or more first reducer assemblies, and one or more second reducer assemblies. Each reducer assembly comprises a first reducer engaging member defining an input, a second reducer engaging member defining an output, and a reference member. The first engaging member simultaneously engages all first reducer engaging members. The second engaging member simultaneously engages all second reducer engaging members. The number of turns applied to the input of each reducer assembly with respect to the number of turns resulting at the output of each reducer assembly defines a reduction ratio for each reducer assembly. All first reducer assemblies provide a common first reduction ratio and all second reducer assemblies provide a common second reduction ratio. 
         [0016]    In certain exemplary embodiments, the first engaging member is co-axial with the second engaging member. 
         [0017]    In certain exemplary embodiments, all reference members are integral with the carrier member and the first reduction ratio is equal to the second reduction ratio. 
         [0018]    In certain exemplary embodiments, each reducer assembly defines a reducer axis which is co-axial with the corresponding first and second reducer engaging members. Each reference member is rotatably coupled to the carrier member about the corresponding reducer axis, and the first reduction ratio is not equal to the second reduction ratio. 
         [0019]    In certain exemplary embodiments, there are an equal number of first and second reducer assemblies. Each first reducer axis is co-axial with a corresponding second reducer axis, and the reference member of each first reducer assembly is integral with the reference member of the corresponding second reducer assembly. 
         [0020]    In certain exemplary embodiments, each reducer assembly further comprises a third reducer engaging member which is fixably coupled to the corresponding reference member and is co-axial with the corresponding reducer axis. The third reducer engaging member of each first reducer assembly engages the third reducer engaging members of one or more second reducer assemblies whereby the reference members of all first reducer assemblies rotate in the same direction, and the reference members of all second reducer assemblies rotate in the opposite direction as the first reducer assemblies. 
         [0021]    In certain exemplary embodiments, there are an equal number of first and second reducer assemblies and each first reducer axis is co-axial with a second reducer axis. 
         [0022]    In certain exemplary embodiments, all first and second reducer axes are parallel, all first reducer axes are spaced from one another, and all second reducer axes are spaced from one another. 
         [0023]    In certain exemplary embodiments, all first reducer axes are at an angle to one another, all second reducer axes are at an angle to one another, and all first and second reducer axes substantially intersect at a common point. 
         [0024]    In certain exemplary embodiments, one or more reducer assemblies are a planetary reducer assembly. 
         [0025]    In certain exemplary embodiments, one or more reducer assemblies are a nutating reducer assembly. 
         [0026]    In certain exemplary embodiments, one or more reducer assemblies are a serial reducer assembly. 
         [0027]    In certain exemplary embodiments, one or more reducer assemblies are a cycloid reducer assembly. 
         [0028]    Certain exemplary embodiments comprise a first engaging member, a second engaging member, a carrier member, and two or more reducer assemblies each comprising a first reducer engaging member defining an input, a second reducer engaging member defining an output, and a reference member. The first engaging member simultaneously engages all first reducer engaging members. The second engaging member simultaneously engages all second reducer engaging members. The number of turns applied to the input of each reducer assembly with respect to the number of turns resulting at the output of each reducer assembly defines a reduction ratio for each reducer assembly. 
         [0029]    In certain exemplary embodiments, all reference members are integral with the carrier member and the reduction ratio provided by all reducer assemblies is common. 
         [0030]    In certain exemplary embodiments, each reducer assembly defines a reducer axis which is co-axial with the corresponding first and second reducer engaging members. Each reducer assembly further comprises a third reducer engaging member which is fixably coupled to the corresponding reference member and is co-axial with the corresponding reducer axis. Each reference member is rotatably coupled to the carrier member about the corresponding reducer axis. The reduction ratio of one or more reducer assemblies is common and unequal to the reduction ratio of the remaining reducer assemblies which is also common. Each third reducer engaging member is engaged with one or more third reducer engaging member from another reducer assembly whereby any two reducer assemblies with a common reduction ratio rotate in the same direction and any two reducer assemblies with an uncommon reduction ratio rotate in opposite directions. 
         [0031]    In certain exemplary embodiments, there are an even number of reducer assemblies. Each reducer assembly defines a reducer axis which is co-axial with the corresponding first and second reducer engaging members. Each reference member is rotatably coupled to the carrier member about the corresponding reducer axis. The reduction ratio of half of the reducer assemblies is common and unequal to the reduction ratio of the remaining reducer assemblies which is also common. The reference member of each reducer assembly is fixably coupled to the reference member of a corresponding reducer assembly which has a co-axial reducer axis and an unequal reduction ratio. 
         [0032]    In certain exemplary embodiments, a method is disclosed. The method provides a first engaging member, a second engaging member, a carrier member, and two or more reducer assemblies each comprising an input portion, an output portion and a reference portion wherein the number of turns applied to each input portion with respect to the number of turns resulting at each output portion defines a reduction ratio for the corresponding reducer assembly. One or more input portions are engaged to the first engaging member and the corresponding output portions are engaged to the second engaging member whereby all backlash is removed for one direction of rotation of the first engaging member. All remaining input portions are engaged to the first engaging member and the corresponding output portions are engaged to the second engaging member whereby all backlash is removed for the opposite direction of rotation of the first engaging member. 
         [0033]    In certain exemplary embodiments, all reference members are integrated with the carrier member and a common reduction ratio is provided for all reducer assemblies. 
         [0034]    In certain exemplary embodiments, each reducer assembly is provided with a reducer axis which is co-axial with the corresponding input and output portions. All reference members are rotatably coupled with the carrier member about the corresponding reducer axis. Reference members are engaged with one another whereby one or more reference members are configured to rotate in one direction and the remaining reference members are configured to rotate in the opposite direction. A common reduction ratio is provided for all reducer assemblies that rotate in a common direction. 
     
