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This paper endeavors to contribute to the field of optimal control via presenting an optimal fuzzy Proportional Derivative (PD) controller for a RPP (Revolute-Prismatic-Prismatic) robot manipulator based on particle swarm optimization and inverse dynamics. The Denavit-Hartenberg approach and the Jacobi method for each of the arms of the robot are employed in order to gain the kinematic equations of the manipulator. Furthermore, the Lagrange method is utilized to obtain the dynamic equations of motion. Hence, in order to control the dynamics of the robot manipulator, inverse dynamics and a fuzzy PD controller optimized via particle swarm optimization are used in this research study. The obtained results of the optimal fuzzy PD controller based on the inverse dynamics are compared to the outcomes of the PD controller, and it is illustrated that the optimal fuzzy PD controller shows better controlling performance in comparison with other controllers.

In the recent years, the robotic arms have been used in the industrial applications, to name but a few, assembly lines [

One of the most crucial problems raised in the study of the direct kinematics of the robot is how the robot's framework is changed when moving [

In order to control the challenging dynamics of the robots, the nonlinearity in the dynamics must be turned into linearity via using techniques such as inverse dynamics. Afterwards, by choosing an appropriate controller, such as PD control, it is feasible to control the challenging dynamics with high precision [

Particle swarm optimization (PSO) is a smart swarm-based technique introduced by Kennedy and Eberhart [

This paper introduced an inverse dynamics based optimal fuzzy Proportional Derivative (PD) controller for a RPP (Revolute-Prismatic-Prismatic) robot manipulator optimized via particle swarm optimization. The Denavit-Hartenberg and Lagrange approaches were utilized for deriving the kinematic and dynamic equations of the manipulator. In order to stabilize the links of the robot, the inverse dynamics approach and proportional-derivative controller were used in this research study. For the gain tuning of the designed controllers, the particle swarm optimization was successfully implemented. The obtained results of the optimal fuzzy PD controller based on the inverse dynamics were compared to the outcomes of the PD controller, and it was illustrated that the optimal fuzzy PD controller shows better controlling performance in comparison with other controllers.

The structure of the paper is as follows. The kinematics and dynamics of the RPP robot manipulator are presented in Section

The RPP robot has three degrees of freedom, which involves one degree of the revolute joint and two degrees of prismatic joint (RPP), where the degree of freedom of the end effector is not considered. The most common method to address the direct kinematic problems is the Denavit-Hartenberg approach. In this method, each member is assigned a number from

The Denavit-Hartenberg parameters with regard to each joint.

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1 | 0 | 0 | | |

2 | 0 | -90 | | 0 |

3 | 0 | 0 | | 0 |

The position of the coordinate systems on each of the joints.

The homogenous transformation matrix is gained through the required transformations and using (

Due to avoiding the computational complexity and providing simplification for nonlinear dynamic equations, the Lagrange method is employed in this study. In the Lagrange approach, the scalar quantities of the kinematic energy and potential energy are calculated and expressed with respect to the general coordinates. Finally, by utilizing the Lagrange approach according to (

By substituting each of the parameters into (

The dynamics of the manipulator's arm is extremely nonlinear, and, hence, designing an efficient controller is a complicated task and of great importance. One of the appropriate approaches to enhance the tracking efficiency of the manipulator's arm is the control method of the computed torque. By using the computed torque control, the linear closed-loop equations are gained, and if the coefficients of the control approach are chosen properly, the stability of the controller would be guaranteed.

The required control torque, which is

The inverse dynamic control approach is a crucial method for the control of the mechanical arms. The inverse dynamic control rule is defined as follows.

The fuzzy system selected for

(I) The inference product engine

(II) The fuzzifier and the trapezoidal membership function in the beginning and end of the range and a triangle in the middle of the range

(III) The defuzzifier of the average of the centers

Therefore, the input membership function is selected as Figure

The fuzzy rules of the system.

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NL | 1 |

Z | 0.5 |

PL | 1 |

The input membership function.

The fuzzy rules of the system are adjusted via setting up the above-mentioned parameters in the MATLAB and using Table

Results and discussions are conducted through the outcomes of the fuzzy PD controller and substituting the outcomes into the inverse dynamic equations of (

The assumed parameters of the PD controller.

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The initial conditions of the location and velocity and the desired values (

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In order to elaborate on the performance of the control approach, a smart optimization method is presented in the following section to optimize the control gains of the fuzzy PD controller.

For the algorithm of the particle swarm optimization, each particle is evaluated among the whole population and if the termination condition is satisfied, the algorithm operation is stopped. However, if the termination condition is not satisfied, the position of each particle is evaluated with respect to its previous position and the position of the best particle among population. In fact, the position and velocity of the particle are being iteratively calculated until the termination condition of the algorithm is satisfied. Consider the general mathematical equations of the PSO as

Two stages can be regarded in order to analyze the performance of this optimization algorithm on the control method of the robot manipulator. In the first stage, the gains of the PD controller are optimized. In the second stage, the gains of the fuzzy PD controller are optimized. The initial assumptions of this problem are as follows.

(1) The number of particles in the initial population is NP=10;

(2) The maximum number of iterations is MI=100;

(3) The impact rate of a particle from its local position is

(4) The impact rate of a particle from the general position of all particles is

(5) The inertia weight is

(6) The objective function (

In the fuzzy control, the heuristic fuzzy parameters (

Design variables and objective functions found by the particle swarm optimization algorithm.

Design variables | | |

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Objective functions | | |

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Convergence diagram of the PSO algorithm over the iteration number.

By regarding and comparing the position, force, and velocity diagrams of each of the links, the obtained results illustrate the superiority of the fuzzy PD controller over the PD controller. It is crucial to note that the input parameters of the controllers and the input coefficients are assumed in an ideal status in this study, and if there exist differences between the real model of the manipulator and its assumed model, it is probable that these controllers show error or instability. First, the position diagrams of each of these controllers are put under analysis.

By comparing the graphs of Figure

The position diagram of each of the links controlled via optimal fuzzy PD and optimal PD controllers, (a) the position of the first link (revolute), (b) the position of the second link (prismatic), and (c) the position of the third link (prismatic).

As it can be found from Figure

The comparison of the controlling force for optimal fuzzy PD and optimal PD controllers, (a) the controlling force of the first link (revolute), (b) the controlling force of the second link (prismatic), and (c) the controlling force of the third link (prismatic).

By comparing the graphs of the controlling force for each of the links, it can be found that the forces of the actuators in both cases have some limits, which illustrates the appropriate performance of the fuzzy PD controller. Further, by using the fuzzy PID controller, the desired state is reached in less time compared to the PD controller.

By comparing the velocity diagrams of each of the links (Figure

The comparison of the velocity diagrams, (a) the velocity of the first link (revolute), (b) the velocity of the second link (prismatic), and (c) the velocity of the third link (prismatic).

This research study presented an optimal fuzzy PD controller for a RPP robot manipulator based on particle swarm optimization and inverse dynamics. Indeed, the Denavit-Hartenberg method and the Jacobi approach for each of the arms of the robot were utilized to obtain its kinematic equations. Moreover, the Lagrange approach was used to gain the dynamic equations of the manipulator. By comparing the results, it was found that the fuzzy PD controller provides a lower settling time and a near-zero steady state error, which resulted in the superiority of the fuzzy PD controller over the PD controller.

The use of classic controllers for the control of the smart autonomous vehicles requires complete knowledge of all the forces and torque. These forces and torque are required to obtain the differential reciprocal and rotational movements. In fact, to obtain these equations, it is needed to address complex equations in a high amount of time or conduct experiments to gain the coefficients.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.