Method for simultaneously measuring displacement and rotation angle of structure by fusing millimeter wave radar and inertial sensor

Abstract

The invention relates to the technical field of structure displacement and rotation angle measurement, in particular to a method for simultaneously measuring structure displacement and rotation angle by fusing a millimeter-wave radar and an inertial sensor, which comprises the following steps of: arranging the millimeter-wave radar and the inertial sensor at a structure response measurement position; based on the millimeter-wave radar, measuring the horizontal displacement of the structure under the consideration of the influence of the rotation angle; based on the acceleration data measured by the inertial sensor, the structural displacement measured by the millimeter wave radar is fused, and the rotation angle of the structure is preliminarily calculated; correcting a direction conversion factor required in the above steps based on the preliminarily calculated rotation angle and angular velocity data measured by an inertial sensor; based on the direction conversion factor, converting the LOS (line of sight) displacement measured by the radar into structural horizontal displacement, and re-optimizing the preliminarily estimated corner data; and constructing an error optimization function, fusing the preliminarily calculated rotation angle and the angular velocity measured by the inertial sensor, and calculating the final rotation angle of the structure.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of structural displacement and rotation angle measurement, in particular to a method for simultaneously measuring structural displacement and rotation angle by fusing millimeter wave radar and inertial sensor. BACKGROUND

[0002] Structural health monitoring plays a crucial role in ensuring the safety, reliability, and maintainability of civil infrastructures such as bridges, wind turbine towers, and high-rise buildings. Among various structural response parameters, displacement and rotation angle measurements are of particular importance as they directly reflect the deformation and motion characteristics of structures under external loads and are key indicators for performance evaluation and safety assessment. Accurate and continuous monitoring of translational and rotational responses not only helps to detect potential damage at an early stage but also provides important data for model updating, health assessment, and predictive maintenance. Therefore, developing high-precision structural displacement and rotation angle synchronous measurement technology has important practical significance and engineering value.

[0003] Currently, various techniques can be used for structural displacement measurement, each with its unique advantages and inherent limitations. Linear variable differential transformers have high measurement accuracy and stability, but are limited in their applicability under field conditions due to their measurement range and requirement for direct physical installation on the structure. Global navigation satellite systems (GNSS) provide absolute positioning suitable for long-term monitoring of large-scale structural displacement; however, their low sampling rate and sensitivity to signal blockage and multipath effects limit their effectiveness in capturing high-frequency dynamic responses. Accelerometers are widely used for dynamic response measurement, allowing displacement measurement through double integration of acceleration data, covering a wide frequency range. However, they are prone to cumulative integration errors and low-frequency drift, especially during long-term monitoring. Vision-based systems enable high spatial resolution non-contact measurement, but are highly sensitive to environmental factors such as lighting conditions, weather changes, and line-of-sight obstruction, which limits their reliability and applicability in complex field scenarios. In recent years, radar-based technology, particularly millimeter wave radar, has received widespread attention due to its remote, all-weather, non-contact operation capabilities. Despite these advantages, radar systems often face challenges such as phase wrapping at large displacements and the need for manual correction to convert line-of-sight (LOS) displacement measurements to actual structural displacement along the true vibration direction.

[0004] In terms of rotation angle measurement, tiltmeters are widely used due to their small size and easy installation. Among them, tiltmeters based on micro-electro-mechanical systems (MEMS) are the most commonly used in structural health monitoring due to their low cost, small size and easy integration into monitoring systems. These sensors mainly measure the tilt angle by detecting the projections of the gravitational acceleration along different axes. In static or low dynamic conditions, this method can produce quite accurate rotation angle measurements. However, under dynamic conditions, their performance can deteriorate significantly. When the structure is subjected to vibration or motion, the accelerometers inside the tiltmeter will also sense the inertial acceleration, which is difficult to separate from the gravity component. This coupling effect will introduce errors in the rotation angle measurement, especially in cases involving high-frequency or transient motion. Gyroscopes provide an alternative rotation angle measurement method by integrating angular velocity, but are prone to drift due to measurement noise and uncertain initial conditions. In addition, rotation angle can be inferred from strain measurements based on virtual work theory; however, this method requires dense step strain sensors and is usually limited by the high noise level in strain data.

