Takagi-Sugeno Fuzzy Modeling and Control for Effective Robotic Manipulator Motion

Robotic manipulators are widely used in applications that require fast and precise motion. Such devices, however, are prompt to nonlinear control issues due to the flexibility in joints and the friction in the motors within the dynamics of their rigid part. To address these issues, the Linear Matrix Inequalities (LMIs) and Parallel Distributed Compensation (PDC) approaches are implemented in the Takagy-Sugeno Fuzzy Model (T-SFM). We propose the following methodology; initially, the state space equations of the nonlinear manipulator model are derived. Next, a Takagy-Sugeno Fuzzy Model (T-SFM) technique is used for linearizing the state space equations of the nonlinear manipulator. The T-SFM controller is developed using the Parallel Distributed Compensation (PDC) method. The prime concept of the designed controller is to compensate for all the fuzzy rules. Furthermore, the Linear Matrix Inequalities (LMIs) are applied to generate adequate cases to ensure stability and control. Convex programming methods are applied to solve the developed LMIs problems. Simulations developed for the proposed model show that the proposed controller stabilized the system with zero tracking error in less than 1.5 s.

https://www.techscience.com/cmc/v71n1/45465/html 2/7 (1) (7) (8) (9) (10) (11) (12) (13) has three main parts: motor, gear, and robot arm [26]. The inertia of the motor, gear and robot arm are denoted by , , and , respectively. The robot arm system is actuated by the input torque of the motor. Besides this, the friction effect is assumed to act on the motor within a nonlinear torsion coefficient . Due to the flexibility effect, the input torque rotates the motor, the gear and robot arm within an unequal angular position , , and , respectively. The flexibility of the gearbox is modeled by a nonlinear torsion stiffness . The gear ratio of the gearbox is assumed one, i.e., . Since this study focused on the nonlinearity of the joints, the flexibility in the robot arm is modeled by a linear spring of torsion stiffness . Besides the effect of flexibility in the gearbox and the robot arm, the damping coefficients in the gearbox and the robot arm are considered and , respectively. The nonlinear torque friction part is considered as the effect of coulomb friction and Striebeck effect. The coulomb friction is assumed as [27].
The total nonlinear torque friction effect is considered as Tustin friction model [28]: where , , and denote static friction, stribeck velocity, and viscous friction, respectively. On the other hand, the nonlinear torsion torque of gear model is assumed as Applying newton second law, the equation of motion for the motor inertia, gear inertia, and arm inertia is Respectively. To get the state space model, assume the following state variables: and the output is the angular velocity of the motor that is assumed . thus, the differential states are, The differential states from Eqs. (7)- (12) and the output formula, i.e., the angular velocity of the motor, can be arranged in state space model format as where The obtained state model in Eq. (13) will be implemented in the next section to be represented by the T-SFM approach. The numerical results of this study are based on the physical parameters of the robot model presented in Tab. 1.
After deriving the state equations of our proposed nonlinear robot model, T-SFM can be implemented to linearize the robot system as explained in the next section.

Implementation of T-SFM
T-SFM is implemented to represent the nonlinear state space model of the robot arm. In addition, the T-SFM has linearized the nonlinear terms in state space equations. The nonlinear robot arm model will be denoted by T-SFM as

Controller Design
The presented design of the controller is closely related to the feature of the derived T-SFM in the previous section. This feature is that the terms of state equations , , , and are not constant. These terms are varying corresponding to the torsion torque and motor friction depends on the velocities of motor and gear of the state variables. This property was the source of the high nonlinearity in the robot arm model. Consequently, for linearization purpose, the nonlinear robot arm system is expressed by the bellow fuzzy model: Model rule 1 If is Small_1 and is Small_2 and is Small_3 and is Small_4 then

Model rule 2
If is Small_1 and is Small_2 and is Small_3 and is Big_4 then

Model rule 3
If is Small_1 and is Small_2 and is Big_3 and is Small_4 theṅ , In this way, the nonlinear robot model can be described in the following general formula Applying the defuzzification principle of fuzzy method, the output of the system is obtained as: where denotes the membership function for the model rule . Hence For the obtained T-S fuzzy model rule, the technique of state feedback is applied to develop the following control rules The PDC technique is implemented in this study to find the solution of the system using the obtained T-SFM. Considering Eqs. (29) and (33), PDC technique is applied to design the controller based on T-S fuzzy approach as follow Our developed mode rules in Eq. (29) can be asymptotically stable by a potential positive matrix Q when satisfying the following conditions: where . Consequently, in Eq. (34) can be calculated by implementing the above LMI conditions.

A Numerical Example
In this section, simulation tests with MatLab R2017b are implemented to verify the performance of the designed controller for the robot arm model of the parameters that are listed in Tab. 1 with initial arm position and angular velocity and , respectively. The control issue of the robot arm system shown in   Fig. 1 is assumed to reach the robot arm a desired position by applying a motor torque input. Furthermore, it is assumed that the robot arm is stabilized at a specific angle. The robot arm model is specified in Eq. (13)

Results
The simulation results of the proposed model are shown in Figs. 2 and 3. In Fig. 2, the results of the robot arm position and velocity are presented for the equilibrium situation where and are equal to zero with time above . The simulation shows the transient response between and . In which, the arm position is reached to zero at time in a linear way. On the other side, the angular velocity of the arm reached its maximum value in opposite direction at time . The dynamic effect of the robot arm is caused by the input torque of the motor presented in Fig. 3. As shown in Fig. 3, to stabilized the robot arm to position, the motor should supply a torque in a short time as an impulse signal. The results of Fig. 3 are useful in determining the suitable features of the potential motor for the proposed robot arm physical parameters. < 0 ,

Conclusion
In this paper, a new control technique of the rotation motion of a nonlinear robot arm manipulator is introduced. The LMI and PDC control approaches are implemented based on TS-FM of the state space equations. The TS-FM has been developed for linearizing the nonlinear parameters in the state space equations in appropriately selected conditions of the operating points. The prime concept of the designed controller is to deduce all the fuzzy rules using the PDC approach in order to compensate for all rules of the fuzzy model. Furthermore, the linear matrix inequalities (LMIs) are applied to generate adequate cases for approving the stability and control purpose issues. The simulation results demonstrate that the proposed control technique stabilized the system with zero tracking error in less than 1.5 s. This is a suitable control performance for nonlinear robot arm models. The main limitation of this work is the non-utilization of optimization techniques such as genetic algorithms to find the optimum boundaries of membership functions that minimize the tracking errors. This will be addressed in the future by using such approaches to deal with the design membership functions in this application considering their types and boundaries.

Funding Statement:
The authors received no specific funding for this research study.

Conflicts of Interest:
The authors declare that they have no conflicts of interest to report regarding the present study.