Solving minimum constraint removal (MCR) problem using a social-force-model-based ant colony algorithm

Accepted Manuscript Title: Solving Minimum Constraint Removal (MCR) Problem Using a Social-Force-Model-Based Ant Colony Algorithm Author: Bo Xu Huaqing Min PII: DOI: Reference:

S1568-4946(16)30099-0 http://dx.doi.org/doi:10.1016/j.asoc.2016.02.047 ASOC 3507

To appear in:

Applied Soft Computing

Received date: Revised date: Accepted date:

5-3-2015 31-10-2015 26-2-2016

Please cite this article as: B. Xu, H. Min, Solving Minimum Constraint Removal (MCR) Problem Using a Social-Force-Model-Based Ant Colony Algorithm, Applied Soft Computing Journal (2016), http://dx.doi.org/10.1016/j.asoc.2016.02.047 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <

1

ip t

Solving Minimum Constraint Removal (MCR) Problem Using a Social-Force-Model-Based Ant Colony Algorithm Bo Xu, Huaqing Min*

an

us

suitable to search for a path with the least obstacles. In other words, it may be necessary to know the minimum obstacle set that obstructs the free path between two points (similar to a road under construction where the developer will remove the minimum existing obstacles in the environment to complete the road construction). When this point is considered, a new problem naturally arises: if there is no obstacle-free path from the initial state to the target state, what is the minimum number of obstacles that need to be removed on any path from the starting point to the target point? Hauser first proposed this problem in 2012 and called it the minimum constraint removal (MCR) problem because the minimum constraints (obstacles) are required to be removed to open up an obstacle-free path in the environment [2]. The objective of a MCR motion-planning problem is to remove the necessary minimum geometric constraints on a free path that connects the starting point and the target point. An MCR motion planning problem describes a problem in which a probabilistic path map motion planner solves an MCR problem in a continuous configuration space by constructing increasingly accurate path maps and efficiently solves the discrete MCR problem on these maps [3]. Using the method that reduced the minimum set cover, Hauser proved that discrete MCR problems were non-deterministic polynomial-time (NP)-hard problems [2]. In addition, at the 2013 Association for the Advancement of Artificial Intelligence conference (AAAI-13), Erickson L H and LaValle S M also proved that discrete MCR problems were NP-hard problems [4]; the result was quite intriguing to them because the corresponding path existing problem could be easily solved through the breadth-first search of the map [5]. A Gorbenko et al. considered the discrete minimum constraint removal motion planning problem that can be used for a motion planning formulation with explanations for failure, and an efficient approach was presented to solve the problem [6]. Athanasios Krontiris et al. proved that searching for solutions in MCR problems is computationally expensive; this leads to approximate methods. And they investigated alternatives in this context and evaluated their performance in terms of such tradeoffs. Solutions that follow a bounded-length approach, seem to provide a good balance between minimizing constraints, computational cost and path length [7]. Discrete MCR problems are NP-hard problems and also route optimization problems [8]. A simple description: searching for the shortest path, P , with the least amount of obstacles from the starting point, s , to t when n vertices are

d

M

Abstract—The minimum constraint removal (MCR) motion planning problem aims to remove the minimum geometric constraints necessary for removing a free path that connects the starting point and the target point. In essence, discrete MCR problems are non-deterministic polynomial-time (NP)-hard problems; there is a “combinatorial explosion” phenomenon in solving such problems on a large scale. Therefore, we are searching for highly efficient approximate solutions. In the present study, an ant colony algorithm was used to solve these problems. The ant colony algorithm was improved based on the social force model during the solving process, such that it was no longer easy for the algorithm to fall into local extreme, and the algorithm was therefore suitable for solving the MCR problem. The results of the simulation experiments demonstrated that the algorithm used in the present study was superior to the exact algorithm and the greedy algorithm in terms of solution quality and running time.

cr

(School of Software Engineering, South China University of Technology, Guangdong 510006, China)

P

Ac ce p

I. INTRODUCTION

te

Index Terms—ant colony algorithm, discrete MCR, motion planning problem, robot path planning, social force model

ATH planning is an important part of robot research and is also an important joint point of robotics and artificial intelligence (AI). The primary objective of path planning is to search for a free path from the initial state (including the location and posture) to the target state (including the location and posture) in an environment where there are some obstacles by a certain analysis standard; the robot will avoid all the obstacles safely and collision-free during its movement process on this path, which is also the shortest [1]. Determining a collision-free free path between two points is one of the most fundamental problems in robot technology. For most robot path-planning problems, the objectives are to search for an obstacle-free path from the initial state to the target state. The obstacles that a robot encounters may be inconvenient but are not insurmountable in most actual situations. For instance, an obstacle in indoor navigation may be a closed gate, which the robot is able to open. The obstacle in automatic outdoor navigation may be a rugged terrain, which is passable for the robot; under such circumstances, the price for overcoming the obstacle is very high, and there is risk to a certain degree; however, it is not impossible. In this situation, it is more

