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Decoupling Research of a Three-dimensional Force Tactile Sensor Based on Radical Basis Function Neural Network [Sensors & Transducers (Canada)](Sensors & Transducers (Canada) Via Acquire Media NewsEdge) Abstract: A decoupling method based on radical basis function neural network (RBFNN) for a novel three-dimensional force flexible tactile sensor is presented in this paper. A numerical model of the tactile sensor is built through finite element analysis, which simulates the mapping between three-dimensional force applied on top surface of the sensor and deformation of the sensor. Furthermore, the RBFNN is applied to approach the nonlinear relationship between the deformation and the three-dimensional force. The row-column resistance values corresponding to the deformation are computed by a mathematical model. At last, the high dimensional nonlinear mapping relationship between resistance and three-dimensional force is also decoupled by RBFNN algorithm. Hence the decoupling system for the tactile sensor is implemented by using RBFNN twice. The decoupling results show that the RBFNN with high nonlinear approximation ability has good performance in decoupling three-dimensional force and satisfies both the decoupling accuracy and real-time requirements of the tactile sensor. Copyright © 2013 IFSA. Keywords: Tactile sensor, Decoupling method, Radical basis function, Neural network, Three-dimensional force. (ProQuest: ... denotes formulae omitted.) 1. Introduction Tactile sense is an important perception for robot to obtain external environment information. Unlike other sensing modalities, it can acquire a lot of useful information from the object directly, such as shape, elasticity, hardness, roughness, and surface vibration signal etc. In view of tactile perception having many unique functions, researching and improving the flexible multi-dimensional tactile sensor become the key issue in recent intelligent robot field and industrial manufacturing areas. The tactile sensor is an essential part of the intelligent bionic robot. Recently, the design of that tactile sensor is mainly attracted to realize the flexibility and the measurement of three-dimensional (3-D) force [1-2]. The design principle is concentrated in piezoelectric [3], capacitor [4-5], optical fiber [6], elastic conductors [7], and conductive rubber [1, 8-9] techniques, from which researchers acquired a series valuable accomplishment. V. Maheshwari et al [10] developed a high-resolution thin-film device to sense texture by touch, both the lateral and height resolution of texture are comparable to the human finger. H. Hu et al [11] developed a flexible capacitive tactile sensor, where PDMS is used as substrate material, 16x16 units can be integrated in 1 mm2 areas. J. G. Rocha et al [2] proposed a capacitive tactile sensor which can detect 3-D force. With the development of the tactile sensor, conductive rubber has been widely used for flexible tactile sensors as its good force-sensitivity and flexibility, but some of that sensors can only measure the Z-directional force or pressure [12-13], and some of them are not flexible enough to be used as bionic robots' skin. In [13], F. Xu et al proposed a flexible tactile sensor based on conductive rubber, which is flexible and can detect pressure accurately. Because coupling relationships generally exist among variables of the 3-D force tactile sensor, the decoupling strategy for the 3-D force takes a very important part in the design of flexible tactile sensors. Researchers pay more attention to the design of decoupling methods of the tactile sensor. J. X. Ding [14] presented a decoupling algorithm based on homotopy theory for 3-D tactile sensor arrays. F. L. Wang [12] decoupled the mapping relationship between resistance and deformation with BPNN. J. M. Kim et al [15] designed a bio-mimetic tactile sensor, which can measure three components forces. That sensor is composed of 100 micro force sensors, and its force capacity is 0.3N in the three-axis. In Sec. 2 of this paper, principles of the tactile sensor are introduced. In Sec. 3, the Finite Element Analysis (FEA) method is used to construct the numerical model of the tactile sensor based on the force-sensitive conductive rubber, and to simulate the mapping between 3-D force and deformation. In Sec. 4, the Radical Basis Function Neural Network (RBFNN) algorithm is applied to realize the mapping from deformation to 3-D force, and then decouple the 3-D force from resistances directly. At last, some conclusions are drawn in Sec. 5. 