Research on Structure Optimization and Measurement Method of a Large-Range Deep Displacement 3D Measuring Sensor

Deep displacement monitoring of rock and soil mass is the focus of current geological hazard research. In the previous works, we proposed a geophysical deep displacement characteristic information detection method by implanting magneto-electric sensing arrays in boreholes, and preliminarily designed the sensor prototype and algorithm of deep displacement three-dimensional (3D) measurement. On this basis, we optimized the structure of the sensing unit through 3D printing and other technologies, and improved the shape and material parameters of the permanent magnet after extensive experiments. Through in-depth analysis of the experimental data, based on the data query algorithm and the polynomial least square curve fitting theory, a new mathematical model for 3D measurement of deep displacement has been proposed. By virtue of it, the output values of mutual inductance voltage, Hall voltage and tilt measuring voltage measured by the sensing units can be converted into the variations of relative horizontal displacement, vertical displacement and axial tilt angle between any two adjacent sensing units in real time, and the measuring errors of horizontal and vertical displacement are tested to be 0–1.5 mm. The combination of structural optimization and measurement method upgrading extends the measurement range of the sensing unit from 0–30 mm to 0–50 mm. It shows that our revised deep displacement 3D measuring sensor can better meet the needs of high-precision monitoring at the initial stage of rock and soil deformation and large deformation monitoring at the rapid change and imminent-sliding stage.


Introduction
Geological disasters refer to rock and soil mass sliding accidents that damage ecological environment resources and seriously threaten human life and property under the influence of natural or human factors, including collapse, landslides, debris flows and so on [1][2][3][4][5][6]. According to the data, 70% of geological disasters are landslides and collapses. Landslide deformation is an integrated reflection of geological structure and internal and external influencing factors of a landslide mass. In view of the actual needs of disaster prevention and mitigation work, we should focus on the monitoring technology of rock and soil mass stability at present [7], which mainly includes surface displacement monitoring and deep displacement monitoring [8]. A large number of scholars' research shows that the initial stage of the landslide is mainly characterized by deformation caused by shear force in the deep part of rock and soil mass, and with the gradual development of landslide, the final measurement algorithm has higher measurement accuracy, larger measurement range, better waterproof, anti-corrosion and pressure resistance performance and can better meet better the needs of high-precision monitoring at the initial stage of rock and soil deformation and large deformation monitoring at the rapid change and imminent-sliding stage.

Sensor Structure and Working Principle
As shown in Figure 1, the magneto-electric deep displacement measurement sensor is composed of control part and measurement part. Data exchange between them is carried out by RS-485 bus. The control unit is the host computer, which transmits the voltage measurement command to the measurement part through RS-485 bus, and receives data such as mutual inductance voltage, Hall voltage and tilt measuring voltage fed back by the measurement part. The received data are input into a deep displacement measurement model to solve the relative tilt angle and relative displacement between any two adjacent units. The host computer can further transmit data to the remote client through wireless network. The proposed sensor is characterized by real-time, dynamic, remote and automatic. The measuring part consists of N identical integrated sensing and measuring units (hereinafter referred to as sensing units). Each sensing unit has the same structure. In the practical engineering, the bottommost sensing unit is buried into the rock and soil fixed layer, numbered 1, serving as the reference point for the entire measurement system. From bottom to top, each sensing unit is numbered 1-N, respectively. When the system works, two adjacent sensing units form a measurement unit, such as: No.1 measurement unit is composed of sensing units 1 and 2, No.2 measurement unit composed of sensing units 2 and 3, ..., with a total of N-1 measurement units for the sensor. When the instrument is installed on site, the upper and lower end faces of two adjacent sensing units can be directly attached or vertically spaced at a certain distance according to monitoring requirements. By measuring the displacement of each layer of rock and soil from N-1 measuring units in turn, the distributed flexible measurement of three-dimensional displacement from the surface to the depth of rock and soil mass can be realized. Figure 2 shows the prototype of the magneto-electric deep displacement 3D measuring sensor, which we designed before [32][33][34][35][36]. Each sensing unit has the same structure. The main body is a solenoid, the outer wall is covered with several layers of thin coils and the inner wall is embedded by a PCB panel, which mainly integrates sinusoidal signal generation, mutual inductance voltage The measuring part consists of N identical integrated sensing and measuring units (hereinafter referred to as sensing units). Each sensing unit has the same structure. In the practical engineering, the bottommost sensing unit is buried into the rock and soil fixed layer, numbered 1, serving as the reference point for the entire measurement system. From bottom to top, each sensing unit is numbered 1-N, respectively. When the system works, two adjacent sensing units form a measurement unit, such as: No.1 measurement unit is composed of sensing units 1 and 2, No.2 measurement unit composed of sensing units 2 and 3, . . . , with a total of N-1 measurement units for the sensor. When the instrument is installed on site, the upper and lower end faces of two adjacent sensing units can be directly attached or vertically spaced at a certain distance according to monitoring requirements. By measuring the displacement of each layer of rock and soil from N-1 measuring units in turn, the distributed flexible measurement of three-dimensional displacement from the surface to the depth of rock and soil mass can be realized. Figure 2 shows the prototype of the magneto-electric deep displacement 3D measuring sensor, which we designed before [32][33][34][35][36]. Each sensing unit has the same structure. The main body is a solenoid, the outer wall is covered with several layers of thin coils and the inner wall is embedded by a PCB panel, which mainly integrates sinusoidal signal generation, mutual inductance voltage measurement, tilt measurement, serial communication and other functions. A small cylindrical permanent magnet with a diameter of 5.6 mm and a height of 18 mm is mounted at the center of the sensing unit's lower end faces. Another PCB panel is mounted on the upper end faces of the sensing unit, and a micro Hall sensor is installed at the center of the panel, beside which a Hall voltage measuring integrated circuit is arranged. As mentioned, when the measurement system works, any two adjacent sensing units works as a measurement unit, for which, the lower sensing unit acts as a reference end while the upper one as a measurement end, and are labeled as Sensing units I and II, respectively. Here we only outline its working principle. More detailed description can refer our published works [32][33][34][35].
