Abstract:
A device for use in conjunction with a coordinate measuring device, a three-dimensional coordinate measuring system and a method of measuring three-dimensional coordinates are described. The device includes a light source mounted substantially within the center of a spherical member having a spherical surface configured such that the distance from the light source to the outer contact surface of the sphere is substantially equal to the radius of the sphere regardless of sphere orientation. The three-dimensional coordinate measuring system further includes a coordinate-measuring device, the coordinate-measuring device configured to track the movement of the light source and to measure either azithmuth angles and zenith angles, or to spherically encode tracking information related to relevant positions of the light source. The present method includes emitting light at a target with a light source, wherein the light source is mounted substantially within the center of a spherical member having a spherical surface configured such that the distance from the light source to the outer contact surface of the sphere is substantially equal to the radius of the sphere regardless of sphere orientation, moving the surface of the spherical member across a device to be measured, tracking said light source with a tracking device, and measuring information related to the relative position of the light source using either angular or spherical encoders.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
   The present application claims priority to U.S. Provisional Patent Application Ser. No. 60/358,930, filed Feb. 22, 2002, the entire contents of which are specifically incorporated herein by reference. 

   BACKGROUND 
   The present disclosure relates to a light-source assembly used in conjunction with tracking, angle-measuring devices to determine coordinates of points or surfaces. 
   An instrument commonly referred to as a laser tracker has the ability to measure the coordinates of a point by sending a laser beam to a retroreflector target that is in contact with the point. A laser tracker is often used in conjunction with a particular type of retroreflector target called the spherically mounted retroreflector (SMR). The SMR comprises a cube-corner mounted within a sphere with the vertex of the cube-corner at the sphere center. A gimbal mechanism within the laser tracker directs a laser beam from the tracker to the SMR. Part of the light retroreflected by the SMR enters the laser tracker and passes onto a position detector. A control system within the laser tracker uses the position of the light on the position detector to adjust the rotation angle of mechanical azimuth and zenith axes of the laser tracker to keep the laser beam centered on the SMR. In this way, the laser beam is able to track an SMR that is moved over the surface of an object of interest. Part of the light retroreflected into the laser tracker passes onto a distance-measuring device such as an interferometer or absolute distance meter (ADM). Angular encoders attached to the mechanical azimuth and zenith axes of the tracker measure the azimuth and zenith angles of the laser beam (with respect to the tracker frame of reference). The one distance and two angles measured by the laser tracker are sufficient to completely specify the three-dimensional location of the SMR. 
   The cube-corner retroreflector embedded within the sphere in the SMR comprises three mutually perpendicular reflecting surfaces. These three reflecting surfaces occupy three of the six faces of a cube. The diagonal of this cube is the axis of symmetry of the cube corner and, hence, also the axis of symmetry of the SMR. To avoid problems caused by optical effects such as vignetting, the axis of symmetry of the SMR must ordinarily be aligned to within about 40 degrees of the incident laser beam. Because of this limitation, it is difficult to make certain types of measurements with an SMR. For example, it is difficult to measure the coordinates of an SMR attached to a machine tool that is moved over a large distance or rotated over a large angle. As another example, it is sometimes desirable to simultaneously measure a target with several widely separated laser trackers to improve measurement accuracy. However, the SMR is impractical for such measurements because of its limited angle of acceptance. Furthermore, in most applications, the SMR is carried by hand over the surfaces of interest. If the SMR is tilted too far from the laser beam, the tracker measurement will be interrupted until the SMR is rotated back into position. If an interferometer is being used to measure the distance, then it will be necessary to move the SMR back to a reference position before resuming the measurement. Such interruptions can be annoying and time consuming. 
