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Adaptive Fuzzy Type-II Controller for Wheeled Mobile Robot with Disturbances and Wheelslips

Adaptive Fuzzy Type-II Controller for Wheeled Mobile Robot with Disturbances and Wheelslips Hindawi Journal of Robotics Volume 2021, Article ID 6946210, 11 pages https://doi.org/10.1155/2021/6946210 Research Article Adaptive Fuzzy Type-II Controller for Wheeled Mobile Robot with Disturbances and Wheelslips 1 2 1 Viet Quoc Ha, Sen Huong-Thi Pham, and Nga Thi-Thuy Vu School of Electrical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam Faculty of Control and Automation, Electric Power University, Hanoi, Vietnam Correspondence should be addressed to Nga i-uy Vu; nga.vuthithuy@hust.edu.vn Received 9 June 2021; Revised 26 July 2021; Accepted 30 August 2021; Published 15 September 2021 Academic Editor: Arturo Buscarino Copyright © 2021 Viet Quoc Ha et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper proposed adaptive fuzzy type-II controllers for the wheeled mobile robot (WMR) systems under conditions of wheel slips and disturbances. e system includes two control loops: outer loop for position tracking and the inner loop for velocity tracking. In each loop, the controller has two parts: the feedback which keeps the system stable and the adaptive type-II fuzzy part which is used to compensate the unknown components that act on the system. e stability of each loop as well as the overall system is proven mathematically based on the Lyapunov theory. Finally, the simulation is setup to verify the effectiveness of the presented algorithm. e simulation results show that, in comparison with the corresponding fuzzy type-I controller, the performance of the adaptive fuzzy type-II controller is better, i.e., the position error is smaller and the velocity is almost smooth under the conditions that the reference trajectory is changed, and the system is affected by wheel slips and external disturbances. “pure rolling without slip.” As mentioned above, these as- 1. Introduction sumptions are not exactly since there are many factors such as unknown external disturbances and wheel slipping due to Wheeled mobile robots (WMRs) are widely used in industry and service robotics. Mobile robots are self-moving vehicles the slippery roads and the wheel. In order to reduce the effect and versatile with many indoor and outdoor applications [1]. of the wheel slips, an adaptive robust control is used for e control problems in the WRMs are quite rich such as trajectory tracking of wheeled mobile robots in the presence path planning, trajectory tracking, and obstacle avoidance. of unknown skidding and slipping [10]. Although this Each problem has different importance in specific appli- controller solved external disturbances, the error tracking is cation. Among these, trajectory tracking control has the role still pretty big. In [11], the adaptive fuzzy output feedback to keep the WMR following the desired trajectory. is controller is proposed to solve the trajectory problem for control problem is not easy because WMRs are under- WMR with the uncertainties and effects of external dis- actuated systems. Moreover, operation of WMR is greatly turbances. In this research, the tracking position errors affected by working conditions such as system uncertainties, converge asymptotically to a small neighborhood near the wheel slips, and external disturbances. erefore, tracking origin with a faster response than achieved by other existing control design for WRM still has the attraction to many controllers, and all of the signals are bounded. A controller researchers. based on the robust dynamic surface control method is Many nonlinear control methods have been used to solve proposed in [12] to eliminate the problems of “explosion of the tracking control problems of nonholonomic mobile complexity.” e controller can become much simpler than robots such as robust adaptive [2], sliding mode control backstepping controller, but the illustrated results are so [3, 4], backstepping control [5, 6], adaptive fuzzy logic poor. An adaptive neural network based on reinforcement control [7], and adaptive neural-network control [8, 9]. learning is presented in [13] for WMR with considering However, all of these are proposed with the assumption of skidding and slipping. In this work, the error tracking is 2 Journal of Robotics (iv) e stability of the closed-loop system with the almost zero, but the structure with four neural networks in the scheme can be the burden for calculation system. proposed controller is proven mathematically based on the Lyapunov theory. is proof for the cascade Recently, the advanced and intelligent control methods are considered as flexible tools to deal with the uncertain system is much harder than the single loop. systems. In [14], the authors propose a strategy that can ensure optimal working under the imperfect dynamic 2. System Model and Adaptive conditions based on the excitation of the hidden dynamics. Controller Design Nevertheless, some strict assumptions need to be satisfied to complete the control strategy. In [15], the disturbance 2.1. System Modeling. Let consider a nonholonomic WMR problems for nonlinear systems are solved by using neural with skidding and slipping as shown in Figure 1. Located at networks. is controller guarantees that all the signals in point G(x , y ) is the center of mass (COM) of the wheel G G the closed-loop system are semiglobally uniformly ultimate mobile robot, and point M(x , y ) indicates the midpoint M M bounded. However, due to the effect of the iterative deri- of the wheel shaft. e distance between G and M is a.r is the vation of virtual control laws, the “complexity explosion” radius of each driving wheel, and the length of wheel mobile problem appears which increases the computational com- robot shaft is 2b. θ is the orientation of the wheel mobile plexity. Another approach to deal with the model parameter robot with respect to the initial frame. uncertainty problem is shown in [16]. In this research, a F and F illustrate the total longitudinal friction force at 1 2 model is developed for corrosion prediction using a neu- the right and left driving wheels, respectively. F is the total rofuzzy expert system. e results show that the neurofuzzy lateral force that acts on the mobile robot, whereas F and ϖ expert systems have better accuracy with fewer number of are the external disturbances and moment at point G input parameters than the neural networks prediction accordingly. model. In [17], a controller based on the fuzzy logic system to According to [20], we can consider the kinematics model approximate unknown parameters in nonlinear systems is of the wheeled mobile robot under slipping and skidding proposed. In this work, the tracking error can converge to a condition as follows: small neighborhood of zero with a fixed time, and all the ⎪ x _ � ϑ cos θ − η _ sin θ, signals of the system are bounded. However, the external ⎧ ⎪ M disturbances are not considered in this research. In [18, 19], ⎨ y _ � ϑ sin θ + η _ cos θ, (1) the interval fuzzy type-II controller is introduced for non- ⎪ linear uncertain systems. It is proved that the fuzzy type-II ⎩ θ � ω, controller handles uncertainties and external disturbances better than the fuzzy type-I controller. where ϑ is the forward velocity and ω is the angular velocity In this paper, an adaptive interval