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         [0035]      FIGS. 1A-1E  depict respectively, a schematic, a perspective view, a cross-sectional side view, an exploded perspective view, and an additional exploded perspective view in accordance with a first exemplary embodiment of an anti-backlash assembly. 
           [0036]      FIGS. 2A-2C  depict respectively, a schematic, a perspective view, and an exploded perspective view in accordance with a second exemplary embodiment of an anti-backlash assembly. 
           [0037]      FIGS. 3A-3C  depict respectively, a schematic, a perspective view, and an exploded perspective view in accordance with a third exemplary embodiment of an anti-backlash assembly. 
           [0038]      FIGS. 4A-4D  depict respectively, a schematic, a perspective view, a cross-sectional side view, and an exploded perspective view in accordance with a first exemplary embodiment of a reducer assembly. 
           [0039]      FIGS. 5A-5D  depict respectively, a schematic, a perspective view, a cross-sectional side view, and an exploded perspective view in accordance with a second exemplary embodiment of a reducer assembly. 
           [0040]      FIGS. 6-7  depict respectively, schematics in accordance with a third and fourth exemplary embodiment of a reducer assembly. 
           [0041]      FIGS. 8A-8D  depict respectively, a schematic, a perspective view, a cross-sectional side view, and an exploded perspective view in accordance with a fifth exemplary embodiment of a reducer assembly. 
           [0042]      FIGS. 9-10  depict respectively, schematics in accordance with a sixth and seventh exemplary embodiment of a reducer assembly. 
           [0043]      FIGS. 11A-11D  depict respectively, a schematic, a perspective view, a cross-sectional side view, and an exploded perspective view in accordance with an eighth exemplary embodiment of a reducer assembly. 
           [0044]      FIG. 12  depicts a schematic in accordance with a ninth exemplary embodiment of a reducer assembly. 
           [0045]      FIGS. 13A-13D  depict respectively, a schematic, a perspective view, a cross-sectional side view, and an exploded perspective view in accordance with a tenth exemplary embodiment of a reducer assembly. 
           [0046]      FIGS. 14-15  depict respectively, schematics in accordance with an eleventh and twelfth exemplary embodiment of a reducer assembly. 
           [0047]      FIGS. 16A-16D  depict respectively, a schematic, a perspective view, a cross-sectional side view, and an exploded perspective view in accordance with a thirteenth exemplary embodiment of a reducer assembly. 
           [0048]      FIGS. 17-32  depict respectively, schematics in accordance with a fourteenth through twenty-ninth exemplary embodiment of a reducer assembly. 
           [0049]      FIGS. 33A-33D  depict respectively, a schematic, a perspective view, a cross-sectional side view, and an exploded perspective view in accordance with a thirtieth exemplary embodiment of a reducer assembly. 
           [0050]      FIGS. 34A-34D  depict respectively, a schematic, an exploded side view, a cross-sectional side view, and an exploded perspective view in accordance with a first exemplary embodiment of an anti-backlash planetary gear. 
           [0051]      FIGS. 35-37  depict respectively, schematics in accordance with a second through fourth exemplary embodiment of an anti-backlash planetary gear. 
           [0052]      FIGS. 38A-38D  depict respectively, a schematic, a cross-sectional side view, an exploded perspective view, and an additional exploded perspective view in accordance with a fifth exemplary embodiment of an anti-backlash planetary gear. 
           [0053]      FIGS. 39A-39C  depict respectively, a schematic, a cross-sectional side view, and an exploded perspective view in accordance with a sixth exemplary embodiment of an anti-backlash planetary gear. 
           [0054]      FIGS. 40-41  depict respectively, schematics in accordance with a seventh and eighth exemplary embodiment of an anti-backlash planetary gear. 
           [0055]      FIGS. 42A-42D  depict respectively, a schematic, a perspective view, a cross-sectional side view, and an exploded perspective view in accordance with a ninth exemplary embodiment of an anti-backlash planetary gear. 
           [0056]      FIGS. 43A-43B  depict respectively, a perspective view, and an exploded perspective view in accordance with a tenth exemplary embodiment of an anti-backlash planetary gear. 
           [0057]      FIGS. 44A-44B  depict respectively, a perspective view, and an exploded perspective view in accordance with a first exemplary embodiment of an anti-backlash star gear. 
           [0058]      FIGS. 45A-45B  depict respectively, a perspective view, and an exploded perspective view in accordance with a second exemplary embodiment of an anti-backlash star gear. 
           [0059]      FIGS. 46A-46E  depict respectively, a schematic, a perspective view, an exploded perspective view, a cross-sectional side view, and an additional exploded perspective view in accordance with a fourth exemplary embodiment of an anti-backlash assembly. 
           [0060]      FIGS. 47A-47F  depict respectively, a schematic, a perspective view, a front view, a rear view, an exploded perspective view, and an additional exploded perspective view in accordance with a fifth exemplary embodiment of an anti-backlash assembly. 
           [0061]      FIGS. 48A-48D  depict respectively, a schematic, a perspective view, an exploded perspective view, and an additional exploded perspective view in accordance with a sixth exemplary embodiment of an anti-backlash assembly. 
           [0062]      FIGS. 49A-49D  depict respectively, a schematic, a perspective view, an exploded perspective view, and a cross-sectional side view in accordance with an eleventh exemplary embodiment of an anti-backlash planetary gear. 
           [0063]      FIGS. 50A-50F  depict respectively, a schematic, a front view, a rear view, a cross-sectional side view, an exploded perspective view, and an additional exploded perspective view in accordance with a twelfth exemplary embodiment of an anti-backlash planetary gear. 
           [0064]      FIGS. 51-53  depict respectively, schematics in accordance with a thirteenth through fifteenth exemplary embodiment of an anti-backlash planetary gear. 
           [0065]      FIGS. 54A-54C  depict respectively, a front perspective view, a rear perspective view, and an exploded perspective view in accordance with a sixteenth exemplary embodiment of an anti-backlash planetary gear. 
           [0066]      FIGS. 55A-55B  depict respectively, a perspective view, and an exploded perspective view in accordance with a third exemplary embodiment of an anti-backlash star gear. 
           [0067]      FIGS. 56A-56B  depict respectively, a perspective view, and an exploded perspective view in accordance with a fourth exemplary embodiment of an anti-backlash star gear. 
       
    
    
     REFERENCE NUMERALS 
       [0000]    
       
           1 —axis 
           2 —reducer axis 
           3 —eccentric axis 
           8 —direction 
           9 —direction 
           11 —main drive assembly 
           12 —carrier assembly 
           13 —main driven assembly 
           14 —drive assembly 
           15 —driven assembly 
           16 —reference assembly 
           17 —reducer assembly 
           18 —stepped gear assembly 
           19 —cycloid assembly 
           21 —first gear 
           22 —second gear 
           31 —first reducer gear 
           32 —second reducer gear 
           33 —third reducer gear 
           34 —fourth reducer gear 
           35 —fifth reducer gear 
           36 —sixth reducer gear 
           38 —first step 
           39 —second step 
           40 —case 
           41 —input shaft 
           42 —output shaft 
           44 —carrier 
           45 —post 
           50 —driven member 
           51 —eccentric shaft 
           52 —eccentric race 
           53 —drive member 
           54 —reference member 
           55 —cycloid shaft 
           56 —cycloid race 
           60 —bearing 
           61 —bearing 
           62 —bearing 
           63 —bearing 
           64 —bearing 
           65 —bearing 
       
