OPTIMIZING THE DESIGN OF A FUZZY PATH PLANNER FOR CAR-LIKE AUTONOMOUS ROBOTS

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1 OTIMIZI THE DESI OF A FUZZY ATH LAER FOR CAR-LIKE AUTOOMOUS ROBOTS I. Baturone 1, F. J. Moreno-Velo 1, S. Sánchez-Solano 1, A. Ollero 1 Instituto de Microelectrónica de Sevilla - Centro acional de Microelectrónica Avda. Reina Mercedes s/n, (Edif. CICA). E-11, Sevilla, Spain Dep. Ingeniería de Sistemas y Automática. E.S.Ingenieros. Camino de los Descubrimientos s/n. 19, Sevilla, SAI The 11th International Conference on Advanced Robotics (ICAR 3), Vol. 3, pp , Coimbra, June 3 - July 3, 3. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are epected to adhere to the terms and constraints invoked by each author s copyright. In most cases, these works may not be reposted without the eplicit permission of the copyright holder.

2 Optimizing the design of a fuzzy path planner for car-like autonomous robots I. Baturone 1, F. J. Moreno-Velo 1, S. Sánchez-Solano 1, A. Ollero 1 Instituto de Microelectrónica de Sevilla (IMSE-CM). Avd. Reina Mercedes, s/n. Edif. CICA, 11, Sevilla, SAI Dep. Ingeniería de Sistemas y Automática. E.S.Ingenieros. Camino Descubrimientos s/n. 19, Sevilla, SAI fuzzy-team@imse.cnm.es Abstract This paper presents methods and tools to design a fuzzy path planner for autonomous non-holonomic vehicles by means of supervised learning. The method combines heuristic knowledge and geometric considerations to obtain a continuous-curvature short path that can be eecuted efficiently by the path tracking controller of the mobile robot. Furthermore, the method minimizes the computer requirements to implement the fuzzy planner. The proposed design method can be easily carried out by means of the Xfuzzy 3. environment developed by some of the authors. The resulting planning strategies have been proven successfully in the Romeo R autonomous vehicle fully designed and built at the Escuela Superior de Ingenieros, University of Seville. 1 Introduction Maneuvering between initial and target configurations is an important function of autonomous navigation of robotic vehicles with non-holonomic configurations. The path planner of a car-like robot has to meet two non-holonomic kinematic constraints: the movement direction must always be tangent to its trajectory and the turning radius is mechanically limited to a minimum value, which is equivalent to say that the vehicle curvature is upper-bounded. Assuming that there is no slipping of the wheels, a simple kinematic model usually employed for car-like robots is the following: = v sinφ y = v cosφ φ = γ v where (,y) are the coordinates of the rear ale midpoint, φ is the robot orientation, v is the speed, and γ is the vehicle curvature. In the absence of obstacles, the shortest paths connecting two given initial and final configurations for a car-like vehicle consist of a finite sequence of two elementary components: arcs of circle (with minimal turning radius) (1) and straight line segments. This was proved by Dubins for vehicles moving only forward, and by Reeds and Sheep for car-like robots going both forward and backward [1] []. In any case, the curvature is discontinuous between two elementary components, so that these shortest paths cannot be followed precisely without stopping at each discontinuity point to reorient the front wheels. To avoid these stops, several authors have proposed continuous-curvature path planners using differential geometric methods. These planners generate clothoids, cubic spirals, B-splines, β-splines, quintic polynomials, etc., which are then followed by using a path-tracking technique based on, for eample, pure-pursuit or predictive control methods [3]. Other authors have proposed fuzzy logic-based planners based on emulating the heuristic knowledge of epert drivers and further tuned or adjusted with learning algorithms [] [5]. Contrary to the geometric-based planners, most of fuzzy path planners does not take into account either shortest path requirements or the robot constraints. This paper describes how geometric and heuristic methods can be combined to optimize the design of a fuzzy path planner that provides continuous-curvature short paths. We will focus on the problem of backing up a vehicle so as to arrive at a desired loading dock (=, y=) with a desired orientation (φ=) (Figure 1). The paper is structured as follows. Section describes the design of a fuzzy path planner whose initial structure (its rule base) is chosen heuristically and whose parameters (related to the membership functions involved) are adjusted by applying heuristics and taking into account the objective (,y) vehicle angle, φ wheel angle Figure 1: The parking problem addressed in this paper. This work has been partially supported by the Spanish CICYT rojects TA99-9-C-1 and TIC1-17.