Sensors 2020, 20, 1689 6 of 20 mm, respectively when the measuring range of ΔR and ΔZ controlled within 0-30 mm for each measuring unit. Finally, it should be mentioned, the three-dimensional angular displacement can be expressed by Euler angles, that is, elevation angle θ0, yaw angle ϕ and roll angle γ. Considering the axisymmetric structure of sensing unit, the measurement of the roll angle γ is of little significance. In this context, the pitch angle θ0 and the yaw angle ϕ correspond to the relative axial tilt angle θ0 and the azimuth angle ϕ between two adjacent sensing units. The relative horizontal displacement ΔR and relative vertical displacement ΔZ can be calculated by combination of the two corresponding independent equations based on the measured mutual inductance voltage Ur and Hall voltage Uh. At the same time, as shown in Figure 2c, the components on the x-axis and y-axis of ∆R can be calculated with ∆R and azimuth angle ϕ, as follows: Therefore, the correspondence between known parameters and unknown parameters is: (∆X, ∆Y) = f (∆R, ϕ) As shown above, ∆X and ∆Y can be calculated by ∆R and ϕ, so unless otherwise specified, the deep displacement three-dimensional parameters to be measured are expressed by the relative horizontal displacement ∆R, relative vertical displacement ∆Z, relative tilt angle (θ0) and relative direction angle (ϕ) between adjacent sensing units. The relative tilt angle θ0 and relative direction angle ϕ can be directly measured by PCB integrated module, therefore the deep displacement measurement model is simplified as applying two independent functions (Equations (2) and (3)) to As shown in Figure 2, driven by the movement of surrounding rock and soil mass, a relative horizontal displacement ∆R, vertical displacement ∆Z and tilt angle θ 0 might synchronously occur between Sensing Units I and II. The mutual inductance voltage U r and Hall voltage U h generated on Sensing unit II will be varied simultaneously due to electromagnetic induction and Hall effect. We will explain it briefly.
If a sinusoidal voltage U i with fixed amplitude and frequency, such as 10 KHz, is applied between the two ends of the solenoid in the sensing unit I, a corresponding mutual inductance voltage U r will be generated across solenoid of sensing unit II, which, as we had proved [32], is directly proportional to the mutual inductance M between Solenoids I and II, with the following relationship where, L is the coefficient of self-inductance. Mutual inductance M, reflecting the magnetic coupling state between Solenoids I and II, is determined by the solenoids' geometry parameter (shape, size, turns of coils, etc.) and the relative position between them. Each solenoid is covered with thick polypropylene plastic pipe (featured as tough and resist to wear, corrosion, pressure, etc.), so generally it will not deform with the sliding of surrounding rock and soil mass, and the variation of M is only related to the relative displacement between two adjacent solenoids. In other words, the mutual inductance voltage U r has a definitive functional relationship with the relative horizontal displacement ∆R, vertical displacement ∆Z and tilt angle θ 0 between Sensing Units I and II. It can be generally expressed as As Figure 2 shows, when Solenoids I and II produce a relative displacement (∆R, ∆Z and θ 0 ), the relative position changes between the permanent magnet on Sensing unit I and the Hall sensor on Sensing unit II, so the magnetic field strength and direction applied on the Hall sensor change accordingly, resulting in a varied output of Hall voltage from the Hall sensor. It can also expressed as Meanwhile, by virtue of the built-in tilt measuring integrated module, the relative axial tilt angle θ 0 between Sensing units I and II can be automatically measured with a measurement error of less than 1 • . Therefore, as long as an effective measurement model can be established that is capable of accurately describing the complex relationship between the output values of U r , U h , θ 0 from sensing units and the measuring displacement parameters ∆R and ∆Z between Sensing Units I and II, the relative horizontal displacement ∆R and vertical displacement ∆Z for each measuring unit can be inversely deduced (the axial tilt angle θ 0 can be directly measured) according to the output values of mutual inductance and Hall voltage.
