Patent Publication Number: US-7901270-B2

Title: Method and apparatus for precision polishing of optical components

Description:
CROSS-REFERENCE TO RELATED PATENT APPLICATIONS 
     This application is a continuation application of copending patent application Ser. No. 11/743,333, filed on May 2, 2007, which claims priority from U.S. provisional patent application Ser. No. 60/746,346 filed May 3, 2006, the entire disclosures of which are incorporated herein by reference. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     This invention was made with government support under one or more of Contract Numbers W31P4Q-05-C-R048 and W31P4Q-04-C-R101 awarded by the Defense Advanced Research Projects agency (DARPA); and Contract Numbers N41756-05-M-1390 and N68936-06-C-0010 awarded by the Navy Engineering Logistics Office and NAVAIR. The government has certain rights in this invention. 
    
    
     This invention relates in one embodiment to a method and apparatus for polishing objects, and more particularly to a method and apparatus for polishing optical elements. 
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     A method and apparatus for correcting surface errors, and for polishing objects comprising a wide variety of materials and shapes including precision optical surfaces and injection mold inserts having plano, concave, convex, spherical, and other complex surfaces. 
     2. Description of Related Art 
     Currently, many optical lenses are made beginning with a “blank” starting part (such blank part being an approximately formed and generally roughly finished piece) in several processing steps. The process steps typically include fine grinding, followed by conventional polishing techniques wherein the surface roughness and surface accuracy of the lens is significantly improved. This prior art process is sufficient for many conventional low-precision lenses. However, when the desired lens has a shape that is not spherical or plano and/or where such conventional methodologies cannot be applied (e.g. to aspherics), or where the lens has very high accuracy requirements, such prior art process is not sufficient. In such circumstances, the method and apparatus of the present invention is advantageous. 
     Heretofore, a number of patents and publications have disclosed methods, apparatus, and compositions for polishing of precision surfaces. United States Patent Application Publication No. US 2004/0229553 A1 of Bechtold et al., which is assigned to the assignee of the present invention and incorporated herein by reference, describes a tool, apparatus, and method for polishing objects. The tool has a rotatable drive wheel engaged with a polishing wheel by use of a polishing foil formed as a flexible belt. The polishing wheel may have a cavity within, the cavity being inflatable using a variety of fluids having a range of physical properties. The polishing wheel is adjustably positionable against an object to be polished by actuating means joined thereto. The apparatus comprises a multi-axis computer controlled machine to which the tool is attached. 
     In some circumstances, is preferable to configure such an apparatus with a large diameter drive pulley, relative to the polishing wheel. This provides a large length of wrap of the polishing belt around the drive pulley so that it does not slip, and it also provides a high belt speed at a relatively low drive pulley rotational speed. However, if one configures the tool and apparatus of the published application of Bechtold et al. with a large driven pulley and a small polishing wheel, it is less capable of polishing deeply concave surfaces. This is because the angle formed by the straight lengths of belt between the drive pulley and the polishing wheel is large, and may even exceed 90 degrees. Thus the polishing wheel and abrasive belt can not be located within an object with a deeply concave surface, since the belt will rub on the edges of the object before the polishing wheel reaches the concave surface. 
     SUMMARY OF THE INVENTION 
     Accordingly, embodiments of the present invention are provided that meet at least one or more of the following objects of the present invention. 
     It is an object of this invention to provide an apparatus and method for precision polishing of objects of a wide variety of materials and shapes. 
     It is an object of this invention to provide a method, a tool, and an apparatus that has the ability to perform polishing of object surfaces that are deeply concave in shape. 
     This invention relates to a method and apparatus for correcting figure errors, and for polishing a wide variety of materials and shapes including but not limited to precision optical surfaces, injection mold cavities, thin film coating dies, and the like. The method of the present invention provides for improving and further finishing of a variety of surfaces, ranging from a relatively rough ground surface to a polished surface. The shape of the surfaces of the objects may include deeply concave surfaces. 
     Typically the part being finished according to the present invention is measured with a coordinate measurement machine (CMM), a surface profilometer, an interferometer, a microscope or some other measuring instrument capable of giving surface roughness and/or profile data. The data from such measurement and analysis is then entered into a machine process-controlling computer that then manipulates the data into process parameters for improving or polishing the desired component by a polishing machine of the present invention. One or more iterations of the process of the present invention may be required to achieve the desired results. 
     In the preferred embodiment, a polishing assembly comprising a driven abrasive belt is attached to and moved by a rotary positioning device of the polishing machine. The part to be improved or polished, whether spherical, aspherical or parabolic in shape, is placed into the work piece spindle of the polishing machine. If such part is not axially symmetrical, it may be held in a braked position in the work piece spindle, or held in a fixture on a table of the machine. The abrasive belt is then compressed against and traversed in a path over the component. Numerous variables are able to be controlled as process parameters, so that the desired finishing results are achieved. 
     In accordance with the present invention, there is provided a polishing apparatus comprising a base for affixing structures thereto, a drive wheel connected to a rotatable shaft, a polishing wheel assembly, a polishing belt, and at least one routing wheel engaged with the polishing belt. The rotatable shaft is disposed in a housing that is joined to the base or formed therein. The polishing wheel assembly includes an elongated arm including a distal end, a proximal end joined to the base, and a rotatable polishing wheel supported at the distal end of the elongated arm. The polishing belt is made with an abrasive outer surface that is applied to the object to perform the polishing, and an inner surface for engagement with the perimeters of the drive wheel and the polishing wheel. The routing wheel is engaged with the outer surface of the polishing belt, such that the contact arc of the polishing belt with the polishing wheel differs from an arc of the polishing wheel perimeter extending from a first tangent line between the drive wheel and the polishing wheel to a second tangent line between the drive wheel and the polishing wheel. The rotating shaft may be driven by an electric, pneumatic, or hydraulic motor, thereby rotating the drive wheel and advancing the polishing belt along the perimeter of the polishing wheel and along the object to be polished. 
     The polishing belt may be supplied from a supply spool, engaged with the polishing wheel, and wound up on a take-up spool. In the preferred embodiment, the polishing belt is a continuous loop of belt. The diameter of the drive wheel is preferably greater than the diameter of the polishing wheel. The contact arc of the polishing belt with the polishing wheel is greater than an arc of the polishing wheel perimeter extending from a first tangent line between the drive wheel and the polishing wheel to a second tangent line between the drive wheel and the polishing wheel. By configuring the polishing apparatus in this manner, the routing of the belt provides a sharper angle between the free spans of belt material. The apparatus is thus much more capable of reaching the recessed surface of a highly concave object such as a parabolic work piece, while having a wider range of tool paths and angles to polish such an object. 