3 the requirement of providing a short path as well as the non-holonomic kinematic constraints of a car-like robot. Section 3 goes further and describes how the design can be optimized by eploiting geometric methods not only to adjust the parameters of the fuzzy path planner but also to select a good initial structure. Section shows how the efficiency of the resulting planner has been eperimentally confirmed with the Romeo R autonomous vehicle designed and built at the Escuela Superior de Ingenieros, University of Seville. A continuous-curvature fuzzy planner The complete fuzzy controller considered to solve the parking problem in Figure 1 consists of two modules: a fuzzy path planner, which decides the vehicle curvature, and a speed planner, which controls the vehicle speed. The latter is very simple since it contains two rules that, taking into account only the y variable, modify the speed from -1m/s to as y approaches to, so as to stop the vehicle at the desired y=. The design of the fuzzy path planner is the problem addressed in the following. A fuzzy path planner designed by translating the knowledge of eperts is difficult to be optimum. Even the knowledge translation itself needs to apply a trial and error depuration process. This process is considerably reduced by using supervised learning algorithms with a set of input-output training data. In the fuzzy planners reported in the literature, these data are provided by either the input-output behavior of an human driver or another controller already proven to be efficient [] [5]. Since these data do not generally provide the shortest paths we will resort to geometric methods to generate them. According to Dubins results, the shortest paths for a car-like vehicle consist of straight line segments and circular arcs of minimum radius. In addition, if the distance travelled in the y direction is wanted to be minimized, the line segments have to form a right angle with the vertical. Figure shows three eamples of these shortest paths. In all of them, the car begins the parking maneuver starting with an orientation of 1º. In path 1, the vehicle starts at final position y path 1 path path 3 starting positions Figure : Some of the ideal shortest parking trajectories. final position y path 1 path path 3 Figure 3: Some smoother trajectories. starting positions = and y large; in path, at large and y large; and in path 3, at large and y short. Since a car-like vehicle can not follow discontinuouscurvature paths without stopping, we have changed heuristically these shortest trajectories into the ones shown in Figure 3. The main difference is that the approaching to the desired configuration (=, y=, φ=) is performed more gradually to avoid oscillations around the y ais. The data corresponding to these smooth trajectories have been employed to train a fuzzy path planner with supervised learning algorithms. As done in [], the initial fuzzy planner considered takes the variables and φ (robot orientation) as input variables that represent the robot configuration, and provides the curvature as the output. For simplicity, we have selected a zero-order Takagi-Sugeno controller whose output is computed as the weighted average of the singleton consequents of the rules. Since we do not apply any knowledge at first, we start from a system with enough compleity to solve our problem. In particular, we consider 7 linguistic labels to cover the input variable and 9 labels for the vehicle angle, both of them represented by gaussian membership functions. The controller contains the 3 possible rules, each one with its own consequent, so that 3 singletons are considered for the output variable. Initially, all the membership functions of the input variables cover the universes of discourse uniformly, and the 3 singleton values are taken equal, thus providing a flat control surface. The CAD tool fsl of the environment Xfuzzy 3. has been employed to carry out the training []. Applying the clustering process supported by fsl on the output space, the number of different rule consequents is reduced from 3 to 3. This allows merging some membership functions of the input variables, thus resulting 5 and 7 labels to cover the and φ variables, as shown in Figure. Therefore, the learned rule base consists of 35 instead of 3 rules, as shown in Figure 5a. The learned control surface is shown in Figure 5b. As required, the provided curvature is continuous (because the membership functions of the input variables always overlap each other) and the robot does not need to stop until reaching y =.

