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Adaptive Charged System Search Approach to Path Planning for Multiple Mobile Robots
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IFAC-PapersOnLine 48-10 (2015) Approach 294–299 Adaptive Charged System Search to Path Planning Adaptive Charged System Search Approach Path Planning forSystem Multiple Mobile Robots to Adaptive Search Approach to Adaptive Charged Charged System Search Approach to Path Path Planning Planning for Multiple Mobile Robots for Multiple Mobile Robots for Multiple Mobile Robots
Radu-Emil Precup*, Emil M. Petriu**, Mircea-Bogdan Radac***, Emil-Ioan Voisan****, Florin Dragan***** Radu-Emil Precup*, Emil M. Petriu**, Mircea-Bogdan Radac***, Emil-Ioan Voisan****, Florin Dragan***** Radu-Emil Radu-Emil Precup*, Precup*, Emil Emil M. M. Petriu**, Petriu**, Mircea-Bogdan Mircea-Bogdan Radac***, Radac***, Emil-Ioan Emil-Ioan Voisan****, Voisan****, Florin Florin Dragan***** Dragan***** * Department of Automation and Applied Informatics, Politehnica University of Timisoara, Bd. * Parvan 2, 300223 Timisoara, (Tel: +40 256 403229; e-mail: [email protected]) *V.Department Department of Automation Automation andRomania Applied Informatics, Informatics, Politehnica University of Timisoara, Timisoara, of and Applied Politehnica University of *V.Department ofofAutomation andRomania Appliedand Informatics, Politehnica University Timisoara, ** School Electrical Engineering Computer Science, University ofofOttawa, Bd. Parvan 2, 300223 Timisoara, (Tel: +40 256 403229; e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (Tel: +40 256 403229; e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (Tel: +40 256Science, 403229; e-mail: [email protected]) 800 King Edward, Ottawa, Ontario, K1N 6N5 Canada (e-mail: [email protected]) ** School of Electrical Engineering and Computer University of Ottawa, ** of Electrical Engineering and Computer Science, University of ** School School of Engineering and Computer Science, University of Ottawa, Ottawa, *** Department of Electrical Automation and Applied Informatics, Politehnica University of Timisoara, 800 King Edward, Ottawa, Ontario, K1N 6N5 Canada (e-mail: [email protected]) 800 King Edward, Ottawa, Ontario, K1N 6N5 Canada (e-mail: [email protected]) 800 King Edward, Ottawa, Ontario, K1N 6N5 Canada (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) *** Department of Automation and Applied Informatics, Politehnica University of Timisoara, *** of Automation and Applied Informatics, Politehnica University of Timisoara, *** Department Department of Automation and Applied Informatics, Politehnica University of Timisoara, **** Department of Automation and Applied Informatics, Politehnica University of Timisoara, Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) **** Department of Automation and Applied Informatics, Politehnica University of Timisoara, **** of Automation and Applied Informatics, Politehnica University of Timisoara, **** Department Department of and Applied Politehnica University ***** Department ofAutomation Automation and AppliedInformatics, Informatics, Politehnica Universityof ofTimisoara, Timisoara, Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) ***** Department of Automation and Applied Informatics, Politehnica University of Timisoara, ***** of and Applied Informatics, Politehnica University ***** Department Department of Automation Automation and AppliedRomania Informatics, Politehnica University of of Timisoara, Timisoara, Bd. V. Parvan 2, 300223 Timisoara, (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected])
Abstract: This paper suggests the application of adaptive Charged System Search (CSS) algorithms to the optimalThis pathpaper planning (PP) of multiple mobile robots. AnCharged off-lineSystem adaptiveSearch CSS-based PP approach to is Abstract: suggests the application of adaptive (CSS) algorithms Abstract: This paper suggests the application of adaptive System Search (CSS) algorithms to Abstract: This paper suggests themultiple application ofplatforms adaptiveAnCharged Charged System Search (CSS) algorithms to proposed and applied to holonomic wheeled in static environments. The adaptive CSS the optimal path planning (PP) of mobile robots. off-line adaptive CSS-based PP approach is the optimal path planning (PP) of mobile robots. An off-line adaptive CSS-based PP approach is the optimaland pathapplied planning (PP) of multiple multiple mobile robots. Anin off-line adaptive CSS-based PP approach is algorithms solve the optimisation problems thatplatforms aim the minimisation of objective functions (o.f.s) proposed to holonomic wheeled static environments. The adaptive CSS proposed and applied to holonomic wheeled platforms in static environments. The adaptive CSS proposed and applied to holonomic wheeled platforms in static environments. The adaptive CSS specific to PP andthe expressed as the weighted sum of four functions that target separate PP objectives. A algorithms solve optimisation problems that aim the minimisation of objective functions (o.f.s) algorithms solve the optimisation problems that aim the minimisation of objective functions (o.f.s) algorithms solve the optimisation problems that aim the minimisation of The objective functions (o.f.s) penalty term is added in certain situations in the first step of the PP approach. specific features of the specific to PP and expressed as the weighted sum of four functions that target separate PP objectives. A specific to PP expressed as the weighted sum of four functions that target separate PP objectives. A specific toCSS PPisand and expressed sum ofthe four functions that targetThe separate PPfeatures objectives. A adaptiveterm algorithms areas situations the weighted adaptation of acceleration, velocity, and separation distance penalty added in certain in the first step of the PP approach. specific of the penalty term is added in certain situations in the first step of the PP approach. The specific features of the penalty term is added in certain situations in the first step of the PP approach. The specific features of the parameters to the iteration index, and the substitution of the worst charged particles’ fitness function adaptive CSS are the adaptation of the acceleration, velocity, and separation distance adaptive CSS algorithms algorithms are the adaptation of the velocity, and distance adaptive algorithms are the and adaptation of the acceleration, acceleration, velocity, and inseparation separation distance values andCSS positions with the best performing particle data. The fitness function thefitness adaptive CSS parameters to the iteration index, the substitution of the worst charged particles’ function parameters to the iteration index, and the substitution of the worst charged particles’ fitness function parameters to the iteration index, and the substitution of theThe worst charged particles’ fitness function algorithms corresponds to o.f., and the search space and agents (charged particles) in the adaptive values and positions with the best performing particle data. fitness function in the adaptive CSS values and positions with the best performing particle data. The fitness function in adaptive CSS values and corresponds positions with the bestsolution performing particle data. The fitness function in the the adaptive CSS CSS algorithms correspond to the space andspace to the mobile robots, respectively. Ain case study and algorithms to the o.f., and the search and agents (charged particles) the adaptive algorithms corresponds to the o.f., and the search space and agents (charged particles) in the adaptive algorithms corresponds tovalidate the o.f.,solution and thespace search space and agents (charged particles) in the adaptive experiments are included the new adaptive CSS-based PP approach and to compare it with nonCSS algorithms correspond to the and to the mobile robots, respectively. A case study and CSS algorithms correspond to solution space and to robots, respectively. A case study and CSS algorithms correspond to the the solution space to the the mobile mobile robots, respectively. A studynonand adaptive CSS-, Particle Swarm Optimizationandand Gravitational Search Algorithm-based PPcase approaches. experiments are included validate the new adaptive CSS-based PP approach and to compare it with experiments are included validate the new adaptive CSS-based PP approach and to compare it with nonexperiments are included validate the new adaptive CSS-based PP approach and to compare it with nonadaptive CSS-, Particle Swarm Optimizationand Gravitational Search Algorithm-based PP approaches. adaptive CSS-, Particle Swarm Optimizationand Search PP © 2015, IFAC (International Federation of Automatic Control) Hosting by Algorithm-based Elsevier All rights reserved. Keywords: Adaptive System Search algorithms, mobile robots,Ltd. obstacles, optimisation adaptive CSS-, Particle Charged Swarm Optimizationand Gravitational Gravitational Search Algorithm-based PP approaches. approaches. problems, path planning, penalty term. Keywords: Adaptive Charged System Search algorithms, mobile robots, obstacles, optimisation Keywords: Adaptive Charged System Keywords:path Adaptive Charged System Search Search algorithms, algorithms, mobile mobile robots, robots, obstacles, obstacles, optimisation optimisation problems, planning, penalty term. problems, path planning, penalty term. problems, path planning, penalty term. The nature-inspired optimisation algorithms produce 1. INTRODUCTION convenient versions of optimisation centralised algorithms to solve PP The nature-inspired algorithms produce The nature-inspired optimisation algorithms produce 1. INTRODUCTION INTRODUCTION 1. The nature-inspired optimisation algorithms produce optimisation problems. The current approaches to optimal Centralised and decentralised algorithms solve path planning convenient versions of centralised algorithms to solve PP 1. INTRODUCTION convenient versions of centralised algorithms to solve PP convenient versions ofThe centralised algorithms tooptimal solve PP that includes nature-inspired algorithms use Particle Swarm (PP) problems for multiple robots. The decentralised optimisation problems. current approaches to PP Centralised and decentralised algorithms solve path planning optimisation problems. The