     In one embodiment, the elongated arm of the polishing wheel assembly is comprised of a housing including a first portion and a second portion. Each of the housing portions comprise a proximal end and a distal end, with each of the distal ends including a socket formed therein. A first bearing is disposed between the socket of the first housing half and the polishing wheel, and a second bearing is disposed between the socket of the second housing half and the polishing wheel. The sockets may be cylindrically shaped for receiving bearings comprised of an outer race, an inner race, and a plurality of rolling members, such as balls (i.e. a ball bearing) or rollers (i.e. a roller bearing or needle bearing). Alternatively, the sockets may be substantially spherical shaped, for receiving bearings that are simply spherical balls. 
     The polishing wheel may be comprised of a rigid interior portion including the first socket and the second socket, and an elastic exterior portion. The perimeter of the polishing wheel may be arcuate shaped. The polishing wheel may have an elongated barrel or cylindrical shape having a ratio of length to diameter greater than one. 
     The polishing apparatus may further include a tensioning wheel engageable with the polishing belt for taking up slack in the belt at the beginning and during a polishing operation. The tensioning wheel may be engaged with the belt by a linear or rotary actuator that deploys the tensioning wheel against the inner or outer surface of the belt. 
     The polishing apparatus may further include a dressing assembly including a stripping surface that is contactable with the outer surface of the polishing belt. The stripping surface may be a stick of material or a rotating wheel that is applied to the polishing belt. In a preferred embodiment, the dressing assembly is comprised of a dressing belt having an outer surface that is the stripping surface. The dressing belt may be stored on and deployed from a supply spool and wound up on a take-up spool after engagement with the polishing belt outer surface. 
     The polishing apparatus may further include a polishing spot measurement tool comprised of a deployable housing containing a light source and a light detector. The tool is preferably retracted when not in use, and deployed by an actuator when a spot measurement is needed. The light source may be a super luminescent light emitting diode (SLED). The measurement is non-contact, i.e. the tool does not touch the surface of the polished spot being measured. 
     The polishing apparatus may further include polishing wheel position measuring device comprising a laser and a photodetector. 
     A computer numerically controlled (CNC) machine may be used to articulate the polishing wheel assembly against the surface of the object to be polished. The CNC machine may include a first linear slide movable along a first axis, disposed upon a machine platform; a second linear slide movable along a second axis, engaged with the first linear slide, with the second axis disposed orthogonally to the first axis; a third linear slide movable along a third axis, with the third axis disposed orthogonally to the first and second axes; and a first rotatable positioning device engaged with the third linear slide, the first rotatable positioning device being rotatable around an axis parallel to the second axis and further comprising a turret head. The base of the polishing apparatus is joined to the turret, thereby enabling the polishing wheel assembly to be articulated against the surface of the object to be polished. The CNC machine may further comprise a rotatable spindle for holding an object to be polished. The spindle may hold the object stationary or rotate the object as the polishing wheel and belt are moved along the surface of the object to be polished. 
     In accordance with the present invention, there is provided a method of polishing objects using an apparatus comprised of a rotary positioning device comprising a turret rotatable around a turret axis; a base for affixing structures thereto, the base mounted on the turret and comprising a plate having a surface defining a plane perpendicular to the turret axis; a drive wheel connected to a rotatable shaft, the drive wheel having a perimeter, and the rotatable shaft disposed in a housing joined to the plate and having a rotational axis that is substantially parallel to the turret axis; and a polishing wheel assembly. The polishing wheel assembly may be comprised of an elongated arm including a proximal end joined to the base, and a distal end; a rotatable polishing wheel supported at the distal end of the elongated arm, the rotatable polishing wheel having a perimeter; and a polishing belt comprising an inner surface and an outer surface, the inner surface engageable with the perimeters of the drive wheel and the polishing wheel. The method is comprised of contacting the outer surface of the polishing belt to a contact region of the surface of the object; and controlling the contact region by rotating the elongated arm around the turret axis. 
     The apparatus of the present invention and associated methods for using the apparatus are advantageous because the apparatus can be adapted for the polishing of a variety of materials and shapes, particularly those objects having deeply concave shapes. As a result of the invention, articles of manufacture such as precision optics, injection mold cavities, and thin film coating dies can be polished with high precision at a high throughput and low cost. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention will be described by reference to the following drawings, in which like numerals refer to like elements, and in which: 
         FIG. 1  is a perspective view of a computer numerically controlled machine that includes the polishing apparatus of the present invention; 
         FIG. 2A  is a front elevation view of a first embodiment of the polishing apparatus; 
         FIG. 2B  is a front elevation view of a second embodiment of the polishing apparatus; 
         FIG. 3A  is a front elevation view of a third embodiment of the polishing apparatus including additional guidance wheels and components to enable dressing of the polishing belt; 
         FIG. 3B  is a front elevation view of a variant of the embodiment of  FIG. 3B  including spools of a secondary belt for dressing of the polishing belt; 
         FIG. 4  is a perspective view that depicts the polishing of an object by the polishing apparatus; 
         FIG. 5  is a front elevation view that depicts the polishing of an object having a concave surface by the polishing apparatus; 
         FIG. 6  is a perspective view of one polishing wheel assembly comprising a pair of spherical ball bearings; 
         FIG. 7  is a side elevation view of the polishing wheel assembly of  FIG. 5 , taken along the line  7 - 7  of  FIG. 6 ; 
         FIG. 8  is a front elevation view of the polishing wheel assembly of  FIG. 5 , taken along the line  8 - 8  of  FIG. 6 ; 
         FIG. 9  is an exploded perspective view of the polishing wheel assembly of  FIG. 6 ; 
         FIG. 10  is a front elevation view of the interior of one housing half of the polishing wheel assembly; 
         FIG. 11  is a cross-sectional view of the polishing wheel assembly, taken along line  11 - 11  of  FIG. 8 ; 
         FIG. 12  is a perspective view of an alternative polishing wheel assembly comprising a pair or race-type bearings; 
         FIG. 13A  is an exploded perspective view of the polishing wheel assembly of  FIG. 12 ; 
         FIG. 13B  is a cross-sectional view of the polishing wheel assembly of  FIG. 12  taken along line  13 B- 13 B of  FIG. 12 ; 
         FIG. 14  is a side elevation view of a first device used to clean the polishing belt during operation of the polishing apparatus; 
         FIG. 15A  is a side elevation view of a second device and a third device used to clean the polishing belt during operation of the polishing apparatus; 
         FIG. 15B  is a top view of the second device of  FIG. 15A , taken along the line  15 B- 15 B of  FIG. 15A ; 
         FIG. 16  is a perspective view of a position measurement device for detecting the polishing wheel of the apparatus to an object to be polished; and 
         FIG. 17  is a front elevation view of the polishing apparatus of  FIG. 3B , depicting a spot testing device of the polishing apparatus. 