[0005] Despite the significant progress in displacement and rotation angle measurement techniques, there are still many challenges in accurately and synchronously measuring both parameters due to signal coupling and mutual interference issues. For example, structural rotation angle can cause significant errors in vision-based displacement measurement and introduce gravity-related components in accelerometer output, distorting the true dynamic response. Therefore, handling displacement and rotation angle separately, ignoring the coupling between them, will lead to cumulative measurement errors and inaccuracies. Recently, multi-sensor fusion techniques have been used to improve measurement accuracy and robustness by complementary sensing modalities, thus overcoming these limitations. Currently, there are various fusion strategies, such as combining accelerometers with strain sensors, vision systems, laser radars or millimeter wave radars for displacement measurement, and integrating accelerometers with gyroscopes or fusing inertial measurement units (IMUs) with GNSS for rotation angle measurement. However, methods that consider the inherent coupling and interaction between displacement and rotation angle and simultaneously measure both are still lacking. SUMMARY

[0006] In view of the defects in the prior art, the present application provides a method for simultaneously measuring structural displacement and rotation angle by fusing millimeter wave radar and inertial sensors.

[0007] To achieve the above-mentioned purpose, the present application provides the following technical solutions:

[0008] A method for simultaneously measuring structural displacement and rotation angle by fusing millimeter wave radar and inertial sensors, characterized in that it comprises the following steps:

[0009] Step one, millimeter wave radar and inertial sensors are arranged at the structural response measurement position, and the LOS displacement and the structural horizontal displacement are measured based on the millimeter wave radar considering the influence of the rotation angle;

[0010] Step two, based on the acceleration data measured by the inertial sensor, fuse the structure displacement measured by the millimeter wave radar to preliminarily calculate the rotation angle of the structure;

[0011] Step three, based on the preliminarily calculated rotation angle and the angular velocity data measured by the inertial sensor, correct the direction conversion factor required in step one and step two;

[0012] Step four, based on the above direction conversion factor, convert the LOS displacement measured by the radar into the structure horizontal displacement, and re-optimize the preliminarily estimated rotation angle data;

[0013] Step five, build an error optimization function to fuse the preliminarily calculated rotation angle and the angular velocity measured by the inertial sensor to calculate the final rotation angle of the structure.

[0014] Specifically, in step one, the specific steps of measuring the structure horizontal displacement by the millimeter wave radar considering the influence of the rotation angle include the following:

[0015] Step 1, the millimeter wave radar transmits a frequency-modulated continuous wave (FMCW) signal:

[0016]

[0017] Where T k (t) represents the FMCW signal, f s represents the starting frequency of the FMCW signal, K represents the frequency slope, i.e. the ratio of the frequency bandwidth B to the signal duration T c , t represents time, and j represents the imaginary unit.

[0018] Receive the reflected echo signal:

[0019]

[0020] Where R k (t) represents the echo signal, δ represents the signal attenuation factor, Δt represents the time delay between the transmitted signal and the reflected signal, the subscripts m and k represent the mth target and the kth time step, respectively, and Q represents the number of all reflected targets received;

[0021] Mix the received signal with the transmitted signal to generate an intermediate frequency signal:

[0022]

[0023] Where IF k (t) represents the intermediate frequency signal, and ω respectively represent the phase and frequency of the intermediate frequency signal;

[0024] Step 2, Fourier transform the intermediate frequency signal, use peak detection algorithm to identify the peak in the amplitude spectrum as the multiple targets detected by radar, and select the target with the largest amplitude for displacement monitoring, then extract the phase information corresponding to the selected target, denoted as According to the characteristics that the phase is limited in the range of [-π, π], a phase unwrapping algorithm is used to process it to obtain the unwrapped phase

[0025] Step 3, based on the unwrapped phase Calculate the time delay Δt s,k :

[0026]

[0027] Step 4, according to the relationship between distance and time delay, the displacement of radar LOS can be obtained by Δt s,k :

[0028]

[0029] Where c represents the speed of light;

[0030] Step 5, initially set the value of the direction conversion factor γ to 0.5, and convert the radar LOS displacement to structural horizontal displacement:

[0031] u k =γu LOS,k (6)

[0032] Where the direction conversion factor γ is further corrected in step three.