In our previous research work [35], we had established such a measurement model. It is an original deep horizontal and vertical displacement joint inversion method called "joint forward simulation-optimization inversion method". It can realize a joint inversion of deep horizontal displacement and vertical displacement for the proposed sensor. The deviations between the experimentally measured and modeling inversed ∆R and ∆Z were tested to be less than 3 mm and 1 mm, respectively when the measuring range of ∆R and ∆Z controlled within 0-30 mm for each measuring unit.
Finally, it should be mentioned, the three-dimensional angular displacement can be expressed by Euler angles, that is, elevation angle θ 0 , yaw angle φ and roll angle γ. Considering the axisymmetric structure of sensing unit, the measurement of the roll angle γ is of little significance. In this context, the pitch angle θ 0 and the yaw angle φ correspond to the relative axial tilt angle θ 0 and the azimuth angle φ between two adjacent sensing units. The relative horizontal displacement ∆R and relative vertical displacement ∆Z can be calculated by combination of the two corresponding independent equations based on the measured mutual inductance voltage U r and Hall voltage U h . At the same time, as shown in Figure 2c, the components on the x-axis and y-axis of ∆R can be calculated with ∆R and azimuth angle φ, as follows: Therefore, the correspondence between known parameters and unknown parameters is: As shown above, ∆X and ∆Y can be calculated by ∆R and φ, so unless otherwise specified, the deep displacement three-dimensional parameters to be measured are expressed by the relative horizontal displacement ∆R, relative vertical displacement ∆Z, relative tilt angle (θ 0 ) and relative direction angle (φ) between adjacent sensing units. The relative tilt angle θ 0 and relative direction angle φ can be directly measured by PCB integrated module, therefore the deep displacement measurement model is simplified as applying two independent functions (Equations (2) and (3)) to solve two independent variables (∆R, ∆Z). In the practical monitoring process, if the surrounding rock and soil slide, the above-mentioned U r , U h , θ 0 and φ can be measured in real time by the proposed 3D sensor.
From them, not only the relative vertical displacement and horizontal displacement can be calculated, but also the relative tilt angle and tilt direction of the two sensing units can be determined, so as to realize the three-dimensional measurement of deep displacement of rock and soil mass.

Sensing Unit Structure Optimization Design
Combined theoretical research and experimental results of our research group, shortcomings of the above structure design are found. First of all, our previous research results show that, based on the above structural design of the deep displacement measurement senor, the measured relative tilt angle of any measurement unit can reach 0-90 • , but the measuring ranges of relative horizontal displacement and vertical displacement can only reach 0-30 mm. It will limit the effectiveness of the proposed system for landslide early warning and monitoring to a certain extent, considering the big measurement range requirements of some large landslides disasters in near and accelerated sliding stage. Secondly, the PCB board and the permanent magnet have been mounted on the upper and lower end faces of the sensing unit respectively. Originally, its purpose is to narrow the initial distance between the permanent magnet and Hall sensor in each measurement unit and increase the displacement measurement range. However, it may affect the durability of the designed sensing unit due to not conducive to water-proof and pressure-proof treatments and increase difficulty in the subsequent sealing packaging and overall assembly of sensors for in-situ monitoring.
In order to adapt to the complex environment of deep rock and soil mass, the sensing unit must have such properties as waterproof and anti-corrosive. The structural optimization design of sensing unit for the sensor is required. Firstly, as shown in Figure 3, the permanent magnet and the Hall sensor are no longer installed on the upper and lower end faces of sensing unit, but are moved a certain distance inside, respectively. In this way, the permanent magnet and Hall sensor do not protrude from the upper and lower end faces of the solenoid. At the same time, since the external PVC plastic pipe is 20 mm higher than the solenoid, the glue can be filled in the gap between the end face of the solenoid and the end face of the external PVC plastic pipe, so that the sensing unit has better waterproof and anti-corrosive properties. Figure 4 shows the actual structure of each component of the sensing unit in details.

Sensing Unit Structure Optimization Design
Combined theoretical research and experimental results of our research group, shortcomings of the above structure design are found. First of all, our previous research results show that, based on the above structural design of the deep displacement measurement senor, the measured relative tilt angle of any measurement unit can reach 0-90°, but the measuring ranges of relative horizontal displacement and vertical displacement can only reach 0-30 mm. It will limit the effectiveness of the proposed system for landslide early warning and monitoring to a certain extent, considering the big measurement range requirements of some large landslides disasters in near and accelerated sliding stage. Secondly, the PCB board and the permanent magnet have been mounted on the upper and lower end faces of the sensing unit respectively. Originally, its purpose is to narrow the initial distance between the permanent magnet and Hall sensor in each measurement unit and increase the displacement measurement range. However, it may affect the durability of the designed sensing unit due to not conducive to water-proof and pressure-proof treatments and increase difficulty in the subsequent sealing packaging and overall assembly of sensors for in-situ monitoring.
