Patent Publication Number: US-11378163-B2

Title: Multistage pericyclic gear reducer

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of priority from CU.S. Provisional Patent Application Ser. No. 63/075,207, filed on Sep. 7, 2020, and entitled “PERICYCLIC GEAR REDUCER,” which is incorporated herein by reference in its entirety. 
    
    
     TECHNICAL FIELD 
     The present disclosure relates to gear reducers and particularly to epicycloidal high-ratio gear reducers. More particularly, the present invention relates to a compact pericycloidal high-ratio gear reducer. 
     BACKGROUND 
     A fundamental requirement of majority of devices that utilize mechanical power is transmission of the rotational movement of a power source, such as an electric motor or an internal combustion engine to a low speed power consumer or end effector. One way to address this requirement may be utilization of conventional gear boxes with multistage gear trains. However, such multistage gear boxes may be relatively large and heavy for applications where high reduction ratios are required. 
     To obtain higher reduction ratios while maintaining the compactness of power transmission systems, gear reducers may be utilized instead of conventional multistage gear boxes. Planetary gear transmissions (PGTs), cycloidal speed reducers (CSRs), worm and gears (WGs), and harmonic drives (HDs) are among the well-known gear reducers that may provide high gear reduction ratios and maintain the compactness of the transmission systems. 
     In comparison with conventional gearboxes with simple gear trains, single-stage PGTs have several advantages and some disadvantages. Coaxial input and output, as well as two degrees of freedom are important benefits which enable the PGTs to provide several gear ratios when several stages of PGTs are mounted consecutively. In short, the advantages of PGTs in comparison with ordinary gear boxes may include compact structure, light weight, small moment of inertia, high power density, high efficiency, coaxial input-output shafts, low backlash, low pitch line velocity in gears, multiplicity in reduction ratios, and multiple degrees of freedom. However, PGTs may have disadvantages, such as having a large number of elements which makes the system overly complex, requiring an exceedingly high precision manufacturing process which makes PGTs expensive, having indeterminate load distribution on gears, high fatigue due to high repetition of loading on teeth of sun gear and bidirectional bending loads on teeth of planet gears, having weak heat transfer capabilities, and finally a narrow range of reduction ratios per stage. 
     In comparison with conventional gearboxes, a CSR may have all the above-mentioned advantages of a PGT, except having multiplicity in reduction ratios, and multi degrees of freedom. A CSR may further have drawbacks, such as non-involute-cut gears, load alternation on gears and bearings and internal speed fluctuation as well as centrifugal forces which altogether increase internal vibrations and probability of fatigue failure. Additionally, it can offer reduction ratios only in a range of 10 to 100, which are limited to integer numbers and getting decimal ratios is impossible. A CSR may also require high precisions and high technology level in production due to complexity in its components&#39; shape and their accurate assembly. The advantages of a CSR may include a high reduction ratio in a single stage, a very compact design, smooth running with moderate noise, low backlash, high shock absorbance, smaller number of parts, high reliability and long life. However, a CSR may have disadvantages, such as internal vibrations and non-steady output velocity, requiring balancing, expensive manufacturing process, limited range and numbers of possible reduction ratios, and indeterminate load distribution on rollers. 
     Compared to PGTs, HDs may have advantages, such as considerably larger reduction ratios per stage, high torque capacity per weight, excellent positioning accuracy and repeatability, compact design, near zero backlash, self-locking property, and high torsion stiffness. On the other hand, HDs may have drawbacks, such as high elasticity requirement in its flex spline, together with nonlinear stiffness damping resulting in a low efficiency, a restricted reduction range: i.e. 30:1 up to 320:1 in normal applications. Furthermore, HDs cannot be back driven. HDs may further require high technology level both in gear production and bearing production. In addition, HDs may be applicable only in small sizes since for larger sizes, larger deflections and stresses may occur, which might go beyond the resistance margins of materials. 
     SUMMARY 
     This summary is intended to provide an overview of the subject matter of the present disclosure and is not intended to identify essential elements or key elements of the subject matter, nor is it intended to be used to determine the scope of the claimed implementations. The proper scope of the present disclosure may be ascertained from the claims set forth below in view of the detailed description and the drawings. 
     According to one or more exemplary embodiments, the present disclosure is directed to a multistage pericyclic gear reducer. An exemplary multistage pericyclic gear reducer may include an input shaft that may be rotatable about a longitudinal axis of an exemplary input shaft, an input external-teeth gear that may be coaxially coupled to and rotatable with an exemplary input shaft, a first middle planet ring gear, and a first driver ring gear that may be meshed with an exemplary input external-teeth gear and rotatable with an exemplary input shaft about a first rotational axis coinciding with an exemplary longitudinal axis of an exemplary input shaft. 
     An exemplary first driver ring gear may be coupled to an exemplary first middle planet ring gear from a first side of an exemplary first middle planet ring gear utilizing a first prismatic joint. An exemplary first rotational axis may be parallel with a first central normal axis of an exemplary first middle planet ring gear. An exemplary first prismatic joint may be disposed on a plane perpendicular to an exemplary first rotational axis. An exemplary first driver ring gear may be configured to transfer the rotational movement of an exemplary input shaft to an exemplary first middle planet ring gear. 
     An exemplary multistage pericyclic gear reducer may further include a first driven wheel, where a first side of the first driven wheel may be coupled to an exemplary first middle planet ring gear from a second side of an exemplary first middle planet ring gear utilizing a second prismatic joint. An exemplary second prismatic joint may be disposed on a plane perpendicular to an exemplary first rotational axis. An exemplary second prismatic joint may be perpendicular to an exemplary first prismatic joint. An exemplary first driven wheel may be rotatable about a second rotational axis, where an exemplary second rotational axis may be parallel with an exemplary first rotational axis. 
     An exemplary multistage pericyclic gear reducer may further include a second middle planet ring gear, where a first side of the second middle planet ring gear may be coupled to a second side of an exemplary first driven wheel utilizing a third prismatic joint. An exemplary third prismatic joint may be disposed on a plane perpendicular to an exemplary first rotational axis. An exemplary multistage pericyclic gear reducer may further include a second driven wheel that may be coupled from a first side of an exemplary second driven wheel to a second side of an exemplary second middle planet ring gear utilizing a fourth prismatic joint. An exemplary fourth prismatic joint may be disposed on a plane perpendicular to an exemplary first rotational axis. An exemplary fourth prismatic joint may be perpendicular to an exemplary third prismatic joint. An exemplary second driven wheel may be rotatable about a third rotational axis, where an exemplary third rotational axis may be parallel with an exemplary second rotational axis. 
     An exemplary multistage pericyclic gear reducer may further include an output external-teeth gear that may extend through and mesh with both an exemplary first middle planet ring gear and an exemplary second middle planet ring gear, and a central output shaft that may be coupled to an exemplary output external-teeth gear. An exemplary central output shaft may be rotatable with an exemplary output external-teeth gear about a central axis. An exemplary central axis may be along and parallel with a longitudinal axis of an exemplary central output shaft, where an exemplary output external-teeth gear may be extended along an exemplary central axis. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The novel features which are believed to be characteristic of the present disclosure, as to its structure, organization, use and method of operation, together with further objectives and advantages thereof, will be better understood from the following drawings in which a presently exemplary embodiment of the present disclosure will now be illustrated by way of example. It is expressly understood, however, that the drawings are for illustration and description only and are not intended as a definition of the limits of the present disclosure. Embodiments of the present disclosure will now be described by way of example in association with the accompanying drawings in which: 
         FIG. 1A  illustrates a right-side exploded view of a pericyclic gear reducer, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 1B  illustrates a left-side exploded view of a pericyclic gear reducer, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 2A  illustrates a perspective view of a middle planet ring gear coupled between a driver ring gear and a driven wheel, consistent with one or more exemplary embodiments of the present disclosure; 
         FIGS. 2B and 2C  illustrate front views of a middle planet ring gear coupled between a driver ring gear and a driven wheel, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 3A  illustrates a perspective view of a middle planet ring gear meshed with an output sun gear, consistent with one or more exemplary embodiments of the present disclosure; 
         FIGS. 3B and 3C  illustrate front views of a middle planet ring gear meshed with an output sun gear, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 4A  illustrates an exploded view of a coaxial pericyclic gear reducer, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 4B  illustrates an exploded view of a coaxial pericyclic gear reducer, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 5A  illustrates a perspective exploded view of a driver ring gear meshed with input external-teeth gear and a middle planet ring gear meshed with output sun gear in a coaxial pericyclic gear reducer, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 5B  illustrates a perspective exploded view of components of a coaxial pericyclic gear reducer with a middle planet ring gear and a driven wheel having offset on the same side of a central axis of the coaxial pericyclic gear reducer, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 6A  illustrates a perspective view of a first counterweight mounted on an input shaft, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 6B  illustrates a front view of a balancing weight mounted on an input shaft, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 6C  illustrates a perspective view of a first counterweight and a second counterweight mounted on an input shaft, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 6D  illustrates an exploded perspective view of two coaxial pericyclic gear reducers coupled to an electric motor, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 7  illustrates a sectional perspective view of an electrical linear actuator coupled to a coaxial pericyclic gear reducer, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 8A  illustrates a graphical chart of gear reduction ratio ranges of exemplary gear reducers, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 8B  illustrates a graph of maximum normalized height of teeth in a single-stage PGT and a graph of maximum normalized height of teeth in an equivalent pericyclic gear reducer, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 8C  illustrates a graph of torque capacity of a pericyclic gear reducer and a graph of an equivalent PGT, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 9  illustrates a two-stage pericyclic gear reducer, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 10A  illustrates an exploded view of a coaxial pericyclic gear reducer, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 10B  illustrates an exploded view of a middle planet ring gear and cages, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 11A  illustrates an exploded view of a coaxial pericyclic gear reducer, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 11B  illustrates an exploded view of a middle planet ring gear mounted between a driver ring gear and a driven wheel, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 12  illustrates a sectional side-view of a three-stage pericyclic gear reducer, consistent with one or more exemplary embodiments of the present disclosure; 
         FIG. 13A  illustrates a dual-speed pericyclic gearbox, consistent with one or more exemplary embodiments of the present disclosure; and 
         FIG. 13B  illustrates front views of output sun gears of a dual-speed pericyclic gearbox, consistent with one or more exemplary embodiments of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     The novel features which are believed to be characteristic of the present disclosure, as to its structure, organization, use and method of operation, together with further objectives and advantages thereof, will be better understood from the following discussion. 
     The present disclosure is directed to exemplary embodiments of a pericyclic gear reducer. An exemplary pericyclic reducer may include a driver ring gear coupled to or integrally formed with an input shaft, a middle wheel disposed between an exemplary driver ring gear and a driven wheel, and an output sun gear disposed within and meshed with an exemplary middle planet ring gear. An exemplary output sun gear may be coupled to or integrally formed with an output shaft. An exemplary driver ring gear and an exemplary driven wheel may be mounted with an offset relative to a central axis of an exemplary pericyclic gear reducer, while respective rotational axes of an exemplary driver ring gear and an exemplary driven wheel may be parallel with offsets in opposite sides of the central axis. An exemplary driver ring gear and an exemplary driven wheel may be coupled at either sides of an exemplary middle planet ring gear utilizing a prismatic joint. A first prismatic joint that may be utilized for coupling an exemplary driver ring gear at a first side of middle planet ring gear may be perpendicular to a second prismatic joint that may be utilized for coupling an exemplary driven wheel at an opposing second side of middle planet ring gear. Such coupling of an exemplary driver ring gear and an exemplary driven wheel may allow for converting a rotational movement of an exemplary input shaft to a spinning and revolving motion of an exemplary middle planet ring gear. 
     An exemplary middle planet ring gear may spin about a normal central axis of an exemplary middle planet ring gear and may further revolve around a longitudinal axis of an exemplary output sun gear on a circular path. In response to such spinning and revolving motion of an exemplary middle planet ring gear, an exemplary output sun gear may rotate about the longitudinal axis of an exemplary output sun gear and may transfer such rotational movement to an output shaft. An exemplary middle planet ring gear may revolve around an exemplary output sun gear on a circular path and every point on a pitch diameter of an exemplary middle planet ring gear may trace a pericycloid curve on an exemplary output sun gear, hence the name pericyclic gear reducer. 
     An exemplary pericyclic gear reducer may be a coaxial pericyclic gear reducer, in which an exemplary input shaft may be coupled or integrally formed with an input external-teeth gear. An exemplary input external-teeth gear may be disposed within and meshed with a driver ring gear. An exemplary driver ring gear may be coupled to an exemplary middle planet ring gear utilizing a prismatic joint from a first side of an exemplary middle planet ring gear. An exemplary middle planet ring gear may be coupled to a driven wheel utilizing a prismatic joint from a second opposing side of an exemplary middle planet ring gear. An exemplary output sun gear may be disposed within and meshed with an exemplary middle planet ring gear and may be rotatable with an exemplary middle planet ring gear. An exemplary output sun gear may further be coupled or integrally formed with an output shaft. An exemplary driver ring gear and an exemplary driven wheel may have parallel axes of rotation with an offset with respect to a central axis of an exemplary coaxial gear reducer. However, the two exemplary offsets may be equal and on opposite sides of an exemplary central axis. In this configuration, an exemplary input shaft and an exemplary output shaft may be parallel and aligned with each other without any offset with respect to central axis of an exemplary coaxial gear reducer. Hence, a distance between axes of an exemplary input shaft and an exemplary driver ring gear may be equal to offset of an exemplary driver ring gear and also equal to offset of an exemplary driven wheel from an exemplary central axis. 
       FIG. 1A  illustrates a right-side exploded view of a pericyclic gear reducer  100 , consistent with one or more exemplary embodiments of the present disclosure.  FIG. 1B  illustrates a left-side exploded view of pericyclic gear reducer  100 , consistent with one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, pericyclic gear reducer  100  may include an input shaft  106  that may be rotatable about a first rotational axis  124 , a middle planet ring gear  102  that may include an internal-teeth gear, an output sun gear  104  that may be positioned inside and meshed with middle planet ring gear  102 . 
     In an exemplary embodiment, input shaft  106  may be an elongated cylindrical shaft extended along the longitudinal axis of input shaft  106 . In an exemplary embodiment, input shaft  106  may be coupled to an external motor, such as an internal combustion engine or an electric motor. In an exemplary embodiment, output sun gear  104  may be an external-teeth gear that may have a smaller diameter in comparison with middle planet ring gear  102 . In an exemplary embodiment, output sun gear  104  may be disposed within a central opening of middle planet ring gear  102 , such that external teeth of output sun gear  104  may mesh with internal teeth of middle planet ring gear  102 . 
     In an exemplary embodiment, pericyclic gear reducer  100  may further include a driver ring gear  108  that may be structured as a first flange coupled or integrally formed with input shaft  106  and rotatable with input shaft  106  about first rotational axis  124  along and parallel with a longitudinal axis of input shaft  106 . In an exemplary embodiment, driver ring gear  108  may be coupled to middle planet ring gear  102  from a first side of middle planet ring gear  102  utilizing a first prismatic joint. In an exemplary embodiment, driver ring gear  108  may assume a rotational motion in response to rotational motion of input shaft  106 . Since driver ring gear  108  is coupled to middle planet ring gear  102  utilizing the first prismatic joint, rotational motion of driver ring gear  108  may urge middle planet ring gear  102  to rotate, as well. In other words, driver ring gear  108  may be configured to transfer the rotational motion of input shaft  106  to middle planet ring gear  102 . 
     In an exemplary embodiment, pericyclic gear reducer  100  may further include a driven wheel  110  that may be structured as a second flange coupled to middle planet ring gear  102  from a second side of middle planet ring gear  102  utilizing a second prismatic joint. In an exemplary embodiment, driven wheel  110  may be rotatable about a second rotational axis  126 . In an exemplary embodiment, the first prismatic joint and the second prismatic joint are perpendicular to each other. 
     In an exemplary embodiment, pericyclic gear reducer  100  may further include a central output shaft  112  that may be coupled or integrally formed with output sun gear  104 . In an exemplary embodiment, central output shaft  112  may be rotatable with output sun gear  104  about a longitudinal axis of central output shaft  112 , which is hereinafter referred to as a central axis  128  of pericyclic gear reducer  100 . In an exemplary embodiment, a plane of rotation of middle planet ring gear may be perpendicular to central axis  128  and the first side and the second side of middle planet ring gear  102  may be opposite each other along central axis  128 . In an exemplary embodiment, driven wheel  110  may include a central hole  111  that may be configured to allow passage of central output shaft  112  through driven wheel  110 . 
     In an exemplary embodiment, first rotational axis  124 , second rotational axis  126 , and central axis  128  may lay on a common plane. Furthermore, in an exemplary embodiment, first rotational axis  124  and central axis  128  may be misaligned or offset by a first distance  130   a  and second rotational axis  126  and central axis  128  may be misaligned or offset by a second distance  130   b . In an exemplary embodiment, first distance  130   a  and second distance  130   b  may be equal, meaning that first rotational axis  124  and second rotational axis  126  may be symmetrically positioned on opposite lateral sides of central axis  128 . In other words, first rotational axis  124  and second rotational axis  126  may have a misalignment by a distance equal to sum of first distance  130   a  and second distance  130   b.    
     In an exemplary embodiment, pericyclic gear reducer  100  may further include a first wall  114  that may include a first hole  117  that may be fitted with a rolling contact bearing, such as first ball bearing  118 . In an exemplary embodiment, first ball bearing  118  may include an inner ring and an outer ring that may be utilized to contain the balls of first ball bearing  118 . In an exemplary embodiment, first hole  117  may be configured to allow input shaft  106  to rotatably pass through first wall  114  while input shaft  106  may lean on first ball bearing  118 . As used herein, input shaft  106  rotatably passing through first wall  114  leaning on first ball bearing  118  may refer to a configuration, where an outer ring of first ball bearing  118  is fixed to first hole  117  and an inner ring of first ball bearing  118  is fixed to input shaft  106 . 
     In an exemplary embodiment, pericyclic gear reducer  100  may further include a second wall  116  that may include a second hole  119  fitted with a rolling contact bearing, such as second ball bearing  120 . In an exemplary embodiment, second hole  119  may be a through-all or open hole that may be configured to allow central output shaft  112  to rotatably pass through second wall  116  while central output shaft  112  may be coupled to second ball bearing  120 . In an exemplary embodiment, second wall  116  may further include a third ball bearing  122  that may be coupled to driven wheel  110 . In an exemplary embodiment, third ball bearing  122  may be housed in a third hole  121  in second wall  116 . In an exemplary embodiment, third hole  121  may be a closed-end hole, in which an inner ring of third ball bearing  122  may be fixed relative to second wall  116  and an outer ring of third ball bearing  122  may be fixed relative to driven wheel  110 . Hence, third ball bearing  122  may rotatably couple driven wheel  110  with second wall  116  and may allow driven wheel  110  to rotate about second rotational axis  126 . In an exemplary embodiment, first wall  114  and second wall  116  may be parallel with each other. In an exemplary embodiment, such parallel configuration and the fixed distance and orientation of first wall  114  and second wall  116  with respect to each other may be maintained by fastening or securing first wall  114  and second wall  116  to a shell (not illustrated for simplicity) that may house pericyclic gear reducer  100 . 
       FIG. 2A  illustrates a perspective view of a middle planet ring gear  202  coupled between a driver ring gear  208  and a driven wheel  210 , consistent with one or more exemplary embodiments of the present disclosure.  FIG. 2B  illustrates a front view of middle planet ring gear  202  coupled between driver ring gear  208  and driven wheel  210 , consistent with one or more exemplary embodiments of the present disclosure. 
     For simplicity and for purpose of describing prismatic joints between middle planet ring gear  202  and driver and driven wheels ( 208 ,  210 ), internal gear teeth of middle planet ring gear  202  are not illustrated in  FIGS. 2A and 2B . In an exemplary embodiment, middle planet ring gear  202  may be structurally similar to middle planet ring gear  102 . Similarly, for simplicity, a central hole of driven wheel  210  is not illustrated in  FIGS. 2A and 2B . In an exemplary embodiment, driven wheel  210  may be structurally similar to driven wheel  110 . In an exemplary embodiment, driver ring gear  208  may be structurally similar to driver ring gear  108  and driver ring gear  208  may be rotatably coupled to an input shaft  206  similar to input shaft  106 . In an exemplary embodiment, input shaft  206  may be configured to transfer a rotational movement of an external rotary actuator, such as an electric motor or an internal combustion engine to driver ring gear  208 . 
     In an exemplary embodiment, driver ring gear  208  may be rotatable about a first rotational axis  224  and driven wheel may rotate about a second rotational axis  226 . In an exemplary embodiment, first rotational axis  224  and second rotational axis  226  may have a parallel misalignment by a distance labeled as “e.” In an exemplary embodiment, middle planet ring gear  202  may include a bulge or protuberance or tongue at either side of middle planet ring gear  202 , namely, a first protuberance  201   a  and a second protuberance  201   b . In an exemplary embodiment, driver ring gear  208  may include a slot or groove, such as a first slot  207  and driven wheel  210  may include a respective slot or groove, such as a second slot  209 . In an exemplary embodiment, first protuberance  201   a  may be slidably coupled to first slot  207  to form a first prismatic joint between middle planet ring gear  202  and driver ring gear  208 . Similarly, second protuberance  201   b  may be slidably coupled to second slot  209  to form a second prismatic joint between middle planet ring gear  202  and driven wheel  210 . In an exemplary embodiment, first protuberance  201   a  may slide within first slot  207  along a first direction, and second protuberance  201   b  may slide within second slot  209  along a second direction. In an exemplary embodiment, the first direction may be perpendicular to the second direction and both the first direction and the second direction are perpendicular to normal central axis  203 . 
     In an exemplary embodiment, due to the eccentricity of driver ring gear  208  and driven wheel  210  with an offset of “e” between their respective rotational axes ( 224 ,  226 ), in response to the rotational movement of input shaft  206 , middle planet ring gear  202  may assume a combined rotational movement that may include a spinning element and a revolving element, as will be discussed. In an exemplary embodiment, the first and second prismatic joints may allow for such combined rotational movement of middle planet ring gear  202  by allowing middle planet ring gear  202  to slide within first slot  207  and second slot  209  as middle planet ring gear  202  rotates. 
     In an exemplary embodiment, a distance between a normal central axis  203  of middle planet ring gear  202  from first rotational axis  224  is labeled as “y” and a distance between normal central axis  203  of middle planet ring gear  202  from second rotational axis  226  is labeled as “x.” In an exemplary embodiment, normal central axis  203  is an axis perpendicular to a plane of middle planet ring gear  202  passing through a central point, the position of which is shown by letter “B.” In an exemplary embodiment, first rotational axis  224  may intersect middle planet ring gear  202  at a point labeled as “C” and second rotational axis  226  may intersect middle planet ring gear  202  at a point labeled as “A.” In an exemplary embodiment, responsive to driver ring gear  208  rotating about first rotational axis  224 , middle planet ring gear  202  may spin about normal central axis  203  while revolving around a point labeled as “O,” such that point “B” may have a circular locus with the origin of point “O” and a diameter of “e.” In an exemplary embodiment, since points C and A are stationary and a triangle CBA has a right angle at point B, the angle at B may be inscribed within a circle with diameter CA. In other words, point B may trace a circle with a diameter of CA in response to a rotational motion of line CB with driver ring gear  208 . 
       FIG. 2C  illustrates a geometrical relationship among center points of middle planet ring gear  202 , driver ring gear  208 , and driven wheel  210 , consistent with one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, angular velocity of revolving motion of middle planet ring gear  202  may be defined as equation (1) below: 
     
