Bio-inspired Optimization of Fuzzy Logic Controllers for Autonomous Mobile Robots Ricardo Martinez-Soto Luis T. Aguilar Ieroham S. Baruch and Oscar Castillo Instituto Politécnico Nacional Department of Automatic Control Tijuana Institute of Technology Ave. del Parque 1310, Mesa de Otay CINVESTAV-IPN Tijuana, México Tijuana 22510 07360 Mexico City, Mexico [email protected], Tijuana, México [email protected] [email protected] [email protected]

Abstract—In this paper we propose the use of a hybrid PSO-GA autonomous mobile robots. We expect better results from the optimization method for automatic design of fuzzy logic hybrid PSO-GA method than those traditional methods like controllers (FLC). The optimal fuzzy logic controllers are used GA and PSO. Also we compare the proposed method with for the trajectory tracking control of autonomous mobile robots. genetic algorithms and particle swarm optimization used The bio-inspired and the evolutionary methods are used to find separately. The bio-inspired methods are used to find the the parameters of the membership functions of the FLC to obtain the optimal controller. Simulation results are obtained with parameters of the membership functions to obtain the optimal Simulink showing the feasibility of the proposed approach. A FLC to obtain a stable trajectory of the autonomous mobile comparison is also made among the proposed Hybrid PSO-GA, robot. GA and PSO to determine if there is a significant difference in the results. Related works. Up to date there are several research papers using genetic algorithms and particle swarm optimization for Keywords: Fuzzy Logic Controllers, Genetic Algorithms, different optimization problems like mathematical functions Particle Swarm Optimization, Autonomous Mobile Robot. [1], [13], [28], fuzzy controllers [17], control parameters [27], the vehicle trajectories [20], scheduling problems [24] and I. INTRODUCTION many more. However, in this paper we used the hybrid PSO- Optimization is a term used to refer to a branch of GA to generate an optimal fuzzy logic controller finding the computational science concerned with finding the “best” best parameters of the membership functions of the trajectory solution to a problem. Here, “best” refers to an acceptable (or tracking control that can be more robust than using the PSO or satisfactory) solution, which may be the absolute best over a GA individually matter. For example, Valdez et al. in [28] set of candidate solutions, or any of the candidate solutions. uses fuzzy logic to integrate the results obtained by PSO and The characteristics and requirements of the problem determine GA applied to benchmark functions. Muhammad et al. in [23] whether the overall best solution can be found [9]. proposed a hybrid method that optimizes the rule base of the Optimization algorithms are search methods, where the goal is k-means method that is used to detect an impostor in a call on to find a solution to an optimization problem, such that a given the Smartphone’s. Marinakis et al. [20] proposed a hybrid quantity is optimized, possibly subject to a set of constraints Genetic-PSO to help in the offspring of the individuals to be [9],[21],[26]. Some optimization methods are based on transmitted to all the population to get the best solution that in populations of solutions [26]. Unlike the classic methods of this case was applied to vehicle routing. Niu et al. [24] improvement for trajectory tracking, in this case each iteration proposed a hybrid PSO using genetic operators inside the PSO of the algorithm has a set of solutions. These methods are to help to optimize the fuzzy ranking number for job based on generating, selecting, combining and replacing a set scheduling. of solutions. Since they maintain and they manipulate a set, instead of a This paper is organized as follows: Section 2 presents unique solution throughout the entire search process, they use the theoretical basis and problem statement. Section 3 more computer time than other metaheuristic methods. This introduces the controller design where a Hybrid PSO-GA is fact can be aggravated because the “convergence” of the used to select the parameters of the membership functions of population requires a great number of iterations. For this the FLC, GA and PSO are also applied for comparison reason a concerted effort has been dedicated to obtaining purposes. Robustness properties of the closed-loop system are methods that are more aggressive and manage to obtain achieved with a fuzzy logic control system using a Takagi- solutions of quality in a nearer horizon. Sugeno model where the angular velocity error and the linear This paper is concerned with bio-inspired optimization velocity error, are considered as the linguistic variables. methods where a hybrid PSO-GA is proposed. In this case, we Section 4 provides a simulation result of the trajectory tracking combine each method in order to obtain the best feature to of the autonomous mobile robot using the controller described design an optimal fuzzy controller (FLC) applied to in Section 3. Finally, Section 5 presents the conclusions. II. THEORETICAL BASIS AND PROBLEM STATEMENT C. Genetic Algorithms(GA)