2. Principles of the Tactile Sensor 2.1. Characteristic of Force-sensitive Conductive Rubber The conductive rubber is made by conductive particles (such as carbon black) which are mixed and dispersed in silicone rubber material [16]. It has many excellent features, such as flexibility, low cost, electricity conducting, and force-sensitivity. If there is no force applied on the conductive rubber, the conductive particles in the interior of the rubber are positioned apart from each other, and the resistance is very large. However, when the conductive rubber is subjected to external force, the resistance value of its own will be changed and become smaller accordingly. This characteristic of the conductive rubber is called force-sensitive property. Recently, many searchers have studied the forcesensitive property of conductive rubber. In paper [17], the authors got the pressure-conductive feature of a rubber at different concentrations of carbon black (Fig. 1), which reflects the conductive rubber's resistivity is changing along with external force. 2.2. Structure of the 3-D Force Tactile Sensor As shown in Fig. 2, a flexible tactile sensor based on conductive rubber with carbon black filler, which can detect 3-D force is designed for our experiments. The electrodes and wires are distributed in the interior of the sensor. They are disposed as two layers in the tactile sensor. The upper electrodes are connected through row-wires at the upper layer, while the lower electrodes are connected by columnwires at lower layer, as shown in Fig. 2. The resistance between a row-wire of the upper layer and a column-wire of a lower layer is called "row-column resistance" [1]. In the tactile sensor, the upper layer array is N1*N2, and the lower layer array is N3XN4. Here, N1 is number of row-wires, N2 is number of electrodes in each row-wire, N3 is number of column-wires and N4 is number of electrodes in each column-wire. We need to keep N4=3*N2, so Eq. (1) is proper to be solved. In practical application, the array size and the number of the electrodes can be adjusted according to the scale and resolution of the señor. In this paper, Nl=10, N2=10, N3=3, N4=30. In the Simulation model, it is assumed that there are 10 row-wires distributing at the upper layer of the tactile sensor and 10 electrodes on each row-wire. There are also 30 column-wires listed on the lower layer, and each of them has 3 electrodes. When a 3-D force exerted on the top surface of the sensor, the deformation of it can be gained by scanning the "row-column resistances", then we can use computation method such as RBFNN algorithm to decouple the 3-D force from deformation and resistances separately, which are introduced in Sec. 4. The detection principle of the 3-D force flexible tactile sensor can be briefly described: if there is a 3-D force loaded on the sensor, the conductive rubber as the sensitive material of the sensor would be deformed. Meanwhile, the coordinate of electrodes on the upper layer changed, which would make the "row-column resistance" transformed. The "rowcolumn resistance" can be measured from the circuit, and the deformation of the electrodes can be solved through a mathematical model (1). Then the mapping between resistance and 3-D force would be searched. Eventually, in this paper, the 3-D force information can be decoupled based on RBFNN. ... (1) where N is the number of the electrode nodes on each row-wire, M is the number of the electrode nodes on each column-wire, i and j are the label of the electrodes on the current row-wire and the columnwire separately, k is the label of the column-wire, xi, yi, and zi represent the 3-D deformation of the i-th node on a row-wire under the 3-D force; xijk , yijk , and zjk are the coordinate difference between the upper electrode nodes and the lower electrode nodes, whose values are already known when we design the sensor when there is no pressure; Rk is the rowcolumn resistance between the current row-wire and the k-th column-wire; g is the resistance rate of the conductive rubber. In our experiment, M=3, N=10. In actual design, the resolution of the tactile sensor could be advanced by reducing the distances between the electrodes and adjusting the size of the sensor array, so that it could meet the actual needs for detecting 3-D force under different conditions. 3. Finite Element Analysis The flexible tactile sensor is composed of forcesensitive conductive rubber. In the simulation of the rubber, there are some problems, such as viscosity, hysteresis, and detection error. In order to overcome those problems, we apply ANSYS advanced parametric design language to produce training and testing samples directly through intelligent analysis, to analyze the feasibility of numerical model, and to optimize the sensor's structure. In the simulation, we assume that the conductive rubber is a linear, isotropic, repetitive and non-viscoelastic material. In this section, a preliminary numerical model of a 3-D force tactile sensor based on Finite Element Analysis (FEA) is constructed. Using FEA technology to optimize the sensor, which verify the feasibility of the design strategy and