       
         
           
             
               
                 
                   
                     θ 
                     . 
                   
                   = 
                   
                     
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     dt 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     1 
                     ) 
                   
                 
               
             
           
         
       
     
     In an exemplary embodiment, angular velocity of spinning motion of middle planet ring gear  202  may be defined as equation (2) below: 
     
       
         
           
             
               
                 
                   
                     φ 
                     . 
                   
                   = 
                   
                     
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       φ 
                     
                     dt 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     2 
                     ) 
                   
                 
               
             
           
         
       
     
     Based on the geometry illustrated in  FIG. 2C , since θ=2φ at any given moment, angular velocity of revolving motion of middle planet ring gear  202  may be twice as angular velocity of the spinning motion of middle planet ring gear  202 . In an exemplary embodiment, both revolving motion and spinning motion of middle planet ring gear  202  may be in same direction. 
     Similarly, in an exemplary embodiment, middle planet ring gear  102  may revolve around central axis  128  which coincides with the longitudinal axis of output sun gear  104 . In other words, middle planet ring gear  102  may revolve around output sun gear  104  and since middle planet ring gear  102  and output sun gear  104  are meshed together, rotational movement of middle planet ring gear  102  may be transferred to output sun gear  104 . Based on the geometry discussed above, in response to a rotational movement of input shaft  106  with a given angular velocity, middle planet ring gear  102  may revolve around output sun gear  104  at an angular velocity twice the given angular velocity of input shaft  106 . 
       FIG. 3A  illustrates a perspective view of middle planet ring gear  102  meshed with output sun gear  104 , consistent with one or more exemplary embodiments of the present disclosure.  FIGS. 3B and 3C  illustrate front views of middle planet ring gear  102  meshed with output sun gear  104 , consistent with one or more exemplary embodiments of the present disclosure. 
     In an exemplary embodiment, the power received via input shaft  106  may be transferred to central output shaft  112  through middle planet ring gear  102 . In an exemplary embodiment, to maintain the contact between middle planet ring gear  102  and output sun gear  104 , as middle planet ring gear  102  is revolving around output sun gear  104 , pitch diameters of middle planet ring gear  102  and output sun gear  104  must follow a relationship defined by equation (3) below:
 
 d   P   =d   S   +e   Equation (3)
 
     In equation (3) above, d P  denotes the pitch diameter of middle planet ring gear  102 , d S  denotes the pitch diameter of output sun gear  104 , and e denotes the offset between driver ring gear  108  and driven wheel  110 . Since, a pitch diameter of an exemplary gear is equal to teeth number of an exemplary gear multiplied by a module of an exemplary gear, equation (3) may be rewritten as equation (4) below: 
     
       
         
           
             
               
                 
                   
                     
                       N 
                       P 
                     
                     - 
                     
                       N 
                       S 
                     
                   
                   = 
                   
                     e 
                     m 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     4 
                     ) 
                   
                 
               
             
           
         
       
     
     In equation (4) above, N P  denotes teeth number of middle planet ring gear  102 , N S  denotes teeth number of output sun gear  104 , and m denotes gear module. Referring to equation (4), it may be understood that the amounts of e and m should be chosen, such that the fraction 
             e   m         
may be a natural number.
 
     In an exemplary embodiment, as middle planet ring gear  102  revolves around output sun gear  104 , a central point of middle planet ring gear  102  that may be labeled as point B may revolve around a point, the position of which is labeled as point O on a circular path with a diameter of e. As mentioned before, e denotes an amount of offset between driver ring gear  108  and driven wheel  110  or as illustrated in  FIGS. 1A and 1B , e denotes the sum of first distance  130   a  and second distance  130   b . As discussed before, in an example, when middle planet ring gear rotates about point B in a clockwise manner at a rotational speed of ω P , its revolving speed around point O may be 2ω P . Consequently, an absolute velocity of point B may be described by equation (5), below: 
     
       
         
           
             
               
                 
                   
                     v 
                     B 
                   
                   = 
                   
                     
                       2 
                       ⁢ 
                       
                         ω 
                         P 
                       
                       × 
                       
                         e 
                         2 
                       
                     
                     = 
                     
                       
                         ω 
                         P 
                       
                       ⁢ 
                       e 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     5 
                     ) 
                   
                 
               
             
           
         
       
     
     An absolute velocity of a contact point of middle planet ring gear  102  and output sun gear  104 , which is designated by point D, may be obtained by equation (6) below:
 
 v   D   =v   B   +v   D/B   Equation (6)
 
     In equation (6) above, v D/B  denotes the velocity of point D relative to point B, which may be defined by equation (7) below: 
     
       
         
           
             
               
                 
                   
                     v 
                     
                       D 
                       ⁢ 
                       
                         / 
                       
                       ⁢ 
                       B 
                     
                   
                   = 
                   
                     - 
                     
                       
                         ω 
                         P 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             e 
                             2 
                           
                           + 
                           
                             
                               d 
                               S 
                             
                             2 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     7 
                     ) 
                   
                 
               
             
           
         
       
     
     By incorporating equations (5) and (7) in equation (6), the velocity of point D may be obtained by equation (8) below: 
     
       
         
           
             
               
                 
                   
                     v 
                     D 
                   
                   = 
                   
                     
                       ω 
                       P 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           e 
                           2 
                         
                         - 
                         
                           
                             d 
                             S 
                           
                           2 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     8 
                     ) 
                   
                 
               
             
           
         
       
     
     In an exemplary embodiment, if the clockwise rotation is considered as a positive direction, the angular velocity of central output shaft  112  may be obtained by equation (9) below: 
     
       
         
           
             
               
                 
                   
                     ω 
                     S 
                   
                   = 
                   
                     
                       
                         - 
                         
                           v 
                           D 
                         
                       
                       
                         ( 
                         
                           
                             d 
                             S 
                           
                           ⁢ 
                           
                             / 
                           
                           ⁢ 
                           2 
                         
                         ) 
                       
                     
                     = 
                     
                       - 
                       
                         
                           ω 
                           P 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               e 
                               
                                 d 
                                 S 
                               
                             
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     9 
                     ) 
                   
                 
               
             
           
         
       
     
     By replacing e in equation (9) above from equation (4) and by further replacing the pitch diameter, d s  with m×N S , a velocity ratio of output sun gear  104  to spinning velocity of middle planet ring gear  102 , i, may be obtained as described by equation (10) below: 
     
       
         
           
             
               
                 
                   i 
                   = 
                   
                     
                       
                         ω 
                         S 
                       
                       
                         ω 
                         P 
                       
                     
                     = 
                     
                       
                         1 
                         - 
                         
                           
                             m 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   N 
                                   P 
                                 
                                 - 
                                 
                                   N 
                                   S 
                                 
                               
                               ) 
                             
                           
                           
                             m 
                             ⁢ 
                             
                               N 
                               S 
                             
                           
                         
                       
                       = 
                       
                         2 
                         - 
                         
                           
                             N 
                             P 
                           
                           
                             N 
                             S 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     10 
                     ) 
                   
                 
               
             
           
         
       
     
     The spinning velocity of middle planet ring gear  102  is equal to rotational speed of input shaft  106 , consequently, “i” in equation (10) above denotes the output-input ratio of pericyclic gear reducer  100 . As evident from equation (10) above, input shaft  106  and output sun gear  104  may rotate in the same direction responsive to N P &lt;2N S , output sun gear  104  may stop responsive to N P =2N S , and input shaft  106  and output sun gear  104  may rotate in opposite directions responsive to N P &gt;2N S . Consequently, for 
               N   P       N   S           
close to 2, high reduction ratios may be attained. In an exemplary embodiment, higher teeth numbers for middle planet ring gear  102  and output sun gear  104  may allow for attaining higher reduction ratios. For example, for a teeth number of 49 for middle planet ring gear  102  and a teeth number of 25 for output sun gear  104 , the velocity ratio will be 1:25. In another example, for a teeth number of 99 for middle planet ring gear  102  and a teeth number of 50 for output sun gear  104 , the velocity ratio will be 1:50. In another example, for a teeth number of 101 for middle planet ring gear  102  and a teeth number of 50 for sun gear  104 , the velocity ratio will be −1:50 which signifies reverse rotation of output shaft  112  with respect to input shaft  106  while keeping high reduction ratio.
 
     As is further evident from equation (10) above, 
               N   P       N   S           
is greater than unity since output sun gear  104  is encircled by middle planet ring gear  102 . Consequently, it may be implied that i is less than 1 or in other words, pericyclic gear reducer  100  may be considered as an absolute speed reducer. In an exemplary embodiment, for a pressure angle of 20° for gears of middle planet ring gear  102  and output sun gear  104 , the maximum value for i must be approximately 0.6 to avoid trimming interference for output sun gear  104  and middle planet ring gear  102 . In an exemplary embodiment, for higher pressure angles of output sun gear  104  and middle planet ring gear  102 , i may slightly increase beyond this limit of 0.6, however, the value of 0.6 may be considered as the upper limit of the output-to-input velocity ratio for gear reducers similar in type with pericyclic gear reducer  100 .
 
       FIG. 4A  illustrates an exploded view of a coaxial pericyclic gear reducer  400 , consistent with one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, coaxial pericyclic gear reducer  400  may include an input shaft  406  that may be rotatable about a central axis  428 , a middle planet ring gear  402  similar to middle planet ring gear  102  that may include a central annular gear with teeth cut in an internal surface of the annular gear, an output sun gear  404  similar to output sun gear  104  that may be positioned inside and meshed with middle planet ring gear  402 . 
     In an exemplary embodiment, coaxial pericyclic gear reducer  400  may further include a driver ring gear  408  that may be configured to be coupled to input shaft  406  via an input external-teeth gear  407 . In an exemplary embodiment, driver ring gear  408  may be similar to driver ring gear  108  however with teeth cut in an internal cylindrical surface of driver ring gear  408 . In an exemplary embodiment, input shaft  406  may be coupled to or integrally formed with input external-teeth gear  407 . In an exemplary embodiment, input external-teeth gear  407  may be disposed within driver ring gear  408  and may mesh with driver ring gear  408 . In an exemplary embodiment, such coupling of input external-teeth gear  407  and driver ring gear  408  may allow for converting the rotational movement of input shaft  406  about central axis  428  to a rotational movement of driver ring gear  408  about a first rotational axis  424 . In an exemplary embodiment, driver ring gear  408  may be coupled to middle planet ring gear  402  from a first side of middle planet ring gear  402  utilizing a first prismatic joint. In an exemplary embodiment, middle planet ring gear  402  may be rotatable with driver ring gear  408  utilizing the first prismatic joint. 
     In an exemplary embodiment, coaxial pericyclic gear reducer  400  may further include a driven wheel  410  structurally similar to driven wheel  110  that may be a flange coupled to middle planet ring gear  402  from a second side of middle planet ring gear  402  utilizing a second prismatic joint. In an exemplary embodiment, driven wheel  410  may be rotatable with middle planet ring gear  402  utilizing the second prismatic joint about a second rotational axis  426 . In an exemplary embodiment, first prismatic joint and the second prismatic joint may make a right angle relative to each other, as described in connection with the embodiment shown in  FIG. 2A . 
     In an exemplary embodiment, coaxial pericyclic gear reducer  400  may further include an output shaft  412  that may be coupled or integrally formed with output sun gear  404 . In an exemplary embodiment, output shaft  412  may be rotatable with output sun gear  404  about central axis  428  of coaxial pericyclic gear reducer  400 . In an exemplary embodiment, the first side and the second side of middle planet ring gear  402  may be opposite each other along central axis  428 . In an exemplary embodiment, driven wheel  410  may include a central hole  411  that may be configured to allow passage of output shaft  412  through driven wheel  410 . 
     In an exemplary embodiment, first rotational axis  424  and central axis  428  may have a first distance  430   a  and second rotational axis  426  and central axis  428  may have a second distance  430   b . In an exemplary embodiment, first distance  430   a  and second distance  430   b  may be equal and opposite to each other, meaning that first rotational axis  424  and second rotational axis  426  may be symmetrically positioned on opposite lateral sides of central axis  428 . In other words, first rotational axis  424  and second rotational axis  426  may have an offset equal to sum of first distance  430   a  and second distance  430   b.    
     In an exemplary embodiment, coaxial pericyclic gear reducer  400  may further include a first wall  414  that may include a first hole that may be fitted with a first ball bearing  418 . In an exemplary embodiment, the first hole may be configured to allow input shaft  406  to rotatably pass through first wall  414  while input shaft  406  may lean on first ball bearing  418 . In an exemplary embodiment, first ball bearing  418  may be aligned with a centerline of first wall  414  and may be coaxial with central axis  428 . In an exemplary embodiment, first wall  414  may further include a fourth ball bearing  423  that may be coupled to driver ring gear  408 . In an exemplary embodiment, fourth ball bearing  423  may be housed in a fourth hole, which may be a blind or closed-end hole, in first wall  414 . In an exemplary embodiment, an inner ring of fourth ball bearing  423  may be fixed to first wall  414  and an outer ring of fourth ball bearing  423  may be fixed to driver ring gear  408 , hence fourth ball bearing  423  may be configured to rotatably couple driver ring gear  408  with first wall  414  and to allow driver ring gear  408  to rotate about first rotational axis  424  while leaning on fourth ball bearing  423 . 
     In an exemplary embodiment, coaxial pericyclic gear reducer  400  may further include a second wall  416  that may include a second hole fitted with a second ball bearing  420 . In an exemplary embodiment, a central normal axis of second ball bearing  420  may be coaxial with central axis  428 . In an exemplary embodiment, the second hole may be configured to allow output shaft  412  to rotatably pass through second wall  416  while output shaft  412  may leaning on second ball bearing  420 . In an exemplary embodiment, second wall  416  may further include a third ball bearing  422  that may be coupled to driven wheel  410 . In an exemplary embodiment, third ball bearing  422  may be housed in a third hole, which may be a blind or closed-end hole, in second wall  416 . In an exemplary embodiment, third ball bearing  422  may facilitate rotatable coupling of driven wheel  410  to second wall  416 . An inner ring of third ball bearing  422  may be fixed to second wall  416  and an outer ring of third ball bearing  422  may be coupled to driven wheel  410  allowing driven wheel  410  to rotate about second rotational axis  426  while being mounted onto second wall  416 . In an exemplary embodiment, first wall  414  and second wall  416  may be parallel with each other. In an exemplary embodiment, such parallel configuration and the fixed distance and orientation of first wall  414  and second wall  416  with respect to each other may be maintained by fastening or securing first wall  414  and second wall  416  to a shell (not illustrated for simplicity) that may house pericyclic gear reducer  400 . 
     In an exemplary embodiment, input shaft  406  may be a hollow shaft configured to allow coaxial passage of output shaft  412  through input shaft  406 . Such hollow configuration of input shaft  406  may allow for output shaft  412  to pass through input shaft  406  such that a first end of output shaft  412  may extend beyond first wall  414  and lean on a fifth ball bearing  421   b  mounted on third sidewall  417 . In an exemplary embodiment, input shaft  406  may further extend beyond first sidewall  414  and may be coupled to a ball bearing  421   a  mounted on third wall  417  adjacent fifth ball bearing  421   b . In an exemplary embodiment, a second opposing end of output shaft  412  may pass through central hole  411  of driven wheel  410  and may lean on second ball bearing  420  on second wall  416 . In an exemplary embodiment, input shaft  406  may pass through walls  414  and  417  along central axis  428  leaning on ball bearings  418  and  421   a.    
     In an exemplary embodiment, such hollow configuration of input shaft  406  and coupling of input shaft  406  with driver ring gear  408  via input external-teeth gear  407  may allow for mounting input shaft  406  and output shaft  412  in a coaxial arrangement, where output shaft  412  may coaxially pass through input shaft  406  and be rotatable within input shaft  406  independent from input shaft  406 . 
     As mentioned before, in an exemplary embodiment, first rotational axis  424  of driver ring gear  408  may have offset  430   a  from central axis  428  and second rotational axis  426  of driven wheel  410  may have offset  430   b  from central axis  428  from an opposite lateral direction. Such offset between first and second rotational axes ( 424 ,  426 ) means that driver ring gear  408  and driven wheel  410  may be mounted in an eccentric and symmetric relationship relative to central axis  428 . 
     In an exemplary embodiment, such eccentricity of driver ring gear  408  from central axis  428  by first distance  430   a  and eccentricity of driven wheel  410  from central axis  428  by second distance  430   b  on an opposite lateral side of central axis  428  and how driver ring gear  408  and driven wheel  410  may be coupled on either sides of middle planet ring gear  402  utilizing mutually perpendicular prismatic joints may urge middle planet ring gear  402  to assume a spinning motion about a normal central axis of middle planet ring gear  402  and a revolving motion about central axis  428  in response to rotational movement of input shaft  406 . As used herein, a normal central axis of middle planet ring gear  402  may refer to as an axis perpendicular to the largest surface of middle planet ring gear  402  passing through center of middle planet ring gear  402 . In an exemplary embodiment, such spinning motion and revolving motion of middle planet ring gear  402  may then be transferred to output sun gear  404 . 
     In an exemplary embodiment, speed reduction ratio of coaxial pericyclic gear reducer  400  may be calculated by equation (11) below: 
     