4 LB LM LS Z RS RM RB y (m) 1-1º º 1º LB LS Z RS RB φ 1-1m m 1m Figure : Membership functions learned for the input variables. The structure of this fuzzy path planner is similar to that reported in [], but with the improvement of providing shorter paths thanks to the use of geometric methods to generate the training data. This is illustrated in Figure a by simulations obtained with the tool fsim of Xfuzzy 3., considering a car-like vehicle with the model in (1), and with a first-order dynamic response. The dark line represents the trajectory provided by the fuzzy controller whose path planner is the one described in this Section while the light line represents the trajectory provided by a fuzzy controller whose path planner has been designed without considering shortest path requirements. As can be seen, the latter controller is not able to reach the target configuration (it requires a longer distance in y). The starting configuration of the robot in these simulations is φ LB LS Z RS RB curvature (m -1 ) (). () LB LM LS Z RS RM RB Z (m) Figure 5: Rule base obtained by learning. The learned control surface. 1. φ (degrees) - - curvature (m -1 ) simulation step Figure : Comparing trajectories. Continuous curvature of the dark trajectory in. =, y=m, and φ=-1º. Figure b shows how the planner designed in this Section makes the robot follow a continuous-curvature path. 3 Further optimizing the fuzzy planner (m) In the previous Section, we have used geometric methods to adjust the parameters of a fuzzy planner with an initial structure selected heuristically. However, geometric methods can also be eploited to select an adequate structure. If we analyze the shortest paths shown in Figure, we obtain that, geometrically: if φ 9 = ± or cosφ = γ = R if cosφ < and [, R] γ = γ R ma if φ > 9 and R γ = γ ma if cosφ > and [ R, ] γ R if φ > 9 and R γ otherwise γ = γ ma = γma = γ ma ()

5 curvature (m -1 ) (m) φ (degrees) Figure : Control surface provided. 1. where R represents the minimal turning radius and its reciprocal, γ ma, the maimum curvature. Therefore, a path planner providing a value of γ according to the equations in () will implement these shortest trajectories. As a matter of fact, the control surface learned in the previous Section (see Figure 5b) fuzzifies the equations in () because the training trajectories were smoother. What becomes apparent is that the curvature to apply should be positive or negative depending on the relation between the variables and φ. Hence, the fuzzy path planner described in the previous Section can be simplified if relational rules are employed instead of grid-based rules. That is, a more adequate structure for the fuzzy planner is obtained if we use rules as the following: (1) IF φ is much greater than f() THE γ is ositivebig () IF φ is slightly greater than f() THE γ is ositive- Small (3) IF φ is slightly smaller than f() THE γ is egativesmall () IF φ is much smaller than f() THE γ is egativebig where φ = f() implements the line in the φ- space that separates positive from negative curvatures. With this idea, the new fuzzy planner considered in this Section is a hierarchical controller with the structure shown in Figure 7a. The so called smoothing rule base implements the previous rules, while the interpolation rule base generates the function f() by means of other rules acting on the variable. Both rule bases implement a zero-order Takagi-Sugeno inference process. With this structure, our initial fuzzy controller to adjust is much simpler than the previous one ( instead of 3 rules). In addition we employ normalized piecewise linear membership functions to cover the input variables of the two rule bases, as shown in Figure 7b. This ensures that the resulting controller will employ only sums and products, which is very simple to implement by real-time and embedded control systems, against the eponential operations required by the controller of the previous Section [7]. This initial fuzzy path planner has also been adjusted with the smooth trajectories of Figure 3. The capabilities of the tool fsl for training hierarchical systems and maintaining constraints like normalized membership functions have been eploited. The control surface provided by the planner after learning is shown in Figure. It is very similar to that in Figure 5b and, hence, the trajectories generated (an eample is shown in Figure 9a) are similar to those generated by the previous controller. The difference is that now the curvature transitions in the different trajectories do not involve many turns in the steering wheel, which is better for control. Figure 9b illustrates an eample of these tran- y (m) φ interpolation f() smoothing γ rule base - Σ rule base + membership degree curvature (m -1 ).. (m) -. input universe of discourse Figure 7: Structure of the further optimized controller. ormalized membership functions employed. -. simulation step Figure 9: A trajectory eample. Its corresponding curvature transition.

