Patent Publication Number: US-8979286-B2

Title: Spherical mechanical linkage and multi-axis trackers

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation-in-part of U.S. patent application Ser. No. 14/034,267, entitled “SPHERICAL MECHANICAL LINKAGE AND MULTI-AXIS TRACKERS,” and filed Sep. 23, 2013, which in turn is a continuation of U.S. application Ser. No. 13/536,932, now U.S. Pat. No. 8,540,382, entitled “SPHERICAL MECHANICAL LINKAGE AND MULTI-AXIS TRACKERS” and filed Jun. 28, 2012, the disclosures of which are incorporated herein by reference. 
    
    
     BACKGROUND 
     Mechanical linkages include components coupled to one another that may transmit movement or force in one or more directions. Various mechanical linkages may convert linear motion to linear motion, rotational motion to rotational motion, rotational motion to linear motion (and its reverse), linear (or rotational) motion to oscillatory motion or sliding motion, and so on. Mechanical linkages have many uses, such as in mechanical systems, automotive systems, aerospace systems, robotics, prosthetics, biomedical devices, solar tracking, photography, cinematography, and others. 
     Spherical mechanical linkages are a type of mechanical linkage wherein the components have axes of movement that intersect at the center of a sphere. Current spherical mechanical linkages have a number of drawbacks. One drawback of current spherical mechanical linkages is that the components may interfere with the placement of long, wide, or otherwise irregularly-shaped payloads. Examples of such payloads include tubes, telescopes, guns, or other long or wide devices. Another drawback of current spherical mechanical linkages is that without counterweights, they may become unbalanced if holding a payload, even a compact payload. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The foregoing aspects and many of the attendant advantages will become more readily appreciated as the same become better understood by reference to the following detailed description, when taken in conjunction with the accompanying drawings, wherein: 
         FIG. 1  depicts an exploded view of several components that may be included in various embodiments of spherical mechanical linkages. 
         FIG. 2  depicts an environmental view of an embodiment of a spherical mechanical linkage. 
         FIGS. 3A and 3B  depict embodiments of bearings that may be included in various embodiments of spherical mechanical linkages. 
         FIG. 4  depicts an environmental view of an embodiment of a multi-axis tracking system that incorporates a spherical mechanical linkage. 
         FIG. 5  depicts an environmental view of an embodiment of a multi-axis tracking system that incorporates a spherical mechanical linkage and a paraboloidal mirror that may be used as a heliostat. 
         FIG. 6  depicts an isometric schematic view of a gearbox that may be incorporated in an embodiment of a multi-axis tracking system. 
         FIG. 7A  depicts an isometric schematic view of an embodiment of an equation-of-time correction mechanism that may be incorporated in an embodiment of a multi-axis tracking system. 
         FIG. 7B  depicts an isometric schematic view of an embodiment of an equation-of-time correction mechanism that may be incorporated in an embodiment of a multi-axis tracking system. 
         FIG. 7C  depicts an isometric schematic view of an embodiment of an equation-of-time correction mechanism that may be incorporated in an embodiment of a multi-axis tracking system. 
         FIG. 8  depicts an environmental view of a multi-axis tracking system that incorporates a spherical mechanical linkage and a square paraboloidal mirror. 
         FIG. 9  depicts an environmental view of a multi-axis tracking system that incorporates a spherical mechanical linkage and a plane mirror. 
         FIG. 10  depicts an environmental view of an embodiment of a multi-axis tracking system that incorporates a spherical mechanical linkage and is mounted on a boom. 
         FIG. 11  depicts a cut-away of an embodiment of a multi-axis tracking system that incorporates a spherical mechanical linkage housed within a drive shaft. 
     
    
    
