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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Distance has been one of the basic factors in manufacturing and control fields, and ultrasonic distance sensors have been widely used as a low-cost measuring tool. However, the propagation of ultrasonic waves is greatly affected by environmental factors such as temperature, humidity and atmospheric pressure. In order to solve the problem of inaccurate measurement, which is significant within industry, this paper presents a novel ultrasonic distance sensor model using networked error correction (NEC) trained on experimental data. This is more accurate than other existing approaches because it uses information from indirect association with neighboring sensors, which has not been considered before. The NEC technique, focusing on optimization of the relationship of the topological structure of sensor arrays, is implemented for the compensation of erroneous measurements caused by the environment. We apply the maximum likelihood method to determine the optimal fusion data set and use a neighbor discovery algorithm to identify neighbor nodes at the top speed. Furthermore, we adopt the NEC optimization algorithm, which takes full advantage of the correlation coefficients for neighbor sensors. The experimental results demonstrate that the ranging errors of the NEC system are within 2.20%; furthermore, the mean absolute percentage error is reduced to 0.01% after three iterations of this method, which means that the proposed method performs extremely well. The optimized method of distance measurement we propose, with the capability of NEC, would bring a significant advantage for intelligent industrial automation.

Distance is one of the most basic factors in manufacturing and control fields. It is used for local positioning, object identification, automation control, human-computer interaction, and so on [

As shown in

The most basic method of UDS is using a simple pulse echo sensor, which acts as both transmitter and receiver. An echo is detected by the receiver after the transmitted pulse propagates outwards. The distance between the transmitter and receiver is obtained by counting the elapsed time from the start of the transmission to the end of the receipt, defined as the time-of-flight (TOF) [_{s}_{s}_{s}_{s}_{0} if nothing is measured. Therefore, there is a zero mode (Z-MODE) in the device meaning that, irrespective of the measured value from the UDS, the distance is regarded as _{0} without any definite errors.

The fact that the industrial environments of sensors are non-ideal implies that it is sometimes difficult to detect the real size of the semi-manufactured products. In order to improve the detection process, many mathematical methods have been used, closely followed by the use of UDS growing in an exponential manner since the 1980s [

As mentioned above, in some of modern industrial cases, these devices have formed a network with wired or wireless methods passing data to a DC. Furthermore, many data mining and mathematical and statistical algorithms have been used for data analysis. A model using data mining algorithms which then processes data based on the same model has been widely employed in industrial fields to gain accurate measurements. In [

In this study, based on the fact that UDS devices have been organized as a network to gather real-time data into a DC, we focus on the mathematical methods to calculate the precise correction value by analysis of the data correlation coefficients

Over all, the contributions of this paper are summarized as the followings:

We have modeled the NEC system on the optimization of the relationship of the topological structure of the sensor arrays, and developed a Two-Step Error Correction Process (TSECP) based on the relations between neighboring nodes.

An abstract model is set up to illustrate the ability of the compensation for error measurements caused by the environments.

TSECP is described with the theory of fusion set. To compare with the previous NEC methods, we have analyzed the design comparisons and the system improvements.

We have designed the UDS with the maximum likelihood method, neighbor discovery discovery algorithm and the NEC optimization algorithm.

We apply the maximum likelihood method to determine the optimal fusion data set and use a neighbor discovery algorithm to identify neighbor nodes at the top speed.

We adopt the NEC optimization algorithm, take full advantage of the correlation coefficients for neighbor sensors.

The experimental results demonstrate that the NEC system has excellent anti-interference performance, and its ranging errors are within 2.2%. Furthermore, the mean absolute percentage error is reduced to 0.01% after three iterations of this method.

Because we take advantage of data correlation coefficients for the first time, the novel NEC model is more accurate than existing approaches. Its superiority over the former methods is obvious: instead of the solo node, all of the topological information of the sensor arrays is involved in error compensation. Observational evidence has frequently been linked to numerical error and propagation of the ultrasonic wave affected by the environment; however, there is little direct trial evidence to train and verify the exact equation of the two parameters. In view of the significant potential value of this issue, further investigation of the equation is warranted.

Recently, distance measurement based on UDS has increased significantly. In [

Moveover, the performance of the UDS arrays system can be greatly improved by adopting data fusion technologies and self-configuring scheduling protocols. In [

Notably, the implementations of UDS arrays have obtained increasingly influence in the industrial world. In [

NEC is the key stone for the implementation of UDS. In [

Moreover, mathematical statistics has been widely used for NEC. In [

The observations above demonstrate that measured values are unavoidably imprecise, because of the instability of the environment. This motivates us to propose an approach of minimizing the error between the measured response and the desired response. This study provides a potential solution based on NEC for the reliable and low-cost distance-measurement applications by using the information from neighboring sensors—which has not been considered before. This also has the potential to shed light on the automation fields.

