Multi-objective Evolutionary Fuzzy Modelling in Mobile Robotics

Transcription

1 Multi-objective Evolutionary Fuzzy Modelling in Mobile Robotics J. M. Lucas Dept. Information and Communications Engineering University of Murcia Murcia, Spain H. Martínez Dept. Information and Communications Engineering University of Murcia Murcia, Spain F. Jiménez Dept. Information and Communications Engineering University of Murcia Murcia, Spain Abstract In some environments, mobile robots need to perform docking tasks in a precise manner. In the application domain of this work, an Autonomous Guided Vehicle (AGV), specifically, a fork-lift truck must often perform docking maneuvers to load pallets in conveyor belts. In these maneuvers, the robot motion should be controlled accurately when the mobile robot is close to the target. We propose a multiobjective evolutionary algorithm in order to find multiple controllers with imposed constraints for docking task in charge of following up an online generated trajectory. Main purpose is to improve some features of docking task as its duration, accuracy and stability, satisfying determined constraints. Keywords: Multi-objective evolutionary algorithms, Path following, Path tracking, Mobile robots, Fuzzy control. 1 Introduction In some environments, mobile robots need to perform docking tasks accurately. The final position and orientation of the robot must be suitable for the tolerances required by the particular task. A robot requires docking when needs to interact with objects in its environment. In this paper, we present an environment where an Autonomous Guided Vehicle (AGV), specifically, a fork-lift truck performs docking maneuvers to load pallets in conveyor belts [13]. The main purpose is performing the docking task with specific requirements as duration and precision, following up an online generated trajectory. We propose a multi-objective evolutionary algorithm based partially on [10] in order to generate and optimize controllers in charge of docking control. In the evolutionary learning process, several objectives and constraints have been handled in a simultaneous manner. This process has been performed by means of the simulation system for mobile robots, called ThinkingCap-II. ew evolved controllers are simulated in that environment allowing the required data acquisition to evaluate the fitness of controller. This paper is organized as follows: Section introduces the most important features of the docking task. Section 3 explains the features of the proposed multi-objective evolutionary algorithm. Section 4 describes the experiments and results, and finally, Section 5 summarizes the most important conclusions. The docking task Docking task can be defined simply as motion from the current position to a desired position and orientation, while following a safe trajectory []. In fact, docking task can be performed as three distinct sub-problems [1]: 1. Moving to the required location while there is no danger of collision.. Moving accurately to come close to the required docking configuration. 714

2 3. Moving to the destination with the required precision, or performing some docking operation. We consider two movements: 1. avigation: The vehicle follows up a free curve (without constraints) towards the waypoint.. Docking: The vehicle follows up a preplanned B-Spline curve heading to the docking point (goal point). Figure 1 shows how the vehicle begins the docking maneuver. Docking points Waypoints Figure 1: Docking maneuver The control problem examined in this work is a path tracking problem. Essentially, the goal is that a mobile robot or an AGV navigates properly along the desired trajectory in a bidimensional environment. Pure Pursuit approach is a path tracking algorithm that works by calculating the curvature that will move a vehicle from its current position to some goal position [5]. The whole point of the algorithm is to choose a goal position that is some distance ahead of the vehicle on the path. An analogy is a car driver who looks a further point in front of the car and afterwards tends to head toward that goal point gradually. In particular, Pure Pursuit approach has been applied in order to calculate the heading-error respect to the preplanned trajectory. The heading-error is an angle denoted by δ, and is calculated by subtracting from heading-value the current orientation of robot (see Figure ). The heading value is calculated by using the look-ahead point in the trajectory. Figure depicts the fork-lift truck robot and B-Spline curve corresponding to the trajectory previously generated. Figure : Calculation the heading-error in the trajectory The value of δ is calculated as follows: Dx = look-ahead point in the path.x - current position of robot.x Dy = look-ahead point in the path.y - current position of robot.y Heading = Arc tangent of Dy / Dx in the range of -π to π δ = Heading - Current Orientation of Robot (where δ is normalized). 3 Description of the multi-objective evolutionary algorithm 3.1 Multi-objective algorithms and Fuzzy control Evolutionary Algorithms (EAs) are stochastic search techniques inspired by the principles of natural selection and evolution of species. EAs were initially extended to solve optimization problems with multiple objectives (often conflicting) [4, 6]. The population-based nature of EAs allows finding several elements of the Pareto optimal set in a single run of algorithm. The fuzzy modelling of learning systems and mechanisms allows applying the fuzzy logic to the control and operation tasks of autonomous mobile robots [15, 16]. Soft computing techniques (such as EAs) are suitable to be applied to applications domains with imprecise data and/or incomplete knowledge of environment such as control problems in mobile robotics. The idea is taking advantage of an evolutionary algorithm in order to evolve the behaviors for control tasks with the aim of performing them in a more accurate and efficient manner. Several works have been proposed 715

