Abstract: The concept of design, manufacture and inspection of the modern industrialized products have been changing

with the advent of computer numeric control machine tools and technologies such as computer aided design (CAD) and computer aided manufacturing (CAM). The dimensional control of manufactured parts is indispensable for the guarantee of the fulfillment of tolerances specified in the design. The Coordinate Measuring Machine (CMM) is recognized as a powerful tool and is frequently used in the inspection of industrialized products. Currently, CMM software is based on algorithms which use the least squares method or the minimum zone method to determine parameters of substitute geometries. This paper presents a new approach based on artificial neural networks to determine parameters of substitute geometric elements, by using coordinate measuring machines, such as: circle, ellipse, sphere, involute profile of spur gear, as well as free form surface. In order to evaluate the performance of the proposal approach, synthetic data were applied to both least square and neural network methods to determine parameters of different substitute geometric elements as for example: circle, sphere and ellipse. Simulation results show that the artificial neural networks are capable to determine parameters of substitute geometries with satisfactory accuracy. Additionally, artificial neural networks can contribute for measuring workpieces with free form surface and complex geometries without requiring numeric algorithms and mathematical modeling of the workpiece to be measured.

1. Introduction

The concept of design, manufacture and inspection of the modern industrialized products have been changing with the advent of computer numeric control machine tools and technologies such as computer aided design (CAD) and computer aided manufacturing (CAM). The dimensional control of manufactured parts is indispensable for the guarantee of the fulfillment of tolerances specified in the design (Silva & Burdekim, 2002). Thus, an inspection system capable of attending to current manufacture requirements must answer to the following requisites: velocity compatible with production speed, capacity to control complex geometries, measurement uncertainty compatible with the tolerances of the part, high flexibility in order to control a great diversity of geometries, and have a high degree of automation (Bosch, 1995). Traditional dimensional inspection techniques have not been capable of answering these requirements in terms of flexibility and high accuracy (Lima Jr, 2007). Coordinate measurement machines (CMMs), which operate according to the coordinate metrology principle, are capable of fulfilling these requirements, satisfactorily.

Therefore, these measuring machines are able to measure parts with complex formats (Webera et al, 2002 and Curran and Phelan, 2004). Despite having the aforementioned characteristics, the CMM is a measurement tool and, as such, may present measurement errors, which may come from various sources, as shown in Figure 1 (Weckenmann, 2001). This figure shows software error which may also influence the measured results. Majority of CMM software incorporate algorithms based on Euclidean geometry and use the least squares or minimum zone methods to obtain substitute geometry parameters. This paper presents a new approach based on artificial neural networks to determine parameters of substitute geometric elements, by using coordinate measuring machines, such as: circle, ellipse, sphere, involute profile of spur gear, as well as free form surface. In order to evaluate the performance of the proposal approach, synthetic data were applied to both least square and neural network methods to determine parameters of different substitute geometries as for example: circle, sphere and ellipse. Additionally, artificial neural networks were applied for measuring workpieces with free form surface and complex geometries.

Figure 1. Some factors that influence the CMM results (Weckenmann, 2001)

2. CMM algorithm based on the least squares method

CMMs determine specific substitute geometry from a set of N points which vary according to the geometry being measured. For example, in order to determine the sphere center and radius, there is need to collect at least four points on its surface. The majority of CMMs use the Least Square Method (LSM) to determine the parameters of substitute geometries (Capello and Semeraro, 1999). The main goal of this method is to determine the curve that best adapts itself to a certain set of measured points, in such a way as to have a minimal sum of the quadratic error. In the following sessions, a mathematical model based on LSM for standard geometric entities such as circle, ellipse and sphere will be presented.

2.1 The circle

The localization of the circle center and the value of its radius are the geometric parameters of the circle, which one desires to determine using Least Square Method (LSM). For such, equation 1 was used (Murthy, 1986).

(1)

2.2 The ellipse
In order to determine the center and radius of an ellipse based on LSM, it will be used Equation 2 (Murthy, 1986).