Page 1 of 9

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <

d

II. AN MCR MOTION-PLANNING PROBLEM

te

The black shadows in Fig. 1(a) represent obstacles, while the rest of the figure represents the free zone. The left top corner and the right bottom corner are the given starting vertex, qs and the target vertex, q g . The robot is required to

Ac ce p

search for the shortest path with the least amount of obstacles to be removed from the starting vertex, qs to the target vertex, q g . The figure shows that there are at least two obstacles ( O3 and

O5 ) on the path connecting the starting

point and the target point that need to be removed. The aforementioned problem is described in a map. One point in each zone is selected as the vertex. If the vertex is in the free zone, it is not marked. If the vertex is in the obstacle zone, the vertex is marked with the number of the obstacle. For instance, if the vertex is marked with 1, the vertex is covered

(a) (b) Fig. 1. MCR motion-planning problem.

ip t

by obstacle 1; if the vertex is marked with 1, 3 and 5, the vertex is in a zone covered by obstacles 1, 3 and 5. Fig. 1(b) shows the full description. The initial vertex is s , and the target vertex is t ; a partition of the continuous space generates a discrete MCR problem on the map. Every node is marked as the subset of the obstacles that “cover” the node. MCR is the minimum obstacle subset that covers all the nodes on the path from the starting point, s , to the target point, t . The mathematical definition of the MCR problem is as follows [3]: Continuous MCR

d -dimensional configuration space C ⊆ R d ; n obstacle zones O1 ,..., On ⊆ C ; these zones are all open

cr

Input.

sets, and the extreme points are q s , q g ∈ C .

us

Definition 1. The definition of the cover function

an

C (q ) : C → 2{1,...,n} is the obstacle set that violates the configuration q : C (q ) = {i | q ∈ Oi } . For a subset of the same configuration space A ⊆ C , the definition of its corresponding cover function, C ( A) , is all the vertices, q , of the alliance cover of C (q ) ( q ∈ A ), or C ( A) = {i | A ∩ Oi ≠ φ} . Definition 2. For one subset S ⊆ {1,..., n} , we define that

M

given. For any solution of a discrete MCR problem, it is necessary to compare the solution with all other feasible paths to determine whether the solution is the optimum. However, there is an exponential number of comparisons; hence, it is impossible to verify any solution within the polynomial time. In essence, MCR problems are N-hard combinational optimization hard problems. The perfect method for solving MCR problems is a global search. However, when n is relatively large, it is almost impossible to accurately find the optimal solution using the global search method. Therefore, researchers have started searching for a high-efficiency approximate solving method. Hauser presented a large amount of new theoretical results of discrete MCR problems in [3] and described two types of search algorithms that proved to be good in practice – greedy algorithms and exact algorithms; in addition, Hauser used these two types of algorithms to solve discrete MCR problems. The motion planner was proved to produce the optimal MCR with a probability close to one due to more time spent; in addition, effective sampling strategies were used to improve its convergence rate. Hauser also provided three application examples: generating failure reasons that could be determined, motion planning in an undetermined environment and repositioning moveable obstacles. However, we believe that after discrete MCR problems have been determined to be NP-complete problems, the focus of the study on discrete MCR problems should be shifted to the use of intelligent heuristic algorithms to solve discrete MCR problems.

2

S − can reach q ' starting from q . If there is a continuous path y ( s ) : [0,1] → C , the path starts from q and ends at

q ' , and it also satisfies C ( y ( s )) ⊆ S ， s ∈ [0,1] . Output. MCR S * is the shortest path between qs and q g . Correspondingly, MCR is a minimum set (not necessarily the only minimum set). Therefore, qs can be reached by S * starting from q g . However, according to the sample method, the discrete MCR is an important sub-problem, which is described as follows: Discrete MCR Output. Figure G = (V , E ) ; cover function C[v ] ; start and end vertices s, t ∈ V . In C[v ] , the subset of

{1,..., n} is used to mark every vertex, v , indicating which constraints are applied to v . * Output. S is the smallest subset of {1,..., n} . In addition, there is a path, P , from s to t in Figure G, and for all the * vertices, v ∈ P , C[v ] ⊆ S The above definitions directly expand similar concepts of continuous MCR to discrete MCR. What interests the researchers is to find a solution for the number of MCR. This new type of MCR problem is significantly different from the previous motion-planning problems – it exhibits the curse of dimensionality and the complexity of combinations in a

Page 2 of 9

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <

0

d

te

Ac ce p

A. Social-Force-Model-Based Acceleration In 2000, Helbing et al. proposed a social-force model based on Newtonian dynamics; the expression of each force was used to describe different motives and effects of an individual [17， 18]. Individual actual behavior and subjective consciousness as well as the effect among individuals and obstacles can all be

