Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 623156, 15 pages

http://dx.doi.org/10.1155/2014/623156

Research Article

Comparison of Three Different Curves Used in Path Planning Problems Based on Particle Swarm Optimizer

J. J. Liang,¹ H. Song,¹ B. Y. Qu,² and Z. F. Liu²

¹School of Electrical Engineering, Zhengzhou University, Zhengzhou 450001, China

²School of Electric and Information Engineering, Zhongyuan University of Technology, Zhengzhou 450007, China

Received 23 December 2013; Accepted 11 March 2014; Published 17 April 2014

Academic Editor: Fang Zong

Copyright © 2014 J. J. Liang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In path planning problems, the most important task is to find a suitable collision-free path which satisfies some certain criteria (the shortest path length, security, feasibility, smoothness, and so on), so defining a suitable curve to describe path is essential. Three different commonly used curves are compared and discussed based on their performance on solving a set of path planning problems. Dynamic multiswarm particle swarm optimizer is employed to optimize the necessary parameters for these curves. The results show that Bezier curve is the most suitable curve for producing path for the certain path planning problems discussed in this paper. Safety criterion is considered as a constrained condition. A new constraint handling method is proposed and compared with other two constraint handling methods. The results show that the new method has a better characteristic to improve the performance of algorithm.

1. Introduction

The mobile robot path planning is an important research field of robotics. One of the most important tasks to realize navigation and control of the robots is path planning. In an environment with obstacles, the aim of path planning is to find a suitable collision-free path, which satisfies some certain optimal criteria (such as the shortest path length, security, and feasibility), for a mobile robot to move from a start position to a target position. Most researches have focused on finding the shortest path, the minimum-time path, or the safest path, but the generated paths may be discontinued. Smoothness of the path is essential for the navigation of mobile robots, because nonsmooth motions have effect on slip [1–5]. So finding a suitable curve to describe the path is a very important task in path planning problems.

In [6], curve with parallel variable-length genetic algorithm has been used to realize path planning problems. Ferguson is another curve which is used commonly with particle swarm optimizer in [7, 8] and particle filter in [9]. Bezier curve is one of the most common curves which is combined with de Casteljau algorithm in [10], genetic algorithm in [11], and particle swarm optimizer in [12] in recent years. These curves which are used to generated curve have their own specific characteristics and requirements, but which one is the best has not been discussed.

The approaches of traditional path planning are artificial potential field [13], neural network [14], algorithm [15], and so on. With the appearance and development of evolutionary computation algorithms, many nature inspired optimization computing methods have been proposed to solve path planning problems, including genetic algorithms [16, 17] and differential evolution [18, 19].

Particle swarm optimization (PSO) which was proposed by Eberhart and Kennedy in 1995 [20, 21] is based on swarm intelligence. It has been applied to many areas successfully such as artificial neural network training [22], path planning problems [23, 24], multiworking modes product-color planning [25], and robust control of 3RPS parallel manipulators [26], for its easiness to use, robustness, and strong ability of global optimization. An improved particle swarm optimizer is applied to solve the path planning problems in this paper.

The robotic path planning problem is to find a suitable path for a mobile robot to move from the start location to the target location, which satisfies some optimum criteria in an environment full of obstacles. In this paper, we define the security and the shortest path as the optimum criteria. The security means no collision between the robot and all the obstacles and the shortest path describes the distance the robot moves from the initial point to the end point [5, 27]. The three times Bezier curve, Ferguson curve, and curve are used to generate the path and their performances are compared. With these curve generating methods the path planning can be transformed into optimizing a few limited anchor points which are used to form the path. Then dynamic multiswarm particle swarm optimizer (DMS-PSO) is employed to optimize the locations of these anchor points.

In the previous work, two different constraint handling methods, dynamic threshold and dynamic balance function, have been tested [28]. Based on the analysis on the weakness of these two constraint handling methods, a novel constraint handling method “dynamic compared ” is proposed to be incorporated intothe algorithm to improve the search efficiency. The experimental result shows that this new method has a better performance on most path planning problems discussed in this paper.

The rest of this paper is organized as follows. The characteristics of the three curves are introduced in detail in Section 2. Section 3 gives a brief introduction on the dynamic multiswarm particle swarm optimizer and the constraint handling mechanisms employed in this work. The experimental setup and the results are presented in Section 4. Conclusions and future work are given in Section 5.

2. Description of Curves

2.1. The Definition and Properties of Bezier Curve

Bezier curve was proposed by the French engineer Pierre Bezier, who used Bezier curve to design for the body of the car in 1962 [29]. In recent years, Bezier curve was applied to various occasions for its advantages on describing both straight line and curve.

