2010 Fourth Asia International Conference on Mathematical/Analytical Modelling and Computer Simulation
A Particle Swarm Optimization Approach to Robotic Drill Route Optimization Asrul Adam, Amar Faiz Zainal Abidin, Zuwairie Ibrahim, Abdul Rashid Husain, Zulkifli Md Yusof and Ismail Ibrahim Faculty of Electrical Engineering Universiti Teknologi Malaysia 81310 UTM Skudai, Johor, Malaysia Email:
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route. The time to complete a route depends on the distance travel by the drilling bit. Meanwhile, the size of the search space is equivalent to the total number of possible routes available for the robotic drill to choose from. Literatures on drilling route optimization problem can be traced back to the year 1996 where Kolahan and Liang presented a case study with variable holes sizes [9]. The authors approached the case study by using Tabu search algorithm before proceeding with an improved algorithm in another journal [10]. Taking Computer Numerical Control (CNC) machine as the subject, Sigl and Mayer had introduced the 2-Opt Heuristic Evolutionary algorithm in optimizing the drilling path [11]. In 2007, Qudeiri et. al. used Genetic Algorithm (GA) in finding the optimized sequence for holes cutting process in CNC machine tool [12]. Recently, Kentli and Alkaya proposed the used of Record-to-Record Travel with Local Exchange Moves (RRTLEM) algorithm in tackling sequence based optimization problems like holes drilling [13]. In spite of numerous literatures on holes drilling optimization, the treatment of the problem using PSO is rarely touched until recently. In 2004, Onwubolu and Clerc initiated the idea of the use of PSO as an optimization method for holes drilling path optimization problem [14]. Noticing the global convergence is not carefully considered in the holes drilling problem, Zhu proposed an improved PSO as a remedy to the problem [15]. In his paper, he offers a case study of a 14 hole-drilling problem where the improved PSO is compared with the original PSO. This paper proposed a model to illustrate the particles in a search space for sequence based problems, like robotic drill route optimization problem. The focal point of this paper is how this model is illustrated and the simplicity of the execution of the model. The remainder of this paper as follows: Section II will briefly introduce PSO and Global PSO (GPSO) algorithm. Section III describes the problem of interest and a case study taken from previous literature. Section IV explains the model proposed before going into its implementation in holes drilling path optimization problem. In Section V, the case study mentioned in Section II is attempted with the
Abstract—Most of the operational time of a PCB Robotic Drill is spent on moving the drill bit between the holes. This operational time can be kept at a minimal level by optimizing the route taken by the robot. An optimized route translates to a minimal cost of operating the robot. This paper proposes a new model that implements Particle Swarm Optimization (PSO) in order to find optimized routing path when using the PCB Robotic Drill. The main task of the PCB Robotic Drill is to drill holes at Printed Circuit Board (PCB). This PCB Robotic Drill will route the drill site by moving the drill bit along Cartesian axes from it’s initial position. Then, the drill bit will return back to the initial position. The drill route consists of a number of potential locations where the holes are going to be drilled. As the number of holes required increases so thus does the complexity to find the optimized route. The proposed model can be used to solve this complex problem with minimal computational time. The result of a case study indicates that the proposed model is capable to find the shortest path for the robot to complete its task. Thus concluded the proposed model can be implemented in any drill route problems. Keywords-particle swarm optimization; route optimization problems
I. I NTRODUCTION PCB Robotic Drill is widely used in electronic industries in order to drill the holes on the PCB. Due to a significant amount of time required for moving the drill bit from one point to another point, holes drill routing optimization problem attracts a great interest among the academicians, researchers and engineers to solve for. One of the way to reduce time taken for the robot to complete the robot task is by optimizing the route chosen by the robot. Route optimization problem is a branch of Travel Salesman Problem (TSP) [1]. Routing optimization also can be applied in other area such as the traffic system [2], robotics [3], [4], scheduling [5], municipal service [6] and manufacturing [7]. PSO algorithm is one of the many available algorithms used to optimize routing time. PSO is an evolutionary optimization concept developed by James Kennedy (Social- Psychologist) and Russell Eberhart (Electrical Engineer) in 1995 [8]. It is a robust stochastic optimization concept based on the movement and intelligence of a swarm [8]. For this routing problem, particles will move around at the search space to find the moving time of each 978-0-7695-4062-7/10 $26.00 © 2010 IEEE DOI 10.1109/AMS.2010.25
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result is compared with its original literature. Section VI concludes the paper finding.