type-2 fuzzy logic of the wheeled mobile robot at points M, ϑ, and ω and are controller is proposed for two-loop control of wheeled mobile calculated as follows: robots with external disturbances. e proposed controller is _ _ r􏼐ϕ + ϕ 􏼑 c _ + c _ expected to allow the error tracking of WMRs to converge to R L R L ϑ � + , zero under the acting of unknown wheel slips, unknown 2 2 (2) bounded external disturbances, and model uncertainties. _ _ r􏼐ϕ − ϕ 􏼑 e main contributions of the proposed algorithm can _ _ c − c R L R L ω � + , be stated as follows: 2b 2b (i) Design the adaptive interval fuzzy type-II controller with c and c are the coordinates of the longitudinal slip of R L for both the inner and outer loop of WMR with the right and left driving wheels, respectively, and η is the uncertainties and external disturbances. coordinate of the lateral slip along the wheel shaft. φ _ and φ _ R L (ii) Under the conditions that the reference trajectory is are the angular velocities of the right and the left wheels, respectively. complex changed, the proposed controller can still We consider a two-wheeled mobile robot with coordi- deal with the tracking problem, and the result nates illustrated in Figure 1 described by the following showed that the error between the reference tra- dynamic models [20]: jectory and the real trajectory is quite small. (iii) Using the adaptive fuzzy type-II controller, the Mv _ + B(v)v + Ev + Q€ c + Cη _ + Gη € + τ � τ, (3) difference between the real velocity and the refer- ence velocity is almost zero. Moreover, in this re- _ _ where v � 􏽨 􏽩 , c � 􏼂 c c 􏼃 , and M � ϕ ϕ R L R L search, the comparison results show that the real m m 11 12 velocity of the proposed controller is smoother than 􏼢 􏼣 m m 21 22 the real velocity of the fuzzy type-I controller. F1 F2 F4 F3 Journal of Robotics 3 Figure 1: Model of the wheeled mobile robot. 2 2 2 2 r a r r m � m � m 􏼠 − 􏼡 − I + 2I 􏼁 , 12 21 G G D 2 2 4b 4b 2 2 2 2 r a r r m + + m � m � m 􏼠 + 􏼡 I + 2I 􏼁 r + I , 11 22 G G D W w 2 2 4b 4b Q Q 1 2 ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ Q � , Q Q 2 1 r a r Q � m 􏼠1 ± 􏼡 ± I + 2I 􏼁 , 1,2 G G D 4 4b 0 1 ar ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ _ _ ⎢ ⎥ ⎣ ⎦ B(v) � m 􏼐ϕ − ϕ 􏼑 , (4) 2 R L 4b − 1 0 0 1 ar c _ − c _ R L ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ E � 􏼠m 􏼡􏼠 􏼡 , 2b 2b − 1 0 ar ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ G � m , 2b − 1 ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ C � m ω, τ � τ , τ . 􏼂 􏼃 R L 2.2. Adaptive Tracking Controller Design for WMR. e 2.2.1. Kinematic Controller Design. In the xOy coordinate, control objective of this work is to design the controller for the position error between the point M(x , y ) and the M M the WMR subjected to unknown wheel slips and model target point C(x , y ) is calculated as follows: C C uncertainties to ensure that the desired reference trajectory e cos θ sin θ x − x p1 C M is tracked. To solve this problem, the adaptive control loop is ⎣ ⎦ ⎡ ⎤ e � � 􏼢 􏼣􏼢 􏼣. (5) − sin θ cos θ y − y proposed with the block diagram shown in Figure 2: p2 C M 2b 4 Journal of Robotics e ~ p u u e m Position Controller Velocity Controller WMR y 2 Type-2 interval MFs Type-2 interval MFs Adaptation ~ ξ ~ e e Adaptation law law . Figure 2: e control scheme for the wheeled mobile robot. e derivative of (5) along with time under the condition det(h) � − e r /2b ⟶ 0 and h is not invertible. To avoid p1 of wheel slips and external disturbances is obtained as [20] this problem, controllers (8) and (10) are modified as follows: e_ cos θ sin θ x _ p1 ⎡ ⎣ ⎤ ⎦ e_ � � 􏼢 􏼣􏼢 􏼣 + hu + d , (6) p p e_ − sin θ cos θ y _ _ cos θ sin θ x p2 ⎧ ⎪ C C ⎪ 􏼌 􏼌 ⎪ ⎡ ⎢ ⎤ ⎥⎡ ⎢ ⎤ ⎥ 􏼌 􏼌 ⎢ ⎥⎢ ⎥ − 1⎢ ⎥⎢ ⎥ ⎪ ⎢ ⎥⎢ ⎥ 􏼌 􏼌 ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎪ ⎢ ⎥⎢ ⎥ 􏼌 􏼌 − h ⎢ ⎥⎢ ⎥, if e ≥ λ, ⎢ ⎥⎢ ⎥ 􏼌 ⎪ ⎣ ⎦⎣ ⎦ 􏼌 p1 (e /b − 1)r/2 − (e /b + 1)r/2 p2 p2 where h � 􏼢 􏼣 and d � ⎪ − sin θ cos θ y _ ⎪ C − e r/2b e r/2b p1 p1 u � (((c _ − c _ )/2b)e ) − (( c _ + c _ )/2) R L p2 R L 􏼢 􏼣 ⎪ cos θ sin θ x _ _ _ _ C − ((( c − c )/2b)e ) − η ⎪ 􏼌 􏼌 R L p1 ⎪ ⎢ ⎥⎢ ⎥ 1 ⎡ ⎢ ⎤ ⎥⎡ ⎢ ⎤ ⎥ 􏼌 􏼌 ⎢ ⎥⎢ ⎥ ⎪ ⎢ ⎥⎢ ⎥ 􏼌 ⎢ ⎥⎢ ⎥ ⎪ ⎢ ⎥⎢ ⎥ 􏼌 􏼌 ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥, if e < λ, ⎪ ⎢ ⎥⎢ ⎥ 􏼌 􏼌 ⎣ ⎦⎣ ⎦ p1 (11) e control u is separated into two components u and 1 ⎩ − sin θ cos θ y _ u , in which u is the feed-forward part used to compensate C 2 1 for the nonlinear component in model (6) and u is the 􏼌 􏼌 􏼌 􏼌 − 1 􏼌 􏼌 ⎧ ⎪ 􏼌 􏼌 − h K e , if e ≥ λ, feedback controller. ⎪ 􏼌 􏼌 1 p p1 u � u � u + u , (7) 1 2 2 ⎪ 􏼌 􏼌 1 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 K e , if e < λ. 1 p 􏼌 p1􏼌 where cos θ sin θ x − 1 Here, λ is a small enough scalar. u � − h 􏼢 􏼣􏼢 􏼣. (8) In fact, the disturbances are unknown (d ≠ 0), and − sin θ cos θ y _ then we cannot apply controller (10) directly. To solve Substituting (7) and (8) into (6), we get this problem, we will design the adaptive interval fuzzy type-II logic controller to approximate control law u as e_ � hu + d . (9) p 2 p follows: First, considering (9) without disturbances (d � 0), we propose the controller: − 1 ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎝ ⎠ u � u􏼐e |φ 􏼑 − h Λ e + Λ 􏽚 e dt , (12) 2 p 1 1 p 2 p ∗ − 1 u � − h K e , (10) 2 1 p with K is the positive scalar. where Λ and Λ are positive scalars and u(e |φ ) 1 1 2 p 1 Substituting (10) into (9), we can achieve the control goal � ξ (e )φ is the output of the interval fuzzy type-II logic p 1 controller in which ξ (e ) � diag ξ (e ) ξ (e ), . . . that is lim e (t) � 0. 􏼐􏽨 11 p 12 t⟶∞ p 1 p p It is noted that when the WMR tracks the desired tra- ξ (e )]), ξ � f / 􏽐 f , where f is the firing in- 1M p 1j 1j 1j 1j j�1 jectory, the errors e and e go to zero. is leads to terval of the jth-rule, M is the number of rules, and φ is p1 p2 1 Journal of Robotics 5 the designed parameter which is calculated by the adaptive law: e_ � − Ae − Λ 􏽚 e dt + B h􏼐u􏼐e ∣ φ 􏼑 − u􏼐e ∣ φ 􏼑􏼑 + B 􏼐Δ + w 􏽥 􏼑 p p 2 p 1 p 1 p 1 1 p _ (13) φ � − 􏽮ξ 􏼐e 􏼑h B e 􏽯. 1 1 p 1 p � − Ae − Λ 􏽚 e dt + B hξ φ 􏽥 + B σ, p 2 p 1 1 1 1 Substituting (12) into (9), it leads to the following results: (16) ∗ ∗ e _ � − Λ e − Λ 􏽚 e dt + h u e ∣ φ − u + hu + d 􏼐 􏼐 􏼑 􏼑 p 1 p 2 p p 1 2 2 p where φ 􏽥 � φ − φ is the estimated errors, σ � Δ + w 􏽥 . 1 1 1 p � − Ae − Λ 􏽚 e dt + B h􏼐u􏼐e ∣ φ 􏼑 − u 􏼑 + B Δ , Assumption 1. With σ ∈ L [0, T], T ∈ (0, +∞), there exists p 2 p 1 p 1 2 1 p 0 the constant c such that σ σ ≤ c . Define a Lyapunov function (14) 1 0 T t t d d where A � Λ + K , B � 􏼢 􏼣 and Δ � 􏽨 􏽩 . p1 p2 1 1 1 p 1 1 1 T T 0 1 ⎜ ⎟ ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ ⎝ ⎠ (17) V � e e + 􏽚 e dt Λ 􏽚 e dt + φ 􏽥 φ 􏽥 . 1 p p 2 p 1 p 1 2 2 2 Defining the estimated error: 0 0 ∗ ∗ w 􏽥 � h􏼐u􏼐e ∣ φ 􏼑 − u 􏼑. (15) 1 2 Its time derivative along the error dynamics (17) is given by Substituting (15) into (14), the following is obtained: t t 1 1 1 1 T T T ⎜ ⎟ ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎛ ⎜ ⎞ ⎟ _ ⎝ ⎠ ⎝ ⎠ _ (18) V � e_ e + e e_ + e Λ 􏽚 e dt + 􏽚 e dt Λ e + φ 􏽥 φ 􏽥 , 1 p p 2 p p 2 p p p p 1 1 2 2 2 2 0 0 t t 1 ⎡ ⎢ ⎤ ⎥ 1 ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ T T T T T T T T T ⎥ ⎢ T T T T T T ⎥ ⎢ ⎛ ⎜ ⎞ ⎟ ⎥ ⎢ ⎛ ⎜ ⎞ ⎟ ⎥ _ ⎢ ⎜ ⎟ ⎥ ⎢ ⎜ ⎟ ⎥ ⎢ ⎝ ⎠ ⎥ ⎢ ⎝ ⎠ ⎥ ⎢ 􏽥 ⎥ ⎢ 􏽥 ⎥ V � ⎢− e A e − 􏽚 edt Λ e + φ ξ h B e + σ B e ⎥ + ⎣− e A e − e Λ 􏽚 e dt + e B hξ φ + e B σ⎦ ⎣ ⎦ 1 p p 2 p 1 1 1 p 1 p p p p 2 p p 1 1 1 p 1 2 2 0 0 t t 1 1 ⎛ ⎜ ⎞ ⎟ ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ _ ⎝ ⎠ ⎝ ⎠ + e Λ 􏽚 e dt + 􏽚 e dt Λ e + φ 􏽥 φ 􏽥 . p 2 p p 2 p 1 1 2 2 0 0 T T T T T T _ _ V � − e Ae + 􏼐σ B e + e B σ􏼑 + 􏼒e B hξ + φ 􏽥 􏼓φ 􏽥 . 