二坐標(biāo)數(shù)控工作臺(tái)設(shè)計(jì),坐標(biāo),數(shù)控,工作臺(tái),設(shè)計(jì) ORIGINAL ARTICLE Postprocessor development of a five-axis machine tool with nutating head and table configuration Chen-Hua She the angle of inclination of the nonorthogonal axis is variable, and the offset vector from the origin of the workpiece to the rotary table is defined. Moreover, the linearization algorithm of the post- processor is developed to ensure the machining accuracy. Awindow-based postprocessor is developed and a graph- ical interface that dynamically displays the surface model and themotionsofalloftheaxesoftheconfiguredmachinetoolis presented to help the user to input relevant parameters correctly. Additionally, the generated NC data are verified usingthecommercialsolidcuttingsoftwareVERICUT 13 and a machining experiment is conducted on a five-axis machine tool with a nutating table to confirm the effective- ness of the proposed postprocessor methodology. 2 Configuration and modeling of five-axis machine tool Most five-axis machine tools have two rotary axes as well as the conventional X, Y and Z axes. Following Sakamoto and Inasaki 14, the configurations of five-axis machine tools can be categorized into three types: spindle-tilting, table-tilting and table/spindle-tilting. Commercial machine tools with the nonorthogonal configuration, as shown in Fig. 1, are also of three types. Figure 1 (a) shows the spindle-tilting type with a nutating head, such as the Makino MAG3 2, which is designed with a rotary axis (C-axis) behind a nutating head that rotates about the B- axis. Figure 1 (b) displays the table-tilting type with a nutating table, such as the Deckel Maho DMU 70 eVolution 15, which has two rotary axes on the table, and one rotary axis (C-axis) is parallel to the Z-axis while the non- orthogonal rotary axis is inclined at an angle to the C-axis. Figure 1 (c) presents the table/spindle-tilting type with a nutating head, such as the Deckel Maho 200P 15,inwhich one rotary table (C-axis) is on the table and the nutating rotary head (B-axis) is on the spindle. Since the authors have already presented the spindle-tilting postprocessor with a nutating head 11, this study focuses on developing the postprocessors with the other two configurations. A five-axis machine tools can be regarded as a mechanism with serially connected links with revolute or prismatic joints. Forward kinematic equations must be established to describe mathematically the motion of the cutting tool in relation to the workpiece. The fundamental coordinate transformation matrices, including the transla- tion matrix Trans and the rotation matrix Rot 16, are introduced. The translation matrix Trans can be expressed as follows: Transa; b; c 100a 010b 001c 0001 2 6 6 4 3 7 7 5 1 where Trans(a, b, c) implies a translation given by the vector a i+b j+c k. The general rotation transformation matrix should be used to describe the rotation of the nutating unit. The coordinate system is assumed to rotate through an angle of f w around any arbitrary vector W=W x i+W y j+W z k; the rotational transformation matrix can be expressed as: Rot W; w W 2 x V w C w W x W y V w C0 W z S w W x W z V w W y S w 0 W x W y V w W z S w W 2 y V w C w W y W z V w C0 W x S w 0 W x W z V w C0 W y S w W y W z V w W x S w W 2 z V w C w 0 0001 2 6 6 4 3 7 7 5 2 Int J Adv Manuf Technol where “C” and “S” are cosine and sine functions, respectively, and Vf w =1Cf w . 