The basic architecture of networked sensors is shown in _{0}, and the error of the node can be found by a simple subtraction. Second, in a time interval, _{1}, _{2}], each node measures an independent series of data points (because of the different objects measured). There is no distinctive function for a node; however, for the nodes in their fixed positions, the errors caused by the environment have the implicit function. In mathematics, an implicit equation is a relation of the form _{1}, …, _{n}^{2} + ^{2} − 1 = 0. While, a distinctive function is a relation of the form

For example, if Nodes _{t}_{t}_{t}_{0}—the error in this point is _{It}_{0}_{t}_{1}, _{2} and _{3} obtaining two sets of raw data, {_{1}, _{2}, _{3}} and {_{1}, _{2}, _{3}} with errors of {_{I}_{1}, _{I}_{2}, _{I}_{3}} and {_{J}_{1}, _{J}_{2}, _{J}_{3}}, (_{1} has no direct relation with _{2} and _{1}, the environments are very similar for both of nodes at the time points, _{1}_{2} and _{3}. The errors caused by the environment have the inner relations that can be concluded from the trends of _{1} is in Z-MODE, _{I}_{1} can be calculated directly, then _{J}_{1} can be calculated from the function, and _{1} can then be found.

In the air with a constant pressure, _{s}_{c}_{0} + _{c}_{0} is the Absolute Temperature, a constant. Using

In addition, _{s}

If _{s}_{s}_{s}_{s}_{s}_{s})_{s}_{s}_{s}

Mathematical statistics has been widely used in the collection, analysis, interpretation and presentation of data in the process of error correction. Variance is used as a measure of discreteness. A set of numbers has a probability distribution, expressing how far the numbers deviate from ^{2} is the variance.

For two variables, covariance is a measure of the probability that they change together. The sign of the covariance shows the tendency in the linear relationship between the variables, and the magnitude of the covariance shows the strength of the linear relation. We define the covariance of the two variances

While the magnitude of

Considering that

An abstract model is set up to illustrate the problem. Assume all of the UDS distributed in a large area, form a matrix and are at time _{ij}

Some of these values could have been corrected by other methods, such as the Z-MODE of a UDS device. Assume that
_{00}, _{01}, _{11}, _{mn}

If _{0}; it is an initial state. Using simple subtraction, the error matrix is easily obtained:

In this paper, the key information is that _{ij}

For _{ij}_{ij}_{ij}

Note that each _{ij}_{ij}

According to the analysis above, the key problem is to find the suitable _{ij}_{ij}

Neighboring nodes are located in fixed static positions and have similar trends with environment changes, so they may have some relations. As shown in

As a simple case, in a time interval, [_{1}, _{2}], where each node of _{i}

From _{i,j}

Of particular note is that the sum of each line in the _{i,j}_{i,j}

Matrix multiplication between _{t}_{i,j}

Based on the NEC model, we can assume that the measured data of _{i}_{j}_{i}_{j}_{i}_{j}_{i}_{j}_{i}_{j}

We introduce fiducial distance measurement, in order to reflect the size of the error between _{i}_{j}_{i}_{i}_{i}_{j}

As mentioned above, _{ij}_{i}_{j}_{ij}_{ij}_{i}_{j}_{ij}_{i}_{j}

_{ij}

In the Learning Mode, three thresholds of _{ij}_{2}, _{1}—can be obtained after repeated trials. Then:

Subsequently, the standardized relationship matrix, _{ij}

The data from one sensor are labeled as valid if the relationship value in the standardized relationship matrix _{ij}_{1}, _{2}, …, _{l}

Based on the relationship matrix and fusion set, the maximum likelihood method has been adopted as the data fusion method. If
_{1}, _{2}, …, _{l}_{1}, _{2}, …, _{l}

As mentioned above, the focus of the study is a novel method of error correction that lends weight to the argument that the accuracy of the UDS system relies heavily on the fitness and robustness of the function involving the association between the various patterns of the error, the position of the nodes, and the real-time temperature. Particularly noteworthy are previous studies on the NEC which, in general, consist of a Learning Mode and an Operating Mode, as shown in

In

Accuracy: instead of single-ended input, the way of input the present NEC system has adopted is differential input, which reduces measurement error, such as cosine error and Abbe error.

Robustness: if one node is broken, previous NEC systems have no choice but to abandon its measured value, because the value is intolerant. However, the present NEC system can rationally estimate the value on the basis of the updated relation matrix.

Renewability: in previous NEC systems, once the Learning Mode is over, the function of each node is be irrevocable, unless starting the Learning Mode over again. However, as shown in

The UDS is composed of three galvanically isolated main parts: a power module, a power amplifier module and a transmitter-receiver module. The power module provides two main functions: driving the sensors and ground protection. The power amplifier module, which is composed of a power amplifier and an inverting amplifier, can be integrated into a differential amplifier. Compared with the single-ended signal and the common-mode signal, the different-mode signal provides the following advantages:

A small signal can be more easily detected by means of controlling a reference voltage.