current approaches to optimal PP Centralised and decentralised algorithms solve path planning optimisation problems. The current approaches to optimal PP Centralised and decentralised algorithms solve path planning Optimization (PSO) (Masehian and Sedighizadeh, 2010; algorithms generate independently collision-free paths for that includes nature-inspired algorithms use Particle Swarm (PP) problems for multiple robots. The decentralised that includes nature-inspired algorithms use Particle Swarm (PP) problems for multiple robots. The decentralised that includes nature-inspired algorithms use Particle Swarm (PP) problems for multiple robots. The decentralised Purcaru et al., (PSO) 2013b;(Masehian Ma et al., and 2014), genetic algorithms each robot to reach the target solvingcollision-free the possible paths collisions Optimization Sedighizadeh, 2010; algorithms generate independently for Optimization (PSO) (Masehian and Sedighizadeh, 2010; algorithms generate independently collision-free paths for (PSO) (Masehian and Sedighizadeh, 2010; algorithms generate independently collision-free paths for Optimization (Roberge et al., 2013; Lim et al., 2014), Gravitational Search between robots. Some popular decentralised algorithms are Purcaru et al., 2013b; Ma et al., 2014), genetic algorithms each robot to reach the target solving the possible collisions Purcaru et al., 2013b; Ma et al., 2014), genetic algorithms each robot to reach the target solving the possible collisions Purcaru et al., 2013b; Ma et al., 2014), genetic algorithms each robot to reach the target solving the possible collisions Algorithms (GSAs) (Purcaru et al., 2013a, 2013b), migration based onrobots. partially observable Markov decision processes (Roberge et al., 2013; Lim et al., 2014), Gravitational Search between Some popular decentralised algorithms are (Roberge et al., Lim et al., 2014), Gravitational Search between robots. Some popular decentralised algorithms are (Roberge et (Vašcák al., 2013; 2013; LimPal’a, et et al.,al., 2014), Gravitational Search between robots. Some popular decentralised algorithms are algorithms and 2012), memetic algorithms (Banerjee et al., 2011) and on the environment-based Algorithms (GSAs) (Purcaru 2013a, 2013b), migration based on partially observable Markov decision processes Algorithms (GSAs) (Purcaru et al., 2013a, 2013b), migration based on partially observable Markov decision processes Algorithms (GSAs) (Purcaru et al., 2013a, 2013b), migration based on partially observable Markov decision processes (Bigaj and Kacprzyk, 2013; Zhang et al., 2014), simulated adaptation and optimisation (Kleiner et al., 2011). The belief algorithms (Vašcák and 2012), memetic algorithms (Banerjee et al., 2011) and on the environment-based algorithms (Vašcák and Pal’a, Pal’a, 2012), memetic algorithms (Banerjee et al., 2011) and on environment-based algorithms (Vašcák Pal’a, 2012), memetic (Banerjee et decentralised al., 2011) for and on the the environment-based annealing (Miao and and Tian, 2013; Maet etal., al.,2014), 2014),algorithms chemical functions are tracking in (Kim et The al., 2013). (Bigaj and Kacprzyk, 2013; Zhang simulated adaptation and optimisation (Kleiner et al., 2011). belief (Bigaj and Kacprzyk, 2013; Zhang et al., 2014), simulated adaptation and optimisation (Kleiner et al., 2011). The belief (Bigaj and(Miao Kacprzyk, 2013; Zhang et al., 2014), simulated adaptation anddecentralised optimisation (Kleiner et in al.,curves 2011). The belief optimisation (Melin et al., 2013), and Charged System Search Genetic algorithms combined with Bézier are used by annealing and Tian, 2013; Ma et al., 2014), chemical functions are for tracking (Kim et al., 2013). annealing (Miao and Tian, 2013; Ma et chemical functions are for tracking in (Kim al., 2013). annealing (Miao and Tian, 2013; MaCharged et al., al., 2014), 2014), chemical functions are decentralised decentralised for tracking in curves (Kim et etare al.,used 2013). (CSS) algorithms (Precup et al., 2014b). Kala and Warwick (2014) in path generation. optimisation (Melin et al., 2013), and System Search Genetic algorithms combined with Bézier by optimisation (Melin et 2013), and Genetic algorithms combined with Bézier (Melin et al., al., et 2013), and Charged Charged System System Search Search Genetic algorithms combined withgeneration. Bézier curves curves are are used used by by optimisation (CSS) algorithms (Precup al., 2014b). Kala and Warwick (2014) in path (Precup et Kala and (2014) path Using algorithms our recently proposed adaptive CSS algorithms The centralised algorithms the individual robots as (CSS) (CSS) algorithms (Precup et al., al., 2014b). 2014b). Kala and Warwick Warwick (2014) in inconsider path generation. generation. (Precup et al., 2014a),proposed this paperadaptive suggestsCSS a new off-line subsystems to algorithms allow for consider global optimisation. Successful recently algorithms The centralised the individual robots as Using Using our our recently proposed adaptive CSS algorithms The centralised algorithms consider the individual robots our recently proposed adaptive CSS algorithms The centralised algorithms consider the individual robots as as Using adaptive CSS-based approach that generates optimal paths for examples of decentralised algorithms are the sampling-based (Precup et al., 2014a), this paper suggests a new off-line subsystems to allow for global optimisation. Successful (Precup et al., this suggests aa new off-line subsystems to allow global optimisation. Successful (Precup et al., 2014a), 2014a), this paper paper suggests new off-line subsystems to allow for for global optimisation. Successful multiple robots. The adaptive CSS algorithms are mapped PP algorithms PRM* and RRT* (Karaman and Frazzoli, adaptive CSS-based approach that generates optimal paths for examples of decentralised algorithms are the sampling-based adaptive CSS-based approach that generates optimal paths for examples of decentralised algorithms are adaptive CSS-based approach that generates optimal paths for examples ofalgorithm decentralised algorithms are the the sampling-based sampling-based onto the optimisation problems by means of several elements: 2011), the based sets of non-colliding paths on multiple robots. The adaptive CSS algorithms are mapped PP algorithms PRM* and RRT* (Karaman and Frazzoli, multiple robots. The adaptive CSS algorithms are mapped PP algorithms PRM* and RRT* (Karaman and Frazzoli, multiple robots. The adaptive CSS algorithms are mapped PP algorithms PRM* and RRT* (Karaman and Frazzoli, the fitness functions are the objective functions (o.f.s), the graphs (Luna and Bekris, 2011), and the algorithm that onto the optimisation problems by means of several elements: 2011), the the algorithm algorithm based based sets sets of of non-colliding non-colliding paths paths on on onto the optimisation problems by means of several elements: 2011), ontofitness thespare optimisation by meansthe of several elements: 2011), the algorithm based sets oftask-allocation non-colliding pathsthat on search is the problems solution space, agents(o.f.s), (charged involves self-organized multi-robot (Sarker et the functions are the objective functions the graphs (Luna and Bekris, 2011), and the algorithm the fitness functions are the functions (o.f.s), the graphs (Luna and Bekris, 2011), and the algorithm that the fitness functions are the objective objective functions (o.f.s), the graphs (Luna and Bekris, 2011), task-allocation and the algorithm that particles) are the mobile robots, and the population of agents al., 2014). search spare is the solution space, the agents (charged involves self-organized multi-robot (Sarker et search spare is the solution space, the agents (charged involves self-organized multi-robot task-allocation (Sarker et search spare is the solution space, the agents (charged involves self-organized multi-robot task-allocation (Sarker et is the set of mobile robots.robots, particles) are the mobile and the population of agents al., 2014). particles) are the robots, al., particles) aremobile the mobile mobile robots, and and the the population population of of agents agents al., 2014). 2014). is the set of robots. is the set of mobile robots. is the set of mobile robots. Copyright © 2015 IFAC 294
2405-8963 ©©2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 IFAC 294 Copyright 2015responsibility IFAC 294Control. Peer review© of International Federation of Automatic Copyright ©under 2015 IFAC 294 10.1016/j.ifacol.2015.08.147
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We are using the notation t for both the algorithm iteration and the time along the planned trajectory in order to reduce the number of indices. We are also using the general notation || U − V ||2 for the Euclidean distance between the vectors U = (U X ,U Y ) ∈ R 2 and V = (V X ,VY ) ∈ R 2 :
This paper considers four PP objectives (Purcaru et al., 2013b; Precup et al., 2014a), aggregated in a set of separate o.f.s, and the optimisation problems aim the minimisation of the scalar o.f.s expressed as their weighted sum. An additional penalty term is added to the o.f.s in certain situations in the first step of the PP approach in order to avoid the robot positions to be inside the obstacles or the line that connects two consecutive robot positions to intersect obstacles.
|| U − V ||2 = (U X − V X ) 2 + (U Y − VY ) 2 .
time moments or iterations allowed for i th agent in the population PRk . But it is needed to eliminate all agents’ paths
that are not collision-free, namely we eliminate the path point sets Ti , PRk ,t ( i ,k ) , i ∈ S cf , PRk ⊂ {1,2,..., N } , where S cf , PRk is the max
set of agents which lead to collision-free paths for the population PRk . The sets Ti , PRk ,t ( i ,k ) actually correspond to max
separate collision-free paths for each robot Rk , i.e., to feasible paths. The total length of each path is Di ,k =
This paper is organised as follows: the PP problem is formulated in the next section. The new off-line adaptive CSS-based optimal PP approach is presented in Section 3. The approach is validated in Section 4 by experimental results and by the comparison with similar approaches. The concluding remarks are pointed out in Section 5.
tmax ( i ,k )
∑ || X t =1
i , PRk
(t ) − X i , PRk (t − 1) ||2 , i ∈ S cf ,PRk .
