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This paper focuses on neural learning from adaptive neural control (ANC) for a class of flexible joint manipulator under the output tracking constraint. To facilitate the design, a new transformed function is introduced to convert the constrained tracking error into unconstrained error variable. Then, a novel adaptive neural dynamic surface control scheme is proposed by combining the neural universal approximation. The proposed control scheme not only decreases the dimension of neural inputs but also reduces the number of neural approximators. Moreover, it can be verified that all the closed-loop signals are uniformly ultimately bounded and the constrained tracking error converges to a small neighborhood around zero in a finite time. Particularly, the reduction of the number of neural input variables simplifies the verification of persistent excitation (PE) condition for neural networks (NNs). Subsequently, the proposed ANC scheme is verified recursively to be capable of acquiring and storing knowledge of unknown system dynamics in constant neural weights. By reusing the stored knowledge, a neural learning controller is developed for better control performance. Simulation results on a single-link flexible joint manipulator and experiment results on Baxter robot are given to illustrate the effectiveness of the proposed scheme.

Due to the great demands in industrial applications, the tracking control problem for flexible joint robot (FJR) manipulator has attracted much attention in recent years. Unlike rigid joint robot, the joint flexibility of FJR results in complex control situation, so that the control problem of FJR becomes much more difficult. In the past few decades, lots of efforts have been made on the research of FJR systems. Based on the model of FJR presented in [

The backstepping control [

Besides, the transient and steady-state tracking performance constraints of system’s output are an important issue that needs to be taken into consideration [

In addition, adaptive neural control of nonlinear system has been widely studied for decades, but most of the traditional works focus on the system stability through online adjustment of neural weights and less works discuss the knowledge acquisition, storage, and utilization of optimal neural weights. To achieve such learning ability, the key problem is to verify the persistent excitation (PE) condition. Thanks to the results in [

This paper focuses on learning from adaptive neural control of flexible joint manipulator with unknown dynamics under the prescribed constraints. A performance function is introduced to transform the constrained tracking error into the unconstrained variable. To avoid the curse of dimensionality of RBF NN, first-order filters are introduced to reduce the number of NN approximators and decrease the dimension of NN inputs. The control law is constructed based on Lyapunov stability, which guarantees the closed-loop stability and the tracking error satisfying the prescribed performance during the transient process. Subsequently, due to the property of DSC and structure features of the considered manipulator, a system decomposition strategy is employed to decompose the stable closed-loop system into two linear time-varying (LTV) perturbed subsystems on the basis of the number of NNs in the whole system. Through the recursive design, the recurrent properties of NN input variables are easily proven. Consequently, with the satisfaction of the PE condition of RBF NNs, the convergence of partial neural weights is verified, and the unknown dynamics of system are approximated accurately in a local region along recurrent orbits. By utilization of the constant neural weights stored, a neural learning controller is developed to achieve the closed-loop stability and better control performance under the prescribed constraints for the same or similar control task. Compared with the existing neural learning results, the proposed neural learning control scheme not only achieves better control performance with specified transient and steady-state constraints but also reduces the dimension of NN inputs and the number of NNs significantly.

This paper is organized as follows. In Section

In this paper, we consider an

The inertia matrix

The Coriolis and centrifugal matrix

The reference trajectory vector

Our goal is to design a neural learning controller, which forces the tracking error vector (i.e.,

In this paper, the output error vector of system (

According to [

On the other hand, it has been shown in [

Consider any continuous recurrent trajectory

In this section, performance function is introduced for describing constraints of system (

Similarly to the traditional backstepping design, we set

It should be pointed out that any errors set in previously traditional design are under unrestricted condition [

It implies that

Transformed function

It can be concluded from (

By combining (

Since

Noting that

It is clear that

Let

Define the unknown dynamics in (

According to the property of RBF NN,

Then the virtual controller

Take

Define

Introduce a new filter variable

Let

According to the property of RBF NN,

Then the control input

Let us construct the following Lyapunov function candidate:

It should be pointed out that, in the adaptive neural backstepping design [

Consider the manipulator model (

See Appendix

In this section, we will show the learning ability of RBF NNs for unknown system dynamics

According to Lemma

Consider the closed-loop system consisting of the flexible joint manipulator model (

From Theorem

It would be shown that the perturbation term

According to Theorem

According to the above analysis and noting

Similarly,

It would be shown that the perturbation term

Using the similar step and choosing

Since the locally accurate NN approximation can be achieved by the constant RBF NN

Consider the closed-loop system consisting of the manipulator model (

See Appendix

For clarity, a block diagram of the proposed schemes is shown in Figure

Block diagram of the proposed learning control scheme.

In this section, to illustrate the effectiveness of the proposed approach, a single-link manipulator system with flexible joint is considered by the following form:

Single-link flexible joint manipulator.

Based on Theorem

According to (

The related simulation results are shown in Figures

Tracking error: LC (-) and ANC (- -).

System output:

State variables

Control input: LC (-) and ANC (- -).

Function approximation:

The convergence of partial NN weights

The convergence of partial NN weights

In order to compare with the difference of tracking performance between different parameter selection in (

Tracking error:

Moreover, in order to validate the effectiveness of the proposed control scheme, the Baxter bimanual robot is used in the experiment, as shown in Figure

Overview of the experimental platform.

The desired motion of the robot’s link.

To verify the effectiveness of neural learning control scheme for Baxter robot, the constant weights

The related results are shown in Figures

Tracking error: LC (-) of Baxter robot and LC (- -) of simulation.

State variables

Control input: LC (-) of Baxter robot and LC (- -) of simulation.

In this paper, we studied learning from adaptive neural dynamic surface control for a class of flexible joint manipulator with unknown dynamics under the prescribed constraint. A novel error transformed function was utilized to transform the constrained tracking problem into the the equivalent unconstrained one so as to facilitate the controller design. Furthermore, by combining DSC method, which was used to reduce the number of NN approximators and decrease the dimension of NN inputs, a novel adaptive neural control scheme was proposed to guarantee the prescribed performance during the transient process. Then, the closed-loop stability and control performance were achieved according to the construction of the Lyapunov function. After the stable control process, since two NNs were used in the controller design, the recurrent property of the NN input variables and the partial PE condition of RBF NNs were proved recursively. Therefore, the locally accurate approximations of unknown system dynamics by RBF NNs were achieved, and the proposed control scheme was verified to be capable of storing the learned knowledge in constant RBF NNs. Finally, the stored knowledge was reused to develop the neural learning controller for the same system model and the same or similar control task, so that the closed-loop stability and better control performance were achieved under the prescribed constraint. Simulation results for a single-link flexible joint manipulator and experiment results for Baxter robot were presented to prove the effectiveness of the proposed control scheme.

Firstly, the derivative of

In addition, since

Since

Similar to the adaptive neural DSC design in the Section

According to Theorem

The authors declare that they have no conflicts of interest.

The authors would like to thank Professor Chenguang Yang for the experiment platform. Moreover, this work was partially supported by the National Natural Science Foundation of China (nos. 61773169, 61374119, 61473121, and 61611130214), the Royal Society Newton Mobility Grant IE150858, the Guangdong Natural Science Foundation under Grants 2017A030313369 and 2017A030313381, the youth talent of Guangdong Province, and the Fundamental Research Funds for the Central Universities.