In order to adapt to the complex environment of deep rock and soil mass, the sensing unit must have such properties as waterproof and anti-corrosive. The structural optimization design of sensing unit for the sensor is required. Firstly, as shown in Figure 3, the permanent magnet and the Hall sensor are no longer installed on the upper and lower end faces of sensing unit, but are moved a certain distance inside, respectively. In this way, the permanent magnet and Hall sensor do not protrude from the upper and lower end faces of the solenoid. At the same time, since the external PVC plastic pipe is 20 mm higher than the solenoid, the glue can be filled in the gap between the end face of the solenoid and the end face of the external PVC plastic pipe, so that the sensing unit has better waterproof and anti-corrosive properties. Figure 4 shows the actual structure of each component of the At the same time, in order to improve stability of the device and ensure the initial positions of three center points of the permanent magnet, Hall sensor induction surface and solenoid surface to be in the same vertical line, we use 3D printing technology to design the fine structure for the placement and fixing of permanent magnets, Hall sensors and PCB boards. The fixing devices of paper [34]. The Hall sensor saturated all the time when the permanent magnet has too strong magnetism because of the measurement range of Hall sensor (SS94A1F, made by Honeywell International, Morris Plains, NJ,, USA.) is −100 Gauss to 100 Gauss. Therefore, the size of the permanent magnet corresponding to the largest location in the figure is selected, as shown in Table  1. However, it should be pointed out that although the structure of the sensing unit has been optimized, the measurement principle remains unchanged.   At the same time, in order to improve stability of the device and ensure the initial positions of three center points of the permanent magnet, Hall sensor induction surface and solenoid surface to be in the same vertical line, we use 3D printing technology to design the fine structure for the placement and fixing of permanent magnets, Hall sensors and PCB boards. The fixing devices of permanent magnet and Hall sensor can not only enhance the robustness and durability of the sensing unit on the one hand, but also prevent the permanent magnet or the Hall sensor from falling or shifting, which will increase the measurement error and even lead the measurement system failing to work. Compared to the original design, the renewed device is robust, stable and convenient for assembly. Secondly, the structure of permanent magnets is optimized. Our research shows that the deep displacement of rock and soil mass can be accurately measured by using mutual inductance effect and Hall effect synthetically. However, the measurement range based on Hall effect is smaller than that based on mutual inductance effect, that is, the system's measurement range of the system is mainly limited by the Hall effect. Under certain conditions, the increase of magnetic field intensity of the permanent magnet can increase the measuring range of Hall voltage. Based on the geometry of mutual inductance solenoids, a large number of experiments have been carried out on permanent magnets with different sizes and magnetic field intensities, and the comparison results are shown in Figure 5. It can be seen in Figure 5 that with the same magnetic brand, as the size of the permanent magnet becomes larger, the actual measurement range of the Hall sensor increases accordingly. Moreover, with the same size of permanent magnet, the measurement range of using the N35 permanent magnet is larger than using the N52 permanent magnetic (for example, 20X5 (N35) is greater than 20X5 (N52)). The conclusion also verified the structural principle of the related sensing model (referred as Hall voltage measuring modeling), which has been detailed in our published paper [34]. The Hall sensor saturated all the time when the permanent magnet has too strong magnetism because of the measurement range of Hall sensor (SS94A1F, made by Honeywell International, Morris Plains, NJ" USA.) is −100 Gauss to 100 Gauss. Therefore, the size of the permanent magnet corresponding to the largest location in the figure is selected, as shown in Table 1. However, it should be pointed out that although the structure of the sensing unit has been optimized, the measurement principle remains unchanged.

Experimental Data Acquisition
The mutual inductance voltage and Hall voltage data are collected as benchmark data for establishing the new mathematical model when calculating the displacements. A photograph of data acquisition experimental platform is shown in Figure 6. It consists of host computer, five-axis motion device, controller of motion device and sensor prototype (mainly includes a measuring unit and its communication setting with the host computer). The host computer communicates with the sensor prototype through RS-485 and communicates with the controller through RS-232. Figure 7 shows schematic diagram of the above experimental device. Sensing unit I is fixed on the smooth surface of the test bench. Three stepper motors (A, B and C) drive sensing unit II to do translational motion in X, Y and Z axes respectively, which can simulate three-dimensional space motion. At the same time, controlled by motor D, the rotation-axis H can drive the sensing unit II to rotate axially (θ 0 ), simulating the case of inclination. The motion controller can receive the command from the host computer to control the motion of five-axis motion device. However, it should be pointed out, for convenience of experiment and data analysis, the relative direction angle φ has been set as 0 in this experiment process, that is, ∆X = ∆R and ∆Y = 0 according to Equations (4) and (5). Therefore, the horizontal displacement ∆R, vertical displacement ∆Z and tilt angle θ 0 between adjacent sensing units can be changed by driving motor A, C and D, respectively, so as to simulate the deep displacement of rock and soil mass.