3 Postprocessor 3.1 Table-tilting type with a nutating table Figure 2 depicts relevant coordinate systems for this configuration. The coordinate system for the workpiece is O w X w Y w Z w while the system O t X t Y t Z t is attached to the cutting tool. Since the two rotary axes are assumed not to intersect each other, a common normal line is mutually perpendicular to both axes. The common normal line inter- sects with the C-axis and B-axis at two points, RC and RB. The offset vector L x i+L y j + L z k is determined from the origin O w to the pivot point RC, whereas the offset vector M x i+M y j+M z k is calculated from the pivot point RC to the pivot point RB. Since the structural elements of the machine tool comprise the C rotary table, the B nutating rotary table, the machine bed, the X linear table, the Y linear table, the Z linear table, the spindle head and the cutting tool. The relative position and orientation of the cutting tool with respect to the workpiece can be determined sequentially starting from the workpiece and ending at the cutting tool and is referred to as the form-shaping function 17. The C B X Y Z a XY Z C B c B C Y X Z b Fig. 1 Configuration for five-axis machine tool with nutating head and table. a spindle-tilting type with a nutating head. b table-tilting type with a nutating table. c table/spindle-tilting type with a nutating head t O t Xt Y t Z B RB w O w X w Y w Z C xyz LLL+ijk RC Offset vector x y z MMM+ijk Offset vector Fig. 2 Coordinate systems of table-tilting type configuration Int J Adv Manuf Technol form-shaping function of this machine tool can be mathematically expressed in matrix form as follows: Trans L x ;L y ;L z C0C1 Rot z;C0 z Trans M x ;M y ;M z C0C1 Rot W; C0 w Trans P x ;P y ;P z C0C1 00 00 10 01 2 6 6 4 3 7 7 5 3 whereP x , P y and P z denote the relative translation distances of the X, Y and Z linear tables, respectively. The terms f z and f w represent the angles of rotation for the C-axis and the B- axis, respectively. The positive rotation is in the direction of an advancing right-hand screw about the +C and +B axes. Equation (3) specifies the form-shaping function matrix of this machine tool and the joint parameters P x , P y , P z , f z and f w should be determined by the inverse kinematics. The first step is to calculate the required rotary angles to yield the tool orientation, and the second is to calculate the required position in relation to the linear axis to determine the position of the centre of the tool tip using the known rotary angles. Whenthe CL data includingthe position of thecentreof the tool tip Q x i Q y j Q z k and the tool orientation K x i K y j K z k are known, the CL data can be expressed in the matrix form as follows: KQ 01 C20C21 K x K y K z 0 Q x Q y Q z 1 2 6 6 4 3 7 7 5 4 Since both Eq. (3) and Eq. (4) represent the same relationship between the cutting tool and the workpiece, the desired joint parameters can be determined by equating these two matrices. Equating the CL data matrix and the form-shaping function matrix, and taking the corresponding elements of the two matrices yield the following equations: K x K y K z 0 2 6 6 4 3 7 7 5 C z W x W z 1 C0 C w C0W y S w C2C3 S z W y W z 1 C0 C w W x S w C2C3 C0S z W x W z 1 C0 C w C0W y S w C z W y W z 1 C0 C w W x S w W 2 z 1 C0 C w C w 0 2 6 6 4 3 7 7 5 5 Q x Q y Q z 1 2 6 6 4 3 7 7 5 P x C z W 2 z 1 C0 C w C w C2C3 S z W x W y 1 C0 C w C0W z S w C2C3C8C9 P y C z W x W y 1 C0 C w W z S w S z W 2 y 1 C0 C w C w hino P z C z W x W z 1 C0 C w C0W y S w C2C3 S z W y W z 1 C0 C w W x S w C2C3 L x C z M x S z M y P x C0S z W 2 x 1 C0 C w C w C2C3 C z W x W y 1 C0 C w C0W z S w C2C3C8C9 P y C0S z W x W y 1 C0 C w W z S w C z W 2 y 1 C0 C w C w hi P z C0S z W x W z 1 C0 C w C0W y S w C2C3 C z W y W z 1 C0 C w W x S w C2C3 L y C0 S z M x C z M y P x W x W z 1 C0 C w W y S w C2C3 P y W y W z 1 C0 C w C0W x S w C2C3 P z W 2 z 1 C0 C w C w L z M z 1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 6 Int J Adv Manuf Technol The joint angles f z and f w should be determined first. Equating the corresponding third element in Eq. (5) yields the following B-axis angle: B w arccos K z C0 W 2 z 1 C0 W 2 z C18C19 0 C20 w C20 7 Notably, there is another possible solution for B-axis angle in the range of C0 C20 w C20 0, which can be obtained as follows: B w C0arccos K z C0 W 2 z 1 C0 W 2 z C18C19 8 If the operating range of the nutating table is in the range between and 0, the solution should be modified as shown in Eq. (8). On the other hand, if the operating