     
    
    
     The present invention will be described in connection with a preferred embodiment, however, it will be understood that there is no intent to limit the invention to the embodiment described. On the contrary, the intent is to cover all alternatives, modifications, and equivalents as may be included within the spirit and scope of the invention as defined by the appended claims. 
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     For a general understanding of the present invention, reference is made to the drawings. In the drawings, like reference numerals have been used throughout to designate identical elements. In describing the present invention, a variety of terms are used in the description: 
     As used herein, the term figure error (or form error) is the measured global deviation from the desired surface shape e.g., a sphere, asphere or polynomial geometric shape. 
     As used herein, the term polishing, when used in reference to a work piece to be finished, is meant to indicate a chemical and/or mechanical process that ablates material from a surface. 
     In the following specification, for the sake of linguistic simplification, only optical components, also known as precision optics, or optics generally, are typically mentioned as the work piece. However, it is to be understood that all lenses, spherical and aspherical, conformal optics, mirrors, plano shapes, injection mold components, coating dies, and other articles of manufacture that require highly polished accurate surfaces are also included in the description, and are to be considered as being within the scope of the present invention. Materials that may be finished using the method and apparatus of the present invention include, but are not limited to brittle amorphous materials such as e.g., glass, ceramics, infrared materials such as quartz, visible light and ultraviolet light transmissive materials, and the like. Also included are metals such as e.g., tool steel, stainless steel, and the like; crystalline materials such as e.g. silicon; and any other work pieces requiring high finish and form specifications. 
       FIG. 1  is a perspective view of a machine that includes the polishing apparatus of the present invention. Machine  10  is preferably a computer numerically controlled (CNC) machine, and is described with respect to the orthogonal X axis  2 , Y axis  4 , and Z axis  6 . A preferred orientation of machine  10  and the polishing apparatus  100  thereof with respect to the X, Y, and Z axes is as shown in  FIG. 1 . However, it is to be understood that machine  10  may be oriented and operated in positions other than as depicted in  FIG. 1  with respect to horizontal X and Y axes  2  and  4  and the vertical Z axis  6 . 
     Machine  10  comprises a machine platform  12  that supports Y-axis linear slide  40 , the motion of which is bi-directional along Y-axis  4  as indicated by arrow  99 . Linear slide  20 , the motion of which is bi-directional along X-axis  2  as indicated by arrow  98 , is mounted upon Y-axis linear slide  40 . These linear slides  20  and  40  are preferably both computer numerically controlled (CNC) positioning devices, providing programmable motion of spindle  80  in the X-Y plane. 
     Machine  10  further comprises vertical slide  60  attached to polishing machine frame or plate  14 , which is joined to platform  12 . The motion of vertical slide  60  is bi-directional along Z-axis  6  as indicated by arrow  97 . Machine  10  is preferably provided with turret  70 , which is mounted upon vertical slide  60 , and which is rotatable around B axis  5  (parallel to Y-axis  4 ) as indicated by bidirectional arcuate arrow  96 . Polishing apparatus  100  of the present invention, to be described subsequently herein, is attached to turret  70 . 
     Machine  10  further comprises a work piece spindle  80  mounted upon linear slide  20 . Spindle  80  is a vertically disposed spindle, the central axis  7  of which may be parallel to Z-axis  6 . Rotatable work piece chucking device  82  is attached to the end of work piece spindle  80 . The work piece  90  to be polished is engaged and held by chuck  82  and may be rotated by spindle  80  around the central rotary axis  7  thereof as indicated by arrow  95 . This spindle  80  is also a positioning/variable speed device which will allow for deterministic finishing of form errors that may not be rotationally symmetric such as astigmatism in a spherical optic through controlled slowing or speeding up of the rotation of spindle  80  during each revolution. The motion of spindle  80  is bidirectionally programmable along the X axis  2  and the Y axis  4 . 
     Machine  10  further comprises a polishing apparatus  100  mounted on turret  70 . A polishing wheel and polishing belt (to be described subsequently herein), which form part of polishing apparatus  100 , may be brought into contact with work piece  90  by downward motion of vertical slide  60 , and by rotary motion of turret  70  as indicated by arcuate arrow  94 . 
     The motion of polishing apparatus  100  with respect to work piece  90 , or work pieces of a variety of different shapes is thus fully programmable, and has great flexibility. Machine  10  may articulate apparatus  100  over the surface of object  90  in a complex path in X-Y-Z space. For example, apparatus  100  may be generally advanced along a linear path, but with a circular motion superimposed on such linear path. Such a tool path is known in the art as a trichordal path. Alternatively, such tool paths may include arcuate, zigzag, sinusoidal, or other combinations of motion so as to enhance the removal rate of material from object  90 , and to prevent the occurrence of any “grooving” effect in the object surface during the polishing thereof. 
     Polishing assembly  100  is preferably configured with a large diameter drive pulley, relative to the polishing wheel. This provides a large length of wrap of the polishing belt around the drive pulley so that it does not slip, and it also provides a high belt speed at a relatively low drive pulley rotational speed. However, in order to enable the polishing apparatus  100  to access the recessed surfaces of deeply concave objects, polishing apparatus  100  is comprised of one or more additional routing wheels to provide the polishing belt with a large angle of wrap around the polishing wheel, and a highly acute angle formed by the two lengths of polishing belt immediately adjacent to the polishing wheel. This feature is best understood with reference to  FIGS. 2A-3B , which depict various embodiments of polishing apparatus  100 . 
       FIG. 2A  is a front elevation view of a first embodiment  101  of the polishing apparatus  100 . Apparatus  101  is comprised of a base  110  for affixing and/or supporting various objects thereto. Base  110  is preferably a rigid metal plate such as e.g., aluminum. Drive wheel  120  of apparatus  110  is joined to a rotatable shaft  122 , which is disposed in a housing  124  that may be joined to the base  110 , or formed therein. Housing  124  comprises a bushing or bearing(s) (not shown) so that shaft  122  is rotatable therein. The housing may be a separate bearing block that is joined to the housing, or if the base is sufficiently thick walled, the bearing housing may be formed in the base itself. Rotatable shaft  122  is driven by motor  126 , or other suitable power transmission means, which may include various gears, pulleys, and other drive components (not shown). Drive wheel  120  is disposed on the outer side of base  110 , while motor  126  and housing  124  (if separate from base  110 ) are disposed on the inner side, with rotatable shaft  122  passing through a hole (not shown) in base  110 . 