       
         
           
             
               
                 
                   i 
                   = 
                   
                     
                       
                         N 
                         
                           D 
                           ⁢ 
                           R 
                         
                       
                       
                         N 
                         IE 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         2 
                         - 
                         
                           
                             N 
                             P 
                           
                           
                             N 
                             S 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     11 
                     ) 
                   
                 
               
             
           
         
       
     
     In equation (11) above, N DR  stands for teeth number of driver ring gear  408 , N IE  stands for teeth number of input external-teeth gear  407 , N P  stands for teeth number of middle planet ring gear  402 , and N S  stands for teeth number of output sun gear  404 . In an exemplary embodiment, 
               N     D   ⁢   R         N   IE           
may be less than 0.7 for avoiding interference and consequently the overall reduction limit for coaxial pericyclic gear reducer  400  may be about 0.4 since it was previously discussed in equation (10) that the maximum value of
 
             2   -       N   P       N   S             
in equation (10) is almost equal to 0.6. In an exemplary embodiment, a practical minimum limit for i in equation (11) may be considered to be 1:200. However, the lower limit of i can converge to zero, theoretically but it will result in very small teeth. In an exemplary embodiment,
 
               N     D   ⁢   R         N   IE           
being less than 0.7 may guarantee avoidance of trimming interface.
 
       FIG. 4B  illustrates an exploded view of a coaxial pericyclic gear reducer  400 ′, consistent with one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, coaxial pericyclic gear reducer  400 ′ may include an input shaft  406 ′ disposed between parallel walls ( 414 ,  417 ) leaning on bearings ( 418 ,  421   a ). In an exemplary embodiment, a ball bearing  421   c  may be mounted at a first end of input shaft  406 ′, where ball bearing  421   c  may be coaxial with central axis  428 . In an exemplary embodiment, ball bearing  421   c  may be rotatably coupled input shaft  406 ′ to output shaft  412 , so that input shaft  406 ′ may be coaxial with output shaft  412 . 
       FIG. 5A  illustrates a perspective exploded view of a driver ring gear  408  meshed with input external-teeth gear  407  and a middle planet ring gear  402  meshed with output sun gear  404  in a coaxial pericyclic gear reducer, consistent with one or more exemplary embodiments of the present disclosure. As mentioned before, in an exemplary embodiment, coaxial pericyclic gear reducer  400  may include the pair of driver ring gear  408  and input external-teeth gear  407  and the pair of middle planet ring gear  402  and output sun gear  404 . In an exemplary embodiment, middle planet ring gear  402  may be mounted with a misalignment relative to central axis  428  by a distance of e/2 and driver ring gear  408  may also be mounted with a misalignment relative to central axis  428  by a distance of e/2. As used herein with respect to  FIG. 5A , the letter “e” may refer to the distance by which first rotational axis  424  and second rotational axis  426  are misaligned or offset. In an exemplary embodiment, the eccentricity of middle planet ring gear  402  and driver ring gear  408  with respect to central axis  428  may be described in terms of pitch diameters by equation (12) below: 
     
       
         
           
             
               
                 
                   
                     e 
                     2 
                   
                   = 
                   
                     
                       
                         
                           d 
                           
                             D 
                             ⁢ 
                             R 
                           
                         
                         - 
                         
                           d 
                           IE 
                         
                       
                       2 
                     
                     = 
                     
                       
                         
                           d 
                           P 
                         
                         - 
                         
                           d 
                           S 
                         
                       
                       2 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     12 
                     ) 
                   
                 
               
             
           
         
       
     
     In equation (12) above, d DR  denotes the pitch diameter of driver ring gear  408 , d IE  denotes the pitch diameter of input external-teeth gear  407 , d P  denotes the pitch diameter of middle planet ring gear  402 , and d S  denotes the pitch diameter of output sun gear  404 . The equation (12) above may be described in terms of modulus (m) and teeth numbers (N) by equation (13) below: 
     
       
         
           
             
               
                 
                   
                     m 
                     P 
                   
                   = 
                   
                     
                       m 
                       
                         D 
                         ⁢ 
                         R 
                       
                     
                     ⁢ 
                     
                       
                         
                           N 
                           
                             D 
                             ⁢ 
                             R 
                           
                         
                         - 
                         
                           N 
                           IE 
                         
                       
                       
                         
                           N 
                           P 
                         
                         - 
                         
                           N 
                           S 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     13 
                     ) 
                   
                 
               
             
           
         
       
     
     In equation (13) above, all gears are considered to be of spur type and m P  denotes the module of middle planet ring gear  402 , N DR  denotes the teeth number of driver ring gear  408 , N IE  denotes the teeth number of input external-teeth gear  407 , N P  denotes the teeth number of middle planet ring gear  402 , N S  denotes the teeth number of output sun gear  404 , and m DR  denotes the module of driver ring gear  408 . 
     In an exemplary embodiment, input external-teeth gear  407  and driver ring gear  408  may include helical gears, while output sun gear  404  and middle planet ring gear  402  may include spur gears. In an exemplary embodiment, for a helix angle of ψ for helical gears, a transverse module may be defined as m t =m n /cos (ψ), where m n  is a normal module selected from a standard table of modules. In an exemplary embodiment for helical input external-teeth gear  407  and driver ring gear  408 , equation (13) may be rewritten as equation (14) below: 
     
       
         
           
             
               
                 
                   
                     m 
                     P 
                   
                   = 
                   
                     
                       m 
                       nDR 
                     
                     ⁢ 
                     
                       
                         
                           N 
                           
                             D 
                             ⁢ 
                             R 
                           
                         
                         - 
                         
                           N 
                           IE 
                         
                       
                       
                         
                           N 
                           P 
                         
                         - 
                         
                           N 
                           S 
                         
                       
                     
                     ⁢ 
                     sec 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       ψ 
                       ) 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     14 
                     ) 
                   
                 
               
             
           
         
       
     
     In equation (14) above, m nDR  denotes a standard normal module selected for driver ring gear  408 . As is evident from equation (14) above, if the helix angle ψ is selected such that the factor 
                   N     D   ⁢   R       -     N   IE           N   P     -     N   S         ⁢   sec   ⁢           ⁢     (   ψ   )           
becomes a proper coefficient, then m P  and m nDR  may be kept at standard values. In an exemplary embodiment, helix angle ψ may be adjusted to any desired value in gear cutting apparatus, such as hobbing, power skiving, scudding, and some milling and shaping machines.
 
     Equation (14) above is written for a configuration where input external-teeth gear  407  and driver ring gear  408  include helical gears while output sun gear  404  and middle planet ring gear  402  include spur gears. In an exemplary embodiment, input external-teeth gear  407 , driver ring gear  408 , output sun gear  404 , and middle planet ring gear  402  may include helical gears and equation (14) may be rewritten as equation (15) below: 
     
       
         
           
             
               
                 
                   
                     m 
                     
                       n 
                       ⁢ 
                       P 
                     
                   
                   = 
                   
                     
                       m 
                       
                         n 
                         ⁢ 
                         D 
                         ⁢ 
                         R 
                       
                     
                     ⁢ 
                     
                       
                         
                           N 
                           
                             D 
                             ⁢ 
                             R 
                           
                         
                         - 
                         
                           N 
                           IE 
                         
                       
                       
                         
                           N 
                           P 
                         
                         - 
                         
                           N 
                           S 
                         
                       
                     
                     × 
                     
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             ψ 
                             P 
                           
                           ) 
                         
                       
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             ψ 
                             
                               D 
                               ⁢ 
                               R 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     15 
                     ) 
                   
                 
               
             
           
         
       
     
     In equation (15) above, m nP  denotes a standard normal module selected for middle planet ring gear  402 , ψ DR  denotes helix angle for driver ring gear  408 , and ψ P  denotes helix angle for middle planet ring gear  402 . 
     In exemplary embodiments, input external-teeth gear  407  and driver ring gear  408  including helical gears may allow for eccentricity adjustments and noise reduction. Specifically, input external-teeth gear  407  and driver ring gear  408  may normally work at high speeds and consequently the noise generated by input external-teeth gear  407  and driver ring gear  408  may be relatively high. Utilizing helical gears in manufacturing input external-teeth gear  407  and driver ring gear  408  may considerably reduce the generated noise. Furthermore, utilizing helical gears in input side may allow designers to choose smaller teeth numbers for input external-teeth gear  407  without facing involute interference with teeth of driver ring gear  408 , which may result in coarser teeth and increment of torque capacity in driver side. 
     As mentioned before, middle planet ring gear  402  may assume a combined rotational motion in response to the rotational movement of input shaft  406 . Such combined rotational movement of middle planet ring gear  402  may include a spinning rotation of middle planet ring gear  402  about normal central axis of middle planet ring gear  402  and a revolving motion of middle planet ring gear  402  about central axis  428  around output sun gear  404 . 
       FIG. 5B  illustrates a perspective exploded view of components of a coaxial pericyclic gear reducer with driver ring gear  408  and driven wheel  410  having offset on the same side of a central axis  428  of the coaxial pericyclic gear reducer, consistent with one or more exemplary embodiments of the present disclosure; As used herein, such assembly may be referred to as a coaxial direct assembly hereinafter. Both the offsets are equal to e/2. In an exemplary embodiment, equations 12 to 15 may be valid for the coaxial direct assembly. To achieve this assembly from the coaxial pericyclic gear reducer  400 , it is enough to rotate either the wall  416  or the wall  414  of coaxial pericyclic gear reducer  400 , half a turn (180°) about the centerline of the gearbox  428  and then secure the walls to the gearbox shell (which is not shown in the figure for simplicity). Coaxial direct assembly can also be applied to pericyclic gear reducer  100  in a similar way. In the coaxial direct assembly, since the driver ring gear  408 , the middle planet wheel  402  and the driven wheel  410  are inline, the revolutionary motion of the middle planet wheel  402  will vanish and hence the speed ratio of the coaxial direct assembly will be quite different than those of pericyclic gear reducer  100  and  400 . In the coaxial direct assembly, the gear train from input external-teeth gear  407  to output shaft gear  404  may no longer be a cycloidal gear train. It will be just a simple train with fixed axes and hence it will not require balancing. Upholding the nomenclature used in equation (11), the velocity ratio from input shaft  406  to output shaft  412  in the coaxial direct assembly would be obtained as below: 
     
       
         
           
             
               
                 
                   
                     i 
                     direct 
                   
                   = 
                   
                     
                       
                         N 
                         IE 
                       
                       
                         N 
                         
                           D 
                           ⁢ 
                           R 
                         
                       
                     
                     × 
                     
                       
                         N 
                         P 
                       
                       
                         N 
                         S 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     16 
                     ) 
                   
                 
               
             
           
         
       
     
     In an exemplary embodiment, revolving motion of middle planet ring gear  402  may create a centrifugal force of a significant amount. Since, the generated centrifugal force may be proportional to the square of the angular velocity of revolutionary motion of middle planet ring gear  402  and considering the angular velocity of revolutionary motion of middle planet ring gear  402  being twice the angular velocity of driver ring gear  408 , the order of the generated centrifugal force in middle planet ring gear  402  may be up to four times larger than the order of centrifugal forces generated in driver ring gear  408  and driven wheel  410 . Considering such centrifugal forces within coaxial pericyclic gear reducer  400 , balancing coaxial pericyclic gear reducer  400  may be considered crucial and if not balanced, coaxial pericyclic gear reducer  400  may experience damaging vibrations at high speeds. 
     In an exemplary embodiment, to address the balancing issue in coaxial pericyclic gear reducer  400 , angular velocity ratio of driver ring gear  408  to input external-teeth gear  407  may be 1:2. In other words, angular velocity of input external-teeth gear  407  may be twice the angular velocity of driver ring gear  408  and therefore the angular velocity of input external-teeth gear  407  may be equal to the angular velocity of revolutionary motion of middle planet ring gear  402 . This way, the rotational motion of input external-teeth gear  407  may become synchronous to the revolving motion of middle planet ring gear  402 . Such synchronization between the rotational movements of input external-teeth gear  407  and middle planet ring gear  402  may allow for balancing coaxial pericyclic gear reducer  400  either by mounting counterweights or by digging cavities on input shaft  406  at precise locations on input shaft  406 . 
       FIG. 6A  illustrates a perspective view of a first counterweight  600  mounted on input shaft  406 , consistent with one or more exemplary embodiments of the present disclosure.  FIG. 6B  illustrates a front view of balancing weight  600  mounted on input shaft  406 , consistent with one or more exemplary embodiments of the present disclosure.  FIG. 6C  illustrates a perspective view of first counterweight  600  and a second counterweight  606  mounted on input shaft  406 , consistent with one or more exemplary embodiments of the present disclosure. 
     In an exemplary embodiment, to achieve a static balance within coaxial pericyclic gear reducer  400 , first counterweight  600  may be mounted on input shaft  406  on the opposite side of the eccentricity of middle planet ring gear  402 . Specifically, if middle planet ring gear  402  has an eccentricity  608  of e/2 relative to central axis  428 , first counterweight  600  may be mounted on input shaft  406  extending outward with respect to central axis  428  in the opposite direction with respect to eccentricity  608 . Simply put, if output sun gear  404  is engaged with a lower portion of internal gear of middle planet ring gear  402 , counterweight  600  may be mounted on a lower half of input shaft  406 . 
     In an exemplary embodiment, for static balancing of coaxial pericyclic gear reducer  400 , the following equation (17) must be satisfied: 
     
       
         
           
             
               
                 
                   
                     
                       M 
                       
                         c 
                         ⁢ 
                         w 
                       
                     
                     ⁢ 
                     d 
                     ⁢ 
                     
                       ω 
                       2 
                     
                   
                   = 
                   
                     
                       
                         M 
                         P 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           2 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ω 
                       2 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     17 
                     ) 
                   
                 
               
             
           
         
       
     
     In equation (17) above, M cw  denotes the mass of first counterweight  600 , M P  denotes the mass of middle planet ring gear  402 , ω denotes angular velocity of the revolving motion of middle planet ring gear  402  (which is equal to angular velocity of input shaft), and e/2 may refer to the eccentricity of middle planet ring gear  402  with respect to central axis  428 . Referring to  FIG. 6B , d denotes a distance between center of mass of counterweight  600  and central axis  428 . 
     In an exemplary embodiment, in addition to static balancing, for dynamic balancing of coaxial pericyclic gear reducer  400 , a second counterweight  606  may be mounted on input shaft  406  on an opposite side of input shaft  406  with respect to first counterweight  600 . In an exemplary embodiment, second counterweight  606  may be lighter than first counterweight  600 . 
     In an exemplary embodiment, a first centrifugal force  612   a  may be exerted on middle planet ring gear  402 , a second centrifugal force  612   b  may be exerted on first counterweight  600 , and a third centrifugal force  612   c  may be exerted on second counterweight  606 . In an exemplary embodiment, masses of first and second counterweights ( 600  and  606 ) as well as their eccentricity with respect to central axis  428  should satisfy the following equations (18) and (19) for both static and dynamic balancing:
 
 F   P   L   1   =F   SC   L   2   Equation (18)
 
 F   P   +F   SC   =F   FC   Equation (19)
 
     In equations (18) and (19) above, F P  denotes first centrifugal force  612   a , F SC  denotes third centrifugal force  612   c , F FC  denotes second centrifugal force  612   b , L 1  denotes a longitudinal distance  614  between first counterweight  600  and middle planet ring gear  402 , and L 2  denotes a distance  616  between second counterweight  606  and first counterweight  600 . In equations (18) and (19) above: 
     
       
         
           
             
               
                 
                   
                     F 
                     P 
                   
                   = 
                   
                     
                       
                         M 
                         P 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           2 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ω 
                       2 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     20 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     F 
                     
                       F 
                       ⁢ 
                       C 
                     
                   
                   = 
                   
                     
                       M 
                       
                         F 
                         ⁢ 
                         C 
                       
                     
                     ⁢ 
                     
                       d 
                       
                         F 
                         ⁢ 
                         C 
                       
                     
                     ⁢ 
                     
                       ω 
                       2 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     21 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     F 
                     
                       S 
                       ⁢ 
                       C 
                     
                   
                   = 
                   
                     
                       M 
                       
                         S 
                         ⁢ 
                         C 
                       
                     
                     ⁢ 
                     
                       d 
                       SC 
                     
                     ⁢ 
                     
                       ω 
                       2 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     22 
                     ) 
                   
                 
               
             
           
         
       
