This paper proposes an adaptive predefined performance neural control scheme for robotic manipulators in the presence of nonlinear dead zone. A neural network (NN) is utilized to estimate the model uncertainties and unknown dynamics. An improved funnel function is designed to guarantee the transient behavior of the tracking error. The proposed funnel function can release the assumption on the conventional funnel control. Then, an adaptive predefined performance neural controller is proposed for robotic manipulators, while the tracking errors fall within a prescribed funnel boundary. The closed-loop system stability is proved via Lyapunov function. Finally, the numerical simulation results based on a 2-DOF robotic manipulator illustrate the control effect of the presented approach.
Robotic manipulators have been widely utilized in industrial applications such as manufacturing industry, aerospace, and military equipment [
In fact, the difficulties in the control design for robotic systems mainly stem from nonlinear terms. To tackle these nonlinear terms, disturbance observer techniques were proposed to reject the unknown disturbance [
On the other hand, as neural networks (NNs) [
Recently, it is well known that the prescribed performance control (PPC) method can be used to quantitatively analyse the transient behavior [
This paper will propose a novel adaptive neural prescribed performance control method for robotic manipulators with unknown dead zone. A novel funnel variable is defined based on the tracking error. The modified funnel variable can release the assumption on the original funnel control. An echo state neural network (ESN) is adopted to estimate the unknown dynamics of robotic manipulators, and the approximation is used in control design to compensate the nonlinear dead zone. Then, an adaptive control scheme for a robotic manipulator is proposed to improve the control performance. Numerical simulation demonstrates the effectiveness of the proposed control approach.
The special contributions of this paper are as follows: A novel funnel function is proposed based on the tracking error, and it can release the limitation on the original funnel function and is used in control design to improve the control performance A neural network is utilized to estimate the nonlinear dead zone, and the approximation is to design a controller, where the dead zone is compensated The effectiveness of the proposed control method is evaluated based on a robotic manipulator by using numerical simulations
The remainder of this paper is organized as follows. Section
This paper considers a
For the matrices
The matrix
The matrix
Nonlinear dead zone.
The dead-zone nonlinearity (see Figure
Using (
The echo state neural network is a novel NN with superior capability to approximate the unknown dynamics. The basic architecture of the ESN is shown in Figure
Basic architecture of the ESN.
The function
Therefore,
Funnel control [
The system Relative degree Minimum phase Known high frequency gain
The controller is given as
Thus, the funnel itself is defined as the set
The gain
According to [
Funnel control.
A novel funnel variable can be given as
In this section, we consider the full state information,
Controller architecture.
The tracking error
The time derivative of (
The Lyapunov function is defined as
Its time derivative is
The second error variable is defined as
Substituting (
An intermediate control signal is chosen as
The time derivative of
According to (
The time derivative of
The Lyapunov function is defined as
The derivative of (
The actual controller can be designed as
In this section, we will employ the Lyapunov function to analyse the convergence of the closed-loop system.
Consider the robotic manipulators (
A Lyapunov function is chosen as
The time derivative of (
Substituting (
Based on (
Using Young’s inequality, one has
Substituting (
From (
The parameter tuning guidelines are given as follows: Select the funnel variables Choose the control gains
In this section, we will employ an example to illustrate the control performance of the developed control method. A diagram of the robotic manipulator system with 2-DOF is shown in Figure
Diagram of the robotic manipulator.
Parameters for the robotic manipulator.
Parameters | Description | Value | Unit |
---|---|---|---|
|
Length of link 1 | 1 |
|
|
Length of link 2 | 0.8 |
|
|
Mass of link 1 | 1 |
|
|
Mass of joint 2 | 1.5 |
|
|
Mass of link 2 | 2 |
|
|
Mass of actuator | 1.5 |
|
The system matrices
The controller parameters are given as
Figures
Output tracking.
Friction compensation.
Control signals.
In this paper, an adaptive predefined performance control for robotic manipulators in the presence of nonlinear dead zone was proposed. A novel funnel variable was designed based on the tracking error. The new error variable was utilized to design a controller that can guarantee the transient response. A neural network was adopted to estimate the unknown dynamics (parameter uncertainties and nonlinear dead zone), and the approximation was utilized in controller design to compensate the unknown dynamics. An adaptive controller based on funnel control was designed for the robotic manipulator. Both the transient response and steady-state performance of the tracking error are guaranteed.
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.
This work was supported by the Project of Science and Technology Department of Shanxi Province (16JK1100).