     
       DESCRIPTION OF EMBODIMENTS 
       [0110]    Wherever possible, the same reference numerals are used throughout the accompanying drawings and descriptions to refer to the same or like parts. Components such as retainers and fasteners that do not substantially contribute to the functionality of the embodiments disclosed herein are neglected for the sake of simplicity. 
         [0111]    Although spur gears and conical gears are used as exemplary engaging members in the accompanying drawings, it is understood that many other means would suffice, such as spiral gears, helical gears, double helical gears, herring-bone gears, roller tooth gears, friction couplings, magnetic couplings, pulleys and belts, or sprockets and chains. 
         [0112]    Although a uniform tooth module is depicted in the accompanying drawings, it is understood that any module will suffice and that any number of different modules may be used as long as all pairs of engaging gears have equivalent modules. In particular, different modules and tooth geometries may be used to satisfy the different torque requirements of the individual members. 
         [0113]    Although a straight tooth profile is depicted in the accompanying drawings, it is understood that any tooth profile will suffice, such as an involute profile with any desired pressure angle. It is also understood that any number of different tooth profiles may be used as long as all pairs of engaging gears have complementary tooth profiles. 
         [0114]    It is understood that a spur gear comprises teeth arranged circumferentially on a cylindrical hub with a substantially uniform tooth profile. It is also understood that a radial gear comprises teeth arranged circumferentially on a circular hub with a non-uniform tooth profile. It is also understood that a conical gear comprises teeth arranged circumferentially on a conical hub with a non-uniform tooth profile. It is also understood that a conical gear may comprise teeth with a substantially uniform tooth profile to allow engagement with a spur gear. 
         [0115]    It is understood that a conical or a radial gear comprise a non-uniform pitch radius. In the accompanying specification, the specified pitch radius of a conical or a radial gear corresponds to its minimum pitch radius. 
         [0116]    It is understood that a ring gear engages on its interior surface and a pinion gear engages on its exterior surface. It is also understood that a conical ring gear engages on the surface that faces toward its rotation axis and a conical pinion gear engages on the surface that faces away from its rotation axis. A conical gear that engages on a surface that is parallel to its rotation axis is referred to as a radial gear. 
         [0117]    Although bearings are used to depict rotatable couplings in the accompanying drawings, it is understood that any other means will suffice, such as roller bearings, plain or journal bearings, thrust bearings, low friction coatings, materials or surface treatments or favorable clearances and lubricants. It is also understood that the male and female members making up a rotatable coupling may often be interchanged without substantially affecting function. It is also understood that rotatably coupling a first member to a second member which is also rotatably coupled to a third member about a common axis, is equivalent to rotatably coupling the first member to the third member. 
         [0118]    Although shafts are used to depict rotational inputs and outputs in the accompanying drawings, it is understood that any other means will suffice, such as eccentric or crank shafts, gears, friction couplings, pulleys, sprockets, female couplings, fastener interfaces such as keyways or threaded holes, or materials, circuits or assemblies providing a magnetic or electrostatic interface. 
         [0119]    Any members that are described as integral in the following description are fixably connected. Although posts are also used to depict fixable connections in the accompanying drawings, it is understood that any other means will suffice, such as welds, fasteners, elongated members of any cross-sectional shape, or forming the integral parts from a single piece of material. 
         [0120]    Although a cooling means is not depicted in the accompanying drawings, a person skilled in the art will appreciate that a cooling means such as cooling fins, a heat conduction system, a splashed lubricant bath, a forced fluid and heat exchange system, or a directed air flow system could be included and is contemplated. 
         [0121]    Although each exemplary embodiment is depicted as a speed reducer, a person skilled in the art will appreciate that a speed reducer may be used to amplify speed by interchanging the roles of the input and output. In fact, the roles of the reference, input and output may all be interchanged to obtain a desired reduction or amplification ratio, or to cause the input and output to rotate in the same or opposite directions. Similarly, if any one is used as an input and the remaining two are used as outputs, then a differential mechanism is obtained. Consequently, reduction, amplification and differential mechanisms are all contemplated. 
         [0122]    In the schematics illustrated in the accompanying drawings, gears are depicted as closed contours, rotatable couplings are depicted as thick parallel lines, stiff members are depicted as solid thick lines, and hidden, out of plane members are depicted using dotted lines. 
         [0123]    A representative sample of embodiments is included in the accompanying figures for exemplary purposes only. A great number of additional tooth geometries, ring and pinion combinations and kinematic arrangements are also contemplated. The scope of the present invention is not limited to the embodiments included but spans all possible combinations anticipated by the specification and claims. 
         [0124]      FIG. 1  illustrates a first exemplary embodiment of an anti-backlash assembly. The anti-backlash assembly provides a low backlash, speed reduced engagement path between a first gear  21  and a second gear  22  whereby high speed rotation of the first gear  21  results in low speed rotation of the second gear  22  with minimal free-play between the first gear  21  and the second gear  22 . 
         [0125]    The anti-backlash assembly comprises a case  40  and a pair of substantially equivalent reducer assemblies  17   a ,  17   b  which each provide a common reduction ratio. The reducer assemblies  17   a ,  17   b  are axially aligned about a common reducer axis  2 . The first gear  21  and second gear  22  each define an axis  1   a ,  1   b , which is parallel and spaced from the reducer axis  2 . The axes  1   a  and  1   b  may be configured to be parallel or co-axial with each other. 
         [0126]    Each reducer assembly  17  comprises a drive assembly  14 , a driven assembly  15 , and a reference assembly  16 , all co-axial with the reducer axis  2 . Each reducer assembly  17  further comprises a stepped gear assembly  18  defining an eccentric axis  3  which is substantially parallel to and spaced from the reducer axis  2 . 
         [0127]    A first reducer gear  31 , a drive member  53  and an eccentric race  52  are integral and combine with a bearing  62  to form the drive assembly  14 . The eccentric race  52  is co-axial with the eccentric axis  3 . A second reducer gear  32  and a fourth reducer gear  34  are integral and combine with a bearing  64  to form the driven assembly  15 . A third reducer gear  33  and a reference member  54  are integral and fixably connected to the case  40  to form the reference assembly  16 . A first step  38  and a second step  39  are integral and fixably combined with a bearing  63  to form the stepped gear assembly  18 . 
         [0128]    Bearing  62  rotatably couples the drive assembly  14  to the reference assembly  16 . Bearing  64  rotatably couples the driven assembly  15  to the reference assembly  16 . Bearing  63  rotatably couples the stepped gear assembly  18  to the drive assembly  14 . 
         [0129]    The first gear  21  simultaneously engages both first reducer gears  31 . Each first step  38  engages the corresponding third reducer gear  33 . Each second step  39  engages the corresponding fourth reducer gear  34 . Both second reducer gears  32  simultaneously engage the second gear  22 . 
         [0130]    Rotating the first gear  21  causes both drive assemblies  14  to rotate. Each drive assembly  14  guides the corresponding stepped gear assembly  18  along an eccentric path. The first step  38  engages the stationary third reducer gear  33 , causing the stepped gear assembly  18  to rotate inside the eccentric race  52  as it circulates around the reducer axis  2 . The second step  39  engages the fourth reducer gear  34 , causing the driven assembly  15  to rotate. Rotation of the two driven assemblies  15  causes the second gear  22  to rotate at a lower rate than the first gear  21 . 
         [0131]      FIG. 1D  illustrates the method for reducing backlash. First the first gear  21  is disengaged from the anti-backlash assembly. Next, the first reducer gear  31  of reducer assembly  17   a  is rotated in one direction  8  while the first reducer gear  31  of reducer assembly  17   b  is rotated in the opposite direction  9  until the teeth of the second reducer gear  32  of reducer assembly  17   a  and the teeth of the second reducer gear  32  of reducer assembly  17   b  come into contact with opposite sides of the teeth of second gear  22 . Finally, the first gear  21  is re-engaged with the anti-backlash assembly. It is understood that the directions  8  and  9  could be reversed without substantially affecting the method. 
         [0132]    For one direction of rotation of first gear  21 , the second reducer gear  32  of reducer assembly  17   a  transmits torque to second gear  22  while the second reducer gear  32  of reducer assembly  17   b  follows the rotational trajectory without contributing to, or impeding the applied torque. For the opposite direction of rotation of first gear  21 , the second reducer gear  32  of reducer assembly  17   b  transmits torque to second gear  22  while the second reducer gear  32  of reducer assembly  17   a  follows the rotational trajectory without contributing to, or impeding the applied torque. A stiff, anti-backlash engagement path is provided between first gear  21  and second gear  22  for each direction of rotation by one of the two reducer assemblies  17 . 
         [0133]      FIG. 2  illustrates a second exemplary embodiment of an anti-backlash assembly. The backlash reduction method is illustrated in  FIG. 2C . The second exemplary embodiment is substantially equivalent to the first exemplary embodiment illustrated in  FIG. 1 , except for the following. In the second exemplary embodiment, the reducer assemblies  17   a ,  17   b  are arranged in a co-planar rather than a co-axial configuration whereby the reducer axes  2   a ,  2   b  are substantially parallel to and spaced from one another. In addition, the two first reducer gears  31  each engage the first gear  21  at a different point on the first gear  21  pitch surface. In addition, the two second reducer gears  32  each engage the second gear  21  at a different point on the second gear  22  pitch surface. 
