CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]     This application is a continuation-in-part of co-pending U.S. patent application Ser. No. 11/142,829 filed on May 31, 2005. The disclosure of the above application is incorporated herein by reference. 
     
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT  
       [0002]     This invention was made with Government support under contract number F33615-99-2-5215 awarded by the United States Air Force. The Government has certain rights in this invention. 
     
    
     FIELD  
       [0003]     The present disclosure relates to multi-axis machines and, more particularly, to systems and methods of dealing with kinematic singularities during machine operation.  
       BACKGROUND  
       [0004]     The statements in this section merely provide background information related to the present disclosure and may not constitute prior art.  
         [0005]     Many machine tool installations are operated under numerical control (NC) to produce high-precision items such as aircraft parts. In such installations, a multi-axis machine may be software-driven to move one or more machine tools relative to a work piece. A multi-axis kinematic linkage of the machine is capable of moving a tool relative to a plurality of predetermined coordinate axes. Specifically, a plurality of translational and/or rotational joints may be operable singly and/or cooperatively to move one or more links to position a tool at a desired location. Occasionally, however, a multi-axis linkage may be moved to a position (“singularity”, or “singular point”) in which one or more degrees of freedom are lost. At a singular point, one or more joints may be incapable of moving a tool as instructed by the software.  
       SUMMARY  
       [0006]     The present disclosure, in one implementation, is directed to a method of operating a multi-axis machine. A kinematic linkage of the machine is monitored to detect an approach by one or more joints of the linkage toward singularity. The monitoring is performed using a manipulator matrix. A degree n of the approached singularity is determined. The one or more joints approaching singularity are identified. The method includes obtaining n virtual joints, replacing the one or more identified joints with the virtual joints in the manipulator matrix to modify the manipulator matrix, and using the modified manipulator matrix to determine position changes for links of the linkage.  
         [0007]     Further areas of applicability will become apparent from the description provided herein. It should be understood that the description and specific examples are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0008]     The drawings described herein are for illustration purposes only and are not intended to limit the scope of the present disclosure in any way.  
         [0009]      FIG. 1  is a diagram of a numerically controlled (NC) processing system in accordance with one implementation of the present disclosure;  
         [0010]      FIG. 2  is a frontal perspective view of a portion of a multi-axis kinematic linkage of a machine controlled in accordance with one implementation of the present disclosure;  
         [0011]      FIG. 3  is a diagram of a singular value decomposition of a manipulator Jacobian matrix in accordance with one implementation of the disclosure; and  
         [0012]      FIGS. 4A and 4B  are a flow diagram of a method of operating a multi-axis machine in accordance with one implementation of the disclosure. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0013]     The following description is merely exemplary in nature and is not intended to limit the present disclosure, application, or uses. It should be understood that throughout the drawings, corresponding reference numerals indicate like or corresponding parts and features.  
         [0014]     In co-pending U.S. patent application Ser. No. 11/142,829, entitled “Kinematic Singular Point Compensation Systems and Methods”, the disclosure of which is incorporated herein by reference, various systems and methods are described for positioning a tip portion of a multi-axis kinematic linkage near a singularity. The present disclosure, in one implementation, is directed to a method of automatically determining in real time (1) whether a machine linkage is approaching a singularity and how close the linkage is to the singularity, (2) which joint(s) of the linkage are approaching the singularity, and (3) which degrees of freedom would be lost at the singularity.  
         [0015]     A numerically controlled (NC) processing system in accordance with one implementation of the present disclosure is indicated generally in  FIG. 1  by reference number  20 . The system  20  includes one or more multi-axis machines  24 , one of which is shown in  FIG. 1 . A machine  24  may be configured, for example, to perform multi-axis machining operations on a work piece (not shown). Additionally or alternatively, the system  20  may include other types of multi-axis machines capable of component placement, for example, a numerically controlled assembly robot.  
         [0016]     The system  20  includes one or more computers  28 , one of which is shown in  FIG. 1 . The computer  28  includes a processor  32  and memory  36 . The computer  28  may be programmed in accordance with one implementation of the disclosure to monitor and control operation of the multi-axis machine  24 . The computer  28  may be used to execute software for adjustably compensating the positioning of the multi-axis machine  24 , as further described below. Software compensation may be used, e.g., to adjust for certain “as-built” conditions of the machine  24  in order to bring operation of the machine  24  within tolerances specified relative to an ideal or “perfect” machine.  
         [0017]     A user interface  40  is provided by which a user may communicate with the system  20 . It should be noted that the present disclosure could be implemented relative to many and various kinds of installations, including but not limited to materials processing installation configurations described in co-pending U.S. patent application Ser. No. 11/142,829. It shall be understood by those skilled in the art that many different types and configurations of multi-axis machines, computers, processors, input-output devices, communication systems, etc., could be included in NC processing systems in accordance with various implementations of the disclosure.  
