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A Cutter Orientation Modification Method for the Reduction of Non-linearity Errors in Five-Axis CNC MachiningABSTRACTIn the machining of sculptured surfaces，five-axis CNC machine tools provide more flexibility to realize the cutter position as its axis orientation spatially changes .Conventional five-axis machining uses straight line segments to connect consecutive machining data points ,and uses linear interpolation to generate command signals for positions between end points，Due to five-axis simultaneous and coupled rotary and linear movements, the actual machining motion trajectory is a non-linear path. The non-linear curve segments deviate from the linearly interpolated straight line segments, resulting in a non-linearity machining error in each machining step. These non-linearity errors, in addition to linearity error, commonly create obstacles to the assurance of high machining precision. In this paper, a novel methodology for solving the non-linearity errors problem in five-axis CNC machining is presented. The propose method is based on the machine type-specific kinematics and the machining motion trajectory. Non-linearity errors are reduced by modifying the cutter orientations without inserting additional machining data points. An off-line processing of a set of tool path data for machining a sculptured surface illustrates that the proposed method increases machining precision. KeywordNon-linear error; Machine kinematics; Machining motion trajectory. INTRODUCTIONIn conventional five-axis machining, a tool path, represented by the cutter locations data (CLDATA), consists of the spatially varying cutter positions and its axis orientations. These CLDATA are generated based solely on the geometrical properties of the machined surfaces and the cutter. These CLDATA are further processed into NC-codes which is specific to a particular machine configuration. Linear interpolation is then used to generate the required commands for positions along line segment connecting the machining data points. The simultaneous linear and rotary movements are involved in five-axis machining since ever new cutter axis orientation requires the motion at least one other axis. There are also coupling effects of the cutter axis will affect the position of the cutter. These simultaneous and coupled movements cause the cutter contract point (CC point) to move in a non-linear manner. As a result, the machining error in each motion step is made up of not only the linear segmentation approximation error but also an additional machining error. As shown in figure 1 for machining is either a concave surface or a convex surface, a line segment is used to connect two consecutive machining data points (the spindle chunk is the machine control point MCP). Linear interpolation generate intermediate positions along the line segment. The desire surface is design curve(either concave or convex). The linear segment approximates to design curve resulting in the linearity error,t. Apart from the linearity error . The non-linear CC point trajectory deviates from the straight line segment (the cutter gage length is constant and MCP is interpolated along the line segment)result in an additional machining error, referred to as the non-linearity error, n. In the case that the desire surface is concave(see figure 1a), the total machining error is difference of the non-linearity error and the linearity error : total=t-n. The non-linearity error, in this case, compensate for the total machining error(AIGP Post-processor,1996;Liu,1994). On the contrary, for the machining of convex surface as shown in figure 1b, the non-linearity error adds onto the linearity error and enlarges the machining error: total=t+n(AIGP Post-processor,1996;Liu,1994). figure1. The multi-axis CNC machining errorConsequently the non-linearity error have caused difficulties for ensuring ultra-precision machining requirements. In the machining of airfoil surface, for example, the machining of the contour surface of airfoil to the edges is problematic. The surface curvature on these area changes abruptly, and thus the cutter orientation varies inconsistently from one cutter to the next. These abrupt cutter orientation variations inconsistently from one cutter location to the next .These abrupt cutter orientation are a typical non-linearity error problem.In order to solve the five-axis CNC machining error problem, efforts have been made to treat non-linearity errors in generate NC codes. Some researchers and postprocessor producers used “linearization processes” for this purpose. The basic function of “linearization processes ” are inserting machining data points between NC codes where the total machining error is out of the specified tolerance range. Takeuchi et al. (1990) inserted points by subdividing the line segment with equally space d interval. Cho et al. (1993) inserted data points by limiting the maximum machining error within the line interval from the start point to the inserted point to be the tolerance. And, both of them set the cutter orientations varying linearly in successive positions. In the Automation Intelligence Generalization Postprocessor (AIGP)(1996), a “linearization processes ” calculates the middle point (MP) between adjacent NC-codes and inserts the MP as an additional data in the NC code. The insertion can be performed further between the consecutive NC-coded until either all points are within the machining tolerance or until a maximum of 63 points are inserted between the consecutive data point. The current post-processors, such as the