         [0134]      FIG. 3  illustrates a third exemplary embodiment of an anti-backlash assembly. The backlash reduction method is illustrated in  FIG. 3C . In  FIG. 3A , the first gear  21  and second gear  22  are depicted as invisible members. The third exemplary embodiment is substantially equivalent to the second exemplary embodiment illustrated in  FIG. 2 , except for the following. In the third exemplary embodiment, the reducer assemblies  17   a ,  17   b  are arranged in a circumferential rather than a co-planer configuration whereby the reducer axes  2   a ,  2   b  are at an angle to one another and substantially intersect at a common point on a central axis  1 . In addition, the central axis  1  is common for the first gear  21  and second gear  22  and is substantially perpendicular to the plane defined by the two reducer axes  2   a ,  2   b . In addition, the first gear  21 , second gear  22 , first reducer gears  31  and second reducer gears  32  are bevel gears rather than spur gears, although spur gears would suffice. 
         [0135]    The method for reducing backlash illustrated in  FIGS. 2C and 3C  is similar to that described for  FIG. 1D . It is understood that the directions  8  and  9  could be reversed without substantially affecting the method. 
         [0136]      FIGS. 4-33  illustrate respectively, a first through thirtieth exemplary embodiment of a reducer assembly  17 . Although the eighth exemplary embodiment illustrated in  FIG. 11  is depicted in the exemplary embodiments of an anti-backlash assembly illustrated in  FIGS. 1-3 , any one of the thirty reducer assembly  17  exemplary embodiments would suffice. It is understood that many of the exemplary reducer assemblies  17  comprise a non-uniform mass distribution and may benefit from a counter-balancing means to reduce vibration. It is also understood that any of the exemplary reducer assemblies  17  may be configured so that the first reducer gear  31  and second reducer gear  32  are on located opposite sides, or on the same side of the reference member  54 . 
         [0137]      FIGS. 4-15  and  29 - 32  illustrate alternative embodiments of a planetary reducer assembly  17 .  FIGS. 16-24  illustrate alternative embodiments of a nutating reducer assembly  17 .  FIGS. 25-28  illustrate alternative embodiments of a serial reducer assembly  17 .  FIG. 33  illustrates a cycloid reducer assembly  17 . 
         [0138]      FIGS. 4-7  illustrate respectively, a first through fourth exemplary embodiment of a reducer assembly  17  comprising a drive member  53  which supports a plurality of stepped gear assemblies  18   a ,  18   b  to provide a counter-balanced orbit gear. Each stepped gear assembly  18  is rotatably coupled to the drive member  53  comprising two eccentric shafts  51   a ,  51   b , each defining an eccentric axis  3   a ,  3   b  which are each parallel to and spaced from the reducer axis  2 . 
         [0139]    The first through third exemplary embodiments of a reducer assembly  17  illustrated in  FIGS. 4-6  respectively, each comprise a ring third reducer gear  33 , a ring fourth reducer gear  34 , a pinion first step  38  and a pinion second step  39 . The second and third exemplary embodiments of a reducer assembly  17  illustrated in  FIGS. 5 and 6  respectively, each comprise stepped gear assemblies  18   a ,  18   b  that overlap one another to provide a counter-balanced orbit gear. 
         [0140]    The fourth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 7  comprises a pinion third reducer gear  33 , a pinion fourth reducer gear  34 , a pinion first step  38  and a pinion second step  39 . 
         [0141]      FIGS. 8-11  illustrate respectively, a fifth through eighth exemplary embodiment of a reducer assembly  17  comprising a drive member  53  which supports a stepped gear assembly  18  which is rotatably coupled to the drive member  53  about an eccentric axis  3  which is parallel to and spaced from the reducer axis  2 , thereby providing an orbit gear. 
         [0142]    The fifth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 8  comprises a ring third reducer gear  33 , a ring fourth reducer gear  34 , a pinion first step  38  and a pinion second step  39 . 
         [0143]    The sixth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 9  comprises a pinion third reducer gear  33 , a pinion fourth reducer gear  34 , a ring first step  38  and a ring second step  39 . 
         [0144]    The seventh exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 10  comprises a ring third reducer gear  33 , a ring fourth reducer gear  34 , a pinion first step  38  and a pinion second step  39 . 
         [0145]    The eighth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 11  comprises a pinion third reducer gear  33 , a pinion fourth reducer gear  34 , a ring first step  38  and a ring second step  39 . In  FIGS. 11B-11D , first reducer gear  31  and second reducer gear  32  are conical gears as depicted in the third exemplary embodiment of an anti-backlash assembly illustrated in  FIG. 3 . 
         [0146]      FIGS. 12-13  illustrate respectively, a ninth and tenth exemplary embodiment of a reducer assembly  17  comprising a drive member  53  comprising two eccentric races  52   a ,  52   b , which are each rotatably coupled to a stepped gear assembly  18   a ,  18   b , by a bearing  63 . Each stepped gear assembly  18   a ,  18   b  engages a third reducer gear  33   a ,  33   b , and a common fourth reducer gear  34  which is sufficiently deep to allow simultaneous engagement with the two stepped gear assemblies  18   a ,  18   b . Alternatively, the common fourth reducer gear  34  could be replaced by two separate gears, one belonging to each reducer assembly  17   a ,  17   b . The reference assembly  16  comprises a split reference member  54   a ,  54   b  joined by a plurality of posts  45  to support the two third reducer gears  33   a ,  33   b  on opposite sides of the drive assembly  14 . Each stepped gear assembly  18   a ,  18   b  is co-axial with an eccentric axis  3   a ,  3   b  which is parallel to and spaced from the reducer axis  2 . The two eccentric axes  3   a ,  3   b  are located on opposite sides of the reducer axis  2  to provide a balanced mass distribution and a counter-balanced orbit gear. 
         [0147]    The ninth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 12  comprises ring third reducer gears  33   a ,  33   b , a ring fourth reducer gear  34 , pinion first steps  38   a ,  38   b , and pinion second steps  39   a ,  39   b.    
         [0148]    The tenth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 13  comprises pinion third reducer gears  33   a ,  33   b , a pinion fourth reducer gear  34 , ring first steps  38   a ,  38   b , and ring second steps  39   a ,  39   b.    
         [0149]      FIGS. 14-15  illustrate respectively, an eleventh and twelfth exemplary embodiment of a reducer assembly  17  comprising a drive member  53  which supports a stepped gear assembly  18  which is rotatably coupled to the drive member  53  about an eccentric axis  3  which is parallel to and spaced from the reducer axis  2 , thereby providing an orbit gear. 
         [0150]    The eleventh exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 14  comprises a ring third reducer gear  33 , a pinion fourth reducer gear  34 , a pinion first step  38  and a ring second step  39 . 
         [0151]    The twelfth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 15  comprises a ring third reducer gear  33 , a ring fourth reducer gear  34 , a pinion first step  38  and a pinion second step  39 . The twelfth exemplary embodiment is substantially equivalent to the fifth exemplary embodiment illustrated in  FIG. 8 , except for the following. The third reducer gear  33 , fourth reducer gear  34 , first step  38  and second step  39  are bevel gears rather than spur gears. It is anticipated that any pair of mating spur gears in any of the first through eleventh exemplary embodiments may be replaced by a pair of mating bevel gears, as depicted in the twelfth exemplary embodiment. 
         [0152]      FIGS. 16-24  illustrate respectively, a thirteenth through twenty-first exemplary embodiment of a reducer assembly  17  comprising a drive member  53  comprising an eccentric shaft  51  or eccentric race  52  defining an eccentric axis  3  which is at an angle to and substantially intersecting the reducer axis  2 . The eccentric shaft  51  or eccentric race  52  supports and is rotatably coupled to a stepped gear assembly  18  by a bearing  63 . Rotation of the drive assembly  14  causes the stepped gear assembly  18  to follow a nutating path as it simultaneously engages the fixed third reducer gear  33  and the fourth reducer gear  34 , causing the driven assembly  15  to rotate, thereby providing a nutating gear. 
         [0153]    The thirteenth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 16  comprises a bevel third reducer gear  33 , a bevel fourth reducer gear  34 , a pinion first step  38  and a pinion second step  39 . 
         [0154]    The fourteenth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 17  comprises a bevel third reducer gear  33 , a bevel fourth reducer gear  34 , a radial first step  38  and a radial second step  39 . 
         [0155]    The fifteenth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 18  comprises a bevel third reducer gear  33 , a bevel fourth reducer gear  34 , a ring first step  38  and a ring second step  39 . 
         [0156]    The sixteenth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 19  comprises a pinion third reducer gear  33 , a pinion fourth reducer gear  34 , a bevel first step  38  and a bevel second step  39 . 
         [0157]    The seventeenth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 20  comprises a radial third reducer gear  33 , a radial fourth reducer gear  34 , a bevel first step  38  and a bevel second step  39 . 
         [0158]    The eighteenth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 21  comprises a ring third reducer gear  33 , a ring fourth reducer gear  34 , a bevel first step  38  and a bevel second step  39 . 
         [0159]    The nineteenth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 22  comprises a bevel third reducer gear  33 , a bevel fourth reducer gear  34 , a bevel first step  38  and a bevel second step  39 . 
         [0160]    The twentieth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 23  comprises a radial third reducer gear  33 , a radial fourth reducer gear  34 , a bevel first step  38  and a bevel second step  39 . The twentieth exemplary embodiment is substantially equivalent to the seventeenth exemplary embodiment illustrated in  FIG. 20 , except for the following. Drive member  53  comprises an eccentric race  52  rather than an eccentric shaft  51  which is rotatably coupled to the stepped gear assembly  18 . Any of the thirteenth through nineteenth exemplary embodiments of a reducer assembly  17  may have their male drive member  53  replaced by a female drive member  53  similar to that depicted in the twentieth exemplary embodiment. 
         [0161]    The twenty-first exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 24  comprises a bevel third reducer gear  33 , a bevel fourth reducer gear  34 , a radial first step  38  and a ring second step  39 . The twenty-first exemplary embodiment comprises a first step  38  which is dissimilar from the second step  39 . It is understood that a nutating reducer assembly  17  may comprise any combination of pinion, radial, ring and bevel gears that may be configured to align correctly and that the particular combination used for the third reducer gear  33  and first step  38  may be different from the particular combination used for the fourth reducer gear  34  and second step  39 . All possible combinations are anticipated. 
