【正文】 is superposition with coordinate system xOy that describes the curved surface S, then the plane equation describing workpiece image isz = 0 （12）For one cutting tool path on plane with length of d, start and end point is Ps(x1,y1,0) and Pe(x2,y2,0) respectively, which should be mapped into a curve segment when engraving on spacial curved surface. Furthermore, the vertical projection’s length of chord length corresponding to curved segment should equal with the straight line segment’s length d. The line equation of plane cutting tool path’s straight line segment can be expressed as The equation of plane P that passes plane cutting tool path’s straight line segment and is vertical with image plane is(y2 ? y1 )x ? (x2 ? x1 )y ? x1 y2 + x2 y1 = 0 (14)Then the intersecting line of plane P and curved surface S to be engraved is the curve C, on which is cutting tool path on curved surface crresponding to that on plane. By this, the start and end point’s coordinates of cutting tool path on curved surface, which corresponding to that on plane, are Ps’ (x1,y1,z1) and Pe’ (x2,y2,z2) respectively, as x1,y1,x2,y2 are known, z1and z2 can be deduced with equation (10). It is easy to validate that the cutting tool path’s mapping method metioned above can suit for arbitrary quadrant of the coordinate system and engraving path in arbitrary direction. Interpolation puting of spacial curveRealtime interpolation puting is necessary to engrave spacial curve segment, whose coordinates ofstart and end point are Ps’ (x1,y1,z1) and Pe’ (x2,y2,z2) respectively, with parallel engraving machine. Firstly, theinterpolation number N should be calculated by interpolation precision, then the coordinates of ith interpolation point can be expressed as (xi , yi , zi ) , where , can be deduced with equation (10). When i=N, it shows the terminal Pe’ of the curve segment. According to actual needs, many interpolation algorithms can be selected during engraving(Xunnian,Y.,2002). Posture planning of cutting toolIn view of the special structure of parallel mechanism and ensuring effect for engraving, the posture of cutting tool should be considered during engraving(YuanShin,L.,1997), namely, the posture of cutting tool for every interpolation point should also be calculated besides its position during the control process. For parallel engraving machine with 6DOF, the rotation around axis ZB is not used in generally, the rotation angle α and β around axis XB and YB will be solved inFollowing. Supposing the base coordinate system of parallel engraving mechanism in initial state is same with that of curved surface(curve), that is, the axial line of cutting tool of cutting tool is vertical with dynamic platform plane of engraving machine and fixed on it, then the rotation angles of dynamic platform plane around XB and YB both are zero, namely, the angle between axial line of cutting tool and XB and YB both are π/2, as shown in (a).(a) initial state(b) during engravingFig. 9. The sketch map of cutting tool’s posture for initial state and during engravingWithout losing generality, now taking arbitrary interpolation point Pi (xi,yi,zi) on curve as sample, as shown in (b), then the tangent plane (parallel with the dynamic platform plane of engraving machine) equation of curved surface through point Pi can be expressed as Translating the general form as Ax + By +Cz ? (Axi + Byi +Czi ) = 0 （16）Where A = y39。u’ z’v’ ? y’v ‘z’u ‘B ‘= z’u‘ x’v’ ? z’v’ x’u’ C = x’u’ y’v’ ? x’v’ y’u’ In addition, the line equation of axis XB and Y is as following respectively y=z=0 and z=x=0 （17）According to the relation between spacial plane and line, the angle θx and θy, representing the angle between dyanmic platform plane of engraving machine and axis XB and YB respectively, satisfyIn formula (18) and (19), lmn1 and lmn2 represent the parameters of symmetrical line eqution of axis XB and YB respectively. Then we can get,when A≥ 0 is θx，otherwise θx。，when B≥ 0 is θy，otherwise θy。Here, θx and θy represent the angle β and α, β and α is the rotation angle of dynamic platform plane rotating around axis YB and XB respectively. Positive value denotes counterclockwise rotation, negative value denotes clockwise rotation. Thus, the posture parameter (α,β,0) of arbitrary interpolation point can be obtained, coupled with position parameter above, the position and posture of arbitray interpolation point on curve can be expressed as Pi(xi,yi,zi,α,β,0). So it can be ensured that the axial line of cutting tool at arbitrary interpolation point will be vertical with the tangent plane of engraving point all the time. Summary of tool path planningOn the basis of preceeding work, table 3 gives the corresponding relationship betweeninterpolation points of straight line cutting tool path on plane and that of curved surface in space. Based on the theory mentioned above, we simulate the cutting tool path on plane and spacial curved surface when engraving Chinese characters “mu”, as shown in . This section mainly resolves some questions about parallel engraving machine engraves on arbitrary spacial curved surface which is expressed with parameter equation, then realizes the mapping of cutting tool path from plane to spacial curve surface in the workspace of engraving machine. In addition, questions about interference and error analysis are not discussed in this section, the reason is that these questions will be involved only considering the radius of cutting tool, these works will be done in the future. 起點 插補點 終點 刀具在平面上的軌跡 刀具在曲面上的軌跡 切平面 位置和形態(tài)Table 3. The corresponding relationship between cutting