Takagi–Sugeno Fuzzy Controller and Sliding Mode Controller for a Nonholonomic Mobile Robot

(is paper focuses on the nonholonomic wheeled mobile robot. We have presented a scheme to develop controllers. Two controllers have been developed.(e first concerns the kinematic behavior, while the second relates to the dynamic behavior of the mobile robot. For the kinematic controller, we have used a Takagi–Sugeno fuzzy system to overcome the nonlinearities present in model, whereas for the second controller, we have used the sliding mode approach. (e sliding surface has the identical structure as the proportional integral controller. (e stability of the system has been proved based on the Lyapunov approach. (e simulation results show the efficiency of the proposed control laws.


Introduction
In the last decades, the path of travel is considered as one of the critical problems in the field of mobile robotics. e trajectory tracking consists of guiding the robot through intermediate points to reach the final destination.
is tracking is carried out under a constraint time, which means that the robot must reach the goal within a predefined time. In the literature, the problem is treated as the tracking of a reference robot that moves to the desired trajectory with a certain rhythm. e real robot must follow precisely the reference and reduce the distance error, by varying its linear and angular velocities [1,2]. ere are many works that have focused on tracking the trajectory of the mobile robots, and they consider the mobile robot as a particle; in this case, the inputs are velocities. eir aims are kinematic models. In [3], the kinematic control law approach supposes that the control signal generates the exact motion commanded. On the contrary, some works consider the kinematic aspect and the dynamic aspect for the mobile robot. In this case, the actuator inputs signals are torques instead of velocities [4]. In [5], Lee et al. suggest a technique for designing the tracking control of wheeled mobile robots based on a new sliding surface with an approach angle. In [6], authors proposed a robust backstepping controller for the uncertain kinematic model of the wheeled mobile robot based on a nonlinear disturbance observer in order to cope with model uncertainties and the external disturbances. Topalov [7] proposed an adaptive fuzzy approach for the kinematic controller. is method was able to decrease the effect of unmodeled disturbances. In [8], a dynamic Petri recurrent fuzzy neural network was proposed. In [9], the proposed controller combines nonlinear time varying feedback with an integral sliding mode controller. e latter is obtained by introducing an integral term in the switching manifold.
In [10], a robust adaptive mobile robot controller is presented using backstepping for kinematics and dynamics motions, and the adaptive process was based on the neural network. In [11], a classical parallel distributed compensation (PDC) control law, based on Takagi-Sugeno fuzzy modeling, is proposed. e controller comprises sixteen rules in which the control gains have been calculated using LMI techniques. In [12], the authors present an adaptive controller with consideration of unknown model parameters.
In [13], the authors suggest a controller of a mobile robot in Cartesian coordinates with an approach angle based on the sliding mode. In [14], the authors combine hybrid backstepping kinematic control with the adaptive integral sliding mode kinetic control of the three-wheeled mobile robot.
Most of the works deal with nonholonomic wheeled mobile robot, which is used for kinematic motion of a classical controller arising from the backstepping method [2,10,12,14,15]. is paper includes two main contributions. First, a new controller based on Takagi-Sugeno fuzzy systems for kinematic motion.
is lastly uses three fuzzy rules. e second contribution consists of developing for the dynamic part a controller based on the sliding mode. e sliding surface, which is based on linear and angular velocities of the robot, has the similar structure as the proportional integral controller. e switching control term of the latter controller combines the two sliding surfaces. e remainder of this paper is organized as follows. Section 2 is devoted to the description of the kinematic and dynamic models of the two-wheeled mobile robot. Section 3 that is reserved to the controllers design includes two subsections, the first is reserved to the development of the new T-S type fuzzy controller of the kinematic behavior, whereas the second is consecrated to the design of the dynamic motion controller using the sliding mode approach. e stability analysis is checked in the both precedent subsections by the Lyapunov approach. en, Section 4 is sacred to the presentation of the simulation results.