(4)
The optimisation problems that give the indices iˆk of the agents which correspond to the minimum length feasible paths are (Purcaru et al., 2013b) iˆk = arg min Di ,k ,
(5)
i∈Scf , PRk
2. PATH PLANNING PROBLEM
and the optimal paths are Tiˆ ,t ( iˆ ,k ) . The elements of the path max k
As considered in (Purcaru et al., 2013b; Precup et al., 2014b), let {Rk | k = 1... p} be a set of p mobile robots in a known
point sets Ti , PRk ,t , defined in (2) and used in the o.f.s (4), are computed as solutions to the following optimisation problems for each robot Rk , k = 1... p :
static environment that includes obstacles. The path for each robot is generated by a different population with the same number of N agents. The individual populations of agents that generate the paths for the robots R1 ...R p are PR1...PRp . The
Xi , PRk (t ) = arg min f i , PRk ( X i , PRk (t )), t = 1...t max (i, k ),
(6)
Xi , PRk ( t )
solution space of the mobile robot is a two-dimensional search space, and all agents in a population are placed in the initial robot’s position for which they need to generate the path. Using the notation X i , PRk (t ) for the agent vector
i = 1...N , k = 1... p,
the first element in Ti , PRk ,t , i.e., X i ,PRk (0) , is known: (7)
Xi , PRk (0) = X i , PRk (0), i = 1...N , k = 1... p,
position that stands for the current position of i th agent, i = 1...N , in the population PRk , k = 1... p, in the search space at the time moment (iteration) t X i , PRk (t ) = ( X i ,PRk (t ), Yi , PRk (t )) ∈ R ,
(3)
As shown in (Purcaru et al., 2013b), each agent’s path is built by connecting with lines all successive points of the Ti , PRk ,tmax ( i ,k ) sets, where t max (i, k ) is the maximum number of
Our PP approach is advantageous in the context of the stateof-the-art because it can be easily generalised to insert other nature-inspired optimisation algorithms. This is emphasized by the comparison of our approach with our previous nonadaptive CSS-, PSO- and GSA-based PP approaches by experiments conducted on the nRobotic platform developed at the Politehnica University of Timisoara, Romania. The difference from the previous paper (Precup et al., 2004b), which already applies CSS to multiple robot PP, and therefore the novelty of the contribution, is the adaptive CSS algorithm that replaces the non-adaptive CSS algorithm in the PP approach. The reader is invited to use (Precup et al., 2014a) for additional details on CSS algorithms.
2
295
and the expressions of the o.f.s are f i , PRk ( X i , PRk (t )) =|| X f − X i , PRk (t ) || 2 + λ 1 /[
(1)
where R is the set of real numbers, X i , PRk (t ) is agent’s
− X i , PRk (t ) || 2 ] + λ 2 /[
position on the X axis and Yi , PRk (t ) is agent’s position on the Y axis, the paths are generated iteratively by means of path point sets Ti , PRk ,t , which are updated at each iteration adding
+ λ 3 /[
the current agent position: Ti , PPRk ,t = {X i , PRk (0), X i , PRk (1),..., X i , PRk (t )}, i = 1...N , k = 1... p, (2)
where X i , PRk (0) = ( X i , PRk (0), Yi , PRk (0)) is the initial position of the agent i in the population PRk . 295
p
∑|Y
j =1, j ≠ k
best , PRj
p
∑| X
j =1, j ≠ k
best , PRj
p
∑ || X
j =1, j ≠ i
best , PRj
(t ) − X i , PRk (t ) |]
(t )
(8)
(t ) − Yi , PRk (t ) |], i = 1...N , k = 1... p.
X f = ( X f , Y f ) ∈ R 2 in (8) is the target (final) point, it is the
same for all agents, X best , PRj (t ) = ( X best ,PRj (t ), Ybest , PRj (t )) ∈ R 2 is the position of the best agent (in terms of the distance between the current point and the target point) in the
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As shown in (Precup et al., 2014a), the relation between the fitness functions g i ,k (t ) in the adaptive CSS algorithm and
population PRj at the time moment t, λ1 > 0 , λ 2 > 0 and λ 3 > 0 the weights. The first term in the o.f.s f i , PRk ( X i , PRk (t ))
the o.f.s f i , PRk ( X i , PRk (t )) is
aims the minimisation of the Euclidian distance between each agent position from the individual populations at a specific time (iteration) t and X f . The second term in f i , PRk ( X i , PRk (t )) aims the maximisation of the distance between the agents of a population and the best agents from all other populations in order to avoid to deal with many collision points between the paths generated by the agents. The third and fourth terms target the maximisation of the distance between the paths of the robots on the X and Y axes and model applications with one axis more important than the other one.
g i ,k (t ) = f i , PRk ( X i , PRk (t )), i = 1...N , k = 1... p ,
where N is the total number of agents, referred to also as charged particles (CPs) specific to the adaptive CSS algorithm. Agents’ performance in the search space is given by their electric charges qi ,k (t ) qi ,k (t ) = [ g i ,k (t ) − best k (t )] /[best k (t ) − worst k (t )], i = 1... N , k = 1... p,
Using our previous results on GSA, PSO and CSS algorithms applied in (Purcaru et al., 2013b; Precup et al., 2014b) to solve the optimisation problems (6) with the o.f.s (8), the next section will give another PP approach based on adaptive CSS algorithms. Other optimisation problems can be defined as well (Acosta et al., 2007; Filip and Leiviskä, 2009; Petelin et al., 2011 Ruano et al., 2014), but they should account for constraints specific to robots in terms of nonlinear models, which describe the environment at certain accuracy (Tomescu et al., 2007; Guerra et al., 2009; González et al., 2011; Precup et al., 2012).
(10)
where the so far best and the worst fitness of all agents in the population PRk are worst k (t ) = max g i ,k (t ), k = 1... p, i =1... N
best k (t ) = min g i ,k (t ), k = 1... p.
(11)
i =1... N
The separation distance between i th and j th agent is rij ,k (t ) = (|| X i , PRk (t ) ||2 − || X j , PRk (t ) ||2 ) /[|| 0.5( X i , PRk (t ) + X j , PRk (t )) − X best , PRk (t ) ||2 + ε],
(12)
where X best ,PRk (t ) is the position of the best current agent,
3. OFF-LINE ADAPTIVE CHARGED SYSTEM SEARCHBASED PATH PLANNING APPROACH
and the relatively small parameter ε > 0 is introduced to avoid the division by zero. The total forces on the X and Y axes, F jX,k and F jY,k , that act on j th agent at the time (iteration) t are (Precup et al., 2014b)
Our off-line adaptive CSS-based optimal PP approach consists of the following steps: Step PP1. Apply the adaptive CSS algorithms such that to solve the optimisation problems defined in (6) with the o.f.s defined in (8) to get the solutions Xi , PRk (t ) .
F jX,k (t ) = q j ,k (t )
N
∑[q
i =1,i ≠ j
i ,k
(t )cij ,k (t )(rij ,k (t )i1 / a 3
+ i2 / rij2,k (t ))( X i , PRk (t ) − X j , PRk (t ))],
Step PP2. Use the solutions Xi , PRk (t ) obtained in the step PP1 and equation (7) as the vector elements of the path point sets Ti , PRk ,t in (2), i.e., with the notation X i , PRk (t ) for Xi , PRk (t ) .
F jY,k (t ) = q j ,k (t )
N
∑[q
i =1,i ≠ j
i ,k
(13)
(t )cij ,k (t )(rij ,k (t )i1 / a 3
+ i2 / rij2,k (t ))(Yi , PRk (t ) − Y j , PRk (t ))], j = 1...N , k = 1... p,
Step PP3. Solve the optimisation problems defined in (5) with the o.f.s (8) to get the optimal paths Tiˆ ,t ( iˆ ,k ) . max
(9)
where the parameters that outline “good” and “bad” agents (from the fitness function point of view) are
k
Since the steps PP2 and PP3 are very simple because they actually compute the minimum length paths from sets of already computed feasible paths, we will present as follows details on the step PP1. A fifth term is added in the o.f.s f i , PRk ( X i , PRk (t )) in (8), namely the penalty term α > 0 , only
⎧− 1, if g i , k (t ) < g j , k (t ), c ij , k (t ) = ⎨ ⎩ 1, otherwise, i1 = 0, i 2 = 1 for rij , k (t ) ≥ a, i1 = 1, i 2 = 0 for rij , k (t ) < a,
(14)
i, j = 1...N , k = 1... p,
in certain situations that should be avoided. Such situations occur when the current i th agent position at iteration t is inside one of the obstacles or the line that connects the position at iteration t with the position at iteration t − 1 intersects an obstacle. α , which does not depend on the optimised variables, avoids collisions that still can occur on the lines connecting the points. Such augmented o.f.s are discussed in (Purcaru et al., 2013a), and they ensure that the performance of the agents that have collisions with the objects is degraded.
and, as considered in (Purcaru et al., 2013b), and each agent is considered as a charged sphere of radius a with uniform volume charge density. The positions and velocities of j th agent in the search space on the X and Y axes, X j , PRk (t ) , Y j , PRk (t ) , v Xj , PRk (t ) and v Yj , PRk (t ) , are updated in terms of the recurrent equations
(Precup et al., 2014b)
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X j , PRk (t + 1) = r j1k a ( F jX,k (t ) / m j ,k (t ))(Δt ) 2 + r j 2 k v v Xj , PRk (t ) Δt + X j ,PRk (t ), Y j , PRk (t + 1) = r j1k a ( F jY,k (t ) / m j ,k (t ))(Δt ) 2
(15)
+ r j 2 k v v Yj , PRk (t ) Δt + Y j , PRk (t ), v Xj , PRk (t + 1) = ( X j , PRk (t + 1) − X j , PRk (t )) / Δt , v Yj , PRk (t + 1) = (Y j , PRk (t + 1) − Y j , PRk (t )) / Δt ,
Fig. 1. Experimental scenario for two robots and obstacles.
where k a > 0 is the acceleration parameter, k v > 0 is the velocity parameter, 0 < r j1 < 1 and 0 < r j 2 < 1 are uniformly
The optimisation problems have been solved for the weights λ1 = 0.4 , λ 2 = 0.3 and λ 3 = 0.3 , and for the penalty term α = 50 . For the sake of comparison, we have considered: (i) the GSA-based PP approach given in (Purcaru et al., 2013b), with the parameters number of agents N = 100 , maximum number of iterations t max = 30, threshold = 10, and initial velocities viX, PRk (0) = −0.2 , viY, PRk (0) = 0.2 , (ii) the PSO-based
distributed random numbers m j ,k (t ) is the mass of j th agent, m j ,k (t ) = q j ,k (t ) , and Δt is the time step set to 1.
