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Procedía Engineering 206 (2017) 1458-1463

Procedía

Engineering

www.elsevier.com/loeate/procedia

International Conference on Industrial Engineering, ICIE 2017

Modeling Optimal Path of Touch Sensor of Coordinate Measuring Machine Based on Traveling Salesman Problem {Solution

O.V. Zakharov^*, A.F. Balaeva, A.V. Kochetkovb

aYuri Gagarin State Technical University, 77, Politecknicheskaya str., Saratov 410054, Russia bPerm NationalResearch Polytechnic University, 29, Khmsomolpr., Perm 614990, R ussia cSamara University, 34, Moscow Highway, Samara 443086, Russia

Abstract

The article justifies the necessity of reducing the measurement time of the part surfaces in the coordinate measuring machine by minimizing the trajectory of the sensor. It is established that the search op the minimal path by the sensor corresponds to the travelling salesman problem that can be solved by exhct or approximate method. The branch aby bound method and ant colony optimization are selected for comparison. We peeform computer simulation for finding the optimal path of the sensor by sontrolling diaferent numbers of points on plane, cylindrical ¡and rpherical surffces. It in e stablishedthat ant co lony algorithm gives close to optimat solution and requires far less time for its near-ch in contrast to the branch and bound algorithm. Application of the ant colony optimization will significa^ty improve the performance of the measurement of tha part surfaces in the ^ordinate measuring machine.

©20 H The Aushors. Punished by Elsevier Ltd.

Peer-review undcr responsibility of the scientific committee of the Infernational Conference on Industrial Engineering Keywords: coordinate measuring machine; optimization; travelling salesman problem; branch and bound method; ant colony optimization.

1. Introduction

Asis known [1-3], th e m easurement accuracy depends on the number of control point;!, chosen by the operator. The moro control points are measured, the higher is tíoe measurement accuracy. However, with the increase in the number of control points, the trajectory of the shnsor and, consequently, the inspection time increase. I° it is necessary to meafure a large range of products, the operator faces the task o° finding the optimal measurement strategy of the surface, the purpose of which is the reduction of measurement time.

* Corresponding author. Tel.: +7-987-324-9353. E-mail address: zov20@mail.ru

1877-7058 © 2017 The Authors. Published by Elsevier Ltd.

Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering. 10.1016/j.proeng.2017.10.661

The measurement strategies, used at enterprises, depend a lot on the operator skills, and the improvement of measurement performance of scientifically-based approaches [1-10] is based on the search of the location of the minimum sufficient number of points needed in order to achieve the required measurement accuracy. The possibility of improving the measurement performance by selecting an optimal sequence of control points that provides the minimum trajectory of the CMM sensor is practically unused. With the aim of improving measurement performance, there was conducted a study based on simulating the optimal path of the sensor in the coordinate measuring machine.

2. Problem statement

To conduct the study it is necessary to choose the most frequently used surfaces as geometric images of real surfaces so that they have a simple analytical description, and do not cause any difficulties during the transition between the Cartesian, cylindrical, and spherical coordinate systems. Hence there were chosen a plane, cylinder, and sphere, described by parametric equations in the Cartesian coordinate system.

To make control points describe the measured surface with the greatest accuracy, the location of these points should be uniform. Uniform distribution of points can be specified either through stochastic simulation of their coordinates according to the law of uniform distribution, or the analytical spacing between the closest points. Since stochastic simulation is justified only when there are a large number of points, analytical approach to the determination of the points coordinates was chosen to conduct the study.

To determine the coordinates of evenly distributed N points on a plane surface and to simplify the calculations, the surface was aligned with the plane XOY and is limited by four planes (xi=-L/2, x2=L/2, yj=-L/2, y2=L/2), whose intersections with the surface form a square with the side L. The points are located in the plane with equal pitch along the OX axis and OY axis respectively hx=hy= L/N1/2. To determine the coordinates of evenly distributed N points on a cylindrical surface, the axis of the cylinder was aligned with the axis OZ, and the surface was limited by the planes z1=-L/2, z2=L/2. The points on the cylindrical surface are located with angular pitch d($=2Tr/N1/2 in the plane XOY and linear pitch hz= L/N1/2'along the axis OZ. The points on the sphere were located with zenith d and azimuth d6=2tt/N1/2 angular pitch.

For computational convenience the centres of square area of the plane, cylinder and sphere are aligned with the origin of the coordinate system OXYZ.