     Apparatus  101  further comprises a polishing wheel assembly  200  comprised of an elongated arm  211  including a distal end  213 , a proximal end  215  joined to the base, and a rotatable polishing wheel  240  supported at the distal end  213  of the elongated arm  211 . In one embodiment, the elongated arm  211  of the polishing wheel assembly  200  is comprised of a housing  210  including a proximal end  212  and a distal end  214 . The rotatable polishing wheel  240  is supported within the housing  210  at the distal end  214  thereof. Housing  210  is joined to base  110  by a support rod  230 . A polishing belt  130  is fitted to apparatus  101  for the purpose of polishing work piece  90 . Polishing belt  130  comprises an inner surface  132  and an outer surface  134 . The inner surface  132  of belt  130  is engageable with the perimeters of the drive wheel  120  and the polishing wheel  240 . The outer surface  134  of belt  130  is embedded with abrasive particles for performing the polishing material removal when contacted with the object  90 . 
     Apparatus  101  further comprises at least one routing wheel  140  that is engaged with the outer surface  134  of the polishing belt  130 . In general, the one or more routing wheels position the free span  131  of polishing belt that is approaching the polishing wheel and the free span  133  of polishing belt that is departing from the polishing wheel at a sharply acute angle to each other, or even substantially parallel to each other. By configuring the polishing apparatus  100 - 104  ( FIGS. 1-3B ) in this manner, the apparatus is this much more capable of reaching the recessed surface of a highly concave object such as a parabolic work piece, while having a wider range of tool paths and angles to polish such an object, as will be explained presently in more detail. This manner of configuring the polishing apparatus is particularly useful when the diameter of the drive wheel  120  is much larger than the polishing wheel  240  as shown in  FIGS. 2A-3B ; or when the drive wheel is offset from the longitudinal axis  298  of the elongated arm  211  of the polishing wheel assembly  200 . 
     Thus in this configuration, when operating apparatus  101 , routing wheel  140  is engaged with the outer surface  134  of the polishing belt  130  such that the contact arc of the polishing belt with the polishing wheel differs the contact arc that would occur if the polishing belt  130  were directly fitted to the drive wheel  120  and the polishing wheel  240  without one or more routing wheels. When no routing wheels are used, the contact arc of the belt on polishing wheel  240  extends from a first tangent line between the drive wheel and the polishing wheel (which would also be the position of the approaching free span of the polishing belt  130 ) to a second tangent line between the drive wheel and the polishing wheel (which would also be the position of the departing free span of polishing belt  130 ). It is beneficial to use one or more routing wheels to reposition the approaching and departing free spans of belt such that they are generally parallel to the z-axis, or such that the highly acute angle formed between them is approximately bisected by a line parallel to the z-axis. This results in a contact arc of the polishing belt  130  on the polishing wheel  240  having a center point at the “six-o&#39;clock” position when the polishing machine is in its neutral starting position as shown in  FIG. 1 , with the longitudinal axis  298  of elongated arm  211  of the polishing wheel assembly  200  disposed vertically. This positioning of the contact arc centered at “six-o&#39;clock” in turn provides the most versatile overall polishing capability, with the maximum possible range of tool paths that can be used on the object to be polished. 
     In one embodiment (not shown), belt  130  may be provided as a reel of material that is wound on a first spool. Belt  130  may be threaded through the polishing apparatus in a manner similar to that shown in apparatus  100 - 104  of  FIGS. 2A-3B , except that the leading end of belt  130  is fastened to a take-up spool. Belt  130  is then used in a polishing operation in a reel-to-reel manner, such that any given location on belt  130  only makes a single pass over work piece  90 . In another embodiment (not shown), belt  130  may be provided in a “cassette tape” like configuration. In operation, a long length of belt is unwound around a first spool, and wound up on a second spool after passage over the work piece. When the stored belt on the first spool is fully unwound and taken up on the second spool, the direction of the winding is reversed, and the belt is reused. In this manner, a large amount of reusable belt length may be provided for polishing. For both of these embodiments, one or more routing wheels are used to position the approaching span  131  of polishing belt  130 , the departing span  133  of polishing belt  130 , and the contact arc of polishing belt  130  on polishing wheel  240  in the optimum manner as described above. 
     In the preferred embodiment, the polishing belt is a continuous loop of belt, and the diameter of the drive wheel is greater than the diameter of the polishing wheel. In this embodiment, the one or more routing wheels are positioned such that the contact arc of the polishing belt  130  with the polishing wheel  240  is greater than the arc of the polishing wheel perimeter that extends from a first tangent line between the drive wheel  120  and the polishing wheel  240  to a second tangent line between the drive wheel  120  and the polishing wheel  240 . 
     Referring again to  FIG. 2A , routing wheel  140  engages with the outer surface  134  of the polishing belt  130  at a contact arc  141  that is disposed between a first tangent line  199  and a second tangent line  198  between the drive wheel  120  and the polishing wheel  240 . In the embodiment  101  depicted in  FIG. 2A , it can be seen that the second tangent line  198  in  FIG. 2A  is actually the portion  131  of the belt  130  between the drive wheel  120  and the polishing wheel  240 . This is in contrast to the embodiment  102  of  FIG. 2B , which is provided with first and second routing wheels, as will be describe subsequently herein. Additionally, in the embodiment  101  depicted in  FIG. 2A , the contact arc of polishing belt  130  on polishing wheel  240  is greater than the arc of polishing wheel  240  that is between first tangent line  199  and a second tangent line  198  between the drive wheel  120  and the polishing wheel  240 . 
     It can be seen that by providing routing wheel  140  engaged with the outer surface  134 , polishing belt  130  is routed so that the angle  197  between the free span portions  131  and  133  of belt  130  is a highly acute angle. For example, angle  197  in  FIG. 2A  is about 10 degrees. In contrast, the angle  196  between tangent lines  199  and  198 , which would be the angle between the free span portions  131  and  133  of belt  130  if routing wheel  140  were not present, is about 22 degrees. Thus the use of at least one routing wheel in routing the belt to provide a sharper angle between the free spans provides an apparatus that is much more capable of reaching the recessed surface of a highly concave object such as a parabolic work piece  92  depicted in  FIG. 2A , while having a wider range of tool paths and angles to polish such an object. 
     In setting up apparatus  101 , the polishing belt  130  may be sized for a snug fit to its path around drive wheel  120 , polishing wheel  240 , routing wheel  140 , and various other routing wheels if such are used. In this instance, polishing belt  130  is stretched slightly in order to fit it over the wheels  120 ,  240 ,  140 , etc. However, it is preferable that belt  130  be sized slightly longer than is needed to fit around drive wheel  120 , polishing wheel  240 , and within routing wheel  140  to enable easy fitting of belt  130 . Support rod  230  of polishing wheel assembly  200  may be provided with slots (not shown) for engagement with fasteners (not shown), so that rod  230  may be slid downwardly away from drive wheel  120  to take up slack in belt  130  and engage it with portions of the perimeters of drive wheel  120  and polishing wheel  240 . 