3 in the domain of the genetic design and optimization of fuzzy logic controllers [3, 8, 9]. another, the following criteria are established: 3. Features of the multi-objective evolutionary algorithm o A feasible individual is better than another unfeasible one. The proposed algorithm is a Pareto-based multiobjective evolutionary algorithm (MOEA) for fuzzy tuning, i.e., it has been designed to find, in a single run, multiple non-dominated solutions according to the Pareto decision strategy. Main features of MOEA are the following: Representation of solutions: An individual in the MOEA corresponds to a fuzzy controller in charge of the docking task. A controller is a collection of fuzzy rules in which each rule is composed by a set of n fuzzy numbers (antecedent) and m real parameters (consequent). In fact, antecedents are trapezoidal fuzzy sets that can be characterized using four parameters and the output fuzzy singletons using just one number for each one. Hence, focusing on learning rules in rule based systems, the Pittsburgh approach [17] has been applied. This approach is characterized by representing an entire fuzzy rule set as chromosome, although in our particular case, only one part of rule base is considered due to the controller symmetry. Initial population: The initial population is composed by controllers that are generated by means of random variations in their domain. Selection and generational replacement: The selection scheme and generational replacement is based on the work presented in [10]. In each iteration of the MOEA, two individuals are picked at random from the population. These individuals are crossed and mutated producing two offspring. After, the best of the first offspring replaces the first parent, and the best of the second offspring replaces to the second parent only if one of the offspring is better than the parent. ote that the population diversity is maintained in populations because an offspring replaces an individual similar to itself (one of their parents). In order to determine if an individual is better than o One unfeasible individual x is better than another one x if: Cx () l Cx (') l max < max i i i i i= 1,..., m maxpopc i 1,..., m i l = i maxpopci l i o where C i (x) l i, i = 1,,m are constraints imposed to the fuzzy models, and maxpopc is the maximum value of C i in the current population used for normalization. One feasible individual is better than another one if the first dominates the second one (Pareto dominance relation). Variation operators: In order to achieve an appropriate exploitation and exploration of the potential solutions in the search space, we apply three types of crossover operators: uniform, arithmetic and BLX-α [7] for parameter variation. Two types of the mutation operators are applied: uniform and non-uniform mutation [14]. 3.3 Identification of objectives and constraints An important issue of MOEA design is the identification of multiple objectives and constraints. Generally, we need to optimize several objectives that often conflict among them. In other words, a trade-off exists between the objectives, where improvement in one objective cannot be achieved without detriment to another. Moreover, we need to satisfy some specific constraints. These are the objectives and/or constraints to minimize: Objective/Constraint 1 - Execution time (): it is the required time (cycles) in order to carry out the docking maneuver. It is equivalent to the number of measures of the heading-error in the trajectory. 716

