The data obtained from the on-site measurement of the profile error of spiral bevel gear teeth are the readings XRK (I, J), Yrk (I, J) and zrk (I, J) of X, y and Z axes when the probe is triggered (for the concave surface of spiral bevel gear wheel k = 1; for the convex surface of spiral bevel gear wheel k = 2), which can be converted into the spatial coordinates (xpk (I, J) of the measuring ball center in the machine coordinate system when the probe is triggered, YPk ( i，j) ，ZPk ( i，j) ) 。 The curved surface formed by the center of the measuring ball and the actual tooth surface are equidistant curved surfaces away from the radius r of the measuring head.

Using the approximate calculation method, the projection length of the vector between the ball center and the discrete point of the corresponding theoretical tooth surface in the normal vector direction of the discrete point minus the ball radius is directly taken as the tooth profile error value. This method is simple to calculate, but it will produce some errors.

In order to improve the accuracy of the measurement results, the surface fitting technology and optimization algorithm are used to calculate the tooth profile error. According to the measured spherical center coordinates (xpk (I, J), ypk (I, J), ZPK (I, J)), the equation of the surface composed of the spherical center is obtained by NURBS surface fitting technology. Suppose the equation of the surface is s (U, V), the normal vector is N2 (U, V), the partial derivative of the surface to parameter u is Su and the partial derivative to parameter V is SV, then:

The fitting surface compensates the measuring ball radius r along the N2 (U, V) direction to obtain the equation of the actual tooth surface. Let the diameter vector of any point on the actual tooth surface be RA, then:

The figure is the schematic diagram of tooth profile error calculation of spiral bevel gear. Point a in the figure is any discrete point on the theoretical tooth surface, and N is the normal vector of the point. In order to calculate the tooth profile error of the actual tooth surface relative to the theoretical tooth surface, it is necessary to find a point A1 on the actual tooth surface, which is located on the normal vector of point a. Let R be the radial vector of point a and R1 be the radial vector of point A1. R1 is a function of surface parameters u and V, then u and V are used as optimization variables, and (ra-r) is used × N → 0 is the optimization objective, and the parameters U1 and V1 corresponding to point A1 can be obtained by using the optimization algorithm. Then the tooth profile error of the actual tooth surface relative to point a of the theoretical tooth surface is:

According to the above method, the tooth profile error value of the actual tooth surface of the spiral bevel gear wheel relative to the theoretical tooth surface along the normal vector direction of each discrete point can be obtained. In order to eliminate the influence of fixture error and random measurement error, multiple teeth can be measured, and the arithmetic average value can be taken as the measurement result.