A. Hybrid PSO-GA Genetic Algorithms (GAs) are adaptive heuristic search algorithms based on the evolutionary ideas of natural selection We propose a hybrid method using particle swarm and genetic processes [2]. The basic principles of GAs were optimization and genetic algorithms to find an optimal FLC first proposed by John Holland in 1975, inspired by the for the trajectory tracking control of the autonomous mobile mechanism of natural selection where stronger individuals are robot. The hybrid method uses the same population/swarm in likely the winners in a competing environment [3], [6], [7]. the two methods in the same iteration obtaining the best GA assumes that the potential solution of any problem is an individual/particle. PSO and GA communicate with each individual and can be represented by a set of parameters. other; every four iterations the best individual/particle is These parameters are regarded as the genes of a chromosome injected into the population of the worst method and vice and can be structured by a string of values in binary form. A versa. We used the maximum number of iteration/generations positive value, generally know as a fitness value, is used to to stop the method keeping only the best FLC obtained [23]. reflect the degree of "goodness" of the chromosome for the The procedure described [30], is used to find an optimal problem which would be highly related with its objective fuzzy logic controller combining the PSO and GA to exploit value. more completely the space of solutions. D. Porblem Statement B. Particle Swarm Optimization (PSO) To test the optimized FLCs obtained with the bio-inspired methods; we used a mobile robot model that we were used in PSO is a population based stochastic optimization other investigations [18], [19]. The model considered is a technique developed by Eberhart and Kennedy in 1995, unicycle mobile robot (Fig. 1), it consists of two driving inspired by social behavior of bird flocking or fish schooling wheels mounted of the same axis and a front free wheel. [10]. PSO shares many similarities with techniques such as Genetic Algorithms (GA) [11]. The system is initialized with a population of random solutions and searches for optima by updating generations. However, unlike the GA, the PSO has no evolution operators such as crossover and mutation.

In the PSO, the potential solutions, called particles, fly C through the problem space by following the current optimum particles [5]. Each particle keeps track of its coordinates in the problem space, which are associated with the best solution Figure 1. Wheeled Mobile Robot. (fitness) it has achieved so far (The fitness value is also stored). This value is called pbest. A unicycle mobile robot is an autonomous, wheeled vehicle Another "best" value that is tracked by the particle swarm capable of performing missions in fixed or uncertain optimizer is the best value, obtained so far by any particle in environments. The robot body is symmetrical around the the neighbors of the particle. This location is called lbest. perpendicular axis and the center of mass is at the geometric When a particle takes all the population as its topological center of the body. It has two driving wheels that are fixed to neighbors, the best value is a global best and is called gbest the axis that passes through center of mass “C” and one [14]. passive wheel prevents the robot from tipping over as it moves on a plane. In what follows, it is assumed that the motion of The particle swarm optimization concept consists of, at the passive wheel can be ignored in the dynamics of the each time step, changing the velocity of (accelerating) each mobile robot represented by the following set of equations particle toward its pbest and lbest locations (local version of [18]: PSO). Acceleration is weighted by a random term, with separate random numbers being generated for acceleration M ()q v + C (q,q )v + Dv = τ + P ()t (1) toward pbest and lbest locations [5]. In the past several years, ⎡cos θ 0 ⎤ PSO has been successfully applied in many research and ⎢ ⎥ ⎡ v ⎤ q = sin θ 0 (2) application areas. It is demonstrated that PSO gets better results ⎢ ⎥ ⎢ w ⎥ ⎢ 0 1 ⎥ ⎣N⎦ in a faster, cheaper way compared with other methods [15], [4]. ⎣ ⎦ υ () Another reason that PSO is attractive is that there are few J q parameters to adjust. One version, with slight variations, works where q = ()x, y,θ T is the vector of the configuration well in a wide variety of applications. Particle swarm coordinates; υ = ()v, w T is the vector of velocities; τ = ()τ ,τ optimization has been used for approaches that can be used 1 2 across a wide range of applications, as well as for specific is the vector of torques applied to the wheels of the robot τ τ applications focused on a specific requirement [15]. where 1 and 2 denote the torques of the right and left wheel, respectively; P ∈ R2 is the uniformly bounded disturbance The Table I show the rule set of the FLC that contain 9 × rules, which govern the input-output relationship of the FLC vector; M ()q ∈ R 2 2 is the positive-definite inertia matrix; and this adopts the Takagi-Sugeno style inference engine [18], C()q,q ϑ is the vector of centripetal and Coriolis forces; and and we use a single point in the outputs (constant values), × D ∈ R 2 2 is a diagonal positive-definite damping matrix. obtained using weighted average defuzzification procedure. Equation (2) represents the kinematics of the system, where To find the best fuzzy controller, we used three bio-inspired ()x,y is the position in the X – Y (world) reference frame; θ methods: Particle Swarm Optimization, Genetic Algorithms and the proposed method the Hybrid PSO-GA method to find is the angle between the heading direction and the x -axis; v the parameters of the membership functions. and w are the linear and angular velocities, respectively. Furthermore, the system (1)-(2) has the following non- TABLE I. FUZZY RULES OF THE FLC holonomic constraint: Negative Zero Positive Negative N N Z y cosθ − xsinθ = 0 (3) Zero N Z P Positive Z P P which corresponds to a no-slip wheel condition preventing the robot from moving sideways [18]. The system (2) fails to meet Once we obtained the FLC design, we set the parameters of Brockett’s necessary condition for feedback stabilization [29], both methods: the GA chromosome has 20 genes of real which implies that no continuous static state-feedback values that represent the two inputs, linear velocity error and controller exists that, stabilizes the closed-loop system around angular velocity error and two outputs constant values left the equilibrium point. The control objective is to design a torque and right torque; and using different values in the fuzzy logic controller τ that ensures genetic operator’s; mutation and single point crossover. We use different values for the cognitive, social parameters of the