Vanguard Custom Post-processor Generator (1996) , the Ominimill Custom Postprocessor(1992),the AIX Numerical Control Post Generator(1996) , are all having the similar “linearization processes ” as in the AIGP. In the current CAD/CAM software. Unigraphics(2001), the UG /post postprocessors inserts data points between adjacent NC-codes, thereby simulating a straight line with series of small curves. The number of the inserted points is determined based on the maximum allowable deviation and an iteration method is used to segment the move. In the extreme case, namely after looping 20 times, if the deviation between the segmented arcs and the line are still out of the specified tolerance limit, the process is aborted. “linearization processes ” discussed above manipulate NC-codes by inserting extra machining data points. Although the produced NC-codes satisfy the machining requirement, they may contain dense sets of non-equally spaced data with constant or linearly varying cutter orientation. Consequently, the linearization process has raised the following problems. In the machining of complex contour surface, the cutter orientation varies from one cutter location to the next. The cutter position changes in this case can not be too small since the machine will produce either jerk motion or random rotary movements. As in an industrial procedure of machining airfoil surface of an impeller, a linearization process was used to reduce the non-linearity errors. Many data points were inserted between a pair of NC-codes. The insertion of many data points caused the cutter position change to be nearly equal to zero while the cutter orientation changed abruptly. As a consequence, the machine rotary movements were rapid with infinite feedrate. Random rotary movements resulted and the workpiece was damaged. The insertion of machining data points can also cause non-constant federate along the cutting curve. The insertion of additional data results in non-equally spaced segment, while acceleration and deceleration steps are required for each segment. Thus, the feederate varies in each segment and may never reach the desired value. The result of varying feederate causes a nonsmooth surface finish and the unreachable feedrate increases overall machining time. In addition, the insertion of constant cutter orientation variation also causes severe roughness around the end points along the surface. Linearly inaccurately since the change in cutter orientation is not necessarily linear. The non-linearity error problem arises from the fact that five-axis machining motion trajectories are non-linear curve segments. The simultaneous and coupled rotary and translation movements generate the non-linearity motion trajectory, and the linear interpolation technique is not capable to curve fit the nonlinear path. One solution to is to design new interpolation methods. Liang et al.(2002)presented a combine 3D linear and circular (3D L&C) interpolation technique. The new 3D L&C interpolation can on-line drive the rotation movement pivot along a pre-designed 3D curve path, so that the CC point motion trajectory is a via a straight line connecting machining data points, thus, the non-linearity error can be eliminated. Five-axis machining movements are kinematically related to the cutter location data. In other word, the non-linear motion trajectory depends on the cutter orientation changes and non-linearity errors are related to the tool path generation. Thus, another solution to the non-linearity error problem can be approached from tool path(CLDATA)generation with the requirements that the machining errors are minimized and there is no interference between the workpiece and the cutter. In tool path generation, various techniques for different surface representations have been used by the CAD/CAM package producers (CLDATA,1996;Unigraphics, 1990)and researchers. Huang and Oliver (1992) . Bedi et al.(1997) presented a principle curvature alignment technique for five-axis machining using a toroidal shaped tool. Liu (1995)presented the single point offset and the double point offset algorithms for five-axis flank milling tool path generation based on differential geometry and analytical geometry. Morishge et al.(1999)presented a tool path generation method for five-axis CNC machining, which applies the C-space(a 3D configuration space)to determine collision-free cutter positions and its orientation. These research work on tool path generation are all based exclusively on the geometric of the machined surfaces and the cutter, without considering the machining-specific machining kinematics. As a result, the generated tool paths(the machining NC-codes transformed from these CLDATA)commonly cause obstacles for meeting the ultra-precision machining requirements, particularly for the cutter orientation generation in five-axis machining. Thus, the problem with present off-line tool path generation approaches is that the real machining kinematics is not directly incorporated. To ensure machining precision, cutter orientation generation should be based not only on the geometry of the machined surfaces but also on the machine type-specific kinematics. In this paper, a novel methodology for solving the non-linearity error problem in five-axis machining is presented. The method optimizes the CLDATA based on machine-specific kinematics and machining motion trajectory, whereby the cutter orientations are modified to reduce the non-linearity errors provided that there is no interference between the cutter and the workpiece. A software program for implementing the proposed method is presented. As