0

where

τi

vi0 (t )ei0 (t ) − vi (t )

τi

,

(1)

cr

f i 0 (t ) = mi

ip t

denoted as f i ; the calculation formula for f i is

represents the adaptation time for ant individual i ;

mi represents the mass of ant individual i ; ei0 (t ) represents 0

vi (t ) represent the desired speed and the actual speed of ant individual i at

the desired moving direction; and vi (t ) and

time t, respectively. Definition 2: The interaction between ant individuals

j

and i , also known as the repulsive force, denoted as f ij (t ) is

M

There are two types of MCR problems – continuous MCR problems and discrete MCR problems [3]. Discrete MCR problems have been proved to be NP-hard problems. The first type of method for solving discrete MCR problems is the accurate global search algorithm; this method can be used to effectively solve discrete MCR problems, and the solutions are relatively accurate. However, this method is not suitable for solving problems on a large scale, as there may be a “combinational explosion” phenomenon [9]. The second type of method is based on AI algorithms, such as finding the solution using greedy algorithms. The advantages of greedy algorithms are that they are easy to design and can often find good approximate solutions. For most problems, greedy algorithms make a decision prematurely; therefore, these algorithms cannot find the optimal solution. Generally, optimal solutions cannot be found using greedy algorithms because not all the feasible solutions are tested. For instance, for all the greedy algorithms for NP-complete problems, optimal solutions are not ensured [10] [11]. After discrete MCR problems have been determined to be NP-complete problems, we believe that the focus of the study on discrete MCR problems should be shifted to use intelligent heuristic algorithms to solve discrete MCR problems. An ant colony algorithm is a type of swarm intelligence algorithm, which has been widely used in recent years [12] [13] [14]. Reshamwala A et al. described the various techniques for the robot path planning using the Ant colony Algorithm. And also provided the brief comparison of the three techniques described in the paper [15]. Yao Y, et al. proposed a heterogeneous feature ant colony optimization (HFACO) algorithm to solve the robot path planning problem, the experiment results show that HFACO can find a better path in a shorter period of time compared to the classical ACO algorithms [16]. But when the swarm scale is relatively large, premature and stagnation phenomena tend to occur in ant colony algorithms, which prevent their further applications, To improve the algorithm for solving discrete MCR problems in terms of efficiency and reliability, the ant colony algorithm (station ant colony optimization (SACO) was improved based on the social force model during the solving process in the present study, such that the algorithm would not easily fall into local extreme, and became suitable for solving discrete MCR problems.

us

III. SOLVING DISCRETE MCR PROBLEMS USING A SOCIAL-FORCE-MODEL-BASED ANT COLONY ALGORITHM

equivalent to the action of a force on the individual, i.e., the driving force, the interaction between individuals and the interaction between the individual and the obstacle. The resultant force of all of these forces is exerted on the individual, and an acceleration is generated. In the present study, we referred to the aforementioned model and used it in the ant colony algorithm. A detailed description is as follows: Definition 1: The driving force reflects the motive of an ant individual that moves to the destination at a desired speed,

an

continuous space. It is necessary to enumerate the subsets of the constraint set (the number of subsets ranges from several hundred to several thousand) to find the optimal solution. Therefore, we are searching for a highly efficient approximate solution.

3

fij (t ) = w(θij ) fijs (t ) f ijs (t ) = A exp[(

where

(2)

ri + r j − d ij B

)nij ,

(3)

ri and rj represent the radii of ants i and j ,

respectively;

d ij = ri − r j represents the distance between

j ; nij = (nij1 , nij2 ) = (ri − rj ) / dij represents the standardized unit vector of ant individual j that points to i ; A and B represent the parameters of the social force model; A determines the intensity of the repulsive force, while B determines the variation speed of the repulsive force with respect to distance; θ ij represents the angle between the

ant individuals i and

movement direction of ant individual i ,

vi (t ) and nij ; and

w(θ ij ) represents the correction factor of the repulsive force between ant individuals. Definition 3: The interaction between an ant individual and an obstacle is similar to that between ant individuals; the calculation formula is as follows: fiw = A exp[

( ri − diw ) ]niw + kg ( ri − diw )niw − kg (ri − diw )(vi .tiw )tiw ,(4) B

Page 3 of 9

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <

niw = (n1iw , n 2iw ) represents the unit vector of the

obstacle that points to ant i , and

tiw = (−n 2iw , n1iw ) ; d iw

represents the distance between the ant individual and the obstacle; and kg represents the amplitude of the repulsive force of the obstacle for ant i and is a control parameter of the social force model. Definition 4: The acceleration ∆vm i (t ) is generated at

mi

vi0 (t )ei0 (t ) − vi (t)