computer to control the motion of five-axis motion device. However, it should be pointed out, for convenience of experiment and data analysis, the relative direction angle ϕ has been set as 0 in this experiment process, that is, ∆X = ∆R and ∆Y = 0 according to Equations (4) and (5). Therefore, the horizontal displacement ∆R, vertical displacement ∆Z and tilt angle θ0 between adjacent sensing units can be changed by driving motor A, C and D, respectively, so as to simulate the deep displacement of rock and soil mass.  For any measurement unit, the reference points taken by the experiment are the bottom edge points of the upper sensing unit and the upper edge points of the lower sensing unit, namely, points P and O, as shown in Figure 3. The data acquisition process of mutual inductance voltage and Hall voltage is as follows. Firstly, the host computer sends instructions to the reference end of measuring unit (Sensing unit I in Figures 2 and 3) to generate sinusoidal waves which will input into the solenoid of Sensing unit I. Secondly, the host computer sends instructions to the measuring end of measuring unit (Sensing unit II in Figures 2 and 3) to measure the mutual inductance voltage Ur which generated in the solenoid of sensing unit II. Thirdly, the host computer sends instructions to the reference end of measuring unit to turn off the sinusoidal wave so as to reduce the interference of Hall measurement. Finally, the host computer sends instructions to the measurement end to measure Hall voltage Uh. When one measurement of Ur and Uh is completed, an instruction to change the horizontal or vertical displacement is sent to the five-axis motion controller. The displacement change is 1 mm each time. The time interval T = 3 s for each data sampling point (one data acquisition of mutual inductance voltage and Hall voltage), and the experimental process and data storage are controlled by the host computer.
According to the different tilt angle θ0, many batches of mutual inductance and Hall voltage data acquisition experiments were carried out. In each batch of data acquisition process, the five-axis motive device moves upward 80 mm vertically and outward 80 mm horizontally. Each data acquisition experiment takes about 8 h. The initial vertical distance between these two sensing units For any measurement unit, the reference points taken by the experiment are the bottom edge points of the upper sensing unit and the upper edge points of the lower sensing unit, namely, points P and O, as shown in Figure 3. The data acquisition process of mutual inductance voltage and Hall voltage is as follows. Firstly, the host computer sends instructions to the reference end of measuring unit (Sensing unit I in Figures 2 and 3) to generate sinusoidal waves which will input into the solenoid of Sensing unit I. Secondly, the host computer sends instructions to the measuring end of measuring unit (Sensing unit II in Figures 2 and 3) to measure the mutual inductance voltage U r which generated in the solenoid of sensing unit II. Thirdly, the host computer sends instructions to the reference end of measuring unit to turn off the sinusoidal wave so as to reduce the interference of Hall measurement. Finally, the host computer sends instructions to the measurement end to measure Hall voltage U h . When one measurement of U r and U h is completed, an instruction to change the horizontal or vertical displacement is sent to the five-axis motion controller. The displacement change is 1 mm each time. The time interval T = 3 s for each data sampling point (one data acquisition of mutual inductance voltage and Hall voltage), and the experimental process and data storage are controlled by the host computer.
According to the different tilt angle θ 0 , many batches of mutual inductance and Hall voltage data acquisition experiments were carried out. In each batch of data acquisition process, the five-axis motive device moves upward 80 mm vertically and outward 80 mm horizontally. Each data acquisition experiment takes about 8 h. The initial vertical distance between these two sensing units can be set flexibly according to experimental requirements. Here it is 15 mm. The whole experiment is controlled by host computer. Figures 8-10 are 3D curves of mutual inductance voltage and Hall voltage data measured at tilt angles of 0, 10 and 20 degrees, respectively. Figures 8a, 9a and 10a show the mutual inductance voltage data. Figures 8b, 9b and 10b show the Hall voltage data. Analysis of Figure 8 shows that, under 0 degree of tilt angle (θ0 = 0 o ), the mutual inductance voltage and Hall voltage measured at a certain time correspond to many (x, y, U) points in the graph, where x is horizontal displacement, y is vertical displacement and U is the corresponding voltage Analysis of Figure 8 shows that, under 0 degree of tilt angle (θ0 = 0 o ), the mutual inductance voltage and Hall voltage measured at a certain time correspond to many (x, y, U) points in the graph, where x is horizontal displacement, y is vertical displacement and U is the corresponding voltage Analysis of Figure 8 shows that, under 0 degree of tilt angle (θ0 = 0 o ), the mutual inductance voltage and Hall voltage measured at a certain time correspond to many (x, y, U) points in the graph, where x is horizontal displacement, y is vertical displacement and U is the corresponding voltage Analysis of Figure 8 shows that, under 0 degree of tilt angle (θ 0 = 0 • ), the mutual inductance voltage and Hall voltage measured at a certain time correspond to many (x, y, U) points in the graph, where x is horizontal displacement, y is vertical displacement and U is the corresponding voltage value. These points can be connected by a smooth curve to form a voltage contour. Thus, all the mutual inductance voltage and Hall voltage data measured at 0 degrees inclination correspond to a set of contours, among them each voltage data corresponds to one voltage contour, and the x-axis