   In most laser trackers today, the accuracy of the distance and angle measurements is approximately proportional to the distance of the SMR from the laser tracker, with the constant of proportionality given in units of micrometers/meter or, equivalently, in parts per million (ppm). State-of-the-art laser trackers in a typical environment have a radial accuracy, determined by the accuracy of the interferometer or ADM, of approximately 2 ppm. Today, these same trackers have an angular accuracy, determined by a variety of factors including the accuracy of the angular encoders, of approximately 10 ppm. In other words, the main source of error in most measurements is angular rather than radial. The angular accuracy of the laser tracker could be improved to approach the radial accuracy by using larger and more expensive angular encoders together with air bearings, and a stiffer, a thermal mechanical structure. However, a laser tracker modified in this way would be noisier, less portable, and much more expensive than trackers available today. Rather than increase the cost of the laser tracker, it would be preferable to reduce cost while maintaining or improving measurement accuracy. 
   To measure three-dimensional coordinates, an alternative to the retroreflector target is a point source of light formed by sending light out of an optical fiber and through a dispersing element. Schultz et al. describe this type of target in patent number U.S. Pat. No. 5,907,395. 
   SUMMARY 
   The above and other disadvantages of the art are overcome and alleviated by the present device for use in conjunction with a coordinate measuring device, the present three-dimensional coordinate measuring system and the present method of measuring three-dimensional coordinates. The device includes a light source mounted substantially within the center of a spherical member having a spherical surface configured such that the distance from the light source to the outer contact surface of the sphere is substantially equal to the radius of the sphere regardless of sphere orientation. The three-dimensional coordinate measuring system further includes a coordinate-measuring device, the coordinate-measuring device configured to track the movement of the light source and to measure either azithmuth angles and zenith angles, or to spherically encode tracking information related to relevant positions of the light source. The present method includes emitting light at a target with a light source, wherein the light source is mounted substantially within the center of a spherical member having a spherical surface configured such that the distance from the light source to the outer contact surface of the sphere is substantially equal to the radius of the sphere regardless of sphere orientation, moving the surface of the spherical member across a device to be measured, tracking said light source with a tracking device, and measuring information related to the relative position of the light source using either angular or spherical encoders. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention. 
       FIG. 1  depicts an embodiment of two trackers that follow the movement of a spherically mounted light source (SML) and determine its three-dimensional coordinates; 
       FIG. 2  depicts in a perspective view the major elements of the tracker and the SML; 
       FIG. 3  depicts in a cross-sectional view the major elements of the tracker beam-steering system; 
       FIG. 4  depicts in a top view the major elements of the tracker; 
       FIG. 5  illustrates in a perspective view the effect of angular error on measurement accuracy; 
       FIG. 6  illustrates in a top view the effect of angular error on measurement accuracy; 
       FIG. 7  depicts in a perspective view a second exemplary embodiment; and 
       FIG. 8  depicts in a cross-sectional view of the major elements of the beam steering system of  FIG. 7 . 
   

   DETAILED DESCRIPTION 
   Reference will now be made in detail to exemplary embodiments, examples of which are illustrated in the accompanying drawings. The present disclosure illustrates a new way of measuring three-dimensional coordinates by means of a new type of target and a new type of coordinate-measuring device. The target comprises a point light source that is mounted in the center of a spherical surface. As the spherically mounted light source (SML)  20  is moved, the two trackers follow the light-source movement, each simultaneously measuring the azimuth and zenith angles of the SML  20 . These two azimuth angles and two zenith angles are sufficient to determine the three-dimensional location of the point light source. The traditional method of measuring angles is to mount angular encoders to the azimuth and zenith axes. An alternative method, given in the first embodiment below, is to use one spherical encoder rather than two angular encoders to reduce cost while improving angular accuracy. 
