3-Fundamental Issues in Tool Path Planning.pdfVIP

3Fundamental Issues in Tool Path Planning There are a number of issues involved in tool path planning for five-axis NC machining. Four fundamental issues are discussed in this chapter, namely, surface representation, machining strip width estimation, optimal tool orien- tation and forward step (kinematics) error. 3.1 Surface Representation Five-axis NC machines are widely used to machine dies, molds, turbine blades, aerospace and automotive parts, etc. These parts usually have complex geom- etry and are represented by parametric surfaces. Representation of surface by parametric equations allows the simple evaluation of differential properties of the surface. Many tool path generation techniques require calculation of certain surface properties such as the surface normal vector and the normal curvature. The close-form formula for commonly used surface properties are given below. Let S ≡ S(u, v) be the required parametric surface, the unit normal vector , n, is computed from the relation n = Su × Sv |Su × Sv| . (3.1) The first fundamental form (or line element), I, is defined by I = dS · dS, = (Sudu+ Svdv) · (Sudu+ Svdv), = Su · Sudu2 + 2Su · Svdudv + Sv · Svdv2, = Edu2 + 2Fdudv +Gdv2, (3.2) where 52 3 Fundamental Issues in Tool Path Planning E = Su · Su, (3.3) F = Su · Sv, (3.4) G = Sv · Sv. (3.5) The second fundamental form, II, is defined by II = ?dn · dS, = ?(nudu+ nvdv) · (Sudu+ Svdv), = ?nu · Sudu2 ? (nu · Sv + nv · Su)dudv ? nv · Svdv2, = n · Suudu2 + (n · Svu + n · Suv)dudv + n · Svvdv2, = n · Suudu2 + 2n · Suvdudv + n · Svvdv2, = edu2 + 2fdudv + gdv2, (3.6) where e = n · Suu, (3.7) f = n · Suv, (3.8) g = n · Svv. (3.9) The normal curvature of S in the direction v = aSu + bSv, is given by k(v) = ea2 + 2fab+ gb2 Ea2 + 2Fab+Gb2 . (3.10) The principal curvatures, which are the maximum and minimum of the normal curvature, are given by kmax = H + √ H2 ?K, (3.11) kmin = H ? √ H2 ?K, (3.12) where K and H are the Gaussian curvature and the mean curvature, respec- tive