Evaluating and optimization of 7-DOF automated fiber placement robotic manipulator performance index based on AdaBoost algorithm

Abstract

In order to meet the work task better, the new performance index of the 7-DOF (degree of freedom) automated fiber placement robotic manipulator is constructed. The evaluation performance index is composed by giving condition number and manipulability different weights, at the same time considering the flexibility and motion capacity in fiber placement process, the proposed new performance index is optimized by AdaBoost algorithm, and the optimal weighting coefficients that constructed the new performance index are obtained, the proposed new performance index is simulated under the condition that the weighting coefficients are optimal, the results show that the proposed new performance index of the 7-DOF automated fiber placement robotic manipulator can satisfy the flexibility and manipulability in fiber placement process.

Keywords

Redundancy fiber placement robotic manipulator manipulability performance index

1. Introduction

The industrial redundant robotic manipulator has been widely used for high flexibility, precise target tracking, good obstacle avoidance and avoidance of singularity [1, 2, 3]. The flexibility of the robotic manipulator is improved by adding the number of DOF [4], the robotic manipulator is called redundant robot that the DOF number of robotic manipulator is more than 6 [5]. The number of robotic DOF is greater, the robotic manipulator is more flexible, and controlling robotic manipulator becomes more complex, so studying on the operating performance of redundant robotic manipulator is more practical.

In the research on the performance index of the robotic manipulability, Yoshikawa [6] used $w=[\det(JJ^{T})]^{1/2}$ as the performance index measuring the overall flexibility of the redundant robotic manipulator, the essence of $w$ is the product of the singular value of the Jacobian matrix, the physical meaning of $w$ is the motion ability of the redundant robotic manipulator in all directions, the value of $w$ is greater, the motion ability of the redundant robotic manipulator is stronger. For the 7-DOF automated fiber placement robotic manipulator, the motion ability in the direction of the placement trajectory is only necessary. Salishbury and Craig [7] used the condition number as the performance index measuring flexibility of the robotic manipulator, and it shows the motion consistency of the robotic manipulator in all directions. The value of condition number is smaller, the motion ability of the robotic manipulator is more consistent in every direction, and the motion speed of the robotic manipulator is more uniform, the flexibility of the robotic manipulator is better too. However, in fiber tow placement process, the flexibility requirements of the 7 DOF automated fiber placement robotic manipulator in every directions are different, and it is not advisable that pursuit flexibility of the robotic manipulator only. On condition number basis, Angles and Rojas [8] improved the condition number and proposed the minimum condition number as performance index measuring flexibility of the robotic manipulator by optimizing the robotic structure. Yao et al. [9] proposed manipulability index based on speed direction, which increased the flexibility of the robotic manipulator in the specific motion direction, and ignored the influence of condition number on the manipulability of robotic manipulator. Klein and Blaho [10] optimized manipulability by minimum singular value of the Jacobian matrix, these are all derived from manipulability and condition number, and which are based on speed level, and the same in essence. Guo and Geng [11] deduced acceleration level performance index based on Hessian matrix employed global performance index which proposed by Gosselin and Angles [12], and ignored the influence of speed.

In this study, new performance index is composed condition number and manipulability, and different weight are given to condition number and manipulability, which according to requirements such as enough flexibility, adaptability and different task in fiber tow placement process of the 7 DOF automated fiber placement robotic manipulator. The optimal performance index can be obtained under condition that the manipulability as large as possible and the condition number as small as possible by adjusting weights. The AdaBoost algorithm does not require prior knowledge, and a higher accuracy rate can be obtained as long as the number of iterations is appropriate. The smallest condition number is found out under condition of meeting manipulability by AdaBoost algorithm, so that the 7 DOF automated fiber placement robotic manipulator is more flexible.

2. Model and Jacobian matrix of the 7-DOF automated fiber placement robotic manipulator

2.1 Topology and parameters of the 7-DOF automated fiber placement robotic manipulator

Composite fiber placement is completed by redundant robotic manipulator, and the development of the redundant robotic manipulator and the development of fiber placement molding technology affect each other [17]. The task of composite fiber placement is complex according to the different shape of the mold, and the motion constraint of the redundant robotic manipulator is also more and more. For all kinds of core modules of different sizes and shapes, the redundant robotic manipulator can be able to complete flexibly.

The 7 DOF automated fiber placement robotic manipulator is composed by a 6 DOF serial robotic manipulator and a revolute mold. It is shown in Fig. 1. According to the principle of equivalent motion, a 7 DOF serial robotic manipulator with a revolute joint in shoulder, three prismatic joints in elbow and three revolute joints in wrist is obtained by fixed the mandrel coordinate system and the base coordinate system together. The D-H parameters are set up in accordance with the standard D-H method [18] and are shown in Table 1.