provide the basis for the theoretical analysis of the new sensor's structure. This paper generates the model in the way of mapped 10x10x5 line meshing, as shown in Fig. 3. In the modeling and meshing progression, the tactile sensor model is set into a cuboid, the size of which is 30 mm x 30 mm x 5 mm. The wires at the upper layer are parallel to the x-axis and the wires at the lower layer are parallel to the y-axis. Considering the features of the conductive rubber, hyperelastic Mooney-Rivlin unit is adopted to establish the relationship between 3-D force and deformation. This paper chooses the HYPER58 unit, in which the Young's modulus is 2.08e5 N/mm2, Poisson's ratio is 0.499, density is le-9 t/mm3, and Mooney constant clO, cOl are 0.3286, 0.08215, respectively. In the simulation, fixed support is exerted on the lower layer, the 3-D force is loaded on the upper surface of the sensor, and then the mapping between the 3-D force and deformation of the optimized tactile sensor is constructed by ANSYS simulation. Through the simulation of the material and structure, we can design the tactile sensor more accurately, and get effective supports to modify the model. To simplify the calculation, we divided the upper surface of the sensor into 6 small areas, and set the 3-D force on them respectively, as shown in Fig. 3. Then we can get the corresponding deformation of the electrode nodes on the lines of the upper layer. The deformations are used as training or testing samples in the RBFNN algorithm. By applying different force on different areas, we get enough samples for the RBFNN. Fig. 4 shows the simulation result when there is a 3-D force loaded on the first area. It demonstrates that the nodes on or near the corresponding area have obvious deformation, while the other nodes which are far from the first area have less or no deformation. In the ANSYS simulation, 100 groups of 3-D force are loaded on 6 areas respectively, and the deformations of the 10 electrodes on each row-wire are gained. We choose the deformation (30-D) of a line as the experimental data for the RBFNN. Therefore, we get 600 samples as the training set. After that, we select one of the 6 areas randomly, and apply 100 different 3-D forces on that area, then gain the deformations of the 10 electrodes on the same line further, which are used as the testing set. Therefore, 600 training samples and 100 testing samples are produced by ANSYS model for the tactile sensor. Each sample includes 3-D force applied on an area of the upper surface and corresponding 30-D deformation of 10 electrodes on a row-wire. In the simulation, the 3-D force is applied along x, y, and z direction at the same time, and we gain the deformation of all the electrodes along x-axis, y-axis, and z-axis correspondingly. The decoupling results of that is the improvement of paper [12], which only gives the z-directional deformation. Through the simulation, we can design the tactile sensor more accurately and decouple the 3-D force more conveniently. 4. Decoupling Algorithm for the Threedimensional Force Based on RBFNN There are 3 main steps to realize the decoupling system for the 3-D force tactile sensor. The first step is to decouple mapping relation between resistance and deformation, which is carefully described in [12] based on BP neural network. The second step is to approach the mapping relationship from deformation to 3-D force. The third step is to decouple the 3-D force from resistance directly, as the resistance can be detected from the acquisition circuit in actual experiment. Step 2 and Step 3 are introduced in this part and both accomplished by the RBFNN algorithm. 4.1. Radius Basis Function Neural Network RBFNN (as is shown in Fig. 5) is a typical local approximate neural network. It has 3 layers, namely input layer, hidden layer and output layer. In the RBF network, the mapping from the input layer to the hidden layer is nonlinear, but the transformation from the hidden layer to the output layer is linear. In this paper, RBFNN is used as an approximate machine to approach the mapping from deformation to 3-D force, and the relationship between resistance and 3-D force separately. The RBFNN translate the original non-linear separable feature space into another high-dimensional space firstly, and make the original problem linearly separable in the new feature space by selecting a reasonable transformation, at last a linear layer is applied to solve the problem. Actually, this paper utilizes RBFNN to solve the high-dimensional interpolation problem. Considering the mapping from m-dimensional input space to one-dimensional output space, the mdimensional vector of the input space is Xp (p=l,2,...,N) and the corresponding target output of the output space is dp (p=l,2,...,N). There are N input-output samples that are consist of the input vector Xp and output vector dp. The