1 p p 1 p p 1 p 1 1 1 1 (19) Using (13), the following result is obtained: Let us consider function V (e , t) as follows: 0 p T T 1 1 1 T T V � e Q e . (23) _ 0 1 p (20) p V � − e (A − I)e − e − σ e − σ + σ σ, 􏼒 􏼓 􏼒 􏼓 1 p p p p 2 2 4 e time derivative of V is given by or V � 2e Q e_ . (24) 0 p 1 p T T (21) V ≤ − e Q e + σ σ. 1 p 1 p As e , h, ξ , and φ 􏽥 are bounded and σ is limited, e_ is p 1 1 p bounded. According to Barbalat’s lemma, V (e , t) ⟶ 0 as By integrating both sides of equation (21) in [0, T], the 0 p t ⟶ ∞, which means that error system (5) is asymptoti- following equation is derived: cally stable under control law (12). T T T T V (T) − V (0)≤ − 􏽚 e Q e dt + 􏽚 σ σdt. (22) 1 1 p 1 p 2.2.2. Dynamic Controller Design. Considering the dynamic equation of WMR (4), multiplying both sides of (4) with − 1 According to Assumption 1, we have ‖σ ‖≤ c and M , the following is obtained: − 1 − 1 − 1 combining with the condition V (T)≥ 0, so 􏽚 e Q e dt is M Mv _ � 􏼐− M B(v)􏼑v + 􏼐− M 􏼑 Ev + Q€ c + Cη _ + Gη € + τ 􏼁 1 p 1 p − 1 bounded in [0, T]. As V has all variables which are + M τ, bounded, V is bounded. (25) 6 Journal of Robotics or 2 1 T T PK + K P + Q − PB 􏼠 − 􏼡B P � 0, (33) 2 2 2 2 2 2 − 1 − 1 v _ � 􏼐− M B(v)􏼑v + M τ + D, (26) − 1 in which ε is positive scalar and Q is a prescribed sym- where D � (− M )(Ev + Q€ c + Cη _ + Gη € + τ ). metric-defined positive matrix. Equation (26) can be shortened as From (28), the following result is obtained: v _ � F(v) + Gu + D, (27) F(v) � − Gu − K 􏽥 e + u. _ (34) v v _ R R − 1 where v � 􏼢 􏼣, v _ � 􏼢 􏼣, F(v) � (− M B(v))v, G � Substituting (30) and (34) into (27), the result is as v v _ L L follows: − 1 M , and the control input u � τ � 􏼂 τ τ 􏼃 � m R L T 􏽥 e � − K 􏽥 e + B G u 􏽥 e|φ 􏼁􏼁 − u + B D − B u , (35) 2 2 2 2 2 s 􏼂 u u 􏼃 . m1 m2 1 0 Define 􏽥 e � v − v � v − u. Considering the space-model ref where B � 􏼢 􏼣. e estimated error is determined as 0 1 equation (27) without disturbance (D � 0), the controller is follows: proposed as follows: ∗ ∗ Γ � G u 􏽥 e ∣ φ 􏼁 − u 􏼁. (36) 2 2 ∗ − 1 u � − G F(v) + Κ 􏽥 e − u _ , (28) m 2 e closed-loop system error can be developed by where K is the positive scalar. substituting (36) into (35) as Substituting (28) into (27) leads to the following result: 􏽥 e � − K 􏽥 e + B G u 􏽥 e ∣ φ 􏼁 − u 􏽥 e ∣ φ 􏼁 􏼁 + B (D + Γ) − B u 2 2 2 2 2 s 􏽥 e + K 􏽥 e � 0. (29) (37) From (29), the control objective is achieved, that is, or lim 􏽥 e(t) � 0. t− >∞ Actually, in practice, disturbance D is different from 􏽥 e � − K 􏽥 e + B Gξ (􏽥 e)φ 􏽥 + B δ − u , (38) 2 2 2 2 2 s zero, so we cannot apply optimal control law (28). To deal where φ � φ − φ and δ � D + Γ. with this problem, the following adaptive interval fuzzy type- 2 2 2 II H∞ controller is presented with the control law as follows: − 1 (30) u � u 􏽥 e ∣ φ 􏼁 − G u , 2.2.3. Overall Stability m 2 s where u(􏽥 e ∣ φ ) � ξ (􏽥 e)φ is the output of the adaptive 2 2 2 Assumption 2. In this work, we assume that the external interval fuzzy type-II and ξ (􏽥 e) � diag([ξ (􏽥 e), 2 21 disturbance D is bounded, so there exists a constant d such 2 T ξ (􏽥 e), . . . , ξ (􏽥 e)]), ξ � f / 􏽐 f in which f is the 22 2N 2i 2i i�1 2i 2i that ε δ δ ≤ d . firing interval of the i-th rule, N is the number of rules, φ is 2 Consider the Lyapunov function the parameter updated by the adaptive law: 1 1 T T 􏽥 􏽥 􏽥 􏽥 T T V � V + e Pe + φ φ . (39) 2 1 2 2 _ 􏽥 􏽥 (31) φ � − 􏽮cξ (e)GB Pe􏽯, 2 2c 2 2 2 and u � 􏼂 u u 􏼃 is H∞ controller: e time of derivative of the Lyapunov function is given s s1 s2 by u � B P􏽥 e, (32) 1 T 1 1 T β T _ _ _ _ _ V � V + 􏽥 e P􏽥 e + 􏽥 e P􏽥 e + φ 􏽥 φ 􏽥 . (40) 2 1 2 2 2 2 c where β and c are the positive constants and P is the solution of the algebraic Riccati-like equation: Substituting (32) and (38) into (40), we have 1 1 1 1 1 T T T T T T T T T T T T T T _ _ _ 􏽥 􏽥 􏽥 􏽥 V � V + 􏼢􏽥 e K P􏽥 e + ξ (􏽥 e)φ GB P􏽥 e + δ B P􏽥 e − 􏽥 e PB B P􏽥 e􏼣 + 􏼢􏽥 e PK 􏽥 e + 􏽥 e PB Gφ ξ (􏽥 e) + 􏽥 e PB δ − 􏽥 e PB B P􏽥 e􏼣 + φ φ 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 β 2 β c 1 2 1 1 T T T T T T T T _ _ � V + 􏽥 e 􏼢PK + K P − PB B P􏼣􏽥 e + 􏼐δ B P􏽥 e + 􏽥 e PB δ􏼑 + 􏼒φ 􏽥 + c􏽥 e PB Gξ (􏽥 e)􏼓φ 􏽥 1 2 2 2 2 2 2 2 2 2 2 2 β 2 c (41) Substituting the adaptive law (31) into (41), we get Journal of Robotics 7 T T Due to 1/2(1/ε􏽥 e PB − εδ) (1/ε􏽥 e PB − εδ)≥ 0 and 1 1 1 2 2 T T T T T _ _ V � V + 􏽥 e 􏼢− Q − PB B P􏼣􏽥 e + 􏼐δ B P􏽥 e + 􏽥 e PB δ􏼑 2 1 2 2 2 2 2 2 combining with the previous results of V , the following is 2 2 1 obtained: 1 1 1 1 1 T 2 T T T _ 1 1 1 � V − 􏽥 e Q 􏽥 e + ε δ δ − 􏼒 􏽥 e PB − εδ􏼓 􏼒 􏽥 e PB − εδ􏼓. T T T 2 T 1 2 2 2 (43) V ≤ − e Q e + σ σ − 􏽥 e Q 􏽥 e + ε δ δ. 2 2 2 ε ε 2 1 p 2 4 2 2 (42) Integrating both of sides (43) in [0, T], the following equation is derived: T T T T 1 1 1 T T T 2 T (44) V (T) − V (0)≤ − 􏽚 e Q e dt − 􏽚 􏽥 e Q 􏽥 edt + 􏽚 σ σdt + 􏽚 ε δ δdt. 2 2 p 1 p 2 2 4 2 0 0 0 0 2 2 From the condition ε ‖δ ‖≤ d ,V (T)≥ 0 and Next, the 4 rules of fuzzy type-II system are chosen for 0 2 e Q e dt> 0, the above inequality is equivalent to both inner and outer loops. e membership function is p 1 p chosen as trapezoidal functions for e , e and 􏽥 e , 􏽥 e . e p1 p2 1 2 T T T 1 1 1 fuzzy sets are constructed as follows: T T 2 T − V (0)≤ − 􏽚 􏽥 e Q 􏽥 edt + 􏽚 σ σdt + 􏽚ε δ δdt (45) 2 4 2 X − 1, 5 − 1, 5 − 0, 5 1, 5 11u 0 0 0 ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ X ⎥ ⎢ − 1, 5 − 1, 5 − 1, 5 0, 5 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 11l ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ or ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ X � ⎢ ⎥ � ⎢ ⎥, ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ X ⎥ ⎢ − 1, 5 − 1, 5 0, 5 1, 5 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 12u ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ T T T 1 1 1 T T 2 T X − 0, 5 − 0, 5 0, 5 1, 5 􏽚 􏽥 e Q 􏽥 edt≤ 􏽚 σ σdt + 􏽚 ε δ δdt + V (0) 12l 2 1 2 4 2 − 3 − 3 − 1 3 0 0 0 X 21u (46) ⎢ ⎥ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ X ⎢ − 3 − 3 1 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 21l ⎥ ⎥ 1 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ T T ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ X � ⎢ ⎥ � ⎢ ⎥, 􏽥 􏽥 ⎢ ⎥ ⎢ ⎥ + 􏽥 e (0)P􏽥 e(0) + φ (0)φ (0). ⎢ ⎥ ⎢ ⎥ 2 ⎢ ⎥ ⎢ ⎥ 2 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ X ⎥ ⎢ − 3 − 3 1 3 ⎥ 2 2c ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 22u ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ X − 1 − 1 3 3 22l e proof is completed, and then the H∞ tracking (49) performance can be achieved for a prescribed attenuation V − 9 − 9 − 3 9 11u ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ level ε. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ V ⎥ ⎢ − 9 − 9 − 9 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 11l ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ V � ⎢ � ⎢ ⎥, ⎢ ⎥ ⎥ 1 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ V ⎥ ⎢ − 9 3 9 9 ⎥ 3. Control Strategy Verification ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 12l ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ V − 3 9 9 9 12l e physical parameters of the WMR are given as 2 2 V − 15 − 15 − 7.5 15 m � 10 kg, m � 2 kg, I � 0.1 kgm , I � 0.05 kgm , 21u G w w D ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ I � 4 kgm , b � 0.3 m, r � 0.15 m, and a � 0.2m. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ G ⎢ ⎥ ⎢ ⎥ ⎢ V ⎥ ⎢ − 15 − 15 − 15 7.5 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 21l ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Input disturbances and uncertainties as well as slipping ⎢ ⎥ ⎢ ⎥ V � ⎢ ⎥ � ⎢ ⎥. ⎢ ⎥ ⎢ ⎥ 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ V ⎥ ⎢ − 15 7.5 15 15 ⎥ ⎢ ⎥ ⎢ ⎥ and skidding parameters are assumed as ⎢ 22u ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ V − 7.5 15 15 15 τ � 􏼂 2 + sin(0.1t) 1 + cos(0.1t) 􏼃 , 22l ⎧ ⎨ [0, 0, 0] (m/sec), for t< 5(sec), T where X , X , V , and V are type-I fuzzy sets. iju ijl iju ijl c _ , c _ , η _ � 􏼂 􏼃 R L e simulation is done with two types of reference [0.3 sin(2t), 0.3 cos(2t), 0.2] (m/sec), for t≥ 5(sec), trajectory: elliptical and trifolium shapes. In each case, the (47) responses of the proposed scheme are compared with the e control gains are given as Λ � 2, Λ � 3, p � 0.25, corresponding fuzzy type-I to evaluate the effectiveness of 1 2 and c � 0.0001p . e feedback gain matrix is selected as the proposed algorithm. follows: 30 0 Case 1. e target C moves on an elliptical trajectory with Q � , 􏼢 􏼣 the following equation: 0 30 6.5678 0 ⎧ ⎪ x � 5 cos􏼒 t􏼓, P � 􏼢 􏼣, (48) C ⎪ 10 0 6.5678 (50) 1 0 T π K � 􏼢 􏼣. y � 5 sin􏼒 t􏼓. 0 1 10 8 Journal of Robotics -2 -4 -6 -6 -4 -2 0 2 4 6 x (m) Reference Trajectory Fuzzy Type-I Controller Adaptive FT2 Controller Figure 3: e output trajectory of adaptive interval fuzzy type-II and T-S fuzzy type-I controller. 12.5 11.5 –10 10.5 –20 9.5 12.8 13 13.2 13.4 13.6 13.8 14 14.2 0 102030405060 Times (s) vr (ref ) vr (real) vl (real) vl (ref ) Figure 4: e real velocity of T-S fuzzy type-II controller for elliptical trajectory. e initial values for system are chosen as Case 2. e reference trajectory is trifolium which is for- [x , y , θ] � [5, − 1, π/2]. mulated as follows: G G e results of the simulation are presented in π π ⎧ ⎪ x � 5 cos􏼒 t􏼓cos􏼒 t􏼓, Figures 3–5. Specifically, Figure 3 describes the reference ⎪ ⎪ 10 30 trajectory, the response of the adaptive interval fuzzy type- (51) II controller, and the response of the fuzzy type-I con- ⎪ π π troller. It is easy to see in Figure 3 that, the trajectory of the ⎩ y � 5 sin􏼒 t􏼓sin􏼒 t􏼓. 10 30 WMR which is controlled by the proposed controller is e initial values for system are chosen as more exact than the fuzzy type-I, i.e., the max error [x , y , θ] � [5, − 1, π/2]. tracking of the proposed controller is 0.03 m and the fuzzy G G e simulation results are shown in Figures 6–8. Similar type-I controller is 0.0806 m. Figures 4 and 5 are the ve- to Case 1, we also see that responses of the proposed locity errors of the adaptive fuzzy type-II and fuzzy type-I controller is better, i.e., the maximum error tracking of the controllers, respectively. e error between the real velocity proposed controller is 0.03 m and this one is 0.0807 m for and the reference velocity of the proposed scheme is nearly fuzzy type-I controller. In Figure 7, the difference between zero (Figure 4), while this error in the fuzzy type-I con- the real velocity and the reference velocity is almost zero, troller is so large (Figure 5), i.e., in Figure 5, the real velocity while the real velocity of the fuzzy type-I controller is af- oscillates around the reference velocity with an amplitude fected by external disturbance, and the oscillation is so large of 0.02 m/s. m/s y (m) Journal of Robotics 9 11.7 11.6 11.5 -10 11.4 11.3 -20 11.2 10 10.05 10.1 10.15 10.2 10.25 10.3 10.35 10.4 10.45 0 102030405060 Times (s) vr (ref ) vr (real) vl (ref ) vl (real) Figure 5: e real velocity of adaptive T-S fuzzy type-I for elliptical trajectory. -5 -3 -2 -1 0123456 x (m) Reference Trajectory Fuzzy Type-l Controller Adaptive FT2 Controller Figure 6: e output trajectory of adaptive interval fuzzy type-II and T-S fuzzy type-I controller for trifolium trajectory. -10 -20 26.4 26.6 26.8 27 27.2 27.4 27.6 27.8 010 20 30 40 50 60 Times (s) vr (ref ) vr (real) vl (real) vl (ref ) Figure 7: e real velocity of adaptive interval fuzzy type-II for trifolium trajectory. m/s m/s y (m) 10 Journal of Robotics 9.4 9.3 9.2 -10 9.1 -20 14.75 14.8 14.85 14.9 14.95 15 15.05 15.1 15.15 15.2 010 20 30 40 50 60 Time (s) vr (ref ) vr (real) vl (ref ) vl (real) Figure 8: e real velocity of T-S fuzzy type-I for trifolium trajectory. Table 1: Numerical comparison between the proposed controller and T-S fuzzy type-I controller. Trifolium Elliptical Tracking error (m) Velocity (m/s) Tracking error (m) Velocity (m/s) Proposed controller Max � 0.03 No oscillation Max � 0.03 No oscillation Fuzzy type-I controller Max � 0.0806 Oscillation Max � 0.0807 Oscillation with an amplitude about 0.1 m/s (Figure 8). From these added into each loop to deal with the uncertainties and simulation results, we can see that the adaptive interval fuzzy disturbances. In the future, we have planned to develop the type-II logic controller can handle unknown wheel slipping one loop control structure to reduce the complexity for the and skidding, uncertainties, and unknown bounded external overall system. disturbance better than the fuzzy type-I logic controller. All of the above analyses are concluded in Table 1. Data Availability e data used to support the findings of this study have been 4. Conclusions deposited in the CRC Press repository (https://www.routledge. In this paper, an adaptive fuzzy type-II controller was com/Mobile-Robots-Inspiration-to-Implementation-Second- proposed for the WMR system under the condition of wheel Edition/Jones-Seiger-Flynn/p/book/9780367447656); Robotics slips and external disturbances. e control system included and Autonomous Systems repository (https://doi.org/10.1016/ the dynamic loop and kinematic loop. For each loop, this j.robot.2016.01.002); International Conference on Control, proposed controller was used to compensate for the un- Automation and Systems repository (DOI: 10.1109/ known uncertainties and disturbances. In this work, the ICCAS.2015.7365010); IEEE Transactions on Industrial Elec- stability of the closed-loop system and the convergence of tronics repository (DOI: 10.1109/TIE.2013.2282594); the the trajectory-tracking errors were mathematically proved Mediterranean Conference on Control and Automation re- based on the Lyapunov theory. e properness of the pository (DOI: 10.1109/MED.2009.5164519); International proposed controller was verified through two types of ref- Journal of Advanced Robotic Systems repository (https://doi. erence trajectory. Also, the responses of the proposed org/10.5772/55059); IEEE/RSJ International Conference on controller were compared with the fuzzy type-I controller’s Intelligent Robots and Systems repository (DOI: 10.1109/ one to prove the effectiveness of the illustrated scheme. e IROS.2010.5651060); IEEE Transactions on Systems, Man, and simulation results showed that the performance of the Cybernetics: Systems repository (DOI: 10.1109/ adaptive fuzzy type-II controller was better than the fuzzy TSMC.2017.2677472); IEEE Transactions on Control Systems type-I controller, i.e., smaller error, smoother velocity, and Technology repository (DOI: 10.1109/TCST.2008.922584); IET smaller oscillation even that the trajectory was complicated. 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Shi, “Adaptive fuzzy output feedback control of a nonholonomic wheeled mobile robot,” IEEE Access, vol. 6, pp. 43414–43424, 2018. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Robotics Hindawi Publishing Corporation