Table 1
Parameters of the 7 DOF automated fiber placement robotic manipulator

Link $i$	$a_{i-1}$ /mm	$\alpha_{i-1}$ / ${}^{\circ}$	$d_{i}$ /mm	$\theta_{i}$ / ${}^{\circ}$	Range of joint
1	$a_{0}$	0	0	$\theta_{1}$	$-$ 180 $\sim$ 180
2	0	0	$d_{1}$	0	$-$ 150 $\sim$ 150
3	0	90	$d_{2}$	$-$ 90	$-$ 110 $\sim$ 110
4	$a_{3}$	90	$d_{3}$	0	$-$ 100 $\sim$ 100
5	0	0	$c$	$\theta_{5}$	$-$ 90 $\sim$ 90
6	0	90	0	$\theta_{6}$	$-$ 120 $\sim$ 120
7	0	$-$ 90	0	$\theta_{7}$	$-$ 270 $\sim$ 270

Where, $i$ - link order number, $a$ - link length, $\alpha$ - link twist, $d$ - link offset, $\theta$ - joint angle.

Figure 1.

The 7 DOF automated fiber placement robotic manipulator’s structure and topology.

2.2 Jacobian matrix of the 7-DOF automated fiber placement robotic manipulator

2.2.1 Motion screw, exponential product

The motion of each joint of a robotic manipulator is produced by the motion screw on the axis of the joint, thus a geometric description of the robotic manipulator kinematics is obtained [19, 20]. Let $\bm{\omega}=(\bm{\omega}_{x},\bm{\omega}_{y},\bm{\omega}_{z})\in R^{3}$ is a unit vector that represents the direction of a rotating axis when a rigid body rotates around a fixed axis; $r$ is a position vector of a point on the axis. Introducing 4 $\times$ 4 matrix $\hat{\xi}$ .

$\displaystyle\hat{\xi}=\begin{bmatrix}\hat{\bm{\omega}}&\bm{v}\\ 0&0\\ \end{bmatrix}$

Where, $\bm{v}=\bm{r}\times\bm{\omega}$ , $\hat{\bm{\omega}}=\begin{bmatrix}0&-\omega_{z}&\omega_{y}\\ \omega_{z}&0&-\omega_{x}\\ -\omega_{y}&\omega_{x}&0\\ \end{bmatrix}$ .

Define operator $\vee$ , $\vee$ can map the 4 $\times$ 4 matrix to 6 dimensional column vector, that is

$\displaystyle\xi=\begin{bmatrix}\hat{\bm{\omega}}&\bm{v}\\ 0&0\\ \end{bmatrix}^{\vee}=\begin{pmatrix}\bm{\omega}\\ \bm{v}\\ \end{pmatrix}$

Let the basic coordinate system of the $n$ joints robotic manipulator is { $S$ }, and the end effector coordinate system is { $T$ }. $g_{ST}$ represents the forward kinematics $n$ joints robotic manipulator. According to the screw theory [21], the product of exponential (POE) formula of a series robotic manipulator forward kinematics with $n$ joints is as follows:

$\displaystyle\bm{g}_{st}(\bm{\theta})=e^{{\theta}_{1}\hat{\xi}_{1}}e^{\theta_{% 2}\hat{\xi}_{2}}\ldots e^{\theta_{n}\hat{\xi}_{n}}\bm{g}_{st}(0)$

2.2.2 Jacobian matrix of the 7-DOF automated fiber placement robotic manipulator

Let the basic coordinate system of the 7 DOF automated fiber placement robotic manipulator is { $S$ }, and the end effector coordinate system is { $T$ }. $g_{ST}$ represent the forward kinematics of 7 DOF automated fiber placement robotic manipulator. According to screw theory and POE formula, the instantaneous speed of the end effector and joint instantaneous speed of the 7 DOF automated fiber placement robotic manipulator can be expressed as follow:

$\displaystyle\dot{\bm{x}}=\dot{\bm{g}}_{st}(\bm{\theta})\bm{g}_{st}^{-1}(\bm{% \theta})=\sum_{i=1}^{n}\frac{\partial\bm{g}_{st}}{\partial\bm{\theta}_{i}}\dot% {\bm{\theta}}_{i}\bm{g}_{st}^{-1}(\bm{\theta})=\sum_{i=1}^{n}\left(\frac{% \partial\bm{g}_{st}}{\partial\bm{\theta}_{i}}\bm{g}_{st}^{-1}(\bm{\theta})% \right)\dot{\bm{\theta}}_{i}=J(\bm{\theta})\dot{\bm{\theta}}\quad i=1,\ldots,7$ (1)

Where, $\bm{x}$ – the end-effector position, $\bm{g}_{st}$ – forward kinematics, $\bm{g}_{st}(0)$ – initial position and posture.