[0033] Specifically, in step two, under the condition that the structure has horizontal vibration, the acceleration measured at the top of the structure is represented as:

[0034] a t,k =gsin(θ k )+a x,k cos(θ k ) (7)

[0035] a r,k =gcos(θ k )+a x,k sin(θ k ) (8)

[0036] Where a t,k and a r,k represent the tangential acceleration and radial acceleration measured by the accelerometer respectively, g is the acceleration of gravity, a x,k is the true horizontal acceleration of the structure, and θ k is the preliminary estimated rotation angle;

[0037] Step one: obtain the structure horizontal displacement u based on radar data k Step two: estimate the structure horizontal acceleration by numerical differentiation Step three: replace a in equation (7) and (8) with Step four: obtain the structure horizontal displacement u based on the corrected direction conversion factor γ x,k The following matrix equation can be obtained:

[0038]

[0039] Further, the preliminary estimated structure rotation angle θ k is:

[0040]

[0041] Specifically, in step three, a direction conversion factor γ is needed to convert the LOS displacement into the structure horizontal displacement, and the initial value of γ is set to 0.5 in step one. The specific steps for correcting the direction conversion factor γ considering the structure rotation angle are as follows:

[0042] Step 1: apply a band-pass filter to the preliminary estimated rotation angle θ k and time-integrate the angular velocity, then pass it through the same band-pass filter;

[0043] Step 2: calculate the root mean square error between the two band-pass filtered rotation angle data;

[0044] Step 3: start with a value of 0.5 for the direction conversion factor γ and gradually increase it to 2 in steps of 0.01, and repeat steps one and two to obtain the preliminary estimated rotation angle;

[0045] Step 4: repeat steps 1 and 2 to obtain the relationship curve between the root mean square error and the direction conversion factor;

[0046] Step 5: the value of γ corresponding to the minimum root mean square error is the final direction conversion factor.

[0047] Preferably, in step 1, the main parameter to be determined is the cutoff frequency of the band-pass filter, which should be selected according to the main vibration frequency of the measured structure.

[0048] Specifically, in step four, based on the corrected direction conversion factor γ, the LOS displacement is converted into the structure horizontal displacement.

[0049] Further, based on the structure horizontal displacement obtained by the corrected direction conversion factor, the acceleration data of the inertial sensor is fused in step two to optimize the preliminary estimated rotation angle θ k .

[0050] Preferably, in step two, for the preliminary estimated rotation angle θ k, the noise in the accelerometer measurement is not considered, and due to the differentiation process, a x,r may contain a large amount of high-frequency noise, therefore, θ k may have considerable noise in the high-frequency range.

[0051] Further, in step five, the preliminary estimated rotation angle θ k and the gyroscope measured angular velocity ω(t) are combined to better measure the structure rotation angle, the specific steps are as follows:

[0052] Step 1, in a given time window [T1, T2], the error function Π E (θ(t)) is defined as follows:

[0053]

[0054] where and represent the measured rotation angle and angular velocity respectively, the first term of equation (11) represents the squared error between the actual angular velocity and the measured angular velocity, and the second term represents the squared error between the actual rotation angle and the measured rotation angle, the parameter λ adjusts the relative weight of the two parts;

[0055] Step 2, because the actual situation and is sampled at a constant time step Δt, equation (11) is discretized using the trapezoidal rule, and in a given time window [(k-N+1)Δt, (k+N+1)Δt], equation (11) is expressed in matrix form as:

[0056]

[0057] where θ、 and are the actual rotation angle vector [θ k-N , θ k+N …, θ k+N ] T , the measured rotation angle vector and the measured angular velocity vector The matrices L c and L a are defined as follows:

[0058]

[0059] The dimensions of L c and L a are (2N-1)×(2N+1) and (2N-1)×(2N-1) respectively, here, L c uses a two-point central difference formula to discretize the derivative of θ;

[0060] Step 3, the optimal rotation angle is obtained by minimizing Π E (θ) using least square method:

[0061] θ * = 2Δt(L T L+λ 2 I) -1 L T L a ω+λ 2 (L T L+λ 2 I) -1 θ=M ω ω+M θ θ (14)

[0062] where θ * is the optimal rotation angle vector L=L a L c Here, only the rotation angle estimate at the center of the window is kept to maximize the estimation accuracy,

[0063]

[0064] where and represent the Nth column of M ω and the (N-1)th column of M θ respectively, and by using the sliding window algorithm, continuous estimation of the rotation angle can be achieved.

