This paper presents a novel redundantly actuated 2R
Various types of mechanisms have been applied in many robotic fields. Nowadays, some parallel manipulators have attracted exhaustive attention from both academia and industry. Compared to series manipulators, parallel manipulators are claimed to possess inherent advantages such as high stiffness, high loading capability, high precision, low error accumulation, fast response speed, and high orientation capability. These characteristics allow the parallel manipulator to be widely used in various alternative fields, such as high-speed machine tools, high-speed pick and place applications, surgical manipulators, and force/torque sensors [
Generally, some performance indices, such as dexterity, stiffness, accuracy, and workspace, will be involved in the design process of the parallel manipulator. Kinematics solutions, establishing the relationship between the joint space and operation workspace including position, velocity, and acceleration, are an extremely key issue. The inverse kinematics solution is to find the joint angles or actuator lengths when the position and orientation of the moving platform are given [
Recently, neural networks are employed to solve the forward kinematics of a parallel manipulator and draw particular interests for numerous researchers, due to their considerable ability to approximate nonlinear mapping functions. Parikh and Lam [
The goal of this paper is to solve the forward kinematics, which promotes further practical applications of the promising redundantly actuated 2R
A five-axis hybrid kinematic machine tool is design to mill a large complex structure component surface in the aerospace field. As shown in Figure
A five-axis hybrid kinematic machine tool.
Schematic diagram of the redundantly actuated parallel manipulator.
As shown in Figure
In this section, the kinematics analysis, mobility, inverse kinematics, and forward kinematics of the redundantly actuated 2R
The mobility determination is the first and foremost issue in designing the motion characteristic of a parallel manipulator. In order to determine the motion pattern of the 2
In Figure
Then, considering the reciprocal product, the wrench system of equation (
Analogously, the twist screw of the second S
The wrench screw of equation (
The twist screw of the third R
The wrench screw of equation (
In a similar approach, the twist screw of the fourth S
The wrench screw of equation (
So far, the constraint wrench screw of the redundantly actuated parallel manipulator can be expressed as
It is noteworthy that the direction vector
By combining equation (
In equation (
Generally, the number of degrees of freedom of the parallel manipulator can be calculated by utilizing the modified
There is neither a constraint couple in the same direction nor a constraint force in collinearity among all the wrench screws of the parallel manipulator; therefore, there is no common constraint, that is,
Thus, the mobility number of the 2R
In summary, the redundantly actuated 2R
Compared with the existing 4-DOF parallel manipulator possessing four actuators [
Figure
In equation (
As illustrated in Figure
It can be seen from equation (
So, the parasitic motion in the movement of the parallel manipulator is related to the workspace parameters, that is,
Equation (
Taking the positive square root of the resulting equation (
If the structure parameters of the parallel manipulator and pose of the moving platform are given, the displacement of the activated joint can be calculated by employing the inverse kinematics equation (
The forward kinematics solution is aimed at determining the position and orientation of the moving platform with respect to the fixed reference frame once the linear displacement of the joint variables is given. In consideration of a rotation axis of the proposed parallel manipulator located at the
The BP neural network is a dynamic system of the topological structure of the digraph, which employs differentiable function as activation function to propagate forward. At the same time, the weight factor
Topological structure of the BP neural network.
The number of the nodes (
Among them,
The differentiable and linear functions are usually employed as alternative activation function of the BP neural network. In this paper, the
Back propagation requires that the activation function used by the artificial nodes be differentiable, so the derivative of the Tan-Sigmoid function can be expressed as
Since the magnitude of different data is not the same, which may result in slower convergence ratios and long training time of the neural network, furthermore, the range of activation function in the output layer of the neural network is limited to one interval, so it is necessary to normalize the target data of the training network to the range [-1, 1]; the normalization criterion can be written as
The learning and training samples are set up by utilizing the inverse solution equation (
After being successfully trained, once all the kinematic parameters are known, the forward kinematics can be directly computed, which is the main goal of this paper. For the sake of convenience, twelve groups of positions and postures are taken as the test data, their inverse solutions are brought into the trained network, taking the first group set of data as an example, and the iterative process is depicted in Figure
The training curve of the BP neural network.
The forward kinematics solution of the parallel manipulator can be solved by minimizing the network performance function. The mean square error (MSE) of the BP neural network can be derived according to the error back propagation equation
All of the selected parameters (
Forward kinematics results of the parallel manipulator by utilizing the BP neutral network.
