2.1. The dimension reduction by PCA

The dimension reduction by PCA is to map the input data of point cloud with n-dimensional features to the output data with m-dimensional features. The m-dimensional features which are regarded as principal components are completely new orthogonal features. The principal components are reconstructed based on the original n-dimensional features of the input data through a series of transformations to achieve the purpose of dimension reduction.

The specific dimension reduction process by PCA is as follows. First, the input data of point cloud are processed centrally. Second, the covariance matrix $\boldsymbol{X}\boldsymbol{X}^{\mathrm{T}}$ of the input data is calculated, and the covariance matrix is decomposed by eigenvalue. Taking the left image of the CCDM captured by the binocular camera as an example, the coordinate value of the black pixel is reduced. Since pixels are two-dimensional information in the image plane, for a pixel point $(u_{i},v_{i})$ , the covariance calculation of the input data $U$ and input data $V$ is as follows.

(1) \begin{align} Cov\left(U,V\right) & =E\left[(U-E(U))(V-E(V))\right]\nonumber\\ & =\frac{1}{n-1}\sum _{i=1}^{n}(u_{i}-\overline{u}) (v_{i}-\overline{v}) \end{align}

where $E(U)$ and $E(V)$ represent mathematical expectations. $\overline{u}$ and $\overline{v}$ represent average values.

For the two-dimensional data of point cloud, the calculation process of the covariance matrix $\boldsymbol{X}\boldsymbol{X}^{\mathrm{T}}$ can be specified as:

(2) \begin{equation} \boldsymbol{X}\boldsymbol{X}^{\mathrm{T}}=\left(\begin{array}{ll} Cov(x_{1},x_{1}) & Cov(x_{1},x_{2})\\ Cov(x_{2},x_{1}) & Cov(x_{2},x_{2}) \end{array}\right) \end{equation}

Through the above steps, the covariance matrix $\boldsymbol{X}\boldsymbol{X}^{\mathrm{T}}$ of the input data of point cloud is shown in Figure 3. The pixel coordinate information of the CCDM is reduced from two dimensions to one dimension. Its retained principal component is $[0.0197,-0.9999]$ .

Figure 3.

Results of the dimension reduction by PCA.

The explained variance ratio $EVR_{i}$ can be used to measure how much information a principal component retains in the input data of point cloud. It is used to quantify the degree to which a principal component explains the variance of the input data of point cloud. The calculation method is:

(3) \begin{equation} EVR_{i}=\frac{\lambda _{i}}{\sum _{j=1}^{n}\lambda _{j}} \end{equation}

where $\lambda _{i}$ is the $i$ th eigenvalue. $n$ is the amount of the input data.

The variance interpretation ratio of the principal component is 0.992. The greater the proportion of variance explained by the principal component, the more variation that the principal component can explain in the input data of point cloud.

Figure 4.

The dimension reduction by PCA. (a) The point cloud reduced to one-dimensional data by PCA (b) Frequency histogram.

Results of the point cloud reduced to one-dimensional data by PCA are shown in Figure 4(a). PCA converts each black pixel point in the original image from a two-dimensional pixel plane coordinate to a one-dimensional coordinate. Because the point cloud is reduced to one dimension, the ordinate in Figure 4(a) no longer has practical physical significance. So, the value of the ordinate is set to 0. The frequency histogram result is shown in Figure 4(b). The horizontal coordinate represents the projected value of the point cloud on the principal component by PCA. The ordinate represents the frequency of each projected value on the principal component. The frequency histogram can more intuitively show the distribution characteristics of point cloud in the direction of principal components.

According to the analysis of histogram results, it can be intuitively seen that the frequency of the CCDM changes obviously at the connection between the inner cable-driven mechanism and the outer cable-driven mechanism. The outer cable-driven mechanism should have more input data than the inner cable-driven mechanism. However, the inhomogeneity of the point cloud will result in the formation of superior neurons more easily in the data of the outer cable-driven mechanism.

Therefore, the weight vector of the outer cable-driven mechanism will be updated frequently, while the update frequency of the weight vector of the inner cable-driven mechanism is lower than that of the outer cable-driven mechanism. This will lead to the final clustering results more inclined to the outer cable-driven mechanism and then affect the detection of the configuration of the inner cable-driven mechanism. Therefore, in this paper, the point cloud of the CCDM is trained by the S-SOM method by PCA. At the place where the intermediate frequency value drops sharply, the threshold is set to distinguish the segment with different diameters that need to be trained independently. At the end of the training by the S-SOM method, the manipulator with two segments with different diameters trained independently is combined into a complete the CCDM.