     DETAILED DESCRIPTION 
     Illustrative Spherical Mechanical Linkage 
       FIG. 1  displays an exploded view of embodiments of the components of an embodiment of a spherical mechanical linkage. This embodiment includes a yoke  102 , a crank  104 , a deflecting member  106 , and a rocking frame  108 . Other embodiments of spherical mechanical linkages may include variations on the yoke  102 , the crank  104 , the deflecting member  106 , or the rocking frame  108 . A number of these component variations are discussed below. 
     In various embodiments, the yoke  102  is a complete annulus or ring, or a section of an annulus or ring. Other geometric configurations for the yoke  102  may be employed as desired. For example, the yoke  102  may include one or more linear segments, one or more curved segments, or both linear and curved segments. The yoke  102  may also include a hollow cylindrical section or hollow polygonal section. The yoke  102  may also include opposing yoke bearing points  112  as also illustrated in  FIG. 1 . In some embodiments, the yoke bearing points  112  include bearings. 
     In various embodiments, the crank  104  may include one or more linear segments, one or more curved segments, or both linear and curved segments. In some embodiments, the crank  104  has symmetry about an axis. This symmetry may be bilateral or rotational symmetry. Other geometric configurations for the crank  104  may be employed as desired. For example, the crank  104  may include a conic section, such as a conical frustum. The crank  104  may also include a crank bearing point  114  as also illustrated in  FIG. 1 . 
     In various embodiments, the deflecting member  106  is a complete annulus or ring, or a section of an annulus or ring. Other geometric configurations for the deflecting member  106  may be employed as desired. For example, the deflecting member  106  may include one or more linear segments, one or more curved segments, or both linear and curved segments. The deflecting member  106  may have a polygonal shape. The cross-section of the deflecting member  106  may be varied as well. For example, the cross-section may be a circle, square, triangle, cruciform, or other shape. The deflecting member  106  may also include a stub shaft  116  which may be rotationally coupled to the crank  104  at the crank bearing point  114  as is illustrated in  FIGS. 1 and 2 . 
     In various embodiments, the rocking frame  108  is a complete annulus or ring, or a section of an annulus or ring. Other geometric configurations for the rocking frame  108  may be employed as desired. For example, the rocking frame  108  may include one or more linear segments, one or more curved segments, or both linear and curved segments. The rocking frame may have a polygonal shape. Still other geometric shapes are possible for the rocking frame. Additionally, the rocking frame  108  and the deflecting member  106  may have the same shape, or different shapes. The rocking frame  108  may include several bearing points  110  as depicted in  FIG. 1  so that the rocking frame  108  may be slideably coupled to the deflecting member  106  as depicted in  FIG. 2 . It will be appreciated that in other embodiments, any number of bearing points  110  may be provided on the rocking frame  108  to enable slideable coupling with the deflecting member  106 . In another embodiment, and as further depicted in  FIG. 1 , the rocking frame  108  may also include stub shafts  118  so that the rocking frame  108  may be rotatably coupled to the yoke  102  at yoke bearing points  112 , as also shown in  FIG. 2 . In still another embodiment, the rocking frame  108  may include one or more hollow segments to permit the deflecting member  106  to pass therethrough. 
       FIG. 2  depicts an embodiment of an assembled spherical mechanical linkage  100  to explain the relationships between the various spherical mechanical linkage components as described above. The yoke  102 , here an embodiment with a semi-annular shape, defines a first axis  1  and a second axis  2 . More specifically, in one embodiment, the second axis  2  is defined by an imaginary line passing through the opposing yoke bearing points  112 , while the first axis  1  is perpendicular to the second axis  2  and located in the same plane as the yoke  102 . The yoke  102  and the crank  104  are rotatably coupled about the first axis  1 . The rocking frame  108  is rotatably coupled via stub shafts  118  to the yoke  102  at yoke bearing points  112  so that the rocking frame  108  may rotate or oscillate about the second axis  2 . The crank  104  is connected to the deflecting member  106  about a third axis  3  via the stub shaft  116  at crank bearing point  114 . The third axis  3  is deflected from the first axis  1  at an angle Δ. The angle Δ may be selected by varying the position of stub shaft  116  and crank bearing point  114 . In some embodiments, the angle Δ is between about 0° and 90°. In some embodiments, the angle Δ is between about 23° and 24°. In other embodiments, the angle Δ is between about 23.4° and 23.5°. The deflecting member  106  is in turn slideably coupled to rocking frame  108  along the bearing points  110 . The deflecting member  106  and the rocking frame  108  are positioned such that they occupy the same imaginary plane  5 . The first axis  1 , the second axis  2 , and the third axis  3  intersect at intersection point  4  in the imaginary plane  5 . A payload axis  6  is defined by a vector normal to the imaginary plane  5  at the intersection point  4 . The payload axis  6  is deflected at an angle β from a plane that is normal to the first axis  1  that also passes through second axis  2 . 
     As discussed above, many variations of the components of the spherical mechanical linkage  100  are possible. It should be appreciated that in the embodiment illustrated in  FIG. 2 , the embodiments of the yoke  102 , the crank  104 , the deflecting member  106 , and the rocking frame  108  included in the embodiment of the spherical mechanical linkage  100  are all substantially symmetric with respect to the first axis  1 . This embodiment is advantageous in that the spherical mechanical linkage  100  is balanced with respect to the first axis  1 . Accordingly, a payload mounted to the spherical linkage will not unbalance the spherical mechanical linkage  100  if the payload&#39;s center of mass is located along the first axis  1 . It should also be appreciated that in this embodiment, the deflecting member  106  and the rocking frame  108  advantageously define a relatively open space, permitting the placement of long or otherwise irregularly shaped payloads therethrough. 
     The entire spherical mechanical linkage  100  may be rotated about the first axis  1  by rotating the yoke  102  about the first axis  1 . The crank  104  may also be rotated about the first axis  1  independently of the rotation of the yoke  102 . As the crank  104  rotates about the first axis  1 , the crank  104  displaces the deflecting member  106 . As the deflecting member  106  is displaced by the crank  104 , it drags the rocking frame  108  with it, causing the rocking frame  108  to rock back and forth about the second axis  2 . Simple trigonometry shows the relationship between a rotation angle α of the crank  104 , the deflection angle β of the payload axis  6 , and the fixed angle Δ of the third axis  3  with respect to the first axis  1 :
 
Tan β=Tan Δ·Sin α
 
Thus, the imaginary plane  5  and the payload axis  6  may be positioned in any desired direction by rotating the yoke  102  and/or the crank  104 .
 