         [0162]      FIGS. 25-28  illustrate respectively, a twenty-second through twenty-fifth exemplary embodiment of a reducer assembly  17  comprising a serial chain of two or more stepped gear assemblies. The first reducer gear  31  and a sixth reducer gear  36  together comprise the first stepped gear assembly in the sequence. The fourth reducer gear  34  and second reducer gear  32  together comprise the final stepped gear assembly in the sequence, thereby providing a serial gear. 
         [0163]    The twenty-second exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 25  comprises a pinion fourth reducer gear  34 , a pinion sixth reducer gear  36 , and one intermediate stepped gear assembly  18  comprising a pinion first step  38  and a pinion second step  39 . 
         [0164]    The twenty-third exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 26  comprises a pinion fourth reducer gear  34 , a pinion sixth reducer gear  36 , and a plurality of intermediate stepped gear assemblies  18   a ,  18   b ,  18   c  each comprising a pinion first step  38   a ,  38   b ,  38   c  and a pinion second step  39   a ,  39   b ,  39   c.    
         [0165]    The twenty-fourth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 27  comprises a ring fourth reducer gear  34 , and a pinion sixth reducer gear  36 . 
         [0166]    The twenty-fifth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 28  comprises a ring fourth reducer gear  34 , a pinion sixth reducer gear  36 , and a plurality of intermediate stepped gear assemblies  18   a ,  18   b ,  18   c  each comprising a ring first step  38   a ,  38   b ,  38   c  and a pinion second step  39   a ,  39   b ,  39   c.    
         [0167]      FIGS. 29-32  illustrate respectively, a twenty-sixth through twenty-ninth exemplary embodiment of a reducer assembly  17  comprising a driven member  50  which supports a plurality of planet gears to provide a planetary gear. Each planet gear is rotatably coupled to the driven member  50  about an eccentric axis  3   a ,  3   b.    
         [0168]    The twenty-sixth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 29  comprises a ring third reducer gear  33 , a pinion fourth reducer gear  34 , and stepped planet gear assemblies each comprising a pinion first step  38  and a pinion second step  39 . The eccentric axes  3   a ,  3   b  are parallel to and spaced from the reducer axis  2 . 
         [0169]    The twenty-seventh exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 30  comprises a ring third reducer gear  33 , a pinion fourth reducer gear  34 , and single stage planet gears each comprising a pinion sixth reducer gear  36 . The eccentric axes  3   a ,  3   b  are parallel to and spaced from the reducer axis  2 . 
         [0170]    The twenty-eighth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 31  comprises a bevel third reducer gear  33 , a bevel fourth reducer gear  34 , and stepped planet gear assemblies each comprising a bevel first step  38  and a bevel second step  39 . The eccentric axes  3   a ,  3   b  are at an angle to and substantially intersect the reducer axis  2 . 
         [0171]    The twenty-ninth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 32  comprises a bevel third reducer gear  33 , a bevel fourth reducer gear  34 , and single stage planet gears each comprising a bevel sixth reducer gear  36 . The eccentric axes  3   a ,  3   b  are at an angle to and substantially intersect the reducer axis  2 . 
         [0172]      FIG. 33  illustrates, a thirtieth exemplary embodiment of a reducer assembly  17  comprising a drive assembly  14 , a driven assembly  15 , a reference assembly  16 , and a cycloid assembly  19 . The drive assembly  14  comprises a drive gear  31 , a drive member  53  and an eccentric shaft  51 . The driven assembly  15  comprises a driven gear  32 , a driven member  50  and a plurality of cycloid races  56 . The reference assembly  16  comprises a ring third reducer gear  33 . The cycloid assembly  19  comprises a pinion first step  38  and a plurality of cycloid shafts  55 . The cycloid assembly  19  is rotatably coupled to the eccentric shaft  51  about an eccentric axis  3  which is parallel to and spaced from the reducer axis  2 . The first step  38  engages the third reducer gear  33  and each cycloid shaft  55  engages a cycloid race  56 , thereby providing a cycloid gear. 
         [0173]      FIGS. 34-43  illustrate respectively, a first through tenth exemplary embodiment of an anti-backlash planetary gear. Each anti-backlash planetary gear comprises a main drive assembly  11 , a main driven assembly  13 , and a carrier assembly  12 , all co-axial with a central axis  1 . 
         [0174]    A first gear  21  and input shaft  41  are integral and combine with a bearing  60  to form the main drive assembly  11 . A second gear  22  and case  40  are integral and form the main driven assembly  13 . A carrier  44 , an output shaft  42 , and the reference assemblies  16  of all reducer assemblies  17  are integral and combine with a bearing  61  to form the carrier assembly  12 . 
         [0175]    Bearing  60  rotatably couples the main drive assembly  11  to the main driven assembly  13 . Bearing  61  rotatably couples the carrier assembly  12  to the main driven assembly  13 . 
         [0176]    The first gear  21  simultaneously engages all first reducer gears  31  and the second gear  22  simultaneously engages all second reducer gears  32 . Rotating the first gear  21  causes all first reducer gears  31  and drive assemblies  14  to rotate. Rotating the drive assembly  14  of each reducer assembly  17  causes the corresponding driven assembly  15  to rotate at a reduced rate. The simultaneous engagement between the second reducer gears  32  and the fixed second gear  22  causes the reducer assemblies  17  to circulate around the central axis  1  and rotate the carrier assembly  12 . 
         [0177]      FIG. 38D  illustrates the method for reducing backlash. First the first gear  21  is disengaged from the anti-backlash planetary gear. Next, the first reducer gears  31  of each reducer assembly  17   a  are rotated in one direction  8  while the first reducer gears  31  of each reducer assembly  17   b  are rotated in the opposite direction  9  until the teeth of the second reducer gears  32  of reducer assembly  17   a  and the teeth of the second reducer gears  32  of reducer assembly  17   b  come into contact with opposite sides of the teeth of second gear  22 . Finally, the first gear  21  is re-engaged with the anti-backlash planetary gear. It is understood that the directions  8  and  9  could be reversed without substantially affecting the method. 
         [0178]    For each direction of rotation of the first gear  21 , half of the reducer assemblies  17  contribute to the torque applied to the second gear  22  while the others track the rotational trajectory without contributing to, or impeding the applied torque. 
         [0179]    In the first through eighth exemplary embodiments of an anti-backlash planetary gear illustrated in  FIGS. 34-41 , the reducer axes  2  are substantially parallel to and spaced from the central axis  1 . In the ninth and tenth exemplary embodiments of an anti-backlash planetary gear illustrated in  FIGS. 42-43 , the reducer axes are at an angle to one another and substantially intersect at a common point on the central axis  1 . 
         [0180]    The first through fourth exemplary embodiments of an anti-backlash planetary gear illustrated in  FIGS. 34-37 , each comprise six reducer assemblies  17   a ,  17   b  integral with the carrier assembly  12 . Although the first exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 4  is depicted in  FIGS. 34-37 , any one of the thirty exemplary reducer assemblies  17  would suffice. 
         [0181]    The fifth through eighth exemplary embodiments of an anti-backlash planetary gear illustrated in  FIGS. 38-41 , each comprise four reducer assemblies  17   a ,  17   b  integral with the carrier assembly  12 . Although the thirteenth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 16  is depicted in  FIG. 38 , and the first exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 4  is depicted in  FIGS. 39-41 , any one of the thirty exemplary reducer assemblies  17  would suffice. 
         [0182]    The first and fifth exemplary embodiments of an anti-backlash planetary gear, illustrated in  FIGS. 34 and 38  respectively, each comprise a pinion first gear  21 , and a ring second gear  22 . 
         [0183]    The second and sixth exemplary embodiments of an anti-backlash planetary gear illustrated in  FIGS. 35 and 39  respectively, each comprise a ring first gear  21 , and a ring second gear  22 . 
         [0184]    The third and seventh exemplary embodiments of an anti-backlash planetary gear illustrated in  FIGS. 36 and 40  respectively, each comprise a pinion first gear  21 , and a pinion second gear  22 . 
         [0185]    The fourth and eighth exemplary embodiments of an anti-backlash planetary gear illustrated in  FIGS. 37 and 41  respectively, each comprise a ring first gear  21 , and a pinion second gear  22 . 
         [0186]    The ninth exemplary embodiment of an anti-backlash planetary gear illustrated in  FIG. 42 , comprises two reducer assemblies  17   a ,  17   b  integral with the carrier assembly  12 . Although the eighth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 11  is depicted in  FIG. 42 , any one of the thirty exemplary reducer assemblies  17  would suffice. 
         [0187]    The tenth exemplary embodiment of an anti-backlash planetary gear illustrated in  FIG. 43 , comprises six reducer assemblies  17   a ,  17   b  integral with the carrier assembly  12 . Although the eighth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 11  is depicted in  FIG. 43 , any one of the thirty exemplary reducer assemblies  17  would suffice. 
         [0188]    The ninth and tenth exemplary embodiments of an anti-backlash planetary gear illustrated in  FIGS. 42 and 43  each comprise a bevel first gear  21 , and a bevel second gear  22 . 
         [0189]      FIGS. 44-45  illustrate respectively, a first and second exemplary embodiment of an anti-backlash star gear. An anti-backlash star gear is equivalent to an anti-backlash planetary gear except for the following. An anti-backlash star gear uses the carrier assembly  12  as the reference and the main driven assembly  13  as the output. In addition, the output shaft  42  is integral with the driven assembly  13  rather than the carrier assembly  12 . In addition, an anti-backlash star gear comprising two co-axial reducer assemblies  17   a ,  17   b  may be configured to comprise a plurality of driven assemblies  13 , each rotating about an axis which substantially intersects, but is not necessarily co-axial with the central axis  1 , as illustrated in  FIG. 45 . 
         [0190]    The first and second exemplary embodiment of an anti-backlash star gear illustrated in  FIGS. 44 and 45  respectively, each comprise two reducer assemblies  17   a ,  17   b  integral with the carrier assembly  12 . Although the tenth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 13  is depicted in  FIG. 44 , and the eighth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 11  is depicted in  FIG. 45 , any one of the thirty exemplary reducer assemblies  17  would suffice. 
         [0191]    The method for reducing backlash illustrated in  FIG. 45B  is similar to that described for  FIG. 1D . It is understood that the directions  8  and  9  could be reversed without substantially affecting the method. 
         [0192]      FIG. 46  illustrates a fourth exemplary embodiment of an anti-backlash assembly. The anti-backlash assembly provides a low backlash, speed reduced engagement path between a first gear  21  and a second gear  22  whereby high speed rotation of the first gear  21  results in low speed rotation of the second gear  22  with minimal free-play between the first gear  21  and the second gear  22 . 