     FIG. 1  shows a perspective view of two trackers  22 ,  24  following the movement of a spherically mounted light source (SML)  20 . The two trackers  22 ,  24  each measure the azimuth and elevation angles of the SML  20  at the same instant in time. Synchronization is provided by a signal sent from one tracker to the other on an electrical wire  26  or a wireless carrier. Whenever the two trackers  22 ,  24  are moved to new positions, it is necessary to perform a procedure to determine the location and orientation of each tracker within a global frame of reference. The procedure includes simultaneously measuring the azimuth and zenith angles from each of the two trackers  22 ,  24  to the SML  20  with the SML  20  at a variety of different distance and angles relative to the two trackers  22 ,  24 . For the sake of simplicity, the origin (x=0, y=0, z=0) and orientation (direction of the x, y, and z axes) of the global frame of reference are set equal to origin and orientation of the first tracker  22 . The position and orientation of the second tracker  24  within the global frame of reference is fully characterized by three translational and two orientational degrees of freedom, for a total of five degrees of freedom. Each simultaneous measurement of the azimuth and zenith angles of the two trackers  22 ,  24  provides information on four degrees of freedom. Locating the SML  20  requires three degrees of freedom, so each simultaneous measurement provides information on one additional degree of freedom. In other words, each measurement provides redundant information. By making simultaneous measurements at a variety of locations, the five degrees of freedom needed to completely specify the relative positions and orientations of the two trackers  22 ,  24  can be determined. The mathematical procedure to do this is to select the five unknown position and orientation values in such a way as to minimize the sum of the squared errors determined from the collected data. The number of measurements required to accurately determine the location and orientation of the second tracker within the global frame of reference will depend on the locations selected for the SML  20  measurements; however, in most cases, 15 to 20 measurements will be sufficient. 
   As shown in  FIGS. 1 and 2 , the SML  20  includes a light source  28  centered within a spherical surface  30 . The SML  20  shown in  FIGS. 1 and 2  includes a metal ball  32  having the shape of a hemisphere with a lip  34  extending along the edges to increase the spherical surface area. However, the particular shape of the metal ball  32  is not important as long as the outer surface is spherical and the source of light  28  is located at the center of the sphere. The reason for locating the light source at the center of the sphere is to make the distance from the light source to the point of contact of the sphere equal to the radius of the sphere regardless of the sphere orientation. This property makes it easy to find the coordinates of a surface  42 . The SML  20  is moved across the surface while the two trackers collect coordinates of the sphere center. The coordinates of the surface are found by moving each of the collected points by a perpendicular distance equal to the sphere radius. 
   Besides measuring surface coordinates, the SML  20  can also be used to determine fiducial points such those established by the kinematic nest  36  or the tooling ball  38  shown in  FIG. 1 . In a properly constructed kinematic nest, the center of the SML  20  returns to essentially the same place each time the sphere is removed and replaced in the nest. In a common type of nest  36 , three pads provide support for the spherical surface and a permanent magnet holds the metal sphere in place. By placing the light source  28  in the center of the spherical surface  30 , measurements of the SML  20  result in consistent coordinate values regardless of the orientation of the SML  20  within the nest  36 . The tooling ball  38  includes a spherical surface  40  mounted on a shaft (not shown). The shaft fits tightly inside a hole in the object under investigation, and the sphere  40  sits a fixed distance above the object&#39;s surface. The fiducial point is defined as the center of the tooling ball sphere  40 . This coordinates of this fiducial point are determined by moving the SML  20  across the surface of the tooling ball while the trackers collect coordinates. The fiducial point is found by locating the center of the spherically distributed collection of points. Again, it is important that the light source be centered within the spherical surface to correctly determine the location of the fiducial point. 
   The light source  28  in the SML  20  can be of any type but should preferably be small so that it appears as a point source of light. It is also preferable that its emission pattern be nearly hemispherical and its optical power be large enough at all angles. A good choice for the light source is the red High-Flux Surface Mount LED, part number HSMB-HR00-R1T20, manufactured by LumiLeds. For this LED, the normalized intensity varies smoothly from a maximum at normal incidence to 10% of this value at a half-angle of 80 degrees. This relatively high optical power at large angles permits good performance of the trackers described in these embodiments to distances of 30 meters over a full angle of 160 degrees (nearly a hemisphere). The LED described above requires an average cuiTent of 20 milliamps, which is provided through a separate power supply box  31  shown in  2 . This box  31  may contain a battery or it may contain power supply components that connect directly to the electrical power mains. The current is modulated by an electrical signal, preferably by a sinusoidal signal with a fixed frequency in the range of 10 kHz to 100 kHz. The depth of modulation is preferably between 80 and 100 percent. 
   Data from the trackers  22 ,  24  are sent to a computer  44  or microprocessor for analysis. Data may be sent over electrical wires  26  as shown in  FIG. 1  or through a wireless carrier. 