A Bezier curve of degree is a parametric curve composed of Bernstein basis polynomials of degree [22]:

In this equation, basis function is a famous times Bernstein polynomial [23], which is defined as

The parameter equation of every point for three times Bezier curve could be generated by formulas (1) and (2) as follows:

where is in the range of . Bezier curve starts at and ends at .

The properties of Bezier curves [22] can be described as follows.

(1)Bezier curves start at the start point and stop at the end point.(2)First derivatives of the start point and the end point are only related to the two near control points and in the same direction of the line of the two points.

The calculation formula is

A complex first-order continuous Bezier curve can be formed by connecting several segments of low-order Bezier curves. Each segment has four control points. Assuming we have two segments, and , in order to ensure the continuousness of the curve after connection, the following equation should be satisfied:

Therefore, in order to meet the property of first-order continuous when using segments of Bezier curves to describe a path, points ( parameters) are needed. The path can be generated using the following:

where represents the start point while stands for the end point. When changes in the interval , we can get a cubic Bezier curve of segment . These segments of cubic Bezier curve constitute the entire path of the curve.

2.2. The Properties of Ferguson Curve

Ferguson curve is also a famous curve which has many excellent properties and plays an important role in the shape description.

Since Ferguson curve is smooth and easy to implement, it is also often used to describe the path in path planning problems. One segment of Ferguson curve can be defined as follows:

and are the start point and the end point of the curve, respectively, and and are control points which control the shape of the curve. represents Ferguson polynomial and is defined as follows:

Assuming that a curve consists of segments of Ferguson curves, these Ferguson curves should satisfy certain requirements. Two-segment Ferguson curves are taken as an example to illustrate the requirements they should satisfy to ensure smooth connection between these two segments. Suppose that the other Ferguson curve is described as follows:

In order to make the path smooth, the curve which is used to describe the path must be first-order continuous, and then and must satisfy

The same as Bezier curve, if segments Ferguson curves are used to generate the path, there will be control points which means that variables are to be optimized.

2.3. The Properties of Curve

The same as the Bezier curve and Ferguson curve, curve which is used in path planning problems for its good properties in describing lines, arcs, and clothoid is also a widely used curve. As shown in [8], first we set two arbitrary combinations as follows:

Here , , and represent coordinates and direction, respectively, while and denote curvature and curvature derivative of the path at one point. A 7th-order polynomial of curve can be formed by the following formulas:

In order to ensure the smoothness of the curve after connecting, the following formula should be satisfied:

In addition, the polynomial has extra six degrees of freedom. In order to reduce the calculation of degrees of freedom, we use Euclidean distance of two terminal configurations to represent some variables of vectors while the other variables of vectors are set to 0. Therefore, coordinate coefficients used to generate the curve can be obtained according to the above formulas as follows:

coordinate coefficients can be obtained by changing into . When segments of curve are used to describe the path, control points are needed. However, the start point and the end point are known in the path planning problems discussed in this paper; thus the number of the control points which are needed to be optimized is . In other words, there are variables to be optimized. Except the location of the control points, the tangent directions of each control point for the path are also controllable, so there are other points to be optimized for segments. Therefore, there are variables for an curve with segments.

So from the above information, we could know that and are the coordinates of every point which should be optimized. What is more, these three curves are smooth and suitable for path planning for robots. If the same number of segments is needed to generate the path, there are parameters to be optimized for the first two curves and variables for curve.

3. Brief Introduction about Algorithm and Constraint Handling Mechanisms

Particle swarm optimizer is an intelligent evolutionary algorithm which is constructed by mimicking the birds’ behavior of preying food [21]. The basic idea of particle swarm optimization algorithm is to find the optimal solution through collaboration among groups and information sharing among individuals.

The idea of dynamic multiswarm based on periodically changed neighborhood structure was firstly proposed by Liang and Suganthan in 2005 [27]. The good information obtained by each subswarm is exchanged among the subswarms and the diversity of the population is increased simultaneously by using the dynamic changing topology. Considering its good performance on complex optimization problems, the dynamic multiswarm particle swarm optimizer (DMS-PSO) is employed to solve the path planning problems in this paper.

The position updating equations of DMS-PSO with crossover can be described as follows [28]:

where represents the position of the th particle in dimension . represents the velocity of the th particle in dimension . is the best position in history of the th particle in dimension . is the predefined maximum value in dimension .

DMS-PSO was firstly used in path planning problems in [5], where the path planning problem has been solved by the following means.

(1)Security and the shortest path criteria are combined into a punitive function with a constant to balance them.(2)Path length is regarded as the objective function, while the security criterion is regarded as a constraint for the shortest path.