4 Objective/Constraint - Root mean square orientation tracking error in the entire trajectory: corresponds to the total orientation error in the entire trajectory and is calculated as follows: RMSEorientationtrackingtotal = 1 E( δ ) where δ is the heading error in each iteration and is the number of iterations (cycles, execution time). Objective 3 - Root mean square orientation tracking error at the end of the trajectory: corresponds to the total orientation error at the end of the trajectory and is calculated as follows: RMSEorientationtrackingfinal = L E( δ) where L is the number of iterations that are considered at the end of the trajectory. Constraint 4 - Root mean square error of control effort in the entire trajectory: represents the variation of the angular velocity (one output of controller). The goal is to minimize the control effort by reducing the oscillations due to abrupt changes in the angular velocity. This variation is calculated as follows: E( ω ω') RMSEcontroleffort = 1 where ω is the current angular velocity and ω is the previous velocity (this is the desired value in order to achieve a minimum variation). Objective/Constraint 5 Root mean square position tracking error: it is the distance from current robot position to the reference path. This position tracking error is calculated for the entire trajectory as follows: RMSEpositiontracking = 1 E( distance) Objective/Constraint 6 Orientation error goal: it is the difference between the final orientation (angle) of robot and the goal point orientation (docking point). ote that the objective 3 is included in the objective/constraint but this one is used to avoid solutions with large heading errors in the entire trajectory. In fact, it is important that the mobile robot follows up the trajectory in an accurate way in last steps while it is reaching the docking point, so it is considered errors in final actions with the purpose of reducing them to minimum value or zero. 4 Experiments and Results ThinkingCap-II (TC-II) is a framework for developing mobile robot applications. It is a joint effort between the University of Murcia (Spain) and the University of Örebro (Sweden), and it is based on previous work on ThinkingCap and BGA architectures [15, 16]. The framework consists on a reference cognitive architecture (largely based on ThinkingCap) that serves as a guide for making the functional decomposition of a robotics system, a software architecture (partially based on BGA) that allows an uniform and reusable way of organizing software components for robotics applications, and a communication infrastructure that allows software modules to communicate in a common way independently of whether they are local or remote. In our application, the mobile robot is an Autonomous Guided Vehicle (AGV), in particular, it is an industrial fork-lift truck that performs docking maneuvers to load/unload pallets in conveyor belts. In TC-II platform, we work with that vehicle, defining the concrete robot corresponding to the autonomous truck and specifying a plane of the industrial plant in which the way-points and docking-points are placed. In order to perform the learning process, several simulations have been executed with the aim of evaluating the performance of evolved fuzzy controllers after docking maneuvers. During the learning process, each controller is evaluated by means of collecting the required data from several simulations in order to calculate the objectives and constraints, using the average values of all simulations. The proposed controller is a Mamdani-type [11, 1] fuzzy controller which is in charge of the docking task. 717

5 It is composed by rules of this type: if δ A then ω = c0, ν = c1 The input fuzzy set represents the heading-error δ (deg) in the trajectory (antecedent) and the two output crisp values (consequents) for angular ω (deg/s) and linear ν (m/s) velocities control. The antecedent is a trapezoidal fuzzy set and consequents are singletons in floating-point. It is assumed that the membership functions are known in advance and are fixed and the sum is 1. In order to build the controller, we use five floating-point numbers for antecedent and three and four for consequents respectively. On the one hand, the fuzzy knowledge base is composed by seven trapezoidal membership functions for the input fuzzy variable. They are labeled by L (negative large), M (negative medium), S (negative small), Z (zero), PS (positive small), PM (positive medium) and PL (positive large). On the other hand, there are seven values for the angular velocity that are denoted by the same previous labels. Finally, there are four values for the linear velocity that are labeled by Z (negative zero), S (negative small), M (negative medium) and F (negative full). The fuzzy controller is composed by seven rules which were obtained heuristically: subsequently, we can make a decision process in order to select one or more compromise solutions. In order to compare the solutions, we used the controller constructed manually (the initial one). The main purpose is to reduce the orientation and position tracking errors and time in the task, but it is necessary to consider the control effort to avoid unstable maneuvers by means of gradual alterations of output control velocity. Table 1 summarizes the obtained results, showing the values of objectives/constraints (O/C objective and constraint at the same time) for each learned controller. These fuzzy controllers are nondominated and feasible according to the specified constraints. In this way, they achieve better path tracking performance than the initial controller (constructed manually), performing the docking maneuvers with larger precision, but they require some more time. Table 1: Results summary Rule 1: if δ L then ω = L, ν = Z Rule : if δ M then ω = M, ν = S Rule 3: if δ S then ω = S, ν = M Rule 4: if δ Z then ω = Z, ν = F Rule 5: if δ PS then ω = PS, ν = M Rule 6: if δ PM then ω = PM, ν = S Rule 7: if δ PL then ω = PL, ν = Z In our experiment, we decided upon the population size of 50 individuals. According to the variation operators, we applied the crossover operation with probability 0.6 and the mutation operation with 0.4. Each type of operator is applied with the same probability in the corresponding category. Constraints limits, listed in order, are: 300, 340, 10,.5 and After learning process, we obtained a set of non-dominated and feasible solutions and 718