     FIG. 2  shows an enlarged perspective view of a tracker  46  and the SML  20 . A section view  3 — 3  through the center of the tracker is shown in  FIG. 3 . The central shaft  48  is bolted or otherwise attached to a solid structure  50 , such as an instrument stand. The azimuth axis assembly  52  comprises azimuth-axis housing  54 , azimuth-axis bearings  56 , azimuth-axis motor stator  58 , and azimuth-axis motor rotor  60 . The azimuth-axis motor stator  58  is attached to the central shaft  48  and includes coil windings that form an electromagnet when electrical current is applied. The azimuth-axis motor rotor  60  is attached to the azimuth-axis housing  54  and includes a series of permanent magnets. These permanent magnets move in the azimuth direction in response to the changing magnetic field produced by the azimuth-axis motor stator  58 . Azimuth-axis bearings  56  allow the azimuth-axis housing  54  to turn smoothly with a minimum of friction. The left zenith-axis shaft  62  passes through the center of the left zenith-axis assembly  64 , and the right zenith-axis shaft  66  passes through the center of the right zenith-axis assembly  68 . Left zenith-axis assembly  64  comprises left zenith-axis housing  70 , left zenith-axis bearing  72 , zenith-axis motor stator  74 , and zenith-axis motor rotor  76 . Right zenith-axis assembly  68  comprises right zenith-axis housing  78  and right zenith-axis bearing  80 . As shown in  FIG. 2 , the mounting plate  82  attaches to the inner portions of the left  62  and right  66  zenith-axis shafts. Left  72  and right  80  zenith-axis bearings allow the rigid structure comprising the left and right zenith-axis shafts and the mounting plate to turn smoothly with low friction. The zenith-axis motor stator  58  is attached to the left zenith-axis housing  70  and includes coil windings that form an electromagnet when electrical current is applied. The zenith-axis motor rotor  60  is attached to the left zenith-axis housing  70  and includes a series of permanent magnets. These permanent magnets move in the zenith direction in response to the changing magnetic field produced by the zenith-axis motor stator  74 . The left  70  and right  78  zenith-axis housings are attached to the azimuth-axis housing  54  by wing supports  84 . 
   The central shaft  48  is attached to the solid structure  50  and is stationary within the global frame of reference described above. The encoder sphere  86 , which is attached to the central shaft  48 , is also stationary in the global frame of reference. Azimuth-axis housing  54 , wing supports  84 , and left  70  and right  78  zenith axis housings all turn in the azimuth direction in response to electrical current applied to the azimuth-axis motor stator  58 . The rigid structure comprising the left  62  and right  66  zenith-axis shafts and the mounting plate  82  all turn in the zenith direction in response to electrical current applied to the zenith-axis motor stator  74 . The mechanical structure shown in  FIG. 3  is an example of a gimbal mount because there is a central point of rotation, also called the gimbal point that is independent of the azimuth and zenith angles. This central point of rotation is found at the intersection of the centerline of the central shaft and the centerlines of the left and right zenith-axis shafts. To route electrical wires to and from the electrical components attached to the mounting plate  82 , horizontal and vertical holes are drilled into the central shaft  48 . Electrical wires are first wrapped in loops around the upper part of the central shaft, then routed through the upper horizontal hole, into the vertical hole, and out of the lower horizontal hole to the electronics box  88  shown in  FIG. 2 . 
   As shown in  FIGS. 2 and 4 , the afocal beam-reducer assembly  90  includes a first positive (convex) lens  92  followed by a second lens  94 , which may be either positive or negative (concave), with the distance between the first  92  and second  94  lenses equal to the sum of the focal lengths of the two lenses. The magnification of the beam-reducer assembly  90  is equal to the ratio of the focal length of the second lens  94  to the focal length of the first lens  92 . For example, if the focal length of the first lens  92  is 50 mm and the focal length of the second lens  94  is −5 mm, then the lenses are separated by 45 mm and the magnification is − 1/10. The diameter of the first lens  92  is generally much smaller than the distance from the gimbal point to the SML  20 . Consequently, the rays of light  96  arriving at the first lens  92  from the SML  20  are nearly parallel, which will result in rays of light  96  emerging from the second lens  94  that are also nearly parallel. If the clear aperture (diameter) of the first lens  92  is 25 mm, then the diameter of the bundle of light rays  96  exiting the second lens  94  will equal the initial diameter divided by the absolute value of the magnification, or 25/10=2.5 mm. The centers of the first  92  and second  94  lenses and the center of the position detector are all aligned with the gimbal point. 