(2)

Once the constants A, B, F, U and V are determined, it is possible to determine the location of the ellipse center ( X_c , Y_c ), as well as the value of the ellipse major radius, R1, and minor radius, R2, with Equations 3 to 6, through,

(3)(4)(5)(6)

2.3. The sphere
The geometric parameters which must be determined for the sphere are: the localization of the sphere center, as well as its respective radius. Based on LSM, Equation 7 was used to determine these geometric parameters (Murthy, 1986).

(7)

Where: X_i and Y_i : coordinates of the collected points on the sphere.
N is the number of measured points;
R is the sphere radius;
X_c and Y_c : coordinates of the sphere center;

3. Development of artificial neural networks to determine parameters of substitute geometric elements

The main component of an artificial neural network (ANN) is the artificial neuron. By the combination of various artificial neurons, an artificial neural network is formed. An ANN is defined as a parallel distributed processor of simple processing units which have a natural inclination to store experimental knowledge and make it available for use (Haykin, 1998). ANNs differ in the learning method used, that is, the way in which the synaptic weights learn the existing relation between the network incoming and outgoing data. Another difference is in the composition of the network topology, where the determination of the number of layers of which the network is comprised is done empirically, as well as the number of neurons in each ANN layer. The artificial neural network has the capacity to learn from its environment and improve its performance through a learning process also known as training of the network. The neural network learns about its environment through an interactive process of adjustments applied to its synaptic weights which store, at the end of the process, the knowledge that the network has acquired from the environment in which it is operating (Haykin, 1998).

The majority of ANN models have some training rule, where the weights of its connections are adjusted according to the presented standards. In other words, they learn through examples. The neural network architecture is typically organized in layers. The neural network goes through a training process starting from known real and numeric cases, acquiring, from then on, the necessary systematic to adequately execute the desired processing of the supplied data. Thus, the neural network is able to extract basic rules from real data, differing from programmed computation, where a set of rigid pre-set rules and algorithms are needed (Haykin, 1998). There are, nowadays, many algorithms used to train ANNs. Back-propagation is a supervised algorithm that uses pairs (input, desired output) to, by means of learning through error correction, adjust the synaptic weights. The goal of this learning is to adjust the parameters (synaptic weights) of the network to find a relation between the input and output data supplied to the network.

In this work the Back-propagation algorithm was used to obtain artificial neural networks (ANNs) to determine parameters of substitute geometric elements such as circle, ellipse and sphere described in section 2. For each of the geometric entities input and output data were generated to be used in the network training. It should be stressed that the topologies of the ANNs developed in this work vary according to the entity in question. Once the network was trained, comparisons were done between the results obtained with ANNs and the least square method using the same input data.

3.1 Artificial neural network for the circle

The aim of this network is to determine the radius, i R , and the position of the circle center, with relation to a reference coordinate system, from five sampled points which are gathering angulary equispaced on the circle circumference. In order to training the network, several set of five sampled points Pi(X_i , Y_i , 0 ) were generated for a circle which position of its center (X_c , Y_c , 0) is known. This was possible by using the equation 8 and selected values for circle radius, R and a_i.

X₁ = Rcos (a₁)

Y₁ = Rsen(a₁)                                                                           (8)

Where, a₁ is the angular position of the point Pi( i X , i Y , 0 )

Figure 2 shows a set of five points Pi( i X , i Y , 0 ), generated by considering a circle which radius, R, is 10 mm and the position circle center is (X_c = 0, Y_c = 0). The points P1 to P5 are defined by considering the angle a_i to be the following values: 0, 90, 180, 270 and 360 degrees, respectively. Other set of points P_i(X_i , Y_i , 0 ) can be obtained by rotating the points P₁ , P₂ , P₃ , P₄ and P₅ , as shown in Figure 2. This is possible by using the Equation 9 and different values of angle . Figure 3 shows two set of points, the blue ones are before rotation and the red ones are obtained when angle  is 18 degree. Therefore, by varying the angle  from zero to 360 degree at step of 18 degree, 21 set of points ( P₁ , P₂ , P₃ , P₄ and P₅ ) are obtained to each circle of radius, R, and center position ( X_c , Y_c , 0).

In order to get data to training and to validate the artificial neural network developed in this work, the approach described above was applied to several defined circles which radius, R, and coordinates of the center position varying as following:

Coordinate X_c : from 0 to 150 mm at step of 25 mm;
Coordinate Y_c : from 0 to 150 mm at step of 25 mm;
R: from 10 to 150 mm at step of 20 mm.

