Abstract:
A method for determining the axis of rotation (AR) of an object ( 12 ) mounted to a rotary stage or platform ( 10 ) in a gauge measurement system by estimating a transformation (δ) between multiple views or measurements of the object ( 12 ) obtained at different poses.

Description:
BACKGROUND OF INVENTION 
     The present invention relates to a method for the reconstruction and recovery of the three-dimensional shape of an imaged object in a gauge measurement system, and more specifically, to a robust and accurate method for estimating the motion of an measured object mounted on a rotary stage. 
     Traditionally, in a gauge measurement system, such as a non-contact optical measurement system or coordinate measuring machine (CMM), an object to be measured is mounted on a rotary stage or platform. A rotary stage or platform is a mounting apparatus which is constrained to permit only movement about a single axis. Multiple views or contact measurements of the object are obtained by imaging or contacting the object at different poses. A calibration process is then utilized to provide a means to coherently merge the data taken at those views or measurements (recorded in local coordinate systems) into a single global coordinate system. 
     Conventional calibrations processes utilize plane intersections to coherently merge local coordinate systems into a global coordinate system. In such processes, a known planar object is mounted on the rotary stage or platform, a series of range images or sets of contact measurements of the object are obtained at different positions of the rotary stage. The data points from each rotary position are each fitted to an associated plane using a simple linear regression scheme. Intersecting pairs of the planes form vectors in space parallel to the axis of rotation. Combining these intersecting vectors yields the axis of rotation. So long as the rotary stage or platform is rigidly secured in relation to the sensor, and the calibration of the sensor does not subsequently change, the axis of rotation is fixed such that any object mounted to the rotary stage or platform is constrained to rotate about the same identified axis. 
     The precision and robustness of any rotational axis estimation scheme depends upon the amount of uncorrelated (useful) data input to the system. Accordingly, there is a need for a method to accurately estimate the axis of rotation for a rotary stage or platform from multiple views or measurements of an object which provides an increase in the amount of data input to the system. 
     SUMMARY OF INVENTION 
     Briefly stated, the present invention provides a method for determining the axis of rotation of an object mounted to a rotary stage or platform in a gauge measurement system by estimating a transformation between multiple views or measurements of the object obtained at different poses. 
    
    
     The foregoing and, other objects, features, and advantages of the invention as well as presently preferred embodiments thereof will become more apparent from the reading of the following description in connection with the accompanying drawings. 
     BRIEF DESCRIPTION OF DRAWINGS 
     In the accompanying drawings which form part of the specification: 
     FIG. 1 is a perspective view of a calibration object mounted on a rotary stage at a first orientation; 
     FIG. 2 is a perspective view of the calibration object of FIG. 1 at a second orientation; 
     FIG. 3 is a perspective view of the calibration object of FIG. 1 at a third orientation; 
     FIG. 4 is a first coordinate system representation of the rotary stage axis of rotation; 
     FIG. 5 is a second coordinate system representation of the rotary stage axis of rotation; and 
     FIG. 6 is a representation of a coordinate transformation in three-dimensional space. 
    
    
     Corresponding reference numerals indicate corresponding parts throughout the several figures of the drawings. 
     DETAILED DESCRIPTION 
     The following detailed description illustrates the invention by way of example and not by way of limitation. The description clearly enables one skilled in the art to make and use the invention, describes several embodiments, adaptations, variations, alternatives, and uses of the invention, including what is presently believed to be the best mode of carrying out the invention. 
     The present invention provides a method for determining the axis of rotation of a rotary stage or platform using a known calibration object and a robust registration scheme. In one embodiment, a Robust Closest Patch (RCP) scheme is utilized to estimate the transformation between multiple poses of the object being measured. Alternatively, an iterative closest patch (ICP) scheme, or other known scheme may be utilized to estimate the transformation between multiple poses of the object  12 . 
     The known calibration object is designed in conjunction with the type of sensor undergoing calibration. Specifically, when mounted on the rotary stage or platform  10 , a large area of the calibration object  12  must be visible to a camera based sensor or accessible to a contact based sensor in a wide range of poses or positions, without being occluded. The calibration object  12  is preferably centered at the suited-point of the sensor and should be large enough to cover the working volume of the sensor, as the accuracy of the calibration will increase with the average arm length of the calibration object  12 , as measured from the center of rotation. To avoid reducing available information identifying the pose of the calibration object  12 , it is preferred that the calibration object  12  not be round, and include a multiple number of large facets to avoid aliasing effects. Further, the facets are either visible to a camera based sensor or accessible to a contact sensor. 
     Each pose of the calibration object  12  in a sequence is defined as P 1 , P 2 , . . . , P k , . . . P n . Each pose of the calibration object  12  differs from the previous pose by a constant rigid body increment δ, such that P 2 =δP 1 , P k =δP k−1 , and P n =δP n−1 . The angle of rotation is preferably selected to be small enough to enforce a small amount of overlap between succeeding poses of the calibration object  12 . From the data for P k  and P k−1  the transformation δ can be estimated independently using the RCP algorithm or any other registration scheme. 
     By combining all the data from each pose of the calibration object  12  with the same set of constraints, and using the data for a joint estimation, a more accurate and stable solution for δ can be found. The transformation δ, as a rigid body transform, can be decomposed into a rotation R around the origin point, and a translation T: 
     
