In this paper, a non-negative adaptive mechanism based on an adaptive nonsingular fast terminal sliding mode control strategy is proposed to have finite time and high-speed trajectory tracking for parallel manipulators with the existence of unknown bounded complex uncertainties and external disturbances. The proposed approach is a hybrid scheme of the online non-negative adaptive mechanism, tracking differentiator, and nonsingular fast terminal sliding mode control (NFTSMC). Based on the online non-negative adaptive mechanism, the proposed control can remove the assumption that the uncertainties and disturbances must be bounded for the NFTSMC controllers. The proposed controller has several advantages such as simple structure, easy implementation, rapid response, chattering-free, high precision, robustness, singularity avoidance, and finite-time convergence. Since all control parameters are online updated via tracking differentiator and non-negative adaptive law, the tracking control performance at high-speed motions can be better in real-time requirement and disturbance rejection ability. Finally, simulation results validate the effectiveness of the proposed method.

In recent years, parallel manipulators have been widely deployed in various industry fields. Examples can be found in flight simulators, machine tools, micro-mechanisms, haptic devices, etc. [

In order to guarantee the performance of the tracking controller, the terminal sliding mode control (TSMC) scheme has been developed and discussed as an efficient methodology [

The online non-negative adaptive mechanism (NAM) is used to estimate the uncertainties and disturbances. Hence, unlike the existing TSMC [

In this approach, the tracking differentiator (TD) is adopted to cope with the high-speed motions. It can be found that the tracking control performance at high-speed motions can be better in a real-time fashion because all control parameters are online updated based on TD and NAM.

The proposed controller has possessed advantages such as simple structure, easy implementation, chattering-free, high precision, robustness, singularity avoidance, and finite-time convergence. Besides, the proposed approach has superior tracking control performance and disturbance rejection ability. The stability and finite-time convergence of the parallel mechanisms are ensured by the Lyapunov theory.

The stability of a robotic system with the NFTSMC controller [

To address this issue, in [

From [

The parallel mechanisms dynamic is represented in a second-order differential equation as

The Euclidean norm of a vector that has

To avoid any possible confusion, a variable vector

Define

Let

Under Assumption 1, modifications of the NFTSMC design have also been proposed in the literature [

In [

In

To address these issues, in this study, an online NAM is used to estimate the upper bounds of complex uncertainties and external disturbances of parallel manipulator systems. The proposed controller does not require prior knowledge about the bounds of uncertainties and disturbances. Besides, the TD is adopted to deal with the transition process and to decrease the initial impulse of the manipulative variable. Combining NFTSMC, TD, and online NAM together in this study, the proposed ANFTSMC law is expressed as

The NAM law of the parameters

Substituting

In combination with the control input

Hence, the parallel mechanisms states converge to the sliding surfaces asymptotically. To show that the system trajectories can move fast to zero in a finite time,

Define

Following the Lyapunov stability theorem, the sliding surface in

The acceleration signals defined by

In this section, the performances of the proposed controller are verified for a 2-DOF parallel mechanisms as shown in

It is noted that the parallel manipulator is influenced by the uncertainties

In order to show the improvement in performance, the ANFTSMC control scheme is compared with the NFTSMC [

To overcome the chattering problem, the

Sliding surfaces & controllers | Parameters | Value |
---|---|---|

Sliding surface | 1.5 | |

9, 7 | ||

ANFTSMC | 20, 5 | |

0.7, 1.1 | ||

0.01 | ||

NFTSMC [ |
30 | |

15 |

The simulation results in case 1 are given in

To show the quantitative comparison, the root square mean error (RSME) of the position tracking errors of the active joints is chosen as the performance index.

Controllers | NFTSMC [ |
ANFTSMC | |
---|---|---|---|

Cases | Active joints | ||

Case 1 | Joint 1 | ||

Joint 2 | |||

Case 2 | Joint 1 | ||

Joint 2 | |||

Case 3 | Joint 1 | ||

Joint 2 |

This paper reports our study on the adaptive nonsingular fast terminal sliding mode finite-time tracking control for parallel manipulators with the existence of complex uncertainties and external disturbances in the case of high-speed motions. The proposed control scheme is successfully designed based on the tracking differentiator, the non-negative adaptive mechanism, and the nonsingular fast terminal sliding mode control. The non-negative adaptive law is employed to handle the real-time estimation of the total of complex uncertainties and external disturbances. This control scheme does not require prior knowledge of bounded uncertainties and disturbances. Simulation results show that the proposed scheme has superior tracking control performance, and the tracking error converges fast to zero in a finite time without any singularity. Possible future work can be to choose the optimal control parameters by using optimization algorithms.