of the contour corresponds to the horizontal displacement, the y-axis corresponds to the vertical displacement, as shown in Figure 11. The contour shows that the single parameter of mutual inductance voltage or Hall voltage can hardly achieve effective deep displacement measurement of rock and soil mass. When Sensing unit II moves a specific displacement relative to Sensing unit I (e.g., ∆R = 10 mm and ∆Z = 10 mm), a corresponding mutual inductance voltage contour and Hall voltage contour can be obtained in Figure 11a,b, respectively. These two contours must have an intersection point (10,10), which is the spatial position of Sensing unit II relative to Sensing unit I. Its horizontal and vertical coordinates correspond to the relative horizontal and vertical displacements between these two sensing units. Figure 12a,b are the contour maps of mutual inductance voltage and Hall voltage calculated respectively from the mutual inductance voltage and Hall voltage data in Figure 10a,b at the inclined angle of 20 degrees. These contours intersect in pairs and can also be used to determine the relative horizontal and vertical displacements between Sensing units I and II. Inspired by this, we attempt to establish the deep displacement measurement model of rock and soil mass by voltage contour modeling. value. These points can be connected by a smooth curve to form a voltage contour. Thus, all the mutual inductance voltage and Hall voltage data measured at 0 degrees inclination correspond to a set of contours, among them each voltage data corresponds to one voltage contour, and the x-axis of the contour corresponds to the horizontal displacement, the y-axis corresponds to the vertical displacement, as shown in Figure 11. The contour shows that the single parameter of mutual inductance voltage or Hall voltage can hardly achieve effective deep displacement measurement of rock and soil mass. When Sensing unit II moves a specific displacement relative to Sensing unit I (e.g., ΔR = 10 mm and ΔZ = 10 mm), a corresponding mutual inductance voltage contour and Hall voltage contour can be obtained in Figure 11a and 11b, respectively. These two contours must have an intersection point (10,10), which is the spatial position of Sensing unit II relative to Sensing unit I. Its horizontal and vertical coordinates correspond to the relative horizontal and vertical displacements between these two sensing units. Figure 12a and 12b are the contour maps of mutual inductance voltage and Hall voltage calculated respectively from the mutual inductance voltage and Hall voltage data in Figure 10a and b at the inclined angle of 20 degrees. These contours intersect in pairs and can also be used to determine the relative horizontal and vertical displacements between Sensing units I and II. Inspired by this, we attempt to establish the deep displacement measurement model of rock and soil mass by voltage contour modeling.   value. These points can be connected by a smooth curve to form a voltage contour. Thus, all the mutual inductance voltage and Hall voltage data measured at 0 degrees inclination correspond to a set of contours, among them each voltage data corresponds to one voltage contour, and the x-axis of the contour corresponds to the horizontal displacement, the y-axis corresponds to the vertical displacement, as shown in Figure 11. The contour shows that the single parameter of mutual inductance voltage or Hall voltage can hardly achieve effective deep displacement measurement of rock and soil mass. When Sensing unit II moves a specific displacement relative to Sensing unit I (e.g., ΔR = 10 mm and ΔZ = 10 mm), a corresponding mutual inductance voltage contour and Hall voltage contour can be obtained in Figure 11a and 11b, respectively. These two contours must have an intersection point (10,10), which is the spatial position of Sensing unit II relative to Sensing unit I. Its horizontal and vertical coordinates correspond to the relative horizontal and vertical displacements between these two sensing units. Figure 12a and 12b are the contour maps of mutual inductance voltage and Hall voltage calculated respectively from the mutual inductance voltage and Hall voltage data in Figure 10a and b at the inclined angle of 20 degrees. These contours intersect in pairs and can also be used to determine the relative horizontal and vertical displacements between Sensing units I and II. Inspired by this, we attempt to establish the deep displacement measurement model of rock and soil mass by voltage contour modeling.  Horizontal displacement/mm Vertical displacement/mm

Data Modeling
In the previous section, it is concluded that the intersection points of mutual inductance voltage contour and Hall voltage contour correspond to the relative displacements of adjacent sensing units. Therefore, the authors establish a specific voltage contour model to measure the deep displacement of rock and soil mass. In order to establish the contour model of mutual inductance voltage and Hall voltage, it is necessary to extract the discrete points of the voltage contour first, and then perform curve fitting on these extracted discrete points to obtain the contour function. Finally, the intersection points of the mutual inductance voltage curve and Hall voltage curve are solved to obtain the relative displacement of the adjacent sensing units.
The calibrated reference voltage data and the corresponding tilt angle θ 0 are stored in two-dimensional array. Each tilt angle θ 0 corresponds to two 80 * 80 two-dimensional arrays of θ-X-Y-U r and θ-X-Y-U h . The row of the array represents the horizontal displacement is n i *1(where n i is the label of the array rows, numbered 0-n, and the voltage data are collected once every 1 mm in the experiment), and the column of the array represents the vertical displacement is m i * 1. Now we will take the mutual inductance voltage contour modeling as an example to introduce the displacement solution process.