2.2. Training parameters of the S-SOM method

The different selection of training parameters directly affects the training results of the S-SOM method. Training parameters include number of weight vectors, initial learning rate, learning rate function, winner neighborhoods, and number of training epochs.

The weight vector of the S-SOM method is initialized. The weight vector initialization mode is random initialization. The random initialization is to assign the random number to the initial weight vector so that the weight vector can be distributed in the whole point cloud. However, setting the position of the initial weight vector in this way makes it difficult for weight vectors far away from the input data of point cloud to be adjusted by the S-SOM method, while the weight vector in the center of point cloud is adjusted several times. This further affects the effect of clustering.

Another way to initialize weight vectors is linear initialization. The linear initialization is achieved by initializing the weight vector to a straight line. Compared with the random initialization, the linear initialization can avoid the problem that the weight vector is difficult to adjust. If there is no information about the point cloud, the random initialization is usually chosen. According to the prior knowledge, the configuration information of the CCDM is closer to the linear structure. Therefore, this paper chooses linear initialization.

For the S-SOM method, the selection of each training parameter will greatly affect the effect of training. Faced with many training parameters of the S-SOM method, the training parameters are usually adjusted blindly according to the clustering results. This will also affect the performance of the S-SOM method and the effect of training. Therefore, this paper uses topological errors and quantization errors to measure the performance of different training parameters.

The proportion of the input data of point cloud containing the wrong topology to the total input data of point cloud is defined as the topological error $\varepsilon _{te}$ , that is:

(4) \begin{equation} \varepsilon _{te}=\frac{1}{N}\sum _{i=1}^{N}\mu (\hat{\boldsymbol{P}}_{i}) \end{equation}

where $\hat{\boldsymbol{P}}_{i}$ is the $i$ th input vector. $N$ indicates the total number of input data of point cloud. $\sum _{i=1}^{N}\mu (\hat{\boldsymbol{P}}_{i})$ indicates the input data of point cloud containing the incorrect topology.

For input data of point cloud $\hat{\boldsymbol{P}}_{i}$ , find the winning neurons and the second winning neurons. They are judged to be adjacent in the linear array structure of the S-SOM method. If they are adjacent, there is no wrong topology, and $\mu (\hat{\boldsymbol{P}}_{i})$ of the input data of point cloud is equal to 0. On the contrary, if they are not adjacent in the linear array structure, the topological relationship of the winning neurons representing the output data is not related. In this case, $\mu (\hat{\boldsymbol{P}}_{i})$ of the input data of point cloud is equal to 1.

The average Euclidean distance of each input mode vector $\hat{\boldsymbol{P}}_{ij}$ and its corresponding weight vector $\hat{\boldsymbol{W}}_{j}$ is defined as the quantization error $\varepsilon _{qe}$ of the S-SOM method.

(5) \begin{equation} \varepsilon _{qe}=\frac{1}{N}\sum _{i=1}^{n_{i}}\sum _{j=1}^{m}\left\| \hat{\boldsymbol{P}}_{ij}-\hat{\boldsymbol{W}}_{j}\right\| \end{equation}

where $N$ is the total number of input data of point cloud. $n_{i}$ is the number of vectors.

With the continuous learning and iteration of the S-SOM method, the variation trend of quantization errors must be smaller and smaller. It can be concluded that the significance of quantization errors is to reflect the clustering effect of the CCDM. When the quantization error is smaller, the distance between the input vector and its corresponding weight vector is closer. The result of the S-SOM method can better represent the configuration information of the CCDM.

The S-SOM method is trained with different weights. To find the appropriate number of weights, a large interval is selected to calculate the corresponding topological errors. Then, according to the results, the interval is narrowed and the experiment is carried out again. The inner and outer cable-driven mechanism of the CCDM is trained independently. To ensure the consistency of the results as much as possible, the same training parameters are used for simultaneous training, as shown in Figure 5.

Figure 5.

Topological errors corresponding to different numbers of weight vectors.

Through the analysis of error results, the topological error is small when the number of weight vectors of the inner and outer cable-driven mechanisms is between 5 and 25. As the number of weight vectors increases, the quantization error becomes smaller and smaller. Therefore, the quantization error corresponding to the quantities of different weights are calculated in the range (1–30) to further determine the appropriate training parameters. The quantization error results are shown in Figure 6.

Figure 6.

Quantization errors corresponding to different numbers of weight vectors.