In these path planning problems, a series of circles are used to represent obstacles, and the safe distance between path and obstacles is set as (which is radius of the circle). The minimum distance between path and obstacles is . If and only if is larger than , the path could be defined as secure. Otherwise, penalty will be imposed. is treated as security penalty function as follows:

where and are the centre of the obstacles and is a collection of all the obstacles in the space. The total cost is calculated as

is a constant value, which is used to balance the proportion of and . A large will lead to local optimum easily while a small will make a collision with obstacles, so choosing a suitable value is difficult.

In order to overcome this drawback, two constraint handling methods have been used to improve the above static constrain in [27]; is the current fitness evaluation times and is the predefined max fitness evaluation times.

Constraint Handling Method 1 (Dynamic Threshold ). One has

Figure 1 describes the dynamic changing process of . is considered to be better than if

Figure 1: Dynamic change of .

We observed that the potential good solutions which locate near to the global optimum but do not satisfy the current constraint will be replaced. In this way, the useful information obtained along the search process may be lost.

Constraint Handling Method 2 (Dynamic Balance Function). This method is similar to the previous static penalty function (17) except that the balance factor is gradually increasing. The dynamic is defined as follows:

is considered to be better than if

The dynamic changing process of with the FEs is presented in Figure 2. is changed with the and its rising trend is gentle and continuous. It is better than the static penalty function, but it is still difficult to control and select a suitable value to avoid losing some potential solutions.

Figure 2: Dynamic change of .

Constraint Handling Method 3 (Dynamic Compared Function, Described with ). Constraint handling methods 1 and 2 improved the feature of algorithm which has been discussed in previous work, but they still have some drawbacks: the first method may lose some potential solutions which have been abandoned for dissatisfying constraint condition in current generation while the second method may not find the best solution for its gentle change. On the other hand, the first one has large space while the second has small space to be improved. A new constraint handling method which has a larger constraint range is introduced to overcome the shortage of the first constraint handling method. It is expected to have a better ability of global search. In this new method, two different , and , are employed to judge if a solution satisfies the constraint. And the mean value of values of current particles is used to control the value of :

Figure 3 provides the possible range of and . is the effective range of and . The new constraint method is generated by the comparison of and . For any two solutions and to be compared, the following comparison criterion is used:

Figure 3: Dynamic change of and (plot in condition of which means the radius of obstacles is 1 and the maximum is 1).

If is equal to 1, is considered to be better than . This constraint handling method overcomes the shortage of the dynamic threshold which may be trapped into local optimum and improves the exploration property of the algorithm.

4. Experimental Setup and Results

From the previous test in the path planning problems, some conclusions have been made that DMS-PSO with crossover outperforms DMS-PSO and PSO with crossover performs better than PSO. So in this task, DMS-PSO with crossover and PSO with crossover are combined with the above three constraint handling methods which are designed to test the characteristics of the curves in path planning problems.

(1)The following six algorithms are used to test characteristics of each curve in path planning problems:

(i)PSO-: basic particle swarm optimizer with dynamic and crossover operator;(ii)PSO-DP: basic particle swarm optimizer with dynamic balance and crossover operator;(iii)PSO-: basic particle swarm optimizer with dynamic compared and crossover operator;(iv)DMS-PSO-: dynamic multiswarm particle swarm optimizer with dynamic and crossover operator;(v)DMS-PSO-DP: dynamic multiswarm particle swarm optimizer with dynamic balance and crossover operator;(vi)DMS-PSO-: dynamic multiswarm particle swarm optimizer with dynamic compared and crossover operator.

(2)Some parameters settings during the experiment are as follows:

 MaxFEs (the max fitness evaluation): 40000; independent runs for every algorithm: 25; population size of PSO: 30; number of subswarms in DMS-PSO: 10; particles in each subswarm in DMS-PSO: 3.

(3)The settings of every curve and the parameters need to be optimized.

In this task, three different curves are used to generate path. The segment of each curve and parameters needed to be optimized are set in detail as follows. Generally speaking, more points make the path more smooth and complex, while fewer points take less time in optimization.

(a)The first case is that the segments of every curve are uniform (), so there are 8 parameters to be optimized in Bezier curve and Ferguson curve while 5 parameters should be optimized in curve.(b)The second case is that the optimized parameters of every curve are equal (8 parameters for all curves), so three segments of curve are used to describe the path.

4.1. Comparison of Best Satisfied Paths for Each Problem

Eight artificial designed path planning problems which have different properties are used to test characteristics of Bezier curve, Ferguson curve, and curve. In order to show how the robot moves in an environment full of obstacles, the following landscapes with the best path of these tested problems are plotted in Figure 4. The yellow circles describe the dangerous distance around the obstacles. What is more, A represents the start point while B stands for the end point.

(a) F1

(b) F2

(c) F3

(d) F4

(e) F5

(f) F6

(g) F7

(h) F8

Figure 4: Landscapes of the test problems.