an application of proposed method, a case study is presented, which shows an increase in machining precision as compared with those processed by the existing AIGPs method.PROPOSED TOOL PATH GENERATION METHODThe machining non-linearity errors depend upon the actual CC point trajectory, since a CC point trajectory is a function of the machine rotary variables, each actual CC point trajectory can be manipulated within the tolerance limit by changing the machine rotary variables, provided that there is no interference between the workpiece and the cutter. Further more, because of the machine rotary variables are kinematically related to the cutter orientation changes, the non-linearity error problem can be approached by manipulating cutter orientations. To proposed method reduces the non-linearity errors by determining the acceptable machine rotary variables employing the machine motion trajectory model, and by modifying the cutter orientation through the machine kinematic relations. It must be emphasized that the machine kinematic properties and motion trajectory are machine type-specific. Hence, the modification of CLDATA has to be carried out in teams of machine variables and subsequent use of the kinematic transformation to determine the modified CLDATA.The procedure of the proposed method starts with the transformation of the CLDATA to machining NC-codes by employing the machine-type specific inverse kinematic model. In teams of machine variables, the actual machining motion trajectory is determined by using the specific machine motion trajectory model. Then, the machining errors are determined. The linearity error is a function of surface local curvature on the cutting curve and the step-forward distance. From the cubic spline cutting curve function, the surface local curvature can be determined. The linearity error for each move then can be computed from the adjacent CC point data and the surface local curvature. By knowing the linearity error, the allowable non-linearity error can be determined as the difference of the linearity error from the specified machining tolerance. Using the machine trajectory model and the line segment equation, the maximum deviation can be determined. By taking sample points on both of the CC non-linear curve and the line segments, the maximum chord deviation is the maximum non-linearity error. In the steps where the maximum non-linearity error exceeds the allowable non-linearity error, the proposed method modifies the machine rotary variable changes. The modification is carried out by increasing/decreasing a machine rotary variable variation a small angle in the plane containing the two original cutter vectors, and by adding/subtracting the angle to the original machine rotary variables. The new rotary variables are then used to calculate the resultant non-linearity error, which in turn is compared again with the allowable non-linearity error. Thus, by using the difference between the allowable non-linearity error and modified non-linearity error as the criterion, the acceptable machine rotary variables can be determined iteratively. Finally, from the modified machine rotary variables, the corresponding cutter orientations can be determined by performing the forward kinematic transformation. In order to avoid interference between the workpiece and the cutter, the rotary angles were adjusted such that the angle changes are less than half of the orientation angle changes are less than one half of the original cutter orientation changes. Comparing to the existing “linearization processes ”, the additional data points are inserted with cutter orientations either varying linearly or as the average variation(i.e., the one half)of the rotary angle change, and the interference problem is not considered. Alternatively, the modified machine rotary variables angle change from the proposed method are smaller than one half of angel change, which thus ensures the corresponding cutter orientation are within the range such that no interference occurs. For a set of CLDATA, the modification procedure can be performed by using the following algorithm.TOOL PATH MODIFICATION ALGORITHM1. Transform the initial CLDATA, into its corresponding machining NC-codes by using the specific machine inverse kinematic model;2. Determine the CC point coordinates by employing the machine motion trajectory model;3. Compute the desire tool path by using cubic spline function based on the CLDATA and calculate the local surface curvatures Kf of the tool path at the machining points;4. Computer the linearity error by using the formula given by Faux and Pratt(1979):t =1/8 Kf (s)2where, Kf-the surface local curvature determined from step(3); s-the segment length between consecutive CC points from step(2).5. Compute the allowable value of the non-linearity error: a,n=tolerance-t.6. Determined the points on the straight line segment and on the machine motion trajectory segment that correspond to the maximum chordal deviation.7. Compute the maximum non-linearity error , max, using the points from step(6);8. Modify the machine rotary angle change ifmaxa,n ,that is, increase or decreaseBm andCm such that the non-linearity error, n1, will satisfy (n1-a,n)0;9. Computer the machining NC-codes of Bm,i+1 and Cm,i+1 based on the machine rotary angle variation from step 8:Bm,i+1= Bm,iBmCm,i+1=Cm,iCmWhere, Bm,i, Cm,i are the i-th rotary variables , Bm,i+1 Cm,i+1 are the i+1-th rotary variables andBm Cm are modified rotary angle changes. The +and sign depends on whether the angles in step i+1 increase or decrease,10. Determine the optimum cutter orientations b