range of the nutating table meets the two possible solutions, the shortest rotational angle movement of the nutating table is usually chosen in the algorithm. Furthermore, equating the corresponding first and second elements in Eq. (5) and solving the simultaneous linear equations for Sf z and Cf z , yield: S z C0K y W x W z 1 C0 C w C0W y S w C2C3 K x W y W z 1 C0 C w W x S w C2C3 W x W z 1 C0 C w C0W y S w 2 W y W z 1 C0 C w W x S w 2 9 C z K x W x W z 1 C0 C w C0W y S w C2C3 K y W y W z 1 C0 C w W x S w C2C3 W x W z 1 C0 C w C0W y S w 2 W y W z 1 C0 C w W x S w 2 10 Since the denominators in Eqs. (9) and (10) are the same and always positive, the C- axis angle can be determined as follows: C z arctan2C0K y W x W z 1 C0 C w C0W y S w C2C3 K x W y W z 1 C0 C w W x S w C2C3 ; K x W x W z 1 C0 C w C0W y S w K y W y W z 1 C0 C w W x S w C0 C20 z C20 11 where arctan2(y,x) is the function that returns angles in the range C0p C20 q C20 p by examining the signs of both y and x 16. In addition, comparing the corresponding elements of the matrix on both sides of Eq. (6) yields three simulta- neous equations in three unknowns P x , P y and P z . The program coordinate system is assumed to coincide with the workpiece coordinate system. Accordingly, the expressions of the desired NC data (denoted as X, Y and Z) can be obtained by considering the two offset vectors L x i+L y j+L z k and M x i+M y j+M z k, and are expressed as follows: X P x L x M x Q x C0 C z M x S z M y L x C0C1C2C3 C z W 2 x 1 C0 C w C w C2C3 S z W x W y 1 C0 C w C0W z S w C2C3C8C9 Q y C0C0S z M x C z M y L y C0S z W 2 x 1 C0 C w C w C2C3 C z W x W y 1 C0 C w C0W z S w Q z C0 M z L z C138W x W z 1 C0 C w W y S w C2C3 L x M x 12 Int J Adv Manuf Technol Y P y L y M y Q x C0 C z M x S z M y L x C0C1C2C3 C z W x W y 1 C0 C w W z S w C2C3 S z W 2 y 1 C0 C w C w hino Q y C0C0S z M x C z M y L y C0C1 C0S z W x W y 1 C0 C w W z S w C2C3 C z W 2 y 1 C0 C w C w Q z C0 M z L z C138W y W z 1 C0 C w C0W x S w C2C3 L y M y 13 Z P z L z M z Q x C0 C z M x S z M y L x C0C1C2C3 C z W x W z 1 C0 C w C0W y S w C2C3 S z W y W z 1 C0 C w W x S w C2C3C8C9 Q y C0C0S z M x C z M y L y C0S z W x W z 1 C0 C w C0W y S w C z W y W z 1 C0 C w W x S w Q z C0 M z L z C138W 2 z 1 C0 C w C w C2C3 L z M z 14 3.2 Table/spindle-tilting type with a nutating head The table/spindle-tilting type configuration has one rotary axis on the table and one nutating rotary axis on the spindle. Figure 3 illustrates two pivot points RC and RB on the C and B axes, respectively. The pivot point RC is located arbitrarily on the C-axis and the pivot point RB is chosen to be the intersection of the nutating rotary B-axis and the axis of the cutting tool. The offset vector L x i +L y j +L z k is calculated from the origin O w to the pivot point RC and the effective tool length, L t , represents the distance between the pivot point RB and the cutter tip centre. The form-shaping function matrix of this configuration can be obtained by the coordinate transformation matrices: Trans L x ;L y ;L z C0C1 Rot z; C0 z Trans P x ;P y ;P z C0C1 Rot W; w 00 00 1 C0 L t 01 2 6 6 4 3 7 7 5 15 Equating Eq. (4) and Eq. (15) leads to the following equations: K x K y K z 0 2 6 6 4 3 7 7 5 C z W x W z 1 C0 C w W y S w C2C3 S z W y W z 1 C0 C w C0W x S w C2C3 C0S z W x W z 1 C0 C w W y S w C z W y W z 1 C0 C w C0W x S w W 2 z 1 C0 C w C w 0 2 6 6 4 3 7 7 5 16 Q x Q y Q z 1 2 6 6 4 3 7 7 5 C0C z W x W z 1 C0 C w W y S w C2C3 S z W y W z 1 C0 C w C0W x S w C2C3 L t C z P x S z P y L x C0C0S z W x W z 1 C0 C w W y S w C2C3 C z W y W z 1 C0 C w C0W x S w C2C3C8C9 L t C0S z P x C z P y L y C0 W 2 z 1 C0 C w C w C2C3 L t P z L z 1 2 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 5 17 Int J Adv Manuf Technol The joint parameters can be evaluated using the same procedure similar to the table-tilting configuration. Notably, the reference driving point of NC data in this configuration is assumedtobethepivotpointRB.Thisdefinitionisadoptedto both the