         [0193]    The anti-backlash assembly comprises a case  40  and a pair of reducer assemblies  17   a ,  17   b  which each provide a different reduction ratio. The reducer assemblies  17   a ,  17   b  are axially aligned about a common reducer axis  2 . The first gear  21  and second gear  22  each define an axis  1   a ,  1   b , which is perpendicular to the reducer axis  2 . The axes  1   a  and  1   b  may or may not be configured to be co-axial with each other. Although the eighth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 11  is depicted in  FIG. 46 , any one of the thirty exemplary reducer assemblies  17  would suffice. 
         [0194]    The two reference assemblies  16  are fixably connected and rotate together as a common unit. Bearing  65  rotatably couples the reference assemblies  16  to the case  40 . The first gear  21  simultaneously engages both first reducer gears  31 . Both second reducer gears  32  simultaneously engage the second gear  22 . 
         [0195]    Rotating the first gear  21  causes both drive assemblies  14  to rotate. The two driven assemblies  15  each rotate at a different rate due to their unequal reduction ratios. This causes the integrated reference members  54  to rotate inside the case  40  while the two second reducer gears  32  advance the second gear  22  at a lower rate than the first gear  21 . 
         [0196]      FIG. 47  illustrates a fifth exemplary embodiment of an anti-backlash assembly. The backlash reduction method is illustrated in  FIG. 47F . The fifth exemplary embodiment is substantially equivalent to the fourth exemplary embodiment illustrated in  FIG. 46 , except for the following. In the fifth exemplary embodiment, the reducer assemblies  17   a ,  17   b  are arranged in a co-planar rather than a co-axial configuration whereby the reducer axes  2   a ,  2   b  are substantially parallel to and spaced from one another. In addition, the two first reducer gears  31  each engage the first gear  21  at a different point on the first gear  21  pitch surface. In addition, the first gear  21 , second gear  22 , first reducer gears  31 , second reducer gears  32 , and fifth reducer gears  35  are spur gears rather than bevel gears, although spur gears would suffice in either exemplary embodiment. 
         [0197]    In addition, the reference assembly  16  of reducer assembly  17   a  is not fixably connected to the reference assembly  16  of reducer assembly  17   b . Instead, a fifth reducer gear  35  is fixably connected to the reference member  53  of each reducer assembly  17 . The fifth reducer gear  35  of reducer assembly  17   a  engages the fifth reducer gear  35  of reducer assembly  17   b  whereby the two reference assemblies counter-rotate with respect to the case  40 . 
         [0198]      FIG. 48  illustrates a sixth exemplary embodiment of an anti-backlash assembly. The backlash reduction method is illustrated in  FIG. 48D . In  FIG. 48A , the first gear  21  and second gear  22  are depicted as invisible members. The sixth exemplary embodiment is substantially equivalent to the fifth exemplary embodiment illustrated in  FIG. 47 , except for the following. In the sixth exemplary embodiment, the reducer assemblies  17   a ,  17   b  are arranged in a circumferential rather than a co-planer configuration whereby the reducer axes  2   a ,  2   b  are at an angle to one another and substantially intersect at a common point on a central axis  1 . In addition, the central axis  1  is common for the first gear  21  and second gear  22  and is substantially perpendicular to the plane defined by the two reducer axes  2   a ,  2   b . In addition, the first gear  21 , second gear  22 , first reducer gears  31 , second reducer gears  32 , and fifth reducer gears  35  are bevel gears rather than spur gears, although spur gears would suffice. 
         [0199]    Although a fixable coupling is depicted in  FIG. 46  and a pair of fifth reducer gears  35  are depicted in  FIGS. 47 and 48  to rotatably engage the reference assemblies  16  of the two reduction assembles  17   a ,  17   b , any rotatable engagement means would suffice. For example a series of gears, a chain and sprocket, or any other engagement means with any desired turn ratio would suffice. 
         [0200]    The method for reducing backlash illustrated in  FIGS. 46E ,  47 F and  48 D is similar to that described for  FIG. 1D . It is understood that the directions  8  and  9  could be reversed without substantially affecting the method. 
         [0201]      FIGS. 49-54  illustrate respectively, an eleventh through sixteenth exemplary embodiment of an anti-backlash planetary gear. Each anti-backlash planetary gear comprises a main drive assembly  11 , a main driven assembly  13 , and a carrier assembly  12 , all co-axial with a central axis  1 . 
         [0202]    A first gear  21  and input shaft  41  are integral and combine with a bearing  60  to form the main drive assembly  11 . A second gear  22  and case  40  are integral and form the main driven assembly  13 . A carrier  44  and an output shaft  42  are integral and combine with a bearing  61  to form the carrier assembly  12 . 
         [0203]    Bearing  60  rotatably couples the main drive assembly  11  to the main driven assembly  13 . Bearing  61  rotatably couples the carrier assembly  12  to the main driven assembly  13 . Bearing  65  rotatably couples each reference assembly  16  to the carrier  44 . 
         [0204]    The first gear  21  simultaneously engages all first reducer gears  31  and the second gear  22  simultaneously engages all second reducer gears  32 . Rotating the first gear  21  causes all first reducer gears  31  and drive assemblies  14  to rotate. Rotating the drive assembly  14  of each reducer assembly  17  causes the corresponding driven assembly  15  to rotate. The two driven assemblies  15  each rotate at a different rate due to their differing reduction ratios. The reference assemblies  16  each rotate inside the carrier  44  while the second reducer gears  32  advance the second gear  22  at a lower rate than the first gear  21 . Fixing the second gear  22  to the case  40  causes the reducer assemblies  17  to circulate around the central axis  1  and rotate the carrier assembly  12 . 
         [0205]    The method for reducing backlash illustrated in  FIG. 50F  is similar to that described for  FIG. 38D . It is understood that the directions  8  and  9  could be reversed without substantially affecting the method. 
         [0206]    In the twelfth through fifteenth exemplary embodiments of an anti-backlash planetary gear illustrated in  FIGS. 50-53 , the reducer axes  2  are substantially parallel to and spaced from the central axis  1 . In the eleventh and sixteenth exemplary embodiments of an anti-backlash planetary gear illustrated in  FIGS. 49 and 54 , the reducer axes are at an angle to one another and substantially intersect at a common point on the central axis  1 . 
         [0207]    The eleventh exemplary embodiment of an anti-backlash planetary gear illustrated in  FIG. 49 , comprises two reducer assemblies  17   a ,  17   b . Although the eighth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 11  is depicted in  FIG. 49 , any one of the thirty exemplary reducer assemblies  17  would suffice. 
         [0208]    The twelfth through fifteenth exemplary embodiments of an anti-backlash planetary gear illustrated in  FIGS. 50-53 , each comprise six reducer assemblies  17   a ,  17   b . Although the eighth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 11  is depicted in  FIGS. 50-53 , any one of the thirty exemplary reducer assemblies  17  would suffice. 
         [0209]    The sixteenth exemplary embodiment of an anti-backlash planetary gear illustrated in  FIG. 54 , comprises six reducer assemblies  17   a ,  17   b . Although the eighth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 11  is depicted in  FIG. 54 , any one of the thirty exemplary reducer assemblies  17  would suffice. 
         [0210]    The eleventh and sixteenth exemplary embodiments of an anti-backlash planetary gear, illustrated in  FIGS. 49 and 54  respectively, each comprise a bevel first gear  21 , and a bevel second gear  22 . 
         [0211]    The twelfth exemplary embodiment of an anti-backlash planetary gear illustrated in  FIG. 50  comprises a pinion first gear  21 , and a ring second gear  22 . 
         [0212]    The thirteenth exemplary embodiment of an anti-backlash planetary gear illustrated in  FIG. 51  comprises a ring first gear  21 , and a ring second gear  22 . 
         [0213]    The fourteenth exemplary embodiment of an anti-backlash planetary gear illustrated in  FIG. 52  comprises a pinion first gear  21 , and a pinion second gear  22 . 
         [0214]    The fifteenth exemplary embodiment of an anti-backlash planetary gear illustrated in  FIG. 53  comprises a ring first gear  21 , and a pinion second gear  22 . 
         [0215]      FIGS. 55-56  illustrate respectively, a third and fourth exemplary embodiment of an anti-backlash star gear. 
         [0216]    The third and fourth exemplary embodiments of an anti-backlash star gear illustrated in  FIGS. 55 and 56  respectively, each comprise two reducer assemblies  17   a ,  17   b . Although the tenth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 13  is depicted in  FIG. 55 , and the eighth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 11  is depicted in  FIG. 56 , any one of the thirty exemplary reducer assemblies  17  would suffice. 
         [0217]    The two reference assemblies  16  are fixably connected and rotate together as a common unit. Bearing  65  rotatably couples the reference assemblies  16  to the case  40 . 
         [0218]    The method for reducing backlash illustrated in  FIG. 56B  is similar to that described for  FIG. 1D . It is understood that the directions  8  and  9  could be reversed without substantially affecting the method. 
         [0219]    In each of the following examples, the reduction ratio RR=Ti/To where Ti is the number of turns applied to the input shaft  41  and To is the number of turns resulting at the output shaft  42 , all with respect to the rotational reference, which is the case  40  for an anti-backlash assembly, the main driven assembly  13  for an anti-backlash planetary gear, and the carrier assembly  12  for an anti-backlash star gear. The reducer ratio Nx=Tix/Tox where Tix is the number of turns applied to the first reducer gear  31  and Tox is the number of turns resulting at the second reducer gear  32 , all with respect to the reference assembly  16 . The “x” subscript is used to indicate values for a particular collection of reducer assemblies  17 . For example, Na, Tia and To a correspond to reducer assemblies  17   a  and Nb, Tib, and Tob correspond to reducer assemblies  17   b . The “x” subscript is absent when N, Ti and To are common for all reducer assemblies  17 . 
         [0220]    RR and Nx are computed from P21, P22, P31, P32, P33, P34, P38 and P39 which are the pitch diameters of the first gear  21 , second gear  22 , first reducer gear  31 , second reducer gear  32 , third reducer gear  33 , fourth reducer gear  34 , first step  38  and second step  39  respectively. A positive value indicates that the input and output turn in the same direction and a negative value indicates that they turn in opposite directions. 
         [0221]    Each example involves reducer assemblies  17  that are either orbit gears or nutating gears so Nx is computed as follows. 
         [0000]    
       