     
     In equations (20) to (22) above, M P  denotes the mass of middle planet ring gear  402 , M FC  denotes the mass of first counterweight  600 , M SC  denotes the mass of second counterweight  606 , d FC  denotes eccentricity  618  of mass center of first counterweight  600 , and d SC  denotes eccentricity  620  of mass center of second counterweight  606 . 
     In an exemplary embodiment, after static and dynamic balancing of coaxial pericyclic gear reducer  400 , coaxial pericyclic gear reducer  400  may be utilized in various applications, such as robotics, automotive, household machines and aerospace. Furthermore, coaxial pericyclic gear reducer  400  may be utilized in wind turbines if it is applied in a back driving manner, that is, the input shaft side is swapped with the output shaft side and the pericyclic gearbox becomes a speed increaser, and a power generator takes the place of the driving motor. 
       FIG. 6D  illustrates an exploded perspective view of two coaxial pericyclic gear reducers ( 630   a  and  630   b ) coupled to an electric motor  632 , consistent with one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, each coaxial pericyclic gear reducer of two coaxial pericyclic gear reducers ( 630   a  and  630   b ) may be structurally similar to coaxial pericyclic gear reducer  400 . In an exemplary embodiment, two coaxial pericyclic gear reducers ( 630   a  and  630   b ) may be similar in shape but not necessarily in size. In other words, any of two coaxial pericyclic gear reducers ( 630   a  and  630   b ) may be larger than the other one, however, all corresponding gears utilized in two coaxial pericyclic gear reducers ( 630   a  and  630   b ) may have equal teeth numbers. 
     In an exemplary embodiment, motor shaft  634  may function as input shafts of both coaxial pericyclic gear reducers ( 630   a  and  630   b ). In an exemplary embodiment, motor shaft  634  may transmit its power via a first external-teeth gear  636   a  to a first driver ring gear  638   a  of first coaxial pericyclic gear reducer  630   a  of two coaxial pericyclic gear reducers ( 630   a  and  630   b ). In an exemplary embodiment, motor shaft  634  may further transmit its power via a second external-teeth gear  636   b  to a second driver ring gear  638   b  of second coaxial pericyclic gear reducer  630   b  of two coaxial pericyclic gear reducers ( 630   a  and  630   b ). In an exemplary embodiment, first external-teeth gear  636   a  and second external-teeth gear  636   b  may be unified or integrally formed with motor shaft  634 . In an exemplary embodiment, first external-teeth gear  636   a  and second external-teeth gear  636   b  may have similar teeth numbers. In an exemplary embodiment, first external-teeth gear  636   a  and second external-teeth gear  636   b  may include helical gears and corresponding helix angles of first external-teeth gear  636   a  and second external-teeth gear  636   b  may be opposite each other to counterbalance respective axial forces of first external-teeth gear  636   a  and second external-teeth gear  636   b.    
     In an exemplary embodiment, first driver ring gear  638   a  and second driver ring gear  638   b  may include internal-teeth ring gears and each external-teeth gear of first external-teeth gear  636   a  and second external-teeth gear  636   b  may mesh with a corresponding one of first driver ring gear  638   a  and second driver ring gear  638   b . In an exemplary embodiment, velocity ratio of such meshing of each external-teeth gear of first external-teeth gear  636   a  and second external-teeth gear  636   b  with a corresponding one of first driver ring gear  638   a  and second driver ring gear  638   b  may be 1:2. 
     In an exemplary embodiment, driver ring gear  638   a  and driver ring gear  638   b  may be initially so mounted that middle planet ring gears  640   a  and  640   b  have misalignments relative to central longitudinal axis of motor shaft  634  in the same direction. Hence middle planet ring gears ( 640   a  and  640   b ) of both pericyclic gear reducers ( 630   a  and  630   b ) may have same-side offsets while middle planet ring gears ( 640   a  and  640   b ) may be revolving around the central longitudinal axis of motor shaft  634 . Consequently, the centrifugal force of both middle planet ring gears ( 640   a  and  640   b ) may be in the same direction. For purpose of balancing, on one side of middle part of motor shaft  634 , a cavity  642  may be cut out or dug so that cavity  642  may counterbalance the centrifugal forces of both driven ring gears ( 640   a  and  640   b ). In an exemplary embodiment, cavity  642  may be in the same side of the misalignment of middle planet ring gears ( 640   a  and  640   b ) with respect to central longitudinal axis of motor shaft  634  and volume and position of cavity  642  may be so determined to counterpoise the centrifugal forces of middle planet ring gears ( 640   a  and  640   b ). In an exemplary embodiment, motor  632  may further include a rotor  644  that may be press fitted on motor shaft  634 . 
     For equally dividing output torque between two pericyclic gear reducers ( 630   a  and  630   b ), output shafts ( 646   a  and  646   b ) of corresponding two coaxial pericyclic gear reducers ( 630   a  and  630   b ) may be connected to each other by utilizing a central spline  650 . In an exemplary embodiment, spline  650  may be centrally inserted from both ends of spline  650  into corresponding central spline holes on output shafts ( 646   a  and  646   b ) to a certain extent and may become fixed to output shafts ( 646   a  and  646   b ) utilizing two screws ( 648   a  and  648   b ). In an exemplary embodiment, motor shaft  634  may have a central hole that may be configured to allow spline  650  to pass through motor shaft  634 , without touching motor shaft  634 . In an exemplary embodiment, spline  650  and the two output shafts ( 646   a  and  646   b ) may act as an integrated shaft which is of simply-supported type lying on two central bearings on end walls ( 652   a  and  652   b ). For simplicity, bearings are not illustrated in  FIG. 6D . 
     In an exemplary embodiment, such configuration of coaxial pericyclic gear reducers ( 630   a  and  630   b ) and motor  644  as an integrated drive train may allow for a complete balancing of the entire drive train. Furthermore, the integrated drive train has a single output shaft of simply-supported type that connects the output shafts of two coaxial pericyclic gear reducers ( 630   a  and  630   b ) to each other. Consequently, the output shaft may become very strong against bending loads. Such configuration of integrated drive train may also reduce the number of bearings of coaxial pericyclic gear reducers ( 630   a  and  630   b ) at both ends of the integrated drive train and consequently may lead to a compact longitudinal size for the integrated drive train. 
     In an exemplary embodiment, coaxial pericyclic gear reducers ( 630   a  and  630   b ) may be back driven, i.e. the power is transmitted from the output side toward the input side. In this case, motor  644  may be replaced by a generator and applied in wind turbine and similar requests. In an exemplary embodiment, motor  644  may include an internal combustion engine or a jet engine to be utilizable in “geared turbo fan engines” in aviation industry and similar applications. For simplicity, exemplary shells of motor  632  and pericyclic gear reducers  630   a  and  630   b  are not illustrated. 
     Specifically, a pericyclic gear reducer, such as coaxial pericyclic gear reducer  400  may be utilized in electrical linear actuators or jack screws. Electric linear actuator is a widely-used device that performs the function of a hydraulic or pneumatic linear actuators. The usual electrical linear actuators consist mainly of an external electric motor, a gearbox (which is usually a reducer) and a screw unit. The screw is driven by output shaft of the gearbox and pushes the main rod of the actuator. It has many advantages over hydraulic linear actuator except that it is much bigger and heavier than that for a given amount of load to be transmitted. In an exemplary embodiment, a pericyclic gear reducer, such as coaxial pericyclic gear reducer  400  may be utilized as the gearbox of an exemplary electrical linear actuator. Such utilization of coaxial pericyclic gear reducer  400  in an exemplary electric linear actuator may make it powerful while keeps its size compact with an internal electric motor. It will be comparable with hydraulic linear actuators regarding shape, size and load capacity. 
     In an exemplary embodiment, as mentioned before, in coaxial pericyclic gear reducer  400 , a high-speed input shaft, such as input shaft  406  may surrounds a low-speed output shaft, such as output shaft  412  while both of input and output shafts ( 406 ,  412 ) are on the input side of coaxial pericyclic gear reducer  400 . Such configuration of input shaft  406  and output shaft  412  may allow an exemplary electrical motor to be located in an exemplary cylindrical body of an exemplary electrical linear actuator. Consequently, a longitudinal space of an exemplary electrical linear actuator may be reduced by utilization of coaxial pericyclic gear reducer  400  since both an exemplary electric motor and an exemplary screw unit may be mounted on the same side of an exemplary electrical linear actuator. Furthermore, in an exemplary embodiment, due to relatively higher reduction ratios that may be achieved in coaxial pericyclic gear reducer  400  in comparison to other available reducers, high speed electrical motors may be utilized in an exemplary electrical linear actuator, which not only decreases the size and weight of an exemplary electrical linear actuator but also may increase the power density of an exemplary electrical linear actuator. 
       FIG. 7  illustrates a sectional perspective view of an electrical linear actuator  700  coupled to a coaxial pericyclic gear reducer  702 , consistent with one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, coaxial pericyclic gear reducer  702  may be structurally similar to coaxial pericyclic gear reducer  400  and may be utilized as a gear reducer to transfer a rotational movement of a rotary actuator such as an electric motor to electrical linear actuator  700 . In an exemplary embodiment, electrical linear actuator  700  may include a jack screw. 
     In an exemplary embodiment, coaxial pericyclic gear reducer  702  may include an output shaft  712  similar to output shaft  412 . In an exemplary embodiment, output shaft  712  of coaxial pericyclic gear reducer  702  may be coupled to a central screw  704  of electrical linear actuator  700 . In an exemplary embodiment, responsive to the rotation of central screw  704 , a central rod  708  of electrical linear actuator  700  may be extended or retracted along a longitudinal axis of electrical linear actuator  700 . As used herein, a longitudinal axis of an object may refer to an axis associated with the longest dimension of that object. 
     In an exemplary embodiment, an input shaft  706  of coaxial pericyclic gear reducer  702  may be configured as a hollow shaft similar to input shaft  406 , through which output shaft  712  may pass. In an exemplary embodiment, input shaft  706  may be a hollow shaft of an electric motor that may be housed within an elongated cylindrical body  710  of electrical linear actuator  700 . In an exemplary embodiment, central rod  708  may be coaxially disposed within input shaft  706  and input shaft  706  may be configured to guide the linear movement of central rod  708 . In an exemplary embodiment, central rod  708  may be coupled to an end-effector, such as a hook  714 . As mentioned before, such utilization of coaxial pericyclic gear reducer  702  may allow for a more compact design of electrical linear actuator  700 , not to mention allowing a considerable increase in the power density of electrical linear actuator  700 . In an exemplary embodiment, input shaft  706  is coupled to or integrally made with input external-teeth gear  707  so that input external-teeth gear  707  is driven by electrical motor  710  directly. There is a central hole in input shaft  706  as well as in input external-teeth gear  707  to let external output shaft  712 , passes through that hole. external output shaft  712  is directly coupled to or integrally made with output sun gear  704  of the exemplary pericyclic gear reducer  702 . In such a configuration of external output shaft  712 , input shaft  706  and the output shaft  712  may be in same side and low-speed output shaft  712  may be surrounded by high-speed input shaft  706 . 
     An exemplary pericyclic gear reducer such as pericyclic gear reducer  100  or coaxial pericyclic gear reducer  400  may have some advantages over classic gearboxes with common gear trains at the same power capacity and reduction ratio; such advantages may include compact structure, light weight, small moment of inertia, high power density, high torque density (due to utilization of internal-external gear meshing), high efficiency, and lower backlash. 
     An exemplary pericyclic gear reducer such as pericyclic gear reducer  100  or coaxial pericyclic gear reducer  400  may have some advantages over other gear reducers, such as planetary gear transmission (PGT), cycloidal speed reducers (CSR), worm and gear (WG) reducers, and harmonic drives (HD). Such advantages over the aforementioned gear reducers may include a wider range of reduction ratios, covering the most convenient range of reduction ratios in industry (i.e. 2.5:1 up to 50:1), more multiplicity in reduction ratios, capability of having both same direction and opposite direction input-output rotations, Both parallel and coaxial input/output shafts assembly, interference-free gear meshing for a wide range of gear ratios, full involute gear meshing (consequently, smooth running with low noise, high contact ratio and steady output velocity), high torque capacity, high efficiency, lower wear, lower cost, lower levels of noise, and maintaining high torque and power capacity in wide range of reduction ratios, excellent adaptability for modular production, high reliability and two alternatives in assembling components (i.e. pericyclic and direct assemblies). Table 1 below, summarizes a thorough comparison between an a single-stage coaxial pericyclic gear reducer such as coaxial pericyclic gear reducer  400  (referred to as “PGR”) with other gear reducers, such as PGT, CSR, WG, and HD. 
     
       
         
           
               
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Relative Feature 
                 Transmission Type 
               
            
           
           
               
               
               
               
               
               
            
               
                 (per Stage) 
                 PGT 
                 CSR 
                 WG 
                 HD 
                 PGR 
               
               
                   
               
               
                 Total Range of 
                 3:1-10:1 
                 11:1-100:1 
                 10:1-150:1 
                 30:1-350:1 
                 2.2:1-200:1 
               
               
                 Reduction Ratio 
                 10:9-3:2 
                   
                   
                   
                 200:199-7:4 
               
               
                 Optimum Range 
                 4:1-6:1 
                 30:1-50:1 
                 10:1-50:1 
                 50:1-200:1 
                 3:1-100:1 
               
               
                 of Reduction Ratio 
                   
                   
                   
                   
                 100:99-3:2 
               
               
                 Multiplicity in Ratio 
                 good 
                 moderate 
                 moderate 
                 moderate 
                 excellent 
               
               
                 Selection Weight 
                 moderate 
                 moderate 
                 high 
                 low 
                 low 
               
               
                 Longitudinal 
                 moderate 
                 excellent 
                 moderate 
                 weak 
                 excellent 
               
               
                 Compactness 
                   
                   
                   
                   
                   
               
               
                 Radial Compactness 
                 good 
                 moderate 
                 weak 
                 moderate 
                 good 
               
               
                 Input-Output Shaft 
                 coaxial 
                 coaxial 
                 skew 
                 Coaxial 
                 coaxial 
               
               
                 Arrangement 
                   
                   
                   
                   
                 &amp; parallel 
               
               
                 Hollow Shaft 
                 conditionally 
                 possible 
                 possible 
                 possible 
                 possible 
               
               
                 Configuration 
                 possible 
                   
                   
                   
                   
               
               
                 Backlash 
                 moderate 
                 low 
                 high 
                 Low 
                 low 
               
               
                 Equivalent Moment 
                 low 
                 moderate 
                 moderate 
                 Low 
                 moderate 
               
               
                 of Inertia on 
                   
                   
                   
                   
                   
               
               
                 Input Shaft 
                   
                   
                   
                   
                   
               
               
                 Number of Gears 
                 4-7 
                 2-3 
                 2 
                 2 
                 2 or 4 
               
               
                 Number of Bearings 
                 7-12 
                 4-6 
                 4 
                 3-5 
                 3-4 
               
               
                 Gear Profile 
                 involute 
                 special 
                 special 
                 special 
                 involute 
               
               
                 Type of Interference 
                 involute 
                 — 
                 — 
                 trochoid &amp; 
                 involute &amp; 
               
               
                 in Gears that 
                 trochoid &amp; 
                   
                   
                 trimming 
                 trimming 
               
               
                 should be Avoided 
                 trimming 
                   
                   
                   
                   
               
               
                 Type of Bearings 
                 ordinary 
                 ordinary 
                 ordinary 
                 Special &amp; 
                 ordinary 
               
               
                   
                   
                   
                   
                 Ordinary 
                   
               
               
                 Efficiency 
                 high 
                 high 
                 low 
                 moderate 
                 excellent 
               
               
                 Pitch Line 
                 low 
                 low 
                 high 
                 very low 
                 low 
               
               
                 Velocity in Gears 
                   
                   
                   
                   
                   
               
               
                 Operational Input 
                 high 
                 moderate 
                 low 
                 moderate 
                 high 
               
               
                 Speed 
                   
                   
                   
                   
                   
               
               
                 Torque Capacity 
                 high 
                 high 
                 moderate 
                 moderate 
                 high 
               
               
                 per Volume 
                   
                   
                   
                   
                   
               
               
                 Power Capacity 
                 high 
                 high 
                 moderate 
                 moderate 
                 high 
               
               
                 per Volume 
                   
                   
                   
                   
                   
               
               
                 Resistance to Initial 
                 moderate 
                 high 
                 low 
                 moderate 
                 moderate 
               
               
                 and Breaking Shock 
                   
                   
                   
                   
                   
               
               
                 Wear 
                 moderate 
                 moderate 
                 very high 
                 Moderate 
                 low 
               
               
                 Contact Stress 
                 high 
                 low 
                 low 
                 high 
                 low 
               
               
                 on Gear&#39;s Teeth 
                   
                   
                   
                   
                   
               
               
                 Elastic Stress 
                 moderate 
                 moderate 
                 moderate 
                 high 
                 low 
               
               
                 on Parts and 
                   
                   
                   
                   
                   
               
               
                 Hysteresis Damping 
                   
                   
                   
                   
                   
               
               
                 Bending Stress on 
                 moderate 
                 very low 
                 moderate 
                 high 
                 moderate 
               
               
                 Gear&#39;s Teeth 
                   
                   
                   
                   
                   
               
               
                 Number of Alternative 
                 3 
                 1 
                 1 
                 1 
                 2 
               
               
                 Assemblies and 
                   
                   
                   
                   
                   
               
               
                 Corresponding Ratios 
                   
                   
                   
                   
                   
               
               
                 Analysis of Internal 
                 indeter- 
                 indeter- 
                 deter- 
                 indeter- 
                 deter- 
               
               
                 Loads and Stresses 
                 minate 
                 minate 
                 minate 
                 minate 
                 minate 
               
               
                 Complexity of 
                 simple 
                 complex 
                 simple 
                 Complex 
                 simple 
               
               
                 Components&#39; Shape 
                   
                   
                   
                   
                   
               
               
                 Tolerances in 
                 tight 
                 tight 
                 moderate 
                 tight 
                 moderate 
               
               
                 Manufacturing 
                   
                   
                   
                   
                   
               
               
                 Materials and Alloys 
                 ordinary 
                 special 
                 special 
                 Special 
                 ordinary 
               
               
                 Relative Performance 
                 weak 
                 weak 
                 good 
                 weak 
                 good 
               
               
                 in Low 
                   
                   
                   
                   
                   
               
               
                 Reduction Ratios 
                   
                   
                   
                   
                   
               
               
                 (in range) 
                   
                   
                   
                   
                   
               
               
                 Relative Performance 
                 good 
                 good 
                 moderate 
                 good 
                 excellent 
               
               
                 in Medium 
                   
                   
                   
                   
                   
               
               
                 Reduction Ratios 
                   
                   
                   
                   
                   
               
               
                 Relative Performance 
                 weak 
                 weak 
                 week 
                 weak 
                 good 
               
               
                 in High 
                   
                   
                   
                   
                   
               
               
                 Reduction Ratios 
                   
                   
                   
                   
                   
               
               
                 (in range) 
                   
                   
                   
                   
                   
               
               
                 Cost 
                 moderate 
                 high 
                 moderate 
                 High 
                 low 
               
               
                 Endurance 
                 moderate 
                 high 
                 low 
                 Low 
                 high 
               
               
                 Reliability 
                 moderate 
                 high 
                 moderate 
                 moderate 
                 high 
               
               
                 Back Driving 
                 possible 
                 impossible 
                 impossible 
                 Impossible 
                 Possible 
               
               
                 (Opposite of 
                   
                   
                   
                   
                   
               
               
                 Self Locking) 
                   
                   
                   
                   
                   
               
               
                 Sign of Output 
                 positive 
                 negative 
                 positive 
                 negative 
                 Possible &amp; 
               
               
                 Direction (relative 
                   
                   
                   
                   
                 negative 
               
               
                 to Input) 
                   
                   
                   
                   
                   
               
               
                 Balancing 
                 moderately 
                 highly 
                 lowly 
                 moderately 
                 highly 
               
               
                 Requirement 
                   
                   
                   
                   
                   
               
               
                 Possibility of 
                 yes 
                 yes 
                 yes 
                 yes 
                 yes 
               
               
                 Static Balancing 
                   
                   
                   
                   
                   
               
               