$\bm{J}(\bm{\theta})=\dot{\bm{x}}/\dot{\bm{\theta}}$ is the Jacobian matrix of the 7 DOF automated fiber placement robotic manipulator in Eq. (1), and expressed the mapping relationship between the end effector speed and joint speed.

The Jacobian matrix of the 7 DOF automated fiber placement robotic manipulator is obtained by replacing the parameters of the 7 DOF automated fiber placement robotic manipulator in Table 1.

$\displaystyle J\!=\!\begin{bmatrix}-d_{3}s_{6}c_{7}\!+\!(c+d_{4})(s_{5}c_{6}c_% {7}\!+\!c_{5}s_{7})&-c_{5}c_{6}c_{7}\!+\!s_{5}s_{7}&s_{5}c_{6}c_{7}\!+\!c_{5}s% _{7}&s_{6}c_{7}&0&0&0\\ d_{3}s_{6}s_{7}\!+\!(c\!+\!d_{4})(s_{5}c_{6}s_{7}\!+\!c_{5}c_{7})&c_{5}c_{6}s_% {7}\!+\!s_{5}c_{7}&-s_{5}c_{6}s_{7}\!+\!c_{5}c_{7}&-s_{6}s_{7}&0&0&0\\ -d_{3}c_{6}\!+\!(c\!+\!d_{4})s_{5}s_{6}&-c_{5}s_{6}&s_{5}s_{6}&c_{6}&0&0&0\\ -c_{5}c_{6}c_{7}\!+\!s_{5}s_{7}&0&0&0&s_{6}c_{7}&-s_{7}&0\\ c_{5}c_{6}s_{7}\!+\!s_{5}c_{7}&0&0&0&-s_{6}s_{7}&-c_{7}&0\\ -c_{5}s_{6}&0&0&0&c_{6}&0&1\\ \end{bmatrix}$ (2)

Where $s_{5}=\sin\theta_{5}$ , $c_{5}=\cos\theta_{5}$ , and so on.

3. Manipulability evaluation index the 7-DOF automated fiber placement robotic manipulator based on AdaBoost algorithm

3.1 Construct manipulability evaluation index the 7-DOF automated fiber placement robotic manipulator

In the fiber tow placement process, the end effector velocity value of 7 DOF automated fiber placement robotic manipulator is constant, the manipulability study based on speed level is more significant than the flexibility of the 7 DOF automated fiber placement robotic manipulator.

The manipulability index [22, 23] based on speed level is $k=\sqrt{\left({\bm{v}^{T}\bm{J}^{-T}\bm{J}^{-1}\bm{v}}\right)^{-1}}$ . Comprehensive performance index is constructed as follow considering the influence of condition number.

$\displaystyle\gamma_{\text{new}}=w_{1}\delta^{\|\bm{J}\|\|\bm{J}^{-1}\|}+w_{2}% \delta^{\left(\bm{v}^{T}\bm{J}^{-T}\bm{J}^{-1}\bm{v}\right)^{-\frac{1}{2}}}$ (3)

Where, $w_{1}$ and $w_{2}$ represent the weighted coefficient respectively, the value of $\delta$ is set according to different conditions, generally 1.2 $\leqslant\delta\leqslant$ 5. So that determined the optimal coefficient $w_{1}$ , $w_{2}$ is the important standard of measuring the merits of comprehensive performance index for which the change trend of the condition number and manipulability is opposite.

3.2 weighted optimization based on AdaBoost algorithm

AdaBoost algorithm is an iterative algorithm based on Boosting algorithm [24]. Its basic idea is to combine multiple “weak” performance indicators by continuous iteration to produce more effective comprehensive performance indicators. In the traditional AdaBoost algorithm, the weight of the based classifier is constant, and the constant weight will reduce the correct rate of classification, D-AdaBoost (dynamic weights AdaBoost) algorithm is proposed.

D-AdaBoost algorithm takes training samples as a bridge to establish the connection between the test sample and the based classifier. D-AdaBoost algorithm is based on the following assumptions:

For a test sample, if a base classifier has a better classification effect to a group of training samples which are similar to it, the base classifier should have a better classification effect to the test sample too. So, for a test sample, D-AdaBoost algorithm looks for a group of training samples which are the most similar to the test sample. The optimal base classifier weights are calculated on the group of training samples, as reference for the base classifier weight of the sample. The recognition rate of a base classifier to the group of samples is higher, which shows the prediction ability to the test sample is higher, the corresponding weight is greater.