         
           
             
               N 
               x 
             
             = 
             
               
                 
                   P 
                   34 
                 
                  
                 
                   P 
                   38 
                 
               
               
                 
                   
                     P 
                     34 
                   
                    
                   
                     P 
                     38 
                   
                 
                 - 
                 
                   
                     P 
                     33 
                   
                    
                   
                     P 
                     39 
                   
                 
               
             
           
         
       
     
         [0222]    For the first exemplary embodiment of an anti-backlash assembly, RR is computed as follows. 
         [0000]    
       
         
           
             RR 
             = 
             
               N 
                
               
                   
               
                
               
                 
                   
                     P 
                     22 
                   
                    
                   
                     P 
                     31 
                   
                 
                 
                   
                     P 
                     21 
                   
                    
                   
                     P 
                     32 
                   
                 
               
             
           
         
       
     
         [0223]    For the fourth and fifth exemplary embodiments of an anti-backlash assembly, RR is computed as follows. 
         [0000]    
       
         
           
             RR 
             = 
             
               
                 ( 
                 
                   
                     
                       2 
                        
                       
                           
                       
                        
                       
                         N 
                         a 
                       
                        
                       
                         N 
                         b 
                       
                     
                     - 
                     
                       ( 
                       
                         
                           N 
                           a 
                         
                         + 
                         
                           N 
                           b 
                         
                       
                       ) 
                     
                   
                   
                     
                       N 
                       a 
                     
                     + 
                     
                       N 
                       b 
                     
                     - 
                     2 
                   
                 
                 ) 
               
                
               
                 ( 
                 
                   
                     
                       P 
                       22 
                     
                      
                     
                       P 
                       31 
                     
                   
                   
                     
                       P 
                       21 
                     
                      
                     
                       P 
                       32 
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0224]    For the fifth, sixth and ninth exemplary embodiments of an anti-backlash planetary gear, RR is computed as follows where the sign of RR depends on whether the first gear  21  and the second gear  22  are ring or pinion gears. 
         [0000]    
       
         
           
             RR 
             = 
             
               ± 
               
                 N 
                  
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         
                           P 
                           22 
                         
                          
                         
                           P 
                           31 
                         
                       
                       
                         
                           P 
                           21 
                         
                          
                         
                           P 
                           32 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0225]    For the eleventh and twelfth exemplary embodiments of an anti-backlash planetary gear, RR is computed as follows where the sign of RR depends on whether the first gear  21  and the second gear  22  are ring or pinion gears. 
         [0000]    
       
         
           
             RR 
             = 
             
               
                 ± 
                 
                   ( 
                   
                     
                       
                         2 
                          
                         
                             
                         
                          
                         
                           N 
                           a 
                         
                          
                         
                           N 
                           b 
                         
                       
                       - 
                       
                         ( 
                         
                           
                             N 
                             a 
                           
                           + 
                           
                             N 
                             b 
                           
                         
                         ) 
                       
                     
                     
                       
                         N 
                         a 
                       
                       + 
                       
                         N 
                         b 
                       
                       - 
                       2 
                     
                   
                   ) 
                 
               
                
               
                 ( 
                 
                   1 
                   + 
                   
                     
                       
                         P 
                         22 
                       
                        
                       
                         P 
                         31 
                       
                     
                     
                       
                         P 
                         21 
                       
                        
                       
                         P 
                         32 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0226]    For the second exemplary embodiment of an anti-backlash star gear, RR is computed as follows. 
         [0000]    
       
         
           
             RR 
             = 
             
               
                 - 
                 N 
               
                
               
                   
               
                
               
                 
                   
                     P 
                     22 
                   
                    
                   
                     P 
                     31 
                   
                 
                 
                   
                     P 
                     21 
                   
                    
                   
                     P 
                     32 
                   
                 
               
             
           
         
       
     
         [0227]    For the fourth exemplary embodiment of an anti-backlash star gear, RR is computed as follows. 
         [0000]    
       
         
           
             RR 
             = 
             
               
                 - 
                 
                   ( 
                   
                     
                       
                         2 
                          
                         
                             
                         
                          
                         
                           N 
                           a 
                         
                          
                         