         [0018]     A portion of a multi-axis kinematic linkage of the machine  24  is indicated generally in  FIG. 2  by reference number  100 . The linkage  100  includes six movable links  108   a - 108   f  interconnected by six joints  112   a - 112   f . Joints  112   a - 112   e  are rotary and joint  112   f  is prismatic. The linkage  100  is operable by the computer  28  to position and control a tip portion  116  along a predefined path during a work process. The tip portion  116  may have, e.g., a cutting tool or other type of tool (not shown) mounted thereon. It should be understood that the linkage  100  is exemplary only. Machines  24  may have linkages more or less complex than the linkage  100  and may include more, fewer and/or different types of links and/or joints and/or may provide more, fewer and/or different degrees of freedom and/or types of degrees of freedom.  
         [0019]     In operation, the linkage  100  is driven by the computer  28  via a drive apparatus (not shown), e.g., one or more electric motors. As a link  108  is driven, the positioning of other links  108  may be affected. Position and orientation of the tip portion  116  may be represented by a vector X, which is related to a link position vector Q by a set of functions f( ) such that X=f(Q). A Jacobian of the function set f( ), defined as J(Q)=(∂x i /∂q j ), maps differential motions of the joints  112  to their respective effects on the position and orientation of the tip portion  116 . Where the foregoing relationship is approximated by linearization, JΔQ=ΔX so that a change in the Cartesian position X of the tip portion  116  is related to a change in the link positions Q. As the linkage  100  approaches a singularity, the Jacobian matrix J(Q) becomes progressively more ill-conditioned. When the singularity is reached, the Jacobian matrix J(Q) becomes rank-deficient.  
         [0020]     In one implementation of a method of operating the linkage  100  in accordance with the disclosure, singular value decomposition (SVD) is used to determine (1) whether the linkage  100  is approaching a singularity and how close the linkage  100  is to the singularity, (2) which joint(s)  112  of the linkage are approaching the singularity, and (3) which degrees of freedom would be lost at the singularity. Generally, singular value decomposition (SVD) can be used to decompose any matrix A into a set of three matrices U, S, and V such that U*S*V T =A, where U and V are ortho-normal and S is diagonal with elements arranged along the diagonal in descending order. Given a matrix A, known algorithms may be used for computing U, S, and V in an essentially efficient and stable manner. The diagonal elements of S may be referred to as singular values of A. The columns of U may be referred to as left singular vectors. The columns of V (i.e., the rows of V T ) may be referred to as right singular vectors.  
         [0021]     A diagram of singular value decomposition of a manipulator Jacobian matrix J in accordance with one implementation of the disclosure is indicated generally in  FIG. 3  by reference number  200 . Referring to  FIGS. 2 and 3 , differential motions of the joints  112  are related to their respective effects on positioning of the tip portion  116  by the manipulator matrix J. Each column  204  of J is associated with a corresponding joint  112 . Singular value decomposition of the matrix J into matrices U, S and V results in left singular vectors  208  in the matrix U which represent the orthogonal degrees of freedom of the linkage  100 , in order of most sensitive  216  to least sensitive  224 . In the matrix V (the transpose of matrix V T ), right singular vectors  228  are obtained which represent the combinations of joint motion that move the linkage  100  in the directions described by U. In the matrix S, singular values  236  of J are obtained which represent a degree of “manipulability” of the linkage  100 . Accordingly, the joint  112  that is represented as dominant in the last column  242  of V (i.e., the last row of matrix V T ) is the most singular joint  112  of the linkage  100 , and the direction represented in the last column  224  of U is a potentially lost degree of freedom in the linkage  100 .  
         [0022]     Additionally, a condition number of the manipulator matrix J may be calculated by dividing a largest singular value  236  by a smallest singular value  236 . The condition number may range from 1 for a perfectly conditioned matrix to infinity for a singular matrix. In some implementations, for numerical stability, a reciprocal of the condition number is used, which ranges from 1 for a perfectly conditioned matrix to 0 for a singular matrix.  
         [0023]     A flow diagram of a method of operating the machine  24  in accordance with one implementation of the disclosure is indicated generally in  FIGS. 4A and 4B  by reference number  300 . In operation  308 , kinematic joint positions for moving the tip portion  116  along a prescribed path are calculated. In operation  312 , a manipulator Jacobian J corresponding to the calculated joint  112  positions is calculated and then decomposed using SVD. In operation  316 , a reciprocal of a condition number is determined. In operation  320 , the condition number reciprocal is compared to a predefined threshold value, e.g., 0.001. If the condition number reciprocal is greater than the threshold value, then it is assumed that the linkage  100  is not approaching any singularities, and control passes to operation  340 .  