Algorithm 1 Global Best PSO (GPSO) Algorithm 1: for each particle do 2: Initialize particle by randomizing its initial position 3: end for 4: for each particle do 5: Calculate particle fitness value 6: if the fitness value is better than the particle’s personal best fitness value (pBest) then 7: set the current value as the new pBest 8: end if 9: end for 10: Choose the particle with the best pBest fitness value of all the particles as the gBest fitness value 11: for each particle do 12: Calculate the particle velocity according to the (2) 13: Update the particle position according to the (1) 14: end for 15: repeat 16: Procedure 4 17: until maximum generation or stop by user
II. PARTICLE S WARM O PTIMIZATION The fundamental concept of PSO is that a swarm can be represented by a population, while a particle can be represented by an individual. Thus, this particle is treated as a point in a multi dimensional space which adjusts its flying according to its own flying experience as well as the flying experience of other particles. The adjustment is based on the basis that a particle is trying to imitate its success and the success of other particles. Each particle will produce two parameters: velocity of particle and the position of ) particle. The particle position for the next generation (sk+1 i can be mathematically written as (1). = ski + vk+1 sk+1 i i
(1)
From (1), the next particle position (sk+1 ) is dictated by its i current position (ski ) and the velocity for the next placement ). This velocity is calculated using (2). (vk+1 i = ωvki + c1 r1 (pbesti − ski ) + c2 r2 (gbesti − ski ) (2) vk+1 i
to define the fitness function. The fitness function can be obtained by modeling the Robotic Drill Route problem into mathematical equations. Assuming the velocity of the drill bit movement of the robotic drill is constant, the time taken for the drill bit to travels across all holes can be mathematically represented by (4) [15].
Referring to (1) and (2), vki is particle i velocity at k generation and ski is particle i position at k generation. r1 and r2 are a random number [0,1]. c1 and c2 (cognitive and social component) are learning factors usually c1 = c2 = 1.42. The personal best (pbest) is the best solution found by each particle in a swarm. Meanwhile, the global best (gbest) is the best solution among the pbest. Particles inside GPSO have their velocity influenced by the cognitive and social components. The gbest and pbest position are used to update the velocity of the particle via (2). Each particle try to be the best among the others by replicating the success of its neighbors and its own. Algorithm 1 summarizes the GPSO. In this application the decrease inertia weight, ω is used. The formula for this decrease inertia weight as shown in (3). ωmax − ωmin × k) (3) ω = ωmax − ( kmax
ttravel =
n n |xi − xj | + |yi − yj |
v
i=1 j=1
× pij
(4)
Equation (4) is then extends to (5), which take into account the drilling time [15]. ttravel =
n n |xi − xj | + |yi − yj | i=1 j=1
v
×pij +n×tdrill (5)
Note that (5) assumes that the drilling time is constant. Most of the literatures treat the fitness function as the distance covered by the drill bit. From (5), the fitness function in distance is mathematically written to (6).
Where ωmax and ωmin is the representation of the maximum and minimum value of inertia weight, respectively. k is the particle at k generation and kmax is its maximum generation. The decrease initial weight, ωmax = 0.9 and ωmin = 0.4.
dtravel =
III. ROBOTIC H OLE D RILL ROUTING P ROBLEM
n n
(|xi − xj | + |yi − yj |) × pij
(6)
i=1 j=1
The main objective of this problem is to reduce the time taken for the PCB Robotic Drill to complete its task. As the movement of drilling bit from one point to another point takes a significant amount of time, the need of route optimization is highly appreciated. The first step in using any optimization techniques is
Equation (4) to (6), symbols t and d are for time and distance. i and j represents hole i and hole j, n is the value of the maximum hole. x and y is the hole’s coordinate (x, y). pij is the verification variable associated to the movement of hole i to hole j [16]. pij =1 means that the drill bit will travel to hole i from hole j. This paper used Zhu’s 14 holes 61
Figure 2.
Figure 1.
Location of selected holes (3 holes)
Location selected holes (14 holes)
drilling problem to benchmark the model performance [15]. Fig. 1 shows the PCB drawing of the problem. Details of the case study is discussed in Section V.
Figure 3.