                           N 
                           b 
                         
                       
                       - 
                       
                         ( 
                         
                           
                             N 
                             a 
                           
                           + 
                           
                             N 
                             b 
                           
                         
                         ) 
                       
                     
                     
                       
                         N 
                         a 
                       
                       + 
                       
                         N 
                         b 
                       
                       - 
                       2 
                     
                   
                   ) 
                 
               
                
               
                 ( 
                 
                   
                     
                       P 
                       22 
                     
                      
                     
                       P 
                       31 
                     
                   
                   
                     
                       P 
                       21 
                     
                      
                     
                       P 
                       32 
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0228]    When performing the backlash reduction method, after disengaging the first gear  21 , each first reducer gear  31  must be advanced by an integer number of gear teeth or it will not be possible to re-engage the first gear  21 . The backlash reduction ratio Δ corresponds to the number of first reducer gear  31  teeth that must be advanced in order to advance the second reducer gear  32  by a single tooth. Advancing the first reducer gear by a single tooth causes the second reducer gear  32  to advance by 1/Δ of a tooth so a large Δ value is desirable because it allows backlash to be removed in smaller increments. Δ is computed as follows, assuming all associated gears have an equivalent tooth module. 
         [0229]    In the first through seventh examples, Δ is computed as follows. 
         [0000]    
       
         
           
             Δ 
             = 
             
                
               
                 N 
                  
                 
                     
                 
                  
                 
                   
                     P 
                     31 
                   
                   
                     P 
                     32 
                   
                 
               
                
             
           
         
       
     
         [0230]    In the eighth through twelfth examples, Δ is computed as follows. 
         [0000]    
       
         
           
             Δ 
             = 
             
                
               
                 
                   ( 
                   
                     
                       
                         N 
                         b 
                       
                        
                       
                         ( 
                         
                           
                             N 
                             a 
                           
                           - 
                           1 
                         
                         ) 
                       
                     
                     
                       
                         N 
                         b 
                       
                       - 
                       1 
                     
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       P 
                       31 
                     
                     
                       P 
                       32 
                     
                   
                   ) 
                 
               
                
             
           
         
       
     
         [0231]    A first example considers the first exemplary embodiment of an anti-backlash assembly illustrated in  FIG. 1 . The reducer gears  17  are orbit gears. The values P21=24, P22=42, P31=36, P32=18, P33=11, P34=10, P38=13 and P39=12 result in N=−65 RR=−228, and Δ=130. 
         [0232]    A second example considers the fifth exemplary embodiment of an anti-backlash planetary gear illustrated in  FIG. 38 . The reducer gears  17  are nutating gears and P38&gt;P33 and P39&gt;P34 to avoid mechanical interference. The values P21=22, P22=72, P31=P32=25, P33=21, P34=22, P38=22 and P39=23 result in N=484, RR=−2,068 and Δ=484. 
         [0233]    A third example considers the sixth exemplary embodiment of an anti-backlash planetary gear illustrated in  FIG. 39 . The reducer gears  17  are counterbalanced orbit gears and P38&lt;P33/2 and P39&lt;P34/2 to avoid mechanical interference. The values P21=P22=72, P31=P32=25, P33=24, P34=23, P38=11 and P39=10 result in N=19, RR=39 and Δ=19.5. 
         [0234]    A fourth example considers the sixth exemplary embodiment of an anti-backlash planetary gear illustrated in  FIG. 39  with the reducer assemblies  17  replaced by the second exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 5 . The reducer gears  17  are counterbalanced orbit gears with overlapping stepped gear assembly  18 . The values P21=P22=72, P31=P32=25, P33=24, P34=23, P38=13 and P39=12 result in N=27, RR=54 and Δ=27.2. 
         [0235]    A fifth example considers the sixth exemplary embodiment of an anti-backlash planetary gear illustrated in  FIG. 39  with the reducer assemblies  17  replaced by the eighth exemplary embodiment of a reducer assembly  17  illustrated in  FIG. 11 . The reducer gears  17  are orbit gears and P38&lt;P33 and P39&lt;P34 to avoid mechanical interference. The values P21=P22=72, P31=P32=25, P33=24, P34=23, P38=15 and P39=14 result in N=38, RR=77 and Δ=38.3. 
         [0236]    A sixth example considers the ninth exemplary embodiment of an anti-backlash planetary gear illustrated in  FIG. 42 . The reducer gears  17  are orbit gears. The values P21=22, P22=30, P31=37, P32=45, P33=11, P34=10, P38=13 and P39=12 result in N=−65 RR=−138, and Δ=53.4. 
         [0237]    A seventh example considers the second exemplary embodiment of an anti-backlash star gear illustrated in  FIG. 45 . The reducer gears  17  are orbit gears. The values P21=22, P22=30, P31=37, P32=45, P33=11, P34=10, P38=13 and P39=12 result in N=−65 RR=73, and Δ=53.4. 
         [0238]    In the following examples, the pitch diameters of all fifth reducer gears  35  are equal, making the following equations independent of the associated pitch diameters. It is also understood that different reduction ratios may be achieved by making the pitch diameter of the fifth reduction gear  35  of reducer assembly  17   a  unequal to the pitch diameter of the fifth reduction gear  35  of reducer assembly  17   b.    
         [0239]    An eighth example considers the fourth exemplary embodiment of an anti-backlash assembly illustrated in  FIG. 46 . The values P21=22, P22=30, P31=42, P32=50, P33a=11, P34a=10, P38a=13, P39a=12, P33b=10, P34b=11, P38b=12 and P39b=13 result in RR=9,829 and Δ=56.3. 
         [0240]    A ninth example considers the fifth exemplary embodiment of an anti-backlash assembly illustrated in  FIG. 47 . The values P21=26, P22=42, P31=36, P32=18, P33a=11, P34a=10, P38a=13, P39a=12, P33b=10, P34b=11, P38b=12 and P39b=13 result in RR=27,723 and Δ=134. 
         [0241]    A tenth example considers the eleventh exemplary embodiment of an anti-backlash planetary gear illustrated in  FIG. 49 . The values P21=22, P22=30, P31=42, P32=50, P33a=11, P34a=10, P38a=13, P39a=12, P33b=10, P34b=11, P38b=12 and P39b=13 result in RR=18,410 and Δ=56.3. 
         [0242]    An eleventh example considers the twelfth exemplary embodiment of an anti-backlash planetary gear illustrated in  FIG. 50 . The values P21=42, P22=96, P31=36, P32=18, P33a=11, P34a=10, P38a=13, P39a=12, P33b=10, P34b=11, P38b=12 and P39b=13 result in RR=47,808 and Δ=134. 
         [0243]    A twelfth example considers the fourth exemplary embodiment of an anti-backlash star gear illustrated in  FIG. 56 . The values P21=22, P22=30, P31=42, P32=50, P33a=11, P34a=10, P38a=13, P39a=12, P33b=10, P34b=11, P38b=12 and P39b=13 result in RR=−9,829 and Δ=56.3. 
         [0244]    The exemplary embodiments disclosed herein have a number of advantageous properties. Certain exemplary embodiments have backlash that may be adjusted during assembly or at any time thereafter to compensate for manufacturing error, manufacturing tolerances and wear and tear. 
         [0245]    Certain exemplary embodiments may be produced with additional clearance between gear teeth to allow higher gear ratios without increased backlash. 
         [0246]    Certain exemplary embodiments provide a high reduction ratio for a given envelope, using moderately sized gears comprising moderate numbers of teeth which are easy to manufacture and have favorable wear properties. 
         [0247]    Certain exemplary embodiments provide a reduction ratio that scales favorably with the outside diameter. 
         [0248]    Certain exemplary embodiments comprise ring and pinion gears which have high contact ratios and high torque capacities. 
         [0249]    Certain exemplary embodiments may be configured such that the input and output shafts turn in the same or in opposite directions. 
         [0250]    Certain exemplary embodiments may comprise spur gears, helical gears, double helical gears, herring-bone gears, roller tooth gears, conical gears, radial gears, or gears with any other tooth geometry. 
         [0251]    Certain exemplary embodiments may comprise friction couplings, magnetic couplings, ratchet wheels, or any other type of engaging members. 
         [0252]    Certain exemplary embodiments provide a non-compliant, bi-directional coupling between the input and output shafts. 
         [0253]    Other advantages are apparent from the disclosure herein.