Adaptive Fuzzy Type-II Controller for Wheeled Mobile Robot with Disturbances and Wheelslips

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Copyright © 2021 Viet Quoc Ha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Journal of Robotics Volume 2021, Article ID 6946210, 11 pages https://doi.org/10.1155/2021/6946210 Research Article Adaptive Fuzzy Type-II Controller for Wheeled Mobile Robot with Disturbances and Wheelslips 1 2 1 Viet Quoc Ha, Sen Huong-Thi Pham, and Nga Thi-Thuy Vu School of Electrical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam Faculty of Control and Automation, Electric Power University, Hanoi, Vietnam Correspondence should be addressed to Nga i-uy Vu; nga.vuthithuy@hust.edu.vn Received 9 June 2021; Revised 26 July 2021; Accepted 30 August 2021; Published 15 September 2021 Academic Editor: Arturo Buscarino Copyright © 2021 Viet Quoc Ha et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper proposed adaptive fuzzy type-II controllers for the wheeled mobile robot (WMR) systems under conditions of wheel slips and disturbances. e system includes two control loops: outer loop for position tracking and the inner loop for velocity tracking. In each loop, the controller has two parts: the feedback which keeps the system stable and the adaptive type-II fuzzy part which is used to compensate the unknown components that act on the system. e stability of each loop as well as the overall system is proven mathematically based on the Lyapunov theory. Finally, the simulation is setup to verify the effectiveness of the presented algorithm. e simulation results show that, in comparison with the corresponding fuzzy type-I controller, the performance of the adaptive fuzzy type-II controller is better, i.e., the position error is smaller and the velocity is almost smooth under the conditions that the reference trajectory is changed, and the system is affected by wheel slips and external disturbances. “pure rolling without slip.” As mentioned above, these as- 1. Introduction sumptions are not exactly since there are many factors such as unknown external disturbances and wheel slipping due to Wheeled mobile robots (WMRs) are widely used in industry and service robotics. Mobile robots are self-moving vehicles the slippery roads and the wheel. In order to reduce the effect and versatile with many indoor and outdoor applications [1]. of the wheel slips, an adaptive robust control is used for e control problems in the WRMs are quite rich such as trajectory tracking of wheeled mobile robots in the presence path planning, trajectory tracking, and obstacle avoidance. of unknown skidding and slipping [10]. Although this Each problem has different importance in specific appli- controller solved external disturbances, the error tracking is cation. Among these, trajectory tracking control has the role still pretty big. In [11], the adaptive fuzzy output feedback to keep the WMR following the desired trajectory. is controller is proposed to solve the trajectory problem for control problem is not easy because WMRs are under- WMR with the uncertainties and effects of external dis- actuated systems. Moreover, operation of WMR is greatly turbances. In this research, the tracking position errors affected by working conditions such as system uncertainties, converge asymptotically to a small neighborhood near the wheel slips, and external disturbances. erefore, tracking origin with a faster response than achieved by other existing control design for WRM still has the attraction to many controllers, and all of the signals are bounded. A controller researchers. based on the robust dynamic surface control method is Many nonlinear control methods have been used to solve proposed in [12] to eliminate the problems of “explosion of the tracking control problems of nonholonomic mobile complexity.” e controller can become much simpler than robots such as robust adaptive [2], sliding mode control backstepping controller, but the illustrated results are so [3, 4], backstepping control [5, 6], adaptive fuzzy logic poor. An adaptive neural network based on reinforcement control [7], and adaptive neural-network control [8, 9]. learning is presented in [13] for WMR with considering However, all of these are proposed with the assumption of skidding and slipping. In this work, the error tracking is 2 Journal of Robotics (iv) e stability of the closed-loop system with the almost zero, but the structure with four neural networks in the scheme can be the burden for calculation system. proposed controller is proven mathematically based on the Lyapunov theory. is proof for the cascade Recently, the advanced and intelligent control methods are considered as flexible tools to deal with the uncertain system is much harder than the single loop. systems. In [14], the authors propose a strategy that can ensure optimal working under the imperfect dynamic 2. System Model and Adaptive conditions based on the excitation of the hidden dynamics. Controller Design Nevertheless, some strict assumptions need to be satisfied to complete the control strategy. In [15], the disturbance 2.1. System Modeling. Let consider a nonholonomic WMR problems for nonlinear systems are solved by using neural with skidding and slipping as shown in Figure 1. Located at networks. is controller guarantees that all the signals in point G(x , y ) is the center of mass (COM) of the wheel G G the closed-loop system are semiglobally uniformly ultimate mobile robot, and point M(x , y ) indicates the midpoint M M bounded. However, due to the effect of the iterative deri- of the wheel shaft. e distance between G and M is a.r is the vation of virtual control laws, the “complexity explosion” radius of each driving wheel, and the length of wheel mobile problem appears which increases the computational com- robot shaft is 2b. θ is the orientation of the wheel mobile plexity. Another approach to deal with the model parameter robot with respect to the initial frame. uncertainty problem is shown in [16]. In this research, a F and F illustrate the total longitudinal friction force at 1 2 model is developed for corrosion prediction using a neu- the right and left driving wheels, respectively. F is the total rofuzzy expert system. e results show that the neurofuzzy lateral force that acts on the mobile robot, whereas F and ϖ expert systems have better accuracy with fewer number of are the external disturbances and moment at point G input parameters than the neural networks prediction accordingly. model. In [17], a controller based on the fuzzy logic system to According to [20], we can consider the kinematics model approximate unknown parameters in nonlinear systems is of the wheeled mobile robot under slipping and skidding proposed. In this work, the tracking error can converge to a condition as follows: small neighborhood of zero with a fixed time, and all the ⎪ x _ � ϑ cos θ − η _ sin θ, signals of the system are bounded. However, the external ⎧ ⎪ M disturbances are not considered in this research. In [18, 19], ⎨ y _ � ϑ sin θ + η _ cos θ, (1) the interval fuzzy type-II controller is introduced for non- ⎪ linear uncertain systems. It is proved that the fuzzy type-II ⎩ θ � ω, controller handles uncertainties and external disturbances better than the fuzzy type-I controller. where ϑ is the forward velocity