main purpose of an interpolation problem is searching a nonlinear mapping function F(X), which satisfies the interpolation conditions. F(X) is also an approximate function from the input vector to the output vector. ... (2) It is supposed that there are m neural nodes in the input layer and one node in the output layer. The hidden layer is consists of the neural nodes whose number is equal to the number of input/training samples. Each nodes of the hidden layer could be used as a radial basis function. The input nodes and hidden nodes are directly connected, there are no weights among them, but the hidden nodes and the output nodes are connected by weight values. Using radial basis functions technology to solve interpolation problem should select appropriate radial basis functions and each radial basis function should correspond to the training data. The form of the radial basis function is ... (3) where çis the nonlinear radial basis function, X is the input vector of the input layer, and Xp is the center of (p function. In this paper, we use Gauss function as the radial basis function ... (4) The definition of interpolation function based on radial basis functions is a linear combination, such as ... (5) If we put the interpolation conditions described in Eq. (2) into Eq. (5), a set of equations about unknown weights {cop, p=l,2...,N} can be written as ... (6) Let ... can be changed into ... (7) We set ... (8) ... (9) ... (10) In (8), (f) represents the N x N matrix which consists of (pip . In (9) and (10), W and d represent the linear weight vector and the desired output vector separately. Then Eq. (7) can be replaced by ... (11) where (f) is the interpolation matrix, if (f) is a nonsingular matrix, W could be solved by (12). ... (12) Radial basis functions satisfy Micchelli theorem. Due to the Micchelli theorem, if all the input vectors {Xd} are different, it ensure that matrix (f) is nonsingular. It includes that if the input samples are different, the weight vector W can be solved. There are two stages in the training process of the RBFNN. The first stage is unsupervised selforganizing learning phase, whose task is to find the center for radial basis functions through selforganized clustering method and determine the spreading constant of the hidden nodes. The second stage is supervised learning phase, the task is to use supervised learning algorithm such as the least squares method to decide the output layer's weights. In the training process of RBFNN, it needs to determine the main parameters that are the radial basis function's center, the spread of the radial basis function, the number of the radial basis, and the weights between the hidden nodes and the output nodes. In this paper, we utilize Gauss function as the radial basis function for the hidden layer and use Kmeans clustering method [18] which is easily accomplished and has good performance in finding the spreading constant to search the center of the radial basis function, that change the input vector into a high-dimension space. Then, recursive least square method [19] is used to decide the weights between the hidden layer and the output layer. At last, the network's output is gained through the linear weights summation of the hidden layer's output. This paper adopts the RBFNN to evaluate the high-dimensional nonlinear mapping between deformation and 3-D force and the relationship between resistance and 3-D force respectively. When we study the decoupling problem of the 3-D force flexible tactile sensor, the sensor model is viewed as a black box, which produces a large number of training data. We train the training samples reversely based on the RBFNN method as far as possible to approximate the original model equation. 4.2. Decoupling 3-D Force from Deformation In the decoupling process, we choose the rowwire at the center of the top layer as a typical example. The decoupling principles of the other rowwires are similar. Fig. 6 shows the selected row-wire has no deformation when there is no pressure loaded. Fig. 7 shows the simution result of that row-wire when there is a 3-D force loaded on the first area shown in Fig 3. Fig. 7 implies that electrodes on or near the corresponding area has obvious deformation, while the other electrodes far from the first area have less or no deformation. The results decoupled by RBF method are match to that of the ANSYS simulation as shown in Fig. 4. In the experiments, 600 training samples and 100 testing samples are gained via the ANSYS simulation for the 3-D force flexible tactile sensor. Each sample is composed of a 3-D force loaded on the upper surface of the tactile sensor and a 30-D deformation of 10 nodes on the row-wire. We use RBFNN to approach the mapping relation between the 30-D deformation and 3-D force. The input vector X=[xi, x2, ... , x30] of our RBFNN is a 30-D deformation, and the output