[0065] Further, in step 3, the parameters λ and N need to be determined, where there are a large number of zero elements in the vectors and which do not participate in angle measurement, so only the non-zero elements are analyzed, where and are denoted as and respectively. For parameter N, as the value of N increases, and both converge, and their limit values are denoted as A θ and A ω respectively, so the minimum N value that satisfies the following condition is selected:

[0066]

[0067] The final measured rotation angle is composed of two parts: the first part comes from the angular velocity, and the second part comes from the initial measured rotation angle, where acts as an integrator and a high-pass filter, and The low-pass filter function is played by the value of lambda, which depends on the characteristics of the gyroscope, and the selected value of lambda should be verified in the laboratory to ensure that the low-frequency drift caused by the angular velocity integration is completely suppressed.

[0068] Compared with the prior art, the application has the beneficial effects that:

[0069] 1、 The application can accurately estimate the structure displacement even in the case of corner, and is more suitable for engineering practical application.

[0070] 2、 The method of radar-assisted estimation of structure corner from accelerometer data considers the influence of structure translation acceleration on the estimation of inclination, and can better estimate the corner.

[0071] 3、 The initial estimated corner and angular velocity are fused through error optimization, and the accuracy of the corner estimation is significantly improved.

[0072] 4、 The adaptive correction of the direction conversion factor is realized by fusing the accelerometer, gyroscope data and structure displacement data measured by the radar.

[0073] 5、 In the application, compared with point scanning sensors such as single-line laser systems, the millimeter wave radar adopts a fan scanning mechanism, and has strong robustness to slight rotation of the sensor. In the fan scanning, the radar covers a wide angle range in a single scanning process, and in this angle range, the radar can obtain target information from multiple directions. Due to this spatial redundancy, even if the radar sensor experiences slight rotation, as long as the target is still within the fan scanning coverage range, the target can still be reliably detected. BRIEF DESCRIPTION OF DRAWINGS

[0074] Figure 1 The flowchart of the application is as follows:

[0075] Figure 2 The schematic diagram of the influence of the structure corner on the radar-based structure displacement measurement in the embodiment of the application is as follows:

[0076] Figure 3 The schematic diagram of the relationship between the accelerometer measurement and the actual structure acceleration in the embodiment of the application is as follows:

[0077] Figure 4 The theoretical derivation and algorithm flowchart of the direction conversion factor correction in the embodiment of the application are as follows:

[0078] Figure 5 The element distribution example diagram in the embodiment of the application when lambda = 0.025 and N = 100 is as follows: And The element distribution example diagram in the embodiment of the application when lambda = 0.025 and N = 100 is as follows:

[0079] Figure 6Fig. 2 is a schematic diagram of the influence of N value on the non-zero elements in and Fig. 3 is a schematic diagram of the influence of N value on the maximum values in

[0080] Figure 7 Fig. 4 is a schematic diagram of the influence of N value on the non-zero elements in and Fig. 5 is a schematic diagram of the influence of N value on the maximum values in

[0081] Figure 8 Fig. 6 is a schematic diagram of the experimental setup on the cantilever beam in the embodiment of the present application;

[0082] Figure 9 Fig. 7 is a schematic diagram of the displacement measurement results of the cantilever beam in the embodiment of the present application;

[0083] Figure 10 Fig. 8 is a schematic diagram of the rotation angle measurement results of the cantilever beam in the embodiment of the present application; DETAILED DESCRIPTION

[0084] The technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor fall within the protection scope of the present application.

[0085] Embodiment 1

[0086] First, in order to better understand the concept of the present application, it is necessary to understand the advantages of radar over point scanning sensors such as single-line laser systems when measuring structure rotation angle and displacement simultaneously. Millimeter wave radar adopts a fan scanning mechanism, which has strong robustness to slight rotation angle of the sensor. In fan scanning, the radar covers a wide angle range in a single scanning process, usually covering ±30° to ±60°, or even wider. Within this angle range, the radar can obtain target information from multiple directions. Due to this spatial redundancy, even if the radar sensor experiences slight rotation angle, as long as the target is still within the fan scanning coverage range, the target can still be reliably detected.

[0087] As Figure 2The structural corner angle is shown to have an impact on radar-based structural displacement measurement. Here, the structural horizontal displacement is the quantity of interest. It is assumed that a target is initially selected from within the radar field of view for displacement measurement, and as long as the target remains within the scan range after deformation, it can continue to be tracked stably. Since the structural corner angle is usually very small, only a few degrees or less, this condition is usually met in practice. In addition, it can be assumed that the distance and line-of-sight direction of the radar to the target remain unchanged, ignoring small changes due to structural movement. Therefore, the structural displacement can be reliably measured without explicitly considering the impact of the structural corner angle.