Since each circle generates 21 set of points Pi(X_i , Y_i , 0 ), after applying these variations in the parameters R, X_c and Y_c 3087 set of points Pi(X_i , Y_i , 0 ) were obtained to be used in the network training process. The network topology was defined in the following way: ten neurons in the input layer, each coordinate represents one neuron as five points are collected in each circle; thirty neurons in the intermediary layer, with sigmoid activation function, and two neurons in the output layer, with linear activation function. This network diagram is shown in Figure 4. The training was carried out and Figure 5 shows the error decrease in the network, that is, as the repetitions happen, the networks learns the relation between the network input and output data.

After the training, various simulations were carried out to check whether the network had a good performance. This was done supplying the network with coordinated points (X_i ,Y_i , 0 ) of circles which were not used in the training, as shown in Figure 6. These circles may be considered holes in a part to be measured, and it is desired to determine these circles' radii, as well as the center localization of each one of them in relation to the part local coordinates system. Table 1 shows, for each circle, the coordinates ( X_i , Y_i , 0 ) of the points P1 to P5 and the coordinates ( X_c , Y_c ) of their centers.

The data shown in Table 1 were applied to determine the coordinates of the circle center and the radius of the circle by using the developed artificial neural network (ANN) and the Least Square Method (LSM). The LSM was applied as it is used in most CMMs algorithms to determine parameters of substitute geometries. Analyzing the results presented in Table 2, it is possible to conclude that the developed artificial neural network can determine parameters of a circle effectively.

3.2. Artificial neural network for the ellipse

In this section an artificial neural network which was developed to determine the parameters of an ellipse is presented. These ellipse parameters are: position of the ellipse center ( X_c , Y_c ) and the major and minor ellipse radius. In order to get data to training and to validate the artificial neural network, developed in this work, the Equation 11 was used for generating a set of ellipses with radius a, b and position of the ellipse center ( X_c , Y_c ) very well defined. For each ellipse are gathering 10 points Pi( X_i , Y_i , 0 ) as shown in Figure 7.

where:
a and b are the minor and major ellipse radius, respectively;

X_c and Y_c : coordinates of the ellipse center;
X_i and Yi : coordinates of points on the ellipse.

It is possible to get different ellipses from that shown in Figure 7 by attributing, in Equation 11, different values for the parameters a and b as well for the position of the ellipse center ( X_c , Y_c ). In this work the following parameters were used to generate ellipses:

• Radius a: from 30 mm to 100 mm with step of 14 mm;
• Radius b: from 15mm to 50 mm with step of 7 mm;
• Coordinate X_c : from 100 mm to 250 mm with step of 25 mm.
• Coordinate Y_c : from 100 mm to 250 mm with step of 25 mm.

In additional, it is possible to get different ellipses by rotating each one around its center. In this case the rotation angle varied from 0 to 360 degree at step of 36 degree. The artificial neural network developed in this work to determine the parameters of an ellipse has the following topology: 20 neurons with linear activation function in the input layer where each neuron represents a coordinate of each point ( X_i , Y_i , 0 ) collected, 30 neurons in the intermediate layer with hyperbolic tangent activation function and 2 neurons in the output layer with linear activation function, which represent the coordinates ( X_c , Y_c ) of the ellipse center. After training, the network was capable to represent the relation between input and
output data and the error was less than 1 x 10-24 as shown in figure 5. After the network was trained, simulations were carried out to check the ANN performance. Table 3 shows the simulations results obtained from both Least Squares Method (LSM) and Artificial Neural Network developed in this work. Observing this table, one may see that the ANNs had a good performance for both ellipses as the maximum error was found to be 0,1µm .

where: NV = Nominal value;
Radius_BR – Ellipse major radius;
Radius_SR – Ellipse minor radius;
ANN= Value obtained by using artificial neural network.
LSM= Value obtained by using least square method.
Error (LSM) =NV - LSM
Error (ANN) = NV –ANN

