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This work presents the design of two control schemes for a Delta/Par4-like parallel robot: augmented PD (APD) controller and augmented nonlinear PD (ANPD) controller. The stability of parallel robot based on nonlinear PD controller has been analyzed and proved based on Lyapunov method. A comparison study between APD and ANPD controllers has been made in terms of performance and accuracy improvement of trajectory tracking. Also, another comparison study has been presented between augmented nonlinear PD (ANPD) controller and nonaugmented nonlinear PD (NANPD) controller in order to show the enhancement of introducing the augmented structure on dynamic performance and trajectory tracking accuracy. The effectiveness of augmented PD controllers (APD and ANPD) and nonaugmented nonlinear PD (NANPD) controller for the considered parallel robot are verified via simulation within the MATLAB environment.

The parallel manipulators are defined as mechanisms with closed-loop kinematic chains, in which the end effector is linked to the base through several independent kinematic chains. Parallel manipulator has the advantages of high speed, high precision, and ability to manipulate heavy loads [

In the last two decades, several structures of modified proportional integral derivative (PID) controllers have been presented in the industrial control application. One of these controllers is the nonlinear PID (NPID), which is introduced by HAN [

Recently, several controllers are proposed to control the parallel manipulator. In [

In the present work, design and stability analysis of augmented-based PD control structure is presented in order to control a redundant Delta/Par4-like parallel manipulator.

Delta/Par4-like robot is a redundantly actuated parallel manipulator and it is much known for the application that needs for high speed and high acceleration. The Delta/Par4-like robot is characterized by light-weight mechanical component common with good structural stiffness. Also, it can achieve velocity and acceleration to reach above 10 m/

Solid work drawing of Delta/Par4-like parallel manipulator.

The main contribution of the work can be summarized by the following points:

Design of augmented-based PD control scheme for controlling a redundant Delta/Par4-like parallel robot.

Performance comparison between augmented-based controllers (APD and ANPD) in terms of dynamic behaviors and accuracy of trajectory tracking.

A Lyapunov-based stability analysis is presented for parallel robot controlled by ANPD controller to prove the asymptotically convergence of both tracking error and error rate to zero as

A comparison study has been made between augmented-based nonlinear PD (ANPD) controller and nonaugmented-based nonlinear PD (NANPD) controller in terms of tracking accuracy and dynamic performance.

The geometric parameters used in describing the dynamic model of Delta/Par4-like robots are depicted in Figure

Geometric parameters of Delta/Par4-like robot.

The Cartesian coordinates

This section presents the simplified direct dynamic model (DDM) of Delta/Par4-like parallel manipulators. To do so, the following assumptions are made [

Neglect the joint friction.

Neglect the inertia of forearms (

The total mass of each forearm

Due to very high acceleration, which reaches up to 100 G, the gravity will not be considered and can be neglected.

Splitting the forearm mass into two parts.

To start the derivation of simplified dynamic model of Delta/Par4-like robot, the equilibrium analysis of the arm is first presented. The actuator torque vector

Secondly, the equilibrium of the traveling plate is analyzed. The motion equation of the traveling plate is given by [

Substituting (

The ANPD controller proposed in the present work is synthesized by replacing the linear PD by the structure of APD controller based on NPD algorithm. The structure of the NPD controller can be described by [

Graphical description of the nonlinear function.

According to general NPD control law of (

The control law of the ANPD controller of Delta/Par4-like robot consists of two parts: the first part describes the dynamic compensation defined by the desired trajectory

This section focuses on the proof of the asymptotic stability of the Delta/Par4-like robot system controlled by the ANPD controller (

A scalar continuous function

Let the function

Graphical interpretation of Lemma

Since

The continuous matrix diagonal

Suppose that there exist class

Graphical description of Lemma

If

Let us candidate a Lyapunov function of the following form:

The time derivative of the Lyapunov function is

Multiplying both sides of (

However, since

Based on Lyapunov function

The block diagram of Delta/Par4-like Robot control scheme based on augmented PD controller is shown in Figure

Block diagram of ANPD controller.

In case of nonaugmented PD controller, the terms

Delta/Par4-like robot is redundant parallel manipulator consisting of four motors and 3-DOF, and its actuators are the RTMB140-100 ETEL, which have a maximum torque 127 N. m and maximum speed 550 RPM and workspace (cylinder of 300 mm radius and 100 mm height) [

Numerical values of system parameters [

The Parameters | Symbols | Values |
---|---|---|

The radius of robot base. | | 0.35 |

Traveling plate radius. | | 0.1 |

Arm length. | | 0.35 |

Forearm length. | | 0.8 |

Traveling plate mass. | | 0.6 |

The inertia of actuating motor. | | 0.003 |

Arm inertia. | | 0.071 |

Forearm Mass. | | 0.3 |

The maximum allowable torque. | | 90 |

In this section, the dynamic model of Delta/par4-like robot and proposed controllers are implemented in Matlab/Simulink (R2017b). The setting of parameters for both ANPD and APD structures is based on the try-and-error procedure and listed in Table

Settings of controller parameters.

Parameters of ANPD | Value | Parameters of APD | Value |
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The desired trajectory in Cartesian space can be described by

The traces of the desired trajectory.

In performance evaluation of accuracy in most robotic applications, the RMSE is used instead of Integral absolute value of error (IAE) or Integral of the square value of error (ISE). However, the RMSE criteria is strongly related to ISE one, where RMSE=

The first part of the simulation establishes a comparison study in performance between augmented PD control structures (APD and ANPD). Figure

Cartesian error dynamics in

Table

The RMSE given by the APD and ANPD controllers.

APD | ANPD | Improvement |
---|---|---|

0.0068 | 0.0013 | 78.27% |

The responses of torques generated by the four robot actuators are shown in Figure

Torque responses for APD and ANPD controllers.

The second part of simulation results focuses on the evaluation of performance based on augmented and nonaugmented nonlinear PD controllers (ANPD and NANPD). Figure

Cartesian error dynamic in

The torque responses generated from the four actuating motors of the parallel robot is depicted in Figure

Torque responses for NANPD and ANPD controllers.

This paper presented two comparison studies. One study is based on the comparison between augmented PD control structures (APD and NAPD controller). The other comparison study is established between augmented nonlinear PD (ANPD) controller and nonaugmented nonlinear PD (NANPD) controller. The assessment of controllers for comparison is made in terms of tracking performance and accuracy for 3-DOF Delta/Par4-like parallel robot. The measure of improvement is calculated in terms of error variance.

The simulated results showed that ANPD controller gives 78.26% improvement in tracking accuracy as compared to that given by APD as indicated by Table

The RMSE given by the NANPD and ANPD controllers.

NANPD | ANPD | Improvement |
---|---|---|

0.0052 | 0.0013 | 75% |

No data were used to support this study.

The authors declare that they have no conflicts of interest.