C.-H. She et al. / Journal of Materials Processing Technology 177 (2006) 138–141 139

system has developed the window-based graphical interface so that the probe path simulation and projected dimensional error together with the die CAD model can be dynamically displayed on the screen. The experimental test is also performed with an industrial sheet metal stamping die on a CMM and demonstrates the effectiveness and practical application of the developed sys- tem.

2. Free-form surface representation

Various mathematical definitions are developed for represent- ing free-form surfaces while the non-uniform rational B-spline (NURBS) surface is chosen as the main modeling geometric entity in this paper owing to its general form representing many analytical surfaces and supported by most CAD/CAM system as well as standard exchange format (IGES). The mathemat- ical definition [10] of a NURBS surface can be expressed as Q(u, v), where u and v are independent parameters, and is given by

nitude of the projected dimensional error can be obtained by the following:

×E, K∗= (Mx  − Qx)Kx  + (My  − Qy )Ky  + (Mz − Qz)Kz

(3)

Note that the positive value of Eq. (3) means the measured point M is outside the design surface, whereas point M is inside the design surface if the value of Eq. (3) is negative. Therefore, the projected dimensional error can help the die maker to identify the discrepancies such as gouges and excess materials between the machined die surface and the designed surface.

4.  Measuring procedure of stamping die

To perform the measuring procedure, the reference coordi- nate system of the stamping die should be established first. Fig. 1 is the schematic diagram for measuring the die, where DDL rep- resents the die datum line, DDH represents the die datum hole

Q(u, v) =

.n

i=0

.n

j=0Bi,j Wi,j Ni,k (u)Mj,l(v)

.m

(1)

and MHZ denotes the maintenance hole zero that is the program

zero point for die manufacturing. The procedures for determin-

i=0

j=0Wi,j Ni,k (u)Mj,l(v)

ing the reference coordinate system are sequentially numbered

from   to   . First, four points of the bottom of the mold base are

where the Bi,j  are control points of a defining polygon net, and

Wi,j are control point weights. Ni,k(u) and Mj,l(v) are the normal- ized basis function in the u and v directions. The surface normal direction n of a given point with specified u and v parameters can be calculated using differential geometry:

inspected to determine the normal vector (Z-axis direction) of the base plane. Second, two DDH points are inspected to determine the X-axis direction. By applying the right-hand rule, the Y-axis direction can be determined automatically. Third, the origin of the reference coordinate system is determined by inspecting the

∂Q(u, v)

n = ∂u ×

∂Q(u, v)

(2)

∂v

MHZ point. Fourth, the points of the common standard sur- face for different forming operations are inspected. These points shown in Fig. 1 are marked as “*”. The purpose of  measuring

3. Probe path generation and dimensional error representation

From Eqs. (1) and (2), the coordinate and surface normal direction of the candidate inspection point can be determined once the u and v parameters have been specified. In real mea- surement, the touch-trigger probe is driven toward the die surface along the normal direction. Traditionally, the inspection pro- grams can only be employed in the specific CMM. As a result, inspection programs generated on one CMM generally cannot be performed on another. Since there are many CMM vendors, development of the inspection program is a labour-intensive and time-consuming job for each CMM. In this situation, the DMIS format program has emerged. DMIS provides a neutral inspec- tion communication format between various CAD systems and the CMM, so the probe path developed in this paper is written in the DMIS format.