No. | Values | Error/10-3 | |||
---|---|---|---|---|---|
1 | Target | 0.7 | 0.1 | 0 | 2.166008 |
Calculated | 0.701021 | 0.083644 | -0.020175 | ||
2 | Target | 0.7 | 0 | 0.1 | 2.264033 |
Calculated | 0.701452 | -0.011170 | 0.124723 | ||
3 | Target | 0.7 | 0.1 | 0.1 | 0.885016 |
Calculated | 0.701227 | 0.092165 | 0.107064 | ||
4 | Target | 0.75 | 0.15 | 0 | 2.871267 |
Calculated | 0.751755 | 0.132945 | -0.029886 | ||
5 | Target | 0.75 | 0 | 0.15 | 2.332283 |
Calculated | 0.753355 | -0.013697 | 0.174175 | ||
6 | Target | 0.75 | 0.15 | 0.15 | 0.668443 |
Calculated | 0.752444 | 0.144920 | 0.155706 | ||
7 | Target | 0.75 | -0.1 | 0.15 | 2.768450 |
Calculated | 0.755368 | -0.104542 | 0.182469 | ||
8 | Target | 0.75 | 0.15 | -0.1 | 5.261575 |
Calculated | 0.752359 | 0.114646 | -0.152260 | ||
9 | Target | 0.75 | 0 | 0.25 | 3.533075 |
Calculated | 0.755474 | -0.039118 | 0.234595 | ||
10 | Target | 0.75 | 0.25 | 0 | 3.746533 |
Calculated | 0.752127 | 0.254188 | -0.044712 | ||
11 | Target | 0.75 | -0.25 | 0.25 | 9.037167 |
Calculated | 0.748228 | -0.346240 | 0.200049 | ||
12 | Target | 0.8 | 0.15 | 0.25 | 1.170758 |
Calculated | 0.800083 | 0.136980 | 0.244722 |
From Table
Since the position and orientation values of the parallel manipulator are not in an order of magnitude, therefore, the improved BP neural network is used to train position
Forward kinematics results of the parallel manipulator by utilizing the improved BP neutral network.
No. | Values | Error/10-3 | |||
---|---|---|---|---|---|
1 | Target | 0.7 | 0.1 | 0 | 0.971667 |
Calculated | 0.689317 | 0.104087 | -0.002263 | ||
2 | Target | 0.7 | 0 | 0.1 | 0.293098 |
Calculated | 0.700993 | -0.002258 | 0.102508 | ||
3 | Target | 0.7 | 0.1 | 0.1 | 0.147458 |
Calculated | 0.700549 | 0.101409 | 0.099082 | ||
4 | Target | 0.75 | 0.15 | 0 | 1.579175 |
Calculated | 0.732423 | 0.157081 | -0.000093 | ||
5 | Target | 0.75 | 0 | 0.15 | 0.496743 |
Calculated | 0.755302 | -0.002327 | 0.151418 | ||
6 | Target | 0.75 | 0.15 | 0.15 | 0.229083 |
Calculated | 0.749549 | 0.152573 | 0.149145 | ||
7 | Target | 0.75 | -0.1 | 0.15 | 0.917767 |
Calculated | 0.758538 | -0.104923 | 0.154916 | ||
8 | Target | 0.75 | 0.15 | -0.1 | 3.714833 |
Calculated | 0.706730 | 0.159786 | -0.104370 | ||
9 | Target | 0.75 | 0 | 0.25 | 1.788858 |
Calculated | 0.732512 | -0.004709 | 0.238476 | ||
10 | Target | 0.75 | 0.25 | 0 | 2.240533 |
Calculated | 0.723416 | 0.253966 | 0.000679 | ||
11 | Target | 0.75 | -0.25 | 0.25 | 4.398025 |
Calculated | 0.697250 | -0.249546 | 0.248398 | ||
12 | Target | 0.8 | 0.15 | 0.25 | 0.471583 |
Calculated | 0.799934 | 0.152385 | 0.244868 |
The results of the improved BP neural network are more accurate than those of the traditional BP neural network. The order of magnitude has basically reached 10-4, and there has been a great improvement compared with the traditional BP neural network. However, the accuracy is still difficult to satisfy the high accuracy requirements of the parallel kinematic machine tool. Therefore, error compensation is needed to further improve the calculation accuracy.
Even though both BP neural network and improved BP neural network approaches yield good results for training and testing for given desired data points, in order to obtain a more satisfactory accuracy, a BP neural network algorithm with position error compensation is adopted; the calculation process in Figure
Select the actuator displacement
Calculate the pose
Solve the actuator displacement
If the delta absolute value
Flowchart of the position compensation.
Analogously, the forward kinematics solution can be derived by employing the BP neural network with position compensation, and the approximation permission accuracy
Forward kinematics results of the parallel manipulator by utilizing the BP neural network with position compensation.