       
         δ= T ( t )× R (ω,θ)  Eqn. (1) 
       
     
     where: 
     ω is the direction of the axis of rotation; 
     θ is the angle of rotation; and 
     t is the translation vector. 
     The direction of the axis of rotation is extracted from the estimated rotation by computing the quaternion equivalent of the rotation matrix. Using a quaternion argument, a point in an (i, j, k) coordinate system, such as the camera coordinate system, can be transformed into a (u, v, ω) coordinate system wherein the ω-axis is parallel to the axis of rotation of the rotary stage  10 . To find a point (x, y, z)=Ω which is on the axis of rotation AR, it is known that any point on the axis of rotation AR will be rotated onto itself. The center of rotation, i.e. any point on the rotation axis, is determined as an invariant point, written using matrix notation as: 
     
       
           RΩ+T =Ω  Eqn. (2) 
       
     
     where Ω is the center of rotation. If Ω is transformed into the (u, v, ω) coordinate system, it is defined as Ω′=(x′, y′, z′), as is seen in FIGS. 4 and 5. 
     To avoid the inversion of a singular matrix, and appropriate rotation is applied to the original three-dimensional space, which reduces the three-dimensional singular problem to a two-dimensional non-singular one. 
     The original coordinate system for an image of the calibration object  12  is defined as C=(i, j, k). Correspondingly, the rotated coordinate system for the object is defined as C′=(U, V, ω) where u, v, and ω form an orthogonal-normal basis, and ω is the unit vector along the axis of rotation AR. The center of rotation in the rotated space is given coordinates Ω′, such that: 
     
       
         Ω= A Ω′  Eqn. (3) 
       
     
     where A is the rotation from the coordinate system C′to C. Using a quaternion argument, it is possible to compute A such that a point in the (i, j, k) coordinate system can be transformed into the (u, v, ω) coordinate system. 
     Replacing equation (3) into equation (2), with results in: 
     
       
           R′Ω′T ′=Ω′  Eqn. (4) 
       
     
     where: 
     
       
           R′=A   t   RA   Eqn. (5) 
       
     
     
       
           T′=A   t   T   Eqn. (6) 
       
     
     and R′is a rotation around the vertical axis. Equation (4) can be rewritten in matrix form as:                    (           cos                 θ           sin                 θ         0               -   sin                   θ           cos                 θ         0           0       0       1         )          [           x   ′               y   ′               z   ′           ]       +     [           t   x   ′               t   y   ′               t   z   ′           ]       =     [           x   ′               y   ′               z   ′           ]             Eqn   .                (   7   )                                  
     which has a solution if and only if t′ z =0 and cos θ≠1. 
     The condition cos θ≠1 is utilized to avoid the identity transform which is a particular rotation that does not yield any axis of rotation. The condition t z =0 agrees with the geometrical intuition of pure rotation, i.e. the trajectories of points rigidly attached to a rotary stage under a rotation are arcs of circles in planes perpendicular to the axis of rotation, as seen in FIG.  6 . However, the rotary stage or platform  10  is likely to be imperfect, hence the parameter t z  is exploited to check the planarity of the motion of the rotary stage or platform, and any t z ≈0 it is set to zero. 
     Since t′ x , t′ y , and θ are known from the solutions to Eqn. 5 and Eqn. 6, it is possible to solve Eqn. 7 for x′ and y′ by assuming z′ is any value. The assumption of z′ having any value is valid, since z′ merely represents the location of the point on the axis or rotation AR, and any point on the axis may be utilized. Solving the above system of equations yields the following analytical solutions:                x   ′     =           t   x   ′          (     1   -     cos                 θ       )       +       t   y   ′        sin                 θ         2        (     1   -     cos                 θ       )                 Eqn   .                (   8   )                                                y   ′     =           t   y   ′          (     1   -     cos                 θ       )       -       t   x   ′        sin                 θ         2        (     1   -     cos                 θ       )                 Eqn   .                (   9   )                                  z′=any value  Eqn. (10) 
     Once x′ and y′ are identified, Eqn. 3 can be solved for the coordinates of point Ω on the axis of rotation AR. Since the direction of the axis of rotation AR is also known, the rotary stage is completely defined mathematically. 
     In an alternative embodiment, starting from the estimated transformation δ, the transformation is represented as a matrix with coordinate points represented as vectors. The direction of the axis of rotation AR and the point T on the axis of rotation AR are invariant to the various parts of the transformation process. In particular, the direction of the axis of rotation AR is an eigenvector corresponding to an eigenvalue of +1.0 of the rotation R since it is invariant to the rotation. Similarly, the point T on the axis of rotation AR is invariant to the transformation, and thus is an eigenvector corresponding to an eigenvalue of +1.0 of the [R |T R*T] transformation. Those of ordinary skill in the art will recognize that there are numerous conventional techniques which may be employed to solve a matrix for eigenvectors having eigenvalues of 1.0, and which may be utilized to identify the direction of the axis of rotation AR and a point T on that axis given the estimated transform δ. 
     An alternative formulation is that there must be two eigenvectors of [R |T R*T] that have eignenvalues of +1. These two eigenvectors represent two points which are invariant to the transformation δ and hence are on the axis of rotation AR. Thus, and points on the axis of rotation AR are defined as a linear combination of these two eigenvectors. 
     In view of the above, it will be seen that the several objects of the invention are achieved and other advantageous results are obtained. As various changes could be made in the above constructions without departing from the scope of the invention, it is intended that all matter contained in the above description or shown in the accompanying drawings shall be interpreted as illustrative and not in a limiting sense.