6 1 φ (º) (m) Figure 1: Trajectory of Figure 9a in the state space. f() sitions: the curvature changes from -. to. and to (m -1 ) without oscillations like those in Figure b. This further improvement is caused by the use of relational instead of grid-based rules. Figure 1 illustrates with a color degradation, the transition between positive (dark grey) and negative (light grey) curvatures corresponding to the control surface in Figure. The separation line, f(), is depicted with a dark line. The trajectory in Figure 9a is also shown in this figure by a grey line. It eplains the non-oscillating curvature transition of Figure 9b. speed commands from our controller. In addition, the motor control card reads the direction and traction encoders of the engines. These measures, together with the information provided by a gyroscope have to be processed to estimate the current position and orientation of the robot, which are the input variables required by our fuzzy path planner. Hence, the robot knows its configuration by odometry. An interface has been developed that includes a wide set of functions (programmed in C++) to work easily with sensors and actuators. Having this interface, we have included the C++ code of our controller which is as simple as 1 if statements involving sum and product operations. The eecution of this control code and all the other required routines can be performed at real time without problems because a control cycle period of 1 ms is enough for our application. Figure 1 and 13 show two eamples of eperimental trajectories followed by Romeo R when starting at different positions and with different orientations. Comparing Figure 1 with Figure 9 we can see that eperimental results do not differ very much from simulated results, despite simulated results use a simple car model with a first-order dynamics. The smooth trajectories employed to train our fuzzy planner ensure that it is robust against small changes of Eperimental results We have verified the behavior of our optimized path planner with the robot Romeo R []. This robot is an electrical vehicle provided with a set of sensors and actuators that make it capable of autonomous navigation (Figure 11). The information collected by the sensors and that required by the actuators is currently centralized by a computer placed at the back of the robot and which also implements the control algorithms. In our control eample, the computer has to govern a motor control card which in turn governs, independently, the steering and traction electrical motors of RomeoR. These electrical motors have to receive, respectively, the curvature and y (m) curvature (m -1 ).. -. (m) Figure 11: The autonomous robot Romeo R t (s) Figure 1: Eperimental results: Trajectory. Its corresponding curvature transition.

7 y (m) (m) curvature (m -1 ) Figure 13: Eperimental results: Trajectory. Its corresponding curvature transition. the operational conditions. However, different dynamic aspects may appear that make it necessary a fine tuning. Since the fuzzy path planner is a two-input fuzzy controller, the simplest adjustment is either to dilate the space and/or to contract the φ space (so that, given an angle, the smooth transition between γ ma and -γ ma is performed at a longer distance in ) or the opposite action (contract the space and/or dilate the φ space). A more comple tuning would involve the controller parameters. An advantage of the optimized fuzzy path planner is that its comple fine tuning is easier than the grid-based controller because it can involve only parameters whose influence on the robot behavior is intuitively understood. 5 Conclusions t (s) Autonomous navigation of non-holonomic vehicles can be efficiently performed by combining basic heuristic knowledge, geometrical reasoning, and simple kinematic models. This paper has presented a method for fuzzy path planning based on the use of heuristics and simple geometric considerations to generate the training data to apply supervised learning algorithms to the fuzzy system. The design method can be carried out by means of the fsl and fsim Xfuzzy 3. verification tools using a simple kinematic model of the vehicle. Further optimization of the path generation has been carried out by the use of relational instead of grid-based rules. The proposed method generates a continuous-curvature short path that can be efficiently eecuted by means of simple path tracking controllers. The eperimental results in the Romeo R autonomous vehicle demonstrate the concordance with the simulation results and the efficiency of the proposed method. The low computational requirements of the proposed planner will allow a new low cost implementation by means of DS embedded control systems in which both path planning and control functions will be implemented in the near future. References [1] L.E. Dubins, On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents, American Journal Math., Vol. 79, pp , [] J.A. Reeds, R.A. Shepp, Optimal path for a car that goes both forward and backward, acific Journal Math., Vol. 15 (), pp , 199. [3] A. Ollero, Robótica: manipuladores y robots móviles, Marcombo Boiareu Eds., 1. [] S.-. Kong, B. Kosko, Comparison of Fuzzy and eural Truck Backer-Upper Control Systems, Chapter 9 in eural etworks and Fuzzy Systems, B. Kosko, rentice Hall, 199. [5] H. Miyata, M. Ohki, M. Ohkita, Self-tuning of fuzzy reasoning by the steepest descent method and its application to a parallel parking, IEICE Trans. Inf. & Syst., Vol. E79D (5), pp.51-59, May 199. [] F. J. Moreno-Velo, I. Baturone, S. Sánchez-Solano, A. Barriga, "Xfuzzy 3.: A Development Environment for Fuzzy Systems", roc. nd IEEE Int. Conf. on Fuzzy Logic and Technology (EUS- FLAT 1), pp. 93-9, Leicester, 1. [7] I. Baturone, S. Sánchez-Solano, Microelectronic design of universal fuzzy controllers, Mathware & Soft Computing, Vol., pp , Dec. 1. [] A. Ollero, B.C. Arrue, J. Ferruz,. Heredia, F. Cuesta, F. Lóez-ichaco, and C. ogales, Control and perception components for autonomous vehicles guidance. Application to the Romeo Vehicles, Control Engineering ractice, ergamon, Vol. 7 (1), pp , October 1999.