vector 0=[FX, Fy, Fz] is a 3-D force. Generally, the maximum deformation ratio of the conductive rubber does not exceed 20 % of the sensor's thickness, and the followings are the range of the actual 3-D force: Fxe[-1N, lN],Fye[-lN, lN],Fze[-6N, 0] Next step is to construct and train the RBFNN. The 600 groups of input vectors and target output vectors in the training samples are used to train the RBFNN. After that, the mapping between 30-D deformation values and 3-D force can be obtained. This means that the model of RBFNN could be applied to decouple the 3-D force as acquired. The RBFNN we build is optimized by K-means [18] method and least square method [19] to improve the accuracy further. Finally, the trained RBFNN model is used to test the performance of decoupling results of 3-D force. In order to simulate the actual situation, the 30-D deformations in the testing sample from ANSYS simulation are put into the trained RBFNN's input layer to solve the 3-D force, and then the decoupling results of 3-D force are gained at the output layer. Good approximate ability of the high nonlinear mapping from deformation to 3-D force is tested by the RBFNN. In the RBFNN, a large spread rang would make a wide receptive field of the radial basis neurons and those neurons are insensitive to input data. In this way, even if the number of neurons is large, the effect of network may be very poor, and the network may show "less fit". On the contrary, if the spread rang is small, which may make the receptive field of the radial basis neuron smaller as some neurons can't be covered, and the neurons are very sensitive to the input. And then, it may lead to a lot of errors. So, if we blindly increase the number of neurons in the hidden layer that would produce "over fitting" phenomenon, which means that the response of the network is very good for the known input but the network may lead to lots of errors for the unknown input. The decoupling errors between the actual 3-D force and the decoupled 3-D force based on RBFNN with different spreads are given in Table 1 respectively. where, 8Fi is the average relative decoupling error of 100 testing samples of Fi (i=x, y, z) are computed by ... (13) In Eq. (13), Fi denotes the decoupling result of the force loaded on the i-axis (i=x, y, z) direction based on RBFNN, F. denotes the actual force h loaded on the i-axis. Table 1 shows that the decoupling results of 3-D force are very good, and with the increase of spread values the decoupling errors of 3-D force decreases quickly. Especially, when spread is 1.5, the best average decoupling errors between the decoupled 3-D force and the original 3-D force are 0.16 %, 0.39 %, 3.44 %, which reflect that the RBNN model we constructed has good functions in decoupling the 3-D force. With comprehensive consideration, we present the decoupling results of 3-D force under the condition with spread being 1. Figs. 8-10 display the decoupling results of Fx, Fy, and Fz, respectively. The relative decoupling errors of 100 testing samples of the 3-D force based on the RBFNN whose spread is 1 are given in Fig. 11. Figs. 8-11 concluded that the decoupled 3-D force match to the original 3-D force very well. From Fig. 11, we know that all of the decoupling results of Fx and Fy are excellent, whose largest relative errors are all less than 0.15 (15 %), and the average relative error of them are 0.90 % and 1.04 %. The average relative error of Fz, 7.56 %, is not so good as that of Fx and Fy, but is still acceptable. Above results verify that the optimized RBFNN has good performance in approaching the nonlinear relationship between deformation and 3-D force, and could decouple 3-D force accurately and quickly. 4.3 Decoupling 3-D force from Resistance We still gain samples about 3-D force and corresponding deformations by ANSYS analysis as before. In this part, the deformations are applied to compute the row-column resistances for the tactile sensor through the mathematical model (1). We put the values of 30-D deformation of the 10 electrodes on a row-wire into Eq. (1), and then the 30 rowcolumn resistance values which denote the resistances between the current row-wire at the top layer and the 30 column-wires at the bottom layer are gained. In actual experiments, when there is a 3-D force loaded on the sensor, the row-column resistance values can be measured by the acquisition circuit. Then, we can use RBFNN algorithm to decouple the high-dimensional nonlinear mapping from resistance to 3-D force information directly. In this section, we still have 600 training samples and 100 testing samples. The 30 resistance values and 3-D force are used as the input vector and desired output vector to train the RBFNN. After that, the mapping relationship from resistance to 3-D force is gained. At last, the 100 testing dataset are applied to test the nonlinear mapping ability of the RBNN. The 30 resistances are put into the trained RBFNN's input layer, and the decoupling results about the 3-D force are obtained directly at the output layer. Fig. 12 shows the average relative decoupling errors of 3-D force from resistance based on RBFNN with different spread values. 