         [0024]     If the condition number reciprocal is less than or equal to the threshold value, then it is assumed that one or more joints  112  of the linkage  100  are approaching singularity. Accordingly, in operation  324 , a degree n of the singularity is determined in the following manner. The smallest singular value  236  that raises the condition number reciprocal above the threshold value is determined. For example, if the singular values  236  are {0.7, 0.4, 0.2, 0.1, 0.0006, 0.0001}, then the smallest singular value that raises the condition number reciprocal above the threshold value 0.001 is 0.2 (0.2/0.5=0.285, but 0.0006/0.7=0.00086) and the degree n of the singularity is 2 (that is, two singular directions are indicated).  
         [0025]     In operation  328 , it is determined which joint(s)  112  are singular and/or nearly singular, by locating the largest element of the last n right singular vector(s)  228 . In operation  332 , n “virtual joint(s)” are obtained by taking the last n left singular vector(s)  208  and multiplying them by the largest singular value  236 . In operation  336 , the column(s)  204  of J associated with the joint(s)  112  determined in operation  328  are replaced with the “virtual joint(s)” obtained in operation  332 .  
         [0026]     In operations  340  through  360 , compensation for the linkage  100  is calculated, e.g., by iterative solution of the system J*dq =dx to convergence, as described in co-pending U.S. patent application Ser. No. 11/142,829. Specifically and for example, in operation  340 , a nominal inverse kinematics calculation is used to obtain an initial link position vector Q i . In operation  344 , an as-built forward kinematics calculation and the link position vector Q i  are used to obtain an actual tip position vector X. In operation  348 , a ΔX vector is calculated, indicating a difference between a desired tip position vector and an actual tip position vector. In operation  352 , a ΔQ vector is calculated using the Jacobian matrix obtained in operation  312  (and as the Jacobian matrix may have been modified in operation  336 ). In operation  356 , the ΔQ vector calculated in operation  352  and the link position vector Q i  are used to obtain a new link position vector Q i+1 . In operation  360 , convergence is tested by comparing an absolute difference between vectors Q i+1  and Q i  to a predetermined convergence criterion ε. If convergence has not yet been reached, the vector Q i+1  is used to update the vector Q i  and control returns to operation  344 . When convergence is reached in operation  360 , it then is determined in operation  364  whether the condition number reciprocal was less than the threshold value, that is, whether the linkage is approaching singularity. If yes, then in operation  368 , value(s) representing the “virtual joint(s)” are removed from the vector Q i+1  which then is used in operation  372  to compensate the non-singular joints  112 .  
         [0027]     Because SVD is performed only once for each point and is not part of the iterative solution process, the foregoing methods and systems are less computationally expensive than linear quadratic regulation (LQR) or damped least squares.  
         [0028]     The foregoing methods and systems have numerous advantages. For example, left singular vectors provide ideal “virtual joint(s)”, in that they move the linkage in exactly the singular direction(s), regardless of the position or complexity of the machine. Another advantage is that by scaling the “virtual joint(s)” by the largest singular value, the resulting condition of the new manipulator Jacobian is optimized, thereby assuring a stable, rapidly converging solution, e.g., in operations  340  through  360  of the foregoing method  300 . Additionally, implementations of the foregoing method require no a priori knowledge of the locations of singularities for the machine, so the same algorithm can be used on a variety of machines without change.  
         [0029]     Implementations of the present disclosure provide a means of improving the positioning accuracy of automation equipment that works near its singularity. Where accuracy can be improved, costs of acquisition of new capital equipment, and maintenance costs for existing equipment, can be lowered. Furthermore, various implementations of the present disclosure can provide a single method that works on several different machine geometries, thereby reducing a need for software development and maintenance to deal with new machine geometries.  
         [0030]     Calibrated machine tools can be provided which are capable of producing more accurate work. Thus higher precision parts can be made with less waste material. When implemented in connection with aircraft production, the production of higher-precision parts can result in reduced weight and improved performance of aircraft. Implementations of the disclosure can result in reduced set-up time and more productive use of machine tools, thereby reducing the cost of machining.  
         [0031]     Various implementations of the foregoing methods and systems make it possible to use software to compensate machines that have complex singularity configurations (e.g., a six-axis revolute robot). Compensation can be provided “on the fly” for a wide range of machine configurations, even without precise a priori knowledge of the location or nature of all singularities for a given machine. The same core compensation software can be used on a wide range of machine configurations without a need to rewrite the software based on machine geometry.
A method of operating a multi-axis machine. The machine linkage is monitored to detect an approach by linkage joint(s) toward singularity. A degree of the approached singularity is determined. The joint(s) approaching singularity are identified. Virtual joints are used to replace the identified joint(s) in a manipulator matrix to modify the manipulator matrix. The modified matrix is used to determine position changes for the linkage links. This method can provide software-based compensation for a wide range of machine configurations, without a priori knowledge of singularities for a given machine.