τi

∆vmi (t ) =

ip t

time t:

+ ∑ fij (t) + ∑ fiW (t ) j ( ≠i )

C. Global and Local Pheromones Combined Updating Strategy The pheromone-updating strategy is a key step in ant colony algorithms. If pheromones are updated too fast, the algorithm will fall into a local optimum and even stagnation. If pheromones are updated too slowly, the convergence rate will be too slow to search for the optimal path. In the present study, a global and local pheromones combined method was used. The formula for global pheromone updating is as follows: after the ant has traversed all the vertices, the information content is updated; the pheromone on path (i , j ) at time t+n is adjusted according to formulas (8)~(10):

W

(5)

mi

τ ij (t + n) = (1 − ρ )τ ij (t ) + ∆τ ij (t ) m

(i, j ) at time t ; tabu k records the set of the vertices that ant k

[τ ij (t )] .[ηij (t )]

∑ [τ

s∈allowedk

α

is

] .[η ij ]

β

pijk (t ) =0, otherwise Here,

(6)

allowed k represents the vertices that the ant is allowed

Ac ce p

to select at the next step; α represents the information heuristic factor; β represents the desired heuristic factor; and

η ij (t )

represents the heuristic function, and its calculation

formula is defined as follows:

ηij (t ) =

1 , Cij

(7)

where Cij represents the number of obstacles between vertices

i and j ; η ij (t ) represents the relative value of the obstacles

j . For ant k , a smaller value of Cij indicates a larger value of η ij (t ) and a

that the ant encounters from vertices i to

larger value of

us

if the ant k use edge(i, j ) ,

(10)

otherwise Lk represents

k th ant completes; ρ k represents the pheromone volatile coefficient; ∆τ ij represents the concentration of pheromones that the k th ant left on path (i, j ) during this cycle, which is calculated using formula (10); and ∆τ ij represents the sum of the concentrations of pheromones that all the ants released on path (i , j ) during this the path length of one cycle that the

, if j ∈ allowed j

d

pijk (t ) =

Q  ∆τ (t ) =  Lk 0  k ij

where Q represents the pheromone intensity;

β

te

α

(9)

k =1

M

k has traversed; Pij (t ) represents the transition probability for ant k from vertex i to vertex j at time t .

(8)

∆τ ij (t ) = ∑ ∆τ ijk (t )

an

B. Heuristic Function Setting n is set as the number of vertices; m is set as the number of ants; τ ij (t ) represents the residual information on path

cr

where

4

Pijk (t ) ; therefore, paths with a small number of

obstacles are favorable. Clearly, the heuristic function represents the desired degree of a small number of obstacles for the ant that moves from vertex i to j .

cycle. In addition, after the ant finds a sub-feasible solution, the pheromones on the path (i , j ) of the sub-feasible solution are also updated locally:

τ ij = (1 − λ )τ ij + λτ 0 , λ ∈ (0,1) where

λ

,

(11)

represents the pheromone volatile coefficient,

τ0

is

1 5 a constant, and τ 0 = , Omin represents the mean 5 ( nOmin ) value of the 5 most recent optimal objectives. D. Population Classification and Target Vertex Selection In the present study, the individuals in the ant colony were first classified into free individuals and non-free individuals using probability. The calculation formula for the probability of an ant becoming a free individual is

ρα = (

Rdα 3.6 ) , 2

(12)

Page 4 of 9

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <

Rdα represents the relative distance between ant α

and the target. When the random number,

rand ≤ ρα , ant α

becomes a free individual; otherwise, ant α becomes a non-free individual. To prevent the algorithm from falling into a local optimum, the probability selection mechanism was used for free individuals. Ant i (i = 1,2,..., m) selects the next vertex according to the transition probability (determined by information content). For non-free individuals, ant i selects the next vertex according to the transition probability (determined by information content) and the acceleration; the ant’s next vertex is calculated using formula (13):

is a coefficient,

ε ∈ (0, 1) .

Record the optimal solutions of all the cycles

addition, pheromones are initiated. The ant is placed at the initial vertex s . Step 2: Population initialization. The population is classified according to (12); the free individual population and non-free population are formed. Step 3: For ants that belong to the free individual

Ac ce p

calculated. Each ant selects the next path vertex according to formula (6); if the pheromones on the path from this vertex to its adjacent vertex are all 0, the ant returns to the last vertex. Step 4: For ants that belong to the non-free individual population, their Cij and transition probability,