These eight path planning problems can be classified into two classes. F1, F2, F3, and F4 are simple problems which have less local optima and are easier to find the shortest path that satisfies the safety criterion. F5, F6, F7, and F8 can be classified into complex problems which have more local optima and make the algorithms be easily trapped into the local optima.

4.2. Comparison Results of the Different Curves

Nonparametric statistical method -test is used to evaluate the difference between two algorithms. For each problem, the results of the best algorithm which obtains the best average value in the 25 independent runs are compared with those of other algorithms by -test method. indicates a rejection of the null hypothesis at the 5% significance level. indicates a failure to reject the null hypothesis at the 5% significance level.

Case 1. Two segments for all curves are used to describe the path, so there are eight points for Bezier curve and Ferguson curve to be optimized while five parameters are needed for curve. The experiment results are listed in Tables 1 to 3.

Table 1: Result of Bezier curve.

Problems
PSO-
PSO-DP
PSO-
DMS-PSO-
DMS-PSO-DP
DMS-PSO-

F1
Mean
14.9058
15.1830
14.8969
14.9294
14.7792
14.7559

Std.
0.2910
0.4454
0.0282
0.4494
0.0500
0.0087

Min
14.6726
14.6638
14.6969
14.6547
14.6566
14.6626

Max
16.9975
16.9843
15.2326
17.1550
15.7071
15.0132

0
1
1
0
0
—

F2
Mean
14.7260
15.6209
15.1451
14.6492
14.7087
14.6486

Std.
0.0034
1.8989
0.1415
0.00039
0.0034
0.0002

Min
14.6533
14.6889
14.8150
14.6251
14.6274
14.6242

Max
14.8920
19.4449
16.2972
14.7009
14.8882
16.3361

0
1
1
0
1
—

F3
Mean
15.9481
16.7858
16.5723
14.9475
16.3286
14.9701

Std.
1.1006
1.1209
0.9196
0.2976
0.4483
0.1306

Min
14.8502
15.7943
15.1810
14.7240
15.6394
14.7290

Max
18.8319
18.7404
19.0984
17.2080
17.2206
15.7963

1
1
1
—
1
0

F4
Mean
14.8243
15.9185
15.4251
14.7366
15.0635
14.7363

Std.
0.0407
1.5323
1.2592

0.2979

Min
14.7430
14.7225
14.7851
14.7150
14.7151
14.7313

Max
15.7812
18.1838
19.8821
14.7713
17.3141
14.7455

1
1
1
0
1
—

F5
Mean
16.5000
16.6254
16.4928
16.3466
16.3800
16.3361

Std.
0.0544
0.1889
0.0048
0.0007
0.0029
0.0012

Min
16.3327
16.3102
16.3666
16.2868
16.3246
16.2827

Max
17.3413
17.9407
16.6432
16.4001
16.5518
16.4123

0
1
1
0
1
—

F6
Mean
16.2485
16.4073
16.2432
15.3892
16.1031
15.2860

Std.
0.3219
0.2167
0.2125
0.2282
0.3366
0.0103

Min
15.3576
15.2832
15.5306
15.2319
15.2461
15.2400

Max
16.6793
16.6649
16.8466
17.2616
16.6399
15.7535

1
1
1
0
1
—

F7
Mean
16.2040
16.4125
16.4490
15.1967
15.9038
15.2458

Std.
0.4474
0.3349
0.2576
0.1794
0.5311
0.2027

Min
15.1081
15.1079
15.1656
15.0355
15.0516
15.0504

Max
16.6948
16.8264
16.9785
16.6090
16.6292
16.9341

1
1
1
—
1
0

F8
Mean
15.0239
15.2377
14.8827
14.6644
14.6787
14.6699

Std.
0.5322
0.6693
0.0905
0.0003
0.0011
0.0003

Min
14.6303
14.6327
14.6697
14.6291
14.6262
14.6296

Max
16.6699
16.6459
16.2430
14.7046
14.7831
14.7093

1
1
1
—
0
0

Some conclusions could be drawn from Table 1 as follows.

(1)DMS-PSO outperforms PSO in all constraint handling methods correspondingly, which shows that DMS-PSO has better global search ability.(2)The result of -test 2 shows that there is no obvious difference between these two algorithms, so this phenomenon is regarded as these two algorithms have the similar performance on these problems. But DMS-PSO- performs better on problems F1, F2, F4, F5, and F6 while DMS-PSO- outperforms on F4, F6, and F7 on average.(3)Compared with the best solutions obtained by DMS-PSO- and DMS-PSO-, the distribution of optimal solutions of DMS-PSO-DP is significantly different on problems F2, F3, F4, F5, F6, and F7 which could be seen from the results of -test 2.