       FIGS. 3A and 3B  depict one embodiment of the bearing points  110  that may be included on the rocking frame  108 . As discussed above with reference to  FIGS. 1  and  2 , the deflecting member  106  and the rocking frame  108  are slideably coupled to one another along the bearing points  110 . The shape of each of the bearing points  110  may be varied as desired, such as to be compatible with the cross-sectional shape of the deflecting member  106 . For example, in  FIG. 3A , the cross-sectional shape of the deflecting member  106  is a square, and in  FIG. 3B , the cross-sectional shape of the deflecting member is a circle. Thus, in  FIG. 3A , the shape of the depicted bearing point  110  is compatible with the square cross-sectional shape of the deflecting member  106 , while the shape of the bearing points  110  shown in  FIG. 3B  is circular so as to be compatible with the circular cross-sectional shape of the deflecting member  106 . 
     Each of the bearing points  110  may contain one or more bearings  109  that contact one or more surfaces of the deflecting member  106  as the deflecting member slides back and forth in the rocking frame  108 . In some embodiments, a bearing point  110  may include enough bearings  109  so that each surface of the deflecting member  106  is contacted by a bearing  109 . For example, for a deflecting member  106  with a triangle cross section, three bearings  109  may be used, one for each cross-section surface of the deflecting member  106 . In some embodiments, a bearing point  110  may have a number of bearings  109  such that fewer than all surfaces of the deflecting member  106  are contacted by a bearing  109 . For instance, in  FIG. 3A , a deflecting member  106  with a four-sided rectangular cross-section is contacted on three sides by bearings  109 . 
     In some embodiments, the bearings  109  include cylindrical or roller bearing elements. In other embodiments, the bearings  109  include ball bearings. In addition, each of the bearing points  110  on the rocking frame  108  may have the same number of bearings  109 , or one or more bearing points  110  may have a different number of bearings  109 . 
     Illustrative Basic Multi-Axis Tracking System 
       FIG. 4  depicts an embodiment of a multi-axis tracking system  200  that includes the embodiment of a spherical mechanical linkage  100  shown in  FIG. 2  mounted on a base  207 . The spherical mechanical linkage  100  may be mounted or affixed to the base  207  at any desired angle. The yoke  102  may be mechanically coupled to a yoke drive shaft  202 . The crank  104  may be mechanically coupled to a crank drive shaft  204 . The yoke drive shaft  202  and crank drive shaft  204  may be concentric with each other about first axis  1 , with the crank drive shaft  204  located inside the yoke drive shaft  202 . The yoke drive shaft  202  and the crank drive shaft  204  may be coupled through a base  207  by means of a base bearing  203 . It will be appreciated that in some embodiments, the yoke drive shaft  202  may include one integral component that rotates at one rate, or may include multiple segments which may rotate at the same or different rates. 
     The base  207  may house one or more means for driving the yoke  102 , the crank  104 , or both about an axis, such as first axis  1 . In the embodiment shown, a yoke motor  208 A drives the yoke  102  and a crank motor  208 B drives the crank  104 . The rotational motion generated by the motors  208 A and  208 B through their respective motor shafts  209 A and  209 B may be transmitted by means of belts  211 A and  211 B. The belt  211 A transmits the rotational motion of the yoke motor shaft  209 A to the yoke drive shaft  202  and to the yoke  102 . The belt  211 B transmits the rotational motion of the crank motor shaft  209 B to the crank drive shaft  204  and to the crank  104 . In further embodiments, the rotational motion produced by one or more motors is transmitted through one or more gear trains coupled to one or more drive shafts. It should be appreciated that other structures for producing and transmitting motion are possible. For example, the rotational motion produced by one or more motors may be transmitted by means of tracks, sprockets, chains, or even other mechanical linkages. In other embodiments, the structures for producing and transmitting motion are housed partially or entirely outside the base  207 . 
     The components of the multi-axis tracking system  200 , as with the components of the spherical mechanical linkage  100 , may have many variations. For example, one motor may drive the yoke drive shaft  202  and another motor may drive the crank drive shaft  204 , as shown in  FIG. 4 . In other embodiments, multiple motors may separately drive yoke drive shaft  202  and crank drive shaft  204 . These motors may be employed in a vertical cylinder formation, wherein one or more motors are hollow to permit the passage of yoke drive shaft  202  and/or crank drive shaft  204  therethrough. 
     Those skilled in the art will appreciate that motor  208  may drive multiple multi-axis tracking systems, for example by driving multiple yoke drive shafts  202  and/or multiple crank drive shafts  204 . It should also be appreciated that in some embodiments, motors are not used. The yoke drive shaft  202  and the crank drive shaft  204  may be driven by any structure capable of producing rotational motion, such as a water wheel or manual power. 
     Although the multi-axis tracking system discussed above incorporates a concentric yoke drive shaft  202  and crank drive shaft  204 , it should be appreciated that yoke drive shaft  202  and crank drive shaft  204  need not be concentric. For example, the yoke drive shaft  202  may be positioned along the first axis  1  at the opposite end of the yoke  102  from crank  104  and crank drive shaft  204 . Moreover, in embodiments where the yoke drive shaft  202  and the crank drive shaft  204  are concentric, it should be appreciated that the yoke drive shaft  202  may pass through the crank drive shaft  204 , or the crank drive shaft  204  may pass through the yoke drive shaft  202 . Still other configurations may be used. 
     A payload may optionally be mounted on a component of the spherical mechanical linkage. In various embodiments, the payload is mounted on the rocking frame  108 , such that the payload axis  6  and thus the payload itself may be pointed in a desired direction by rotating one or more of the linkage members about an axis. As discussed above, it should be appreciated that because the central area in the rocking frame  108  is relatively open, payloads such as parabolic mirrors, flat mirrors, solar cell panels, cameras, spotlights, radar antennas, telescopes, photodetectors, firearms, or any other suitable object can be easily mounted. 
     Basic multi-axis tracking systems, such as the embodiment of the multi-axis tracking system  200  shown in  FIG. 4 , may prompt many variations with many applications. Several of these variations and their applications are discussed below, though it should be appreciated that further variations are possible and that the described examples are not exhaustive. Components that are new in these variations will be illustrated by new reference numbers. Only new or modified components will be discussed in greater detail as they relate to each example variation. 
     Illustrative Variations: Solar Trackers 
     Various embodiments of multi-axis tracking systems may be used as solar trackers, such as heliostats and coelostats. These embodiments have many applications, such as residential and industrial illumination, spectacular light show displays, intense or concentrated lighting or illumination for solar power generation, and use as a scientific tool in the general field of solar experimentation. For solar power generation, “solar farms” may be constructed containing large arrays of heliostats. In solar tracker embodiments, an embodiment of a spherical mechanical linkage  100 , such as that shown in  FIG. 2 , may be mounted and the yoke  102  and the crank  104  may be driven such that the payload axis  6  follows the position of the sun in the sky. 
       FIG. 5  depicts an embodiment of a heliostat  300 . The heliostat  300  includes an embodiment of a spherical mechanical linkage  100  mounted on a base  207 . This particular embodiment of a spherical mechanical linkage  100  includes a yoke  102  including several linear sections, a symmetric crank  104 , and a partially annular deflecting member  106  and rocking frame  108 . The crank  104  may be coupled to the deflecting member  106  such that the third axis  3  intersects with the second axis  2  at an angle Δ substantially equal to a planet&#39;s angle of obliquity. A planet&#39;s angle of obliquity, also called axial tilt, is the angle a planet&#39;s axis of rotation makes with respect to a line perpendicular to its orbital plane, as determined by the right-hand rule. For example, Earth&#39;s angle of obliquity is currently approximately 23.45°. It should be appreciated that other variations on the components of the spherical mechanical linkage  100 , discussed above with respect to  FIG. 1 , may be used with this heliostat  300  as well as with other heliostat embodiments. 
     As discussed above, the spherical mechanical linkage  100  may be mounted on the base  207  such that the first axis  1  is substantially parallel to a planet&#39;s axis of rotation. Accordingly, in this embodiment, the base  207  has been modified such that when the spherical mechanical linkage is mounted on the base  207 , the first axis  1  is substantially parallel to the Earth&#39;s axis of rotation. Thus the crank  102  and the yoke  104  may be driven such that the payload axis  6  tracks the sun&#39;s position in the sky. The sun&#39;s position in the sky to an observer on a planet may vary based on the observer&#39;s location on the planet, the time of day, the time of year, the planet&#39;s axial tilt, and the equation of time anomaly. 
     One component of the sun&#39;s observed position in the sky is the solar hour angle, or right ascension. On Earth, for example, the sun&#39;s solar hour angle varies periodically with a period of approximately 24 hours, or one mean solar day. Accordingly, in one embodiment of the heliostat  300 , the spherical mechanical linkage  100  is mounted on the base such that the first axis  1  is substantially parallel to a planet&#39;s axis of rotation. The movement of the yoke  102  may be used to account for this component of the sun&#39;s position in the sky. For example, the yoke  102  may be driven at a rate of about one revolution (360 degrees) about the first axis  1  per mean solar day, or about 365.2422 revolutions per year. 
     The sun&#39;s seasonal declination is another component of the sun&#39;s observed position in the sky. The sun&#39;s declination is the angle between the rays of the sun and the plane of a planet&#39;s equator. The sun&#39;s declination varies throughout the year. This variation is periodic and approximately sinusoidal. On Earth, the amplitude of this variation is about 23.45° and the period of this variation is about one sidereal year. The construction and movement of the crank  104  may be used to account for this component of the sun&#39;s position in the sky. For example, the angle Δ of the deflecting member shaft  116  with respect to the first axis  1  may be approximately 23.45°. The crank  104  may be driven at a rate of about one revolution (360 degrees) about the first axis  1  per mean sidereal day, or about 366.2422 revolutions per year. 
     Combining the rotation of the yoke  102  and the crank  104  provides one way to track both the hour angle and seasonal declination components of the sun&#39;s position in the sky. As the yoke  102  rotates about the first axis  1 , it takes the crank  104 , the deflecting member  106 , and the rocking frame  108  with it. Thus, the payload axis  6  rotates about the first axis  1 , following the sun as the sun&#39;s hour angle changes throughout the day. The crank  104  may be driven about the first axis  1  by a motor coupled to a crank drive shaft  204 . As the crank  104  rotates about the first axis  1 , it causes the deflecting member  106  and rocking frame  108  to oscillate about the second axis  2  by about ±23.45° at the rate of one cycle per year. Thus, the payload axis  6  oscillates about the second axis  2 , following the sun as the sun&#39;s declination changes throughout the year. 
       FIG. 6  depicts an embodiment of a gearbox that may be advantageously used with the heliostat  300  illustrated in  FIG. 5  or with any other solar tracker embodiment. This gearbox includes three gear trains: one gear train for increasing torque, one gear train for revolving the yoke  102  and the crank  104  at their respective rates, and an equation-of-time correction mechanism  250  for advancing or retarding the motion of the yoke  102 . In reference to the heliostat  300  illustrated in  FIG. 5 , this motor and gear train configuration may be housed inside the base  207 . Preferably, the gear train for increasing torque and the gear train for revolving the yoke  102  and the crank  104  at their respective rates are both stabilized by providing caging for the gear trains. This caging, which may be affixed to or integral to the base  207 , is not depicted so as not to obscure the principles of the present disclosure. The yoke drive shaft  202  and the crank drive shaft  204  may be arranged such that they are concentric about the first axis  1 , and coupled to the base  207  through the base bearing  203 . 
     A motor  208  drives a motor drive shaft  210  on which motor gear  209  is mounted. In one embodiment, the motor  208  is a stepper motor with a clock circuit and drives the motor gear  209  at a rate of approximately one revolution per second. Gear ratios may be chosen so that the output of motor  208  is geared down by a factor of about 86,400, the number of seconds in a mean solar day. Thus, the yoke shaft gear  231  (and the attached yoke drive shaft  202  and the first yoke gear  212 ) may be driven at a rate of about one revolution per mean solar day. Accordingly, the following gear ratios may be selected so that one revolution per second of the motor gear  209  produces one revolution per mean solar day of the yoke shaft gear  229 B: 
     