The length of the sensor path for each surface depends on the strategy of the control points bypass, which is a sequence of control points. To get full information about the location of all the surface points, it is necessary to make the sensor touch each control point at least once. Upon completion of a full bypass of the surface, the sensor returns to the starting point. In this case, the sensor trajectory is a closed broken line which defines the duration of the measurement. Having presented the path of the sensor as a graph whose nodes are the control points and ribs of the graph are the trajectories between two points, we get the Hamiltonian circuit. Then the problem of path optimization is reduced to finding the Hamiltonian circuit which has the minimum length, and represents the travelling salesman problem.

Let us describe the travelling salesman problem as an integer linear programming problem. Let us represent the movements of the probe from the i-th control point to the j-th in the form of a displacement matrix Lj, where i, j e 1, n is for a certain number of control points n e N . Let us represent the direction of movement from i-th point to j-th in the form of a number matrix lj e (0,1} for i, j e 1, n, where i = 1, if the bypass path has a rib (i, j), and lij = 0 otherwise. Then the problem is reduced to finding a solution l when the path will be minimal:

G(X) = Z Z Ljlj ^ min (1)

i =1 j=1 J J

Depending on the surface, elements of the displacement matrix Lij are determined by the expressions: • for the plane:

Lij =yj(Xj - Xi )2 + (yj - y )2 + (Zj - zi )2 (2)

for the cylinder:

-*ij)2 + (Zj -Zi)2 (3)

• for the sphere:

xixj + yiyj + zizj

Lu = R • ar cos ——---2-(4)

where (x, y, zi) and (x,, y, zj) are the coordinates of control points i and j of the surface.

Currently there have been developed different methods for solving the travelling salesman problem, which are classified according to the way of finding solutions into exact and approximate (heuristic and meta-heuristic). In comparison with the approximate methods, exact ones allow finding the best solution, but require more time for calculation. Approximate methods require less time for calculation, but give only an approximate, and often a local solution.

The problem of method choice and their utility for the path searching of the sensor in coordinate measuring machine is to establish the significance of the influence of the difference in the paths on the performance of the measurement compared with the difference in time spent for calculation of these paths. To solve this problem there was chosen the branch and bound method (BNB) [4] attributable to the exact methods, and the ant colony optimization (ACO) [10], attributable to the meta-heuristic methods. Both of these methods are widely known and well-studied.

To compare the length of the paths, obtained from the solution of the travelling salesman problem by BNB and ACO, it is necessary to have a base path. For this one needs to set the strategy of constructing the base path of the sensor for each surface, taking into account geometrical features of construction of the control points.

Since the points form a square grid of squares on the plane, then as a starting point it is useful to choose a point with coordinates (-L/2+hx, -L/2+hy, 0) in the Cartesian coordinate system. The next point offsets from the previous by one pitch hx along the axis OX. Having passed the first row of points along OX, the sensor switches to the first point of the next row with coordinates (-L/2+hx, -L/2+2hy, 0) sequentially touching each point in the second row and so on until it reaches the last point of the last row. After that, the sensor returns to its initial position opposite the starting point of the first row, and, hence, closing the cycle of motion. Hence, we obtained the base path of the sensor for the planar surface.

To bypass the cylindrical surface, there is selected a starting point with coordinates (R, 0, -L/2) in the cylindrical coordinate system. The next point of the path offsets by an angular step 2/Nxy with coordinates (R, 2/Nxy, -L/2) until the sensor does not pass all the points of the first row of the circle, then it switches to the next row and the bypass of the points continues until the sensor reaches the last point of the last row and then returns to the starting position.

The bypass of the spherical surface begins at the point with coordinates (-R, 0, 0) in spherical coordinate system representing the lower pole of the sphere. Then it goes to the nearest row with the angular pitch d<p and goes round the row of points with angular pitch din the clockwise direction. Having bypassed all the points of the row, the sensor switches to the next row with angular pitch d9. After going through all the points of the spherical surface, the sensor returns to the starting position, forming the base path.

3. Results of computer simulation

For computer simulation we specified initial conditions. The areas of planar, cylindrical, and spherical surfaces are equal to £=40000 mm2 for any number of points. The following surface parameters correspond to this condition: a) the side of the square bounding the plane is L=200 mm2; b) the length of the cylinder is R=31.8 mm, cylinder height is L=200 mm; c) the radius of the sphere is R=56.4 mm. The radius of the sensor is assumed to be r=2 mm. The magnitude of the sensor withdrawal is equal to 2r. To reduce the time for the calculation of the optimal path in

programs implementing the ACO and BNB, the values of the distances in the distance matrix are rounded to tenths. Since the coordinates of the points remain constant, the rounding does not affect the positioning accuracy, but substantially reduce the size of intermediate data during the calculation. Basing on geometric features of the surfaces, we defined the minimum number of measurement points for each surface: for the plane - 3 points; for cylinder - 6 points; for the sphere - 6 points. The maximum number of points for the plane and cylinder is 256 points, for the sphere - 266 points.