                 Possibility of Fully 
                 yes 
                 no 
                 yes 
                 yes 
                 yes 
               
               
                 Dynamic Balancing 
                   
                   
                   
                   
                   
               
               
                 Output Speed 
                 no 
                 yes 
                 no 
                 yes 
                 no 
               
               
                 Fluctuation 
                   
                   
                   
                   
                   
               
               
                 Rippling of 
                 no 
                 yes 
                 no 
                 yes 
                 no 
               
               
                 Internal forces 
                   
                   
                   
                   
                   
               
               
                 Noise Level 
                 moderate 
                 moderate 
                 low 
                 moderate 
                 low 
               
               
                 Heat Exchange 
                 moderate 
                 high 
                 low 
                 moderate 
                 high 
               
               
                 Density 
                   
                   
                   
                   
                   
               
               
                 Size Limitation 
                 Not 
                 Not 
                 Not 
                 Limited 
                 Not 
               
               
                   
                 Limited 
                 Limited 
                 Limited 
                   
                 Limited 
               
               
                 Possible Power 
                 series 
                 Series &amp; 
                 series 
                 series 
                 Series &amp; 
               
               
                 Sequence in 
                   
                 Parallel 
                   
                   
                 Parallel 
               
               
                 Multistage Assembly 
                   
                   
                   
                   
                   
               
               
                 Adaptability to 
                 moderate 
                 moderate 
                 moderate 
                 good 
                 excellent 
               
               
                 Modular Production 
               
               
                   
               
            
           
         
       
     
     In an exemplary embodiment, as evident from equation (11), a coaxial pericyclic gear reducer, such as coaxial pericyclic gear reducer  400  may have no lower limit on the speed reduction ratio, theoretically. However, in practice, some restrictions may be encountered. First, N DR /N IE  in equation (11) must be chosen to be ½ for balancing purposes. The second limitation is that N P  and N S  are integers and hence the fraction N P /N S  cannot take any desirable value. The third limitation is the module limitation, which was thoroughly discussed in previous paragraphs in connection with  FIG. 5A . Finally, the fourth limitation is related to the geometry of gears to avoid different kinds of interferences in gears. For example, for standard 20° pressure angle spur gears, the two conditions: 1.1&lt;N P /N S &lt;1.6 and N IE , N S &gt;21 should be held concurrently to circumvent trimming and involute interferences, respectively 1 . Here, the lower and upper limits (1.1 and 1.6) may correspond to very large and very small teeth numbers, respectively. In an exemplary embodiment, N IE , N S &gt;21 may apply for spur gears and in case helical gears are utilized, this limit may decrease as the helix angle increases.  1  The lower and upper limits (1.1 and 1.6) correspond to very large and very small teeth numbers respectively. 
     In spite of the four limitations mentioned above, there may be so many choices in practice for N P  and N S  with standard moduli to get a variety of reductive ratios. By varying N IE , N DR , N P  and N S  as well as proper selection of gears&#39; moduli and helix angles, almost every desired ratio in ratio range may be achievable in an exemplary pericyclic gear reducer, such as coaxial pericyclic gear reducer  400 . 
     In a single stage planetary gear transmission in a high reduction mode, where an exemplary ring gear of the single stage planetary gear transmission is fixed and the exemplary sun gear may act as the input. An exemplary carrier of planet gears is connected to an exemplary output shaft, an exemplary reduction ratio may be obtained by equation (23) below:
 
Reduction ratio=(1+ N   Ring   /N   Sun ) −1   Equation (23)
 