3.3. Artificial neural network for the sphere

For this geometry, various spheres were generated by using equation 12 and attributing different values for the sphere center coordinates ( X_c , Y_c , Z_c ) as well as for the respective sphere radius R. The same procedure used in the development of the circle Artificial Neural Network (ANN) was applied. Thus, spheres with radii varying from 50 mm to 250 mm at step of 50 mm were generated. The center coordinates of each sphere varied in the following way: X_c from 25 mm to 250 mm at step of 28,125 mm; Y_c from 25mm to 250 mm at step of 28,125 mm and Z_c from 50 mm to 250 at step of 50 mm. For each one of the generated spheres, 5 coordinated points Pi ( X_i , Y_i , Z_i ) were collected. These points served as input data for the ANN, and the target of the neural network are the sphere center coordinates ( X_c , Y_c , Z_c ), determined from the collected coordinated points Pi ( X_i , Y_i , Z_i ) .

(X − X_c)² + (Y −Y_c)² + (Z − Z_c)² = R                                                (12)

The network used for this geometric entity has the following topology: 15 neurons in the input layer, where each coordinate of the five points Pi ( X_i , Y_i , Z_i ), that are collected on the sphere, is represented by one neuron; 30 neurons in the intermediary layer with sigmoid activation function and 3 neurons in the output layer with linear activation function to represent the coordinates X_c , Y_c and Z_c . The algorithm used in the training was, also here, back-propagation. After the network was trained simulations were carried out to check the ANN performance, and the results are shown in Table 3. Also, in Table 4 the results obtained by using the LSM are presented. In the simulation three spheres were used, where the position of the center and the radius of each sphere were known. Departing from the centers and radius, 5 coordinated points Pi ( X_i , Y_i , Z_i ), were generated for each one of the spheres. Therefore, these points served as network input data for the simulation. As can be seen in Table 4, the results obtained by the ANN were satisfactory compared to the LSM.

where: NV = Reference value;
Error (LSM) = NV - LSM
Error (ANN) = NV -ANN
X_c , Y_c and Z_c : coordinates of the circle center

4. Application of artificial neural networks to analyze the involute profile of spur gear

On a practical level, a gear production system aims to produce gears that meet the design specifications. One method for checking whether the gear geometric parameters meet those defined in the design consists of measuring the gear after its manufacturing process. For example, if the measured geometric parameter is the involute profile, then the nominal or design profile is compared with the actual profile measured on the part. Coordinate measuring machine (CMM) may be used for measuring the involute profile of the gear tooth, however, this machine normally does not have, incorporated into its software, a module dedicated to measuring gears. Therefore, it is necessary to implement a specific module in the CMM software to measure gears. It is worth to note that is not an easy task. In order to analyze the involute profile from coordinated points Pi( X_i , Y_i ,0), an Artificial Neural Network (ANN) was developed to determine the differences or errors, if any, existent between the nominal and actual involute profiles after the manufacturing process of a spur gear. For this ANN, it was considered as nominal data a spur gear defined by the following geometric parameters: pressure angle, =20º; module, m = 5mm and number of teeth, Z = 20 . From these parameters it was possible to obtain other gear parameters, such as diameters: external, d_e , internal, d_i , primitive, dp, and base, d_b , (SHIGLEY, 2003). Figure 9 shows a representation of the nominal gear with 20 teeth.

The involute profiles of the teeth shown in Figure 9 were obtained from the equations 13 to 16, which depend on the evolving angle and this depends on the pressure angle  and radius, r, measured from the base diameter.

With the use of equations 13 to 16, it was possible to obtain nominal curve of the gear involute profile shown in Figure 8. Thus, the generated profile is considered the nominal one. However, it is known that, in practice, a manufactured gear may not be in conformity with the dimensions specified in the design. In this work, it was developed an Artificial Neural Network (ANN) which is capable of checking whether the involute profile conforms to the design specifications.

The nominal involute profile, which is shown in Figure 9 was obtained using the equations 13 to 16 and by considering the following data: pressure angle,  = 20⁰ ; module, m = 5mm and number of teeth, Z = 20 . These data were the same used for obtaining Figure 9. Once defined the nominal involute, it is necessary to generate, by simulation, an involute profile with errors. In order to simulate these errors, a modification was made in one of the gear parameters, in this case the pressure angle  . When this angle changes, the geometric shape of the tooth also changes, as seen in Figure 9.