Mobile Robot Modeling
In this section, we are interested in the modeling of the robot, which is composed of two driving wheels and a drive shaft in the center, as shown in Figure 1. Indeed, the Section 2.1 is reserved for kinematic modeling, while Section 2.2 concerns dynamic modeling.
We define the current position (x c , y c ) and the heading angle θ, which constitute the coordinates of the middle point of the mobile robot and the angle between the heading direction and the x-axis to describe the current posture position of the mobile robot. Figure 1 depicts the current posture position of a two wheels mobile robot in Cartesian frame coordinates. e nonholonomic constraint of a wheeled mobile robot is given by the following equation:

Fuzzy Kinematic Model of Robot.
Based on the Newton-Euler equations [16] and the previous hypotheses, the state equations of the mobile robot are represented by the following equations' system [17]: where (x, y), v, and ω represent, respectively, the instantaneous position coordinates of point C of the mobile robot in the global Cartesian frame and the measurements at point C of the linear and angular speeds of the robot. e state variables of mobile robot are q � x y θ T : where v d and w d represent, respectively, the desired linear and angular velocity. e state kinematic model of the mobile robot in Cartesian frame coordinates is given by the following expression: with In order to develop a T-S fuzzy controller, which stabilizes the system and allows the robot to follow the desired path, we need a fuzzy model. In this context, we proceed to determine a fuzzy model of the robot. e posture vector error is not specified in the global frame coordinate system, but quite as a vector error in the local frame coordinate system of the robot: q e (t) � e 1 e 2 e 3 T . e posture vector error q e is computed based on the actual posture vector q(t) � x y θ T and the reference posture vector Figure 1: Representation of the navigation environment. where e relation between the local frame and the global frame, as shown in Figure 2, is given by the following equation: where Equation (8) allows transforming the magnitudes described in the global coordinate system to the local coordinate system: However, by differentiating equation (10), which contains the linear speed and the angular speed terms, we obtain the derivative of the error vector, which is expressed by the following equation: e posture error model can be rewritten as follows: We note that equation (12) contains trigonometric nonlinearities which are cos (e 3 ) and sin (e 3 ). However, the nonlinearities depend on the error e 3 , whose range of variation is from −pi/2 to pi/2. e advantage of the T-S type fuzzy approach is that it allows describing the nonlinear model by linear submodels. Indeed, each submodel represents a local linear relation between the inputs and the outputs and all the nonlinearities are reported in the premises of the fuzzy rules [18].
Based on the theory of T-S fuzzy systems, the nonlinear model (12) can be transformed into three local models, which are inferred by fuzzy rules. e three local models are described by the following systems of equations: From the weights assigned to each rule, the state vector of the fuzzy models is inferred as follows (which corresponds to a barycentric aggregation). e member ship function for the error e 3 is given in Figure 3. e rules of the local models are given by the following expression: e T-S fuzzy model of equation (12) is given by the following equation: where μ i and A i and B i represent, respectively, the weight assigned to each rule and the matrices associated to the local model. With, where C(q, _ q) is the centripetal and Coriolis matrix, F( _ q) is the friction force, τ represents the torque vector, and A T (q) � 0: where m and J g represent, respectively, the mass and the moment inertia of the wheeled mobile robot. L and r represent, respectively, the distance separating the two driving wheels and the wheel radius. Without considering disturbances and uncertainties, the latest equation becomes as where e expressions of linear and angular velocities of the mobile robot, (v, w), depend on the left and right linear velocities of the motors. ey are expressed by the following equations:

Design of Robot Controllers
In this work, we consider the kinematic and dynamic behavior of the robot. e purpose of the control design is to allow the robot to follow the virtual robot. e latter represents the reference robot and provides the desired path defined by the following vector: q d (t) � x d y d θ d T . e architecture of the control scheme of the robot, includes six blocks, as shown in Figure 4. e first block generates the desired states, whereas the second block transforms the error from the local frame into the general frame. e third and fourth blocks are reserved, respectively, for kinetic and dynamic controllers. e fifth and sixth blocks, respectively, describe the behavior of the kinematic and dynamic models of the robot.