In the selection of the number of weight vectors, it is necessary to synthesize the results of topological errors and quantization errors. When the number of weight vectors is small, the topological error is 0, but the quantization error is large. It is difficult to reflect the information of the CCDM. When the number of weight vectors is larger, the quantization error is smaller, but the topological error is larger. Too many weight vectors will cause the weight vectors distributed along the diameter direction of the manipulator when training the point cloud of the outer cable-driven mechanism. It is not in line with the expected objective of the experiment. To better determine the selection of the number of weight vectors of the inner and outer cable-driven mechanisms, many influencing factors are considered. When it is determined to train the inner cable-driven mechanism, the number of weight vectors is selected to be 8. When training the outer cable-driven mechanism, the number of weight vectors is selected to be 12.

When selecting the learning rate function, it is necessary to meet the change law that the learning rate gradually decreases with the increase of the number of iterations. Therefore, in the selection process, the appropriate monotone decreasing function is selected as the learning rate function. The learning rate function selected in this paper is:

(6) \begin{equation} \eta (t)=10\cdot \eta _{0}\times \left(1+\frac{t}{\tau }\right) \end{equation}

where $\eta _{0}$ is the initial learning rate. $t$ is the current number of iterations. $\tau$ is a time constant, and $\tau =T/2$ . $T$ is the total number of iterations.

The design of the winner neighborhood is the same as the learning rate function. It should also continue to decay with the increase of training epochs, showing a decreasing trend in general. Therefore, the formula (6) is also used for the design of the winner neighborhood, so that the S-SOM method can gradually converge with the continuous training process. This design can effectively ensure that the winner neighborhood is large at the initial stage of training, allowing multiple nodes in the S-SOM method to adapt to the input signal simultaneously. This can speed up the convergence of the network in the initial training. As the training process continues, the winner neighborhood decreases to only a few or a single node, thus fine-tuning the topology of the network.

In addition, the design of the winner neighborhood and the determination of the learning rate also affect the training effect of the S-SOM method. Therefore, the influence of different sizes of winner neighborhoods and learning rates on the S-SOM method is measured by quantization errors. Three sets of simulations are carried out for the inner and outer cable-driven mechanisms, respectively, in which the size of the winner neighborhood is 2, 5, and 7. In the simulation result analysis of the S-SOM method, the solid line is used to represent the simulation result of the inner cable-driven mechanism, and the dashed line is used to represent the simulation result of the outer cable-driven mechanism. The analysis of simulation results is shown in Figure 7.

According to the analysis of simulation results, it can be seen that both a large learning rate and a small learning rate will lead to the accuracy of the clustering results of the CCDM. A large learning rate will increase the difficulty of training and produce a large quantization error. However, a small learning rate will cause the input data of point cloud to fall into the local learning situation, thus increasing the quantization error.

According to the analysis of simulation results, the learning rate in this paper is set to 0.15. By adjusting the size of the winner neighborhood, it can be concluded that the smaller the winner neighborhood, the smaller the quantization error. This is because larger winner neighborhoods cluster more input data of point cloud to the same point during training. This will affect the location distribution of clustering results, which is contrary to the original intention of obtaining the configuration of the CCDM. Therefore, according to the analysis of simulation results, the size of the winner neighborhood is set to 2. By determining the training parameters of the S-SOM method, the number of training epochs is selected by quantization errors and topological errors. The simulation results are shown in Figure 8.

It can be concluded from the simulation results that the topological error will decrease to 0 in fewer training epochs. Although the quantization error will decrease with the increase in the number of training epochs, the changing trend will be gradual with the increase in the number of training epochs. The amount of variation in quantization errors no longer changes significantly. Therefore, the choice of the final number of training epochs is 20.

Figure 7.

Quantization errors corresponding to different learning rates and winner neighborhoods.

Figure 8.

Quantization errors and topological errors corresponding to different training epochs.

In addition, since the left and right images of the CCDM are trained separately, the final training result may not reflect the position information of the same point in space well. In the Section 2 of this paper, the image has been stereo rectification. This means that the same point on the CCDM can match the projected pixel points in the left and right image, that is, the longitudinal value of the projected pixel points in the left and right image is equal. This will result in additional errors due to the training of the S-SOM method. This error is defined in this paper as a rectification error $\varepsilon _{re}$ , which is used to measure the difference in the ordinate values of the corresponding pixel in the left and right images.

(7) \begin{equation} \varepsilon _{re}=\frac{1}{m}\sum _{i=1}^{m}\left| v_{i}^{l}-v_{i}^{r}\right| \end{equation}

where $m$ is the number of output weight vectors. $v_{i}^{l}$ is the ordinate value of the $i$ th projection pixel in the left image. $v_{i}^{r}$ is the ordinate value of the $i$ th projected pixel in the right image.

Figure 9.

Weight vectors of left and right images of the CCDM.