spindle-tilting and table/spindle-tilting type config- urations, and is consistent with those used in most of the commercial post-processor software packages. The complete analytical equations for NC data can be expressed as: B w arccos K z C0 W 2 z 1 C0 W 2 z C18C19 0 C20 w C20 18 C z arctan2 C0K y W x W z 1 C0 C w W y S w C2C3 K x W y W z 1 C0 C w C0W x S w C2C3 ; C0 C0 K y W x S w C0W y W z 1 C0 C w K x W y S w W x W z 1 C0 C w C1 C0 C20 z C20 19 X L x P x L t W x W z 1 C0 C w L t W y S w S z L y C0 Q y C0C1 C0 C z L x C0Q x L x 20 Y L y P y L t W y W z 1 C0 C w C0L t W x S w C0 C z L y C0 Q y C0C1 C0 S z L x C0 Q x L y 21 Z L z P z L t W 2 z 1 C0 C w L t C w Q z 22 3.3 Linearization problem Theoretically, the CAD/CAM system generates the CL data based on the assumption that the cutting tool moves linearly between two successive points. However, the actual tool motion trajectory with respect to the workpiece is not linear and becomes curved since the linear and rotary axes move simultaneously. The curved path deviates from the linearly interpolated straight line path between successive path points, and this problem is known as the linearization problem. An algorithm must be developed to solve this problem. Assume that P n , P n+1 and P n+2 are three continuous adjacent points in CL data, plotted in Fig. 4. The vector of P n in matrix form can be expressed as Q n,x Q n,y Q n,z K n,x K n,y K n,z , where Q n,x , Q n,y and Q n,z are the components of the position of the center of the tool tip, and K n,x , K n,y and K n,z are the components of the tool orientation. The corresponding machine NC code of P n is M n =X n Y n Z n B n C n . As the five axes move simultaneously from the current position P n to the subsequent position P n+1 , each axis is assumed to move linearly between the specified points 18. Therefore, each point in the actual curved path can be expressed as follows: M m;t M n t M n1 C0 M n 23 where t is a dummy time coordinate 0 C20 t C20 1. The corresponding CL data P m,t for M m,t can be determined by the forward kinematics equations, e.g. Eqs. (5) and (6) for the table-tilting type and Eqs. (16) and (17) for the table/ spindle-tilting type. Moreover, each point in the ideal linear tool path can be determined as follows: P n;t P n t P n1 C0 P n 24 w O w X w Y w Z t O t X t Y t Z C B x y z L LL+i j k RC RB Offset vector t L Fig. 3 Coordinate systems of table/spindle-tilting type configuration Int J Adv Manuf Technol , nn PM , , nt nt PM , , mt mt MP 11 , nn+ PM 1, 1, , nt nt+ PM 1, 1, , mt mt+ MP , n+2 n+2 PM Interpolated tool path Actual curved tool path Ideal linear tool path ,nt d Fig. 4 Linearization problem in multi-axis machining a b Fig. 5 Initiating dialog for the developed postprocessor. a table-tilting type. b table/spindle-tilting type Int J Adv Manuf Technol The distance between P m,t and P n,t denoted as d n,t forms a chordal deviation. If the maximum deviation (d n,t ) max exceeds the prescribed tolerance, then the additional interpolated CL data P n,t should be inserted into the original CL data. Theoretically, the numerical iterative method for calculating (d n,t ) max must be adopted. Practically, the middle point, t = 0.5, is often selected as the candidate point 10. After the intermediate point P n,t has been inserted, the corresponding machine NC code can be generated. 4 Discussion 1. The main characteristic of the nutating rotary axis configuration is the continuous motion between the horizontal and vertical positions in a single setup on the same machine. In the current configuration of the commercial machine tool, the angle of inclination of the nutating rotary axis is 45 degrees. This fact can be explained by the equations derived above. The table- tilting type is used as an example. Equation (5) represents the tool orientation in relation to the workpiece. The tool orientation relative to the work- piece in the initial position, where the table surface is horizontal, can be determined by substituting