This kind of situation may happen in practice in the following situation: one desire to produce a gear with a defined pressure angle, but because of errors from the manufacturing process, the pressure angle is different of that specified in the design. Figure 10 shows this situation, where the gear defined in the design has a pressure angle equal to  = 20⁰ (blue curve) and after the manufacturing process the gear has a pressure angle equal to  =14,5º (red curve).

The ANN developed in this research has the aim to determine the existing difference, if any, between the nominal involute profile and the actual profile. This network has the following topology:

- 4 neurons with linear activation function at the input layer, where 2 neurons represent the coordinates X_nominal and Y_nominal of a point which is on the nominal involute profile and the other 2 neurons represent the coordinates  X_actual and  Y_actual of a point which is on the actual involute profile. These coordinated points have in common the coordinate X as the aim is to evaluate the difference, if any, between the coordinates Y of the nominal and actual involute profiles. This comparison is done only for the region where the nominal and actual involute profile have the same interval of the coordinate X, as shown in Figure 9. In this figure, the blue and red curves represent the actual and nominal involute profiles, respectively.

- 20 neurons in the intermediary layer with hyperbolic tangent activation function;
- 5 neurons with linear activation function in the output layer, where 4 neurons are the same as in the input layer, described above, and the other one represents Delta(Y ) , that is the difference between the coordinate Y of the points which are on the nominal and actual involute profile, respectively.

In order to get simulation data for training the artificial neural network (ANN), different involute profiles were generated by considering the following pressure angles, (  ) : 14,5º; 17º; 20º and 25º. For each involute profile were collected 50 coordinate points P_i( X_i , Y_i ). It is important to note that the comparisons between the involute profiles took place by considering that the nominal involute profile had a pressure angle, = 20º . Table 5 shows an example of the points used for training the ANN, which are on the nominal and actual involutes shown in Figure 9.

where: Delta( Y ) = Y_nominal - Y_actual = Y_20º - Y_14,5º

: The table data are presented in mm.

After determining input and output data the ANN was trained by using the back propagation algorithm. Figure 10 shows that the ANN error was found to be, after 250 iterations, 2,14*10^-13 . This error is the difference between the expected and actual output of the ANN. Therefore, it can be concluded that the network was successfully trained.

In order to verify the ANN performance various simulations were performed after its training process. In one of the simulations, it was considered that one of the gears should have its pressure angle defined as = 20º , however, in practice, as consequence of manufacturing process errors that pressure angle was equal to  =17º . It should be stressed that in each of the simulations were used points that were not in the set of points
used for training the neural network. Table 6 shows points used in this simulation, where the difference between the involute profiles, in analysis, is known beforehand. These points were collected as shown in Figure 11. Table 7 shows the results obtained with the ANN for this set of points.

After the simulation, the results show that the ANN developed in this research is capable of checking whether the gear involute profile meets the design specifications.

5. Application of artificial neural network for evaluation of 3D surfaces

The following situation will be considered in this section: a part must be manufactured and its profile, a 3D surface, must be that shown in Figure 12. This profile is mathematically defined by Equation 15.

where: A² = 10 e B² = 5
X , Y and Z : coordinates of a point on the surface

In practice, when a part is manufactured it is possible that the final dimensions, after the manufacturing process, to be different from those defined in the design. This difference may be attributed to errors from the manufacturing process, caused by geometric and dynamic errors of the machines tool. In order to determine the existing difference between the nominal and actual surfaces, from coordinate points P_i(X_i,Y_i,Z_i), an artificial neural network (ANN) was developed as part of this research work. The nominal surface was defined by equation 15 using the values of the parameters A² =10 and B² = 5. The actual surface was, also, defined by equation 15, but with parameters A² and B² defined by the equations 16 and 17, respectively.

The value of the angle (y ) , in the simulation process, represents the influence of eventual errors from the manufacturing process. This was done in order to make it easier to generate surfaces which are different from the nominal surface. Therefore, it is possible to develop and train an ANN capable of determining the difference between the nominal and actual surfaces.