[0088] To accurately understand the technical basis on which the present application is based, the basic working principle of a frequency-modulated continuous wave (FMCW) millimeter wave radar is briefly introduced: the millimeter wave radar transmits an FMCW signal:

[0089]

[0090] where T k (t) represents the FMCW signal, f s represents the starting frequency of the FMCW signal, K represents the frequency slope, i.e., the ratio of the frequency bandwidth B to the signal duration T c , t represents time, and j represents the imaginary unit.

[0091] and receives a reflected echo signal:

[0092]

[0093] where R k (t) represents the echo signal, δ represents the signal attenuation factor, and Δt represents the time delay between the transmitted signal and the reflected signal. Subscripts m and k represent the mth target and the kth time step, respectively, and Q represents the number of all reflected targets received.

[0094] The received signal is mixed with the transmitted signal to generate an intermediate frequency signal:

[0095]

[0096] where IF k (t) represents the intermediate frequency signal, and ω represent the phase and frequency of the intermediate frequency signal, respectively.

[0097] Then, the intermediate frequency signal is subjected to a Fourier transform, a peak detection algorithm is used to identify the peak in the amplitude spectrum, as the target detected by the radar, and the target with the largest amplitude is selected for displacement monitoring. Next, the phase corresponding to the selected target is extracted as In view of the fact that this phase is limited to the range [-π, π], a phase unwrapping algorithm is used to process it to obtain the unwrapped phase

[0098] Based on the unwrapped phase Calculate the time delay Δt s,k .

[0099]

[0100] Finally, according to the relationship between distance and time delay, Δt s,k Get the displacement between the radar and the target line of sight (LOS),

[0101]

[0102] Here, c represents the speed of light.

[0103] The initial value of the direction conversion factor γ is set to 0.5, and the LOS displacement is converted into the structural horizontal displacement through the direction conversion factor.

[0104] u k =γu LOS,k (6)

[0105] like Figure 3 Figure 2 shows the relationship between the accelerometer measurement results and the actual structural acceleration installed on the top of the structure.

[0106] a t,k =gsin(θ k )+a x,k cos(θ k ) (7)

[0107] a r,k =gcos(θ k )+a x,k sin(θ k ) (8)

[0108] where a t,k and a r,k They represent the tangential acceleration and radial acceleration measured by the accelerometer, g is the acceleration due to gravity, and a x,k is the true horizontal acceleration of the structure.

[0109] The present invention can measure the horizontal displacement of the structure through radar measurement, and estimate the value of acceleration in the x direction through numerical differentiation based on the radar displacement. By using replace Figure 3 a in the formula x,k , we can get the following matrix equation:

[0110]

[0111] Further, estimate the preliminary measured rotation angle θ k is:

[0112]

[0113] Next, because the direction conversion factor γ is introduced when converting the LOS displacement to the horizontal displacement when measuring the structure horizontal displacement based on radar, it is initially set to 0.5. The correction steps of the direction conversion factor γ are as follows: Figure 4

[0114] 1. Estimate the preliminary rotation angle θ k Apply band-pass filtering. At the same time, time-integrate the angular velocity, and then pass it through the same band-pass filter.

[0115] 2. Calculate the root mean square error between the two band-pass filtered rotation angle data.

[0116] 3. Start with the value of the direction conversion factor γ as 0.5, and gradually increase it to 2 with a step of 0.01. Repeat steps 1 and 2 to obtain the preliminary estimated rotation angle.

[0117] 4. Repeat steps 1 and 2 to obtain the relationship curve between the root mean square error and the direction conversion factor.

[0118] 5. The value of γ corresponding to the minimum root mean square error is the final direction conversion factor.

[0119] Optimize the structure horizontal displacement measured by radar based on the corrected direction conversion factor γ. Based on this structure horizontal displacement, fuse the acceleration data of the inertial sensor again to optimize the preliminary estimated rotation angle θ k .

[0120] Since the preliminary estimated rotation angle θ k of the structure does not consider the noise in the accelerometer measurement, and due to the differentiation process, a x,r may contain a large amount of high-frequency noise. Therefore, θ k may have considerable noise in the high-frequency range.