These three points show that although DMS-PSO- and DMS-PSO- have the same characteristic on -test 2, DMS-PSO- overcomes the drawback of DMS-PSO- which is easy to be trapped into local optimum on average. Compared with other algorithms, the feature of DMS-PSO- stands out on path planning problems where Bezier curve is used to generate path. The result also tells that Bezier curve is suitable on path planning problems for its stable feature when we employ evolutionary algorithm to optimize its parameters.

Table 2 gives us the following information.

(1)DMS-PSO has a better global search ability compared with PSO on the whole.(2)All best solutions about Ferguson curve spread in DMS-PSO- and DMS-PSO-, while the best results of problems F1, F6, F7, and F8 accept DMS-PSO-DP in the distribution of optimal solutions for 25 independent runs. The result of problem F6 has no difference with DMS-PSO-DP, DMS-PSO-, and DMS-PSO- on the distribution of 25 independent runs though there is much difference between these four algorithms on the mean value.(3)The constraint handling method is little better than dynamic balance function and dynamic compared when path is produced by Ferguson curve.(4)Except for problems F1, DMS-PSO- outperforms obviously than other algorithms on the whole no matter on the result of mean value or t-test 2.

In a word, and are suitable and DMS-PSO- and DMS-PSO- are smart choices when Ferguson curve is applied in path planning problems especially on complex problems.

Table 2: Result of Ferguson curve.

Problems
PSO-
PSO-DP
PSO-
DMS-PSO-
DMS-PSO-DP
DMS-PSO-

F1
Mean
15.4745
14.9230
15.8539
14.6720
14.7393
15.0440

Std.
1.0221
0.3217
0.8098

0.1175
0.1913

Min
14.6617
14.6565
14.7062
14.6626
14.6532
14.6601

Max
17.3684
16.3953
17.5340
14.6830
16.3743
16.3967

1
1
1
—
0
1

F2
Mean
14.6541
14.8765
14.7016
14.6399
14.6550
14.6395

Std.

0.9416
0.0013

Min
14.6335
14.6344
14.6415
14.6320
14.6309
14.6245

Max
14.6787
19.5190
14.8158
14.6481
14.6860
14.6531

1
1
1
0
1
—

F3
Mean
17.6613
16.8770
15.1573
14.8069
16.3085
15.7254

Std.
1.0504
1.1620
0.9308
0.0133
0.4618
0.0223

Min
15.5203
15.5515
14.6219
14.6170
15.9185
15.3465

Max
19.6494
19.4469
19.7183
15.2238
17.8514
15.9672

1
1
1
—
1
1

F4
Mean
14.7474
15.4290
14.7654
14.7361
14.7525
14.7364

Std.

1.0917

Min
14.7356
14.7337
14.7378
14.7291
14.7373
14.7288

Max
14.7714
18.3901
14.8401
14.7497
14.7928
14.7488

0
1
1
—
1
0

F5
Mean
18.9011
17.8758
17.8816
17.7712
17.5169
16.8374

Std.
19.3752
0.9286
4.7299
4.3717
0.3750
0.7757

Min
16.4034
16.4773
16.6205
16.4129
16.9094
16.3204

Max
28.7950
19.3831
26.4327
25.2178
18.6831
17.8547

1
1
1
1
1
—

F6
Mean
20.4977
17.2938
21.4213
16.7939
17.0746
16.5777

Std.
21.7799
7.5871
23.4940
5.1557
1.6796
0.3785

Min
17.4722
15.7391
15.4976
15.2908
15.7525
15.6083

Max
29.3754
27.9776
29.9679
22.5225
21.5453
17.9068

1
1
1
0
0
—

F7
Mean
23.5527
17.7175
19.9267
17.3071
18.5039
19.3598

Std.
27.7538
0.5265
35.7375
15.6542
1.9334
32.9936

Min
17.8796
16.8212
15.1296
15.0326
16.8277
15.6002

Max
30.0131
20.5929
29.5523
28.4035
21.4855
30.3578

1
1
1
—
0
0

F8
Mean
18.9859
15.9555
17.1540
15.4485
15.6366
18.54158

Std.
8.6686
0.6078
11.4477
0.0218
0.6935
3.5024

Min
17.4710
15.3587
15.4066
15.1707
15.4429
16.2014

Max
28.6708
17.9973
28.8277
15.7448
19.6240
26.7133

1
1
1
—
0
1

Table 3: Result of curve ().