   The position detector assembly  98  includes a position detector  100  mounted on a circuit board  102 , with electrical wires running from the circuit board  102  to additional electrical components within the electronics box  88 . Although any type of position detector  100  can be used, including a lateral-effect cell or a photosensitive array, a quadrant detector is may be desired because of its high sensitivity, high speed, low cost, and simplicity. A good choice for the quadrant detector is the SPOT-4D manufactured by UDT Sensors, Inc. The quadrant detector comprises four separate electrical detector segments closely spaced into quadrants. The electrical signal from each of the four detector segments is attached to its own transimpedance amplifier. The electrical signal from each transimpedance amplifier is sent through a band-pass filter and then into an amplitude meter. The amplitude meter has the ability to measure the amplitude of the sinusoidal signal at the modulation frequency of the light source while rejecting other signals. The amplitude meter may he implemented either with analog circuitry or with analog-to-digital converters followed by digital signal processing hardware. The left-right position of the spot of light that strikes the quadrant detector is found by calculating the relative voltage difference between the two leftmost detectors and the two rightmost detectors. Similarly, the up-down position of the spot of light that strikes the quadrant detector is found by calculating the relative voltage difference between the two uppermost and the two lowermost detectors. An electrical control system located within the electronics box uses the calculated left-right and up-down positions to adjust the rotation angles of the azimuth and zenith axis assemblies to keep the spot of light centered on the quadrant detector. 
   The line connecting the gimbal point with the center of the position detector  100  and the centers of the first  92  and second  94  lenses of the beam reducer  90  is referred to as the tracker optical axis  104 . The spherical encoder assembly is used to measure the azimuth angle and zenith angle of the tracker optical axis relative to a frame of reference that is fixed with respect to the central shaft and encoder sphere. The spherical encoder can be absolute or incremental, reflective or transmissive, and imaging or diffracting. The present embodiment describes a spherical encoder of the absolute, reflective, imaging type. 
   The spherical encoder assembly  106  comprises the encoder sphere  86 , the beam-illumination assembly  108 , and the beam-imaging assembly  110 . The encoder sphere  86  is centered on the gimbal point. The surface of the encoder sphere  86  is covered with regularly spaced, diffusely reflecting squares, referred to as fiducial squares, whose edges are curved to conform to the spherical surface. The fiducial squares are placed on a dark (absorptive) background and are centered on the sphere surface at predetermined azimuth and zenith angles. Additional, smaller squares, referred to as indexing squares, are placed between the fiducial squares. There are several potential indexing locations between each pair of fiducial squares. Within these potential indexing locations, some of the indexing squares are made to reflect light, thereby making them logical ones, while other squares are made to absorb light, making them logical zeros. The logical ones and zeroes at any particular location on the surface of the sphere are decoded in binary to determine the identity of the corresponding fiducial squares. 