Our adaptive CSS algorithms consist of the following stages: Stage 1. This stage initializes the adaptive CSS algorithm’s population and parameters, namely the parameters in the optimisation problems, the maximum number of iterations t max and p populations of agents.
PP approach given in (Purcaru et al., 2013b), with the parameters N = 100 , t max = 30, threshold = 10, initial velocities viX, PRk (0) = −0.2 and viY, PRk (0) = 0.2 , acceleration constants c1 = 1.49 and c2 = 1.49 , linear decrease of inertia weight w with the advance of PSO algorithm’s iterations within wmin ≤ w ≤ wmax , wmin = 0.0001 and wmax = 0.5 , and (iii) the non-adaptive CSS-based PP approach given in (Precup et al., 2014b), with the parameters N = 100 , initial velocities and t max = 30, viX, PRk (0) = −0.2
Stage 2. The algorithm discovers the extent of the search space in terms of running the search with no modifications of k a and k v , so no constraints are applied to CPs’ movements. This stage accounts for the first 20% out of t max iterations.
Stage 3. The search is run using the linear modifications of k a and k v according to (Precup et al., 2014a) k a = 3(1 − t / t max ), k v = 0.5(1 + t / t max ) .
viY, PRk (0) = 0.2 , a = 1 , ε = 0.01 , and modifications of k a and
k v with respect to the iteration index in terms of (16). For a fair comparison, the parameters of our adaptive CSS-based PP approach are identical to those of the non-adaptive CSSbased PP approach, and ε 0 = 0.01 in (17).
(16)
In addition, the parameter ε is reduced starting with the preset value ε 0 > 0 according to the law ε = ε 0 [1 − (t − 0.2t max ) /(0.8k max )] .
The analysis of the computational complexity of the PP approaches based on the number of agents has been carried out by the computation of the execution time (e.t.) versus the number of agents N. The e.t.s have been measured for all approaches on a computer with Intel I7 @ 2.2 GHz. The average results for the best five runs of the optimisation algorithms have been computed for all optimisation algorithms. The results for two robots and N = 100 are: 4.9415 s for the adaptive CSS-based PP approach, 5.895 s for the GSA-based PP approach, 1.6789 s for the PSO-based PP approach, and 2.7453 s for the non-adaptive CSS-based PP approach. The results for three robots and N = 100 are: 7.5236 s for the adaptive CSS-based PP approach, 10.423 s for the GSA-based PP approach, 3.7941 s for the PSO-based PP approach, and 5.374 s for the non-adaptive CSS-based PP approach. Therefore, the smallest e.t.s are obtained for PSO because of the reduced number of primitive operations.
(17)
The next 40% out of t max iterations are assigned to this stage. Stage 4. The last 40% of the t max iterations of the search process refine the obtained results using the last value of ε obtained during previous stage. At each run the agents’ positions with the worst fitness is reset to the agents’ positions with the best fitness. Stage 5. The CPs’ positions are mapped onto the variables of the optimisation problems, and the o.f.s are evaluated using the simulated robots in nRobotic to evaluate and validate the solutions to the optimisation problems. 4. EXPERIMENTAL VALIDATION The validation of the new PP approach has been carried out by a set of experiments conducted on the nRobotic platform using several experimental scenarios, different missions with at least two robots on holonomic wheeled platforms, different initial robots’ positions and target points and accounting for the presence of obstacles. A typical experimental scenario for two robots is illustrated in Fig. 1.
The evolutions of the o.f.s defined in (8) are illustrated in Fig. 2 for the best agents during the maximum number of iterations t max . The results correspond to one of the
experimental scenarios with two robots and to one of the populations of agents, i.e., PR1 . Fig. 2 shows that the adaptive CSS-based PP approach exhibits the fastest convergence. 297
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improvement by a reduced number of evaluations of the o.f.s using the iterative experiment-based estimation of the gradients. Sensitivity and robustness analyses will be conducted. ACKNOWLEDGEMENTS
Fig. 2. O.f.s versus iteration number for the best agents in the population PR1 considering four PP approaches. The comparisons of the PP approaches depend on the optimisation algorithms, and they can be extended in terms of different collision avoidance algorithms (Klančar and Škrjanc, 2010; Purcaru et al., 2013a). However, the conclusions will be different for other robots and process models (Bošnak et al., 2012a, 2012b; Triharminto et al., 2013; Bolla et al., 2014) because the PP approaches set the reference inputs for the control loops involved in mobile robot tracking problems.
This work was supported by grants from the Partnerships in priority areas – PN II program of the Romanian Ministry of Education and Research (MEdC) – the Executive Agency for Higher Education, Research, Development and Innovation Funding (UEFISCDI), project numbers PN-II-PT-PCCA2013-4-0544 and PN-II-PT-PCCA-2013-4-0070, by a grant from the Partnerships in priority areas – PN II program of the Romanian National Authority for Scientific Research ANCS, CNDI – UEFISCDI, project number PN-II-PT-PCCA-20113.2-0732, by a grant of the Romanian National Authority for Scientific Research, CNCS – UEFISCDI, project number PN-II-ID-PCE-2011-3-0109, and by a grant from the NSERC of Canada. REFERENCES Acosta, J., Nebot, À., Villar, P., Fuertes, J.M. (2007). Optimization of fuzzy partitions for inductive reasoning using genetic algorithms. International Journal of Systems Science, 38 (12), 991-1011. Banerjee, A.G., Chowdhury, S., Losert, W., and Gupta, S.K. (2011). Real-time path planning for coordinated transport of multiple particles using optical tweezers. IEEE Transactions on Automation Science and Engineering, 9 (4), 669-678. Bigaj, P. and Kacprzyk, J. (2013). A memetic algorithm based procedure for a global path planning of a movement constrained mobile robot. Proceedings of 2013 IEEE Congress on Evolutionary Computation, Cancun, Mexico, 135-141. Bolla, K., Johanyák, Z.C., Kovács, T., and Fazekas, G. (2014). Local center of gravity based gathering algorithm for fat robots. In Kóczy, L.T., Pozna, C.R., and Kacprzyk, J. (eds.), Issues and Challenges of Intelligent Systems and Computational Intelligence, Studies in Computational Intelligence, 530, 175-183, SpringerVerlag, Berlin, Heidelberg. Bošnak, M., Matko, D., and Blažič, S. (2012a). Quadrocopter hovering using position-estimation information from inertial sensors and a high-delay video system. Journal of Intelligent and Robotic Systems, 67 (1), 43-60. Bošnak, M., Matko, D., and Blažič, S. (2012b). Quadrocopter control using an on-board video system with off-board processing. Robotics and Autonomous Systems, 60 (4), 657-667. Filip, F.-G. and Leiviskä, K. (2009). Large-scale complex systems. In Nof, S.Y. (ed.), Springer Handbook of Automation, 619-638, Springer-Verlag, Berlin, Heidelberg. González, A., Mata, W., Villaseñor, L., Aquino, R., Simó, J.E., Chávez, M., and Crespo, A. (2011). μDDS: A middleware for real-time wireless embedded systems. Journal of Intelligent and Robotic Systems, 64 (3-4), 489-503.
The results are sensitive to the initial conditions. Different results have been obtained for other initial conditions, but the results Fig. 2 is extracted from the set of experimental results as representative results with average performance. The results are also sensitive to the weights and to the parameters in the optimisation algorithms. Therefore, natureoptimisation algorithms with a reduced parametric sensitivity should be developed. Separate experiments have also been conducted for three and four robots. As pointed out in (Purcaru et al., 2013b) for GSA and PSO, the e.t.s versus the number of iterations are increasing for the CSS and adaptive CSS algorithms. Since the algorithms include random elements, the statistical significance of the results in Fig. 2 is not established. As previously outlined, these results are given as the average results for the best five runs of the optimisation algorithms. The algorithms exhibit the same performance for the best ten runs, but more experiments (for example, 100) for the computation of confidence intervals were run as the future online implementation on the nRobotic platform using several computers is targeted. 6. CONCLUSIONS This paper has proposed a new off-line adaptive CSS-based PP approach that produces optimal collision-free paths for multiple mobile robots in static environments. Holonomic wheeled platforms have been considered. The experimental results and the comparisons with three similar approaches that use other nature-inspired optimisation algorithms show the improvement of the convergence speed. But the computational complexity is average, so the performance is overall in the average. Since the results are encouraging for being further deepened, future research will be focused on the performance 298
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Precup, R.-E., David, R.-C., Petriu, E.M., Preitl, S., and Radac, M.-B. (2014a). Novel adaptive charged system search algorithm for optimal tuning of fuzzy controllers. Expert Systems with Applications, 41 (4), 1168-1175. Precup, R.-E., Petriu, E.M., Fedorovici, L.-O., Radac, M.-B., and Dragan, F. (2014b). Multi-robot charged system search-based optimal path planning in static environments. Proceedings of 2014 IEEE MultiConference on Systems and Control, Antibes, France, 1912-1917. Precup, R.-E., Tomescu, M.-L., Radac, M.-B., Petriu, E.M., Preitl, S., and Dragos, C.-A. (2012). Iterative performance improvement of fuzzy control systems for three tank systems. Expert Systems with Applications, 39 (9), 8288-8299. Purcaru, C., Precup, R.-E., Iercan, D., Fedorovici, L.-O., David, R.-C., and Dragan, F. (2013a). Optimal robot path planning using gravitational search algorithm. International Journal of Artificial Intelligence, 10 (S13), 1-20. Purcaru, C., Precup, R.-E., Iercan, D., Fedorovici, L.-O., Petriu, E.M., and Voisan, E.-I. (2013b). Multi-robot GSA- and PSO-based optimal path planning in static environments. Proceedings of 9th International Workshop on Robot Motion and Control, Wasowo, Poland, 197-202. Roberge, V., Tarbouchi, M., and Labonte, G. (2013). Comparison of parallel genetic algorithm and particle swarm optimization for real-time UAV path planning. IEEE Transactions on Industrial Informatics, 9 (1), 132141. Ruano, A.E., Madureira, G., Barros, O., Khosravani, H.R., Graça Ruano, M., and Ferreira, P.M. (2014). Seismic detection using support vector machines. Neurocomputing, 135, 273-283. Sarker, M.O.F., Dahl, T.S., Arcaute, E., and Christensen, K. (2014). Local interactions over global broadcasts for improved task allocation in self-organized multi-robot systems. Robotics and Autonomous Systems, 62 (10), 1453-1462. Tomescu, M.-L., Preitl, S., Precup, R.-E., and Tar, J.K. (2007). Stability analysis method for fuzzy control systems dedicated controlling nonlinear processes. Acta Polytechnica Hungarica, 4 (3), 127-141. Triharminto, H.H., Adji, T.B., and Setiawan, N.A. (2013). 3D dynamic UAV path planning for interception of moving target in cluttered environment. International Journal of Artificial Intelligence, 10 (S13), 154-163. Vašcák, J. and Pal’a, M. (2012). Adaptation of fuzzy cognitive maps for navigation purposes by migration algorithms. International Journal of Artificial Intelligence, 8 (S12), 20-37. Zhang, X., Chen, J., Xin, B., and Peng, Z. (2014). A memetic algorithm for path planning of curvature-constrained UAVs performing surveillance of multiple ground targets. Chinese Journal of Aeronautics, 27 (3), 622-633.