Reach target vertex or maximum iteration

k ij

P (t ) , are

calculated. In addition, the social driving force and the acceleration of each ant are calculated. Each ant selects the next path vertex according to formula (13). If the pheromones on the path from this vertex to its adjacent vertex are all 0, the ant returns to the last vertex. Step 5: The optimal solution of the objective function of this cycle is calculated. The pheromones on the path of the optimal solution of this cycle are updated globally according to formula (8). In addition, the pheromones on the path of the optimal solution of this cycle are updated locally according to formula (11). The optimal solutions of all the cycles are recorded. Step 6: If all the ants converge to a path that reaches the target vertex t , or the maximum iteration number, the cycle is terminated; otherwise, return to Step 3. Step 7: Output the solution. Fig.2. shows flow chart for the proposed algorithm

N

Y

d Pijk (t ) , are

te

population, their Cij and transition probability,

Pijk (t )

Global and Local Pheromones Combined Updating Strategy

M

t max ; social force model parameters, A , B and kg . In

and

Calculate optimal solution of the objective function

(13)

E. SACO Algorithm for Solving MCR Problems The steps of the algorithm are as follows: Step 1: The parameters of the algorithm are set: ant colony size, m ; ant’s adaptation time, τ ; maximum iteration number,

Cij

us

ε

Calculate

an

where

Population initialization

cr

probij (t ) = Pijk (t ) + ε∆vmi (t ) ,

Set algorithm parameters

ip t

where

5

Output the solution .

Fig.2. SACO Algorithm for Solving MCR Problems IV. EXPERIMENTAL RESULTS AND ANALYSIS A. Parameter Setting and Experimental Data To verify the effect of the improved algorithm, many simulation experiments were performed in the present study. Under different types of experimental environments, the exact algorithm, the greedy algorithm and the SACO algorithm used in the present study were used to find solutions. The parameter settings for the simulation experiment for the SACO TABLE I COMPARISON OF THE EXPERIMENTAL RESULTS FOR THE 3 ALGORITHMS

vertices=25 edges=80 subsets=3 vertices=25 edges=80 subsets=5 vertices=25 edges=80 subsets=10

Time(s)

Cost

Exact Greedy SACO Exact Greedy SACO Exact

Found solution True True True True True True True

0.016 0.016 0.016 0.016 0.015 0.016 0.016

1 1 1 2 2 2 4

Greedy

True

0.032

4

SACO

True

0.032

4

Cover ['O1'] ['O1'] ['O1'] ['O2', 'O3'] ['O2', 'O3'] ['O2', 'O3'] ['O2', 'O3', 'O8', 'O9'] ['O2', 'O3', 'O8', 'O9'] ['O2', 'O3', 'O8', 'O9']

Page 5 of 9

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <

Exact Greedy SACO Exact

0.108 0.090 0.092 0.187

1 1 1 3

Greedy

True

0.091

3

SACO

True

0.094

3

Exact

True

0.187

5

Greedy

True

0.091

5

SACO

True

0.094

5

Cover ['O1'] ['O1'] ['O1'] ['O1', 'O4', 'O5'] ['O1', 'O4', 'O5'] ['O1', 'O4', 'O5'] ['O1', 'O10', 'O4', 'O5', 'O9'] ['O1', 'O10', 'O4', 'O5', 'O9'] ['O1', 'O10', 'O4', 'O5', 'O9']

m = 30 ; maximum tmax =50; weights of pheromone and obstruction information, α = 1 and β = 2 ; constant, C = 100 ; ants’ adaptation time, τ = 0.1 ; social force model parameters, A = 2 × 10 3 , B = 0.08 and 5 −1 −1 kg = 2.4 × 10 kg .m .s . An example of data format used algorithm are as follows: number of ants,

M

iteration number,

ip t

Cost

cr

vertices=441 edges =1680 subsets=10

Time(s)

us

vertices=441 edges =1680 subsets=3 vertices=441 edges =1680 subsets=5

Found solution True True True True

and the detailed settings were as follows: (a) V=25, E=80, S=3, 5, 10; (b) V=441, E =1680, S=3, 5, 10. The 3 algorithms were run separately. Tables I and II list the experiment results. In the output experiment data, vertices represent vertices; edges represent edges; subsets represent the number of obstacles; cost represents the expense for removing the obstacles, which is denoted as the number of removed obstacles in the present study; cover represents the number of obstacles that need to be removed. Tables I and II demonstrate that in the 2 groups of experiments, the 3 algorithms could all be used to find the optimal solution, and there were no significant differences among the 3 algorithms in terms of running time. In particular, there was no significant difference between our SACO algorithm and the greedy algorithm, indicating that the advantages of our SACO algorithm were not significant for small-scale MCR problems. Experiment 2 – Random Rectangular Obstacle Model: In experiment 2, a random rectangular model was used. Different numbers of random obstacles were used to evaluate the exact algorithm, the greedy algorithm and the SACO algorithm. Fig. 3 [3] [10] shows that rectangular obstacles were randomly sampled from the grid in the model. The 3 algorithms were run separately. Fig. 4 presents the experimental results. Fig. 3 demonstrates that for the random rectangular obstacle model, the running times of the 3 algorithms were basically linearly correlated with the number of obstacles. The running time depended on the number of obstacles. However, what is peculiar is that different algorithms exhibited different phenomena. The running time of the exact algorithm increased with an increasing number of obstacles, and the increasing rate was much faster than those of the greedy algorithm and the SACO algorithm. There was no significant difference between the greedy algorithm and the SACO algorithm. The experimental results revealed that compared with other algorithms, the time needed for the SACO algorithm to find the optimal solution was the shortest, and the SACO algorithm was also relatively stable. Experiment 3 – Independent Obstacle Model: To test the adaptability and stability of the SACO algorithm for solving large-scale complex discrete MCR problems, a 50*50*50 grid was used in experiment 3. In addition, an independent obstacle model was used to evaluate the greedy algorithm and the SACO algorithm (the processing time of the exact algorithm was too