No. | Values | Error | |||
---|---|---|---|---|---|
1 | Target | 0.7 | 0.1 | 0 | |
Calculated | 0.7 | 0.099999 | 0 | ||
2 | Target | 0.7 | 0 | 0.1 | |
Calculated | 0.7 | 0 | 0.1 | ||
3 | Target | 0.7 | 0.1 | 0.1 | |
Calculated | 0.7 | 0.099999 | 0.099999 | ||
4 | Target | 0.75 | 0.15 | 0 | |
Calculated | 0.75 | 0.149999 | 0 | ||
5 | Target | 0.75 | 0 | 0.15 | |
Calculated | 0.75 | 0 | 0.15 | ||
6 | Target | 0.75 | 0.15 | 0.15 | |
Calculated | 0.75 | 0.149999 | 0.149999 | ||
7 | Target | 0.75 | -0.1 | 0.15 | |
Calculated | 0.75 | -0.099999 | 0.15 | ||
8 | Target | 0.75 | 0.15 | -0.1 | |
Calculated | 0.75 | 0.149999 | -0.099999 | ||
9 | Target | 0.75 | 0 | 0.25 | |
Calculated | 0.75 | 0 | 0.249999 | ||
10 | Target | 0.75 | 0.25 | 0 | |
Calculated | 0.749994 | 0.249994 | 0 | ||
11 | Target | 0.75 | -0.25 | 0.25 | |
Calculated | 0.749995 | -0.249994 | 0.249991 | ||
12 | Target | 0.8 | 0.15 | 0.25 | |
Calculated | 0.8 | 0.15 | 0.249998 |
From Table
Compared with three neural network strategies, as shown in Figure
Comparison analysis of the mean square error.
However, as depicted in Figure
Bar chart comparison of the three output parameters.
The determination of joint and workspace is one of the most important issues of the redundant actuation parallel manipulator, which directly reflects the comprehensive performance of the parallel manipulator; its analysis results provide a theoretical basis for the design and application of the 2R
The experimental prototype of the parallel manipulator.
For convenience, we summarize the main loop algorithm procedure as follows.
Forward_kinematics = BP_compensation ( break; record (
In the actual movement process of the parallel manipulator, the following limitations should be considered:
Limitation of the actuated joints Joint angle constraints Under consideration, the rotation angle Singularity configuration constraint At a singularity configuration, the parallel manipulator may lose one or more degrees of freedom, which will seriously affect the kinematic performance of the parallel manipulator, and then those configurations should be avoided. So the joints are kept away from the singularity configuration, and the rank of the constraint screw system is limited to be three, that is,
As has been described beforehand, the architectural parameters and the maximum and minimum limit values of constraint conditions are shown in Table
Architecture parameters of the proposed mechanism.
Parameter | Value | Parameter | Value |
---|---|---|---|
0.15 | [-60,60] | ||
0.25 | [-60,60] | ||
[0.3,0.8] | [-45,45] | ||
[0.2,1.0] | [-45,45] |
With consideration of four actuators, the joint workspace can be combined by (
The three-dimensional view of the parallel manipulator.
The two-dimensional view of the joint space of the parallel manipulator.
Simultaneously, the boundary of the workspace can be formulated by adopting the BP neural network with position compensation when given the feasible actuator joint variable displacement, which denotes the available utmost range of the end-effecter of the parallel manipulator. Figure
The operation workspace of the parallel manipulator.
The two-dimensional view of the operation workspace of the parallel manipulator.
In this paper, a novel 1T2R redundantly actuated 2R The mobility of the proposed redundantly actuated 2R The inverse kinematic position of the redundantly actuated parallel manipulator has been sequentially analyzed, which lay the basis for the BP neural network training samples. The forward kinematics solution problem of the redundantly actuated parallel manipulator can be effectively solved by the BP neural network with position compensation with high accuracy, and the calculation results are in good accordance with the high precision requirement of the milling process, which lay a foundation for further dynamic modeling and real-time control of the experimental prototype The joint space and workspace are depicted and constructed. The analytic results illustrate that the proposed manipulator is a good candidate for engineering application. The proposed methodology to realize two rotations and one translation can also extended to other parallel manipulators including redundantly actuated and nonredundantly actuated parallel manipulators with a complicated forward kinematics problem as well. In the future works, we will focus on the dynamic characteristic analysis, error identification and compensation, sensitivity analysis, and real-time control
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
The authors would like to acknowledge the financial support of the Fundamental Research Funds for the Central Universities under Grant Nos. 2018JBZ007, 2018YJS136, and 2017YJS158; China Scholarship Council (CSC) under Grant No. 201807090079; the Natural Sciences and Engineering Research Council of Canada (NSERC); and the York Research Chairs (YRC) program. Meanwhile author Haiqiang Zhang is grateful to Advanced Robotics and Mechatronics Laboratory and the science librarian John Dupuis in York University.