If the angle ( ) is null, the manufactured surface will have the same dimensions and form as specified in design. Figure 13 shows the nominal and actual surfaces, by considering that ( ) =10 degree. In order to determine the difference between the nominal and actual surfaces the following procedure was applied: first, the actual surface is measured by scanning or digitalizing process in order to get a set of points P_ia(X_ia,Y_ia,Z_ia) on the actual surface. Second, the mathematical model that defines the nominal surface is used for each pairs (X_ia,Y_ia) to determine the set of nominal points P_in(X_in,Y_in,Z_in). Finally, it is possible to determine the error, Z_i, between the coordinates Z_ia and Z_in of the actual and nominal surfaces,
respectively.

The artificial neural network (ANN) used to check the difference between the nominal and actual surfaces has the following topology:

• Input layer: 6 neurons with linear activation function where 3 neurons represent the coordinates of a point ( X_nominal, Y_nominal, Z_nominal )  which is on the nominal surface and the other 3 neurons are the coordinates of a point ( X_actual, Y_actual,  Z_actual )  which is on the actual surface. Remembering that the points have in common the coordinates X and Y.
• Intermediary layers: 15 neurons with hyperbolic tangent activation function in the first layer and 10 neurons with hyperbolic tangent activation function in the second intermediary layer.

Output layer: 7 neurons with linear activation function where 3 neurons represent the coordinates of a point (X_nominal, Y_nominal, Z_nominal ) which is on the nominal surface and the other 3 neurons are the coordinates of a point ( X_actual, Y_actual,  Z_actual ) which is on the actual surface and 1 neuron represents the error, Z_i, between the coordinates Z_ia and Z_in of the actual and nominal surfaces, respectively.

Various surfaces were generated by using different values of  such as: 5º; 10º 20º and 30º. After the neural network was defined, it was then trained using the Back-propagation algorithm. Some data used in the network training process is shown in table 8. In this table the actual surface is defined with =10º . After the neural network was defined, it was then trained, and afterwards various simulations were performed to check the ANN performance. In these simulations were considered points which were not in the ANN training data. The simulation was carried out by using 25 points which were collected on the actual surface. Table 9 shows some of the points used in the simulation, where the actual surface has = 25,6º Figure 15 shows the actual and nominal surfaces for the situation in question. Also, in Table 9 are shown the results obtained by the ANN. The maximum error was found to be 0.2 µm, which proves that the ANN has a good performance.

where: Delta (Z ) = Delta (Y)_nominal - Delta (Y)_actual

Other simulations were carried out and the results are shown in Table 10. It is worth to note that each value of the angle ( ) generate a different actual surface.

Figure 14 – Distribution of simulation points for  = 25,6º and 25 collected points

7. Conclusions

This work presented an alternative approach based on artificial neural networks to determine parameters of substitute geometric elements by using coordinate measuring machines (CMMs). Differently from the current computational systems of the most CMMs, which are based on Euclidean geometry and numeric methods such as least squares and minimum zones, the approach proposed in this work uses artificial neural networks (ANNs).
Simulation results obtained by using ANNs to determine the parameters of geometries entities such as circle, ellipse and sphere were found to be as effective as those obtained by using least square method. After the promising results obtained by appling the proposed approach for geometries considered simple, the ANNs were applied to complex geometries such as: involute profile of the tooth of a spur gear and free-form surface. In both cases, the results obtained by using the ANNs were considered effective.

The results, obtained in this work, showed that the artificial neural networks are an important and powerful tool to be applied in coordinate measuring machines and are capable to determine parameters of substitute geometries with satisfactory accuracy, as the maximum error of the ANNs was bellow 1 µm. Additionally, artificial neural networks can contribute for measuring workpieces with free form surface and complex geometries without requiring numeric algorithms and mathematical modeling of the workpiece to be measured . Further research work is to be developed in order to apply this proposed approach in reverse engineering applications.

8. References

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Authored By

José Carlos de Lima Júnior, This e-mail address is being protected from spambots. You need JavaScript enabled to view it

João Bosco de Aquino Silva, This e-mail address is being protected from spambots. You need JavaScript enabled to view it

Federal University of Paraiba

Department of Mechanical Engineering

58059 - 900 – Joao Pessoa – PB, Brazil