PSO and inertial value to balance the swarm. For the hybrid ()− ()= lim qd t q t 0, (4) t→∞ PSO-GA method we apply the same parameter configuration used in GA and PSO. The three bio-inspired methods use the

same space of solutions (population) that we use to find the for any continuously, differentiable, bounded desired optimal values of the parameters of the membership functions. ∈ 3 trajectory qd R while attenuating external disturbances. Table II shows the parameters of the membership III. FUZZY LOGIC CONTROL DESIGN functions, the minimal and the maximum values in the search range for the Hybrid PSO-GA, GA and PSO to find the best In this section a fuzzy logic controller is designed to force fuzzy controller system for the mobile robot. the real velocities of the mobile robot (1) and (2) to satisfy the control objective (4). TABLE II. PARAMETERS OF THE MEMBERSHIP FUNCTIONS We design a Takagi-Sugeno fuzzy logic controller for the autonomous mobile robot, using linguistic variables in the MF Type Point Min Value Max Value a -50 -50 input and mathematical functions in the output [8], [12], [18], b -50 -50 ()ϑ () Trapezoidal [19], [22], [25]. The linear d and the angular wd c -15 -5.05 velocity errors were taken as input variables and the right ()τ d -1.5 -0.5 1 a -5 -1.75 ()τ and left 2 torques as the outputs. The membership Triangular b 0 0 functions used in the input are trapezoidal for the Negative (N) c 1.75 5 a 0.5 1.5 and Positive (P) and triangular for the Zero (C) linguistics b 5.05 15 Trapezoidal terms. The interval used for this fuzzy controller is [-50 50]. c 50 50 Fig. 2 shows the input variables. d 50 50

IV. SIMULATION RESULTS In this section we present the main results of the optimal FLC obtained by the bio-inspired method for the trajectory tracking control of the autonomous mobile robot.

A. Optimal FLC obtained by the Hybrid PSO-GA method Table III presents the main results of the FLC obtained with (a) (b) () () the hybrid PSO-GA method showing in the first row the best Figure 2. (a) Linear velocity error ev . (b) Angular velocity error ew . result.

TABLE III. RESULTS OF THE FLC OBTAINED BY HYBRID PSO-GA. B. Optimal FLC obtained by GA method Swarm/ Iter./ Time Average No C1 C2 Inertia Cross. Mut. Table 4 presents the main results of the FLC obtained with Pop. Gener. Exec. error the genetic algorithms, showing in the first row the best result. 1 70 60 0.1771 0.6982 0.1148 0.8 0.3 1:09:06 0.136014 2 25 100 0.5301 0.5209 0.3777 0.7 0.1 0:35:11 0.136083 3 30 60 0.2558 0.8882 0.9414 0.8 0.1 0:40:50 0.136086 TABLE IV. SIMULATION RESULTS OF TH FLC OBTAINED BY GA 4 200 70 0.4825 0.9969 0.3133 0.8 0.1 3:15:53 0.136096 % GA Average No Ind. Gen. Cross Mut 5 80 80 0.3135 0.9564 0.7126 0.9 0.3 1:28:09 0.136104 Rem Time error 6 60 80 0.8295 0.9765 0.4300 0.8 0.2 1:06:33 0.136168 1 150 50 0.7 0.6 0.3 0:51:23 0.154186 7 70 200 0.9435 0.5308 0.8614 0.8 0.1 6:15:59 0.136184 2 50 60 0.7 0.6 0.4 0:21:30 0.154245 8 90 60 0.5341 0.1247 0.7840 0.9 0.4 1:55:47 0.136188 3 200 100 0.7 0.4 0.1 2:19:46 0.154342 9 200 100 0.1059 0.5469 0.3858 0.8 0.1 20:02:35 0.136199 4 100 80 0.7 0.8 0.3 0:55:04 0.154346 10 45 200 0.6037 0.7407 0.6511 0.7 0.1 5:34:11 0.136449 5 70 60 0.7 0.8 0.4 0:29:23 0.154396 6 90 60 0.7 0.9 0.4 0:37:37 0.154484 7 70 50 0.7 0.8 0.3 0:24:16 0.154608 Fig. 3 shows the particles behavior of the Hybrid PSO-GA 8 100 25 0.7 0.8 0.3 0:17:47 0.155028 giving the best FLC to control the robot mobile. 9 80 80 0.7 0.9 0.3 0:26:53 0.155034 10 30 40 0.7 0.7 0.2 0:08:27 0.155034

Fig. 5 shows the evolution of the giving the best FLC to control the robot mobile.