       
         
           
               
               
               
               
               
             
               
                   
                   
               
               
                   
                 Driving Gear 
                 Driven Gear 
                 Ratio 
                 Output Shaft 
               
               
                   
                   
               
             
            
               
                   
                 209     
                 221A 
                 1:4 
                 221C 
               
               
                   
                 221B 
                 223A 
                 1:5 
                 223C 
               
               
                   
                 223B 
                 225A 
                 1:6 
                 225C 
               
               
                   
                 225B 
                 227A 
                 1:8 
                 227C 
               
               
                   
                 227B 
                 229A 
                 1:9 
                 229A 
               
               
                   
                 229B 
                 231     
                  1:10 
                 202     
               
               
                   
                   
               
            
           
         
       
     
     As the motor  208  is geared down by a factor of 86,400, the output torque on the yoke shaft  202  is increased by a factor of 86,400. An advantage afforded by the added torque is that a relatively low-torque motor  208  may be able to drive a heavy yoke drive shaft  202  and/or a yoke  102  with a heavy payload attached thereto. 
     The first yoke gear  212  may be affixed to yoke drive shaft  202  such that the first yoke gear  212  turns at the same rate as the yoke drive shaft  202 . The first yoke gear  212 , the second yoke gear  216 , the second crank gear  218 , and the first crank gear  214  may be selected such that when the yoke drive shaft  202  and first yoke gear  212  are driven at a rate of about one revolution per mean solar day, or about 365.2422 revolutions per year, the crank drive shaft  204  is driven at a rate of about one revolution per sidereal day, or 366.2422 revolutions per year. Thus, as the yoke shaft  202  and first yoke gear  212  rotate at a rate of once per mean solar day, gears may be selected so that the crank drive shaft  214  is driven at the slightly faster rate of:
 
366.2422/365.2422=1.002,737,909 rev/mean solar day=1 rev/sidereal day.
 