and ω is the angular velocity In this paper, an adaptive interval type-2 fuzzy logic of the wheeled mobile robot at points M, ϑ, and ω and are controller is proposed for two-loop control of wheeled mobile calculated as follows: robots with external disturbances. e proposed controller is _ _ r􏼐ϕ + ϕ 􏼑 c _ + c _ expected to allow the error tracking of WMRs to converge to R L R L ϑ � + , zero under the acting of unknown wheel slips, unknown 2 2 (2) bounded external disturbances, and model uncertainties. _ _ r􏼐ϕ − ϕ 􏼑 e main contributions of the proposed algorithm can _ _ c − c R L R L ω � + , be stated as follows: 2b 2b (i) Design the adaptive interval fuzzy type-II controller with c and c are the coordinates of the longitudinal slip of R L for both the inner and outer loop of WMR with the right and left driving wheels, respectively, and η is the uncertainties and external disturbances. coordinate of the lateral slip along the wheel shaft. φ _ and φ _ R L (ii) Under the conditions that the reference trajectory is are the angular velocities of the right and the left wheels, respectively. complex changed, the proposed controller can still We consider a two-wheeled mobile robot with coordi- deal with the tracking problem, and the result nates illustrated in Figure 1 described by the following showed that the error between the reference tra- dynamic models [20]: jectory and the real trajectory is quite small. (iii) Using the adaptive fuzzy type-II controller, the Mv _ + B(v)v + Ev + Q€ c + Cη _ + Gη € + τ � τ, (3) difference between the real velocity and the refer- ence velocity is almost zero. Moreover, in this re- _ _ where v � 􏽨 􏽩 , c � 􏼂 c c 􏼃 , and M � ϕ ϕ R L R L search, the comparison results show that the real m m 11 12 velocity of the proposed controller is smoother than 􏼢 􏼣 m m 21 22 the real velocity of the fuzzy type-I controller. F1 F2 F4 F3 Journal of Robotics 3 Figure 1: Model of the wheeled mobile robot. 2 2 2 2 r a r r m � m � m 􏼠 − 􏼡 − I + 2I 􏼁 , 12 21 G G D 2 2 4b 4b 2 2 2 2 r a r r m + + m � m � m 􏼠 + 􏼡 I + 2I 􏼁 r + I , 11 22 G G D W w 2 2 4b 4b Q Q 1 2 ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ Q � , Q Q 2 1 r a r Q � m 􏼠1 ± 􏼡 ± I + 2I 􏼁 , 1,2 G G D 4 4b 0 1 ar ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ _ _ ⎢ ⎥ ⎣ ⎦ B(v) � m 􏼐ϕ − ϕ 􏼑 , (4) 2 R L 4b − 1 0 0 1 ar c _ − c _ R L ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ E � 􏼠m 􏼡􏼠 􏼡 , 2b 2b − 1 0 ar ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ G � m , 2b − 1 ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ C � m ω, τ � τ , τ . 􏼂 􏼃 R L 2.2. Adaptive Tracking Controller Design for WMR. e 2.2.1. Kinematic Controller Design. In the xOy coordinate, control objective of this work is to design the controller for the position error between the point M(x , y ) and the M M the WMR subjected to unknown wheel slips and model target point C(x , y ) is calculated as follows: C C uncertainties to ensure that the desired reference trajectory e cos θ sin θ x − x p1 C M is tracked. To solve this problem, the adaptive control loop is ⎣ ⎦ ⎡ ⎤ e � � 􏼢 􏼣􏼢 􏼣. (5) − sin θ cos θ y − y proposed with the block diagram shown in Figure 2: p2 C M 2b 4 Journal of Robotics e ~ p u u e m Position Controller Velocity Controller WMR y 2 Type-2 interval MFs Type-2 interval MFs Adaptation ~ ξ ~ e e Adaptation law law . Figure 2: e control scheme for the wheeled mobile robot. e derivative of (5) along with time under the condition det(h) � − e r /2b ⟶ 0 and h is not invertible. To avoid p1 of wheel slips and external disturbances is obtained as [20] this problem, controllers (8) and (10) are modified as follows: e_ cos θ sin θ x _ p1 ⎡ ⎣ ⎤ ⎦ e_ � � 􏼢 􏼣􏼢 􏼣 + hu + d , (6) p p e_ − sin θ cos θ y _ _ cos θ sin θ x p2 ⎧ ⎪ C C ⎪ 􏼌 􏼌 ⎪ ⎡ ⎢ ⎤ ⎥⎡ ⎢ ⎤ ⎥ 􏼌 􏼌 ⎢ ⎥⎢ ⎥ − 1⎢ ⎥⎢ ⎥ ⎪ ⎢ ⎥⎢ ⎥ 􏼌 􏼌 ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎪ ⎢ ⎥⎢ ⎥ 􏼌 􏼌 − h ⎢ ⎥⎢ ⎥, if e ≥ λ, ⎢ ⎥⎢ ⎥ 􏼌 ⎪ ⎣ ⎦⎣ ⎦ 􏼌 p1 (e /b − 1)r/2 − (e /b + 1)r/2 p2 p2 where h � 􏼢 􏼣 and d � ⎪ − sin θ cos θ y _ ⎪ C − e r/2b e r/2b p1 p1 u � (((c _ − c _ )/2b)e ) − (( c _ + c _ )/2) R L p2 R L 􏼢 􏼣 ⎪ cos θ sin θ x _ _ _ _ C − ((( c − c )/2b)e ) − η ⎪ 􏼌 􏼌 R L p1 ⎪ ⎢ ⎥⎢ ⎥ 1 ⎡ ⎢ ⎤ ⎥⎡ ⎢ ⎤ ⎥ 􏼌 􏼌 ⎢ ⎥⎢ ⎥ ⎪ ⎢ ⎥⎢ ⎥ 􏼌 ⎢ ⎥⎢ ⎥ ⎪ ⎢ ⎥⎢ ⎥ 􏼌 􏼌 ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥, if e < λ, ⎪ ⎢ ⎥⎢ ⎥ 􏼌 􏼌 ⎣ ⎦⎣ ⎦ p1 (11) e control u is separated into two components u and 1 ⎩ − sin θ cos θ y _ u , in which u is the feed-forward part used to compensate C 2 1 for the nonlinear component in model (6) and u is the 􏼌 􏼌 􏼌 􏼌 − 1 􏼌 􏼌 ⎧ ⎪ 􏼌 􏼌 − h K e , if e ≥ λ, feedback controller. ⎪ 􏼌 􏼌 1 p p1 u � u � u + u , (7) 1 2 2 ⎪ 􏼌 􏼌 1 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 K e , if e < λ. 1 p 􏼌 p1􏼌 where cos θ sin θ x − 1 Here, λ is a small enough scalar. u � − h 􏼢 􏼣􏼢 􏼣. (8) In fact, the disturbances are unknown (d ≠ 0), and − sin θ cos θ y _ then we cannot apply controller (10) directly. To solve Substituting (7) and (8) into (6), we get this problem, we will design the adaptive interval fuzzy type-II logic controller to approximate control law u as e_ � hu + d . (9) p 2 p follows: First, considering (9) without disturbances (d � 0), we propose the controller: − 1 ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎝ ⎠ u � u􏼐e |φ 􏼑 − h Λ e + Λ 􏽚 e dt , (12) 2 p 1 1 p 2 p ∗ − 1 u � − h K e , (10) 2 1 p with K is the positive scalar. where Λ and Λ are positive scalars and u(e |φ ) 1 1 2 p 1 Substituting (10) into (9), we can achieve the control goal � ξ (e )φ is the output of the interval fuzzy type-II logic p 1 controller in which ξ (e ) � diag ξ (e ) ξ (e ), . . . that is lim e (t) � 0. 􏼐􏽨 11 p 12 t⟶∞ p 1 p p It is noted that when the WMR tracks the desired tra- ξ (e )]), ξ � f / 􏽐 f , where f is the firing in- 1M p 1j 1j 1j 1j j�1 jectory, the errors e and e go to zero. is leads to terval of the jth-rule, M is the number of rules, and φ is p1 p2 1 Journal of Robotics 5 the designed parameter which is calculated by the adaptive law: e_ � − Ae − Λ 􏽚 e dt + B h􏼐u􏼐e ∣ φ 􏼑 − u􏼐e ∣ φ 􏼑􏼑 + B 􏼐Δ + w 􏽥 􏼑 p p 2 p 1 p 1 p 1 1 p _ (13) φ � − 􏽮ξ 􏼐e 􏼑h B e 􏽯. 1 1 p 1 p � − Ae − Λ 􏽚 e dt + B hξ φ 􏽥 + B σ, p 2 p 1 1 1 1 Substituting (12) into (9), it leads to the following results: (16) ∗ ∗ e _ � − Λ e − Λ 􏽚 e dt + h u e ∣ φ − u + hu + d 􏼐 􏼐 􏼑 􏼑 p 1 p 2 p p 1 2 2 p where φ 􏽥 � φ − φ is the estimated errors, σ � Δ + w 􏽥 . 1 1 1 p � − Ae − Λ 􏽚 e dt + B h􏼐u􏼐e ∣ φ 􏼑 − u 􏼑 + B Δ , Assumption 1. With σ ∈ L [0, T], T ∈ (0, +∞), there exists p 2 p 1 p 1 2 1 p 0 the constant c such that σ σ ≤ c . Define a Lyapunov function (14) 1 0 T t t d d where A � Λ + K , B � 􏼢 􏼣 and Δ � 􏽨 􏽩 . p1 p2 1 1 1 p 1 1 1 T T 0 1 ⎜ ⎟ ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ ⎝ ⎠ (17) V � e e + 􏽚 e dt Λ 􏽚 e dt + φ 􏽥 φ 􏽥 . 1 p p 2 p 1 p 1 2 2 2 Defining the estimated error: 0 0 ∗ ∗ w 􏽥 � h􏼐u􏼐e ∣ φ 􏼑 − u 􏼑. (15) 1 2 Its time derivative along the error dynamics (17) is given by Substituting (15) into (14), the following is obtained: t t 1 1 1 1 T T T ⎜ ⎟ ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎛ ⎜ ⎞ ⎟ _ ⎝ ⎠ ⎝ ⎠ _ (18) V � e_ e + e e_ + e Λ 􏽚 e dt + 􏽚 e dt Λ e + φ 􏽥 φ 􏽥 , 1 p p 2 p p 2 p p p p 1 1 2 2 2 2 0 0 t t 1 ⎡ ⎢ ⎤ ⎥ 1 ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ T T T T T T T T T ⎥ ⎢ T T T T T T ⎥ ⎢ ⎛ ⎜ ⎞ ⎟ ⎥ ⎢ ⎛ ⎜ ⎞ ⎟ ⎥ _ ⎢ ⎜ ⎟ ⎥ ⎢ ⎜ ⎟ ⎥ ⎢ ⎝ ⎠ ⎥ ⎢ ⎝ ⎠ ⎥ ⎢ 􏽥 ⎥ ⎢ 􏽥 ⎥ V � ⎢− e A e − 􏽚 edt Λ e + φ ξ h B e + σ B e ⎥ + ⎣− e A e − e Λ 􏽚 e dt + e B hξ φ + e B σ⎦ ⎣ ⎦ 1 p p 2 p 1 1 1 p 1 p p p p 2 p p 1 1 1 p 1 2 2 0 0 t t 1 1 ⎛ ⎜ ⎞ ⎟ ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ _ ⎝ ⎠ ⎝ ⎠ + e Λ 􏽚 e dt + 􏽚 e dt Λ e + φ 􏽥 φ 􏽥 . p 2 p p 2 p 1 1 2 2 0 0 T T T T T T _ _ V � − e Ae + 􏼐σ B e + e B σ􏼑 + 􏼒e B hξ + φ 􏽥 􏼓φ 􏽥 . 