Figure 3. Behavior of hybrid PSO-GA for the optimized controller.

Fig. 4 shows a) the stability of the position errors ex, ey and etheta and b) shows the control result of the autonomous robot mobile using the optimized FLC obtained with the Hybrid Figure 5. Evolution of the genetic algorithm for the optimized controller. PSO-GA. Fig. 6 shows a) the stability of the position errors ex, ey and etheta and b) shows the control result of the autonomous robot mobile using the optimized FLC obtained with the GA.

a)

a)

b) b)

Figure 4. (a) Position errors of x, y and theta; (b) closed-loop response of Figure 6. (a) Position errors of x, y and theta; (b) closed-loop response of the autonomous mobile robot with the optimized FLC. the autonomous mobile robot with the optimized FLC. C. Optimal FLC obtained by PSO method Table V presents the main results of the FLC obtained with the particle swarm optimization method, showing in the third row the best result. b) TABLE V. SIMULATION RESULTS OF THE FLC OBTAINED BY PSO Max PSO Average No Swarm C1 C2 Inertia Iter. Time error 1 100 50 0.8612 0.2107 0.4291 0:23:59 0.20995 2 100 150 0.7548 0.896 0.8498 0:19:22 0.16190 3 80 250 0.5093 0.5838 0.795 0:59:30 0.16132 Figure 7. (a) Position errors of x, y and theta; (b) closed-loop response of 4 110 220 0.3935 0.1791 0.7216 1:45:28 0.16647 the autonomous mobile robot with the optimized FLC. 5 90 150 0.7799 0.8207 0.3702 1:04:45 0.16474 6 150 80 0.8137 0.8331 0.8847 0:15:14 0.16314 7 150 200 0.1062 0.3237 0.629 2:13:04 0.17241 D. Comparisson results of the optimization methods 8 60 200 0.9517 0.8052 0.7738 0:35:57 0.16205 In this section we present a comparison using the average 9 80 100 0.8347 0.1461 0.2874 0:44:12 0.19508 errors obtained by the autonomous mobile robot using the 10 50 130 0.7374 0.1772 0.4664 0:40:25 0.21352 optimal controller of each optimization method.

Fig. 6 shows the particles behavior of the PSO giving the Table VI presents the comparison results of the Best FLC obtained with the GA, PSO and the hybrid PSO-GA method best FLC to control the robot mobile. applied to the autonomous mobile robot.

TABLE VI. COMPARISON OF THE SIMULATION RESULTS No. Method Time exec Average error 1 GA 0:51:23 0.15418663 2 PSO 0:59:30 0.16132949 3 Hybrid PSO-GA 1:09:06 0.13601414

In Table VI we show the comparison of the optimization methods and we can observe a significant difference in the results obtained by hybrid PSO-GA with respect to the GA and PSO results. We apply a t-Student test using the Minitab 15.1 to compare the proposed method with each GA and PSO individually for the mobile robot.

Figure 6 Particles behavior of the PSO for the optimized controller. Table VII shows the t-Student test for the comparison of

the GA and PSO with the hybrid PSO-GA for the Fig. 7 shows a) the stability of the position errors e , e and autonomous mobile robot. x y etheta and b) shows the control result of the autonomous robot mobile using the optimized FLC obtained with the PSO. TABLE VII. RESULT OF THE T-STUDENT TEST APPLIED TO HYBRID PSO- GA FOR AUTONOMOUS MOBILE ROBOT.

Estimate 95% CI Optimization T- N Mean StDev SE Mean for for DF Method Value difference difference Hybrid 10 0.136157 0.000118 0.000037 -0.01866 - PSO-GA -0.01841 11 162.08 GA 10 0.154571 0.000339 0.00011 -0.01816 Hybrid 10 0.136157 0.000118 0.000037 -0.05583 PSO-GA -0.0409 -6.2 9 PSO 10 0.1771 0.0209 0.0066 -0.02598

Since the t-Value is sufficiently high, we can conclude that a) there exists sufficient statistical evidence to say that the hybrid PSO-GA is better than GA and PSO for the autonomous mobile robot.

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