     An example of a gear configuration for driving the yoke drive shaft  202  at a rate of about one revolution per mean solar day and for driving the crank drive shaft  204  at a rate of about one revolution per sidereal day is as follows: 
                                             Reference Number   # Gear Teeth                          212   79           214   82           216   49           218   51                        
This gear configuration comes very close to the exact ratio of revolutions per sidereal day to revolutions per mean solar day:
 
79·51/49·82=1.002,737,680 rev/mean solar day
 
The difference between the exact ratio and the ratio achieved by this example gear configuration is only 0.000,000,229 rev/day, or approximately 2 arc-minutes of error per year.
 
     Other gear sets or types of motors may be employed as desired or required to vary the gear ratio. For example, a synchronous motor  208  turning motor gear  209  at one revolution per minute may be geared down by a factor of 1,440 (2×2×3×4×5×6)=1,440, the number of minutes in a mean solar day) so that yoke drive shaft  202  is driven at a rate of one revolution per mean solar day. In this embodiment, the torque on the yoke drive shaft  202  and the crank drive shaft  204  is increased by a factor of roughly 1,440. Those skilled in the art will appreciate that any configuration of gears and/or shafts may be employed as desired or required to increase or decrease torque, increase or decrease rotational velocity, or vary the gear ratio. Any number of idler gears may be added to gear configurations as well. One or more motors operating at the same or different speeds may also be used. 
     In addition to the solar hour angle and the seasonal declination, there is a third component of the sun&#39;s position in the sky called the “equation of time.” Because of the eccentricity of Earth&#39;s orbit and Earth&#39;s axial tilt, the sun does not trace a path in the sky at a uniform rate, creating differences in clock time and sundial time. The equation-of-time anomaly is the difference between the local mean time (clock time) and the local apparent time (sundial time). The difference between local mean time and local apparent time may be modeled by the algebraic summation of two independent functions, each of which is approximately sinusoidal. The first is the eccentricity error, which for Earth has a magnitude of 1.918° and has one cycle per year. The second is the obliquity error, which for Earth has a magnitude of 2.47° and has two cycles per year. The two sinusoids are not in phase with one another: the eccentricity sinusoid starts its cycle at aphelion, while the obliquity sinusoid starts its cycles at the vernal and autumnal equinoxes. Further information on the equation-of-time anomaly may be found in U.S. Pat. No. 4,368,962 to Hultberg, titled “SOLAR TRACKING APPARATUS AND SYSTEM,” and issued on Jan. 18, 1983. The disclosure of this patent is hereby incorporated by reference in its entirety. 
     Thus, returning to  FIG. 5  and  FIG. 6 , if the yoke  102  is driven at a uniform rate by the motor  208 , the payload axis  6  may point slightly ahead of the sun&#39;s position in the sky or slightly behind the sun&#39;s position in the sky. Accordingly, solar tracker embodiments and other multi-axis tracker embodiments may include an equation-of-time correction mechanism coupled to the yoke  102  and/or the yoke drive shaft  202  so that the yoke  102  does not rotate about the first axis  1  at a uniform rate. An equation-of-time correction mechanism  250  may include means for advancing or retarding the rotation of the yoke  102  so that the yoke  102  and the segment of the yoke drive shaft  202  to which the yoke  102  is coupled rotate about the first axis  1  at a non-uniform rate of about one revolution per mean solar day. 
       FIG. 7A  shows one embodiment of an equation-of-time correction mechanism  250  that may be advantageously included with an embodiment of a solar tracker, such as the heliostat  300  illustrated in  FIG. 5 , to improve the accuracy of the solar tracker such that the payload axis  6  more accurately traces the observed position of the sun in the sky. Eccentrics may be coupled to the yoke drive shaft  202  in order to advance or retard the rotation of the yoke  102  about the first axis  1  to correct the eccentricity and obliquity errors described above. Though not shown, the gears of these and other equation-of-time correction mechanisms may be caged to stabilize the movement of the gears. This caging is not depicted, so as not to obscure the principles of the present disclosure. 
     The yoke drive shaft  202  is attached to a plate  233 . A central gear  230  is mounted on the crank drive shaft  204  such that it rotates with the same angular velocity as the crank drive shaft  204 . The central gear  230  drives an idler gear  240 , which is rotatably coupled to the plate  233  by a shaft  242 . The idler gear  240  drives an outer gear  244 , which is rotatably coupled to the plate  233  by means of a shaft  246 . The outer gear  244  may have the same number of teeth as the central gear  230 . An eccentric cylinder  241  is connected to the outer gear  244  by means of a pin  245 , such that the eccentric cylinder  241  rotates with the outer gear  244  only when the pin  245  engages the outer gear  244 . The eccentric cylinder  241  includes a lower portion concentric with outer gear  244  and an upper portion that is offset from the center of the outer gear  244 . The upper portion of the eccentric cylinder  241  engages the plate  235 . In one embodiment, the center of the upper portion of the eccentric cylinder  241  is offset from the shaft  246  such that the rotation of the yoke drive shaft  202  is retarded or advanced by approximately ±1.918°, one cycle per year. 
     The central gear  230  also drives an idler gear  232 , which rotates about a shaft  234  rotatably coupled to the plate  235 . In some embodiments, the idler gear  232  is a relatively large gear, advantageously improving the accuracy of the equation-of-time correction mechanism  250 . The idler gear  232  in turn drives an outer gear  236 , which is rotatably coupled to the plate  235 . A shaft  238  is rotatably coupled to the center of the outer gear  236 . An eccentric cylinder  239  is connected to the outer gear  236  by means of a pin  247 , such that the eccentric cylinder  239  rotates with the outer gear  236  only when the pin  247  engages the outer gear  236 . The eccentric cylinder  239  includes a lower portion concentric with outer gear  236  and an upper portion that is offset from the center of the outer gear  236 . The upper portion of the eccentric cylinder  239  engages the plate  237 . In one embodiment, the center of the upper portion of the eccentric cylinder  239  is offset from the shaft  238  such that the rotation of the yoke drive shaft  202  is retarded or advanced by approximately ±2.47°, two cycles per year, with outer gear  236  having half the number of teeth as central gear  230 . 
