Controllers design based on Takagi-Sugeno fuzzy systems and sliding mode for a non-holonomic mobile robot

: This paper is interesting to a nonholonomic wheeled mobile robot. We have presented a scheme to develop controllers. Two controllers have been developed. The 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. The sliding surface has the identical structure as the proportional integral controller. The stability of system has been proved based on Lyapunov approach. The 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. The trajectory tracking consists of guiding the robot through intermediate points to reach the final destination. This tracking is done 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. The real robot must follow precisely that of reference and reduce the distance error, by varying its linear and angular velocities [1], [2]. There are a many works that have focused on tracking the trajectory of the mobile robots, they consider the mobile robot as a particle, in this case the inputs are velocities. Their aims are kinematic models. In [10], the kinematic control law approach supposes that the control signal generates the exact motion commanded. On the other hand, 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 [3]. In [5], Jun-Ku 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 [7], authors proposed a robust backstepping controller for 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. Venelinov [9] proposed an adaptive fuzzy approach for kinematic controller. This method was able to decrease the effect of unmodeled disturbances. In [11], a dynamic Petri recurrent fuzzy neural network was proposed. In [4], the proposed controller combines nonlinear time varying feedback with an integral sliding mode controller. The latter is obtained by introducing an integral term in the switching manifold. In [12], a robust adaptive mobile robot controller is presented using backstepping for kinematics and dynamics motions, the adaptive process was based on the neural network. In [8] a classical parallel distributed compensation (PDC) control law, based on Takagi-Sugeno fuzzy modeling, is proposed. The controller comprises sixteen rules in which the control gains have been calculated using LMI techniques. In [19] authors, present an adaptive controller with consideration of unknown model parameters. In [20] authors Suggest a controller of a mobile robot in Cartesian coordinates with an approach angle based on the sliding mode. In [22] 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, use for kinematic motion a classical controller arising from the backstepping method [2,12,19,21,22]. This paper includes two main contributions. First a new controller based on Takagi-Sugeno fuzzy systems for kinematic motion. This last uses three fuzzy rules. The second contribution consists in developing for the dynamic part a controller based on the sliding mode. The sliding surface, which is based on linear and angular velocities of the robot, has the similar structure as the proportional integral controller. The switching control term of the latter controller combines the two sliding surfaces. This paper is organized as follows. The next section is devoted to the description of the kinematic and dynamic models of the two-wheeled mobile robot. The third section 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. The stability analysis is chekked in the both precedent subsection by the Lyapunov approach. Then, the fourth section 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 first subsection is reserved for kinematic modeling while the second subsection concerns dynamic modeling.

Kinematic model
Based on the Newton-Euler equations [13] and the previous hypothese, the state equations of mobile robot are represented by the following equations system [6]: cos where (x, y), and 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. The state variables of mobile robot are: and represent respectively, the desired linear and angular velocity. The state kinematic model of mobile robot in Cartesian frame coordinates is given by the following expression:

Dynamic model
The dynamic equation of the wheeled mobile robot is given by the following equation: where, ( , ) is the centripetal and Coriolis matrix, ( ) is the friction force, represents the torque vector, , ( ) = 0, and represent respectively the mass and the moment inertia of the wheeled mobile robot. and represent respectively the wheel radius and the distance separating the two driving wheels. Without considering disturbances and uncertainties, the latest equation becomes as: The expressions of linear and angular velocities of the mobile robot, (v, w), depend on the left and right linear velocities of the motors. They are expressed by the following equations:

Design of Robot controllers
In this work, we consider the kinematic and dynamic behavior of the robot. The purpose of the control design is to allow the robot to follow the virtual robot. The latter represents the reference robot and provides the desired path defined by the following vector: Therefore, the linear and the angular reference velocities and can be generated by regularly continuous control inputs.
The 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: The posture vector error ! is computed basing on the actual posture vector ( ) The relation between local frame and global frame, as shown in figure 2, is given by the following equation: The architecture, of the control scheme of the robot, includes six blocks, as shown in Figure 3. The first block generates the desired states whereas the second block transforms the error from the local frame into the general frame. The third and fourth blocks are reserved respectively for kinetic and dynamic controllers. The fifth and sixth blocks respectively describe the behavior of the kinematic and dynamic models of the robot.

Fuzzy Kinematic controller
In this subsection, we are interested in the search for a controller, which guarantees the convergence of the kinematic errors towards zero in the local coordinate system. However, by differentiating equation (13), 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: Our goal is to find a control law that stabilizes the system and allows the robot to follow the desired path. For this reason, we use the fuzzy system. The advantage of the T-S type fuzzy approach is that it allows describing the nonlinear model by linear sub-models. Indeed, each sub-model represents a local linear relation between the inputs and the outputs and all the nonlinearities are reported in the premises of the fuzzy rules (Morère, 2001). We note that equation (16) contains trigonometric nonlinearities which are cos (e3) and sin (e3). However, the nonlinearities depend on the error e3, whose range of variation is -pi / 2 to pi / 2. Based on the theory of T-S fuzzy systems, the nonlinear model (16) can be transformed into three local models, which are inferred by fuzzy rules. The 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). The member ship function for the error e3 is given by figure 4.
The T-S fuzzy model of equation (16)

Stability analysis
To check the stability of the robot, we use Lyapunov's theory. However, we choose the following Lyapunov candidate function.
The derivative of Luapunov function is : If we choose the following linear and angular velocities: The 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 qe 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. In this case, we define two sliding surfaces. The first surface depends on linear velocity while the second uses angular velocity, Where sv and sw are given respectively by equations (28 ) and (29 ), However, the derivatives of the sliding surfaces ( ) The equivalent control law eq u is computed =0by recognizing that 0 = S & which is a necessary condition for the state trajectory to stay in the sliding surface [17], [18]. The derivative of the sliding surface is: Thus substituting (33) for (35), we obtain:     2  3   3  2  3  2  2  3  2  2  2  1  3  3  1  1  1   2  3   3  2  3  2  2  3  2  2  2  1  3  3  1  1  1 eql eqr 2 2  2  3   3  2  3  2  2  3  2  2  2  1  3  3  1  1  1  eql   2  3   3  2  3  2  2  3  2  2  2  1  3  3  1 The switching control term is generally chooses as: , with 0 η f . 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. The Lyapunov candidate function is choosing as: 1 2 The derivative can be expressed as : Basis on the equation (37)

Simulation results
In this part, we present the simulation results of the nonholonomic wheeled mobile robot. The parameters of the robot (see figure 1) The initial speeds have been choose as 38 . 0 , The following figures 5, 6 show respectively the evolution of the robot along the x and y axis (the trajectories and the reference)      Figures 8,9 and 10: show that the errors e1, e2 and e3 rapidly tend towards zero. This clearly shows that the system converges towards the desired trajectories in a very short time. This 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. The first concerns the kinematic behavior while the second relates to the dynamic behavior of the mobile robot. To overcome the nonlinearities present in 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. The proposed sliding surface has the same structure as the proportional integral controller. Lyapunov approach has been used to prove stability of system. The results presented in section 4 show the efficiency of the proposed control laws.