   The beam-illumination assembly  108  comprises the light source  112 , fiber-coupling optics  114 , source ferrule  116 , single-mode optical fiber  118 , mandrel  120 , launch ferrule  122 , collimating lens  124 , and beam block  126 . The purpose of the beam-illumination assembly  108  is to illuminate a small part of the surface of the encoder sphere  86  with visible light that varies smoothly over the illuminated region of the sphere  86 . The light source  112  is preferably a high-flux LED with a coherence length that is small enough to eliminate the deleterious effects of speckle in the light reflected off the sphere surface. A good choice for the light source is the red SnapLED  70 , part number HPWT-FHOO, manufactured by LumiLeds. This LED emits light that is nearly Gaussian in its transverse spatial profile, thereby helping to obtain high-efficiency coupling into the single-mode optical fiber  118 . The fiber-coupling optics  114  comprises one or more lenses. To obtain efficient coupling, these lenses place the beam waist of the focused light near the entrance face of the optical fiber  118 , and they adjust the diameter of the beam waist to approximately equal the mode field diameter of the fiber. The optical fiber  118  is epoxied within the source ferrule  116 , and the end of the fiber is polished. Not far from the source end of the optical fiber, the optical fiber  118  is wrapped several times about a mandrel  120  (cylinder). The purpose of this wrapping is to improve the smoothness of the spatial profile within the optical fiber  118 . The wrapping of the optical fiber  118  into a small loop tends to remove those portions of the electric field within the fiber that do not conform to the lowest-order mode, which is a smooth mode having a nearly Gaussian transverse profile. In other words, the optical fiber  118  and mandrel wrap  120  act as a spatial filter that removes undesirable field components. The end of the optical fiber  118  is epoxied into the launch ferrule  122 , and the end of the fiber is polished smooth. Light diverges from the optical fiber with a smooth, consistent transverse profile. It is collimated by the collimating lens  124  and strikes the surface of the encoder sphere  86 . The surface regions of the encoder sphere  86  are designed to be absorptive or diffusely reflective. However, a fraction of the reflected light is unavoidably specular, and a beam block is provided to absorb this light. An effective beam block can be constructed out of absorbing (neutral density) glass with an antireflective coating on the front surface. 
   The beam-imaging assembly  110  comprises the imaging lens  128 , photosensitive array  130 , and circuit board  132 . The imaging lens  128  forms an image on the photosensitive array  130  corresponding approximately to the illuminated pattern on the surface of the encoder sphere  86 . The photosensitive array  130  may be a CCD, CMOS, CID, or similar device. The pattern on the photosensitive array  130  is analyzed by a computer, microprocessor, or digital signal processor (DSP) located on the circuit board  132  or in the electronics box  88 . The optical axis of the imaging lens  134  is aligned with the tracker gimbal point. 
   The first step in converting the pattern on the photosensitive array  130  into an azimuth and elevation angle is to correct the electrical charge stored in each pixel of the photosensitive array  130  to account for variations in the amount of light reaching each pixel and being converted into a stored electrical charge. These variations have several causes: (1) non-uniform (approximately Gaussian) illumination pattern, (2) non-uniform radiance emitted across the surface of the sphere, and (3) non-uniform responsivity of pixels in the photosensitive array. These combined effects can be characterized by rotating the mounting plate over a small angular extent and noting the response of the each pixel to reflective and non-reflective regions. From these observations, it is possible to determine how each pixel in image space responds to the illumination incident at the corresponding point in object space. (Points in object space lie along the sphere surface.) The second step in converting the pattern into azimuth and zenith angles is to perform a numerical ray trace computation to find correction values for the centroid of the fiducial squares at each position on the photosensitive array. This correction accounts for defocus and distortion resulting from the curvature of the spherical surface, as well as aberrations from the optical components. When the corrections obtained from the two steps above are applied to the pattern on the photosensitive array, the azimuth and zenith angles are determined to relatively high accuracy. The accuracy can be improved further by carrying out an experimental procedure in which the indicated angle is compared to a known angular deviation. Such an angular deviation can be found by placing a retroreflector on a micrometer stage whose position transverse to the tracker is measured with a highly accurate linear encoder. This procedure can detect angular deviations of less than one microradian. 