an

TABLE II COMPARISON OF THE EXPERIMENTAL RESULTS FOR THE 3 ALGORITHMS

6

in the experiments is shown below:

Ac ce p

te

d

Begin param start := 1; // starting point param goal := 4; // target point set V := 1 2 3 4; // vertex set E := (1,2) (2,1) (1,3) (3,1) (2,3) (3,2) (2,4) (4,2) (3,4) (4,3); // edge set S := A B C D E F G; // obstacle set SV := (1,A) (1,B) (1,E) (2,B) (2,E) (2,G) (4,B) (3,C) (3,A) (3,G); // indicating that the obstacle covers some vertex End

Python language was used to write the experimental data generation program, grid_data.py, which was then used to generate the date files necessary for the experiments. In addition, 3 experiments were compared. B. Results and Analysis Experiment 1 – Small-Scale Discrete MCR Problems: In the experiment, a 10*10 grid was used. The obstacles were generated randomly. There were 2 groups of experimental data,

Fig. 3. Random rectangular obstacle model.

Fig. 5. Random rectangular obstacle model[0].

Page 6 of 9

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <

7

3.5

0.25

3

0.2 Exact Greedy SACO

0.15 0.1

time(s)

time(s)

2.5 Exact

2

Greedy SACO

1.5 1

0.05

0.5 0

0 10

50

100

150

200

250

300

350

400

450

10

500

50

100

150

200

4

Exact

0.3

Greedy SACO

0.2

3 2.5 2 1.5 1 0.5

0.1

0 10

0 250

300

350

400

450

500

50

100

an

200

us

time(s)

0.4

time(s)

0.5

150

400

450

500

cr

3.5

100

350

ip t

4.5 0.6

50

300

(e) Vertices=2601, edges=10200

(a) Vertices=121, edges=440

10

250 subsets(n)

subsets(n)

subsets(n)

(b) Vertices=441, edges=1680

150

200

250

300

350

Exact Greedy SACO

400

450

500

subsets(n)

(f) Vertices=3721, edges=14640

Fig. 4. Comparison among the running times of the 3 algorithms.

1 0.8 time(s)

long and was thus omitted). Fig. 5 shows the independent obstacle model [3] [10]. In this model, rectangular obstacles were randomly sampled from the grid. M indicates that each vertex was covered by M independent obstacles (Dr. Hauser K conducted extensive research on M; please see [3] for details). In the experiment, M=0.5, 1 and 2; obstacles, subsets=10, 20, 50, 100, 200, 300, 400 and 500. The 3 algorithms were run separately. Fig. 6 presents the experimental results.

M

1.2

Exact

0.6

Greedy SACO

d

0.4

0 10

50

100

150

200

250

300

350

subsets(n)

te

0.2

400

450

500

30 25

10

50

100

150

200

250

300

350

400

450

time(s)

20 Greedy

15

SACO

10 5

Exact Greedy SACO

0 10

20

50

100

200

300

400

500

subsets(n)

(a) Vertices=125000, edges=735000, M=0.5

500

subsets(n)

40 35 30

(d) Vertices=1681, edges=6560 time(s)

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

Ac ce p

time(s)

(c) Vertices=961, edges=3720

25 Greedy SACO

20 15 10 5 0 10

20

50

100

200

300

400

500

subsets(n)

(b) Vertices=125000, edges=735000, M=1

Page 7 of 9

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <

Greedy SACO

200 150 100 50 0 10

20

50

100

200

300

400

500

subsets(n)

(c) Vertices=125000, edges=735000, M=2 Fig. 6. Comparison among the running times of the 3 algorithms.