Problems
PSO-
PSO-DP
PSO-
DMS-PSO-
DMS-PSO-DP
DMS-PSO-

F1
Mean
15.1590
15.3283
15.2681
14.9161
15.1670
14.8622

Std.
0.3263
0.3658
0.4875
0.2284
0.1987
0.2007

Min
14.6853
14.6728
14.6932
14.6608
14.6801
14.6614

Max
16.98428
16.72636
17.6536
16.4115
16.2402
16.9015

1
1
1
0
1
—

F2
Mean
15.0589
15.0396
15.7143
14.7317
14.9698
14.9601

Std.
0.4267
0.2972
1.4186
0.0902
0.1457
0.4676

Min
14.6589
14.6626
14.7605
14.65536
14.6640
14.6555

Max
16.2536
16.3979
19.6726
16.1707
16.2063
17.2056

1
1
1
—
1
0

F3
Mean
16.9309
16.7421
16.6452
16.2764
16.7070
15.8886

Std.
0.3723
0.5148
0.8288
0.8953
0.3677
0.2801

Min
15.6267
15.6372
15.6718
15.6049
15.7619
15.6068

Max
17.4024
17.3621
19.6036
18.6914
17.3590
17.3939

1
1
1
0
1
—

F4
Mean
15.1204
15.6049
15.0711
15.8886
15.3697
14.9027

Std.
0.2261
0.7872
0.0990
0.2801
0.4204
0.4031

Min
14.7725
14.7932
14.7791
15.6068
14.7799
14.7403

Max
15.8243
18.1527
15.9261
17.3939
16.7863
17.9495

1
1
1
—
1
0

F5
Mean
16.5948
16.4176
16.61561
16.4206
16.4134
16.4329

Std.
0.1251

0.1206
0.0097

0.0102

Min
16.4121
16.4029
16.4001
16.3722
16.3787
16.3842

Max
17.6328
16.4500
17.5537
16.8877
16.4457
16.9001

1
0
1
0
—
0

F6
Mean
16.6868
16.6765
16.7663
16.5985
16.5424
16.0209

Std.
0.0188
0.0731
0.0481
0.1177
0.1248
0.3553

Min
16.1306
15.4302
15.8551
15.4915
15.4417
15.3031

Max
16.9352
16.9246
16.9671
16.9530
16.7489
16.8915

1
1
1
1
1
—

F7
Mean
16.5802
16.3344
16.4030
16.2061
15.9442
15.6069

Std.
0.2244
0.45243
0.3946
0.4480
0.4248
0.3488

Min
15.2971
15.2954
15.3358
15.2914
15.2947
15.2917

Max
16.9642
16.8365
16.9896
16.8971
16.8196
16.8491

1
1
1
1
0
—

F8
Mean
15.4772
15.0808
15.5568
15.1574
15.0251
15.0064

Std.
0.5238
0.0665
0.4020
0.2922

Min
15.0152
14.9885
15.0526
14.9821
14.9887
14.9863

Max
16.9498
16.3030
16.9771
16.9514
15.0988
15.0621

1
0
1
0
1
—

From Table 3, we could observe the following.

(1)Dynamic multiswarm has improved the search ability of traditional particle swarm optimizer which means that DMS-PSO with crossover performs better than PSO under all constraint handling methods.(2)DMS-PSO- has a better mean value than other algorithms on the whole.(3)DMS-PSO- is better than DMS-PSO-+ and DMS-PSO-DP except for F2, F4, and F5 while DMS-PSO- is similar to DMS-PSO- on problem F2 and DMS-PSO-DP+ on problem F5 on average.(4)Except for problems F6 and F7, DMS-PSO- and DMS-PSO- have similar performance.(5)Although DMS-PSO-DP is better than other algorithms on F3, DMS-PSO-, DMS-PSO-+, and DMS-PSO-DP have the same acceptance which means they have no difference under 25 independent runs at the 5% significance level.

Generally speaking, DMS-PSO- possesses a good feature so that it could be applied in path planning problems when curve is used to describe path.

Case 2. Eight parameters are satisfied to generate path; Table 4 is the result (the result concludes curve only).

Table 4: Result of curve ().