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IFAC-PapersOnLine 48-10 (2015) Approach 294–299 Adaptive Charged System Search to Path Planning Adaptive Charged System Search Approach Path Planning forSystem Multiple Mobile Robots to Adaptive Search Approach to Adaptive Charged Charged System Search Approach to Path Path Planning Planning for Multiple Mobile Robots for Multiple Mobile Robots for Multiple Mobile Robots
Radu-Emil Precup*, Emil M. Petriu**, Mircea-Bogdan Radac***, Emil-Ioan Voisan****, Florin Dragan***** Radu-Emil Precup*, Emil M. Petriu**, Mircea-Bogdan Radac***, Emil-Ioan Voisan****, Florin Dragan***** Radu-Emil Radu-Emil Precup*, Precup*, Emil Emil M. M. Petriu**, Petriu**, Mircea-Bogdan Mircea-Bogdan Radac***, Radac***, Emil-Ioan Emil-Ioan Voisan****, Voisan****, Florin Florin Dragan***** Dragan***** * Department of Automation and Applied Informatics, Politehnica University of Timisoara, Bd. * Parvan 2, 300223 Timisoara, (Tel: +40 256 403229; e-mail: [email protected]) *V.Department Department of Automation Automation andRomania Applied Informatics, Informatics, Politehnica University of Timisoara, Timisoara, of and Applied Politehnica University of *V.Department ofofAutomation andRomania Appliedand Informatics, Politehnica University Timisoara, ** School Electrical Engineering Computer Science, University ofofOttawa, Bd. Parvan 2, 300223 Timisoara, (Tel: +40 256 403229; e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (Tel: +40 256 403229; e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (Tel: +40 256Science, 403229; e-mail: [email protected]) 800 King Edward, Ottawa, Ontario, K1N 6N5 Canada (e-mail: [email protected]) ** School of Electrical Engineering and Computer University of Ottawa, ** of Electrical Engineering and Computer Science, University of ** School School of Engineering and Computer Science, University of Ottawa, Ottawa, *** Department of Electrical Automation and Applied Informatics, Politehnica University of Timisoara, 800 King Edward, Ottawa, Ontario, K1N 6N5 Canada (e-mail: [email protected]) 800 King Edward, Ottawa, Ontario, K1N 6N5 Canada (e-mail: [email protected]) 800 King Edward, Ottawa, Ontario, K1N 6N5 Canada (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) *** Department of Automation and Applied Informatics, Politehnica University of Timisoara, *** of Automation and Applied Informatics, Politehnica University of Timisoara, *** Department Department of Automation and Applied Informatics, Politehnica University of Timisoara, **** Department of Automation and Applied Informatics, Politehnica University of Timisoara, Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) **** Department of Automation and Applied Informatics, Politehnica University of Timisoara, **** of Automation and Applied Informatics, Politehnica University of Timisoara, **** Department Department of and Applied Politehnica University ***** Department ofAutomation Automation and AppliedInformatics, Informatics, Politehnica Universityof ofTimisoara, Timisoara, Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) ***** Department of Automation and Applied Informatics, Politehnica University of Timisoara, ***** of and Applied Informatics, Politehnica University ***** Department Department of Automation Automation and AppliedRomania Informatics, Politehnica University of of Timisoara, Timisoara, Bd. V. Parvan 2, 300223 Timisoara, (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected]) Bd. V. Parvan 2, 300223 Timisoara, Romania (e-mail: [email protected])
Abstract: This paper suggests the application of adaptive Charged System Search (CSS) algorithms to the optimalThis pathpaper planning (PP) of multiple mobile robots. AnCharged off-lineSystem adaptiveSearch CSS-based PP approach to is Abstract: suggests the application of adaptive (CSS) algorithms Abstract: This paper suggests the application of adaptive System Search (CSS) algorithms to Abstract: This paper suggests themultiple application ofplatforms adaptiveAnCharged Charged System Search (CSS) algorithms to proposed and applied to holonomic wheeled in static environments. The adaptive CSS the optimal path planning (PP) of mobile robots. off-line adaptive CSS-based PP approach is the optimal path planning (PP) of mobile robots. An off-line adaptive CSS-based PP approach is the optimaland pathapplied planning (PP) of multiple multiple mobile robots. Anin off-line adaptive CSS-based PP approach is algorithms solve the optimisation problems thatplatforms aim the minimisation of objective functions (o.f.s) proposed to holonomic wheeled static environments. The adaptive CSS proposed and applied to holonomic wheeled platforms in static environments. The adaptive CSS proposed and applied to holonomic wheeled platforms in static environments. The adaptive CSS specific to PP andthe expressed as the weighted sum of four functions that target separate PP objectives. A algorithms solve optimisation problems that aim the minimisation of objective functions (o.f.s) algorithms solve the optimisation problems that aim the minimisation of objective functions (o.f.s) algorithms solve the optimisation problems that aim the minimisation of The objective functions (o.f.s) penalty term is added in certain situations in the first step of the PP approach. specific features of the specific to PP and expressed as the weighted sum of four functions that target separate PP objectives. A specific to PP expressed as the weighted sum of four functions that target separate PP objectives. A specific toCSS PPisand and expressed sum ofthe four functions that targetThe separate PPfeatures objectives. A adaptiveterm algorithms areas situations the weighted adaptation of acceleration, velocity, and separation distance penalty added in certain in the first step of the PP approach. specific of the penalty term is added in certain situations in the first step of the PP approach. The specific features of the penalty term is added in certain situations in the first step of the PP approach. The specific features of the parameters to the iteration index, and the substitution of the worst charged particles’ fitness function adaptive CSS are the adaptation of the acceleration, velocity, and separation distance adaptive CSS algorithms algorithms are the adaptation of the velocity, and distance adaptive algorithms are the and adaptation of the acceleration, acceleration, velocity, and inseparation separation distance values andCSS positions with the best performing particle data. The fitness function thefitness adaptive CSS parameters to the iteration index, the substitution of the worst charged particles’ function parameters to the iteration index, and the substitution of the worst charged particles’ fitness function parameters to the iteration index, and the substitution of theThe worst charged particles’ fitness function algorithms corresponds to o.f., and the search space and agents (charged particles) in the adaptive values and positions with the best performing particle data. fitness function in the adaptive CSS values and positions with the best performing particle data. The fitness function in adaptive CSS values and corresponds positions with the bestsolution performing particle data. The fitness function in the the adaptive CSS CSS algorithms correspond to the space andspace to the mobile robots, respectively. Ain case study and algorithms to the o.f., and the search and agents (charged particles) the adaptive algorithms corresponds to the o.f., and the search space and agents (charged particles) in the adaptive algorithms corresponds tovalidate the o.f.,solution and thespace search space and agents (charged particles) in the adaptive experiments are included the new adaptive CSS-based PP approach and to compare it with nonCSS algorithms correspond to the and to the mobile robots, respectively. A case study and CSS algorithms correspond to solution space and to robots, respectively. A case study and CSS algorithms correspond to the the solution space to the the mobile mobile robots, respectively. A studynonand adaptive CSS-, Particle Swarm Optimizationandand Gravitational Search Algorithm-based PPcase approaches. experiments are included validate the new adaptive CSS-based PP approach and to compare it with experiments are included validate the new adaptive CSS-based PP approach and to compare it with nonexperiments are included validate the new adaptive CSS-based PP approach and to compare it with nonadaptive CSS-, Particle Swarm Optimizationand Gravitational Search Algorithm-based PP approaches. adaptive CSS-, Particle Swarm Optimizationand Search PP © 2015, IFAC (International Federation of Automatic Control) Hosting by Algorithm-based Elsevier All rights reserved. Keywords: Adaptive System Search algorithms, mobile robots,Ltd. obstacles, optimisation adaptive CSS-, Particle Charged Swarm Optimizationand Gravitational Gravitational Search Algorithm-based PP approaches. approaches. problems, path planning, penalty term. Keywords: Adaptive Charged System Search algorithms, mobile robots, obstacles, optimisation Keywords: Adaptive Charged System Keywords:path Adaptive Charged System Search Search algorithms, algorithms, mobile mobile robots, robots, obstacles, obstacles, optimisation optimisation problems, planning, penalty term. problems, path planning, penalty term. problems, path planning, penalty term. The nature-inspired optimisation algorithms produce 1. INTRODUCTION convenient versions of optimisation centralised algorithms to solve PP The nature-inspired algorithms produce The nature-inspired optimisation algorithms produce 1. INTRODUCTION INTRODUCTION 1. The nature-inspired optimisation algorithms produce optimisation problems. The current approaches to optimal Centralised and decentralised algorithms solve path planning convenient versions of centralised algorithms to solve PP 1. INTRODUCTION convenient versions of centralised algorithms to solve PP convenient versions ofThe centralised algorithms tooptimal solve PP that includes nature-inspired algorithms use Particle Swarm (PP) problems for multiple robots. The decentralised optimisation problems. current approaches to PP