     However, it is preferable to provide an actuator that is operatively connected to a tensioning wheel to take up the slack of belt  130 . The actuator may be controlled by the process control computer that runs the overall machine  10  (see  FIG. 1 ). The actuator is preferably a simple linear actuator, but may also be a rotary actuator, or other cam-type mechanism that displaces the tensioning wheel toward polishing belt  130 . The tensioning wheel may be disposed such that it engages with the inner surface  132  or the outer surface  134  of polishing belt  130 , depending upon the overall design of the polishing apparatus. One exemplary belt tensioning arrangement is depicted in  FIG. 2A . Linear actuator  150  is operatively connected to tensioning wheel  152  by rod and yoke  154 . Linear actuator  150  deploys and retracts tensioning wheel  152  to and from belt  130 , as indicated by bidirectional arrow  195 , providing belt tension when deployed, and slack when retracted. 
       FIG. 2B  is a front elevation view of a second embodiment  102  of the polishing apparatus  100 . Polishing apparatus  102  is similar to polishing apparatus  101  of  FIG. 2A , except that polishing apparatus  102  comprises two routing wheels  140  and  142  that guide belt  130  to provide a narrow angle between free spans  131  and  133  thereof. Additionally, elongated arm  211  is comprised of a linear actuator  232 , which extends and retracts polishing wheel assembly  200  as indicated by arrow  194  in order to take up or provide slack in belt  130 . It will be apparent that arm  230  could be provided as shown in  FIG. 2A , with a linear actuator  150  and tension wheel  152  for the purpose of controlling belt slack. It can also be seen that the routing wheels  140  and  142  are positioned such that the contact arc of the polishing belt  130  with the polishing wheel  240  is greater than the arc of the polishing wheel perimeter that extends from the first tangent line  198  between the drive wheel  120  and the polishing wheel  240  to the second tangent line  199  between the drive wheel  120  and the polishing wheel  240 . 
       FIGS. 3A and 3B  are front elevation views of a third embodiment  103  and a fourth embodiment  104  of the polishing apparatus  100 , including additional routing wheels and components to enable dressing of the polishing belt. Referring first to  FIG. 3A , polishing apparatus  103  is housed in an enclosure  105 , which is comprised of base  110 , a horizontally extending wall  106 , and cover  108  ( FIG. 1 ), the shape of which is matched to mate with the outer edge of wall  106 . Enclosure  105  isolates most of the moving polishing belt and wheels from the machine operator, thereby providing greater operator safety. 
     Polishing apparatus  103  is similar to polishing apparatus  101  and  102  of  FIGS. 2A and 2B , and comprises two routing wheels  140  and  142  that guide belt  130 ; actuator  232 , which extends and retracts polishing wheel assembly  200 ; and actuator  150 , which is operatively connected to tensioning wheel  152 . Belt  130  may thus be provided with additional slack. Actuator  232  may be used to take up the majority of the belt slack after polishing belt  130  is fitted to apparatus  103 , with actuator  150  used to take up the remaining small amount of slack in order to engage belt  130  with drive wheel  120 , polishing wheel  240 , and the various guide wheels and other belt devices. Actuator  232  is operatively connected to polishing wheel assembly  200  by rod  234 . 
     Apparatus  103  may further include routing wheels  148  and  149 , the position of which may be adjusted in the horizontal direction to provide adjustability of the angle between belt free spans  131  and  133 . In this embodiment, because of the particular arrangement of the belt  130  around routing wheels  140 ,  142 , and  144 , and tensioning wheel  152 , routing wheels  148  and  149  are engaged with the inner surface of polishing belt  130 . Apparatus  103  may further include guide wheels  144  and  146  to provide the capability of guiding belt  130  past or through additional belt treating devices, such as belt dresser  300  to be described subsequently herein. 
       FIG. 3B  is a front elevation view of a variant of the embodiment of  FIG. 3A  including spools of a secondary belt for dressing of the polishing belt. Polishing apparatus  104  is similar to polishing apparatus  103  of  FIG. 3A  and may comprise a first routing wheel  140  that guides belt  130 ; actuator  232 , which extends and retracts polishing wheel assembly  200 ; and actuator  150 , which is operatively connected to tensioning wheel  152 . Apparatus  104  may further include routing wheels  148  and  149 . In this embodiment, routing wheels  148  and  149  are engaged with the outer surface  134  of polishing belt  130 , thereby providing substantially parallel and more narrowly spaced belt free spans  131  and  133 . Apparatus  103  may further include guide wheels  144  and  146  to provide the capability of guiding belt  130  past or through additional belt treating devices, such as belt dresser  301  to be described subsequently herein. 
       FIG. 4  is a perspective view that depicts the polishing of an object by the polishing apparatus, and  FIG. 5  is a front elevation view that depicts the polishing of an object having a concave surface by the polishing apparatus. Polishing belt  130  is wrapped around and engaged with a portion of the perimeter of polishing wheel  240 , such that the motion of belt  130  as indicated by arrows  193  and  192 , result in the rotation of polishing wheel  240  as indicated by arcuate arrow  299 . Polishing wheel  240  preferably has an elastic outer surface and deforms when pressed onto work piece  90 . Polishing belt  130  thus contacts work piece  90  in a generally elliptical spot at contact region  91 . 
     The size and shape of the spot, and the rate of material removal at contact region  91  depends upon a number of operating parameters, and can be measured by certain features of the apparatus, as will be described subsequently herein. The location of contact region  91  on work piece  90  is determined by the motion of polishing apparatus  100  in the x and y planes  2  and  4  (see  FIG. 1 ), the rotation of the polishing apparatus  100  around axis  5  (see  FIG. 1 ) as indicated by arcuate arrow  94 , the positioning of work piece  90  along the z axis  6  (see  FIG. 1 ) and the rotation of work piece  90  by spindle  80  as indicated by arrow  95 . 
     Referring again to  FIGS. 3A and 4 , the polishing machine  10  may further comprise liquid supply tubes for delivering liquid materials to the polishing assembly and the work piece. Supply tube  32  may be used to deliver a liquid stream of coolant, such as water, or a typical machine tool coolant liquid. In one preferred embodiment, tube  32  is used to deliver a mist of air and fine water droplets. Supply tube  34  may be used to deliver a stream of abrasive polishing slurry, such as cerium oxide. Such a slurry may be used when polishing belt is not provided with abrasive particles on the outside surface or impregnated therein. In a further embodiment, a third fluid delivery tube (not shown) is provided so that machine  10  has the capability of providing abrasive slurry, liquid coolant, and an air/liquid mist coolant to the contact region  91  of the work piece. 
     With regard to the polishing apparatus  100 - 104  of  FIGS. 1-3B , the actuators  152  and  232 , and other actuators subsequently described herein may be pneumatic or hydraulic cylinders, rodless cylinders, stepper motors or other electromechanical actuators used to provide linear motion, either directly, or through rotational motion converted to linear motion such as by a cam or a coupled rod. Such actuators may be further provided with position sensing means, and/or position control means, and communication means for control thereof by an external process controller. 