Firstly, according to the mutual inductance voltage U r and tilt angle θ 0 measured by the microcomputer and uploaded to the host computer, the corresponding array θ-X-Y-U r is searched, the first row of the array (x r1 = 0 × 1 = 0) is traversed and the value of the column corresponding to U r is found (y r1 = m i × 1), thus getting the first contour discrete point. Then, traverse each row of the array to get a set of discrete points of the contour: When traversing a row of array, if the corresponding voltage U r is intermediate between the two adjacent voltage values in the array, that is, U r1 ≥ U r ≥ U r2 , where U r1 corresponds to the coordinate (x r1 , y r1 ), U r2 to (x r1 , y r2 ) and y r2 − y r1 = 1 (the displacement step value is 1 mm), then it is assumed that in the interval [y r1 , y r2 ], there is a linear relationship between the vertical displacement and the voltage, i.e., y r = y r1 + (U r1 − U r )/(U r1 − U r2 ). By substituting (U r1 , y r1 ), (U r2 , y r2 ) and Ur into this expression, the values of y r can be solved, and the discrete point (x r , y r ) can be obtained.
According to the mutual inductance voltage, Hall voltage and tilt angle data uploaded by the microcomputer, find the corresponding arrays of θ 0 -X-Y-U r and θ 0 -X-Y-U h , traverse the arrays, and get the discrete points of the mutual inductance voltage contour and Hall voltage contour. To obtain the relative displacement of adjacent sensing units, it is required to perform curve fitting on the obtained discrete points to obtain a voltage contour function. The authors use a polynomial least squares method to fit the curve function.

Experiment and Analysis
First of all, we will conduct experiments to simulate the three-dimensional displacement of

Experiment and Analysis
First of all, we will conduct experiments to simulate the three-dimensional displacement of Sensing unit II relative to Sensing unit I in the deep rock and soil mass. Applying the new

Experiment and Analysis
First of all, we will conduct experiments to simulate the three-dimensional displacement of Sensing unit II relative to Sensing unit I in the deep rock and soil mass. Applying the new measurement model, the measured values of mutual inductance voltage, Hall voltage and tilt measuring voltage of between two adjacent sensing units can be solved as the relative horizontal displacement, vertical displacement and tilt angle in real-time. In the verification experiment, Sensing unit II is moved to be displaced relative to Sensing unit I through the motion controller, the displacement is recorded, and the mutual inductance voltage and Hall voltage at these corresponding points are measured. Experiments are carried out under different inclination angles and displacements. Among them, the change range of tilt angle is 0 • -20 • , that of horizontal displacement is 0-80 mm and that of vertical displacement is 0-80 mm.
We set up the 3D displacement measurement platform as shown in Figures 6 and 7 to simulate the deformation of the measured object, and collect the mutual inductance voltage data and Hall voltage data as the benchmark data to establish the mathematical model when calculating the displacement. The experimental platform is composed of host computer, five-axis motion device with its controller and sensor prototype (measuring unit). Figure 15 shows a block diagram for solving the displacement based on the measured voltage. In the diagram, the discrete points of mutual inductance voltage contour and Hall voltage contour are obtained by inquiring database arrays according to the mutual inductance voltage, Hall voltage and tilt angle measured by Sensing unit II. Then these discrete points are fitted by polynomial least square method. Finally, the intersection points of the two fitting curves are obtained, and the solutions to x and f (x) are the horizontal and vertical displacements of these two adjacent sensing units.