     The eccentric cylinder  239  and  241  may be selectively coupled and decoupled from the rest of the equation-of-time correction mechanism  250 . A bar  248 A may be connected to the eccentric cylinder  239 , with a pin  247  passing therethrough and engaging one of the holes on outer gear  236 . Likewise, a bar  249 A may be connected to the eccentric cylinder  241 , with a pin  247  passing therethrough and engaging one of the holes on outer gear  236 . The pin  245  may be removed so as to decouple the eccentric cylinder  241  from the outer gear  244 . Likewise, the pin  247  may be removed so as to decouple the eccentric cylinder  239  from the outer gear  236 . When decoupled in this way, the outer gears  236  and  244  may be freely rotated so that the position of their respective eccentric cylinders  239  and  241  will line up with the cycles of the obliquity error and eccentricity error, respectively. The pins  245  and  247  may be replaced to recouple the eccentric cylinders  241  and  239 , respectively, to the rest of the equation-of-time correction mechanism  250 , such that the eccentric cylinders  241  and  239  rotate with outer gears  244  and  236 , respectively. 
       FIG. 7B  depicts another embodiment of an equation-of-time correction mechanism  250 ′. The components of the equation-of-time correction mechanism  250 ′ are largely similar to the components of the equation-of-time correction mechanism  250  in  FIG. 7A , with like reference numbers assigned to like components. 
     In the equation-of-time correction mechanism  250 ′, the eccentric cylinder  239  is connected by a screw  248 C (or other equivalent structure) to a disk  248 B. The disk  248 B may include a number of circumferentially-arranged holes adapted to fit a pin  247 . The bar  248 A is attached to the shaft  238 , and may also include a hole adapted to fit the pin  247 . The disk  248 B may be coupled to the bar  248 A (and thus the shaft  238 ) by inserting the pin  247  through the hole in the bar  248 A and a hole in the disk  248 B. The pin  247  may also be removed to decouple the disk  248 B from the bar  248 A. If the disk  248 B is decoupled from the bar  248 A, it may be rotated freely to cause the eccentric cylinder  239  connected by the screw  248 B to rotate as well. The disk  248 B thus provides a convenient way to line up the eccentric cylinder  239  with the cycle of the obliquity error. 
     Another disk  249 B may also be provided, connected by a screw  249 C (or equivalent structure) to the eccentric cylinder  241 . The disk  249 B may include a number of circumferentially-arranged holes adapted to fit a pin  245 . The bar  249 A is attached to the shaft  246 , and the bar  249 A may also include a hole adapted to fit the pin  245 . The disk  249 B may be coupled to the bar  249 A (and thus the shaft  246 ) by inserting the pin  245  through the hole in the bar  249 A and a hole in the disk  249 B. The pin  245  may also be removed to decouple the disk  249 B from the bar  249 A. If the disk  249 B is decoupled from the bar  249 A, it may be rotated freely to cause the eccentric cylinder  241  connected by the screw  249 C to rotate as well. The disk  249 B thus provides a convenient way to line up the eccentric cylinder  241  with the cycle of the eccentricity error. 
       FIG. 7C  depicts another embodiment of components included (at least partially) in the base  207 , including yet another embodiment of an equation-of-time correction mechanism  250 ″. The components of the base  207  in this embodiment, as depicted in  FIG. 7C , are largely similar to the components illustrated in the base  207  of  FIG. 6  including the equation-of-time correction mechanism  250 , with like reference numbers assigned to like components. Accordingly, the description of the components included in the base  207  as now depicted in  FIG. 7C , including a new equation-of-time correction mechanism  250 ″, will focus on the differences of this embodiment. 
     Generally,  FIG. 7C  illustrates additional components that interface with other elements from the base  207  illustrated in  FIG. 6 . In the base  207  (not shown in  FIG. 7C , but shown in  FIG. 6 ), a set screw  260  is provided for locking and unlocking the gear  214  to the crank drive shaft  204 . In addition, as further shown in  FIG. 7C , the base  207  now includes a dial plate  261  graduated into 365 divisions from 0 to 364, or a convenient fraction or multiple thereof and with relative proportional calibrations, which corresponds to the number of Julian days in a calendar year. The first mark on the dial plate  261  corresponds to the date of perihelion, which occurs around the 3 rd  or 4 th  of January depending on the year. Accordingly, the second mark on the dial plate  261 , in this embodiment, corresponds to the day after perihelion, and so on. As noted above, the dial plate  261  may include different marking/numbering schemes in other embodiments. For example, the dial plate  261  could be divided into 73 marks of 5 days per mark. In other embodiments, the dial plate  261  could be divided into any random number of divisions so that the calibrations are divided proportionally into 365 days. 
     As further shown in  FIG. 7C , the base  207  now also includes a dial plate marker  262  attached to the non-moving gear box frame or base  207  (not shown in  FIG. 7C , but shown in  FIG. 6 ), for pointing to corresponding marks on the dial plate  261  which is rotatable with the crank drive shaft  204  (and thus crank  104 ) when the set screw  260  is disengaged. Finally,  FIG. 7C  depicts the yoke shaft  202  as two pieces  202 A and  202 B. The lower portion of the yoke shaft  202  is denoted by  202 A, while the upper portion is denoted by  202 B. 
     In one embodiment of the equation-of-time mechanism  250 ″ of  FIG. 7C , the eccentric cylinders  241  and  239  may be factory set and locked as follows: The eccentric cylinder  241  is set to zero degrees. The eccentric cylinder  241  corrects for the “eccentricity” error and revolves once per year starting at perihelion. The Julian day of perihelion in 2014 occurred on January 4 th  at 12:00 UT1 or about JD 2456662.000000. The eccentric cylinder  239  is rotated and set at about 74 degrees, and more particularly to about 74.18 degrees where 74.18 degrees=(number of days from perihelion to the vernal equinox)×(360 degrees)/(365 days). The eccentric cylinder  239  corrects for the “obliquity” error and revolves twice per year starting at the vernal equinox. The Julian day of the vernal equinox in 2014 occurred on March 20 th  at 16:57 UT1 or about JD 2456737.208333. Julian dates are reckoned from zero on Jan. 1, 4713 BCE. The Julian dates for perihelion and the vernal equinox for a given year can be obtained from the U.S. Naval Observatory web site. 