Since one specific interference source homogeneously affects both sides of the differential signal to a large extent, the different-mode signal is virtually immune to electro-magnetic interference.

Because the different-mode system does not need to build a Virtual Ground at any point between ground and power when processing a bipolar signal in a single-supply system, the fidelity of the signal is much better than the single-ended signal and the common-mode signal.

The transmitter-receiver module is the most essential part of the application of the signal transduction. In the transmitter mode, the transmitters generate an ultrasonic beam; after the ultrasonic beam hits the target and rebounds, the addition of the two waveforms gives the return echo. In the receiver mode, the receiver acts as a mechanical detector of the reflected wave and outputs the echo signal already received. In

Data of characteristic parameters measured by different sensors, even of the same type, are different to some degree. This deviation has two main causes: (1) sensor accuracy and (2) the mathematical algorithm adopted in data processing. The most important parameter in the UDS system is the distance data measured by the sensors, so we apply the same index parameter to multi-sensor measurement for the following reasons:

Using many sensors with different performance and accuracy can complement each other's advantages and cover disadvantages.

The redundancy configuration of sensors can improve reliability and measurement accuracy.

In _{2}_{2}_{2}

The proposed circuit for a 4 × 4 sensor array is represented in _{L}

The senor network can be modeled as a weighted directed graph (

The theoretical neighbor of Node

{

The non-forward neighbor of Node _{0}(

{

The one-step-forward neighbor of Node

(_{1}(_{0}(

The two-steps-forward neighbor of Node _{2}(

{_{1}(

∃

Under the precondition that the topological structure of the wireless sensor network is known, we can obtain the neighborhood discovery algorithm of the UDS as shown in Algorithm 1.

_{0},

_{1},

_{2},

_{1},

_{2})

_{0};

_{1};

_{2};

_{1}: The one-step-forwarding neighbor of Node

_{2}: The two-step-forwarding neighbor of Node

_{0}(

_{1}(

_{2}(

_{0}(

_{1}(

_{2}(

_{1}(

_{2}(

_{2}(

_{1}(

_{1}(

_{0}(

_{2}(

_{2}(

_{0}(

In order to simplify the UDS system, the experiments are based on a sample of sensor arrays, represented as matrices on the order of twenty by twenty. It makes the comparison of initial data between the ideal response, response in the absence of noise, and response with signal noise. The height of the semi-finished products is 50 cm, so the default distance of the parallel lines with none of the products passing by in the workshop that we used for experiments is also 50 cm. Therefore, the ideal value matrix has elements with value either 50 cm or 100 cm. Considering that interference highly depends on the position of the sensors in this model, we have obtained the initial data.

In

In

From

Select a set of data for which the correlation coefficients are large enough.

Correct this value of the arbitrary node based on the set data chosen in the first step.

In

In

In

A novel method based on NEC for obtaining a satisfactory distance measurement has been proposed in this work. It can provide a high-speed and high-accuracy solution for distance measurement, which is vital in most industrial fields. One of the most obvious flaws of the previous methods is that none of them has considered the indirect association among neighboring sensors, which leads to less accuracy and extra cost. Therefore, one of the main aims of this study was to propose a novel method based on NEC that can provide a high-speed and high-accuracy solution for distance measurement. We applied the maximum likelihood method to determine the optimal fusion data set and used a neighbor discovery algorithm to identify neighboring nodes at high speed; furthermore, we adopted the NEC optimization algorithm, which takes full advantage of the correlation coefficient among neighboring sensors. The experimental results demonstrate that the ranging errors of the NEC system are within 2.20%, and the mean absolute percentage error is reduced to 0.01% after three iterations of this method.

Our proposed method has several strengths. However, several limitations merit comment. The most critical of them is that we are unable to establish and then test the exact cause and effect equation of the numerical error and the propagation of the ultrasonic wave affected by the environment, such as temperature, humidity and atmospheric pressure. Nevertheless, in spite of this limitation, we believe this study opens new paths of investigation in distance measurement using a UDS and, to some extent, modifies the manner in which we understand the method of error compensation, because new insights into it will likely generate novel methods to measure distance more effectively and quickly than previously. There is, therefore, a great need for further research in this area.

This work has been partially supported by the project “Science and Technology Plan of Shandong Province, China (No.2012GB020108)” and the project “National Science Foundation of China (No. 61070022)”.

The authors declare no conflicts of interest. We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, and there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in or the review of the manuscript.

Scenario of production lines. (

Block diagram of the basic architecture.

Schematic diagram of the confidence distance measurement.

Design comparison and system improvements based on the previous NEC studies.

View of the design of the UDS.

Architecture of the sensor arrays.

The equivalent reduced circuit when an element (EBA) is selected. (

Comparison of initial data.

Initial data errors.

Correlation coefficient between a node in Not Z-MODE and the nodes in Z-MODE, obtained in the Learning Mode.

First-time corrected error values by applying a UDS.

Second-time corrected values by applying a UDS.

Mean absolute percentage error along with the increasing of iterations.