[0121] Further, combine the preliminary estimated rotation angle θ k and the angular velocity ω(t) measured by the gyroscope to better measure the structure rotation angle, and the specific steps are as follows:

[0122] 1. In a given time window [T1, T2], the error function Π E (θ(t)) is defined as follows:

[0123]

[0124] where​ and denote the measured rotation angle and angular velocity, respectively. The first term in equation (11) represents the squared error between the actual angular velocity and the measured angular velocity, and the second term represents the squared error between the actual rotation angle and the measured rotation angle. The parameter λ adjusts the relative weight of the two parts.

[0125] 2. Because of the actual situation and are sampled at constant time steps Δt, equation (11) is discretized using the trapezoidal rule, and in a given time window [(k-N+1)Δt, (k+N+1)Δt], equation (11) is expressed in matrix form as:

[0126]

[0127] where θ and are the actual rotation angle vector [θ k-N , θ k+N …, θ k+N ] T , the measured rotation angle vector and the measured angular velocity vector The matrices L c and L a are defined as follows:

[0128]

[0129] L c and L a have dimensions (2N-1)×(2N+1) and (2N-1)×(2N-1), respectively. Here, L c The derivative of θ is discretized using the two-point central difference formula.

[0130] 3. By minimizing Π E (θ), the optimal rotation angle is obtained using the least squares method:

[0131] θ * = 2Δt(L T L+λ 2 I) -1 (L T L a ω+λ 2 (L T L+λ 2 I) -1 θ = M ω ω+M θ θ (14)

[0132] where θ * is the optimal rotation angle vector L = L a L c Here, only the corner estimate at the center of the window is kept to maximize the estimation accuracy,

[0133]

[0134] where, and denote the Nth column of M ω and the (N-1)th column of M θ respectively. By employing the sliding window algorithm, continuous estimation of the corner can be achieved.

[0135] Further, in step 3, the parameters λ and N need to be determined. There are a large number of zero elements in the vectors and which do not participate in the angle measurement, such as Figure 5 . Therefore, only the non-zero elements are analyzed.

[0136] Figure 6 shows the influence of different N values on the non-zero elements of and . As shown in Figure 6 (a), the element values near the center of the vector are relatively large, while the elements on both sides are close to zero. As the N value increases, the central region of Figure 6 becomes stable, and the newly added elements at both ends can be ignored. Similar trends also appear in (b)

[0137] Further analysis of the influence of N values on the maximum values of and (labeled as and respectively) is shown in Figure 7 . As the N value increases, and converge, and their limit values are labeled as A θ and A ω respectively. Therefore, the minimum N value that satisfies the following condition is selected:

[0138]

[0139] The final measured corner angle is composed of two parts: the first part comes from the angular velocity, and the second part comes from the initial measured corner angle. Where, acts as an integrator and a high-pass filter, while The low-pass filter acts. The value of λ depends on the characteristics of the gyroscope and needs to be verified in laboratory tests to ensure that the chosen value of λ ensures complete suppression of the low-frequency drift caused by the integration of the angular velocity.

[0140] Example 2

[0141] Figures 8 to 10 A structural displacement measurement method according to an embodiment of the present application and the actual verification results using the method are shown.

[0142] The tip displacement of a cantilever beam with a total length of 600 mm was measured in a real experiment. During the test, the cantilever beam was manually excited by applying impacts, i.e. multiple pushes and releases of the beam to induce forced and free vibrations.

[0143] As Figure 8 shown, to measure the displacement of the cantilever beam tip, a millimeter-wave radar and an IMU sensor were installed on top of the cantilever beam. The radar sensor was mounted on the longitudinal surface of the cantilever beam, emitting a linear frequency-modulated signal in a direction perpendicular to the nearby wall and receiving the signal reflected from the nearby wall. The millimeter-wave radar and the IMU sensor collected data simultaneously at a sampling rate of 100 Hz.

[0144] As Figure 9 shown, the displacement measurement results in this experimental setup. The displacement measured based on the radar shows good agreement with the reference displacement measurement results, with a root mean square error (RMSE) of only 0.277 mm.

[0145] As Figure 10 shown, the rotation angle measurement results in this experimental setup are shown. The rotation angle measured by the IMU sensor’s built-in algorithm exhibits similar trends to the vision-based reference measurement; however, significant differences were observed in the experiment, with a RMSE of up to 0.434°. In contrast, the rotation angle measured by the proposed technique shows good agreement with the reference rotation angle measurement results, yielding a RMSE of only 0.057°. Compared to the built-in algorithm of the IMU, the error is reduced by 97%. The results show that the presence of the rotation angle motion of the cantilever beam tip has a negligible effect on the accuracy of the radar-based displacement measurement.