To determine the values of parameters of the ant colony algorithm, there was conducted pre-testing, as a result of which we adopted the following initial conditions: the number of iterations is m=50; evaporation rate of the pheromone is p=0.3; the degree of influence of the pheromone on the choice of the path is a=7; the degree of influence of distance to the nearest point on the choice of the path is p=6; number of ants is equal to the number of control points.

Program for path search of the sensor using ACO algorithm was run five times for each value of the number of points, and from the obtained results there was chosen the best length of the path.

As an example Figure 1 shows the base path and the routes of the sensor, obtained using ACO for 64 points on the planar surface (Fig. 1 a), on the cylindrical surface (Fig. 1 b) and on the cylindrical surface (Fig. 1 c).

Fig. 1. Route of the sensor with 64 control points on the cylindrical surface: dashed line - without optimization; solid line - after optimization: a) for the ant colony algorithm; b) after the optimization using the branch and bound algorithm.

After simulation for the plane, we obtained dependencies of the path of the sensor (Fig. 2 a) and calculation time

The graph in Fig. 2 shows that the implementation of search algorithms allows finding the path 2 times shorter than the base one, thus reducing the time of measurement of the coordinate measuring machines. The solution obtained by the ACO algorithm is at most 10% worse than the solution found using the BNB algorithm. The graph in Figure 1,b shows that starting with 30 points, there is a significant gap in calculation time using the BNB and ACO algorithms, which increases with the increase in the number of control points. Therefore, the application of the BNB method is suitable for surface control with 30 points. The versatility of CMM, which implies a choice in the range of up to several hundred points, and a minor difference in calculation results of the used algorithms, allow us to conclude that for surface control approximated by plane it is advisable to use the ant colony algorithm.

(Fig. 2 b)

Fig. 2. The graph of dependences for a planar surface: a) of the path distance of the sensor on the number of control points - Le, mm; b) of the calculation time of an optimal path on the number of control points - Te: 1 - the base path; 2 - the branch and bound method; 3 - ant colony

optimization.

The above conclusion about utility of the ant colony algorithm in order to control planar surfaces is true for the cylindrical surface, which is confirmed by the graphs in Fig. 3.

Lî, mm Tî, s

4 16 36 64 100 144 196 256 4 16 36 64 100 144 196 256

N. points N. points

Fig. 3. The dependency graph of control points bypass on the cylindrical surface: a) of the path distance of the sensor on the number of control points - Le, mm; b) of the calculation time of an optimal path on the number of control points - Te: 1 - the base path; 2 - the branch and bound

method; 3 - ant colony optimization.

As a result of simulation for the spherical surface, Fig. 4.

Li, nun

— 1 --3

6 26 62 114 182 266 N, points

were obtained dependency graphs, which are presented in

Te, s. 1?00

6 S14 ]S2

N, points

Fig. 4. The dependency graph of control points bypass on the spherical surface: a) of the path distance of the sensor on the number control points - Le, mm; b) of the calculation time of an optimal path on the number of control points - Te: 1 - the base path; 2 - the branch and bound method;

3 - ant colony optimization.

From the graphs in Fig. 4, it can be seen that the optimal path found by using the BNB algorithm almost does not differ from the base path. For the path obtained by ACO algorithm there occurs a slight deterioration with the increase in the number of control points. This is because with the increase in the number of points and their distribution with equal zenith and azimuth angular pitches lead to the concentration of points around the poles of the sphere. And linear pitch between the nearest points of the circle near the pole becomes substantially less than the linear pitch between the circles, where the points are located. Therefore, the optimal path obtained by using the BNB, coincides with the base path, obtained by a complete bypass of all points of one circle, and only then the sensor moves to the next circle with points. Thus, the best option is already predetermined by the base path. In the case of another distribution of points on a spherical surface, when the path selection for the next point is not clearly expressed, we should expect the result obtained for the planar and cylindrical surfaces.

4. Conclusions

As a result of the study of the productivity improvement of measuring by reducing the sensor path of the CMM on the basis of computer simulation, it was found that the sensor path represents a Hamiltonian cycle, and the problem of finding the minimum Hamiltonian cycle is the travelling salesman problem. This problem can be solved with exact and approximate methods. The comparison of the exact branch and bound method and the approximate method of the ant colony showed that the ACO allows finding a solution which is not more than 10% worse than optimum, requiring far less time for searching solutions. Based on this, the method of ant colony is the most appropriate for finding the optimal path of the sensor and can be recommended for implementation of software development of automatic surface control by the coordinate measuring machine.

Acknowledgements

The study was performed by a grant from the Russian Science Foundation (project №16-19-10204). References

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