V. CONCLUDING REMARKS

In the present study, an ant colony algorithm was used to solve a discrete MCR problem. During the solving process, the social force model was used to improve the ant colony algorithm, such that it would not easily fall into local extreme and became suitable for solving the MCR problem. The results of the simulation experiments revealed that our algorithm was superior to the exact algorithm and greedy algorithm in terms of solution quality and running time. In the MCR motion planning problem, this paper is mainly considering the searching for highly efficient approximate solutions, has not considered the influence of other aspects, therefore, on the basis of the present work, the next step will take multi-goal variants, incorporating movement costs and constraints and other factors into consideration. After discrete MCR problems have been determined to be NP-complete problems, the focus of the study on discrete MCR problems should shift to using AI heuristic algorithms to solve discrete MCR problems. We will study other AI algorithms for solving MCR problems in the future.

d

M

Fig. 6 demonstrates that for relatively large-scale discrete MCR problems, the greedy algorithm was not very stable. Fig. 6(a) demonstrates that with an increasing number of obstacles, the greedy algorithm did not linearly increase or decrease; instead, the algorithm exhibited a fluctuating phenomenon, which was related to the greedy characteristics of the greedy algorithm. The running time of the SACO algorithm was shorter than that of the greedy algorithm, and the SACO algorithm decreased linearly with an increasing number of obstacles. Fig. 6(b) and (c) indicate that compared with the greedy algorithm, the time difference was much more significant in the SACO algorithm; the SACO algorithm could converge to the optimal solution within a shorter time and was relatively stable. The above tests indicate that the advantages of the SACO algorithm were more significant for solving large-scale MCR problems; the SACO algorithm could meet the requirements of large-scale discrete MCR optimization – it was more reliable and more efficient.

ip t

250

cr

time(s)

300

us

350

In addition, when n is relatively large, it is almost impossible to use the global search method to find the optimal solution. Although the cost of greedy algorithms is less than that of exact algorithms, greedy algorithms may not be able to find the optimal solution. The advantage of greedy algorithms lies in their ease of design, and these algorithms can often find good approximate solutions. For most problems, greedy algorithms tend to make a decision prematurely and thus cannot reach the optimal solution. Generally, greedy algorithms cannot find the optimal solution because these algorithms do not test all the feasible solutions. For instance, all the greedy algorithms for solving NP-complete problems are not ensured to find the optimal solution. Ant colony algorithms generally can find the optimal solution or an approximate optimal solution; however, these algorithms are slow and easily fall into a local optimum. Therefore, it is necessary to improve ant colony algorithms. Hence, we applied an ant colony algorithm in an actual discrete MCR problem and improved and perfected the algorithm based on the actual situations; we used the social force model to improve the ant colony algorithm. The experiments also confirmed the effectiveness and feasibility of the improved algorithm.

an

400

8

te

C. Comparative Analysis of the Algorithms Finally, we summarize and compare the algorithms for solving discrete MCR problems (Table III).

O (E | 2|n )

Ac ce p

TABLE III COMPARISON OF THE ALGORITHMS FOR SOLVING DISCRETE MCR PROBLEMS Time Algorit complexi Advantages Disadvantages hms ty Exact Used to find optimal Very slow speed O(| E | 2 n ) solutions Greedy

O (| E | n)

Easy to design.

SACO

O ( N c .n 2 .m ) Used to find optimal or

approximate optimal solutions

In Table III

For NP-complete problems, optimal solutions are not guaranteed to be found Slow speed; local optimum

N c is cycle times， m is the number of ants. In

the computational complexity theory, discrete MCR problems are classic hard problems. The comparison among several algorithms for solving MCR problems reveals that exact algorithms can accurately find the optimal solution; however, these algorithms are very time-consuming. In the worst-case,

O(| E | *2 n ) time, greedy search runs in O (| E | *n) time. SACO runs in O ( N c * n 2 * m) time.

exact search runs in

ACKNOWLEDGMENT This research is supported by the National Natural Science Foundation of China (Grant No. 61272382). Thanks to Dr Hauser K of Indiana University, Associate， Professor An Bo of Nanyang Technological University, Singapore, Dr Smith and Dr Xiao Fangxiong of South China University of Technology, China.

REFERENCES [1] [2]

J. Canny, The Complexity of Robot Motion Planning. Cambridge, MA: MIT press, 1988. K. Hauser, “The minimum constraint removal problem with three robotics applications,” in Proceedings of Workshop on the Algorithmic Foundations of Robotics, 2012.