Sensors 2020, 20,1689 15 of 20 measurement model, the measured values of mutual inductance voltage, Hall voltage and tilt measuring voltage of between two adjacent sensing units can be solved as the relative horizontal displacement, vertical displacement and tilt angle in real-time. In the verification experiment, Sensing unit II is moved to be displaced relative to Sensing unit I through the motion controller, the displacement is recorded, and the mutual inductance voltage and Hall voltage at these corresponding points are measured. Experiments are carried out under different inclination angles and displacements. Among them, the change range of tilt angle is 0°-20°, that of horizontal displacement is 0-80 mm and that of vertical displacement is 0-80 mm. We set up the 3D displacement measurement platform as shown in Figures 6 and 7 to simulate the deformation of the measured object, and collect the mutual inductance voltage data and Hall voltage data as the benchmark data to establish the mathematical model when calculating the displacement. The experimental platform is composed of host computer, five-axis motion device with its controller and sensor prototype (measuring unit). Figure 15 shows a block diagram for solving the displacement based on the measured voltage. In the diagram, the discrete points of mutual inductance voltage contour and Hall voltage contour are obtained by inquiring database arrays according to the mutual inductance voltage, Hall voltage and tilt angle measured by Sensing unit II. Then these discrete points are fitted by polynomial least square method. Finally, the intersection points of the two fitting curves are obtained, and the solutions to x and f(x) are the horizontal and vertical displacements of these two adjacent sensing units.  To fully evaluate the validity and accuracy of the new measurement model, we compare the experimental testing results with the theoretical modeling calculation results under different tilt angles (θ0). Under the same tilt angle, the variation ranges of horizontal displacement and vertical displacement both are 0-50 mm, and the variation interval is 1 mm. Tables 2, 3 Tables 2, 3   To fully evaluate the validity and accuracy of the new measurement model, we compare the experimental testing results with the theoretical modeling calculation results under different tilt angles (θ 0 ). Under the same tilt angle, the variation ranges of horizontal displacement and vertical displacement both are 0-50 mm, and the variation interval is 1 mm. Tables 2-4 show the comparison results under the conditions of 0 • , 10 • and 20 • of θ 0 . Considering limitation of article length, to each table only 18 groups of combined data of [x, y] are randomly selected for comparative analysis. In each table, x and y represents the horizontal displacement and vertical displacement between Sensing units I and II, respectively. The first column of [x, y] are the experimental testing values, the second column are the theoretical calculated values, and the third column are the absolute errors between them. We made statistical analysis on the data in Tables 2-4. It indicates that when θ 0 = 0 • , the relative and absolute mean errors of horizontal displacement (x) are −0.352 mm and 0.605 mm, and the relative and absolute mean errors of vertical displacement (y) are 0.210 mm and 0.388 mm, respectively. When θ 0 = 10 • , they are −0.236 mm and 0.528 mm for x, 0.158 mm and 0.482 mm for y, respectively. When θ 0 = 20 • , they are 0.197 mm and0.582 mm for x, and 0.033 mm and 0.394 mm for y, respectively. Meanwhile, under each table, the absolute errors of horizontal displacement and vertical displacement are all less than 1.5 mm. Combined with more detailed verification experiment results carried out by specifying more tilt angles, while the measured x and y continuously changed, some common conclusions can be drawn: (1) Variations of tilt angle has little effect on the measuring errors of horizontal and vertical displacements (x, y); (2) when the measurement ranges of horizontal and vertical displacement are 0-50 mm, the absolute measurement errors of them both are less than 1.5 mm; (3) the measurement accuracy ofvertical displacement is slightly higher than that of horizontal displacement.

Conclusions
This paper mainly expands the effective measurement range of the three-dimensional displacement measurement sensor designed in the previous research, optimizes the sensor structure, redesigns the data acquisition experiment and rebuilds the mathematical model in view of the complex environment inside the rock and soil. Based on the theoretical analysis and experimental results, the following conclusions are obtained.
As described above, we optimized the structure of the sensing unit through 3D printing and other technologies, and improved the shape and material parameters of the permanent magnet through a large number of experiments. The results show that the improved sensor has better waterproof, anticorrosive and compressive-resistance properties, and is more suitable for the actual needs of deep displacement measurement of rock and soil mass. More importantly, the combination of structural optimization and updated measurement model broadens the measurement range of the sensing unit from 0-30 mm to 0-50 mm.
According to the different tilt angle, many batches of mutual inductance and Hall voltage data acquisition experiments have been conducted. Through in-depth analysis of the experimental data, based on the data query algorithm and the polynomial least square curve fitting theory, a new mathematical model for 3D measurement of deep displacement has been proposed. By virtue of it, the output values of mutual inductance voltage, Hall voltage and tilt measuring voltage measured by the sensing units can be converted into the changes of relative horizontal displacement, vertical displacement and axial tilt angle between any two adjacent sensing units in real time. Analyzing the results, the measuring errors of horizontal displacement and vertical displacement are tested to be 0-1.5 mm. In our published paper [35], the measurement model realized the joint inversion of deep horizontal displacement and vertical displacement for the proposed 3D sensor with measuring errors of 0-3 mm. Compared with the previous model, the measurement error of this new model has reduced by half.
It shows that our revised deep displacement 3D measuring sensor, based on the specific flexible array structure design of sensing units, after structural optimization and measurement algorithm improvement, has such advantages as higher measurement accuracy, larger measurement range, better waterproof, anti-corrosion and pressure resistance performance. It can better meet the needs of high-precision monitoring at the initial stage of rock and soil deformation and large deformation monitoring at the rapid change and imminent-sliding stage. It is more competitive in deep displacement 3D monitoring and geo-hazard prediction and early warming than most of the existing deep displacement monitoring instruments, such as inclinometers, settlement gauges, extensometers and optical fiber sensors.
In order to be more consistent with the actual landslide state and more accurate in early warning and prediction of landslide accidents, the measurement range of the device should still be considered in the future research. The research group is considering using different materials and different shapes of permanent magnets to increase the measurement range of the system. At the same time, we consider to reduce the measurement error and improve the system accuracy by optimizing the modeling scheme and making interference analysis, such as temperature, conductivity, geotechnical geology, water content and magnetoelectric interface.