   In the present (first) embodiment, a spherical encoder  86  is used to measure the azimuth and zenith angles. In the second embodiment that follows, angular encoders are attached to both mechanical axes. The use of angular encoders is well established, but the spherical encoders introduced here make possible higher accuracy at a reduced system cost. Unlike angular encoders, which measure the angles of the mechanical axes, the spherical encoder directly measures the angle of the mounting plate. Because of this, the deleterious effects of bearing wobble and structure bending are greatly reduced in the spherical encoder, permitting a less expensive mechanical assembly to be constructed. Furthermore, the spherical encoder described above can be made to be more accurate than the compact angular encoders commonly available today. For example, suppose that fiducial squares are placed approximately 3 degrees (approximately 1/20 radian) apart on the surface of the sphere and that four of these squares map into a photosensitive array having 1000×1000 pixels. For systems of this sort, the centroid of a square can usually be located to an accuracy of 1/100 of a pixel. A given centroid is therefore known to an accuracy of 4 squares*3 degrees/square*0.01745 radian/degree/(1000 pixels*100 parts/pixel)=2 microradians. If the centroid location is calculated for three of the four squares across the array and if the error is random so that it reduces as the square root of the number of measurements, then the accuracy of the azimuth and zenith angles is 2 microradians/sqrt(3)=1.2 microradians. This accuracy is comparable to the accuracy obtained with displacement interferometers that measure over large ranges in a relatively good (thermally stable) environment. If desired, the number of pixels in the photosensitive array  130  can be reduced while the number of fiducial squares on the sphere surface is increased (in proportion) without affecting angular accuracy. 
   We now consider the accuracy of a measurement in which two trackers are used in combination to determine the coordinates of an SML  20  through the measurement of angles.  FIG. 5  represents in perspective view the effect of angular error. Angular errors are approximately proportional to the distance from the tracker gimbal point to the center of the SML  20 . For example, the calculation above suggests that the constant of proportionality is approximately 1.2 microradians, which is equivalent to 1.2 micrometers/meter or 1.2 ppm. The vertex  136  of each cone  138  in  FIG. 5  corresponds to the location of the gimbal point of a tracker. The central line  140  that runs through each cone represents the true angle that a perfect tracker would measure. The region out to the surface of the cone represents the uncertainty in the angle measurement. The SML  20  is centered at the intersection of the two lines that run through the cone centers. The overall error in the measurement of the coordinates of the SML  20  is found by intersecting the two cones. This intersection (overlap) region  142  is shown in  FIG. 5 . In this, the y axis is pointed from one tracker to the other. The x axis is perpendicular to the y axis and lies in the plane that contains the lines at the centers of the cones. 
     FIG. 6  represents in top view the effect of angular error. This view exposes the maximum error within the intersection region. It can be shown that the coordinate measurement is most accurate when the angle A shown in  FIG. 5  is approximately 90 degrees. In this case, the maximum error is approximately equal to the angular error (1.2 micrometers per meter in this case) times the distance (from the tracker to the SML  20 ) times sqrt(2). For example, for a target located 5 meters from each tracker, the maximum error is approximately 5(1.2)(sqrt(2))=6.6 micrometers. In practice, there are some additional errors in the measurement (such as offset errors and compensation errors) that increase the overall error to some degree. Relatively good measurement accuracy can be maintained over angular separations A from 45 degrees to 135 degrees. At these extremes, the maximum angular measurement error increases (relative to the 90-degree case) by a factor of approximately 2. 
   Second Embodiment 
     FIG. 7  shows a second exemplary embodiment, which is similar to the first embodiment except that the spherical encoder  106  has been replaced with two angular encoders  144 ,  146 . A section view  8 — 8  drawn through the center of the tracker  148  is shown in  FIG. 8 . The azimuth encoder  144  comprises the azimuth-encoder disk  150  and the azimuth read-head assembly  152 . The azimuth-encoder disk  150  is attached to the stationary central axis shaft  154 , and the azimuth read-head assembly  152  is attached to the rotating azimuth-axis frame  153 . The zenith encoder  146  comprises the zenith-encoder disk  156  and the zenith read-head assembly  158 . The zenith-encoder disk  156  is attached to the zenith-axis shaft  160 , which rotates in the zenith direction. The zenith read-head assembly  158  is attached to the zenith-axis frame  162 , which does not rotate in the zenith direction. The mechanical structure, motors, and bearings are essentially identical to the components of  FIG. 3 . As shown in  FIG. 7 , the size of the mounting plate has been greatly reduced from the first embodiment. Potential advantages of the second embodiment relative to the first are better availability of optical encoders and reduced cost for systems in which the highest performance is not essential. 
   It will be apparent to those skilled in the art that various modifications and variations can be made in the spherically mounted light source and tracker of the present invention without departing from the spirit or scope of the invention.