Page 8 of 9

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <

ip t

[13] T. Stutzle and M. Dorigo, “A short convergence proof for a class of ant colony optimization algorithms,” IEEE Trans. Evolut. Comput., vol. 6, no. 4, pp. 358-365, 2002. [14] M. Dorigo, G. D. Garo, and L. M. Gambardella, “Ant algorithms for discrete optimization,” Artif. Life, vol. 5, no. 2, pp. 137-172, 1999. [15] Reshamwala A, Vinchurkar D P. “Robot Path Planning using An Ant Colony Optimization Approach: A Survey”. International Journal of Advanced Research in Artificial Intelligence, vol 2, no 3,.:pp.65-71,2013 [16] Yao Y, Ni Q, Qing L V, et al. “A novel heterogeneous feature ant colony optimization and its application on robot path planning”, IEEE Congress on. Evolutionary Computation (CEC), IEEE, Sendai, pp. 522-528, 2015. [17] D Helbing，P Molnar. "Social Force Model for Pedestrian Dynamics", Physical Review, vol 21, pp:4282-4286, 1995 [18] D. Helbing, I. Farkas, and T. Vicsek, “Simulating dynamical features of escape panic,” Nature, vol. 407, pp. 487-490, 2000.

us

cr

Bo. Xu was born in 1982; he is a Ph.D. candidate at School of Software Engineering, South China University of Technology. He is a Senior Member of China Computer Federation (CCF), member of Computer Applications Professional Committee of CCF, IEEE member and ACM member. He received the M.A. degrees in Computer Science and Technology from Hunan University, Hunan, China in 2009. His research interests include computational intelligence and intelligence robot Huaqing Min* was born in 1956; he is a Professor and Doctoral supervisor at School of Software Engineering, South China University of Technology, Guangdong, China. Research fields: intelligence robot, intelligence software, automatic system.

an

K. Hauser, “The minimum constraint removal problem with three robotics applications,” Int. J. Robot. Res., vol. 33, no. 1, pp. 5-17, 2014. [4] L. H. Erickson and S. M. LaValle, “A simple, but NP-hard, motion planning problem,” in Proceedings of the Twenty-Seventh AAAI Conference on Artificial Intelligence (AAAI-13), Urbana, USA, 2013, pp. 1388-1393. [5] Z. McCarthy, T. Bretl, and S. Hutchinson, “Proving path non-existence using sampling and alpha shapes,” in Proceedings of the 2012 IEEE International Conference on Robotics and Automation (ICRA), 2012, pp. 2563-2569. [6] Gorbenko A, Popov V. “The discrete minimum constraint removal motion planning problem”, International Conference on Numerical Analysis and Applied Mathematics (ICNAAM), Rhodes, Greece, 2015. [7] A. Krontiris , K. E. Bekris. “Computational Tradeoffs of Search Methods for Minimum Constraint Removal Paths ” In Symposium on Combinatorial Search (SoCS), Dead Sea, Israel, 2015. [8] M. Göbelbecker, T. Keller, P. Eyerich, M. Brenner, and B. Nebel, “Coming up with good excuses: what to do when no plan can be found,” in Proceedings of the International Conference on Automated Planning and Scheduling, 2010, pp. 81-88. [9] J. Johnson and K. Hauser, “Optimal longitudinal control planning with moving obstacles,” in Proceedings of 2013 IEEE Intelligent Vehicles Symposium (IV), 2013, pp. 605-611. [10] K. Hauser, “On responsiveness, safety, and completeness in real-time motion planning,” Autonomous Robots, vol. 32, no. 1, pp. 35-48, 2012. [11] K. Hauser, “Minimum constraint displacement motion planning,” in Proceedings of Robotics: Science and Systems (RSS), Berlin, Germany, 2013. [12] M. Dorigo and L. M. Gambardella, “Ant colony system: A cooperative learning approach to the traveling salesman problem,” IEEE Trans. Evolut. Comput., vol. 1, no. 1, pp. 53-66, 1997.

M

[3]

9

Ac ce p

te

d

The minimum constraint removal (MCR) motion planning problem aims to remove the minimum geometric constraints necessary for removing a free path that connects the starting point and the target point. In essence, discrete MCR problems are non-deterministic polynomial-time (NP)-hard problems; there is a “combinatorial explosion” phenomenon in solving such problems on a large scale. Therefore, we are searching for highly efficient approximate solutions. In the present study, an ant colony algorithm was used to solve these problems. The ant colony algorithm was improved based on the social force model during the solving process, such that it was no longer easy for the algorithm to fall into local extreme, and the algorithm was therefore suitable for solving the MCR problem. The results of the simulation experiments demonstrated that the algorithm used in the present study was superior to the exact algorithm and the greedy algorithm in terms of solution quality and running time.

In the present study, an ant colony algorithm was used to solve a discrete MCR problem. During the solving process, the social force model was used to improve the ant colony algorithm, such that it would not easily fall into local extreme and became suitable for solving the MCR problem. The results of the simulation experiments revealed that our algorithm was superior to the exact algorithm and greedy algorithm in terms of solution quality and running time. After discrete MCR problems have been determined to be NP-complete problems, the focus of the study on discrete MCR problems should shift to using AI heuristic algorithms to solve discrete MCR problems. We will study other AI algorithms for solving MCR problems in the future.

Page 9 of 9