     In one embodiment, a sun tracker user may set the crank drive shaft  204  and the crank  104  to the correct angle for a particular chosen day of start-up. For example, the user may first obtain the correct Julian day for the user&#39;s chosen day of start-up from the U.S. Naval Observatory web site. The user may then disengage the set screw  260 , thus free wheeling the crank drive shaft  204  (and thus the crank  104 ). The user then rotates the crank  104  or crank drive shaft  204  until the dial plate marker  262  indicates the difference between the chosen Julian date (corresponding to the user&#39;s chosen day of start-up) and the Julian date of perihelion for the start up year. When the set screw is disengaged and the crank or crank drive shaft  204  is rotated, the equation-of-time correction mechanism  250 ″ will gyrate through its advance-retard motions until it reaches the above-noted difference between the Julian date corresponding to the chosen start-up day and the Julian date of perihelion for the start up year. Next, the user resets the set screw  260 . At the correct time of day, when the sun tracker points to the sun, the user then may start the tracker motor. The use of Julian dates eliminates problems associated with leap years. 
     It should be further recognized that the components of the equation-of-time correction mechanism  250 ″ are largely similar to the components illustrated in the equation-of-time correction mechanism  250  of  FIG. 7A , with like reference numbers assigned to like components except that the equation-of-time mechanism  250 ″ does not include the holes in the outer gears  236  and  244 , the bars  248 A and  249 A, and the pins  245  and  247  as shown in the embodiment of the equation-of-time mechanism depicted in  FIG. 7A . Instead, the equation-of-time mechanism  250 ″ includes the additional elements noted above. 
     It should be appreciated that in these embodiments of the equation-of-time correction mechanisms  250 ,  250 ′, and  250 ″, only the motion of the yoke drive shaft  202  (or a segment thereof) and the yoke  102  is modified. The motor  208  continues to drive the crank drive shaft  204  and the crank  104  at the uniform rate of one revolution per sidereal day. The advancement and retardation of the rotational motion of the yoke  102  may advantageously cause the payload axis  6  to track the sun&#39;s position in the sky with yet greater accuracy. 
     As discussed above, the heliostat  300  as shown in  FIG. 5  (and other solar tracker embodiments) may be equipped with a gear train such as that shown in  FIG. 6  and  FIG. 7A ,  7 B, or  7 C. Thus, the payload axis  6  can very accurately track the movement of the sun through the sky by accounting for the three components of the sun&#39;s position in the sky described above. Returning again to  FIG. 5 , a paraboloidal mirror  201  may be mounted as a payload on the rocking frame  108  to take advantage of this accurate tracking for solar energy collection. The paraboloidal mirror  201  may have a focal point along the payload axis  6  that is identical to intersection point  4 . The diameter of the paraboloidal mirror  201  may be substantially equal to four times its focal length, i.e. the distance along the payload axis  6  from the center of the paraboloidal mirror  201  to the intersection point  4 . An energy receiver  205 , such as a thermal receiver, photovoltaic element, or collection optics, may be positioned at the intersection point  4  in order to receive the solar energy reflected by the paraboloidal mirror  201 . This configuration provides a very small image of the sun on the energy receiver  205 , concentrating a high amount of solar energy on the energy receiver  205 . It should be appreciated that a single energy receiver  205  can receive energy reflected by other sources, such as, for example, mirrors equipped on other heliostats. The energy receiver  205  may optionally be thermally coupled to a thermally conductive energy conduit  206  for converting or transmitting the collected solar energy. In some embodiments, the energy conduit is made of a solid material with a high melting point, such as tungsten. 
     In some embodiments, the energy conduit  206  is a thermally conductive conduit  206  to transmit thermal energy. The thermally conductive conduit  206  may include one or more heat pipes. The heat pipes may, for example, include a sealed length of tubing containing a heat transfer fluid. In some embodiments, the energy conduit  206  is made of a solid material with a high melting point, such as tungsten. The heat transfer fluid may be, for example, water or a molten salt. Thermal energy may be carried off through the thermally conductive conduit  206 . The thermal energy may be transmitted through heat exchangers to drive turbines in order to rotate electric generators. In one embodiment, the generated electrical energy could be used to power an electric motor to drive the yoke  102  or the crank  104  about the first axis  1 . Other applications for the collected thermal energy are possible. For example, the collected thermal energy may be used to melt metals, generate air conditioning by driving a Stirling engine compressor, provide space heating, or distill seawater, among other applications. 
     Many solar tracker variations are possible. For example,  FIG. 8  depicts a heliostat  300 ′. Many of the components are similar to those in the heliostat  300  shown in  FIG. 5 . The heliostat  300 ′ in  FIG. 8  may include a similar gearbox with equation-of-time correction mechanism as shown in  FIGS. 6 and 7 . Unlike the heliostat  300  in  FIG. 5 , however, the heliostat  300 ′ in  FIG. 8  incorporates a square paraboloidal mirror  201 ′. A square paraboloidal mirror  201 ′ may be advantageously used to increase the insolation area without increasing the diameter of the yoke  102  or the rocking frame  108 . The square paraboloidal mirror  201 ′ may also be advantageously employed as a way to center the mass of the moving components of the heliostat  300 ′ with respect to the first axis  1 . 
     Another solar tracker variation, a coelostat  350 , is shown in  FIG. 9 . The coelostat  350  may include a similar gearbox with equation-of-time correction mechanism as shown in  FIGS. 6 and 7 . Central gears  320  may be rigidly attached to the yoke  102  so as not to rotate about the second axis  2 . Idler gears  310  may be rotationally attached to the rocking frame  108 , such that the idler gears  310  rock back and forth with the rocking frame around the central gears  320 . The movement of the idler gears  310  causes the toothed mirror attachments  330  to rock back and forth as well. In one embodiment, the idler gears  310  have n teeth, the central gears  320  have 2n teeth, and the toothed mirror attachments  330  have 4n teeth. If the rocking frame  108  and idler gears  310  rock back and forth by ±23.45° once per year, the plane mirror  301  (depicted herein as partially transparent so as not to obscure the other parts of the coelostat  350 ) and the toothed mirror attachments  330  will rock back and forth by ±11.73° per year. Those skilled in the art will appreciate that this configuration causes the surface of the plane mirror  201 ″ to bisect the angle between the payload axis  6  and the first axis  1 , thus causing the sun&#39;s rays (incident along payload axis  6 ) to be reflected along the first axis  1  throughout the year. 
