Journal of Intelligent Learning Systems and Applications, 2012, 4, 291302 http://dx.doi.org/10.4236/jilsa.2012.44031 Published Online November 2012 (http://www.SciRP.org/journal/jilsa) 291 Interval Type2 Fuzzy Logic Control of Mobile Robots Nesrine Baklouti1, Robert John2, Adel M. Alimi1 1REGIM: Research Group on Intelligent Machines, National Engineering School of Sfax (ENIS), Sfax, Tunisia; 2Centre for Compu tational Intelligence, De Montfort University, Leicester, UK. Email: nesrine.baklouti@gmail.com, rij@dmu.ac.uk, adel.alimi@enis.rnu.tn Received May 9th, 2012; revised July 31st, 2012; accepted August 7th, 2012 ABSTRACT Navigation of autonomous mobile robots in dynamic and unknown environments needs to take into account different kinds of uncertainties. Type1 fuzzy logic research has been largely used in the control of mobile robots. However, type1 fuzzy control presents limitations in handling those uncertainties as it uses precise fuzzy sets. Indeed type1 fuzzy sets cannot deal with linguistic and numerical uncertainties associated with either the mechanical aspect of robots, or with dynamic changing environment or with knowledge used in the phase of conception of a fuzzy system. Recently many researchers have applied type2 fuzzy logic to improve performance. As control using type2 fuzzy sets represents a new generation of fuzzy controllers in mobile robotic issue, it is interesting to present the performances that can offer type2 fuzzy sets by regards to type1 fuzzy sets. The paper presented deep and new comparisons between the two sides of fuzzy logic and demonstrated the great interest in controlling mobile robot using type2 fuzzy logic. We deal with the design of new controllers for mobile robots using type2 fuzzy logic in the navigation process in unknown and dy namic environments. The dynamicity of the environment is depicted by the presence of other dynamic robots. The per formances of the proposed controllers are represented by both simulations and experimental results, and discussed over graphical paths and numerical analysis. Keywords: Type2 Fuzzy Logic; Control; Motion Planning; Mobile Robots 1. Introduction Mobile robotic navigation in a dynamic and unknown environment is a local path planning problem based on sensory information with no knowledge about the form and location of obstacles. Precise endeffector and posi tioning accuracy are required in the navigation process. Due to their mechanic conception, Robots, by their very nature, have significant uncertainties in their readings and movements. These errors can be summarized as fol lows: Sensor measurements are usually noisy due to the instruments. Generally a robot is equipped with in ternal sensors like encoder sensors and coupling sen sors and external sensors as laser sensors, infrared sensors and cameras. Readings extracted from those sensors are generally noisy [1], There are assembly errors that include linear and an gular errors produced during the assembly of the va rious robot mechanical components, Indeed, there are uncertainties provoked from a varia tion in the coupling of actuators characteristics with environmental conditions. Measurement and actuator create endeffector positioning errors. The resolution of encoders and stepper motors are examples of such errors. Although these errors are in the most cases very small, they can be amplified to cause many er rors that can affect the accuracy of the system [1], Real environments are not ideal, so they can produce random errors that change unpredictably. Those er rors are called nonsystematic errors and are generally caused by irregularities or roughness of the floor. In order to overcome those uncertainties and to de velop a robust, flexible and online planner, type1 fuzzy logic has been used [25]. However type1 fuzzy logic cannot fully handle the stated uncertainties because it uses precise type1 fuzzy sets which don’t necessarily cope well with all sources of vagueness and uncertainty. There are at least three sources of uncertainty in type1 fuzzy logic control: According to Mendel [6] “words mean different things to different people”, so while designing a fuzzy con troller experts are unlikely to agree on the member ship functions, Consequents in the rules can be uncertain, And the database used in the conception of a fuzzy controller is generally not ideal and contains errors, even if the database has been constructed by an ex pert. Copyright © 2012 SciRes. JILSA