     Suitable polishing belts may include the two piece polishing foil type belts comprised of an elastic inner band, and an abrasive ring outer band as disclosed in the aforementioned United States Patent Application Publication No. US 2004/0229553 A1 of Bechtold et al. 
     Alternatively, the polishing belt  130  may also be of unitary construction. Such a belt may be a solid band comprising multiple layers, including a structural layer of resin and fiber that provides structural strength and wear resistance needed to run on the various wheels without breaking or wearing; and an abrasive layer adhered or coated on the exterior, which provides the abrasive material used to polish the work piece  90 . Examples of suitable single band polishing belts include belts comprising diamond, alumina, and/or silicon carbide particles. In one embodiment, a belt made of TRIZACT® abrasive manufactured and sold by the 3M Corporation (Minnesota Mining and Manufacturing) of St. Paul, Minn. is used. Depending upon the scale of polishing apparatus  100 , the width of polishing belt  130  may vary from about 0.125 inch to about 4 inches. In preferred embodiments, the width of belt  130  is between about 0.375 inch and 1 inch. 
     Depending upon the particular setup of polishing apparatus  100 , the circumference of belt  130  may be between about 20 inches and about 60 inches, although if polishing apparatus  100  is scaled up or down, belt  130  may need to be dimensioned outside of this range. A large circumference of belt  130  is advantageous from the standpoint of “abrasive capacity,” i.e. belt  130  has a greater surface area to perform the polishing of the work piece  90 , and thus does not wear out as quickly with the finishing of the work piece. 
     In the applicants&#39; aforementioned U.S. provisional patent application Ser. No. 60/746,346, the applicants have described and shown in  FIGS. 6-13  one embodiment of a polishing wheel assembly comprised of a first portion and a second portion, each of the housing portions comprise a proximal end and a distal end, with a polishing wheel disposed between bearings held in the distal ends of the housing portions. The entire disclosure of this provisional patent application is incorporated herein by reference. 
       FIGS. 6-11  and  FIGS. 12-13B  of the instant application depict two alternative embodiments of polishing wheel assemblies that can be used with the applicants&#39; polishing apparatus. More specifically,  FIG. 6  is a perspective view of a first alternative polishing wheel assembly;  FIG. 7  is a side elevation view of the polishing wheel assembly of  FIG. 6 , taken along the line  7 - 7  of  FIG. 6 ;  FIG. 8  is a front elevation view of the polishing wheel assembly of  FIG. 6 , taken along the line  8 - 8  of  FIG. 6 ;  FIG. 9  is an exploded perspective view of the polishing wheel assembly of  FIG. 6 ;  FIG. 10  is a front elevation view of the interior of one housing half of the polishing wheel assembly taken along line  10 - 10  of  FIG. 9 ;  FIG. 11  is a cross-sectional view of the polishing wheel assembly, taken along line  11 - 11  of  FIG. 8 . 
     Referring first to  FIGS. 6-8 , polishing wheel assembly  201  is comprised of a housing  250  comprising a first portion  251  and a second portion  252 . First housing portion  251  comprises a proximal end  253  and a distal end  255 , and second housing portion  252  comprises a proximal end  254  and a distal end  256 . Referring also to  FIGS. 9-11 , each of the distal ends  255  and  256  include sockets  257  and  258  formed therein for holding ball bearings  261  and  262  therein. Polishing wheel  240  may be comprised of a rigid interior portion  242  including a first socket  241  and a second socket  243 , and an elastic exterior portion  246 , which may be formed of suitable elastomers such as rubber, polyurethane, or silicone, or a harder or higher durometer polymer. It is preferable that such elastomeric material be of a Shore A durometer between about 10 and about 90, with the particular durometer depending upon the polishing application. The elastic exterior portion  246  of polishing wheel  240  may be provided with a flat surface  279 , or a generally arcuate surface  279 A, which may be a spherical surface. Polishing belt  130 , which is under tension, conforms to this surface  279 / 279 A as it wraps around the perimeter of polishing wheel  240  during a polishing operation. This wrapping action results in better tracking of polishing belt  130  on polishing wheel  240 . In addition, the radius of curvature and/or the curvature profile (spherical, elliptical, hyperbolic, etc.) of an arcuate surface  279 A partially determines the tool spot size during a polishing operation, in combination with the hardness of elastic wheel portion  246  and the application force of polishing wheel  240  against the work piece  90  ( FIG. 5 ). 
     A first ball bearing  261  is disposed between the socket  257  of the first housing portion  251  and the first socket  241  of the polishing wheel  240 ; and a second ball bearing  262  is disposed between the socket  258  of the second housing portion  252  and the second socket  243  of the polishing wheel  240 . Polishing wheel  240  is thus suspended and rotatable between housing portions  251  and  252  by virtue of being supported by ball bearings  261  and  262 . 
     The applicants&#39; polishing wheel  240  is precisely suspended and runs with minimal friction or vibration, which is advantageous in performing highly precise polishing operations with the applicants&#39; polishing assembly. In the embodiment of  FIGS. 6-11 , in which a pair of ball bearings are used, this precise suspension of the polishing wheel  240  is achieved by certain features of the housing portions  251  and  252 , the polishing wheel  240 , and the applicants&#39; preferred process for fabricating the overall polishing wheel assembly. 
     The process comprises the first step of providing the housing portions  251  and  252 , wherein when housing portions  251  and  252  are fastened to each other, the bearing sockets  257  and  258  are positioned opposite of each other. In one preferred embodiment depicted herein, housing portions  251  and  252  are made as half portions that are identical to each other. Referring to  FIGS. 9 and 10 , it can be seen that when identical housing halves  251  and  252  are turned to face each other, the housing halves can be joined together. Dowel pins  259  of one housing half are mated with dowel holes  260  of the opposite housing half to accurately align the two housing halves when they are joined to each other. Through holes  263 , which are adjacent to hex sockets  264 , are provided to accept threaded screws and hex nuts (not shown) for joining halves  251  and  252  to each other. 
     Wheel  240  and ball bearings  261  and  262  are also provided to form the wheel assembly  200 . Referring in particular to  FIG. 11 , in an intermediate fabrication step, a moldable bearing material is provided in liquid or putty form, such bearing material being flowed into the sockets  257  and  258  of housing portions  251  and  252 , and into the sockets  241  and  243  of polishing wheel  240 . The ball bearings  261  and  262 , coated with a slight amount of release agent such as a silicone oil, are placed in the sockets  241  and  243  of the polishing wheel  240 , and then the wheel  240  and bearings  261  and  262  are placed in housing half  251  with bearing  261  in socket  257 . Housing half  252  is temporarily fastened to housing half  251  with bearing  262  in socket  258 . Any excess bearing material may be vented out through holes  265  and  266  of housing portions  251  and  252 . Polishing wheel  240  may also be also provided with recesses for displacement of excess bearing material within wheel  240 . The moldable bearing material  270  is allowed to cure into a hardened state. 