Modeling the sequence (a–b–c)
IV. M ODELING IN PSO The need to visualize the PSO algorithm is important for a person to grasp and appreciate how PSO algorithm works. Most of the literatures available gave heavy emphasize on the performance of their proposed model but rarely touched how the model is represented in robotic hole route problem. This paper will explains how the model can be graphically represented using GPSO. The GPSO algorithm is chosen due to its simplicity and faster converging period compared to Local Best PSO (LPSO). The proposed model suggests that the GPSO can be model in a multi-dimension search according to the maximum sequence required for a problem. In holes drilling path optimization problem, the sequence value is equal to the number of drilling holes required. Given a 3-holes drilling problem as shown Fig. 2, there are six possible routes that the drill bit can travel in completing its task. This routes are (a-b-c), (a-c-b), (b-a-c), (b-c-a), (c-a-b) and (c-b-a). The robot will visits all holes one at a time, begins at initial condition s(0,0) and ends at initial condition s(0,0). To adapt with the case study, the cost (in distance) of initial condition s(0,0) to the first hole and the cost of last hole to initial position are ignored. Each particle in PSO will choose one of the possible route. For this example, let a particle travels in (a-b-c) sequences like in Fig. 3(a), this particle representation in a search space is as shown in Fig. 3(b). After the particle updates its position using (2) and assuming the new particle position is (b-c-a), the movement can be presented in the search space as in Fig. 4 below: From what can been seen, modeling the particles using this
Figure 4.
PSO Basic approached of the implementation of the model
method gives a clearer idea of the particle location in a search space at a given time. Each particle movement can be seen by simulating the model over a period of time. Combining all these particles motions together, the swarm convergence can be seen.
V. I MPLEMENTATION AND E XPERIMENTAL R ESULTS The Zhu’s 14 holes drilling problem has been chosen due to its complexity. The objective of this problem is to drill 14 holes scattered around the PCB. These holes have the same hole size. Table I shows the difference in value of the PSO parameters chosen by Zhu and us. From Table I, Zhu had
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Table I C OMPARISION THE PSO PARAMETERS USED BETWEEN Z HU AND US
Number of Particles, i Number of Iteration/Generation, k Number of Computations Inertia Weight, ω Cognitive Component, c1 Social Component, c2 r1 and r2
Zhu’s 100 10000 50 0,0.5,1.0 Not Available Not Available Not Available
Ours’ 50 5000 50 0.9∼0.4 (Decrease inertia weight) 1.42 1.42 random number, [0,1]
Table II DATA OBTAINED FROM EXPERIMENT
i. The least generation number while global convergence ii. The average generation number while global convergence iii. Length of optimal solution (Best found) iv. Average fitness after computing 50 generations
Zhu’s Global Convergence PSO Standard PSO ω ω 0.0 0.5 1.0 0.0 0.5 1.0 70 601 93 118
Ours ω 0.9∼0.4 7
1784
3549
2104
-
-
1415
353
280
280
280
-
-
280
280.00
305.7
307.3
289.6
358
359
300.7
292.29
Table III E XPERIMENTAL DATA OBTAINED
Best Found Mean Mode
Fitness 280.00 292.29 298.02
Note Holes Sequence: 2 - 3 - 4 - 7 - 8 - 13 - 14 - 10 - 11 - 12 - 9 - 6 - 5 - 1 Mean=[280.00(1)+287.06(2)+298.02(44)+307.52(1)+311.12(1)+315.74(1)]/50 Holes Sequence: 2 - 3 - 4 - 1 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 - 14
the result, the proposed model is able to solve the robotic drill route optimization problem at par if not better compared to the previous literature. The simplicity and the ability to visualize of the model will be the comparative advantage of this model. For the future work, the extend from this finding is to tackle the problem using binary PSO. It is expected to give a better average fitness due to the nature of the problem is in discrete.
used more particles and the PSO is computed at a greater period. Note also that Zhu had used different value of inertia, ω. Although, the proposed model computed using less computation resources, the overall result of the simulation is as good as Zhu’s. This can be see in Table II. The proposed model able to obtain a better average fitness in a faster period (average generation time). Table III summarizes the data obtained from the experiment. The proposed model able to find the best found path (fitness of 280.00) on its 38th attempted. The average of the computation is 292.29. The most frequent solution suggested by the model is the sequence of 2 - 3 - 4 - 1 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 - 14 (44 out of 50 times) with fitness of 298.02.
ACKNOWLEDGMENT The first author is indebted to Universiti Teknologi Malaysia for granting him financial and the opportunity to undergo this research. This research was financially supported by a ScienceFund grant (Vot 79361) from the Ministry of Science, Technology, and Innovation (MOSTI), Malaysia.
VI. C ONCLUSION In this study has introduced one of the implementation of PSO to robotic drill route optimization problem. The proposed model is illustrated a novelty of PSO modeling for this problem. By using the decrease inertia weight the proposed model is able to balanced the exploration and exploitation of particles in PSO as a result by using minimum number of particles the convergence speed is fast than other. Based on 63
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