1 p p 1 p p 1 p 1 1 1 1 (19) Using (13), the following result is obtained: Let us consider function V (e , t) as follows: 0 p T T 1 1 1 T T V � e Q e . (23) _ 0 1 p (20) p V � − e (A − I)e − e − σ e − σ + σ σ, 􏼒 􏼓 􏼒 􏼓 1 p p p p 2 2 4 e time derivative of V is given by or V � 2e Q e_ . (24) 0 p 1 p T T (21) V ≤ − e Q e + σ σ. 1 p 1 p As e , h, ξ , and φ 􏽥 are bounded and σ is limited, e_ is p 1 1 p bounded. According to Barbalat’s lemma, V (e , t) ⟶ 0 as By integrating both sides of equation (21) in [0, T], the 0 p t ⟶ ∞, which means that error system (5) is asymptoti- following equation is derived: cally stable under control law (12). T T T T V (T) − V (0)≤ − 􏽚 e Q e dt + 􏽚 σ σdt. (22) 1 1 p 1 p 2.2.2. Dynamic Controller Design. Considering the dynamic equation of WMR (4), multiplying both sides of (4) with − 1 According to Assumption 1, we have ‖σ ‖≤ c and M , the following is obtained: − 1 − 1 − 1 combining with the condition V (T)≥ 0, so 􏽚 e Q e dt is M Mv _ � 􏼐− M B(v)􏼑v + 􏼐− M 􏼑 Ev + Q€ c + Cη _ + Gη € + τ 􏼁 1 p 1 p − 1 bounded in [0, T]. As V has all variables which are + M τ, bounded, V is bounded. (25) 6 Journal of Robotics or 2 1 T T PK + K P + Q − PB 􏼠 − 􏼡B P � 0, (33) 2 2 2 2 2 2 − 1 − 1 v _ � 􏼐− M B(v)􏼑v + M τ + D, (26) − 1 in which ε is positive scalar and Q is a prescribed sym- where D � (− M )(Ev + Q€ c + Cη _ + Gη € + τ ). metric-defined positive matrix. Equation (26) can be shortened as From (28), the following result is obtained: v _ � F(v) + Gu + D, (27) F(v) � − Gu − K 􏽥 e + u. _ (34) v v _ R R − 1 where v � 􏼢 􏼣, v _ � 􏼢 􏼣, F(v) � (− M B(v))v, G � Substituting (30) and (34) into (27), the result is as v v _ L L follows: − 1 M , and the control input u � τ � 􏼂 τ τ 􏼃 � m R L T 􏽥 e � − K 􏽥 e + B G u 􏽥 e|φ 􏼁􏼁 − u + B D − B u , (35) 2 2 2 2 2 s 􏼂 u u 􏼃 . m1 m2 1 0 Define 􏽥 e � v − v � v − u. Considering the space-model ref where B � 􏼢 􏼣. e estimated error is determined as 0 1 equation (27) without disturbance (D � 0), the controller is follows: proposed as follows: ∗ ∗ Γ � G u 􏽥 e ∣ φ 􏼁 − u 􏼁. (36) 2 2 ∗ − 1 u � − G F(v) + Κ 􏽥 e − u _ , (28) m 2 e closed-loop system error can be developed by where K is the positive scalar. substituting (36) into (35) as Substituting (28) into (27) leads to the following result: 􏽥 e � − K 􏽥 e + B G u 􏽥 e ∣ φ 􏼁 − u 􏽥 e ∣ φ 􏼁 􏼁 + B (D + Γ) − B u 2 2 2 2 2 s 􏽥 e + K 􏽥 e � 0. (29) (37) From (29), the control objective is achieved, that is, or lim 􏽥 e(t) � 0. t− >∞ Actually, in practice, disturbance D is different from 􏽥 e � − K 􏽥 e + B Gξ (􏽥 e)φ 􏽥 + B δ − u , (38) 2 2 2 2 2 s zero, so we cannot apply optimal control law (28). To deal where φ � φ − φ and δ � D + Γ. with this problem, the following adaptive interval fuzzy type- 2 2 2 II H∞ controller is presented with the control law as follows: − 1 (30) u � u 􏽥 e ∣ φ 􏼁 − G u , 2.2.3. Overall Stability m 2 s where u(􏽥 e ∣ φ ) � ξ (􏽥 e)φ is the output of the adaptive 2 2 2 Assumption 2. In this work, we assume that the external interval fuzzy type-II and ξ (􏽥 e) � diag([ξ (􏽥 e), 2 21 disturbance D is bounded, so there exists a constant d such 2 T ξ (􏽥 e), . . . , ξ (􏽥 e)]), ξ � f / 􏽐 f in which f is the 22 2N 2i 2i i�1 2i 2i that ε δ δ ≤ d . firing interval of the i-th rule, N is the number of rules, φ is 2 Consider the Lyapunov function the parameter updated by the adaptive law: 1 1 T T 􏽥 􏽥 􏽥 􏽥 T T V � V + e Pe + φ φ . (39) 2 1 2 2 _ 􏽥 􏽥 (31) φ � − 􏽮cξ (e)GB Pe􏽯, 2 2c 2 2 2 and u � 􏼂 u u 􏼃 is H∞ controller: e time of derivative of the Lyapunov function is given s s1 s2 by u � B P􏽥 e, (32) 1 T 1 1 T β T _ _ _ _ _ V � V + 􏽥 e P􏽥 e + 􏽥 e P􏽥 e + φ 􏽥 φ 􏽥 . (40) 2 1 2 2 2 2 c where β and c are the positive constants and P is the solution of the algebraic Riccati-like equation: Substituting (32) and (38) into (40), we have 1 1 1 1 1 T T T T T T T T T T T T T T _ _ _ 􏽥 􏽥 􏽥 􏽥 V � V + 􏼢􏽥 e K P􏽥 e + ξ (􏽥 e)φ GB P􏽥 e + δ B P􏽥 e − 􏽥 e PB B P􏽥 e􏼣 + 􏼢􏽥 e PK 􏽥 e + 􏽥 e PB Gφ ξ (􏽥 e) + 􏽥 e PB δ − 􏽥 e PB B P􏽥 e􏼣 + φ φ 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 β 2 β c 1 2 1 1 T T T T T T T T _ _ � V + 􏽥 e 􏼢PK + K P − PB B P􏼣􏽥 e + 􏼐δ B P􏽥 e + 􏽥 e PB δ􏼑 + 􏼒φ 􏽥 + c􏽥 e PB Gξ (􏽥 e)􏼓φ 􏽥 1 2 2 2 2 2 2 2 2 2 2 2 β 2 c (41) Substituting the adaptive law (31) into (41), we get Journal of Robotics 7 T T Due to 1/2(1/ε􏽥 e PB − εδ) (1/ε􏽥 e PB − εδ)≥ 0 and 1 1 1 2 2 T T T T T _ _ V � V + 􏽥 e 􏼢− Q − PB B P􏼣􏽥 e + 􏼐δ B P􏽥 e + 􏽥 e PB δ􏼑 2 1 2 2 2 2 2 2 combining with the previous results of V , the following is 2 2 1 obtained: 1 1 1 1 1 T 2 T T T _ 1 1 1 � V − 􏽥 e Q 􏽥 e + ε δ δ − 􏼒 􏽥 e PB − εδ􏼓 􏼒 􏽥 e PB − εδ􏼓. T T T 2 T 1 2 2 2 (43) V ≤ − e Q e + σ σ − 􏽥 e Q 􏽥 e + ε δ δ. 2 2 2 ε ε 2 1 p 2 4 2 2 (42) Integrating both of sides (43) in [0, T], the following equation is derived: T T T T 1 1 1 T T T 2 T (44) V (T) − V (0)≤ − 􏽚 e Q e dt − 􏽚 􏽥 e Q 􏽥 edt + 􏽚 σ σdt + 􏽚 ε δ δdt. 2 2 p 1 p 2 2 4 2 0 0 0 0 2 2 From the condition ε ‖δ ‖≤ d ,V (T)≥ 0 and Next, the 4 rules of fuzzy type-II system are chosen for 0 2 e Q e dt> 0, the above inequality is equivalent to both inner and outer loops. e membership function is p 1 p chosen as trapezoidal functions for e , e and 􏽥 e , 􏽥 e . e p1 p2 1 2 T T T 1 1 1 fuzzy sets are constructed as follows: T T 2 T − V (0)≤ − 􏽚 􏽥 e Q 􏽥 edt + 􏽚 σ σdt + 􏽚ε δ δdt (45) 2 4 2 X − 1, 5 − 1, 5 − 0, 5 1, 5 11u 0 0 0 ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ X ⎥ ⎢ − 1, 5 − 1, 5 − 1, 5 0, 5 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 11l ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ or ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ X � ⎢ ⎥ � ⎢ ⎥, ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ X ⎥ ⎢ − 1, 5 − 1, 5 0, 5 1, 5 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 12u ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ T T T 1 1 1 T T 2 T X − 0, 5 − 0, 5 0, 5 1, 5 􏽚 􏽥 e Q 􏽥 edt≤ 􏽚 σ σdt + 􏽚 ε δ δdt + V (0) 12l 2 1 2 4 2 − 3 − 3 − 1 3 0 0 0 X 21u (46) ⎢ ⎥ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ X ⎢ − 3 − 3 1 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 21l ⎥ ⎥ 1 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ T T ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ X � ⎢ ⎥ � ⎢ ⎥, 􏽥 􏽥 ⎢ ⎥ ⎢ ⎥ + 􏽥 e (0)P􏽥 e(0) + φ (0)φ (0). ⎢ ⎥ ⎢ ⎥ 2 ⎢ ⎥ ⎢ ⎥ 2 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ X ⎥ ⎢ − 3 − 3 1 3 ⎥ 2 2c ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 22u ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ X − 1 − 1 3 3 22l e proof is completed, and then the H∞ tracking (49) performance can be achieved for a prescribed attenuation V − 9 − 9 − 3 9 11u ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ level ε. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ V ⎥ ⎢ − 9 − 9 − 9 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 11l ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ V � ⎢ � ⎢ ⎥, ⎢ ⎥ ⎥ 1 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ V ⎥ ⎢ − 9 3 9 9 ⎥ 3. Control Strategy Verification ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 12l ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ V − 3 9 9 9 12l e physical parameters of the WMR are given as 2 2 V − 15 − 15 − 7.5 15 m � 10 kg, m � 2 kg, I � 0.1 kgm , I � 0.05 kgm , 21u G w w D ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ I � 4 kgm , b � 0.3 m, r � 0.15 m, and a � 0.2m. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ G ⎢ ⎥ ⎢ ⎥ ⎢ V ⎥ ⎢ − 15 − 15 − 15 7.5 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 21l ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Input disturbances and uncertainties as well as slipping ⎢ ⎥ ⎢ ⎥ V � ⎢ ⎥ � ⎢ ⎥. ⎢ ⎥ ⎢ ⎥ 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ V ⎥ ⎢ − 15 7.5 15 15 ⎥ ⎢ ⎥ ⎢ ⎥ and skidding parameters are assumed as ⎢ 22u ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ V − 7.5 15 15 15 τ � 􏼂 2 + sin(0.1t) 1 + cos(0.1t) 􏼃 , 22l ⎧ ⎨ [0, 0, 0] (m/sec), for t< 5(sec), T where X , X , V , and V are type-I fuzzy sets. iju ijl iju ijl c _ , c _ , η _ � 􏼂 􏼃 R L e simulation is done with two types of reference [0.3 sin(2t), 0.3 cos(2t), 0.2] (m/sec), for t≥ 5(sec), trajectory: elliptical and trifolium shapes. In each case, the (47) responses of the proposed scheme are compared with the e control gains are given as Λ � 2, Λ � 3, p � 0.25, corresponding fuzzy type-I to evaluate the effectiveness of 1 2 and c � 0.0001p . e feedback gain matrix is selected as the proposed algorithm. follows: 30 0 Case 1. e target C moves on an elliptical trajectory with Q � , 􏼢 􏼣 the following equation: 0 30 6.5678 0 ⎧ ⎪ x � 5 cos􏼒 t􏼓, P � 􏼢 􏼣, (48) C ⎪ 10 0 6.5678 (50) 1 0 T π K � 􏼢 􏼣. y � 5 sin􏼒 t􏼓. 0 1 10 8 Journal of Robotics -2 -4 -6 -6 -4 -2 0 2 4 6 x (m) Reference Trajectory Fuzzy Type-I Controller Adaptive FT2 Controller Figure 3: e output trajectory of adaptive interval fuzzy type-II and T-S fuzzy type-I controller. 12.5 11.5 –10 10.5 –20 9.5 12.8 13 13.2 13.4 13.6 13.8 14 14.2 0 102030405060 Times (s) vr (ref ) vr (real) vl (real) vl (ref ) Figure 4: e real velocity of T-S fuzzy type-II controller for elliptical trajectory. e initial values for system are chosen as Case 2. e reference trajectory is trifolium which is for- [x , y , θ] � [5, − 1, π/2]. mulated as follows: G G e results of the simulation are presented in π π ⎧ ⎪ x � 5 cos􏼒 t􏼓cos􏼒 t􏼓, Figures 3–5. Specifically, Figure 3 describes the reference ⎪ ⎪ 10 30 trajectory, the response of the adaptive interval fuzzy type- (51) II controller, and the response of the fuzzy type-I con- ⎪ π π troller. It is easy to see in Figure 3 that, the trajectory of the ⎩ y � 5 sin􏼒 t􏼓sin􏼒 t􏼓. 10 30 WMR which is controlled by the proposed controller is e initial values for system are chosen as more exact than the fuzzy type-I, i.e., the max error [x , y , θ] � [5, − 1, π/2]. tracking of the proposed controller is 0.03 m and the fuzzy G G e simulation results are shown in Figures 6–8. Similar type-I controller is 0.0806 m. Figures 4 and 5 are the ve- to Case 1, we also see that responses of the proposed locity errors of the adaptive fuzzy type-II and fuzzy type-I controller is better, i.e., the maximum error tracking of the controllers, respectively. e error between the real velocity proposed controller is 0.03 m and this one is 0.0807 m for and the reference velocity of the proposed scheme is nearly fuzzy type-I controller. In Figure 7, the difference between zero (Figure 4), while this error in the fuzzy type-I con- the real velocity and the reference velocity is almost zero, troller is so large (Figure 5), i.e., in Figure 5, the real velocity while the real velocity of the fuzzy type-I controller is af- oscillates around the reference velocity with an amplitude fected by external disturbance, and the oscillation is so large of 0.02 m/s. m/s y (m) Journal of Robotics 9 11.7 11.6 11.5 -10 11.4 11.3 -20 11.2 10 10.05 10.1 10.15 10.2 10.25 10.3 10.35 10.4 10.45 0 102030405060 Times (s) vr (ref ) vr (real) vl (ref ) vl (real) Figure 5: e real velocity of adaptive T-S fuzzy type-I for elliptical trajectory. -5 -3 -2 -1 0123456 x (m) Reference Trajectory Fuzzy Type-l Controller Adaptive FT2 Controller Figure 6: e output trajectory of adaptive interval fuzzy type-II and T-S fuzzy type-I controller for trifolium trajectory. -10 -20 26.4 26.6 26.8 27 27.2 27.4 27.6 27.8 010 20 30 40 50 60 Times (s) vr (ref ) vr (real) vl (real) vl (ref ) Figure 7: e real velocity of adaptive interval fuzzy type-II for trifolium trajectory. m/s m/s y (m) 10 Journal of Robotics 9.4 9.3 9.2 -10 9.1 -20 14.75 14.8 14.85 14.9 14.95 15 15.05 15.1 15.15 15.2 010 20 30 40 50 60 Time (s) vr (ref ) vr (real) vl (ref ) vl (real) Figure 8: e real velocity of T-S fuzzy type-I for trifolium trajectory. Table 1: Numerical comparison between the proposed controller and T-S fuzzy type-I controller. Trifolium Elliptical Tracking error (m) Velocity (m/s) Tracking error (m) Velocity (m/s) Proposed controller Max � 0.03 No oscillation Max � 0.03 No oscillation Fuzzy type-I controller Max � 0.0806 Oscillation Max � 0.0807 Oscillation with an amplitude about 0.1 m/s (Figure 8). From these added into each loop to deal with the uncertainties and simulation results, we can see that the adaptive interval fuzzy disturbances. In the future, we have planned to develop the type-II logic controller can handle unknown wheel slipping one loop control structure to reduce the complexity for the and skidding, uncertainties, and unknown bounded external overall system. disturbance better than the fuzzy type-I logic controller. All of the above analyses are concluded in Table 1. Data Availability e data used to support the findings of this study have been 4. Conclusions deposited in the CRC Press repository (https://www.routledge. In this paper, an adaptive fuzzy type-II controller was com/Mobile-Robots-Inspiration-to-Implementation-Second- proposed for the WMR system under the condition of wheel Edition/Jones-Seiger-Flynn/p/book/9780367447656); Robotics slips and external disturbances. e control system included and Autonomous Systems repository (https://doi.org/10.1016/ the dynamic loop and kinematic loop. For each loop, this j.robot.2016.01.002); International Conference on Control, proposed controller was used to compensate for the un- Automation and Systems repository (DOI: 10.1109/ known uncertainties and disturbances. In this work, the ICCAS.2015.7365010); IEEE Transactions on Industrial Elec- stability of the closed-loop system and the convergence of tronics repository (DOI: 10.1109/TIE.2013.2282594); the the trajectory-tracking errors were mathematically proved Mediterranean Conference on Control and Automation re- based on the Lyapunov theory. e properness of the pository (DOI: 10.1109/MED.2009.5164519); International proposed controller was verified through two types of ref- Journal of Advanced Robotic Systems repository (https://doi. erence trajectory. Also, the responses of the proposed org/10.5772/55059); IEEE/RSJ International Conference on controller were compared with the fuzzy type-I controller’s Intelligent Robots and Systems repository (DOI: 10.1109/ one to prove the effectiveness of the illustrated scheme. e IROS.2010.5651060); IEEE Transactions on Systems, Man, and simulation results showed that the performance of the Cybernetics: Systems repository (DOI: 10.1109/ adaptive fuzzy type-II controller was better than the fuzzy TSMC.2017.2677472); IEEE Transactions on Control Systems type-I controller, i.e., smaller error, smoother velocity, and Technology repository (DOI: 10.1109/TCST.2008.922584); IET smaller oscillation even that the trajectory was complicated. 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Published: Sep 15, 2021

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