     In equation (23) above, N Ring  stands for the teeth number of an exemplary ring gear and N Sun  stands for the teeth number of an exemplary input sun gear of an exemplary planetary gear transmission. Comparing equation (23) with equation (11), it is evident that in both planetary gear transmission and pericyclic gear reducer  100 , the upper limit of reduction ratio cannot approach ½ since the fractions N Ring /N Sun  and N P /N S  may approach to unity which would be impossible because the external gears should be as large as the internal gears in the exemplary reducers. In this regard, pericyclic gear reducer  100  may have a better condition than an equivalent planetary gear transmission, since in pericyclic gear reducer  100 , only one external gear, such as output sun gear  104  is located within an internal gear, such as middle planet ring gear  102 , while in an exemplary planetary gear transmission, a chain consisting of three external inline gears are located within the internal gear. In other words, in an exemplary planetary gear transmission the fraction N Ring /N Sun  should take the minimum value of 2 to allow for exemplary planet gears be disposed between an exemplary input sun gear and an exemplary ring gear. On the contrary, in pericyclic gear reducer  100 , the fraction Np/Ns may take a minimum value of 1.1 or even less than 1.1 without encountering any interferences. Hence a minimum value of reduction ratios in pericyclic gear reducer  400  and an exemplary planetary gear transmission may be 2.2 and 3, respectively based on equations 11 and 23. However, the efficient reduction ratio of pericyclic gear reducer  400  may start from 3 instead of 2.2 since the ratios between 2.2 and 3 correspond to small size of teeth and hence low torque capacity. The efficient reduction ratio of an exemplary planetary gear transmission starts from 4 (instead of 3) since the ratios between 3 and 4 correspond to small size planet gears and hence low torque capacity. On the other side of the ratio spectrum, the upper bands of reduction ratio in an exemplary planetary gear transmission cannot exceed 10 since approaching this ratio results in an exceedingly small input sun gear and decrement of planet numbers, as well as very small teeth size to avoid involute interference between sun and planets. This significantly decreases the torque capacity of the transmission and should be avoided in practice, except in special cases. Pericyclic gear reducer  400 , on the other hand, has no limitation on its upper band of ratio range except that the higher ratios yield to higher number of gear teeth and consequently the smaller size of teeth, if the overall size of the gearbox is kept to be unchanged. This is a common drawback of all speed reducers (including an exemplary planetary gear transmission, SCR, W&amp;G HD and PGR) since it weakens the gears in torque transmission in high reduction ratios. Regardless of that, pericyclic gear reducer  400  may efficiently work in reduction ratios up to 100:1. 
     Another benefit of an exemplary pericyclic gear reducer, such as pericyclic gear reducer  100  or coaxial pericyclic gear reducer  400 , in comparison with an exemplary planetary gear transmission is the multiplicity in available ratios. This may be achieved just by changing two parameters, i.e., the teeth numbers of middle planet ring gear  402  and output sun gear  404 . Theoretically, infinite numbers of ratios may be found for both an exemplary pericyclic gear reducer and an exemplary planetary gear transmission in their reduction range but in an exemplary planetary gear transmission much more parameters should be adjusted to obtain desired ratios. A gear reduction ratio for an exemplary planetary gear transmission may depend on the relative sizes of an exemplary input sun gear, exemplary planetary gears, an exemplary ring gear, and an exemplary carrier arm of an exemplary planetary gear transmission. In an exemplary pericyclic gear reducer, such as pericyclic gear reducer  100  or coaxial pericyclic gear reducer  400 , various reduction ratios may be achieved by adjusting the two parameters, namely, the teeth numbers of output sun gear ( 104 ,  404 ) and middle planet ring gear ( 102 ,  402 ). In exemplary embodiments, such configuration of pericyclic gear reducer  100  or coaxial pericyclic gear reducer  400  may offer excellent adaptability for modular production. In other words, in various pericyclic gearboxes of same power class, by having most of parts including shells, bearings, input shafts, driver ring gear and driven wheel to be the same, only by differing the output sun and planet gears ( 102 ,  104  or  402 ,  404 ) various gearboxes may be produced with different ratios. 
     In a comparative example, an exemplary pericyclic gear reducer and an exemplary planetary gear transmission with reduction ratios ranging from 3 to 10 may be compared. Since an exemplary pericyclic gear reducer such as coaxial pericyclic gear reducer  400  may take infinite numbers of available ratios between 3 and 10, for the sake of comparison, some restrictions may be applied on coaxial pericyclic gear reducer  400  as follows. In this comparative example, driver ring gear  408 , input external-teeth gear  407 , output sun gear  404 , and middle planet ring gear  402  may fulfill the following limitations: N DR =38, N IE =19, normal pressure angles of driver ring gear  408 , and input external-teeth gear  407  equal to 20°, helix angles of driver ring gear  408  and input external-teeth gear  407  equal to 18°, and normal modules of driver ring gear  408  and input external-teeth gear  407  equal to 0.9. In this comparative example, output sun gear  404  and middle planet ring gear  402  may include spur gears with pressure angles of 20°. In addition, moduli of output sun gear  404  and middle planet ring gear  402  may be of standard values between 0.3 and 3. Finally, no profile shift is permitted in any of the gears. Despite assuming these very curbing conditions, surprisingly, 268 different combinations of reduction ratios between 3 and 10 and standard moduli between 0.3 and 3 may be achieved just by varying the teeth number of sun gear  404  and middle planet gear  402 . 
     A similar comparison may be made between an exemplary pericyclic gear reducer and CSR and HD. Although, CSRs and HDs have wider domain of available ratios in comparison with PGTs, they have fewer numbers of available ratios than those of PGTs and PGR. SCRs and WG may only have integer ratios and HDs may have pure integers and integers plus 0.5 ratios. The reduction range of SCRs and HDs may have a lower limit of 10 and 30, respectively while the ratio range of coaxial pericyclic gear reducer  400  may start from 2.2. The upper limits of reduction ratio range for CSR, HD and coaxial pericyclic gear reducer  400  have no restriction theoretically. However, they may take the operational values of 100:1, 320:1 and 200:1, respectively. For reduction ratios with larger values, the torque density will be decreased remarkably in CSR, HD and coaxial pericyclic gear reducer as well as WG since the gear teeth size in all of them decreases fatefully. 
       FIG. 8A  illustrates a graphical chart of gear reduction ratio ranges of exemplary gear reducers, consistent with one or more exemplary embodiments of the present disclosure. As evident in  FIG. 8A , an exemplary pericyclic gear reducer may cover most of the ratio ranges of PGT, CSR and HD and hence it may be utilized instead of those in most applications. There is a discontinuity in ratio range of an exemplary pericyclic gear reducer and an exemplary PGT in the vicinity of 2 since the fraction N P /N S  and N Ring /N Sun  in related ratio formula (equations 11 and 23) approaches to unity. It would be impossible since the internal and external gears in both PGT and an exemplary pericyclic gear reducer must encounter trimming and trochoid (or tip) interferences. The ratio ranges of exemplary planetary gear transmission and exemplary pericyclic gear reducer have two portions. The left-hand side portion is related to alternative assemblies of exemplary planetary gear transmission and exemplary pericyclic gear reducer (coaxial direct assembly for pericyclic gear reducer). 
     In an exemplary embodiment, an exemplary pericyclic gear reducer may inherently be capable to transmit high torques, similar to CSR and HD, due to the presence of a pair of internal-external gears in its structures. However, an exemplary pericyclic gear reducer may include big and coarse gear teeth in a wide range of reduction ratios that may guarantee the high torque capacity of an exemplary pericyclic gear reducer. While in SCR and HD, the gear teeth size is proportionally decreased with the increment of reduction ratio. 
     Internal to external gear meshing makes remarkably high resistance to contact stress between gears, since the contact takes place between a concave surface and a convex surface. In case of SCR and HD, the analysis of contact stress is somehow complicated because their gears&#39; profile is not involute. However, in case of a PGT and an exemplary pericyclic gear reducer, which may have involute teeth profiles, the contact stress may be analyzed. The force capacity between two meshing involute gears is proportional to geometry factor, I, where I=u/(u±1). In the geometry factor, u is the teeth number fraction of two meshing gears, which is defined as the ratio of the teeth number of the larger gear to that of smaller one. The + and − signs account for external-external and external-internal gear meshing, respectively. Since u≥1, the geometry factor, I for external-internal gear meshing would be much larger than the geometry factor for external-external gear meshing and consequently the force capacity of external-internal gear meshing will be much larger than the force capacity of external-external gear meshing regarding Hertzian contact stress (or pitting failure). An exemplary PGT may include both external-external and external-internal gear meshing between sun-planet and planet-ring gears, respectively. Hence the limiting factor in an exemplary PGT for load capacity of meshing gears would be related to geometry factor of sun-planet meshing which is significantly lesser than the geometry factor of planet-ring meshing. This becomes more severe in big reduction ratios of PGTs since their input sun gear becomes too small and consequently the geometry factor, “I” decreases. However, an exemplary pericyclic gear reducer may include only an external-internal gear pair and consequently the geometry factor, I for an exemplary pericyclic gear reducer may be much larger than the critical geometry factor of an exemplary PGT. Therefore, the corresponding load capacity of meshing gears in an exemplary pericyclic gear reducer may also be much larger than the load capacity of an exemplary PGT, considering Hertzian contact stress or pitting failure. It can be shown that an exemplary pericyclic gear reducer may have a geometry factor in a range of 2 to 5, while an exemplary PGT may have a geometry factor in a range of 0.5 to 0.8. It means that a gear meshing with same teeth size in an exemplary pericyclic gear reducer may have a load capacity that is averagely four times larger than the load capacity of a gear meshing in an exemplary PGT and such load capacity between two meshing gears directly affects the torque capacity. Therefore, an exemplary single pair of sun-planet gears in an exemplary pericyclic gear reducer, such as middle planet ring gear  402  and output sun gear  404  of coaxial pericyclic gear reducer  400  may handle the torque transmission of multiple pairs of sun-planet gears in an exemplary PGT. Such capability may significantly reduce the overall number of parts and simplify the manufacturing process and decrease the production cost of an exemplary pericyclic gear reducer in comparison with an equivalent PGT. 
     Other than the geometry factor, teeth size is another important factor that may directly affect the torque capacity of two meshing gears. In a gearbox, when the input power is constant, as the reduction ratio increases the output torque is proportionally multiplied. Consequently, stronger gears are required for high reduction ratios in final stages of a gear transmission. However, all the mentioned gear reducers, WG, PGT, CSR, HD and an exemplary pericyclic gear reducer may have an adverse behavior, meaning that when the ratio of the gearbox increases and its radial size is kept constant, the teeth size has to become smaller. One benefit of an exemplary pericyclic gear reducer may be that an exemplary pericyclic gear reducer may be less sensitive to such undesirable effect. It may be shown that the exemplary pericyclic reducer, such as the coaxial pericyclic gear reducer  400  may maintain its big teeth sizes in ratio ranges 1:3 down to 1:50. 
     Specifically, in an exemplary pericyclic gear reducer, such as the coaxial pericyclic gear reducer  400  within the effective reduction ratio range of 2.5:1 up to 150:1, an exemplary middle planet ring gear to an exemplary output sun gear size ratio (N P /N S ) may vary between 6/5 and ½. Most ratios within this range may be considered as golden ratios, since both extremities of this range are far away from interference margins. In addition, N s /N P &gt;½ may cause a significant increase in the geometry factor, which consequently increases the teeth strength. In other words, an exemplary pericyclic gear reducer may have a much wider range of effective reduction ratios and a much higher torque capacity in that range of reduction ratio, in comparison with equivalent exemplary planetary gear reducer. 
       FIG. 8B  illustrates a graph  800  of maximum normalized height of teeth in a single-stage PGT and a graph  802  of maximum normalized height of teeth in a coaxial pericyclic gear reducer, consistent with one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, graph  802  is obtained for a pericyclic gear reducer that may be structurally similar to pericyclic gear reducer  400  and graph  800  is obtained for a single-stage PGT that may have the same radial size with the pericyclic gear reducer. In an exemplary embodiment, both the single-stage PGT and the pericyclic gear reducer may include 20° pressure angle spur gears. In an exemplary embodiment, graphs  800  and  802  may be continuous enveloping curves which pass from the points representing maximum possible teeth height which is equivalent to minimum number of teeth. As evident from graph  800 , the gear teeth of the PGT becomes smaller and smaller beyond the ratio 5:1. As evident from graph  802 , the pericyclic gear reducer holds its teeth coarseness in a wide range of ratio range i.e., from 3:1 to 50:1, which implies that the gear teeth of pericyclic gear reducer keeps its high stiffness against bending loads in that wide range and hence it may be of high torque capacity within that range. Fortunately, the above-mentioned ratio range (i.e., 3:1 to 50:1) is a golden range since it matches with most of industrial applications. On the other hand, the torque capacity of the PGT decreases intensively at reduction ratios more than 6:1 due to having very small and tiny gear teeth. 
       FIG. 8C  illustrates a graph ( 804 ,  804 ′) of torque capacity of a pericyclic gear reducer and a graph ( 806 ,  806 ′) of an equivalent PGT, consistent with one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, the PGT is assumed to have maximum number of planets to have its maximum torque capacity. The torque capacity of gears is obtained from AGMA procedure which is based on two criteria: teeth bending resistance criterion (designated by  804 ′ and  806 ′) and teeth pitting resistance criterion (designated by  804  and  806 ). The comparison is performed in reduction ratio range 2.5 up to 10. It is obvious that the torque capacity of pericyclic gear reducer is absolutely higher than its counterpart PGT in regards of pitting resistance (or Hertzian stress) criterion. But in regards of teeth bending criterion, the pericyclic gear reducer is a little weaker than its counterpart PGT in torque transmission in reduction ratios between 2.5 and 9. This comparative weakness of the pericyclic gear reducer is due to lack of multiple planets instead of single and also its shorter arm of output gear. Advantageously, increasing the bending resistance of gear teeth is much easier than increasing the pitting resistance of those. Hence the pericyclic gear reduced will fit in with high torque applications if some modifications is made on its teeth to heighten their bending resistance. In reduction ratios over 10:1 the pericyclic gear reducer, dominantly has higher torque capacity than its counterpart PGT. 
     In an exemplary embodiment, an exemplary pericyclic gear reducer, such as pericyclic gear reducer  100  or coaxial pericyclic gear reducer  400  may be configured as a multistage pericyclic reducer. An exemplary multistage pericyclic reducer may be a combination of two or more single stage pericyclic reducers such as pericyclic gear reducer  100  or coaxial pericyclic gear reducer  400 . For example, an exemplary multistage pericyclic reducer may be a combination of at least one of pericyclic gear reducer  100  and coaxial pericyclic gear reducer  400  with another gear reducer that may be structured similar to pericyclic gear reducer  100 . 
       FIG. 9  illustrates a two-stage pericyclic gear reducer  900 , consistent with one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, two-stage pericyclic gear reducer  900  may include a first stage  900   a  and a second stage  900   b . In an exemplary embodiment, first stage  900   a  may be an exemplary coaxial pericyclic gear reducer structurally similar to coaxial pericyclic gear reducer  400 . In an exemplary embodiment, second stage  900   b  may be an exemplary pericyclic gear reducer structurally similar to pericyclic gear reducer  100 . 
     In an exemplary embodiment, first stage  900   a  may include an input shaft  906  that may be coupled to a driver ring gear  908  via an input external-teeth gear  907 . In an exemplary embodiment, driver ring gear  908  may be similar to driver ring gear  408  and may be structured as an internal gear, in which input external-teeth gear  907  may be disposed. In an exemplary embodiment, input external teeth gear  907  may be similar to input external-teeth gear  407  and may be coupled to and mesh with driver ring gear  908 . In an exemplary embodiment, first stage  900   a  may further include a first middle planet ring gear  902   a  that may be coupled to driver ring gear  908  via a prismatic joint. In an exemplary embodiment, a coupler wheel  910  of first stage  900   a  may be a driven wheel similar to driven wheel  410 , since an exemplary driven wheel of first stage  900   a  may further function as a coupling member utilized for coupling first stage  900   a  and second stage  900   b , as will be discussed. 
     In an exemplary embodiment, coupler wheel  910  may be coupled to an opposing side of first middle planet ring gear  902   a  with respect to driver ring gear  908 , such that first middle planet ring gear  902   a  may be disposed longitudinally between coupler wheel  910  and driver ring gear  908 . In an exemplary embodiment, driver ring gear  908  and coupler wheel  910  may be mounted eccentric with an offset between a rotational axis of driver ring gear  908  and a rotational axis of coupler wheel  910  similar to what was described in connection with driver ring gear  408  and driven wheel  410 . In an exemplary embodiment, first stage  900   a  may further include an output sun gear  904  similar to output sun gear  404 . In an exemplary embodiment, output sun gear  904  may be an external-teeth gear that may mesh with first middle planet ring gear  902   a.    
     In an exemplary embodiment, second stage  900   b  may include a second middle planet ring gear  902   b  that may be coupled between coupler wheel  910  and driven wheel  911 . In an exemplary embodiment, second middle planet ring gear  902   b  may be similar to first middle planet ring gear  901   a  regarding teeth number and teeth moduli. In an exemplary embodiment, coupler wheel  910  may be configured to function as a driver ring gear similar to driver ring gear  108  that may be coupled to second middle planet ring gear  902   b  with a prismatic joint. Specifically, in an exemplary embodiment, coupler wheel  910  may include at least one of a protuberance and slot of a prismatic joint on one side and at least one of a protuberance and a slot of another prismatic joint on the other opposing side. In an exemplary embodiment, these two protuberances and slots may be perpendicular to each other for purpose of balancing which will be discussed. In an exemplary embodiment, coupler wheel  910  may be rotatably coupled to intermediate wall  917 . In an exemplary embodiment, intermediate wall  917  may be parallel and concentric with input-wall  914  and output-wall  916 . In an exemplary embodiment, three walls ( 914 ,  916  and  917 ) may be fixed and secured with respect to each other by a gearbox shell (which is not illustrated in  FIG. 9  for simplicity). In an exemplary embodiment, the aforementioned gearbox shell may house multi-stage pericyclic gear reducer  900 . Such configuration of protuberances and slots on either side of coupler wheel  910  may allow for connecting coupler wheel  910  with first middle planet ring gear  902   a  from one side and with second middle planet ring gear  902   b  from the other opposing side. 
     In an exemplary embodiment, output sun gear  904  may be extended along a central axis of multistage pericyclic gear reducer  900  through a central hole  909  of coupler wheel  910  into second stage  900   b . In an exemplary embodiment, output sun gear  904  may not contact with internal surface of central hole  909 . In other words, output sun gear  904  may be coupled to and meshed with both first middle planet ring gear  902   a  and second middle planet ring gear  902   b  and may be configured to transfer the rotational movement of first middle ring gear  902   a  and second middle ring gear  902   b  to a central output shaft  912 . Hence, first and second middle planet wheels ( 902   a  and  902   b ) may have same number of teeth because first and second middle planet wheels ( 902   a  and  902   b ) may be meshed with a common output sun gear  904 . 
     In an exemplary embodiment, first middle planet ring gear  902   a  and second middle planet ring gear  902   b  may be mounted with an offset between first middle planet ring gear  902   a  and second middle planet ring gear  902   b , such that, output sun gear  904  may engage a top portion of first middle planet ring gear  902   a  and an opposing bottom portion of second middle planet ring gear  902   b , which is why coupler wheel  910  should have a pair of perpendicular prismatic joints on both sides of coupler wheel  910 . In an exemplary embodiment, such configuration of output sun gear  904 , first middle planet ring gear  902   a , and second middle planet ring gear  902   b  may eliminate the need for static balancing of double-stage pericyclic gear reducer  900 , since two middle wheels  902   a  and  902   b  negate their centrifugal forces. 
     In an exemplary embodiment, other augmentations of single stage pericyclic gear reducers, such as pericyclic gear reducer  100  to the right-hand side of double-stage pericyclic gear reducer  900  is also possible to attain a multi-stage pericyclic gear reducer. An exemplary multistage pericyclic gear reducer, such as multistage pericyclic gear reducer  900  may allow for further increasing the torque and power capacity in comparison with single stage pericyclic gear reducers. As mentioned before, an exemplary multistage pericyclic gear reducer, such as multistage pericyclic gear reducer  900  may further eliminate the need for balancing, since a multistage pericyclic gear reducer, such as multistage pericyclic gear reducer  900  may inherently be balanced. 
     In an exemplary embodiment, since such a multistage pericyclic gear reducer will be characteristically balanced, there is no need to keep N DR /N IE  in equation 11 equal to 2:1. This will offer more varieties in determining both the teeth size and ratio of the driver annular gear pair ( 407  and  408 ). Not keeping N DR /N IE  in equation 11 equal to 2:1 in an exemplary multi-stage pericyclic gear reducer may also extend the lower limit of reduction ratio of the pericyclic gear reducer from 2.2:1 down to 2:1. 
       FIG. 10A  illustrates an exploded view of a coaxial pericyclic gear reducer  1000 , consistent with one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, coaxial pericyclic gear reducer  1000  may be similar to coaxial pericyclic gear reducer  400  and may include an input shaft  1406  similar to input shaft  406  that may be rotatable about a central axis  1428 , a middle planet ring gear  1402  that may include an annular internal-teeth gear  1010  similar to middle planet ring gear  402 , an output sun gear  1404  similar to output sun gear  404  that may be an external-teeth gear positioned inside and meshed with middle planet ring gear  1402 . 
     In an exemplary embodiment, coaxial pericyclic gear reducer  1000  may further include a driver ring gear  1408  similar to driver ring gear  408  that may be configured to be coupled to input shaft  1406  via an input external-teeth gear  1407  similar to input external-teeth gear  407 . In an exemplary embodiment, input shaft  1406  may be coupled to or integrally formed with input external-teeth gear  1407 . In an exemplary embodiment, input external-teeth gear  1407  may be disposed within driver ring gear  1408  and may mesh with driver ring gear  1408 . In an exemplary embodiment, such coupling of input external-teeth gear  1407  and driver ring gear  1408  may allow for converting the rotational movement of input shaft  1406  about central axis  1428  to a rotational movement of driver ring gear  1408  about a first rotational axis  1424 . In an exemplary embodiment, driver ring gear  1408  may be coupled to middle planet ring gear  1402  from a first side of middle planet ring gear  1402  utilizing a first prismatic joint. In an exemplary embodiment, driver ring gear  1408  may be configured to drive a rotational movement of middle planet ring gear  1402 . 
     In an exemplary embodiment, coaxial pericyclic gear reducer  1000  may further include a driven wheel  1410  structurally similar to driven wheel  410  that may be a flange coupled to middle planet ring gear  1402  from a second side of middle planet ring gear  1402  utilizing a second prismatic joint. In an exemplary embodiment, driven wheel  1410  may be rotatable with middle planet ring gear  1402  about a second rotational axis  1426 . 
     In an exemplary embodiment, coaxial pericyclic gear reducer  1000  may further include an output shaft  412  similar to output shaft  412  that may be coupled or integrally formed with output sun gear  1404 . In an exemplary embodiment, output shaft  1412  may be rotatable with output sun gear  1404  about central axis  1428  of coaxial pericyclic gear reducer  1000 . 
     In an exemplary embodiment, the first prismatic joint may include a pair of parallel grooves ( 1002   a ,  1002   b ) attached to or integrally formed on driver ring gear  1408 . In an exemplary embodiment, each groove of pair of parallel grooves ( 1002   a ,  1002   b ) may be extended along a longitudinal plane that may be perpendicular to a front face  1004  of driver ring gear  1408 . In an exemplary embodiment, front face  1004  may include an annular surface of driver ring gear  1408  perpendicular to central axis  1428  and facing middle planet ring gear  1402 . As used herein, a longitudinal plane of an object is a plane associated with the longest dimension of that object. In an exemplary embodiment, pair of parallel grooves ( 1002   a ,  1002   b ) may face each other and towards a center of driver ring gear  1408 . In other words, pair of parallel grooves ( 1002   a ,  1002   b ) may be opposite each other and disposed on driver ring gear  1408  with respect to central line  1428 . In an exemplary embodiment, each groove of pair of parallel grooves ( 1002   a ,  1002   b ) may be a straight groove that may either have an arc section or a rectangular cross-section to allow balls or rollers to moveably fit inside. 