Problems
PSO-
PSO-DP
PSO-
DMS-PSO-
DMS-PSO-DP
DMS-PSO-

F1
Mean
15.1269
15.2927
15.8549
14.8915
15.2233
14.8945

Std.
0.2000
0.3962
0.6258
0.1053
0.2890
0.1533

Min
14.6743
14.6922
14.8753
14.6599
14.6810
14.6746

Max
16.4027
16.9510
17.1939
15.7512
16.5228
16.4695

1
1
1
—
1
0

F2
Mean
15.1828
15.8735
17.3363
14.8915
15.5943
14.7678

Std.
0.4999
0.5877
2.4413
0.1053
0.4150
0.0323

Min
14.6786
14.6978
15.0084
14.6599
14.6649
14.6682

Max
17.1109
17.5539
20.7949
15.7512
17.0959
15.5873

1
1
1
0
1
—

F3
Mean
16.9903
17.3458
17.8328
16.2678
17.13297
15.3092

Std.
0.3941
0.34185
1.5596
1.7428
0.1416
0.7615

Min
15.1838
15.7395
15.8694
14.7341
15.9525
14.7554

Max
17.6611
18.7940
20.4837
18.8251
17.4754
17.1826

1
1
1
1
1
—

F4
Mean
15.1503
16.2299
16.5532
14.7421
16.1149
15.0292

Std.
0.4168
1.0923
1.9814

1.2537
1.8730

Min
14.7527
14.8084
14.9267
14.6988
14.7411
14.7161

Max
17.4512
18.0449
19.4632
14.8068
18.0394
21.5966

1
1
1
—
1
0

F5
Mean
16.8508
16.6488
16.9656
16.4054
16.5114
16.3798

Std.
0.2432
0.1425
0.1367
0.0569
0.0180
0.0037

Min
16.4201
16.3486
16.5099
16.3191
16.3300
16.3254

Max
17.8626
17.4785
17.9822
17.5422
16.9302
16.6216

1
1
1
0
1
—

F6
Mean
16.6472
16.5453
16.9163
16.2758
16.2548
15.7087

Std.
0.0727
0.1016
0.0788
0.3668
0.2496
0.32256

Min
15.4516
15.5077
16.6653
15.2211
15.2660
15.2387

Max
16.9744
16.9084
18.13603
17.0442
16.73439
16.7263

1
1
1
1
1
—

F7
Mean
16.6997
16.4166
16.8786
15.8356
16.1770
15.6007

Std.
0.0129
0.3524
0.0291
0.5831
0.2656
0.3273

Min
16.5565
15.1092
16.5886
15.0152
15.0720
15.0145

Max
16.8951
16.9036
17.3592
16.7376
16.7343
16.7801

1
1
1
0
1
—

F8
Mean
15.1227
14.9834
15.5447
14.6521
14.7546
14.6624

Std.
0.3083
0.2759
0.5752

0.0216
0.0016

Min
14.6780
14.6695
14.7240
14.6327
14.6510
14.6180

Max
16.6986
16.6473
18.1863
14.7619
15.3003
14.7838

1
1
1
—
1
0

The following information is given from Table 4.

(1)DMS-PSO is absolutely better than PSO in three-segment curve no matter the result of mean value or the null hypothesis at the 5% significance level.(2)When DMS-PSO- is better than DMS-PSO- on the mean value, the previous accepts the latter on all problems which means they have no difference.(3)When DMS-PSO- is better than DMS-PSO-+ on the mean value, the previous rejects the latter on F3 and F6.(4)DMS-PSO- and DMS-PSO- are better than DMS-PSO-DP on all problems.

So, when three-segment curve is used to generate path, it is clear to ensure DMS-PSO performs better than PSO. The constraint handling method dynamic threshold and dynamic compared are better than dynamic balance function and dynamic compared overcomes the drawback of dynamic threshold and outperforms it.

Case 3. The comparison of best result under each curve is as follows.

Having compared the best result of every curve, when two segments of Bezier curve, Ferguson curve, and curve are used to describe path, we could observe the following.

(1)Ferguson curve performs better than Bezier curve and curve on problems F1 to F4 which are easy to find global optimum and has a worse feature on problems F5 to F8 which are described as complex problems.(2)Although curve is not so good as Ferguson curve and Bezier curve on simple problem, it outperforms Ferguson curve on complex problem which means curve possesses a better search ability.(3)Bezier curve is better in producing path compared with Ferguson curve and curve on the whole.

When three-segment curve is applied in these problems, conclusions could be made as follows.

(1)Compared with two segments, three-segment has a small range of the distribution between the best and worst solutions on the whole.(2)There is no difference between problems F1, F2, F5, and F7 no matter it is two-segment or three-segment curve.(3)Three-segment curve outperforms on problems F3, F4, F6, and F8 in generating path obviously.(4)On complex problems which are easily trapped into local optimum, three-segment curve is much better than two-segment curve which means that the previous has obvious difference compared to the latter such as F6, F8, F3, and F4.(5)Three-segment curve performs better than Ferguson on complex problems but obviously expresses an inferior characteristic in generating path compared with Bezier curve.

From so many points of discussion from Tables 1 to 5, conclusions could be made that DMS-PSO overcomes the drawback of PSO which is easy to fall into local optimal and premature. Dynamic compared overcomes the drawback and inherits the advantage of dynamic threshold and shows better constraint characteristics than dynamic balance function, which makes it show good binding properties in path planning problems.

Table 5: Best result of every curve.