The training process of the base classifier of D-AdaBoost algorithm is exactly the same as that of AdaBoost algorithm. The main steps are as follows:

firstly, given the sample space ( $x$ , $y$ ), the training samples are divided into $m$ group, the weight of each group is 1/ $m$ ; then iterate by boosting algorithm, the combination properties of the training individual weight are poor and can be increased, so the weight distribution of training samples is updated according to the combined results. Finally, a group of comprehensive performance indicators $P_{1},P_{2},P_{3}\ldots P_{n}$ can be obtained, each index has different weight, and the good performance indicators are of the great weight. Then the product of above indicators and weight are weighted to get the stronger performance index, and the algorithm flow is shown in Fig. 2.

Figure 2.

The AdaBoost algorithm flow.

In order to solve $w_{1}$ and $w_{2}$ , Eq. (3) is simplified as follow:

$\displaystyle\gamma_{\text{new}}=w_{1}\gamma_{1}+w_{2}\gamma_{2}$ $\displaystyle\gamma_{1}=\delta^{\|\bm{J}\|\|\bm{J}^{-1}\|}$ (4) $\displaystyle\gamma_{2}=\delta^{\left(\bm{v}^{T}\bm{J}^{-T}\bm{J}^{-1}\bm{v}% \right)^{-\frac{1}{2}}}$

$\gamma_{1}$ and $\gamma_{2}$ are defined the virtual flexibility index, so the above problem is simplified to determine a set of optimal $w_{1}$ and $w_{2}$ under the condition that manipulability is as maximum as possible and the condition number is smaller, smaller condition number and larger manipulability can be ensured when the comprehensive flexibility index varies with time, $w_{1}$ and $w_{2}$ can be obtained by the AdaBoost algorithm.

Firstly, the flexibility index $\gamma_{1}\ldots\gamma_{n}$ are regarded as weak classifier, the strong flexibility index can be obtained by combining the flexibility index $\gamma_{1}\ldots\gamma_{n}$ with the AdaBoost algorithm, the condition number and manipulability as an iterative condition, finally, $w_{1}$ and $w_{2}$ are 0.235 and 0.965 respectively.

4. Simulation analysis and validation

According to structure parameters of the 7 DOF automated fiber placement robotic manipulator in Table 1, take the velocity of robotic end effector $v$ as (0.04 0.02 0) ${}^{T}$ m/s; take the initial position $\theta$ as (0.687 $-$ 1.514 1.728 $-$ 0.748 $-$ 0.516 2.378 0.523) ${}^{T}$ ; the simulation time $t=$ 5 s, the sampling period $dt=$ 50 ms, set the maximum condition number is 18, take $\delta=$ 3.

The simulation results of Eq. (3.2) are shown in Figs 3 and 4. Figure 3 shows the change trend that evaluation index $\gamma_{\text{new}}$ , $\gamma_{1}$ , $\gamma_{2}$ with condition number. Figure 4 shows the change trend that evaluation index $\gamma_{\text{new}}$ , $\gamma_{1}$ , $\gamma_{2}$ with operability.

Figure 3.

The condition number simulation.

Figure 4.

The manipulability simulation.

As shown in Fig. 3, during period of 1.5 s $\leqslant t\leqslant$ 3 s, $\gamma_{2}$ is larger, and more than the maximum condition number 18, while $\gamma_{\text{new}}$ reduces the condition number largely, and does not exceed the specified maximum conditions number. as shown in Fig. 4, during period of 1.0 s $\leqslant t\leqslant$ 3.0 s, $\gamma_{\text{new}}$ reduced condition number, the change of $\gamma_{2}$ is smaller than the change of $\gamma_{1}$ , and still meeting the performance requirements of the 7 DOF automated fiber placement robotic manipulator. In 3 s $\leqslant t\leqslant$ 5 s, $\gamma_{\text{new}}$ makes the condition number smaller, makes the manipulability larger, the only disadvantage is that the manipulability of $\gamma_{\text{new}}$ is a certain degree of volatility in 0 s $\leqslant t\leqslant$ 0.5 s, which should be caused by joint fluctuation of the 7 DOF automated fiber placement robotic manipulator, and it’s not the $\gamma_{\text{new}}$ itself, all in all, $\gamma_{\text{new}}$ has some advantages compared to $\gamma_{1}$ and $\gamma_{2}$ .

5. Conclusion

(1)

The model of the 7 DOF automated fiber placement robotic manipulator is established, the Jacobian matrix of 7 DOF automated fiber placement robotic manipulator is derived.

(2)

The comprehensive evaluation index of the 7 DOF automated fiber placement robotic manipulator is constructed. The comprehensive evaluation index is optimized by AdaBoost algorithm, and the corresponding coefficient is obtained.

(3)

The effectiveness of the 7 DOF automated fiber placement robotic manipulator’s comprehensive evaluation index is proved by simulation analysis.

Footnotes

Acknowledgments

The authors would like to thank the Henan Province Science and Technology Project (172102210094) for financial support.

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