[0146] It is apparent for the person skilled in the art that the present application is not limited to the details of the above-described exemplary embodiments, but that it can be implemented in other embodiments without departing from the spirit or essential characteristics of the present application. Consequently, the embodiments are to be considered as illustrative and not restrictive and the scope of the application is to be determined only by the appended claims and their equivalents, and therefore all changes which come within the meaning and range of equivalency of the claims are to be embraced by the application. Any reference signs in the claims should not be construed as limiting the claims concerned.

Claims

1. A method for simultaneously measuring structural displacement and rotation angle by integrating millimeter-wave radar and inertial sensor, characterized in that: The steps include: Step 1: Deploy a millimeter-wave radar and an inertial sensor at the structural response measurement location, and measure the line-of-sight (LOS) displacement and the horizontal displacement of the structure based on the millimeter-wave radar while considering the influence of the rotation angle; Step 2: Based on the acceleration data measured by the inertial sensor, the structural displacement measured by the millimeter-wave radar is integrated to preliminarily calculate the rotation angle of the structure; Step 3: Based on the initially calculated rotation angle and the angular velocity data measured by the inertial sensor, the direction conversion factors required in Step 1 and Step 2 are corrected; Step 4: Based on the above direction conversion factor, the LOS displacement measured by the radar is converted into the horizontal displacement of the structure, and the acceleration data of the inertial sensor is integrated with the acceleration data of the inertial sensor in the same way as in step 2 to re-optimize the preliminary estimated angle data; Step 5: Construct an error optimization function to fuse the initially calculated rotation angle and the angular velocity measured by the inertial sensor to calculate the final rotation angle of the structure.

2. The method for simultaneously measuring structural displacement and rotation angle by integrating millimeter-wave radar and inertial sensor according to claim 1, characterized in that: In step 1, the specific steps of measuring the horizontal displacement of the structure based on the millimeter wave radar while considering the influence of the rotation angle include the following: Step 1: The millimeter wave radar transmits a frequency modulated continuous wave (FMCW) signal: Among them, T k (t) represents the FMCW signal, f s Indicates the starting frequency of the FMCW signal, K represents the frequency slope, that is, the frequency bandwidth B and the signal duration T c The ratio of , t represents time, and j represents the imaginary unit. Receive reflected echo signal: Among them, R k (t) represents the echo signal, δ represents the signal attenuation factor, Δt represents the time delay between the transmitted signal and the reflected signal, subscripts m and k represent the mth target and the kth time step respectively, and Q represents the number of all reflected targets received; Mix the received signal with the transmitted signal to generate an intermediate frequency signal: Among them, IF k (t) represents the intermediate frequency signal, and ω represent the phase and frequency of the intermediate frequency signal respectively; Step 2: Perform Fourier transform on the intermediate frequency signal, use peak detection algorithm to identify the peak in the amplitude spectrum as multiple targets detected by the radar, and select the target with the largest amplitude for displacement monitoring, and then extract the phase information corresponding to the selected target, which is recorded as In view of the fact that the phase is limited to the range of [-π,π], the phase unwrapping algorithm is used to process it and the unwrapped phase is obtained. Step 3: Based on the unwrapped phase Calculate the time delay Δt s,k : Step 4: Based on the relationship between distance and time delay, the displacement of the radar line of sight (LOS) can be expressed as Δt s,k get: Where c represents the speed of light; Step 5: Initially set the direction conversion factor γ to 0.5 and convert the radar LOS displacement into the horizontal displacement of the structure: in k =γu LOS,k (6) The direction conversion factor γ is further calibrated in step three.

