A novel exponential varying-parameter neural network (EVPNN) is presented and investigated to solve the inverse redundancy scheme of the mobile manipulators via quadratic programming (QP). To suspend the phenomenon of drifting free joints and guarantee high convergent precision of the end effector, the EVPNN model is applied to trajectory planning of mobile manipulators. Firstly, the repetitive motion scheme for mobile manipulators is formulated into a QP index. Secondly, the QP index is transformed into a time-varying matrix equation. Finally, the proposed EVPNN method is used to solve the QP index via the matrix equation. Theoretical analysis and simulations illustrate that the EVPNN solver has an exponential convergent speed and strong robustness in mobile manipulator applications. Comparative simulation results demonstrate that the EVPNN possesses a superior convergent rate and accuracy than the traditional ZNN solver in repetitive trajectory planning with a mobile manipulator.

Robotic arms have attracted increasing attention in engineering applications. Various algorithms and methodologies have been investigated for the kinematics of the robotic arms. Among the existed robot arms, redundant manipulators have played an enormous role in industrial control for repeatable dull work, such as equipping [

To obtain widely flexible operation space for special tasks, algorithms about kinematics of the mobile manipulators have been studied by industry, military, and aerospace control [

Kinematic control of the robot arm via neural networks is a popular trend for different trajectory tracking. To remedy the drifting joint phenomenon, an extended motion scheme at the joint-velocity level has been proposed [

However, the position rehabilitation of each joint of the robotic arms is an important direction in robotic kinematics, which can avoid joint-physical limitation and realize repeatable motion task. This is our main motivation for the present research. To satisfy the faster convergent requirement, different from the fixed parameter neural network model such as ZNN, an exponential varying-parameter neural network (EVPNN) is constructed in this paper. It is noted as varying-parameter neural network because the scaling parameter of the EVPNN is varying with time. It is necessary to point out that the proposed EVPNN is prompt to solve complex online optimization, such as trajectory planning of a mobile manipulator.

The remainder of this paper is organized as follows. Section

A novel EVPNN is presented and analyzed to solve the repetitive trajectory tracking of mobile manipulators under external noises. It is the first time to construct such an EVPNN model for solving this inverse redundancy scheme.

Theoretical analysis proves that the novel EVPNN can reduce to zero in exponential convergent rate and obtain high convergent precision.

Simulation comparisons between the EVPNN and the ZNN illustrate the exponential convergent rate, higher convergent accuracy, and strong robustness of the EVPNN when both neural solutions are applied to realize the repeatable motion of mobile manipulators.

Kinematic analysis of the manipulator is demonstrated in experiments with the seven-DOF (degree-of-freedom) mobile-base PA10 robot. PA10 robotic arm in [

For mobile manipulators, the issue of kinematics can be described as studying the relation between the movement of each joint angle and the pose of the end effector without considering torques for the motor system. The forward kinematic equation is as follows:

According to the definition of equation (

We now consider a mobile platform with three Swedish wheels, and the geometric analysis of the mobile base in the global coordinate system is depicted in Figure

The structure of the mobile platform in the global coordinate system.

As for the PA10 robot (

Evaluate the derivative of equation (

This is simplified as

With regard to a redundant mobile PA10 robot, when performing a series of complex tasks repeatedly, the path crossed by the manipulator must be closed, which means each joint angle of the end effector must eventually return to the original position. On the contrary, we aim to research a method to minimize the joint displacement between the current and initial status under the above condition. Therefore, consider the following repetitive motion optimal scheme:

Note that

Using the relative Lagrange theorem can solve the above QP (quadratic programming) problem. Firstly, we set

That is,

To approximate the solution of (

Y. N. Zhang in [

With the improvement of (

There are three regulable constants

Linear type,

Biexponential type,

Bipolar sigmoid type:

At last, applying (

Figure

The realization process of model (

Given

Firstly, a nonnegative Lyapunov function

Considering

Given time-varying matrices

When adopting a specific linear activation function, equation (

The convergence rate is

When adopting the nonlinear activation function in (

In order to reflect the superexponential convergence of the model under the nonlinear activation function, we compare

Let

If

Due to the existences of various kinds of noises, a robust compensator is designed based on the control theory in this section.

Even in the environment of external interference

For solving QP problem (

Then, equation (

Taking equation (

Letting

For the convenience of research, we further get

On account of the uncertainty of

If

If

There is no doubt that

Since the previous assumption is that the upper bound of

Therefore, every

In this section, for testing the reliability and accuracy of two models (

In addition, considering the existence of external interference

In these simulations, we set

The end effector is expected to track a star-shaped path firstly. The relevant parameters of the EVPNN are

The corresponding simulation results of the EVPNN without noise are shown in Figures

Simulation results of EVPNN model (

Synthesized positioning errors of the end effector by the EVPNN model and ZNN model when completing the task of the star-shaped path. (a) EVPNN model. (b) ZNN model.

To demonstrate the robustness of model (

Desired path and actual trajectory synthesized by two models in the presence of vector-form noise. (a) Disturbed EVPNN model (

For further validating the model, this section is designed to track a cardioid path by the PA10 manipulator. The desired trajectory is functioned from the

In this task, we choose the bipolar sigmoid function as the nonlinear activated function with parameter

Simulation results of EVPNN model (

Comparing Figures

Synthesized positioning errors of the end effector by the EVPNN model and ZNN model when completing the task of the cardioid path. (a) EVPNN model. (b) ZNN model.

All of the above are simulated in an undisturbed environment. However, external interference is added in the following experiments. As seen from Figure

Desired path and actual trajectory synthesized by two models in the presence of vector-form noise. (a) Disturbed EVPNN model (

This paper is aimed at solving the problem of repetitive motion of the PA10 manipulator with a mobile base. After analysis, it can be transformed into a quadratic programming problem mathematically. Then, the key point of this paper is to propose an improved QP solver that is EVPNN model (

The source code and source data can be provided by contacting with the corresponding author.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This study was partly supported by the National Natural Science Foundation of China (Grant nos. 61803338, 61972357, and 61672337), the Zhejiang Provincial Natural Science Foundation of China (Grant no. LGG18F020011), and Zhejiang Key R&D Program (Grant no. 2019C03135).