Centralised and decentralised algorithms solve path planning optimisation problems. The current approaches to optimal PP Centralised and decentralised algorithms solve path planning optimisation problems. The current approaches to optimal PP Centralised and decentralised algorithms solve path planning Optimization (PSO) (Masehian and Sedighizadeh, 2010; algorithms generate independently collision-free paths for that includes nature-inspired algorithms use Particle Swarm (PP) problems for multiple robots. The decentralised that includes nature-inspired algorithms use Particle Swarm (PP) problems for multiple robots. The decentralised that includes nature-inspired algorithms use Particle Swarm (PP) problems for multiple robots. The decentralised Purcaru et al., (PSO) 2013b;(Masehian Ma et al., and 2014), genetic algorithms each robot to reach the target solvingcollision-free the possible paths collisions Optimization Sedighizadeh, 2010; algorithms generate independently for Optimization (PSO) (Masehian and Sedighizadeh, 2010; algorithms generate independently collision-free paths for (PSO) (Masehian and Sedighizadeh, 2010; algorithms generate independently collision-free paths for Optimization (Roberge et al., 2013; Lim et al., 2014), Gravitational Search between robots. Some popular decentralised algorithms are Purcaru et al., 2013b; Ma et al., 2014), genetic algorithms each robot to reach the target solving the possible collisions Purcaru et al., 2013b; Ma et al., 2014), genetic algorithms each robot to reach the target solving the possible collisions Purcaru et al., 2013b; Ma et al., 2014), genetic algorithms each robot to reach the target solving the possible collisions Algorithms (GSAs) (Purcaru et al., 2013a, 2013b), migration based onrobots. partially observable Markov decision processes (Roberge et al., 2013; Lim et al., 2014), Gravitational Search between Some popular decentralised algorithms are (Roberge et al., Lim et al., 2014), Gravitational Search between robots. Some popular decentralised algorithms are (Roberge et (Vašcák al., 2013; 2013; LimPal’a, et et al.,al., 2014), Gravitational Search between robots. Some popular decentralised algorithms are algorithms and 2012), memetic algorithms (Banerjee et al., 2011) and on the environment-based Algorithms (GSAs) (Purcaru 2013a, 2013b), migration based on partially observable Markov decision processes Algorithms (GSAs) (Purcaru et al., 2013a, 2013b), migration based on partially observable Markov decision processes Algorithms (GSAs) (Purcaru et al., 2013a, 2013b), migration based on partially observable Markov decision processes (Bigaj and Kacprzyk, 2013; Zhang et al., 2014), simulated adaptation and optimisation (Kleiner et al., 2011). The belief algorithms (Vašcák and 2012), memetic algorithms (Banerjee et al., 2011) and on the environment-based algorithms (Vašcák and Pal’a, Pal’a, 2012), memetic algorithms (Banerjee et al., 2011) and on environment-based algorithms (Vašcák Pal’a, 2012), memetic (Banerjee et decentralised al., 2011) for and on the the environment-based annealing (Miao and and Tian, 2013; Maet etal., al.,2014), 2014),algorithms chemical functions are tracking in (Kim et The al., 2013). (Bigaj and Kacprzyk, 2013; Zhang simulated adaptation and optimisation (Kleiner et al., 2011). belief (Bigaj and Kacprzyk, 2013; Zhang et al., 2014), simulated adaptation and optimisation (Kleiner et al., 2011). The belief (Bigaj and(Miao Kacprzyk, 2013; Zhang et al., 2014), simulated adaptation anddecentralised optimisation (Kleiner et in al.,curves 2011). The belief optimisation (Melin et al., 2013), and Charged System Search Genetic algorithms combined with Bézier are used by annealing and Tian, 2013; Ma et al., 2014), chemical functions are for tracking (Kim et al., 2013). annealing (Miao and Tian, 2013; Ma et chemical functions are for tracking in (Kim al., 2013). annealing (Miao and Tian, 2013; MaCharged et al., al., 2014), 2014), chemical functions are decentralised decentralised for tracking in curves (Kim et etare al.,used 2013). (CSS) algorithms (Precup et al., 2014b). Kala and Warwick (2014) in path generation. optimisation (Melin et al., 2013), and System Search Genetic algorithms combined with Bézier by optimisation (Melin et 2013), and Genetic algorithms combined with Bézier (Melin et al., al., et 2013), and Charged Charged System System Search Search Genetic algorithms combined withgeneration. Bézier curves curves are are used used by by optimisation (CSS) algorithms (Precup al., 2014b). Kala and Warwick (2014) in path (Precup et Kala and (2014) path Using algorithms our recently proposed adaptive CSS algorithms The centralised algorithms the individual robots as (CSS) (CSS) algorithms (Precup et al., al., 2014b). 2014b). Kala and Warwick Warwick (2014) in inconsider path generation. generation. (Precup et al., 2014a),proposed this paperadaptive suggestsCSS a new off-line subsystems to algorithms allow for consider global optimisation. Successful recently algorithms The centralised the individual robots as Using Using our our recently proposed adaptive CSS algorithms The centralised algorithms consider the individual robots our recently proposed adaptive CSS algorithms The centralised algorithms consider the individual robots as as Using adaptive CSS-based approach that generates optimal paths for examples of decentralised algorithms are the sampling-based (Precup et al., 2014a), this paper suggests a new off-line subsystems to allow for global optimisation. Successful (Precup et al., this suggests aa new off-line subsystems to allow global optimisation. Successful (Precup et al., 2014a), 2014a), this paper paper suggests new off-line subsystems to allow for for global optimisation. Successful multiple robots. The adaptive CSS algorithms are mapped PP algorithms PRM* and RRT* (Karaman and Frazzoli, adaptive CSS-based approach that generates optimal paths for examples of decentralised algorithms are the sampling-based adaptive CSS-based approach that generates optimal paths for examples of decentralised algorithms are adaptive CSS-based approach that generates optimal paths for examples ofalgorithm decentralised algorithms are the the sampling-based sampling-based onto the optimisation problems by means of several elements: 2011), the based sets of non-colliding paths on multiple robots. The adaptive CSS algorithms are mapped PP algorithms PRM* and RRT* (Karaman and Frazzoli, multiple robots. The adaptive CSS algorithms are mapped PP algorithms PRM* and RRT* (Karaman and Frazzoli, multiple robots. The adaptive CSS algorithms are mapped PP algorithms PRM* and RRT* (Karaman and Frazzoli, the fitness functions are the objective functions (o.f.s), the graphs (Luna and Bekris, 2011), and the algorithm that onto the optimisation problems by means of several elements: 2011), the the algorithm algorithm based based sets sets of of non-colliding non-colliding paths paths on on onto the optimisation problems by means of several elements: 2011), ontofitness thespare optimisation by meansthe of several elements: 2011), the algorithm based sets oftask-allocation non-colliding pathsthat on search is the problems solution space, agents(o.f.s), (charged involves self-organized multi-robot (Sarker et the functions are the objective functions the graphs (Luna and Bekris, 2011), and the algorithm the fitness functions are the functions (o.f.s), the graphs (Luna and Bekris, 2011), and the algorithm that the fitness functions are the objective objective functions (o.f.s), the graphs (Luna and Bekris, 2011), task-allocation and the algorithm that particles) are the mobile robots, and the population of agents al., 2014). search spare is the solution space, the agents (charged involves self-organized multi-robot (Sarker et search spare is the solution space, the agents (charged involves self-organized multi-robot task-allocation (Sarker et search spare is the solution space, the agents (charged involves self-organized multi-robot task-allocation (Sarker et is the set of mobile robots.robots, particles) are the mobile and the population of agents al., 2014). particles) are the robots, al., particles) aremobile the mobile mobile robots, and and the the population population of of agents agents al., 2014). 2014). is the set of robots. is the set of mobile robots. is the set of mobile robots. Copyright © 2015 IFAC 294
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We are using the notation t for both the algorithm iteration and the time along the planned trajectory in order to reduce the number of indices. We are also using the general notation || U − V ||2 for the Euclidean distance between the vectors U = (U X ,U Y ) ∈ R 2 and V = (V X ,VY ) ∈ R 2 :
This paper considers four PP objectives (Purcaru et al., 2013b; Precup et al., 2014a), aggregated in a set of separate o.f.s, and the optimisation problems aim the minimisation of the scalar o.f.s expressed as their weighted sum. An additional penalty term is added to the o.f.s in certain situations in the first step of the PP approach in order to avoid the robot positions to be inside the obstacles or the line that connects two consecutive robot positions to intersect obstacles.
|| U − V ||2 = (U X − V X ) 2 + (U Y − VY ) 2 .
time moments or iterations allowed for i th agent in the population PRk . But it is needed to eliminate all agents’ paths
that are not collision-free, namely we eliminate the path point sets Ti , PRk ,t ( i ,k ) , i ∈ S cf , PRk ⊂ {1,2,..., N } , where S cf , PRk is the max
set of agents which lead to collision-free paths for the population PRk . The sets Ti , PRk ,t ( i ,k ) actually correspond to max
separate collision-free paths for each robot Rk , i.e., to feasible paths. The total length of each path is Di ,k =
This paper is organised as follows: the PP problem is formulated in the next section. The new off-line adaptive CSS-based optimal PP approach is presented in Section 3. The approach is validated in Section 4 by experimental results and by the comparison with similar approaches. The concluding remarks are pointed out in Section 5.
tmax ( i ,k )
∑ || X t =1
i , PRk
(t ) − X i , PRk (t − 1) ||2 , i ∈ S cf ,PRk .