8Fi (i=x, y, z) has the same meaning as that in Table 1 and Eq. (13). From Fig. 12, it can be seen that with the decrease of spread values, the decoupling errors of 3-D force decreases quickly, which is contrary to that in Table 1. Those phenomenons conclude that the best results may accompany with different spread values under different conditions, and there is an optimal spread value. In Fig. 12, the best results are gained when spread value is 1. The best values of 8FX, 8Fy and ôFz are 4.11 %, 0.11 % and 1.03 %, respectively. These best results in Fig. 12 are much better than those in Table 1 whose spread value is 1.5. This strongly implies that the optimized RBFNN also has good ability in decoupling nonlinear mapping from resistance information to 3-D force. As the resistance can be gained from the acquisition circuit of the sensor, decoupling the 3-D force from resistance directly has great significance in actual application. It means that we can decouple the 3-D force by computation method through detecting the peripheral circuit directly, and make the experiment more simple, convenient, and accurate. The decoupling results show that the 3-D force can be accurately and quickly detected through RBFNN algorithm, which satisfies the real-time requirements of the multi-dimensional tactile sensor and improves the performance of the flexible tactile sensor further. The detailed decoupling results are presented in Figs. 13-15 when spread value is 0.5. The three figures are the decoupled results of Fx, Fy, and Fz respectively, which indicate that the decoupling absolute errors of the 3-D force represented by Fx-Error, Fy-Error, and Fz-Error are very small. Especially, most of the errors between decoupled Fy (Fz) and original Fy (Fz) are nearly zero, which mean that the decoupled results based on RBFNN algorithm match to the actual 3-D force very well. Fig. 16 shows the relative decoupling errors of 100 testing samples of the 3-D force when spread is 0.5, and the average relative decoupling errors of 3-D force from resistance are 5.18 %, 0.41 %, and 1.79%. All of the decoupling results above imply that RBFNN is a very efficient method to decouple the 3-D force and approximate the high-dimensional nonlinear relationship between the 30 resistances and the 3-D force. If we gain the precision resistance values from the external circuit, the 3-D force can be decoupled quickly and accurately based on RBFNN algorithm. 5. Conclusions This paper proposes a decoupling method for a novel 3-D force flexible tactile sensor. The method based on Radical Basis Function Nueral Network (RBFNN) algorithm is proposed. At first the finite element analysis is used to construct the numerical model of the tactile sensor and compute the deformation induced by 3-D force applying on the top surface of the sensor. Then, the RBFNN is trained by the samples produced via the numerical model. At last, the RBFNN is used to découlé 3-D force from deformation and resistance respectively. All of the numerical experiments show that the improved RBFNN algorithm deals the decoupling process very well. Analysis about the dcoupling results indicates that the RBFNN with high nonlinear mapping ability has high performance in both accuracy and anti-noise ability, which also satisfies the rea-time requirement of decoupling for the 3-D force tactile sensor. Acknowledgements The paper is supported by Natural Science Research of Colleges of Anhui (KJ2012B037), NSF of Anhui (No. 1408085QF123), and NSFC (Nos. 61175070, 61201076, and 61301060). Junxiang Ding thanks the support from NSF of Anhui (No. 1308085QF104). 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Lee, Study of a reasonable initial center selection method applied to a K-means clustering, IEICE Transactions on Information and Systems, Vol. E96-D, Issue 8,2013, pp. 1727-1733. [19] . S. Haykin, Neural Networks and Learning Machines, Third Edition, Prentice Hall, 2008. l'2,3 Feilu WANG,2 Xin SUN,2 Yubing WANG, 4 Junxiang DING, 2Hongqing PAN, 2Quanjun Song,2 Yong YU, 1,2'*Feng SHUANG 1 Department of Automation, University of Science and Technology of China, Hefei, 230027, China 2 Institute of Intelligent Machines, Chinese Academy of Sciences, Hefei, 230031, China 3 School of Electronics and Information Engineering, Anhui Jianzhu University, Hefei, 230601, China 4 New Star Research Institute of Applied Technology in Hefei City, Hefei, 230031, China 8 E-mail: [email protected] Received: 28 October 2013 /Accepted: 26 November 2013 /Published: 30 November 2013 (c) 2013 International Frequency Sensor Association |