Interval Type2 Fuzzy Logic Control of Mobile Robots 292 Then in mobile robot control by fuzzy logic, all the cited forms of uncertainties will be multiplied over fuzzi fication, inference and defuzzification. Those errors can degrade the performance of the whole robot controller. Type2 fuzzy logic has been used by researchers to try and overcome some of these difficulties [68]. And since many researchers have explored the use of type2 fuzzy logic controllers in various applications [916]. In fact, type2 fuzzy sets were initially introduced by Zadeh [17]. Firstly a general type2 fuzzy set was defined, where it represents a 3D set in which each membership grade is a type1 fuzzy set bounded in [0,1]. Due to the complexity of the join (OR) and meet (AND) operations performing the inference part [18] and typereduction in the defuzzi fication part [19], the application of general type2 fuzzy sets has been limited. Hence, a simplified version of general type2 set called interval type2 fuzzy set is used more widely [6,20]. This kind of set has membership grades that are crisp interval sets bounded in [0,1]. The uncertainty here is represented as a 2D bounded region that is called the Footprint of Uncertainty. Various re searchers have explored the advantages of interval type2 fuzzy sets [21,22]. Moreover, a geometric approach has been introduced by Coupland and John [23] distinguish ing between fuzzy logic over discrete and continuous do main. But this approach is not fast enough in control ap plications [24]. In the rest of this paper we treat only in terval type2 fuzzy sets over discrete domains. In mobile robotics, some researchers have explored the control of mobile robots using interval type2 fuzzy logic [21,2429]. As Hagras states in [21] control using type2 fuzzy sets represents a new generation of fuzzy control lers. In [25] Hagras presented an interval type2 fuzzy logic controller to command a robot in indoor and out door unstructured environment. A robot was tested under different sources of nonsystematic errors. The results showed that type2 fuzzy logic outperforms its type1 counterpart. This was shown through robot paths and control surfaces. In [27], an interval type2 fuzzy logic was proposed for the control of a robot tracking a mobile object in the context of robot soccer games. In this game the robot has to track a ball. To evaluate the performance of the type2 fuzzy logic against its type1 counterpart, graphical paths analysis were presented showing the way the player reaches the position of the ball. Also, an addi tional test was made presenting the ability of type2 con troller to track the ball with less standard deviation error than its type1 counterpart. In this paper we propose specific aspects of control of mobile robots in unknown and dynamic environments using type2 fuzzy logic. The dynamicity of the envi ronment is depicted by the presence of other dynamic robots. The performances of the proposed controllers are represented by both simulations and experimental results, and discussed over graphical paths and numerical analy sis. This paper has essentially two parts: In the first part we designed an IT2TSK fuzzy logic controller for avoi ding obstacles using simulations. In the second part we designed a Mamdani interval type2 (IT2) fuzzy logic controller for wall following behavior using the robot Khepera II from LAMI [30]. The remainder of the paper is organized as follows: Section II introduces type2 fuzzy logic. Sections III presents the proposed Interval type2 fuzzy logic controller for obstacle avoidance be havior. Next section presents the conception of an Inter val type2 fuzzy logic for a wall following behavior. Simulations and experimental results in both applications are presented and are discussed. Finally, some conclu sions are pointed out in Section VI. 2. Type2 Fuzzy Logic 2.1. Overview on Type2 Fuzzy Sets Type1 fuzzy sets are certain and crisp, whereas type2 fuzzy sets are themselves fuzzy. Type2 fuzzy sets were first introduced by Zadeh [17]. An Interval T2 fuzzy set is described as in the following definition, where in (1) all ,1 Axu , 0, Axu 1 . 1, 0,1 XX xXuJ AxuJ (1) where denotes the union of all admissible and . An IT2 FS is represented by a bounded region limit ed by two membership functions, where corresponding to each primary MF (which is in [0,1]), a secondary MF is used to the primary one. The Uncertainty in the primary membership function consists of the union of all mem bership functions. This Uncertainty represents a bounded region that we call the Footprint of Uncertainty UFO , i.e., FOU X xX uJ (2) The FOU represents a complete description of an IT2 FS. It uses an UMF Upper Membership Function, and LMF Lower Membership Function; The F UM and the LMF of are two T1 MFs that bound the FOU . UMF represents the upper bound of FOU and is denoted A FOU , and the LMF represent the lower bound of and is denoted A : A A FOU AxX FOU AxX (3) For an IT2 FS, , XAA xx , X . In the rest of this paper, we will use only the IT2 fuzzy sets in the design of our work. Copyright © 2012 SciRes. JILSA
Interval Type2 Fuzzy Logic Control of Mobile Robots 293 2.2. Interval Type2 Fuzzy Logic Controller In fact, a type2 fuzzy logic system or controller uses the same familiar notions as used in a type1 fuzzy logic controller as membership functions, rules, tnorms opera tions, fuzzification, inference, defuzzification. A type1 fuzzy logic system consists basically of three blocs; fuz zification, inference and defuzzification as presented in Figure 1. A type2 fuzzy logic system is very similar to type1, where it follows the same methodology, but the only difference is in the third block where we no longer speak of only defuzzification but we speak about a type reducer and defuzzification parts that constitute both the output processing block. This difference is mainly asso ciated with the nature of the membership functions, where typereducer is needed due to the added degree in the kind of fuzzy sets. Figure 2 presents a type2 fuzzy logic system. Today, the two most popular fuzzy logic systems used by engineers in control are the Mamdani and TSK sys tems. 2.2.1. Mamdani Type2 Fuzzy Logic Controller 2.2.1.1. Fuzzification In this part, we must first define the fuzzy sets of all in puts’ system. Those memberships can contain one or se veral type2 fuzzy sets. Second, the fuzzifier maps in puts into the associated fuzzy sets to determine the de gree of membership of each input variable. We consider only singleton fuzzification for which the inputs are crisp values. 2.2.1.2. Inference This block expresses the relationship that exists between the input variables (expressed as linguistic variables) and the output variables (also expressed as linguistic vari ables). As in type1 fuzzy logic, in the design of a type2 fuzzy logic we generally have IFTHEN rules. The for Figure 1. Type1 fuzzy logic. Figure 2. Type2 fuzzy logic. mulation of rules is the same. The only distinction be tween type1 and type2 is associated with the nature of the membership functions. The inference engine com bines rules and makes a combination between input type2 fuzzy sets and output type2 fuzzy sets. This is ensured by searching unions and intersections of type2 sets, as well as compositions of type2 relations. For a type2 fuzzy logic with inputs and 11 p ,, p XxX and one output Y, and with M rules. The th rule has the following form: 11 :IFisand andis THEN is1 p RxF x yGl M F The firing strength of the ith rule is as in (4). The re sult of the input and antecedent operations is an interval type1 set. 11 ,[,] , iiii pp iii ii pp FFFF Fxfxfxff xx x (4) where i F and i p F designed respectively upper and lower membership grades of i F and 1 1 ii p ii p ip FF i xx F Fxx (5) Since generally we use the meet operation under product or minimum tnorm. So, at each value of x the intersection and union operations are referred to as the meet and join operations, respectively. 2.2.1.3. Type Reducer and Defuzzification In a type1 fuzzy logic system the output of the inference engine corresponding to each fired rule is a type1 set. The defuzzifier combines those output sets to obtain a single output set. Using one of the existing methods of defuzzification, for example the centroid of sets, the de fuzzifier searches the centroid of the obtained set to ob tain finally a crisp output. In a type2 fuzzy logic, since this kind of system deal with type2 sets, then it is nec essary to have a type reducer block to map a T2 FS into a T1 FS, and then defuzzification, as usual, maps that T1 FS into a crisp number. We can consider that the de fuzzification block of a T1 fuzzy logic is replaced by the output processing block in a T2 fuzzy logic. That block consists of typereducer followed by defuzzification. In fact, type Reducer was proposed by Karnik and Mendel [7]. For now, there are five different typereduction me thods. Karnik and Mendel [19] defined the centroid of an IT2 FS which is an IT1 FS that is ensured using the Ex tension Principle. This IT1 FS is characterized by its left and right end points l and r , which can be written Copyright © 2012 SciRes. JILSA
Interval Type2 Fuzzy Logic Control of Mobile Robots 294 in the following equation: 111 1 ,,, 1 , 1 M M MMMM rr Mii lr iM vwwvwwfffi i yy w (6) This equation is derived from the consequent centroid sets . , MM r ww 2.2.2. Interval Type 2 TSK Fuzzy Logic Contro ller The differences between a type1 TSK fuzzy logic con troller [31] and a Mamdani T1 fuzzy logic consist essen tially of the definition of outputs and then on the conse quent part of rules. Consider we have a first order type1 TSK fuzzy logic with inputs 11 p,, p XxX, one output Y, and with M rules. The th rule can be expressed as: 11 011 :IFisand andis THEN p pp RxFxF yccx cx A type2 TSK fuzzy logic controller or system (T2 TSK fuzzy logic) was firstly introduced by Liang and Mendel [32]. Although TSK type1 fuzzy systems have received a lot attention, the literature on TSK type 2 fuzzy systems is few. Liang and Mendel applied type2 TSK systems in channel equalization of channels [12]. Where, according to them, there are three models of T2 TSK fuzzy logics depending on the kind of the antece dent and consequent part of rules, to have: T2 TSK Model I, T2 TSKModel II and T2 TSKModel III. We can see in Tables 1 and 2 the difference between those models. where are the consequent parameters, , ii cC , Y are the outputs of the th rule, 1 j jp are type2 fuzzy sets and 1 j jp are type1 fuzzy sets. The firing strength of the ith rule i Wx with meet operation under product or minimum tnorm is an interval type1 set expressed as follows : 1 1 1 1 , i p ii p i ii iip F F ip FF Wx wxwx wx xx wx x (7) The final output is also an interval type1 set and is calculated as follows: Table 1. Models of T2 TSK FLS. Model Antecedents Consequents Model I Type2 fuzzy sets type1 fuzzy sets Model II Type2 fuzzy sets crisp numbers Model III Type1 fuzzy sets type1 fuzzy sets Table 2. Rules of T2 TSK FLS. TSK FLS Rules l Type1 11 011 IFisand and THEN ll pp lll l pp Fxis yccxLcx F T2 Model I 11 011 IFand andis THEN pp pp xisFx F YCCx Cx T2 Model II 11 011 IFisand andis THEN pp pp xFxF yccx cx T2 Model III 11 011 IFisand andis THEN pp pp xF xF YCCx Cx 11 1 11 1 ,,, ,,, 1 MM lr Mii i MM M yywwi i YYYWWy y wy w (8) where i i Y , and , thus for each rule we will obtain l and . Since all sets are crisp, the Equation (9) results to: ,,1 iii lr YyyiM yr y 11 11 ; MM ii ii ll rl ii lr MM ii lr ii wy wy yy ww (9) And the defuzzified output is: 2 lr yyy (10) 3. Interval Type2 TSK Fuzzy Logic for an Obstacle Avoidance Behavior 3.1. Conception of Type1 Behavior We designed a zeroorder TSK type1 fuzzy logic con troller for the navigation of a mobile robot in dynamic and unknown environment for obstacle avoidance be havior. The purpose of the controller is to perform the navigation in unknown and dynamic environments for polygonal mobile robot. In this behavior we defined six sensor measurements inputs,as are shown in Figure 3 where those distances represent respectively the zones: central, left, right, lateral left, lateral right and back. Each input expressed the distance to the nearest obstacle in the zone of vision of its corre spondent sensor and is defined by two MFs Near and Far, which are represented by trapezoidal membership func tions as in Figure 4. ,,, ,and clrllrla dddddd Copyright © 2012 SciRes. JILSA
Interval Type2 Fuzzy Logic Control of Mobile Robots 295 The outputs of the behavior are the left and right wheels’ rotational velocities noted respectively l V and r. We defined 64 TSK rules deduced by authors’ hu man expertise. For obstacles near the robot, we have widely used the rotation without translation to avoid ob stacles instantly. An example of rules is expressed in Equation (11), where we defined the case of robot finding obstacles in his whole left side, so to guarantee avoiding static or dynamic obstacles the robot turn little to his right. V IFis Far andis Near andis Far andisNear andisFar andis Far THEN ,0.5,0 clr lllr a lr ddd ddd VV (11) Figure 3. Representation of robot sensor positions. (a) (b) (c) Figure 4. Membership functions of typ e1 fuzz y logic inputs . 3.2. Conception of Type2 Behavior In the conception of type2 fuzzy logic for obstacle avoi dance behavior, we extended the proposed type1 con troller to a type2 one by adding uncertainties in both the antecedent and the consequent parts of each rule. The main idea here consists in spreading the membership functions’ values of the antecedent part by ± A%, and the consequent part by ± C%. Therefore by this, our type2 fuzzy logic is an IT2 TSK fuzzy logic Model I. So, the six inputs of the behavior were extended by ± A%. The following figures, Figures 57, show those inputs for ± A% = ± 5%. The outputs of the controller are the same as in type1 fuzzy logic r V and l V. We have chosen to fix the spread of the consequent parameter “C” equal to 1 rad/s. We have defined thus 64 TSK T2 rules that are repre sented in Table 3 and are as the following form: 123 45 :IFis andis andis andis andis andis THEN ,, clr ll rl a lr lr RdFdFd dF dF dF VV CC 6 F Figure 5. MFs of the frontal input with ± 5%. Figure 6. MFs of the lateral input with ± 5%. Figure 7. MFs of the back input with ± 5%. Copyright © 2012 SciRes. JILSA
Interval Type2 Fuzzy Logic Control of Mobile Robots 296 where , 6 are type1 fuzzy sets, and are the consequent parameters for and . , lr CC 1,, 6j 1, 4 In the inference engine, we use the meet operation un der the product tnorm. 3.3. Simulation Results Several simulations were tested while varying the spread of the antecedent parameters ± A%. The results were implemented in Matlab 6.5 under the operating system Windows XP. We used an environment with dimensions of 189 × 190 containing arbitrary complex obstacles, using the SIMROBOT toolbox software [33]. A wheeled mobile robot is considered under the assumptions of nonslipping and pure rolling. Its kinematics can be de rived using Figure 8. This robot has two degrees of freedom: ytranslation and either xtranslation or zrotation. Its dimensions are taken into account when it navigates in an arbitrary com plex environment. The actuated inverse velocity solution of this robot is as the following equation 1 2 2 2 2 1 1 1 1 x y z B Wab ba B Wab ba b B ll ll ll ll Rl (12) Table 3. Rules of the T2 FLC. Rules Inputs Outputs c d l d r d ll d rl d a d , lr VV 1 L L L L L L [1.85, 1.85] ± C% 2 P P P P P P [0.95, −0.95] ± C% . . . . . . . . 64 P P P P P P [0.71, −0.71] ± C% Figure 8. Kinematic of the Simrobot robot. where: , B y : are translational velocities of the robot body in ms, : is the robot zrotational velocity in rad s, 12 W , W : are wheel rotational velocities in rads , : is actuated wheel radius in and : are distances of wheels from robot's axes in . More details about the robot used kinematics can be found in [33]. Rm, ab ll m To prove the efficiency of T2 fuzzy sets, we firstly choose to test the robot navigation in a complex place, like a narrow passageway. And, as we have not made any learning study on the adequate footprint of uncertainty (FOU) to our T2 fuzzy logic, we make simulations as follows: We tested T1 fuzzy logic and T2 fuzzy logic for different spread or (FOU), from the same initial position and for the same number of steps, which were re spectively chosen to “[105 179]” and “110 steps”. Re sults are presented in the following part; Figure 9 re presents the initial robot position. Figure 10 shows the trajectory generated by the robot using T1 fuzzy logic controller. From this figure, we notice that while passing through the narrow passage, the robot makes several oscillations to attempt finally, at the end of 110 steps, the position [140 178]. Sure our T1 fuzzy logic controller is not perfect and may contains some issues of errors. This explains well the obtained oscillations in the resulted trajectory where several rules give opposite and acute outputs reflecting an unstable navigation. Figure 9. Initial chosen robot position (position = [105 179]). Figure 10. TSK T1 fuzzy logic path for 110 steps. Copyright © 2012 SciRes. JILSA
Interval Type2 Fuzzy Logic Control of Mobile Robots 297 Figures 11(a)(f) represent the simulation results of the T2 TSK fuzzy logic for different deviations of the antecedent part ± A%, which correspond respectively to 5, 10, 15, 20, 30 and 40. It is clear from those figures the ability of T2 fuzzy logics in avoiding obstacles or walls as T1 fuzzy logic, but the most important things to high light are that all the generated trajectories by T2 fuzzy logics are clearly smoother and contain less oscillations and deviations than the one generated by type1 fuzzy logic in Figure 10. Thus T2 fuzzy sets can reduce uncer tainties coming from not perfect tuning or noisy database. Besides, we remark that for the same number of steps, the robot in all T2 fuzzy logics browsed more land than in T1 fuzzy logic. So T2 controllers are more rapid and allow the robot to arrive faster to a given destination point. To have a quantitative comparison, and to demonstrate well the smoothness of the obtained T2 trajectories, we choose to concentrate only on the narrow passageway; we propose to visualize the angular velocities generated by all T1 and T2 controllers, all from the same initial position [105 179] to almost the same final one [140 178] which represents the end of the passage way as shown in Figure 12, while the number of steps is varying depend ing on the adopted controller to arrive quickly or slowly to the final position. Thus, we obtain those results; Fig ure 13 represents the generated angular velocity by T1 fuzzy logic, where the positive values represent the left robot turnings and negative ones represent the right turn (a) (b) (c) (d) (e) (f) Figure 11. TSK T2 FL paths for 110 steps. (a) A = 5; (b) A = 10; (c) A = 15; (d) A = 20; (e) A = 30; (f) A = 40. Figure 12. Initial and final chosen robot positions. Figure 13. Simulation generated by TSK T1 FLC. ings. We notice from the figure that the robot continu ously oscillates right and left along the whole passage, reflecting by thus the lack of stability and smoothness of T1 fuzzy logic. Whereas Figures 14(a)(f) denote the ge nerated angular velocity by T2 fuzzy logics for different deviations A, which correspond respectively to 5, 10, 15, 20, 30 and 40. We remark clearly from those Figures 13 and 14, that all T2 generated trajectories contain less peaks and oscillations than the T1 generated trajectory. From those figures we extracted in Table 4 some re sults; we calculated first number of steps taken by each simulation. We can remark that navigating from the same initial position to the same final one takes from T1 fuzzy logic the biggest number of steps which is 110 steps than all the other T2 fuzzy logics that takes 93 steps for A = 5 to only 19 steps for A = 40. So, T2 fuzzy logics are faster than T1 fuzzy logic. Besides, in the table we extracted a smoothness meas urement: the angular velocity smoothness Index (AVSI). AVSI represents the average accumulative angular ve locities made by the robot simulation during k steps. Moreover, we calculate the Mean Square Error (MSE) expressing the error between the actual generated output Copyright © 2012 SciRes. JILSA
Interval Type2 Fuzzy Logic Control of Mobile Robots Copyright © 2012 SciRes. JILSA 298 (a) (b) (c) (d) (e) (f) Figure 14. Simulations generated by TSK T2 FLCs for different A. (a) A = 5; (b) A = 10; (c) A = 15; (d) A = 20; (e) A = 30; (f) A = 40. Table 4. Simulation results given by T1 and T2 controllers. TSK FLC T1 T2 A = 5 T2 A = 10 T2 A = 15 T2 A = 20 T2 A = 30 T2 A = 40 Steps 110 93 58 49 38 36 19 AVSI (˚/s) 2.6437 2.5582 2.5522 2.4100 2.1224 1.2763 0.3242 MSE 57.08 37.895 33.541 32.075 19.073 19.061 2.0678
Interval Type2 Fuzzy Logic Control of Mobile Robots 299 from the controller and the expected desired output in the passage that normally might be a straight line path. The equations of these measurements are given as follows: 1 2 1 AVSI 1 MSE k i i k idi i Vk yy k (13) It can be seen also from Table 4 that: while the pa rameter “A” is increasing, the navigator becomes smoo ther and with less oscillations. This is expressed mainly with a degradation in AVSI and MSE measurements (A = 40, AVSI = 0.3242˚/s; MSE = 2.0678).Sure there is a limit of this decrease, but this point is not discussed in this paper. Nevertheless, this remark is very interesting as an optimization point in future work. The most impor tant thing to highlight from the table is that all the T2 generated trajectories represent better results than the T1 one in terms of AVSI or MSE. Secondly, to prove the efficiency of type2 fuzzy sets in the setting of local avoiding obstacles, we tested the controllers towards moving obstacles in dynamic and unknown environment. For T2 controller, we choose to test only an example of a T2 controller with a spread equal to A = 20. The dynamicity of the environment is manifested by the presence of several robots. We can see in Figures 15(a) and (b) the ability of robots in both T1 and T2 controllers in avoiding obstacles and also in avoiding themselves. Although we tested in both figures the robots from same initial positions and for the same number of steps, we can see that we have not the same robots trajectories and this since the controllers did not give the same instant outputs. Also, it is clear from both figures the smoothness, the rapidity of T2 controller in relation to T1 one, where robots in T2 figure browse more land with smooth turnings. To enhance the found result, we presented another simulation in a different restricted complex place, where the robot is supposed to be front with a corner as in Fig ures 16(a) and (b). We can see from the different gener ated trajectories that the T2 fuzzy logic with A = 20 pre sents the smoothest path essentially in the corner part. Where T2 robot turns slowly towards the corner in a way it seems following the wall. Whereas T1 robot presents sharp turns in a way it turns back in other direction. 4. Interval Type2 Fuzzy Logic for a Wall Following Behavior In this section we presented a type2 mamdani fuzzy logic for a wall following behavior for the miniature mo bile robot kheperaII from KTeam [30]. In reality, this robot presents a good example of existing uncertainties, (a) (b) Figure 15. TSK T1 T2 paths (A = 20) from the same initial robots positions. (a) (b) Figure 16. Simulations generated by T1 and T2 FLCs against a corner. where it is faced to a large amount of vagueness. The kheperaII robot has 8 infrared proximity sensors. Each of them has a maximum range of measurement of about 5cm with accuracy and resolution depending on the mea sured distance. Those sensors are imprecise and present several kinds of errors like ambient light, color, shape and intensity of the detected obstacle. It was proved in [34] that the total error in the distance estimation by an infrared sensor depends on the uncertainty in the read ings and the uncertainty in the angle of incidence of the sensor. In our type2 fuzzy controller we have three in puts, two outputs and eight rules, that is deduced from its corresponding type1 fuzzy logic by spreading the ante cedent and consequence parts by a footprint of uncer tainty. The robot have to navigate forward in the envi ronment until it detects an unexpected obstacle or a wall, follows its contour on the right side with little turns. To detect the wall, the robot used its six frontal proximity sensors which are grouped two by two to constitute the three inputs. Each of them is defined by two fuzzy sets that are presented in Figure 17. The outputs are the left and right wheels’ rotational velocities of khepera Vr and Vl. The both are defined with two fuzzy sets and are pre sented in Figure 18. Whereas, in the inference part we conceived eight rules which are deduced from human expertise. Concer ning the implementation part, we used the kMatlab rou tines [30] to interact with Khepera robot over a serial connection. The two controllers are compared in same conditions, where robots are fixed approximately at the same initial position and then tested during the same number of steps Copyright © 2012 SciRes. JILSA
Interval Type2 Fuzzy Logic Control of Mobile Robots 300 chosen equal to 130 steps. Figure 19 shows the robot in the worked environment. As presented in previous simu lations, we extracted the angular velocities generated by T1 fuzzy logic and T2 fuzzy logic illustrated respectively in Figures 20 and 21. We remark from these trajectories that type2 path presents less deviation with a smoothness measurement index AVSI equal to 1.0934 than type1 path that pre sents an AVSI equal to 0.5554. Figure 22 presents paths generated by both controllers. We noted that in a part of T1 trajectory, the robot touched the wall and did not keep enough distance to it. Whereas T2 robot has kept ap proximately 8 cm to the wall. Besides, the T2 controller has traveled 22.8 cm more in land than its equivalent T1 controller. Thus, type1 fuzzy sets may not be robust en ough to handle uncertainties caused by infrared meas ures. 5. Conclusion In this paper we presented T2 controllers for mobile ro bot navigation. We can highlight from above results that type2 fuzzy sets are very interesting in control of mobile Figure 17. Inputs MFs of the wall following behavior. Figure 18. Outputs’ MF of the wall following behavior. Figure 19. The robot in the worked environment. Figure 20. Angular velocities generated by T1 FLC. Figure 21. Angular velocities generated by IT2 FLC. Figure 22. Paths generated by T1 and T2 FLCs. Copyright © 2012 SciRes. JILSA
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