(4)
The optimisation problems that give the indices iˆk of the agents which correspond to the minimum length feasible paths are (Purcaru et al., 2013b) iˆk = arg min Di ,k ,
(5)
i∈Scf , PRk
2. PATH PLANNING PROBLEM
and the optimal paths are Tiˆ ,t ( iˆ ,k ) . The elements of the path max k
As considered in (Purcaru et al., 2013b; Precup et al., 2014b), let {Rk | k = 1... p} be a set of p mobile robots in a known
point sets Ti , PRk ,t , defined in (2) and used in the o.f.s (4), are computed as solutions to the following optimisation problems for each robot Rk , k = 1... p :
static environment that includes obstacles. The path for each robot is generated by a different population with the same number of N agents. The individual populations of agents that generate the paths for the robots R1 ...R p are PR1...PRp . The
Xi , PRk (t ) = arg min f i , PRk ( X i , PRk (t )), t = 1...t max (i, k ),
(6)
Xi , PRk ( t )
solution space of the mobile robot is a two-dimensional search space, and all agents in a population are placed in the initial robot’s position for which they need to generate the path. Using the notation X i , PRk (t ) for the agent vector
i = 1...N , k = 1... p,
the first element in Ti , PRk ,t , i.e., X i ,PRk (0) , is known: (7)
Xi , PRk (0) = X i , PRk (0), i = 1...N , k = 1... p,
position that stands for the current position of i th agent, i = 1...N , in the population PRk , k = 1... p, in the search space at the time moment (iteration) t X i , PRk (t ) = ( X i ,PRk (t ), Yi , PRk (t )) ∈ R ,
(3)
As shown in (Purcaru et al., 2013b), each agent’s path is built by connecting with lines all successive points of the Ti , PRk ,tmax ( i ,k ) sets, where t max (i, k ) is the maximum number of
Our PP approach is advantageous in the context of the stateof-the-art because it can be easily generalised to insert other nature-inspired optimisation algorithms. This is emphasized by the comparison of our approach with our previous nonadaptive CSS-, PSO- and GSA-based PP approaches by experiments conducted on the nRobotic platform developed at the Politehnica University of Timisoara, Romania. The difference from the previous paper (Precup et al., 2004b), which already applies CSS to multiple robot PP, and therefore the novelty of the contribution, is the adaptive CSS algorithm that replaces the non-adaptive CSS algorithm in the PP approach. The reader is invited to use (Precup et al., 2014a) for additional details on CSS algorithms.
2
295
and the expressions of the o.f.s are f i , PRk ( X i , PRk (t )) =|| X f − X i , PRk (t ) || 2 + λ 1 /[
(1)
where R is the set of real numbers, X i , PRk (t ) is agent’s
− X i , PRk (t ) || 2 ] + λ 2 /[
position on the X axis and Yi , PRk (t ) is agent’s position on the Y axis, the paths are generated iteratively by means of path point sets Ti , PRk ,t , which are updated at each iteration adding
+ λ 3 /[
the current agent position: Ti , PPRk ,t = {X i , PRk (0), X i , PRk (1),..., X i , PRk (t )}, i = 1...N , k = 1... p, (2)
where X i , PRk (0) = ( X i , PRk (0), Yi , PRk (0)) is the initial position of the agent i in the population PRk . 295
p
∑|Y
j =1, j ≠ k
best , PRj
p
∑| X
j =1, j ≠ k
best , PRj
p
∑ || X
j =1, j ≠ i
best , PRj
(t ) − X i , PRk (t ) |]
(t )
(8)
(t ) − Yi , PRk (t ) |], i = 1...N , k = 1... p.
X f = ( X f , Y f ) ∈ R 2 in (8) is the target (final) point, it is the
same for all agents, X best , PRj (t ) = ( X best ,PRj (t ), Ybest , PRj (t )) ∈ R 2 is the position of the best agent (in terms of the distance between the current point and the target point) in the
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As shown in (Precup et al., 2014a), the relation between the fitness functions g i ,k (t ) in the adaptive CSS algorithm and
population PRj at the time moment t, λ1 > 0 , λ 2 > 0 and λ 3 > 0 the weights. The first term in the o.f.s f i , PRk ( X i , PRk (t ))
the o.f.s f i , PRk ( X i , PRk (t )) is
aims the minimisation of the Euclidian distance between each agent position from the individual populations at a specific time (iteration) t and X f . The second term in f i , PRk ( X i , PRk (t )) aims the maximisation of the distance between the agents of a population and the best agents from all other populations in order to avoid to deal with many collision points between the paths generated by the agents. The third and fourth terms target the maximisation of the distance between the paths of the robots on the X and Y axes and model applications with one axis more important than the other one.
g i ,k (t ) = f i , PRk ( X i , PRk (t )), i = 1...N , k = 1... p ,
where N is the total number of agents, referred to also as charged particles (CPs) specific to the adaptive CSS algorithm. Agents’ performance in the search space is given by their electric charges qi ,k (t ) qi ,k (t ) = [ g i ,k (t ) − best k (t )] /[best k (t ) − worst k (t )], i = 1... N , k = 1... p,
Using our previous results on GSA, PSO and CSS algorithms applied in (Purcaru et al., 2013b; Precup et al., 2014b) to solve the optimisation problems (6) with the o.f.s (8), the next section will give another PP approach based on adaptive CSS algorithms. Other optimisation problems can be defined as well (Acosta et al., 2007; Filip and Leiviskä, 2009; Petelin et al., 2011 Ruano et al., 2014), but they should account for constraints specific to robots in terms of nonlinear models, which describe the environment at certain accuracy (Tomescu et al., 2007; Guerra et al., 2009; González et al., 2011; Precup et al., 2012).
(10)
where the so far best and the worst fitness of all agents in the population PRk are worst k (t ) = max g i ,k (t ), k = 1... p, i =1... N
best k (t ) = min g i ,k (t ), k = 1... p.
(11)
i =1... N
The separation distance between i th and j th agent is rij ,k (t ) = (|| X i , PRk (t ) ||2 − || X j , PRk (t ) ||2 ) /[|| 0.5( X i , PRk (t ) + X j , PRk (t )) − X best , PRk (t ) ||2 + ε],
(12)
where X best ,PRk (t ) is the position of the best current agent,
3. OFF-LINE ADAPTIVE CHARGED SYSTEM SEARCHBASED PATH PLANNING APPROACH
and the relatively small parameter ε > 0 is introduced to avoid the division by zero. The total forces on the X and Y axes, F jX,k and F jY,k , that act on j th agent at the time (iteration) t are (Precup et al., 2014b)
Our off-line adaptive CSS-based optimal PP approach consists of the following steps: Step PP1. Apply the adaptive CSS algorithms such that to solve the optimisation problems defined in (6) with the o.f.s defined in (8) to get the solutions Xi , PRk (t ) .
F jX,k (t ) = q j ,k (t )
N
∑[q
i =1,i ≠ j
i ,k
(t )cij ,k (t )(rij ,k (t )i1 / a 3
+ i2 / rij2,k (t ))( X i , PRk (t ) − X j , PRk (t ))],
Step PP2. Use the solutions Xi , PRk (t ) obtained in the step PP1 and equation (7) as the vector elements of the path point sets Ti , PRk ,t in (2), i.e., with the notation X i , PRk (t ) for Xi , PRk (t ) .
F jY,k (t ) = q j ,k (t )
N
∑[q
i =1,i ≠ j
i ,k
(13)
(t )cij ,k (t )(rij ,k (t )i1 / a 3
+ i2 / rij2,k (t ))(Yi , PRk (t ) − Y j , PRk (t ))], j = 1...N , k = 1... p,
Step PP3. Solve the optimisation problems defined in (5) with the o.f.s (8) to get the optimal paths Tiˆ ,t ( iˆ ,k ) . max
(9)
where the parameters that outline “good” and “bad” agents (from the fitness function point of view) are
k
Since the steps PP2 and PP3 are very simple because they actually compute the minimum length paths from sets of already computed feasible paths, we will present as follows details on the step PP1. A fifth term is added in the o.f.s f i , PRk ( X i , PRk (t )) in (8), namely the penalty term α > 0 , only
⎧− 1, if g i , k (t ) < g j , k (t ), c ij , k (t ) = ⎨ ⎩ 1, otherwise, i1 = 0, i 2 = 1 for rij , k (t ) ≥ a, i1 = 1, i 2 = 0 for rij , k (t ) < a,
(14)
i, j = 1...N , k = 1... p,
in certain situations that should be avoided. Such situations occur when the current i th agent position at iteration t is inside one of the obstacles or the line that connects the position at iteration t with the position at iteration t − 1 intersects an obstacle. α , which does not depend on the optimised variables, avoids collisions that still can occur on the lines connecting the points. Such augmented o.f.s are discussed in (Purcaru et al., 2013a), and they ensure that the performance of the agents that have collisions with the objects is degraded.
and, as considered in (Purcaru et al., 2013b), and each agent is considered as a charged sphere of radius a with uniform volume charge density. The positions and velocities of j th agent in the search space on the X and Y axes, X j , PRk (t ) , Y j , PRk (t ) , v Xj , PRk (t ) and v Yj , PRk (t ) , are updated in terms of the recurrent equations
(Precup et al., 2014b)
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297
X j , PRk (t + 1) = r j1k a ( F jX,k (t ) / m j ,k (t ))(Δt ) 2 + r j 2 k v v Xj , PRk (t ) Δt + X j ,PRk (t ), Y j , PRk (t + 1) = r j1k a ( F jY,k (t ) / m j ,k (t ))(Δt ) 2
(15)
+ r j 2 k v v Yj , PRk (t ) Δt + Y j , PRk (t ), v Xj , PRk (t + 1) = ( X j , PRk (t + 1) − X j , PRk (t )) / Δt , v Yj , PRk (t + 1) = (Y j , PRk (t + 1) − Y j , PRk (t )) / Δt ,
Fig. 1. Experimental scenario for two robots and obstacles.
where k a > 0 is the acceleration parameter, k v > 0 is the velocity parameter, 0 < r j1 < 1 and 0 < r j 2 < 1 are uniformly
The optimisation problems have been solved for the weights λ1 = 0.4 , λ 2 = 0.3 and λ 3 = 0.3 , and for the penalty term α = 50 . For the sake of comparison, we have considered: (i) the GSA-based PP approach given in (Purcaru et al., 2013b), with the parameters number of agents N = 100 , maximum number of iterations t max = 30, threshold = 10, and initial velocities viX, PRk (0) = −0.2 , viY, PRk (0) = 0.2 , (ii) the PSO-based
distributed random numbers m j ,k (t ) is the mass of j th agent, m j ,k (t ) = q j ,k (t ) , and Δt is the time step set to 1.