     Precise alignment of polishing wheel  240  within housing  250  may be further attained by the placement of a pair of O-rings (not shown) in O-ring grooves in housing portions  251  and  252 , and O-ring grooves in wheel rim  242  as described in the applicants&#39; aforementioned provisional patent application 60/746,346. These O-rings help to maintain a uniform separation gap between the polishing wheel  240  and the housing portions  251  and  252 . 
     The moldable bearing material is preferably a hard, low friction material with self-lubricating properties. One preferred moldable bearing material is MOGLICE®, which is manufactured by Diamant Metallplastic GMBH of Moenchengaldbach West Germany. This material is provided in liquid or putty form, and cures into a hard solid material with a low coefficient of friction. One particular preferred formulation of MOGLICE® is Moglice Putty Hard, which is a no-slump putty that can be applied to vertical or overhead surfaces without running or dripping. 
     After the moldable bearing material  270  has fully cured, the housing portions  251  and  252  are separated from each other. Any small excess material that has extruded into the gap between the polishing wheel  240  and the housing portions  251  and  252  is trimmed or machined away. Additionally, the circular ring of contact between the ball bearings  261  and  262  and the sockets  241  and  243  of the polishing wheel  240  may also be machined slightly to be slightly recessed from the cured bearing material  270 , so that the ball bearings  261  and  262  only run on the cured bearing material  270  when the polishing wheel assembly  201  is reassembled. 
     Thus by using the components of the applicants&#39; polishing wheel assembly, along with a low-friction moldable bearing material, a highly precise, smoothly running durable assembly is made, without the requirement that the individual housing portions  251  and  252  and the polishing wheel  240  be made with high precision, which would make such components more costly. 
     Additional features of the housing portions  251  and  252  are now described. Referring to  FIG. 10 , housing portions  251  and  252  are provided with upper and lower recesses  275  and  277 , which serve to improve melt flow and reduce the material usage when the housing portions are made of molded plastic or composite material. Additionally, when housing portions  251  and  252  are fitted together, the upper recesses  275  form a rectangular cavity which can accept support bar  234  (see  FIG. 6 ), which is used to joint the polishing wheel assembly to base  110  (see  FIG. 3 ). 
     Housing portions  251  and  252  may be made of any suitable rigid structural material. The housing portions are preferably made of a molded polymer material. In one preferred embodiment, housing portions  251  and  252  are made of glass fiber-reinforced acrylonitrile butadiene styrene (ABS.) Housing portions  251  and  252  may also be made of a metal such as aluminum, steel, stainless steel, or brass. 
       FIG. 12  is a perspective view of a second alternative polishing wheel assembly comprising a pair or race-type bearings;  FIG. 13A  is an exploded perspective view of the polishing wheel assembly of  FIG. 12 ; and  FIG. 13B  is a cross-sectional view of the polishing wheel assembly of  FIG. 12  taken along line  13 B- 13 B of  FIG. 12 . Polishing wheel assembly  202  is comprised of a housing  280  comprising a first portion  281  and a second portion  282 . First housing portion  281  comprises a proximal end  853  and a distal end  285 , and second housing portion  282  comprises a proximal end  284  and a distal end  286 . Each of the distal ends  285  and  286  include generally cylindrical sockets  287  and  288  formed therein for holding race-type bearings  291  and  292  therein. 
     Bearings  291  and  292  are comprised respectively of inner races  293  and  294 , outer races  295  and  296 , and a plurality of rolling members (not shown) contained within the bearing races. Polishing wheel  220  may be comprised of a rigid interior portion or hub  222 , and an elastic exterior portion  226 , which may be formed of suitable elastomers as described previously herein for polishing wheel  240 . Polishing wheel  220  is further comprised of spindle  224 , which is provided with a close-tolerance fit within hub  222  and within the inner races  293 / 294  of bearings  291 / 292 , in order to enable smooth rotation of wheel  240 . The perimeter of the polishing wheel  220  may be arcuate shaped, as described previously herein. The polishing wheel  220  may have an elongated barrel or cylindrical shape having a ratio of length to diameter greater than one. 
     The polishing apparatus may further include a dressing assembly including a stripping surface that is contactable with the outer surface of the polishing belt. The stripping surface may be a bar or stick of material or a rotating wheel that is applied to the polishing belt. In a preferred embodiment, the dressing assembly is comprised of a dressing belt having an outer surface that is the stripping surface. The dressing belt may be stored on and deployed from a supply spool and wound up on a take-up spool after engagement with the polishing belt outer surface. 
       FIG. 14  is a side elevation view of a first device  300  (also shown in  FIG. 3A ) that is used to dress the polishing belt  130  during operation of the polishing apparatus. As used herein, the term “dressing a polishing belt” is meant to indicate a cleaning of the belt  130 , wherein accumulated particles of material from the work piece  90  and/or a polishing slurry (if used) are dislodged from the rough abrasive outer surface  134  of the belt, so that the belt maintains its abrasive polishing capability. Referring to  FIG. 14 , dressing device  300  is comprised of an abrasive bar  302  that provides a stripping surface. Bar  302  is operatively connected to an actuator  304 , and backing wheel  306 . Abrasive bar  302  is movable by actuator  304  as indicated by bidirectional arrow  399 . In  FIG. 14 , abrasive bar  302  is shown deployed against the abrasive surface  134  of belt  130 . At the startup of the polishing process, prior to the contacting of the polishing belt with the work piece, belt  130  may be dressed by abrasive bar  304 . The bar is then retracted from the belt and polishing proceeds. The belt may be further dressed intermittently during the polishing process. 
     In an alternate embodiment, backing wheel  306  may be replaced by a backing block  308  (see  FIG. 15A ). Backing block  308  is partially rotatable around mounting pin  310 , so that surface  311  of backing block  310  aligns with belt  130  and abrasive bar  302 . In one embodiment, abrasive bar  302  may be made of alumina or silicon carbide. 
       FIG. 15A  is a side elevation view of a second device  301  (also shown in  FIG. 3B ) that may be configured in at least two ways to clean the polishing belt during operation of the polishing apparatus, and  FIG. 15B  is a top view of the device of  FIG. 15A , taken along the line  15 B- 15 B of  FIG. 15A . In one configuration, dressing device  301  is comprised of dressing wheel  312  that provides a stripping surface against the outer surface  134  of belt  130 . Dressing wheel  312  is operatively connected by shaft  314  to motor  316 . During a belt dressing operation, dressing wheel  312  is rotated by motor  316  as indicated by arrow  397 . Alternatively, wheel  312  may be driven by the motion of the belt  130 , similar to methods used in brake dressing. In this embodiment, an adjustable tension or drag is applied to the wheel  312  or shaft  314 , which minimizes its ability to rotate as fast as belt  130  would spin wheel  312  if no drag were present. 