3. The method for simultaneously measuring structural displacement and rotation angle by integrating millimeter-wave radar and inertial sensor according to claim 2, characterized in that: In step 2, the acceleration measured at the top of the structure in the presence of horizontal vibration is expressed as: a t,k =gsin(θ k )+a x,k cos(θ k ) (7) a r,k =gcos(θ k )+a x,k sin(θ k ) (8) where a t,k and a r,k They represent the tangential acceleration and radial acceleration measured by the accelerometer, g is the acceleration due to gravity, and a x,k is the true horizontal acceleration of the structure, θ k is the initial estimated turning angle; In step 1, the horizontal displacement u of the structure is obtained based on the radar data k , numerically differentiate it to estimate the horizontal acceleration of the structure use Replace a in equations (7) and (8) x,k , we can get the following matrix equation: Furthermore, the rotation angle θ of the structure is preliminarily estimated k for:

4. The method for simultaneously measuring structural displacement and rotation angle by integrating millimeter-wave radar and inertial sensor according to claim 3, characterized in that: In step three, a direction conversion factor γ is required to convert the LOS displacement into the horizontal displacement of the structure. In step one, the initial value of γ is set to 0.

5. The specific steps for correcting the direction conversion factor γ considering the structural rotation are as follows: Step 1: Preliminary estimation of the rotation angle θ k Apply bandpass filtering and, at the same time, time-integrate the angular velocity and then pass it through the same bandpass filter; Step 2: Calculate the root mean square error between the two bandpass filtered rotation angle data; Step 3: Start the value of the direction conversion factor γ from 0.5 and gradually increase it to 2 with a step size of 0.

01. Repeat steps 1 and 2 to obtain a preliminary estimated turning angle. Step 4: Repeat steps 1 and 2 to obtain a relationship curve between the root mean square error and the direction conversion factor; In step 5, the γ value corresponding to the minimum root mean square error is the final direction conversion factor.

5. The method for simultaneously measuring structural displacement and rotation angle by integrating millimeter-wave radar and inertial sensor according to claim 4, characterized in that: In step 1, the main parameter that needs to be determined is the cutoff frequency of the bandpass filter, which should be chosen according to the main vibration frequencies of the measured structure.

6. The method for simultaneously measuring structural displacement and rotation angle by integrating millimeter-wave radar and inertial sensor according to claim 4, characterized in that: In step 5, combined with the initial estimated rotation angle θ k The angular velocity ω(t) measured by the gyroscope is used to better measure the structural rotation angle. The specific steps are as follows: Step 1: The error function Π of the actual rotation angle (θ(t)) within a given time window [T1, T2] E (θ(t)) is defined as follows: in and Denote the measured rotation angle and angular velocity respectively. The first term of equation (11) represents the square error between the actual angular velocity and the measured angular velocity, and the second term represents the square error between the actual rotation angle and the measured rotation angle. The parameter λ adjusts the relative weight of these two parts. Step 2: Because of the actual situation and The sample is taken at a constant time step Δt, and the trapezoidal rule is used to discretize Equation (11). In a given time window [(k-N+1)Δt, (k+N+1)Δt], Equation (11) is expressed in matrix form as follows: Among them, and They are the actual rotation angle vector [θ k-N ,θ k+N …,θ k+N ] T , measure the angle vector and the measured angular velocity vector Matrix L c and L a The definition is as follows: L c and L a The dimensions are (2N-1)×(2N+1) and (2N-1)×(2N-1), respectively. Here, L c Use the two-point central difference formula to discretize the derivative of θ; Step 3: By minimizing π E (θ), and use the least squares method to obtain the optimal rotation angle: i * =2Δt(L T L+λ 2 I) -1 L T L a ω+λ 2 (L T L+λ 2 I) -1 θ=M ω ω+M θ i (14) Among them, θ * is the optimal rotation vector L=L a L c , where only the corner estimate at the center of the window is retained To maximize the estimation accuracy, in, and Represents M ω The Nth column and M θ By adopting the sliding window algorithm, the continuous estimation of the turning angle can be achieved.

7. The method for simultaneously measuring structural displacement and rotation angle by integrating millimeter-wave radar and inertial sensor according to claim 6, characterized in that: In step 3, the parameters λ and N need to be determined, where the vector and There are a lot of zero elements in , which do not participate in the angle measurement, so only the non-zero elements are analyzed, among which, and The maximum values ​​of and For parameter N, as the value of N increases, and All converge, and their limit values ​​are marked as A θ and A ω , so the minimum N value that satisfies the following conditions is selected: The final measured rotation angle consists of two parts: the first part comes from the angular velocity, and the second part comes from the initial measured rotation angle, where It plays the role of integrator and high-pass filter, and It acts as a low-pass filter. The λ value depends on the characteristics of the gyroscope and needs to be verified in laboratory tests to ensure that the selected λ value can completely suppress the low-frequency drift caused by the angular velocity integral.

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