       FIG. 10B  illustrates an exploded view of middle planet ring gear  1402  and cages ( 1016 ,  1032 ), consistent with one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, the first prismatic joint may further include a corresponding first pair of parallel grooves ( 1006   a ,  1006   b ) on middle planet ring gear  1402 . In an exemplary embodiment, middle planet ring gear  1402  may be configured to have a rectangular body  1008 , where a normal axis of rectangular body  1008  may be parallel with central axis  1428 . In an exemplary embodiment, middle planet ring gear  1402  may further include a cylindrical internal-teeth gear  1010  similar to middle planet ring gear  402  cut in the middle of rectangular body  1008  of middle planet ring gear  1402 . 
     In an exemplary embodiment, such rectangular shape of middle planet ring gear  1402  may allow for lightening the weight of middle planet ring gear  1402 , which may result in less centrifugal force exerted on middle planet ring gear  1402  in motion. In an exemplary embodiment, rectangular body  1008  may further include cut out portions or cavities ( 1007   a - 1007   d ) that may allow for further decreasing the weight of middle planet ring gear  1402 . 
     In an exemplary embodiment, rectangular body  1008  may include a pair of parallel lateral sides ( 1012   a ,  1012   b ) and a pair of parallel top and bottom sides ( 1014   a ,  1014   b ), where parallel lateral sides ( 1012   a ,  1012   b ) are perpendicular to parallel top and bottom sides ( 1014   a ,  1014   b ). In an exemplary embodiment, first pair of parallel grooves ( 1006   a ,  1006   b ) may be formed on top lateral side  1014   a  and bottom lateral side  1014   b  of middle planet ring gear  1402  and each groove of first pair of parallel grooves ( 1006   a ,  1006   b ) may correspond to each groove of pair of parallel grooves ( 1002   a ,  1002   b ). For example, groove  1006   a  may be formed on top lateral side  1014   a  of rectangular body  1008  and may correspond to groove  1002   a , meaning that groove  1006   a  may face and extend parallel with groove  1002   a . In an exemplary embodiment, a plurality of balls or rollers may be disposed and sandwiched between groove  1006   a  and groove  1002   a , such that the plurality of balls or rollers may be moveably contained between groove  1006   a  and groove  1002   a . Similarly, groove  1006   b  may correspond to groove  1002   b , meaning that groove  1006   b  may face and extend parallel with groove  1002   b . In an exemplary embodiment, a plurality of balls or rollers may be disposed and sandwiched between groove  1006   b  and groove  1002   b , such that the plurality of balls or rollers may be moveably contained between groove  1006   b  and groove  1002   b . In other words, the first prismatic joint between middle planet ring gear  1402  and driver ring gear  1408  may include a rolling-contact prismatic joint made up of pair of parallel grooves ( 1002   a ,  1002   b ) on driver ring gear  1408 , corresponding first pair of parallel grooves ( 1006   a ,  1006   b ) on middle planet ring gear  1402 , and a plurality of balls or rollers disposed between each corresponding pair of grooves. 
     In an exemplary embodiment, as middle planet ring gear  1402  moves with respect to driver ring gear  1408 , balls or rollers disposed between groove  1002   a  and groove  1006   a , and balls or rollers disposed between groove  1002   b  and groove  1006   b  may facilitate a sliding movement of middle planet ring gear  1402  relative to driver ring gear  1408  with less friction, lower backlash, and higher speed compared to what a tongue-and-slit prismatic joint could have allowed as discussed in connection with  FIG. 4 . 
     In an exemplary embodiment, the first prismatic joint may further include a first cage  1016  that may include a first plate  1020  extended perpendicular to central axis  1428  and parallel with rectangular body  1008 . In an exemplary embodiment, first cage  1016  may be disposed within a small gap between middle planet ring gear  1402  and driver ring gear  1408 . In an exemplary embodiment, such disposal of first cage  1016  within the small gap may allow for first cage  1016  to be slidable within the gap relative to middle planet ring gear  1402  and driver ring gear  1408  without friction. In an exemplary embodiment, first cage  1016  may further include a pair of extended lips ( 1018   a ,  1018   b ), where each extended lip of pair of extended lips ( 1018   a ,  1018   b ) may extend from a respective edge of first plate  1020  perpendicular to a plane of first plate  1020  and parallel with central axis  1428 . In an exemplary embodiment, extended lip  1018   a  may be positioned between grooves  1002   a  and  1006   a , and extended lip  1018   b  may be positioned between grooves  1002   b  and  1006   b.    
     In an exemplary embodiment, each extended lip of pair of extended lips ( 1018   a ,  1018   b ) may include a plurality of holes that may be configured to encompass the plurality of balls or rollers of the first prismatic joint. For example, extended lip  1018   a  may include holes  1024   a  that may be configured to encompass the plurality of balls or rollers disposed between grooves  1002   a  and  1006   a , and extended lip  1018   b  may include holes  1024   b  that may be configured to encompass the plurality of balls or rollers disposed between grooves  1002   b  and  1006   b . In an exemplary embodiment, each hole of holes  1024   a  and  1024   b  may have a little larger diameter compared to each ball or roller of the plurality of balls or rollers disposed between the corresponding grooves ( 1002   a ,  1002   b ,  1006   a ,  1006   b ) to allow the plurality of balls or rollers to freely role inside holes  1024   a  and  1024   b.    
     In an exemplary embodiment, first cage  1016  may further include a first oblong oval hole  1022  that may extend on first plate  1020  perpendicular to central axis  1428  between pair of extended lips ( 1018   a ,  1018   b ) and parallel with the second pair of parallel grooves ( 1030   a ,  1030   b ). In an exemplary embodiment, first oblong oval hole  1022  may be configured to allow for output shaft  1412  to pass through first cage  1016 . In an exemplary embodiment, first oblong oval hole  1022  may have a width  1017  equal to the diameter of output shaft  1412 , such that lateral edges of first oblong oval hole  1022  may be tangential to an outer surface of output shaft  1412 . Such configuration of first oblong oval hole  1022  may prevent any lateral movements of first cage  1016  relative to output shaft  1412 . 
     In an exemplary embodiment, the second prismatic joint may include a pair of parallel grooves ( 1026   a ,  1026   b ) attached to or integrally formed on of driven wheel  1410 . In an exemplary embodiment, each groove of pair of parallel grooves ( 1026   a ,  1026   b ) may be extended along a longitudinal plane that may be perpendicular to a front face  1028  of driven wheel  1410 . In an exemplary embodiment, front face  1028  may include an annular surface of driven wheel  1410  perpendicular to central axis  1428  and facing middle planet ring gear  1402 . In an exemplary embodiment, pair of parallel grooves ( 1026   a ,  1026   b ) may face each other and towards a center of driven wheel  1410 . In an exemplary embodiment, each groove of pair of parallel grooves ( 1026   a ,  1026   b ) may be a straight groove that may either have an arc section or a rectangular cross-section to allow balls or rollers to moveably fit inside. 
     In an exemplary embodiment, the second prismatic joint may further include a corresponding second pair of parallel grooves ( 1030   a ,  1030   b ) on respective pair of parallel lateral sides ( 1012   a ,  1012   b ) of middle planet ring gear  1402 . In an exemplary embodiment, each groove of second pair of parallel grooves ( 1030   a ,  1030   b ) may correspond to each groove of pair of parallel grooves ( 1026   a ,  1026   b ). For example, groove  1030   a  may be formed on first lateral side  1012   a  of rectangular body  1008  and may correspond to groove  1026   a , meaning that groove  1030   a  may face and extend parallel with groove  1026   a . In an exemplary embodiment, a plurality of balls or rollers may be disposed and sandwiched between groove  1030   a  and groove  1026   a , such that the plurality of balls or rollers may be moveably contained between groove  1030   a  and groove  1026   a . Similarly, groove  1030   b  may correspond to groove  1026   b , meaning that groove  1030   b  may face and extend parallel with groove  1026   b . In an exemplary embodiment, a plurality of balls or rollers may be disposed and sandwiched between groove  1030   b  and groove  1026   b , such that the plurality of balls or rollers may be moveably contained between groove  1030   b  and groove  1026   b . In other words, the second prismatic joint between middle planet ring gear  1402  and driven wheel  1410  may include a rolling-contact prismatic joint made up of pair of parallel grooves ( 1026   a ,  1026   b ) on driven wheel  1410 , corresponding second pair of parallel grooves ( 1030   a ,  1030   b ) on middle planet ring gear  1402 , and a plurality of balls or rollers disposed between each corresponding pair of grooves. 
     In an exemplary embodiment, as middle planet ring gear  1402  moves with respect to driven wheel  1410 , balls or rollers disposed between groove  1026   a  and groove  1030   a , and balls or rollers disposed between groove  1026   b  and groove  1030   b  may facilitate a sliding movement of middle planet ring gear  1402  relative to driven wheel  1410  with less friction and lower backlash compared to what a tongue-and-slit prismatic joint could have allowed as discussed in connection with  FIG. 4 . 
     In an exemplary embodiment, the second prismatic joint may further include a second cage  1032  that may include a second plate  1034  extended perpendicular to central axis  1428  and parallel with rectangular body  1008  of middle planet ring gear  1402 . In an exemplary embodiment, second cage  1032  may be disposed within a small gap between middle planet ring gear  1402  and driven wheel  1410 . In an exemplary embodiment, such disposal of second cage  1032  within the small gap may allow for second cage  1032  to be slidable within the gap relative to middle planet ring gear  1402  and driven wheel  1410  without friction. In an exemplary embodiment, second cage  1032  may further include a pair of extended lips ( 1036   a ,  1036   b ), where each extended lip of pair of extended lips ( 1036   a ,  1036   b ) may extend from respective edges of second plate  1034  perpendicular to a plane of second plate  1034  and parallel with central axis  1428 . In an exemplary embodiment, extended lip  1036   a  may be positioned between grooves  1026   a  and  1030   a , and extended lip  1036   b  may be positioned between grooves  1026   b  and  1030   b.    
     In an exemplary embodiment, each extended lip of pair of extended lips ( 1036   a ,  1036   b ) may include a plurality of holes that may be configured to encompass the plurality of balls or rollers of the second prismatic joint. For example, extended lip  1036   a  may include holes  1038   a  that may be configured to encompass the plurality of balls or rollers disposed between grooves  1026   a  and  1030   a , and extended lip  1036   b  may include holes  1038   b  that may be configured to encompass the plurality of balls or rollers disposed between grooves  1026   b  and  1030   b.    
     In an exemplary embodiment, second cage  1032  may further include a second oblong oval hole  1040  that may extend on second plate  1034  perpendicular to central axis  1428  between pair of extended lips ( 1036   a ,  1036   b ) and parallel with the first pair of grooves ( 1006   a  and  1006   b ). In an exemplary embodiment, second oblong oval hole  1040  may be configured to allow for output shaft  1412  to pass through second cage  1032 . In an exemplary embodiment, second oblong oval hole  1040  may have a width  1041  equal to the diameter of output shaft  1412 , such that lateral edges of second oblong oval hole  1040  may be tangential to an outer surface of output shaft  1412 . Such configuration of second oblong oval hole  1040  may prevent any lateral movements of second cage  1032  relative to output shaft  1412 . 
     In an exemplary embodiment, the first prismatic joint and the second prismatic joint are perpendicular to each other. In other words, pair of parallel grooves ( 1002   a ,  1002   b ) may be perpendicular to pair of parallel grooves ( 1026   a ,  1026   b ) and first pair of parallel grooves ( 1006   a ,  1006   b ) may be perpendicular to second pair of parallel grooves ( 1030   a ,  1030   b ). Furthermore, the largest diameter of first oblong oval hole  1022  may be perpendicular to the largest diameter of second oblong oval hole  1040 , while a plane of first oblong oval hole  1022  may be parallel with a plane of second oblong oval hole  1040 . 
     In exemplary embodiments, utilization of such rolling-contact prismatic joints to couple driver ring gear  1408  and driven wheel  1410  to either side of middle planet ring gear  1402 , where instead of protrusions that may slide within corresponding grooves, a combination of rollers/balls and grooves are utilized, may allow for making coaxial pericyclic gear reducer  1000  with less friction, lower backlash, and capable of operation at high speeds. However, in the embodiments illustrated in  FIGS. 10A and 10B  such utilization of grooves ( 1006   a  and  1006   b ) on the edges of middle planet ring gear  1402  may lead to a slight increase in the radial size of coaxial pericyclic gear reducer  1000 , which may be addressed by forming the grooves on the faces of middle planet ring gear  1402 , which will be discussed in the following paragraphs. 
       FIG. 11A  illustrates an exploded view of a coaxial pericyclic gear reducer  2000 , consistent with one or more exemplary embodiments of the present disclosure.  FIG. 11B  illustrates an exploded view of a middle planet ring gear  2402  mounted between a driver ring gear  2408  and a driven wheel  2410 , consistent with one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, coaxial pericyclic gear reducer  2000  may be similar to pericyclic gear reducer ( 1000  and  400 ). In an exemplary embodiment, coaxial pericyclic gear reducer  2000  may include an input shaft  2406  similar to input shaft  406  that may be coupled to or integrally formed with an input external-teeth gear  2407  similar to input external-teeth gear  407 . In an exemplary embodiment, input external-teeth gear  2407  may mesh with an annular internal-teeth gear of a driver ring gear  2408  similar to driver ring gear  408 . In an exemplary embodiment, coaxial pericyclic gear reducer  2000  may further include a middle planet ring gear  2402  similar to middle planet ring gear  402  that may include an annular internal-teeth gear that may mesh with an output sun gear  2404  similar to output sun gear  404 . In an exemplary embodiment, output sun gear  404  may further be coupled to or integrally formed with an output shaft  2412  similar to output shaft  412 . In an exemplary embodiment, coaxial pericyclic gear reducer  2000  may further include a driven wheel  2410  similar to driven wheel  410  that may be coupled to and rotatable with middle planet ring gear  2402 . 
     In an exemplary embodiment, driver ring gear  2408  may be coupled to a first side wall  2414  of coaxial pericyclic gear reducer  2000  from one side utilizing a first thrust bearing  2114  and driver ring gear  2408  may further be coupled to middle planet ring gear  2402  form the other opposing side. In an exemplary embodiment, driven wheel  2410  may further be coupled to a second side wall  2416  of coaxial pericyclic gear reducer  2000  utilizing a second thrust bearing  2116 . In an exemplary embodiment, such utilization of thrust bearings ( 2114  and  2116 ) instead of ordinary ball bearings may allow for a more radially compact design of coaxial pericyclic gear reducer  2000 . 
     In an exemplary embodiment, driver ring gear  2408  may be coupled to a first side of middle planet ring gear  2402  utilizing a first prismatic joint and driven wheel  2410  may be coupled to an opposing second side of middle planet ring gear  2402  utilizing a second prismatic joint. In an exemplary embodiment, middle planet ring gear  2402  may include a central annular internal-teeth gear  2010  similar to annular internal-teeth gear  1010  of middle planet ring gear  1402 . 
     In an exemplary embodiment, middle planet ring gear  2402  may further include a pair of parallel lateral grooves ( 2030   a ,  2030   b ) on the first face of middle planet wheel  2402 , and a pair of upper and lower parallel grooves ( 2006   a ,  2006   b ) on the second face of the middle planet wheel  2402  opposite to the first face of the middle planet wheel  2402 . In an exemplary embodiment, pair of parallel lateral grooves ( 2030   a ,  2030   b ) and pair of parallel upper and lower grooves ( 2006   a ,  2006   b ) may either be attached to planetary ring gear  2402  or may be integrally formed with internal-teeth gear  2010 . 
     In an exemplary embodiment, driver ring gear  2408  may include a pair of parallel grooves ( 2002   a ,  2002   b ) that may be formed on a front annular surface  2004  of driver ring gear  2408 . Each groove of pair of grooves ( 2002   a ,  2002   b ) may be aligned with a corresponding groove of pair of parallel upper and lower grooves ( 2006   a ,  2006   b ). In an exemplary embodiment, a first cage  2016  may be disposed between driver ring gear  2408  and middle planet ring gear  2402 . In an exemplary embodiment, each pair of corresponding grooves, for example, grooves  2002   a  and  2006   a  may be configured to allow for a plurality of balls or rollers to be disposed between grooves  2002   a  and  2006   a . Responsive to the plurality of balls or roller being disposed or sandwiched between grooves  2002   a  and  2006   a , a gap may be formed between driver ring gear  2408  and middle planet ring gear  2402 . In an exemplary embodiment, first cage  2016  may have a thickness less than the aforementioned gap between driver ring gear  2408  and middle planet ring gear  2402 , which may allow for sandwiching first cage  2016  between driver ring gear  2408  and middle planet ring gear  2402 . Such configuration of first cage  2016  may allow for first cage to freely slide between driver ring gear  2408  and middle planet ring gear  2402  and all the axial force may be exerted on the balls or rollers within grooves  2002   a  and  2006   a  and not on first cage  2016 . 
     In an exemplary embodiment, first cage  2016  may include two parallel rows of holes ( 2024   a ,  2024   b ) in upper and lower portions of first cage  2016 , where an upper row of holes  2024   a  may be aligned with groove  2002   a  and groove  2006   a , and a lower row of holes  2024   b  may be aligned with groove  2002   b  and groove  2006   b . In an exemplary embodiment, a plurality of balls or rollers may be disposed between groove  2002   a  and groove  2006   a  and encircled by upper row of holes  2024   a , and another plurality of balls or rollers may be disposed between groove  2002   b  and groove  2006   b  and encircled by lower row of holes  2024   b . In an exemplary embodiment, holes  2024   a  and  2024   b  may have a little larger diameter compared to the balls or rollers to allow the balls or rollers to freely role within corresponding holes  2024   a  and  2024   b . In an exemplary embodiment, upper row of holes  2024   a  may be configured to retain and guide the plurality of balls or rollers between groove  2002   a  and groove  2006   a , while lower row of holes  2024   b  may be configured to retain and guide the plurality of balls or rollers between groove  2002   b  and groove  2006   b.    
     In an exemplary embodiment, first cage  2016  may further include a first oblong oval hole  2022  similar to first oblong oval hole  1022  of first cage  1016 . In an exemplary embodiment, first oblong oval hole  2022  may be fitted with a first journal bearing  2216 . In an exemplary embodiment, first journal bearing  2216  may include a central hole that may be rotatably coupled to output shaft  2412 . In an exemplary embodiment, first journal bearing  2216  may be rotatable about a central axis  2428 . In an exemplary embodiment, first journal bearing  2216  may further include two flat surfaces symmetrically disposed on lateral sides of the central hole of first journal bearing  2216 . In an exemplary embodiment, inner edges of first oblong oval hole  2022  may contact respective flat surfaces of first journal bearing  2216  and may have prismatic sliding movements relative to first journal bearing  2216 . Such configuration of first journal bearing  2216  may allow for output shaft  2412  to rotatably pass through first oblong oval hole  2022  without the outer surface of output shaft  2412  being in direct contact with the inner edges of first oblong oval hole  2022 , which may prevent wearing of the inner edges of first oblong oval hole  2022 . 
     In an exemplary embodiment, driven wheel  2410  may include a pair of parallel grooves ( 2026   a ,  2026   b ) formed on a front surface  2028  of driven wheel  2410 . In an exemplary embodiment, each groove of parallel grooves ( 2026   a ,  2026   b ) may be aligned with a corresponding groove of pair of parallel lateral grooves ( 2030   a ,  2030   b ). In an exemplary embodiment, a second cage  2032  may be disposed between middle planet ring gear  2402  and driven wheel  2410 . In an exemplary embodiment, each pair of corresponding grooves, for example, grooves  2026   a  and  2030   a  may be configured to allow for a plurality of balls or rollers to be disposed between grooves  2026   a  and  2030   a . Responsive to the plurality of balls or roller being disposed or sandwiched between grooves  2026   a  and  2030   a , a gap may be formed between front surface  2028  and middle planet ring gear  2402 . In an exemplary embodiment, second cage  2032  may have a thickness less than the aforementioned gap between middle planet ring gear  2402  and driven wheel  2410 , which may allow for sandwiching second cage  2032  between middle planet ring gear  2402  and driven wheel  2410 . Such configuration of second cage  2032  may allow for second cage  2032  to freely slide between middle planet ring gear  2402  and driven wheel  2410  and all the axial force may be exerted on the balls or rollers within grooves  2026   a  and  2030   a  and not on second cage  2032 . 
     In an exemplary embodiment, second cage  2032  may include a first series of lateral holes  2038   a  aligned with groove  2026   a  and groove  2030   a , and a second series of lateral holes  2038   b  aligned with groove  2026   b  and groove  2030   b . In an exemplary embodiment, a plurality of balls or rollers may be disposed between groove  2026   a  and groove  2030   a  and encircled by first series of lateral holes  2038   a . In an exemplary embodiment, a plurality of balls or rollers may further be disposed between groove  2026   b  and groove  2030   b  and encircled by second series of lateral holes  2038   b . In an exemplary embodiment, first series of lateral holes  2038   a  may be configured to retain and guide the plurality of balls or rollers between groove  2026   a  and groove  2030   a , while second series of lateral holes  2038   b  may be configured to retain and guide the plurality of balls or rollers between groove  2026   b  and groove  2030   b . In an exemplary embodiment, second cage  2032  may further include a second oblong oval hole  2040  similar to second oblong oval hole  1040  of second cage  1032 . In an exemplary embodiment, second oblong oval hole  2040  may be fitted with a second journal bearing  2232 . In an exemplary embodiment, second journal bearing  2232  may be configured to allow for output shaft  2412  to rotatably pass through second oblong oval hole  2040 , while facilitating a relative sliding motion between second cage  2032  and output shaft  2412  along second oblong oval hole  2040 . In an exemplary embodiment, second journal bearing  2232  may include a central hole that may be rotatably coupled to output shaft  2412 . In an exemplary embodiment, second journal bearing  2232  may be rotatable about a central axis  2428 . In an exemplary embodiment, second journal bearing  2232  may further include two flat surfaces symmetrically disposed on lateral sides of the central hole of second journal bearing  2232 . In an exemplary embodiment, inner edges of second oblong oval hole  2040  may contact respective flat surfaces of second journal bearing  2232  and may have prismatic sliding movements relative to second journal bearing  2232 . Such configuration of second journal bearing  2232  may allow for output shaft  2412  to rotatably pass through second oblong oval hole  2040  without the outer surface of output shaft  2412  being in direct contact with the inner edges of second oblong oval hole  2040 , which may prevent wearing of the inner edges of second oblong oval hole  2040 . In an exemplary embodiment, first oblong oval hole  2022  may be perpendicular to second oblong oval hole  2040 . 
     In an exemplary embodiment, first journal bearing  2216  and second journal bearing  2232  may be made of brass to reduce the friction while journal bearings ( 2216 ,  2232 ) are rotating on steel output shaft  2412 . Consequently, first journal bearing  2216  and second journal bearing  2232  may lean on output shaft  2412  and the cages  2016  and  2032  may lean on journal bearings  2216  and  2232 . Hence the cages  2016  and  2032  may not be in direct contact with output shaft  2412  and may be worn to much less degree in comparison with two cages  1416  and  1432  due to their larger surface of contact with the journal bearings  2216  and  2232 . 
     In an exemplary embodiment, pair of parallel upper and lower grooves ( 2006   a ,  2006   b ) may be perpendicular to pair of parallel lateral grooves ( 2030   a ,  2030   b ), and parallel grooves ( 2002   a ,  2002   b ) on driver ring gear  2408  may be perpendicular to pair of parallel grooves ( 2026   a ,  2026   b ) on driven wheel  2410 . In an exemplary embodiment, front surface  2004  of driver ring gear  2408  may refer to an annular surface of driver ring gear  2408  facing middle planet ring gear  2402  and a front annular surface  2028  of driven wheel  2410  may refer to an annular surface of driven wheel  2410  facing middle planet ring gear  2402 . In an exemplary embodiment, pair of parallel grooves ( 2002   a ,  2002   b ) on driver ring gear  2408  may extend on a plane parallel with a plane of rotation of driver ring gear  2408 , and pair of parallel grooves ( 2026   a ,  2026   b ) on driven wheel  2410  may extend on a plane parallel with a plane of rotation of driven wheel  2410 . 
       FIG. 12  illustrates a sectional side-view of a three-stage pericyclic gear reducer  3000 , consistent with one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, three-stage pericyclic gear reducer  3000  may include three stages ( 3000   a ,  3000   b , and  3000   c ) that may be structurally similar to pericyclic gear reducer  2000 . In an exemplary embodiment, thrust bearings  3023  and  3022  may be utilized for coupling driver ring gear  3008  to wall  3014  and driven wheel  3010  to wall  3016  respectively to reduce radial dimensions of three-stage pericyclic gear reducer  3000 . In an exemplary embodiment, an input shaft  3006  may transfer the rotational movement of an electromotor to driver ring gear  3008  by utilizing an external-teeth input gear  3007  meshed with driver ring gear  3008 . In an exemplary embodiment, input shaft  3006  may be rotatably coupled to wall  3014  utilizing bearing  3018  and to wall  3017  utilizing bearing  3021   a . In an exemplary embodiment, all walls ( 3014 ,  3016 ,  3017 ,  3017   a ,  3017   b ) may be secured in position parallel with and at predetermined distances from each other utilizing shells that encompasses all walls ( 3014 ,  3016 ,  3017 ,  3017   a ,  3017   b ). For simplicity the aforementioned shells are not illustrated. 