     Dressing wheel  312  and motor  316  are movable by actuator  304  as indicated by bidirectional arrow  396 . In  FIG. 15A , dressing wheel  312  is shown slightly retracted from the abrasive surface  134  of belt  130 . At the startup of the polishing process, prior to the contacting of the polishing belt  130  with the work piece  90 , belt  130  may be dressed by dressing wheel  312  by deploying dressing wheel  312  against the abrasive surface  134  of belt  130 . The inner surface  132  of belt  130  runs against surface  311  of backing block  310  during the dressing operation. The dressing wheel  312  is then retracted from the belt and polishing proceeds. The belt  130  may be further dressed intermittently during the polishing process. Suitable materials for dressing wheel  312  include porous ceramic materials such as alumina, which is typically used in cylindrical grinding stones. 
       FIG. 15A  also depicts a second configuration of dressing device  301  which includes a dressing belt that is unwound from a supply spool, applied to the polishing belt to dress it, and wound up on a windup spool. The dressing belt provides a stripping surface against the outer surface  134  of belt  130 . In this embodiment, object  312  is not a dressing wheel. Instead, object  312  is a rectangular block of material with an upper surface that is substantially the same shape as the lower surface of backing block  310 . At the startup of the polishing process, prior to the contacting of the polishing belt  130  with the work piece  90 , belt  130  may be dressed between backing block  310  and dressing block  312 . Dressing block  312  is deployed upwardly, displacing dressing belt  320  against the abrasive surface  134  of belt  130 . The inner surface  132  of belt  130  runs against surface  311  of backing block  310  during the dressing operation, while the outer surface  134  of belt  130  runs against a corresponding portion of dressing belt  320  that functions as a stripping surface. The dressing block  312  is then retracted from the belt and polishing proceeds. 
     The belt  130  may be further dressed intermittently during the polishing process. A fresh section of stripping surface on dressing belt  320  can be provided by rotationally indexing windup spool  322  and supply spool  324  as indicated by arcuate arrows  395  and  394 . Referring also to  FIG. 3A , windup spool  322  may be driven by suitable means, such as by motor  326 , speed reducer  328 , and drive belt  330  that is engaged with windup spool  322 . 
     In a further embodiment, polishing apparatus is provided with a vacuum tube  318 , which is located in close proximity to the dressing device  301 . Vacuum tube  318  evacuates any particulate matter that is dislodged from belt  130  during the dressing operation. 
       FIG. 16  is a perspective view of an position measurement device for aligning and/or detecting the position of the polishing wheel of the apparatus to an object to be polished. Position measuring device  340  is comprised of a U-shaped housing  342  that includes a horizontal base  344 , a first upright housing member  346 , and a second upright housing member  348 . A laser  350  is contained in first upright housing member  346 , and a photodetector is contained in second upright housing member  348 . In performing an position measuring operation, position measuring device  340  is placed upon the upper surface of spindle  82 . Laser  350  emits laser beam  351  toward photodetector  352 . When the path from laser  350  to photodetector  352  is unobstructed, photodetector  352  detects laser beam  351 . Polishing assembly  100  may be slowly lowered by CNC machine  10  (see  FIG. 1 ) such that the lower edge  135  of polishing belt  130  breaks laser beam  351 . This interruption of beam  351  is detected by photodetector  352 , and the precise location of the polishing spot portion of polishing belt  130  is thus detected, and can then be programmed into the system to perform a polishing operation. In one embodiment, position measuring device  340  may be purchased as a fully assembled unit, such as a Mida series laser manufactured by Marposs S.p.A. of Bentivoglio, Italy. 
       FIG. 17  is a front elevation view of the polishing apparatus of  FIG. 3B , depicting a spot testing device of the polishing apparatus. Referring to  FIGS. 3B and 17 , polishing apparatus  104  (or apparatus  100 - 103  of  FIGS. 1-3A ) may further include a polishing spot measurement tool  360  comprised of a “light pen”  361  including deployable housing  362  containing a light source and a light detector. Deployable housing may be deployed and retracted by actuator  364 . When tool  360  is not in use, deployable housing  362  is retracted up to the horizontal position along the lower portion of base  110 , as shown in  FIG. 3B . The measurement is a non-contact, i.e. the light pen does not touch the surface of the polished spot being measured. 
     Spot testing device can be used to perform a spot measurement as follows. When apparatus  104  is fully set up for polishing, the polishing belt  130  running on polishing wheel  240  is contacted briefly with the work piece  90  under precisely controlled conditions, thereby making a slight generally elliptical shaped polished test spot on work piece  90 . Apparatus  104  is then raised, withdrawing polishing belt  130  from contact with work piece  90 , and apparatus  104  is rotated by turret  70  as indicated by arcuate arrow  395 . Actuator  364  deploys housing  362  via motion of rod  366  as indicated by arrow  394  and arcuate arrow  393  to a substantially vertical position. Housing  362  is positioned so that the distal end  363  thereof is proximate to or directly above the test spot in work piece  90 . The light source (not shown) within housing  362  is energized, and light beam  368  is directed to or near the test spot. Light is reflected back from work piece  90 , back into housing  362  to a light detector contained therein. 
     The spectral content of back reflected light varies from the region outside of the spot, and within the spot, and is also dependent upon the depth of removal within the spot. By scanning the light pen  361  over the region of the test spot using a control program of CNC machine  10  ( FIG. 1 ), the shape and spatial depth variation of the spot can be measured. The overall rate of material removal of the apparatus  104  can thus be calculated, and used to program an overall deterministic finishing process for the work piece  90  to be executed by CNC machine  10 . 
     When the spot measurement is completed, the housing  362  of light pen  361  is retracted back to the horizontal position along the lower portion of base  110 , as shown in  FIG. 3B , as indicated by arrows  391  and  392 . Apparatus  104  is rotated back to a polishing position by turret  70  as indicated by arcuate arrow  390 . 
     In one embodiment, light pen  361  may be purchased as a fully assembled unit, such as a Model CHR150, manufactured by Stil S. A. of Aix en Provence, France. 
     It is, therefore, apparent that there has been provided, in accordance with the present invention, an apparatus and methods for polishing of optics and other objects having high precision surfaces. While this invention has been described in conjunction with preferred embodiments thereof, it is evident that many alternatives, modifications, and variations will be apparent to those skilled in the art. Accordingly, it is intended to embrace all such alternatives, modifications and variations that fall within the spirit and broad scope of the appended claims.