Our adaptive CSS algorithms consist of the following stages: Stage 1. This stage initializes the adaptive CSS algorithm’s population and parameters, namely the parameters in the optimisation problems, the maximum number of iterations t max and p populations of agents.
PP approach given in (Purcaru et al., 2013b), with the parameters N = 100 , t max = 30, threshold = 10, initial velocities viX, PRk (0) = −0.2 and viY, PRk (0) = 0.2 , acceleration constants c1 = 1.49 and c2 = 1.49 , linear decrease of inertia weight w with the advance of PSO algorithm’s iterations within wmin ≤ w ≤ wmax , wmin = 0.0001 and wmax = 0.5 , and (iii) the non-adaptive CSS-based PP approach given in (Precup et al., 2014b), with the parameters N = 100 , initial velocities and t max = 30, viX, PRk (0) = −0.2
Stage 2. The algorithm discovers the extent of the search space in terms of running the search with no modifications of k a and k v , so no constraints are applied to CPs’ movements. This stage accounts for the first 20% out of t max iterations.
Stage 3. The search is run using the linear modifications of k a and k v according to (Precup et al., 2014a) k a = 3(1 − t / t max ), k v = 0.5(1 + t / t max ) .
viY, PRk (0) = 0.2 , a = 1 , ε = 0.01 , and modifications of k a and
k v with respect to the iteration index in terms of (16). For a fair comparison, the parameters of our adaptive CSS-based PP approach are identical to those of the non-adaptive CSSbased PP approach, and ε 0 = 0.01 in (17).
(16)
In addition, the parameter ε is reduced starting with the preset value ε 0 > 0 according to the law ε = ε 0 [1 − (t − 0.2t max ) /(0.8k max )] .
The analysis of the computational complexity of the PP approaches based on the number of agents has been carried out by the computation of the execution time (e.t.) versus the number of agents N. The e.t.s have been measured for all approaches on a computer with Intel I7 @ 2.2 GHz. The average results for the best five runs of the optimisation algorithms have been computed for all optimisation algorithms. The results for two robots and N = 100 are: 4.9415 s for the adaptive CSS-based PP approach, 5.895 s for the GSA-based PP approach, 1.6789 s for the PSO-based PP approach, and 2.7453 s for the non-adaptive CSS-based PP approach. The results for three robots and N = 100 are: 7.5236 s for the adaptive CSS-based PP approach, 10.423 s for the GSA-based PP approach, 3.7941 s for the PSO-based PP approach, and 5.374 s for the non-adaptive CSS-based PP approach. Therefore, the smallest e.t.s are obtained for PSO because of the reduced number of primitive operations.
(17)
The next 40% out of t max iterations are assigned to this stage. Stage 4. The last 40% of the t max iterations of the search process refine the obtained results using the last value of ε obtained during previous stage. At each run the agents’ positions with the worst fitness is reset to the agents’ positions with the best fitness. Stage 5. The CPs’ positions are mapped onto the variables of the optimisation problems, and the o.f.s are evaluated using the simulated robots in nRobotic to evaluate and validate the solutions to the optimisation problems. 4. EXPERIMENTAL VALIDATION The validation of the new PP approach has been carried out by a set of experiments conducted on the nRobotic platform using several experimental scenarios, different missions with at least two robots on holonomic wheeled platforms, different initial robots’ positions and target points and accounting for the presence of obstacles. A typical experimental scenario for two robots is illustrated in Fig. 1.
The evolutions of the o.f.s defined in (8) are illustrated in Fig. 2 for the best agents during the maximum number of iterations t max . The results correspond to one of the
experimental scenarios with two robots and to one of the populations of agents, i.e., PR1 . Fig. 2 shows that the adaptive CSS-based PP approach exhibits the fastest convergence. 297
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improvement by a reduced number of evaluations of the o.f.s using the iterative experiment-based estimation of the gradients. Sensitivity and robustness analyses will be conducted. ACKNOWLEDGEMENTS
Fig. 2. O.f.s versus iteration number for the best agents in the population PR1 considering four PP approaches. The comparisons of the PP approaches depend on the optimisation algorithms, and they can be extended in terms of different collision avoidance algorithms (Klančar and Škrjanc, 2010; Purcaru et al., 2013a). However, the conclusions will be different for other robots and process models (Bošnak et al., 2012a, 2012b; Triharminto et al., 2013; Bolla et al., 2014) because the PP approaches set the reference inputs for the control loops involved in mobile robot tracking problems.
This work was supported by grants from the Partnerships in priority areas – PN II program of the Romanian Ministry of Education and Research (MEdC) – the Executive Agency for Higher Education, Research, Development and Innovation Funding (UEFISCDI), project numbers PN-II-PT-PCCA2013-4-0544 and PN-II-PT-PCCA-2013-4-0070, by a grant from the Partnerships in priority areas – PN II program of the Romanian National Authority for Scientific Research ANCS, CNDI – UEFISCDI, project number PN-II-PT-PCCA-20113.2-0732, by a grant of the Romanian National Authority for Scientific Research, CNCS – UEFISCDI, project number PN-II-ID-PCE-2011-3-0109, and by a grant from the NSERC of Canada. REFERENCES Acosta, J., Nebot, À., Villar, P., Fuertes, J.M. (2007). Optimization of fuzzy partitions for inductive reasoning using genetic algorithms. International Journal of Systems Science, 38 (12), 991-1011. Banerjee, A.G., Chowdhury, S., Losert, W., and Gupta, S.K. (2011). Real-time path planning for coordinated transport of multiple particles using optical tweezers. IEEE Transactions on Automation Science and Engineering, 9 (4), 669-678. Bigaj, P. and Kacprzyk, J. (2013). A memetic algorithm based procedure for a global path planning of a movement constrained mobile robot. Proceedings of 2013 IEEE Congress on Evolutionary Computation, Cancun, Mexico, 135-141. Bolla, K., Johanyák, Z.C., Kovács, T., and Fazekas, G. (2014). Local center of gravity based gathering algorithm for fat robots. In Kóczy, L.T., Pozna, C.R., and Kacprzyk, J. (eds.), Issues and Challenges of Intelligent Systems and Computational Intelligence, Studies in Computational Intelligence, 530, 175-183, SpringerVerlag, Berlin, Heidelberg. Bošnak, M., Matko, D., and Blažič, S. (2012a). Quadrocopter hovering using position-estimation information from inertial sensors and a high-delay video system. Journal of Intelligent and Robotic Systems, 67 (1), 43-60. Bošnak, M., Matko, D., and Blažič, S. (2012b). Quadrocopter control using an on-board video system with off-board processing. Robotics and Autonomous Systems, 60 (4), 657-667. Filip, F.-G. and Leiviskä, K. (2009). Large-scale complex systems. In Nof, S.Y. (ed.), Springer Handbook of Automation, 619-638, Springer-Verlag, Berlin, Heidelberg. González, A., Mata, W., Villaseñor, L., Aquino, R., Simó, J.E., Chávez, M., and Crespo, A. (2011). μDDS: A middleware for real-time wireless embedded systems. Journal of Intelligent and Robotic Systems, 64 (3-4), 489-503.
The results are sensitive to the initial conditions. Different results have been obtained for other initial conditions, but the results Fig. 2 is extracted from the set of experimental results as representative results with average performance. The results are also sensitive to the weights and to the parameters in the optimisation algorithms. Therefore, natureoptimisation algorithms with a reduced parametric sensitivity should be developed. Separate experiments have also been conducted for three and four robots. As pointed out in (Purcaru et al., 2013b) for GSA and PSO, the e.t.s versus the number of iterations are increasing for the CSS and adaptive CSS algorithms. Since the algorithms include random elements, the statistical significance of the results in Fig. 2 is not established. As previously outlined, these results are given as the average results for the best five runs of the optimisation algorithms. The algorithms exhibit the same performance for the best ten runs, but more experiments (for example, 100) for the computation of confidence intervals were run as the future online implementation on the nRobotic platform using several computers is targeted. 6. CONCLUSIONS This paper has proposed a new off-line adaptive CSS-based PP approach that produces optimal collision-free paths for multiple mobile robots in static environments. Holonomic wheeled platforms have been considered. The experimental results and the comparisons with three similar approaches that use other nature-inspired optimisation algorithms show the improvement of the convergence speed. But the computational complexity is average, so the performance is overall in the average. Since the results are encouraging for being further deepened, future research will be focused on the performance 298
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