To eliminate the error of matching, the clustering results of the right image are corrected based on the clustering points of the CCDM in the left image. Firstly, a central skeleton line representing the shape of the CCDM in each image is generated by the output weight vector trained by the S-SOM method. Secondly, the right image weight vector corresponding to the left image weight vector is found by way of a polar search. Thirdly, the correction step of the output weight vector of the CCDM is completed. The corrected weight vector is used as the input coordinates for the subsequent calculation of spatial coordinates. The weight vector of the left and right image of the CCDM before and after correction is shown in Figure 9.

3.1. Experimental system

To verify the feasibility of the S-SOM method for detecting the configuration of the CCDM, the CCDM prototype system is designed, as shown in Figure 10. The prototype system includes the CCDM, driving mechanism, controller, power supply, digital transmitter, binocular camera, and computer. The prototype system controls the motor to drive the corresponding cable to complete the configuration change of the CCDM. The contraction and bending of the CCDM are also realized.

Figure 10.

Experimental system.

The prototype system detects the configuration information of the CCDM by the binocular camera. The binocular camera is connected to a computer, and the configurations of the CCDM in different motion states are obtained from the computer.

The motor is controlled by the controller and the digital converter, and the cable is pulled to make the CCDM start to move. When the inner cable-driven mechanism and the outer cable-driven mechanism are driven by cables, the CCDM gradually bends. To verify the feasibility of the S-SOM method, a configuration detection experiment of the CCDM is designed. During the gradual bending movement of the CCDM, three different configurations are selected to detect. The experimental results are shown in Figure 11.

Figure 11.

Detection results of the three configurations.

3.2. Analysis of experimental results

In order to process the experimental data of the CCDM obtained by the S-SOM method, the scatter plot is simulated into a curve by polynomial. As shown in Figure 12, the three smooth curves in the figure are respectively obtained by fitting the scatter plot drawn by the S-SOM method. The shape information of the curve reflects the configuration information of the CCDM.

The marker detection method is a traditional method used to detect the configuration of the CCDM. The marker detection method has quite high precision and can be used as a comparison of the S-SOM method in this paper. As shown in Figure 12, a black circular marker is pasted on each joint of the CCDM, and the center position of the marker is identified by the binocular vision, so as to detect the configuration of the CCDM.

Figure 12.

Polynomial fitting of the curve obtained from the S-SOM method.

Because the discrete point information obtained by the marker detection method corresponds to the joint of the CCDM. In other words, the joint points of ten inner and ten outer cable-driven mechanism joints are obtained. However, there are eight weight vectors for training of the inner cable-driven mechanism and twelve weight vectors for training of the outer cable-driven mechanism. Although the number of discrete points obtained by the two methods is the same, the corresponding position of the CCDM is not the same. Therefore, the distance difference of corresponding discrete points cannot be directly compared to measure the feasibility of the S-SOM method. Therefore, in this paper, 20 discrete points obtained by the S-SOM method and the marker detection method are fitted, respectively. Two curves describing the configuration information of the CCDM are obtained. The same number of point pairs are sampled at the upper distance between the two curves. The corresponding three-dimensional coordinate values are obtained. In this paper, the average point-to-point distance error is designed to quantify the accuracy of the CCDM configuration obtained by the S-SOM method. The calculation formula is as follows:

(8) \begin{equation} \varepsilon _{de}=\frac{1}{n}\sum _{i=1}^{n}d_{i} \end{equation}

where $n$ indicates the number of pairs of sampling points. $d_{i}$ is the distance between pairs of sampling points in group $i$ .

The starting and ending points of curves obtained by the two methods are different. The curvature of the outer cable-driven mechanism is small, and the variation along the x-axis is small. Therefore, in this paper, the first set of point pairs is set at the position of the configuration curve $y=-30$ of each the CCDM. The remaining pairs are evenly arranged according to the length of the cable, totaling 20 pairs. The difference in the configuration curves of the CCDM obtained by the two methods is quantified by the average point-to-point distance error $\varepsilon _{de}$ .

As shown in Figure 13, when the CCDM is at its initial configuration A, the curve obtained by using the S-SOM method is compared with that obtained by using markers. The average point-to-point distance error is $\varepsilon _{de}^{\mathrm{A}}=1.52\mathrm{mm}$ . When the CCDM is at its initial configuration B, the curve obtained by using the S-SOM method is compared with that obtained by using markers, and the average point-to-point distance error is $\varepsilon _{de}^{\mathrm{B}}=2.34\mathrm{mm}$ . When the CCDM is at its initial configuration C, the curve obtained by using the S-SOM method is compared with that obtained by using markers, and the average point-to-point distance error is $\varepsilon _{de}^{\mathrm{C}}=2.57\mathrm{mm}$ .

Figure 13.

Configuration curve of the CCDM conclusion.