4.1. Problem of probing the tooth

Firstly, the probing obstacle is in zones 1 and 2 when the information is entered by the tridimensional measuring machine (see Figure 3). Indeed, we saw that the segment (ab) is undefined after having transferred the coordinates of the center of the ball. That is, there are no coordinates for probing this one (Figure 3-1), and the same goes for zone 2; when we did the transfer, no results were interpreted because the segment (ab) is unknown. To solve this problem of probing, we apply the system and notions of fuzzy logic.

#### 4.2. The principle of determination of the measured point corrected

As regards the measurement with high accuracy on CMMs, we specify the probe path. For this, we propose a new algorithm for the compensation of the radius of the tip of the stylus in a process of scanning the surface of the tooth by CMMs. The proposed algorithm is dedicated to the measurement of high definition. It is done to calculate the normal vector.

Figure 3. Problem of probing the tooth of a wheel.

The proposed compensation method consists of the following steps [1]:


measured points Oi. However, as a first approximation, the ideal points of the stylus ball in contact with the measuring surface are evaluated midway on the arc: AiAi+1. Indeed, some

Fuzzy Logic Applications in Metrology Processes http://dx.doi.org/10.5772/intechopen.79381 47

Figure 4. Analyzes the geometry of the sweep path for the determination of the measured point corrected.

In the first estimate, a preliminary point Si was chosen to the bow AiAi+1. Then, we take into account the mutual position of the neighboring points, Oi and Oi<sup>m</sup>,…, Oi<sup>l</sup> and Oi+1,…Oi+n. (where m and n are a number of previous and following points, respectively). An angular adjustment Δα<sup>i</sup> is obtained to improve the new position of the ideal point of contact with the flank and the ball and thus correctly calculate the points which are the closest Pi. Calculations of Δα<sup>i</sup> can rely on artificial intelligence techniques that are based on the rules of fuzzy logic. In the experimental implementation, we use the method of correction the measurement of the points of stylus (stylus tip envelop method). We opted for a fuzzy logic algorithm to compute Δαi. The entry of this system of logic is summarized in Table 2 with two components: Δαi, Azi, and Δki. We define the first magnitude Δzi which is the distance between point Oi and the point of intersection with the line (Oi1Oi+1). The second component, Δki, is the distance between point Oi and the point of intersection with the line (Oi<sup>1</sup> Oi<sup>2</sup>). These elements define the input

The choice of input and output variables depends on the control we want to achieve and the available parameters. In our chapter, we can consider the entries xm and ym which are the Cartesian coordinates of the points entered by the three-dimensional measuring machine in the course of scanning the surface of the right or left flank by the probing system. This choice

adjustments of the corrected measured point may be essential.

is intuitive and based on the experience of the operator.

values and output (see Figure 4).

4.3.1. Inputs


The calculation of angles Δα<sup>i</sup> can be achieved by exploiting a known basic variety or other-rule artificial intelligence techniques [2, 3]. In the experimental implementation of this method, we opted for the calculation of the angles Δα<sup>i</sup> with a fuzzy logic algorithm [4].

#### 4.3. Analysis of the geometry of the probe trajectory

We consider point Oi as one of the data points describing the position of the center of the ball of the spherical stylus registered by the CMMs (see Figure 4) [5]. We take the previous additional points Oi<sup>1</sup> and the following point Oi+1. Considering the external envelope tip of the stylus at the Oi point, it can be said that the ball of the stylus is always in contact with the material of the gears and that no part can be at the limit of the tip of the stylus; the point of contact of the stylus ball with the measuring surface is on the arc AiAi+1. The points Ai and Ai+1 have points of intersection of the three circles that have the centers Oi, Oi<sup>1</sup> and Oi+1, respectively [6].

All three circles have a radius R equal to the radius of the stylus ball, with which the preliminary calculations of the CMMs are made, according to the qualification of the probe system. All points of the arc are selected to transfer the corrected measured points associated with the

Figure 4. Analyzes the geometry of the sweep path for the determination of the measured point corrected.

measured points Oi. However, as a first approximation, the ideal points of the stylus ball in contact with the measuring surface are evaluated midway on the arc: AiAi+1. Indeed, some adjustments of the corrected measured point may be essential.

In the first estimate, a preliminary point Si was chosen to the bow AiAi+1. Then, we take into account the mutual position of the neighboring points, Oi and Oi<sup>m</sup>,…, Oi<sup>l</sup> and Oi+1,…Oi+n. (where m and n are a number of previous and following points, respectively). An angular adjustment Δα<sup>i</sup> is obtained to improve the new position of the ideal point of contact with the flank and the ball and thus correctly calculate the points which are the closest Pi. Calculations of Δα<sup>i</sup> can rely on artificial intelligence techniques that are based on the rules of fuzzy logic. In the experimental implementation, we use the method of correction the measurement of the points of stylus (stylus tip envelop method). We opted for a fuzzy logic algorithm to compute Δαi. The entry of this system of logic is summarized in Table 2 with two components: Δαi, Azi, and Δki. We define the first magnitude Δzi which is the distance between point Oi and the point of intersection with the line (Oi1Oi+1). The second component, Δki, is the distance between point Oi and the point of intersection with the line (Oi<sup>1</sup> Oi<sup>2</sup>). These elements define the input values and output (see Figure 4).

#### 4.3.1. Inputs

The proposed compensation method consists of the following steps [1]:

46 Fuzzy Logic Based in Optimization Methods and Control Systems and Its Applications

• contour of the sample defines a bow per ball, for each measured point,

opted for the calculation of the angles Δα<sup>i</sup> with a fuzzy logic algorithm [4].

measured by the spherical stylus tip,

Figure 3. Problem of probing the tooth of a wheel.

located in the middle of the arc,

4.3. Analysis of the geometry of the probe trajectory

previous point,

• realization of a series of high-density measurements on the characteristic of geometry

• calculating the points of intersection Ai and Ai+1 for each arc, with the next and the

• for each arc, the estimate of the point of contact with Si as the characteristic of this point is

• determination of angular compensation using the fuzzy logic knowledge base and the

The calculation of angles Δα<sup>i</sup> can be achieved by exploiting a known basic variety or other-rule artificial intelligence techniques [2, 3]. In the experimental implementation of this method, we

We consider point Oi as one of the data points describing the position of the center of the ball of the spherical stylus registered by the CMMs (see Figure 4) [5]. We take the previous additional points Oi<sup>1</sup> and the following point Oi+1. Considering the external envelope tip of the stylus at the Oi point, it can be said that the ball of the stylus is always in contact with the material of the gears and that no part can be at the limit of the tip of the stylus; the point of contact of the stylus ball with the measuring surface is on the arc AiAi+1. The points Ai and Ai+1 have points of

All three circles have a radius R equal to the radius of the stylus ball, with which the preliminary calculations of the CMMs are made, according to the qualification of the probe system. All points of the arc are selected to transfer the corrected measured points associated with the

intersection of the three circles that have the centers Oi, Oi<sup>1</sup> and Oi+1, respectively [6].

application of compensation based on the corresponding angular adjustment.

The choice of input and output variables depends on the control we want to achieve and the available parameters. In our chapter, we can consider the entries xm and ym which are the Cartesian coordinates of the points entered by the three-dimensional measuring machine in the course of scanning the surface of the right or left flank by the probing system. This choice is intuitive and based on the experience of the operator.

### 4.3.2. Output

The outputs are based on the problem that was posed; anyway, we can find one or more outputs and so on. Finally, it is lucid that the outputs in our work are two: xm and ym.