f z =f w = 0 into Eq. (5), and is given by 0 0 1 0 T . The nutating rotary axis is assumed to rotate around X-axis by an angle so that the components of the vector W are W x =0, W y =S and W z =C. Substituting the above conditions into Eq. (5), and setting f z =0 and K x K y K z 0 T =0 100 T for the table surface in the vertical position yields the following equation: 0 C01 0 0 2 6 6 4 3 7 7 5 SS w C0SC 1 C0 C w C 2 1 C0 C w C w 0 2 6 6 4 3 7 7 5 25 The solutions to Eq. (25) for and f w are =/4 and f w =. Therefore, the table surface can be set in the vertical position when the table is rotated through an angle about the nutating B rotary axis at an angle of inclination of /4. 2. The nutating units on the five-axis machine tools can enhance the flexibility of the machining strategy. How- ever, the CL data considered are limited. Equation (7) Fig. 6 Implementation dialog for generating NC data for table-tilting type configuration Int J Adv Manuf Technol shows that the condition K z C0W 2 z 1C0W 2 z C12 C12 C12 C12 C12 C12 C20 1shouldbesatisfied. When the nutating axis is set at an angle of 45 degrees, i.e. =/4 and W z =C45, K z is in the range 0 K z 1. Consequently, the CL data generated by the CAD/CAM system can not be machined in this configuration if K z is a negative value. 3. The derived analytical equation of NC data is a general form, which can be reduced to the orthogonal config- uration. The table-tilting type is chosen as an example. If the vector W is in the X-axis direction where W x =1 and W y =W z =0, the configuration reduces to the CA table-tilting type. The analytical equation of the NC data, specifying for example Y-axis values, agrees with those presented elsewhere 8 and can be expressed as follows: Y Q x C0 L x S z C w Q y C0 L y C0C1 C z C w C0 Q z C0 L z S w L y 26 Notably, in the cited work, the two rotary axes are assumed continuously to intersect each other, and the offset vector M 0i 0j 0k is used to derive the above equation. 4. Based on the solutions for f z and f w , the cutting tool may traverse the singular point where f w =0,Q x =Q y =0, Q z =1 and f z is undefined 12, 18. This singular position occurs when f w =0 and the C-axis axis is parallel to the cutting tool axis. As displayed in Fig. 4,if the current position P n+1 is the singular point, any value of f z is theoretically acceptable since f z is undefined. The next point P n+2 should be read further to ensure that f z varies linearly between two successive points. The value of f z at position P n+1 can be determined by considering a linear change between P n and P n+2 . 5. Feedrate control is an important issue in practical multi- axis machining. Most controllers, such as FANUC and Cincinnati Milacron, apply the feed rate number (FRN) and G93 code to control the feedrate. FRN is determined by the feedrate on the workpiece divided Fig. 7 Implementation dialog for generating NC data for table/spindle-tilting type configuration Int J Adv Manuf Technol by the span length of the resulting path. When rotational movements are combined with two or more linear axes movements, determinations for the path length become very complex. In most cases, an adequate approximation of the true path length can be determined by using only the linear displacement 19. 5 Implementation and verification 5.1 Software implementation The aforementioned postprocessor methodology was imple- mented in the Windows-XP environment using the Borland Fig. 8 Snapshot of VERICUT simulation for table-tilting type configuration Fig. 9 Snapshot of VERICUT simulation for table/spindle-tilt- ing type configuration Int J Adv Manuf Technol C+ Builder programming language and the OpenGL graphics library. A semi-sphere with a radius of 35 mm is used to illustrate machining. The CL data are generated by the commercial CAD/CAM system, PowerMILL 20. Two typical configurations of the machine tool, the table-tilting and the table/spindle-tilting, were tested. Figure 5 (a) shows the initiating dia