6 4 Conclusions We presented an approach to path tracking controller tuning based on soft computing techniques. This paper remarks some results in the combination of Pareto-based multi-objective evolutionary algorithms (MOEA), handling constraints and fuzzy logic controller tuning and design. The MOEA was applied to tune the fuzzy controller for the docking maneuver of an Autonomous Ground Vehicle. The proposed MOEA allows obtaining a set of solutions for the controller designer, reducing the hard work to design or optimize controllers manually. Moreover, multiple objectives and constraints have been handled at the same time with the purpose of obtaining solutions which are non-dominated and feasible according to specified constraints. One of the advantages is reducing complexity because we need a single run of MOEA for tuning process. Hence, human intervention is only required at the end of the execution to choose one of the multiple non-dominated and feasible solutions found by the MOEA. Objectives are often in conflict, in our case, the designer could select a compromise solution that requires more time but with a larger accuracy and stability, for example, in cases in which fragile merchandise are transported. Finally, the performance of learned fuzzy controllers was compared to a controller obtained manually. References [1] R.C. Arkin and D. MacKenzie, Temporal coordination of perceptual algorithms for mobile robot navigation, IEEE Trans. on Robotics and Automation, vol. 10, o. 3, pp , [] R.C. Arkin and R.R. Murphy, Autonomous navigation in a manufacturing environment, IEEE Trans. on Robotics and Automation, vol. 6, o. 4, pp , [3] P. Bonissone, P. Khedkar and Y. Chen, Genetic algorithms for automated tuning of fuzzy controllers: a transportation application, IEEE Conference on Fuzzy Systems (FUZZ-IEEE'96), pp , [4] C.A. Coello Coello, D.A. Van Veldhuizen and G.B. Lamont, Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic Publishers, ew York, 00. [5] R.C. Coulter, Implementation of the Pure Pursuit Path Tracking Algorithm, Technical Report, Robotics Institute, Carnegie Mellon University, 199. [6] K. Deb, Multi-objective Optimization Using Evolutionary Algorithms, John Wiley & Sons, 001. [7] L.J. Eshelman and J.D. Schaffer, Real-coded genetic algorithm and interval schemata, Foundations of Genetic Algorithms II, Ed. L. Darrell Whitley (Morgan Kaufmann), pp , [8] F. Hoffmann, Evolutionary algorithms for fuzzy control system design, Proc. IEEE, vol. 89, no. 9, pp , 001. [9] A. Homaifar, D. Battle and E. Tunstel, Soft computing-based design and control for mobile robot path tracking, Proc. IEEE on Computational Intelligence in Robotics and Automation (CIRA '99.), pp , [10]F. Jiménez, A.F. Gómez Skarmeta, H. Roubos and R. Babuska, Accurate, transparent, and compact fuzzy models for function approximation and dynamic modeling through multi-objective evolutionary optimization, First International Conference on Evolutionary Multi- Criterion Optimization, Springer-Verlag, pp , 001. [11]E.H. Mamdani, Applications of fuzzy algorithms for control a simple dynamic plant, Proc. IEEE, vol. 11, no. 1, pp , [1]E.H. Mamdani and S. Assilian, An experiment in linguistic synthesis with fuzzy logic controller, International Journal of Man- Machine Studies, vol. 7, pp. 1 13, [13]H. Martínez-Barberá, J.P. Cánovas, M.A. Zamora and A.F. Gómez-Skarmeta, i-fork: a flexible AGV system using topological and grid maps, Proc. IEEE Conference on Robotics and Automation, pp , 003. [14]Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, 3 rd eds., Springer-Verlag, London, [15]A. Saffiotti, The use of fuzzy logic for autonomous robot navigation, Soft Computing, vol. 1, no. 4, pp , [16]A. Saffiotti, K. Konolige and E.H. Ruspini, A multivaluted-logic approach to integrating planning and control, Artificial Intelligence, vol. 76, no. 1-, pp , [17]S.F. Smith, A learning System based on genetic adaptive algorithms, Doctoral Dissertation, Department of Computer Science, University of Pittsburgh,