Problems
Bezier curve
Ferguson curve
curve
curve

F1
Mean
14.7559
14.6720
14.8622
14.8915

Std.
0.0087

0.2007
0.1053

Min
14.6626
14.6626
14.6614
14.6599

Max
15.0132
14.6830
16.9015
15.7512

F2
Mean
14.6486
14.6395
14.7317
14.7678

Std.
0.0002

0.0902
0.0323

Min
14.6242
14.6245
14.65536
14.6682

Max
16.3361
14.6531
16.1707
15.5873

F3
Mean
14.9475
14.8069
15.8886
15.3092

Std.
0.2976
0.0133
0.2801
0.7615

Min
14.7240
14.6170
15.6068
14.7554

Max
17.2080
15.2238
17.3939
17.1826

F4
Mean
14.7363
14.7361
15.8886
14.7421

Std.

0.2801

Min
14.7313
14.7291
15.6068
14.6988

Max
14.7455
14.7497
17.3939
14.8068

F5
Mean
16.3361
16.8374
16.4134
16.3798

Std.
0.0012
0.7757

0.0037

Min
16.2827
16.3204
16.3787
16.3254

Max
16.4123
17.8547
16.4457
16.6216

F6
Mean
15.2860
16.5777
16.0209
15.7087

Std.
0.0103
0.3785
0.3553
0.32256

Min
15.2400
15.6083
15.3031
15.2387

Max
15.7535
17.9068
16.8915
16.7263

F7
Mean
15.1967
17.3071
15.6069
15.6007

Std.
0.1794
15.6542
0.3488
0.3273

Min
15.0355
15.0326
15.2917
15.0145

Max
16.6090
28.4035
16.8491
16.7801

F8
Mean
14.6644
15.4485
15.0064
14.6521

Std.
0.0003
0.0218

Min
14.6291
15.1707
14.9863
14.6327

Max
14.7046
15.7448
15.0621
14.7619

When all curves are composed by the same number of segments, curve outperforms Ferguson curve on complex problems specifically but is worse than Bezier curve for all problems. Fewer points would be optimized when curve is used to describe path, so less time is needed.

Three-segment curve is better than two-segment one for generating path because it has more anchor points to control and can generate a more flexible path, especially on complex problems.

Bezier curve expresses better performance on path planning problems compared with Ferguson curve and curve. The most possible reason may be that Bezier curve is easier to change the shape of the path via adjustment of a fixed number of anchor points than the other two curves. So Bezier curve is the most suitable curve to produce path in this paper and DMS-PSO with crossover combined with dynamic compared is the best choice to optimize path in path planning problems.

In order to show the property of Bezier curve and DMS-PSO with crossover combined with dynamic compared , complex problem F6 is an example to show how particles learn from their neighborhood and avoid being trapped into local optimum. The yellow circles describe the dangerous distance around the obstacles, red paths mean the current local paths, and blue path is the best path satisfying some certain criteria. The processes of iteration are in Figure 5.

(a) Initial population

(b) Population after 10 iterations

(c) Population after 400 iterations

(d) Population after 700 iterations

Figure 5: Processes of iteration.

Figure 5 shows the search process which could be seen that although the robot always runs into obstacles in the first 400 iterations, it is far away from obstacles step by step. After 700 iterations, solutions are converged into the best path gradually which shows that DMS-PSO with crossover combined with dynamic compared has a good ability of global search in early stage and global convergence in latter stage.

5. Conclusion

In order to solve path planning problems in static environment, suitable curves and algorithms with constraint mechanisms are designed in this paper. Three curves are compared under six algorithms, and the results have proved that DMS-PSO has a better ability of global search than PSO again. At the same time, the analyses of three constraint handling methods and curves are carried on. Firstly, dynamic constraint methods are designed well for path planning problems compared with the previous work where static constraint is used. Then the dynamic compared possesses a better feature than dynamic threshold and dynamic balance function which has overcome the drawback of dynamic threshold which might lose the previous good solutions found in the search process, which are near to the global optimum but do not satisfy the current constraint. So dynamic compared is more suitable to be applied in path planning problems. From the results, we could observe that curve outperforms Ferguson curve when the same segment is used to describe path, especially on the distribution of solutions for complex problems. What is more, compared with two segments, three segments of curve are more suitable to generate path for the reason that more points may make the path more flexible and easier to change the direction of path. The most important is that Bezier curve outperforms curve and Ferguson curve no matter on simple or complex problems and it improves the solutions further, which is more likely depending on the property of its flexible shape changed by adjusting a fixed number of anchor points. So when PSO and its improved versions are used to solve path planning problems, Bezier curve possesses a higher status. For the limitation of experimental conditions, only the path length and security criteria are compared in this paper. Bezier curve will be used to evaluate more criteria in path planning problems as well as in the condition of dynamic environment where the obstacles are changed with time in the future.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported partially by National Natural Science Foundation of China (61305080, U1304602), Postdoctoral Science Foundation of China (Grant 20100480859), Specialized Research Fund for the Doctoral Program of Higher Education (20114101110005), Scientific and Technological Project of Henan Province (132102210521, 122300410264), and Key Foundation of Henan Educational Committee (14A410001).

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