     Those skilled in the art will appreciate that the geometry of any mirror or other payload mounted on a solar tracker may be varied as desired in two or three dimensions. For example, the geometry of the mirror or other payload may be chosen so as to be compatible with an energy conduit, as shown in  FIG. 5  and  FIG. 8 , or with components of the spherical mechanical linkage, as shown in  FIG. 9 . As shown in  FIG. 5 ,  FIG. 8 , and  FIG. 9 , the mirrors mounted on the rocking frame  108  of each solar tracker may have notches. For example, in  FIG. 5  and  FIG. 8 , the mirrors  201  and  201 ′ have notches such that their rotational movement is not obstructed by the energy conduit  206 . Likewise, in  FIG. 9 , the plane mirror  301  includes notches permitting the movement of the deflecting member  106  therethrough. 
     Illustrative Variation: Boom-Mounted Multi-Axis Tracker 
       FIG. 10  depicts an embodiment of a multi-axis tracker  400  wherein the base  207  is mounted on boom arms  404  and  406 . In this embodiment, a payload  402 , such as a still camera, video camera, or spotlight, may be coupled to the rocking frame  108 . The shape of the rocking frame  108  may be chosen as desired to facilitate coupling with the payload  402 . For example, the rocking frame  108  may include several straight portions along with several curved portions to facilitate mounting, for example, a cylindrical payload  402 . 
     In this embodiment, the yoke motor  208 A is mounted on the base  207  and is coupled to the yoke drive shaft  202 . The crank motor  208 B is connected to the yoke drive shaft  202 , which is rotationally coupled to the yoke motor  208 A. The crank motor  208 B is connected directly to crank  104 . The deflecting member  106  is coupled to the crank  104  by the stub shaft  116 . The crank  104  is connected to the deflecting member  106  about the third axis  3  via a stub shaft  116  on the deflecting member  106 . The deflecting member  106  is also slideably coupled to the rocking frame  108  through the bearing points  110 . 
     As the yoke motor  208  drives the yoke drive shaft  202  about the first axis  1 , the crank motor  208 B and the yoke  102  turn with the yoke drive shaft  202  in a “panning” motion about the first axis  1 . A “tilting” motion may be provided by the movement of the rocking frame  108  about the second axis  2 . The rocking frame  108  may be turned by the movement of the deflecting member  106 , which is coupled to the crank  104 , which is driven by the crank motor  208 B. 
     Illustrative Variation: Hollow Shaft 
       FIG. 11  depicts an embodiment of a hollow shaft multi-axis tracking system  500  in which the yoke  102  and crank drive shaft  204  are hollow cylinders coupled to a crank  104 . The yoke  102  and the crank drive shaft  204  have an inner diameter greater than the diameter of the deflecting member  106 . The yoke  102  is driven by the motion of the yoke drive belt  211 A coupled to the yoke motor  208 A. The crank drive shaft  204  is driven by the motion of the crank drive belt  211 B coupled to the crank motor  208 B. As in other embodiments, the yoke  102  and the crank drive shaft  204  are free to rotate independently about the first axis  1 , such that the crank  104  is free to rotate about the first axis  1  independently of the rotation of the yoke  102 . 
     This embodiment has many uses. For example, this embodiment could be configured as a heliostat by providing motors and gears as discussed with reference to  FIGS. 5 through 8 . A mirror could be mounted on rocking frame  108  to deflect sunlight through the shaft and into, for example, the interior of a building. With an appropriate payload mounted on rocking frame  108 , this embodiment could also be used in, for example, a camera obscura (with a mirror mounted on the rocking frame  108  and lenses to direct light onto a screen), a periscope (with a telescope mounted on the rocking frame  108 ), or a search light (with a light source mounted on the rocking frame  108 ). Those skilled in the art will recognize that an appropriately small hollow shaft multi-axis tracking system  500  may be mounted on the distal end of an endoscope and used to control a miniaturized camera or light source in small spaces, for example, inside of a human or animal patient. 
     Conditional language, such as, among others, “can,” “could,” “might,” or “may,” unless specifically stated otherwise, or otherwise understood within the context as used, is generally intended to convey that certain embodiments include, while other embodiments do not include, certain features, elements and/or steps. Thus, such conditional language is not generally intended to imply that features, elements and/or steps are in any way required for one or more embodiments. As used herein, the terms “comprises,” “comprising,” “includes,” “including,” “has,” “having” or any other variation thereof, are intended to cover a non-exclusive inclusion. For example, a process, method, article, or apparatus that comprises a list of elements is not necessarily limited to only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Also, the term “or” is used in its inclusive sense (and not in its exclusive sense) so that when used, for example, to connect a list of elements, the term “or” means one, some, or all of the elements in the list. 
     Although certain preferred embodiments and examples are disclosed herein, inventive subject matter extends beyond the specifically disclosed embodiments to other alternative embodiments and/or uses, and to modifications and equivalents thereof. Thus, the scope of the disclosure is not limited by any of the particular embodiments described herein. For example, in any method disclosed herein, the acts or operations of the method can be performed in any suitable sequence and are not necessarily limited to any particular disclosed sequence. Various operations can be described as multiple discrete operations in turn, in a manner that can be helpful in understanding certain embodiments; however, the order of description should not be construed to imply that these operations are order dependent. Additionally, the structures described herein can be embodied as integrated components or as separate components. For purposes of comparing various embodiments, certain aspects and advantages of these embodiments are described. Not necessarily all such aspects or advantages are achieved by any particular embodiment. Thus, for example, various embodiments can be carried out in a manner that achieves or optimizes one advantage or group of advantages as taught herein without necessarily achieving other aspects or advantages as can also be taught or suggested herein. No single feature (or group of features) is necessary or indispensable for each embodiment. All modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims.