Uncertainty Modeling with Interval Tye-2 Fuzzy Logic Systems in Mobile Robotics Ondrej Linda, Student Member, IEEE, Milos Manic, Senior Member, IEEE bstract Interval Tye-2 Fuzzy Logic Systems (IT2 FLSs) have been commonly attributed with the caability to model and coe with dynamic uncertainties. However, the interretation of this uncertainty modeling using the IT2 FLSs have been rarely addressed or taken into consideration during the design of the resective fuzzy controller. This aer extends the reviously roosed method for incororating the exerimentally measured inut uncertainty into the design of the IT2 FLS. Two novel uncertainty quantifiers are roosed to track the uncertainty modeling throughout the inference rocess: the antecedent uncertainty and the consequent uncertainty quantifiers. Further, the new IT2 FLS design method was used to design a wallfollowing navigation controller for an autonomous mobile robot. It is demonstrated that the new IT2 FLS design offers imroved uncertainty modeling, when comared to classical design methodologies. It was shown that the modeled inut uncertainty is more accurately reflected in the system outut as the geometry of the tye-reduced interval centroid. This uncertainty model rovides valuable information about the uncertainty associated with the outut decision and can be used for more informer decision making. Index Terms Interval Tye-2 Fuzzy Logic Systems, Uncertainty Modeling, Centroid, Tye-Reduction. T I. INTRODUCTION YE-2 Fuzzy Logic Systems (T2 FLSs), roosed by Zadeh [1], have received increased attention of many researchers in the ast decade [2]-[4]. T2 FLSs have been alied in many engineering areas, demonstrating their ability to outerform Tye-1 (T1) FLSs mainly in the resence of dynamic uncertainties [5]-[7]. The major difference between the T1 and T2 FLSs is in the model of individual Fuzzy Sets (FSs), which use membershi degrees that are themselves FSs. The most commonly used kind of T2 FLS is the Interval T2 (IT2) FLS, which uses interval membershi degrees. Many researchers argue in favor of IT2 FLSs because of their otential to model and minimize the effects of dynamic uncertainties [4], [6], [8]. Tyically, the erformance of IT2 FLSs in various alications is comared to their T1 counterarts demonstrating imrovements when noise and uncertainty are introduced into the system. The imroved erformance can be attributed to the Footrint of Uncertainty (FOU) of the IT2 FSs, which rovide additional dimension for designing the fuzzy membershi functions. The inference rocess of IT2 FLSs results in an outut IT2 FS. This IT2 FS must be first tye-reduced into its interval T1 centroid, which can then be defuzzified into the final outut value [9]. Many researchers associate the geometrical roerties of the interval centroid with the uncertainty about the system s outut [10]-[14]. For instance, Wu and Mendel state in [13] that: the length of the tye-reduced set can therefore be used to measure the extent of the outut s uncertainty. Other researchers used the width of the interval centroid to create an uncertainty bounds on the system s outut for redicting micro milling cutting forces [12]. However, the correctness of such outut uncertainty interretation has not been reviously addressed. This aer focuses on the analyzing the interretation of uncertainty modeling with IT2 FLSs. reviously, a method for incororating the exerimentally measured inut uncertainty into the design of the IT2 FLS was roosed [15]. This design method first extracts the inut uncertainty distribution from the inut data and then incororates the measured uncertainty into the FOU of the IT2 FSs. This resented aer extends the revious work in two major ways: i) two additional uncertainty quantifiers for tracking the uncertainty modeling throughout the inference rocess are roosed, and ii) the novel design method is used to imlement an IT2 FLS for wall-following behavior of an autonomous mobile robot. The imlemented wall-following behavior on real mobile robot validates the resented conclusions in real oerational settings. It is shown that the modeled inut uncertainty is more accurately reflected in the system outut as the geometry of the tye-reduced interval centroid. The rest of the aer is organized as follows. Section II rovides background review on IT2 FSs and FLSs. The novel uncertainty quantifiers for IT2 FLSs are introduced in Section III. The design of the IT2 FLSs for autonomous wallfollowing mobile behavior is exlained in Section IV. Exerimental results are shown in Section V and the aer is concluded in Section VI. II. INTERVL TYE-2 FUZZY LOGIC SYSTEMS This section rovides brief overview of IT2 FLSs. n IT2 FLS emloys one or more IT2 FSs. n IT2 FS ~ can be exressed as [4]: ~ 1/( x, u) J [0,1] xx uj x Here, x and u are the rimary and secondary variables, X is the domain of variable x and J x is the rimary membershi of x. In the secial case of IT2 FSs, all secondary grades of fuzzy x (1)
set ~ are equal to 1. By instantiating the variable x into a secific value x, the vertical slice of the IT2 FS is obtained: ~ ( x x, u) ~ 1/ u J [0,1] (2) uj x x This vertical slice is an interval T1 fuzzy set. The domain of the rimary membershis J x defines the FOU of ~. The FOU of an IT2 FS ~ can be bounded by its uer and lower membershi functions (see Fig. 1): FOU ~ ) (3) ( ) ( ~, ~ xx This reresentation constitutes a substantial simlification when comared to the general T2 FSs. Here, only two T1 fuzzy membershi functions (the uer ( ~ ) and the lower ( ~ ) boundary of the FOU) are sufficient to comletely describe the IT2 FS. The simlification removes much of the comutational burden of general T2 FLSs. The fuzzy inference rocess then uses interval join and meet oerations to calculate the outut fuzzy sets based on the set of rovided linguistic rules [4]. In order to obtain a cris outut value y, the resulting outut IT2 FS B ~ must be first tye reduced and then defuzzified. The interval centroid is an interval T1 FS that can be described by its left and right end oints y l and y r. These endoints can be comuted for instance by the Karnik-Mendel (KM) iterative rocedure [9]. Using the boundary values of the tye-reduced interval centroid the final cris defuzzified value y can be obtained as the mean of the centroid interval: ( y l yr ) y (4) 2 The combined result of the outut rocessing stage of the IT2 FLS is the cris outut value y accomanied by the interval centroid. The resence of the centroid constitutes one of the main advantages of IT2 FLSs when comared to T1 FLSs. The geometrical roerties of the interval centroid (e.g. its width) can rovide additional information commonly interreted as a measure uncertainty associated with the outut value. III. UNCERTINTY MODELING WITH IT2 FLSS In order to analyze the uncertainty modeling of IT2 FLSs, the joint inut uncertainty quantifier has been reviously roosed in [15]. In this Section, two novel uncertainty quantifiers are introduced. The uncertainty quantifiers are intended to track the uncertainty in the IT2 FLS through the individual stes of the fuzzy inference rocess: fuzzification, rule firing, rule aggregation and tye-reduction. The IT2 FLS is commonly attributed with the caability to model and minimize the effects of dynamic uncertainties [13]. The uncertainty is modeled by the FOUs of the IT2 FSs and it is translated into the interval centroid via the fuzzy inference rocess. Therefore, the IT2 FLS erforms a functional maing between the modeled inut uncertainties (i.e. the FOUs) and the outut uncertainty (i.e. the interval centroid). nalyzing the interretation of this uncertainty maing is imortant since the resence of reliable uncertainty measure associated with the outut variable constitutes valuable information.. Joint Inut Uncertainty Quantifier The joint inut uncertainty quantifier was reviously roosed in [15]. Consider an IT2 FS ~ described by its FOU using the uer and the lower membershi functions ~ and ( ) as in (4). The inut uncertainty u ~ associated ~ x with the fuzzification of an inut value x by fuzzy set ~ can be exressed as the width of the firing interval: u ~ ( ~ x x) ~ ( ) (5) In the first ste of the IT2 fuzzy inference rocess the inut variables are fuzzified using the inut IT2 FSs. The roosed joint inut uncertainty quantifier U I (x) associated with the fuzzification of an inut vector x can be exressed as the aggregated inut uncertainty from all firing intervals over all inut variables: M 1 j1 Fig. 1 Interval Tye-2 fuzzy set. M ~ ( x ) j 1 j1 I U u j ( x ) ~ j ( x ) ~ (6) Here, is the dimensionality of the inut vector x and M is the number of IT2 FSs for the secific dimension. B. ntecedent Uncertainty Quantifier The next ste in the fuzzy inference rocess is calculating the rule firing strength of each fuzzy rule by combining the membershi degrees of all rule antecedents. s the fuzzified Fig. 2 The IT2 FLS and the uncertainty quantifiers.
inut values are transformed into the rule antecedents, the joint inut uncertainty U I (x) is transformed into the roosed antecedent uncertainty quantifier U (x). This roosed measure of uncertainty is calculated as the cumulative uncertainty associated with the rule firing strengths: U K k ( x ) k ( x 1 1 k1 ) (7) Here, K is the number of fuzzy rules in the rule base and symbol denotes the t-norm oerator (e.g. minimum). C. Consequent Uncertainty Quantifier The fuzzy inference rocess follows with the rule aggregation ste. Tyically, multile fuzzy rules share common consequents. The roosed consequent uncertainty quantifier U C rovides an insight into how much aggregated uncertainty from all fuzzy rules will be alied to the rule consequents. The consequent uncertainty U C (x) can be exressed as follows: U C C m1 K m j1 1 j K m ( x ) j1 1 j ( x ) Here, C denotes the number of distinct rule consequents, K m is the number of fuzzy rules contributing to the m th consequent and symbol denotes the secific t-conorm oerator (e.g. maximum). D. Outut Uncertainty Quantifier Finally, the outut rocessing stage transforms the outut fuzzy set into the outut value y. s reviously demonstrated by other authors, the outut uncertainty U O (x) (also called the uncertainty interval [13]) is comuted as the width of the interval centroid: O U ( x ) y y (9) The values of y l and y r denote the left and right boundaries of the interval centroid. These values can be obtained using the iterative Karnik-Mendel algorithm. lternatively, an aroximate outut uncertainty can be calculated using the Wu-Mendel s uncertainty bounds method [13]. The calculation of the roosed uncertainty quantifiers is summarized in Fig. 2. It is trivial to show, that in accordance with the fundamental design rinciles of T2 FLS, when all r l (8) Fig. 3 Lego NXT autonomous mobile robot equied with two sonar sensors. sources of uncertainty disaear, the uncertainty quantifiers all reduce to zero. IV. DESIGN OF WLL-FOLLOWING IT2 FLS BSED ON MESURED INUT UNCERTINTY novel design methodology for incororating the measured inut uncertainties into the design of IT2 FLS was reviously roosed in [15]. In this aer, the roosed design method is used to imlement a wall-following navigation behavior for an autonomous mobile robot. The robotic latform consists of an autonomous Lego NXT mobile robot equied with two sonar range finders. The task of the mobile robot was to autonomously navigate in an unstructured indoor environment while negotiating obstacles. The navigation is controlled by an IT2 FLS that is imlemented on a lato CU remotely communicating with the mobile robot over a Bluetooth connection. The inuts of the IT2 FLS are connected to the sonar sensors. The calculated outut signal is roortional to the ower alied to the robot s differential motor drives. The Lego NXT robotic latform is deicted in Fig. 3. The design method can be summarized in three stes as follows: Ste 1: Measure the uncertainty distribution of the inuts by calibrating the inut sensors. Ste 2: Construct a T1 FLS with the desired control behavior. Ste 3: Extend the T1 FLS into the IT2 FLS by creating the FOUs of the IT2 FSs via fusing the measured inut uncertainty with the T1 fuzzy membershi functions. The first ste of the roosed design methodology is to measure the inut uncertainty of the actual inut data. The inut data are samled from a air of sonar range finders mounted on the mobile robot. In this alication, the distribution of inut uncertainty can be exerimentally measured by calibrating the sensors resonse against known ground truth values. The received sonar measurements can be considered subject to many kinds of uncertainties, e.g. due to (a) (b) (c) Fig. 4 The measured sensor uncertainties (a), the interolated continuous inut uncertainty distribution (b), and the inut T1 fuzzy sets (c).
(a) (b) (c) Fig. 5 The FOUs of the inut fuzzy sets with the embedded joint inut uncertainty for the left (a) and right (b) sonar sensors and the outut fuzzy sets (c). TBLE I FUZZY RULE TBLE & x 2 Small Medium Large x 1 Small Zero ositive ositive Medium Negative Zero ositive Large Negative Negative Zero the beam width, signal attenuation, variable reflexivity of surrounding materials or manufacturing defects in the sonic emitter or receiver [16]. First, the available range of inut values is discretized into M samles. The sonar sensors are laced at the calibrated distance from the reflective surface (e.g. wall). set of inut measurements is obtained for each calibrated distance. The standard deviation of the sensory readings at articular distance samle is comuted and stored as the amount of uncertainty associated with that articular inut value. The set of M measured standard deviation values defines the samled inut uncertainty for each robot s sonar sensor. n illustrative examle is shown in Fig. 4(a). The continuous distribution of inut uncertainty f u (x) for variable x can be obtained by alying linear interolation between the calibration oints. The interolated uncertainty distribution for both robot s sonar sensors is deicted in Fig. 4(b). The second ste in the roosed design methodology is the construction the initial T1 FLS. Here, it is assumed that the user has already constructed the initial T1 FLS, using one of the well-established design methodologies [4]. In this work, the Gaussian rincial membershi functions are considered. The major criteria for selecting the mean (m i ) and the standard i * deviation ( ) of rincial membershi function i is sufficient overla between neighboring fuzzy sets ensuring continuity of the outut variable [17]. In the resented case study three equidistantly saced Gaussian rincial membershi functions with identical standard deviations were considered. The resective T1 FSs are deicted in Fig. 4(c). The fuzzy rules of the T1 FLS are listed in Table I. In the third and final ste, the T1 FLS is extended into the IT2 FLS. The FOU of the inut IT2 FSs is constructed by fusing the exerimentally measured distribution of inut uncertainty f u (x) with the corresonding rincial membershi functions. Firstly, a maing between the range of measured standard deviations and the normalized uncertainty domain of the IT2 FSs must be established. This is achieved by scaling the amlitude of the measured uncertainty distribution by the maximum required inut uncertainty ~. The lower and the uer membershi functions ~ and ( ) of the inut u IT2 FS ~ can then obtained as follows: ~ x * fu ~ min, 1. 0 (10) 2 * fu ~ max ( x),0. 0 (11) 2 In order to achieve admissible design of the inut IT2 FSs, the values of both uer and lower membershi functions are bounded in the interval [0, 1]. The resulting FOU of the inut IT2 FSs together with the lotted distributions of the joint inut uncertainty quantifier U I (x) for both sonar sensors are deicted in Fig. 5(a)-(b). Here, the left and the right sonar inuts are denoted as inuts x 1 and x 2. It can be observed that the distribution of the joint inut uncertainty quantifier U I (x) accurately reflects the actual distribution of uncertainty in the inut data firing the IT2 FLS. The oututs of the IT2 FLS are designed as IT2 FS with uncertain mean to account for outut uncertainties in the system as deicted in Fig. 5(c). Figures 6(a)-6(d) deict the distribution of the uncertainty quantifiers U I, U, U C and U O in the inut domain. Furthermore, the outut control surface is shown in Fig. 6(e). V. EXERIMENTL RESULTS This section resents exerimental testing of the imlemented IT2 FLS for autonomous mobile robot wall- (a) (b) (c) (d) (e) Fig. 6 Distribution of the joint inut uncertainty (a), the antecedent uncertainty (b), the consequent uncertainty (c), the outut uncertainty (d) and the outut control surface (e) for the roosed wall-following IT2 FLS.
(a) (b) (c) Fig. 7 The inut signals (a), the outut signal and its uncertainty (b), and the uncertainty quantifiers (c) during an autonomous navigation of a mobile robot using the common IT2 FLS. following behavior. Both the common and the roosed IT2 FLS design methodology were imlemented and exerimentally comared. The common IT2 FLS used IT2 FSs with uncertain mean. The roosed IT2 FLS design was imlemented according to the descrition given in the revious Section with the inut uncertainty distribution as in Fig. 4(a). The robot s sonar sensors were also artificially augmented with random noise signal following the distribution on Fig. 4(a). It should be noted here that the subject of this exerimental study is mainly to analyze the uncertainty modeling erformance rather than measuring minor differences in the robot s trajectory. Both FLCs succeeded in satisfactory robot navigation but featured significantly different uncertainty modeling erformance. First, the recorded signals from robot with the common IT2 FLS are lotted in Fig. 7. It is easy to see when the robot aroached an obstacle as well as when it negotiated a narrow corridor (i.e. Event in Fig. 7(a)). The outut of the common IT2 FLS with the calculated outut uncertainty U O is deicted in Fig. 7(b). Negative and ositive values of the outut signal show robot s steering to the right and to the left while negotiating the sensed obstacles. The roosed uncertainty quantifiers U I, U, U C are lotted in Fig 7(c). main observation can be made as follows. The amlitude of the calculated outut uncertainty does not reflect the uncertainty in the sensory inuts (noise amlitude). In the labeled Event, high inut uncertainty was resent in both robot s inuts (negotiating a narrow corridor). However, because the distribution of the inut uncertainty was not taken into account during the design of the common IT2 FLS, the outut uncertainty U O does not reflect the increased inut uncertainty accordingly. Fig. 8 lots the measured sonar readings, the robot s control signal with the associated outut uncertainty U O and the calculated uncertainty quantifiers U I, U, U C for the roosed IT2 FLS. Two cases with increased inut uncertainty are labeled as Event B and Event C in Fig 8(a), where the robot was negotiating narrow corridors. The following two observations can be made. First, the outut uncertainty U O now more accurately reflects the increased uncertainty in the inut sensory readings (manifested as increased noise amlitude). For examle, the outut uncertainty signal lotted in Fig. 8(b) clearly marks Event B and Event C as events with increased uncertainty associated with the calculated outut. Second, as it was already demonstrated before, the roosed uncertainty quantifiers U I, U, U C feature strong ositive correlation with the outut uncertainty U O and rovide a good aroximation of the outut uncertainty. VI. CONCLUSION This aer extended the revious work on uncertainty modeling with IT2 FSs and IT2 FLSs. Two novel uncertainty quantifiers were roosed to track the uncertainty roagation throughout the inference rocess: antecedent uncertainty and
(a) (b) (c) Fig. 8 The inut signals (a), the outut signal and its uncertainty (b), and the uncertainty quantifiers (c) during an autonomous navigation of a mobile robot using the roosed uncertainty-based design of IT2 FLS. consequent uncertainty. The novel design methodology for constructing IT2 FLSs based on measured distribution of inut uncertainty was used to design a wall-following behavior for an autonomous mobile robot. The exerimental results verified that the roosed design methodology for IT2 FLS offers more accurate interretation of the uncertainty associated with the outut of the IT2 FLS. Such additional information can be further utilized for imroved control of the mobile robot (e.g. reduce robot s seed in more uncertain conditions). REFERENCES [1] L.. Zadeh, The Concet of a Linguistic Variable and its roximate Reasoning - II, in Information Sciences, No. 8,. 301-357, 1975. [2] S. Couland, R. John, Geometric Tye-1 and Tye-2 Fuzzy Logic Systems, in IEEE Trans. on Fuzzy Systems, vol. 15, no. 1,. 3-15, February 2007. [3] N. N. Karnik, J. M. Mendel, Tye-2 Fuzzy Logic Systems, in IEEE Trans. on Fuzzy Systems, vol. 7, no. 6,. 643-658, December 1999. [4] J. M. Mendel, Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions, rentice-hall, Uer Saddle River, NJ, 2001. [5] M. Biglarbegian, W. Melek, J. M. Mendel, On the robustness of Tye-1 and Interval Tye-2 fuzzy logic systems in modeling, in Information Sciences, vol. 181, issue: 7,. 1325-1347, ril 2011. [6] H.. Hagras, Hierarchical Tye-2 Fuzzy Logic Control rchitecture for utonomous Mobile Robots, in IEEE Trans. Fuzzy Systems, vol. 12, no. 4,. 524-539, 2004. [7] O. Linda, M. Manic, Interval Tye-2 Fuzzy Voter Design for Fault Tolerant Systems, in Information Sciences, in ress, 2011. [8] O. Linda, M. Manic, Comarative nalysis of Tye-1 and Tye-2 Fuzzy Control in Context of Learning Behaviors for Mobile Robotics, in roc. IEEE IECON 10, 36th nnual Conference of the IEEE Industrial Electronics Society, Glendale, rizona, US, Nov. 7-10, 2010. [9] N. Karnik, J. M. Mendel, Centroid of a tye-2 fuzzy set, in Information Sciences, vol. 132,. 195-220, 2001. [10] L. Li, W.-H. Lin, H. Liu, Tye-2 fuzzy logic aroach for short-term traffic forecasting, in IEEE roc. Intel. Trans. Syst., vol. 153, no. 1, March, 2006. [11] J. M. Mendel, Tye-2 Fuzzistics for Symmetric Interval Tye-2 Fuzzy Sets: art 1, Forward roblems, in IEEE Trans. on Fuzzy Systems, vol. 14, no. 6,. 781-791, December 2006. [12] Q. Ren, L. Baron, K. Jemielniak, M. Balazinski, Modeling of Dynamic Micromilling Cutting Forces Using Tye-2 Fuzzy Rule-Based System, in roc. IEEE World Congress on Comutational Intelligence, Barcelona, Sain,. 2311-2317, July 18-23, 2010. [13] H. Wu, J. M. Mendel, Uncertainty Bounds and Their Use in the Design of Interval Tye-2 Fuzzy Logic Systems, in IEEE Transaction of Fuzzy Systems, vol. 10, no. 5,. 622-639, October, 2002. [14] D. Wu, J. M. Mendel, Uncertainty measures for interval tye-2 fuzzy sets, in Information Sciences, No. 177,. 5378-5393, 2007. [15] O. Linda, M. Manic, Uncertainty Modeling for Interval Tye-2 Fuzzy Logic System Based on Sensor Characteristics, in roc. 2011 IEEE Symosium Series on Comutational Intelligence,. 31-37, ril 2011. [16] S. Guadarrama,. Ruiz-Mayor, roximate robotic maing from sonar data by modeling ercetions with antonyms, in Information Sciences, vol. 180, issue: 21,. 4164-4188, Nov. 2010. [17] D. Wu, J. M. Mendel, On the Continuity of Tye-1 and Interval Tye-2 Fuzzy Logic Systems, in IEEE Trans. on Fuzzy Systems, vol. 19, no. 1,. 179-192, Feb. 2011.