     In an exemplary embodiment, driver ring gear  3008  may be rotatably coupled to wall  3014  utilizing a thrust bearing  3023 . In an exemplary embodiment, driver ring gear  3008  may have an offset equal to e/2 from the central axis of three-stage pericyclic gear reducer  3000 . In an exemplary embodiment, a first stage  3000   a  of three stages ( 3000   a ,  3000   b , and  3000   c ) may include driver ring gear  3008  similar to driver ring gear  2408 , a coupler wheel  3010   a  and a middle planet ring gear  3002   a  similar to middle planet ring gear  2402 . In an exemplary embodiment, a second stage  3000   b  of three stages ( 3000   a ,  3000   b , and  3000   c ) may include middle planet gear  3002   b  and a coupler wheel  3010   b . In an exemplary embodiment, coupler wheel  3010   a  may be configured to function as a driver ring gear similar to driver ring gear  2408  for second stage  3000   b  and coupler wheel  3010   b  may be configured to function as a driven wheel similar to driven wheel  2410  for second stage  3000   b . In an exemplary embodiment, a third stage  3000   c  of three stages ( 3000   a ,  3000   b , and  3000   c ) may include middle planet ring gear  3002   c  and a driven wheel  3010  similar to driven wheel  2410 . In an exemplary embodiment, coupler wheel  3010   b  may be configured to function as a driver ring gear for third stage  3000   c.    
     In an exemplary embodiment, three-stage pericyclic gear reducer  3000  may further include an output shaft  3012 . In an exemplary embodiment, output shaft  3012  may include an elongated output sun gear  3004  that may be coupled to or integrally made with external shaft  3012 . In an exemplary embodiment, output sun gear  3004  may be extended through the central holes of the coupler wheels  3010   a  and  3010   b  and may be meshed with middle planet ring gears ( 3002   a ,  3002   b , and  3002   c ). In an exemplary embodiment, output shaft  3012  may lean on bearing  3020  from a first end of output shaft  3012  and may lean on bearing  3021   b  that may be centrally disposed within input shaft  3006 . In an exemplary embodiment, the offsets of middle planet ring gears  3002   a  and  3002   c  from the central axis may be opposite the offset of middle planet ring gear  3002   b  with respect to the central axis. In an exemplary embodiment, the thickness or the mass of middle planet ring gear  3002   b  may be twice as the thickness and mass of middle planet ring gears  3002   a  and  3002   c . Consequently, the centrifugal force of middle planet ring gear  3002   b  may be twice as the centrifugal forces of middle planet ring gears  3002   a  and  3002   c . In an exemplary embodiment, the centrifugal force of middle planet ring gear  3002   b  may have an opposite direction with respect to the centrifugal forces of middle planet ring gears  3002   a  and  3002   c  and these centrifugal forces may be considered as vectors, that may be exerted along axes perpendicular to central axis of the exemplary tree-stage pericyclic gear reducer. The centrifugal force vector of middle planet ring gear  3002   b  may be in the middle of the centrifugal force vectors of middle planet ring gears  3002   a  and  3002   c  and in opposite direction of centrifugal forces of middle planet ring gears  3002   a  and  3002   c . As a result, three-stage pericyclic gear reducer  3000  may have a complete static and dynamic balance. 
     In an exemplary embodiment, three-stage pericyclic gear reducer  3000  may offer more multiplicity in overall reduction ratios compared to a single-stage coaxial pericyclic gear reducer, such as coaxial pericyclic gear reducer  400 , since there is no need to keep the gear ratio of 2:1 between external-teeth input gear  3007  and driver ring gear  3008  for balancing purposes. Furthermore, selection of teeth size and module of gears in external-teeth input gear  3007  and driver ring gear  3008  may not be as limited as those for a single-stage coaxial pericyclic gear reducer, such as coaxial pericyclic gear reducer  400 . 
     In an exemplary embodiment, all three stages ( 3000   a ,  3000   b , and  3000   c ) may include pairs of cages ( 3032   a ,  3016   a ,  3032   b ,  3016   b ,  3032   c , and  3016   c ) for holding balls or rollers of their respective rolling-contact prismatic joints on either side of respective middle planet ring gears ( 3002   a ,  3002   b , and  3002   c ) structurally similar to pair of cages  2016  and  2032 . In an exemplary embodiment, cages ( 3032   a ,  3016   a ,  3032   b ,  3016   b ,  3032   c , and  3016   c ) may be mounted on journal bearings (JBa, JBb, JBc, JBd, JBe and JBf) structurally similar to journal bearings  2216  and  2232 . (Journal bearing JBb is hidden under the cage  3032   a ). In an exemplary embodiment, journal bearings (JBa, JBb, JBc, JBd, JBe and JBf) may be mounted on output shaft  3012  utilizing annular gears (a, b, c, and d). In an exemplary embodiment, annular gears (a, b, c, and d) may have internal teeth conforming to external teeth of output sun gear  3004  and may be fitted on it like a sheath. Such annular gears (a, b, c, and d) may be moveable on output sun gear  3004  along the main central axis and may be fixed in place by screws or other fastening members. In an exemplary embodiment, journal bearings (JBa, JBb, JBc, JBd, JBe and JBf) may freely rotate about the central axis on outer surface of annular gears (a, b, c, and d). The annular gears (a, b, c, and d) may have shoulders on their outer surface to prevent axial movement of journal bearings (JBa, JBb, JBc, JBd, JBe and JBf). Utilization of such fastened annular gears (a, b, c, and d) on output sun gear  3004  may significantly increase the bending strength of teeth of output sun gear  3004 . 
     In an exemplary embodiment, a multistage coaxial pericyclic gear reducer may have three stages, such as three-stage pericyclic gear reducer  3000  or a two-stage pericyclic gear reducer, such as two-stage pericyclic gear reducer  900 . A pericyclic gear reducer with two stages, such as two-stage pericyclic gear reducer  900  may only be statically balanced, however, a multistage coaxial pericyclic gear reducer with at least three stages may be statically and dynamically balanced. In an exemplary embodiment, a multistage coaxial pericyclic gear reducer may have more than three stages. 
     According to one or more exemplary embodiments, the present disclosure is directed to a multistage pericyclic gear reducer, such as two-stage pericyclic gear reducer  900  or three-stage pericyclic gear reducer  3000 . An exemplary multistage pericyclic gear reducer may include an input shaft that may be rotatable about a longitudinal axis of an exemplary input shaft, an input external-teeth gear that may be coaxially coupled to and rotatable with an exemplary input shaft, a first middle planet ring gear, and a first driver ring gear that may be meshed with an exemplary input external-teeth gear and rotatable with an exemplary input shaft about a first rotational axis coinciding with an exemplary longitudinal axis of an exemplary input shaft. 
     An exemplary first driver ring gear may be coupled to an exemplary first middle planet ring gear from a first side of an exemplary first middle planet ring gear utilizing a first prismatic joint. An exemplary first rotational axis may be parallel with a first central normal axis of an exemplary first middle planet ring gear. An exemplary first prismatic joint may be disposed on a plane perpendicular to an exemplary first rotational axis. An exemplary first driver ring gear may be configured to transfer the rotational movement of an exemplary input shaft to an exemplary first middle planet ring gear. 
     An exemplary multistage pericyclic gear reducer may further include a first driven wheel, where a first side of the first driven wheel may be coupled to an exemplary first middle planet ring gear from a second side of an exemplary first middle planet ring gear utilizing a second prismatic joint. An exemplary second prismatic joint may be disposed on a plane perpendicular to an exemplary first rotational axis. An exemplary second prismatic joint may be perpendicular to an exemplary first prismatic joint. An exemplary first driven wheel may be rotatable about a second rotational axis, where an exemplary second rotational axis may be parallel with an exemplary first rotational axis. 
     An exemplary multistage pericyclic gear reducer may further include a second middle planet ring gear, where a first side of the second middle planet ring gear may be coupled to a second side of an exemplary first driven wheel utilizing a third prismatic joint. An exemplary third prismatic joint may be disposed on a plane perpendicular to an exemplary first rotational axis. An exemplary multistage pericyclic gear reducer may further include a second driven wheel that may be coupled from a first side of an exemplary second driven wheel to a second side of an exemplary second middle planet ring gear utilizing a fourth prismatic joint. An exemplary fourth prismatic joint may be disposed on a plane perpendicular to an exemplary first rotational axis. An exemplary fourth prismatic joint may be perpendicular to an exemplary third prismatic joint. An exemplary second driven wheel may be rotatable about a third rotational axis, where an exemplary third rotational axis may be parallel with an exemplary second rotational axis. 
     An exemplary multistage pericyclic gear reducer may further include an output external-teeth gear that may extend through and mesh with both an exemplary first middle planet ring gear and an exemplary second middle planet ring gear, and a central output shaft that may be coupled to an exemplary output external-teeth gear. An exemplary central output shaft may be rotatable with an exemplary output external-teeth gear about a central axis. An exemplary central axis may be along and parallel with a longitudinal axis of an exemplary central output shaft, where an exemplary output external-teeth gear may be extended along an exemplary central axis. 
     An exemplary multistage pericyclic gear reducer may further include a third middle planet ring gear, where a first side of the third middle planet ring gear may be coupled to a second side of an exemplary second driven wheel utilizing a fifth prismatic joint. An exemplary fifth prismatic joint may be disposed on a plane perpendicular to an exemplary first rotational axis. An exemplary multistage pericyclic gear reducer may further include a third driven wheel that may be coupled to a second side of an exemplary third middle planet ring gear utilizing a sixth prismatic joint. An exemplary sixth prismatic joint may be disposed on a plane perpendicular to an exemplary first rotational axis. An exemplary sixth prismatic joint may be perpendicular to an exemplary fifth prismatic joint. An exemplary third driven wheel may be rotatable about a fourth rotational axis, where an exemplary fourth rotational axis may be parallel with an exemplary third rotational axis. An exemplary output external-teeth gear may further extend through and mesh with an exemplary third middle planet ring gear. 
     An exemplary first side of the first middle planet gear may be opposite an exemplary second side of an exemplary first middle planet gear along an exemplary central axis. An exemplary first side of an exemplary second middle planet gear may be opposite an exemplary second side of an exemplary second middle planet gear along an exemplary central axis. An exemplary first side of an exemplary third middle planet gear may be opposite the second side of an exemplary third middle planet gear along an exemplary central axis. 
     An exemplary first side of an exemplary first driven wheel may be opposite an exemplary second side of an exemplary first driven along an exemplary central axis. An exemplary first side of an exemplary second driven wheel may be opposite an exemplary second side of an exemplary second driven wheel along an exemplary central axis. 
     An exemplary first rotational axis, an exemplary second rotational axis, and an exemplary central axis may be parallel with each other and laid on a single plane. An exemplary first rotational axis and an exemplary second rotational axis may be misaligned and are symmetrically disposed on either side of an exemplary central axis with equal respective distances from an exemplary central axis. 
     An exemplary second rotational axis, an exemplary third rotational axis, and an exemplary central axis may be parallel with each other and laid on a single plane. An exemplary second rotational axis and an exemplary third rotational axis may be misaligned and symmetrically disposed on either side of an exemplary central axis with equal respective distances from an exemplary central axis. 
     An exemplary third rotational axis, an exemplary fourth rotational axis, and an exemplary central axis may be parallel with each other and laid on a single plane. An exemplary third rotational axis and an exemplary fourth rotational axis may be misaligned and symmetrically disposed on either side of an exemplary central axis with equal respective distances from an exemplary central axis. 
     An exemplary first prismatic joint may include a first tongue and a first groove, where one of an exemplary first tongue and an exemplary first groove may be formed on an exemplary first side of an exemplary middle planet ring gear and the other corresponding one of an exemplary first tongue and an exemplary first groove may be formed on an exemplary first driver ring gear. An exemplary first tongue may be engaged with and slidably moveable within an exemplary first groove along a first direction. 
     An exemplary second prismatic joint may include a second tongue and a second groove, where one of an exemplary second tongue and an exemplary second groove formed on an exemplary second side of an exemplary first middle planet ring gear and the other corresponding one of an exemplary second tongue and an exemplary second groove formed on an exemplary first side of an exemplary first driven wheel. An exemplary second tongue may be engaged with and slidably moveable within an exemplary second groove along a second direction, where an exemplary second direction may be perpendicular to an exemplary first direction. 
     An exemplary third prismatic joint may include a third tongue and a third groove, where one of an exemplary third tongue and an exemplary third groove may be formed on the first side of an exemplary second middle planet ring gear and the other corresponding one of an exemplary third tongue and an exemplary third groove may be formed on the second side of an exemplary first driven wheel. An exemplary third tongue may be engaged with and slidably moveable within an exemplary third groove along a third direction. 
     An exemplary fourth prismatic joint may include a fourth tongue and a fourth groove, where one of an exemplary fourth tongue and an exemplary fourth groove may be formed on an exemplary second side of an exemplary second middle planet ring gear and the other corresponding one of an exemplary fourth tongue and an exemplary fourth groove may be formed on an exemplary first side of an exemplary second driven wheel. An exemplary fourth tongue may be engaged with and slidably moveable within an exemplary fourth groove along a fourth direction, where an exemplary fourth direction may be perpendicular to an exemplary third direction. 
     An exemplary fifth prismatic joint may include a fifth tongue and a fifth groove, where one of an exemplary fifth tongue and an exemplary fifth groove may be formed on an exemplary first side of an exemplary third middle planet ring gear and the other corresponding one of an exemplary fifth tongue and an exemplary fifth groove may be formed on an exemplary second side of an exemplary second driven wheel. An exemplary fifth tongue may be engaged with and slidably moveable within an exemplary fifth groove along a fifth direction. 
     An exemplary sixth prismatic joint may include a sixth tongue and a sixth groove, where one of an exemplary sixth tongue and an exemplary sixth groove may be formed on an exemplary second side of an exemplary third middle planet ring gear and the other corresponding one of an exemplary sixth tongue and an exemplary sixth groove formed on an exemplary third driven wheel. An exemplary sixth tongue may be engaged with and slidably moveable within an exemplary sixth groove along a sixth direction, where an exemplary sixth direction may be perpendicular to an exemplary fifth direction. 
       FIG. 13A  illustrates a dual-speed pericyclic gearbox  4000 , consistent with one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, dual-speed pericyclic gearbox  4000  may include a first stage  4000   a  that may be similar to coaxial pericyclic gear reducer  400  coupled to a second stage  4000   b  that may be similar to pericyclic gear reducer  100 . In an exemplary embodiment, a driver ring gear  4008  of first stage  4000   a  may be structurally similar to driver ring gear  408  and may be rotatably coupled to a wall  4016  of first stage  4000   a  by utilizing a thrust bearing  4022 . In an exemplary embodiment, such coupling of driver ring gear  4008  to wall  4016  utilizing thrust bearing  4022  may allow for reducing radial size of dual-speed pericyclic gearbox  4000  in contrast with when a driver ring gear such as driver ring gear  408  may be coupled to a wall, such as sidewall  414  utilizing a radial rolling contact bearing, such as fourth rolling contact bearing  423 . 
     In an exemplary embodiment, a middle planet ring gear  4002   a  of first stage  4000   a  and a middle planet ring gear  4002   b  of second stage  4000   b  may have different teeth numbers and modules. In an exemplary embodiment, middle planet ring gear  4002   a  may be coupled to a corresponding output sun gear  4004   a  and middle planet ring gear  4002   b  may be coupled to a corresponding output sun gear  4004   b . In an exemplary embodiment, dual-speed pericyclic gearbox  4000  may further include an output shaft  4012  that may be configured as a spline, on which output sun gears ( 4004   a ,  4004   b ) may be mounted and may be slidably moveable along a longitudinal axis of output shaft  4012 . 
       FIG. 13B  illustrates front views of output sun gears ( 4004   a ,  4004   b ) of dual-speed pericyclic gearbox  4000 , consistent with one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, output sun gear  4004   a  may include a spline hole  4400   a  that may include a plurality of teeth  4402   a  that may conform to teeth of output shaft  4012 . In an exemplary embodiment, number of plurality of teeth  4402   a  may be less than the number of teeth or grooves on output shaft  4012 . In an exemplary embodiment, plurality of teeth  4402   a  may be meshed or interlocked with the teeth of output shaft  4012 , such that output sun gear  4004   a  may be rotatable with output shaft  4012  and thereby may transfer the torque to output shaft  4012 . In an exemplary embodiment, output sun gear  4004   b  may include a spline hole  4400   b  that may include a plurality of teeth  4402   b  that may conform to teeth of output shaft  4012 . In an exemplary embodiment, number of plurality of teeth  4402   b  may be less than the number of teeth or grooves on output shaft  4012 . In an exemplary embodiment, plurality of teeth  4402   b  may be meshed or interlocked with the teeth of output shaft  4012 , such that output sun gear  4004   b  may be rotatable with output shaft  4012  and thereby may transfer the torque to output shaft  4012 . 
     In an exemplary embodiment, output shaft  4012  may lean on rolling contact bearing  4020  form one end and may lean on rolling contact bearing  4021   b  from the other opposing end. In an exemplary embodiment, rolling contact bearing  4021   b  may be coaxially disposed within an input shaft  4006  of dual-speed pericyclic gearbox  4000 . 
     In an exemplary embodiment, dual-speed pericyclic gearbox  4000  may further include shift forks (M 1  and M 2 ). In an exemplary embodiment, shift fork M 1  may include an elongated portion that may be configured as an elongated cylindrical sector M 12  disposable within one of spline grooves on output shaft  4012  and a protruded portion M 14  that may radially protrude from elongated cylindrical sector M 12  and a yoke M 13  on its left hand side radially protrude from elongated cylindrical sector. In an exemplary embodiment, yoke M 13  may be configured to selectively engage an internal portion  4404   a  of output sun gear  4004   a  and to selectively move output sun gear  4004   a  along longitudinal axis of output shaft  4012 . In an exemplary embodiment, shift fork M 2  may include an elongated portion that may be configured as an elongated cylindrical sector M 22  disposable within one of spline grooves on output shaft  4012  and a protruded portion M 24  that may radially protrude from elongated cylindrical sector M 22  and a yoke M 23  on its left hand side radially protrude from elongated cylindrical sector. In an exemplary embodiment, yoke M 23  may be configured to selectively engage an internal portion  4404   b  of output sun gear  4004   b  and to selectively move output sun gear  4004   b  along longitudinal axis of output shaft  4012 . In an exemplary embodiment, shift forks (M 1  and M 2 ) may be configured such that at any given instant, only one of output sun gears ( 4004   a ,  4004   b ) may be engaged with a respective middle planet ring gear of middle planet ring gears ( 4002   a ,  4002   b ). Consequently, for each output sun gear of output sun gears ( 4004   a ,  4004   b ) engaged with each respective middle planet ring gear of middle planet ring gears ( 4002   a ,  4002   b ), a different conversion ratio may be obtained from dual-speed pericyclic gearbox  4000 . 
     In an exemplary embodiment, rolling contact bearing  4020  may either be directly coupled with output shaft  4012  or rolling contact bearing  4020  may be mounted on an outer surface of a sheath N. In an exemplary embodiment, sheath N may be coaxially engaged and rotatable with output shaft  4012 . In an exemplary embodiment, sheath N may be configured to allow shift forks (M 1  and M 2 ) to pass through sheath N. In an exemplary embodiment, shift forks (M 1  and M 2 ) may pass through sheath N and may be coupled to respective output sun gears ( 4004   a ,  4004   b ) by utilizing end yokes (M 13  and M 23 ). In an exemplary embodiment, shift forks (M 1  and M 2 ) may be axially slidable within respective spline grooves of output shaft  4012  and thereby move their corresponding output sun gear of output sun gears ( 4004   a ,  4004   b ) along output shaft  4012 . In an exemplary embodiment, each shift fork of shift forks (M 1  and M 2 ) may be configured to engage or disengage each output sun gear of respective output sun gears ( 4004   a ,  4004   b ) with a corresponding middle planet ring gear of middle planet ring gears ( 4002   a ,  4002   b ) by moving each output sun gear of respective output sun gears ( 4004   a ,  4004   b ) along output shaft  4012 . In an exemplary embodiment, responsive to each one of output sun gears ( 4004   a ,  4004   b ) being engage with a corresponding one of middle planet ring gears ( 4002   a ,  4002   b ), the other output sun gear of output sun gears ( 4004   a ,  4004   b ) may be disengaged from the other corresponding one of middle planet ring gears ( 4002   a ,  4002   b ). For example, when output sun gear  4004   a  needs to be engaged with middle planet ring gear  4004   a  to obtain a first gear ratio, shift fork M 1  may be utilized to move output sun gear  4004   a  into a position where output sun gear  4004   a  may mesh with middle planet ring gear  4002   a , and shift fork M 2  may be utilized to move output sun gear  4004   b  to a position where output sun gear  4004   b  may be disengaged from middle planet ring gear  4002   b . similarly, when output sun gear  4004   b  needs to be engaged with middle planet ring gear  4004   b  to obtain a second gear ratio, shift fork M 2  may be utilized to move output sun gear  4004   b  into a position where output sun gear  4004   b  may mesh with middle planet ring gear  4002   b , and shift fork M 1  may be utilized to move output sun gear  4004   a  to a position where output sun gear  4004   a  may be disengaged from middle planet ring gear  4002   a.    
     In an exemplary embodiment, to prevent the risk of both output sun gears ( 4004   a ,  4004   b ) being simultaneously engaged with corresponding middle planet ring gears ( 4002   a ,  4002   b ), an index cylinder Z may be mounted on the spline of output shaft  4012  to one side of sheath N and bearing  4020 . In an exemplary embodiment, index cylinder Z may not have any internal teeth and consequently, index cylinder Z may be rotatable about the longitudinal axis of output shaft  4012  relative to output shaft  4012 . In an exemplary embodiment, index cylinder Z may include a slit which is a section cut that may be configured to conform with protruded portions M 14  and M 24 . In an exemplary embodiment, the slit on index cylinder Z may be configured to allow only one of protruded portions M 14  or M 24  to longitudinally move inside the slit and hence only one of shift forks M 1  and M 2  may move along output shaft  4012  and engage corresponding output sun gear to corresponding middle planet gear. 
     In an exemplary embodiment, a pericyclic gearbox may be configured as a multi-speed pericyclic gear box that may have several output sun gears engageable with several corresponding middle planet gears to provide multiple gear ratios. In an exemplary embodiment, dual-speed pericyclic gearbox  4000  is just an example of a multi-speed pericyclic gearbox and as mentioned before, an exemplary multi-speed pericyclic gearbox may include more than two stages. 
     The embodiments have been described above with the aid of functional building blocks illustrating the implementation of specified functions and relationships thereof. The boundaries of these functional building blocks have been arbitrarily defined herein for the convenience of the description. Alternate boundaries can be defined so long as the specified functions and relationships thereof are appropriately performed. 
     The foregoing description of the specific embodiments will so fully reveal the general nature of the disclosure that others can, by applying knowledge within the skill of the art, readily modify and/or adapt for various applications such specific embodiments, without undue experimentation, without departing from the general concept of the present disclosure. Therefore, such adaptations and modifications are intended to be within the meaning and range of equivalents of the disclosed embodiments, based on the teaching and guidance presented herein. It is to be understood that the phraseology or terminology herein is for description and not of limitation, such that the terminology or phraseology of the present specification is to be interpreted by the skilled artisan in light of the teachings and guidance. 
     The breadth and scope of the present disclosure should not be limited by any of the above-described exemplary embodiments but should be defined only in accordance with the following claims and their equivalents. 
     Throughout this specification and the claims which follow, unless the context requires otherwise, the word “comprise”, and variations such as “comprises” or “comprising”, will be understood to imply the inclusion of a stated integer or step or group of integers or steps but not to the exclusion of any other integer or step or group of integers or steps. 
     Moreover, the word “substantially” when used with an adjective or adverb is intended to enhance the scope of the particular characteristic, e.g., substantially planar is intended to mean planar, nearly planar and/or exhibiting characteristics associated with a planar element. Further use of relative terms such as “vertical”, “horizontal”, “up”, “down”, and “side-to-side” are used in a relative sense to the normal orientation of the apparatus.