Fuzzy Kinematic Controller.
In this section, we are interested in the search for a T-S type fuzzy controller, which guarantees the convergence of the kinematic errors towards zero in the local coordinate system and allows the robot to follow the desired path.
Based on the T-S fuzzy model (14), the rules for the local controllers are given by the following expressions: e global T-S fuzzy controller is given by the following equation: If e 3 � 0, then μ 1 � 1 and μ 2 � μ 3 � 0, so Mathematical Problems in Engineering

Stability Analysis.
To check the stability of the robot, we use Lyapunov's theory. However, we choose the following Lyapunov candidate function: e derivative of Lyapunov function is as follows. So, So, If we choose the following linear and angular velocities, Equation (26) becomes If e 3 � 0, then μ 1 � 1 and e derivative of the Lyapunov function is negative and the stability of the system is guarantee.

Dynamic Controller Based on Sliding Mode.
In this section, we are interested in the development of a controller, which guarantees the convergence of the posture error q e towards zero for any arbitrary reference trajectory. However, we have developed a controller based on the sliding mode approach because the latter is considered a robust approach [19,20]. In this case, we define two sliding surfaces. e first surface depends on linear velocity, while the second uses angular velocity, S � s v s w T : where s v and s w are given, respectively, by equations (31) and (32), However, the derivatives of the sliding surfaces s v (t) and s w (t) are given by the following expressions: _ S w (t) � _ e w (t) + k w e w (t).
(33) e dynamic motion of the robot is described by equation (7) which can be transformed as Equation (34) can be written as where B � (M) − 1 B Based on the sliding mode theory, the controller includes two terms which are known as equivalent control law and switching control. e global control law is expressed as e equivalent control law u eq is computed by recognizing that _ S � 0 which is a necessary condition for the state trajectory to stay in the sliding surface [19,20]. e derivative of the sliding surface is with _ e e � ( _ V c − _ V m ) and e e � (V c − V m ). us, substituting (35) for (37), we obtain  with However, the equivalent control law u eq � τ eq can be computed as Finally, we obtain (43) e switching control term is generally chooses as u s � −ηsign(S), with η≻0. is term can be canceled when the system reaches the sliding surface [20]. In this paper, the switching control law is chosen as follows:

Reaching Condition and Stability Analysis.
To verify the reaching condition, we need to just check the following condition. e Lyapunov candidate function is chosen as e derivative can be expressed as Based on equation (40), We note k 1s � (k s rm)/2 and k 2s � (k s rJ g )/2L. So,

Simulation Results
In this part, we present the simulation results of the nonholonomic wheeled mobile robot. e parameters of the robot (see Figure 1) are m � 17 kg, J � 0.537 kg · m 2 , L � 0.24 m, and r � 0.1 m and initial position x o y o θ � 5 5 π/12 . e reference speed has been chosen as the variable as shown by the following equation:      Figures 5-7 show, respectively, the evolution in time of mass, inertia, and radius of the robot. Figures 8 and 9 show, respectively, the evolution in time of the robot along the x-and y-axis (the trajectories of robot and its reference). Figures 10 and 11 show, respectively, the behavior evolution of linear speed v and the error speed ev � v r − v. Figures 8-10 show that all the signals follow quickly the reference. e curves of the reference signals and the responses overlap in a very short time after start-up. Figure 12 shows the evolution the angular error e 3 .    Figures 13 and 14 show, respectively, the behavior evolution of errors x 2ref − x 2 and x 1ref − x 1 . Figures 15 and 16 show, respectively, the behavior evolution of errors e 1 and e 2 . Figures 12-16 show that all the errors rapidly tend towards zero. is clearly shows that the system converges towards the desired trajectories in a very short time. is proves the effectiveness of the proposed command.

Conclusion
In this paper, we have presented a scheme to control a nonholonomic robot. We have proposed two controllers. e first concerns the kinematic behavior, while the second relates to the dynamic behavior of the mobile robot. To overcome the nonlinearities present in the model, we have used a Takagi-Sugeno fuzzy system for the kinematic controller. For the second controller, we have used the sliding mode technique, which is known as a robust technique. e proposed sliding surface has the same structure as the proportional integral controller. e Lyapunov approach has been used to prove stability of the system. e results presented in Section 4 show the efficiency of the proposed control laws.

Data Availability
ere is no specific data used to support the conclusions of the study. All the necessary information exists in the paper. Mathematical Problems in Engineering 9