A low complexity self-calibration method for an imu array

Abstract

The present application belongs to the technical field of calibrating inertial measurement unit, and particularly relates to a low-complexity self-calibration method for IMU array. The present application firstly aligns the coordinate system of IMU on the IMU array by using iterative closest point algorithm to obtain the initial attitude of each IMU, and then establishes signal model for the accelerometer and gyroscope respectively, and solves the geometric distance of each IMU by using maximum likelihood estimation. The present application can effectively solve the problems of sensor installation error and unknown physical position of sensor chip internal sensing elements, thereby improving the applicability and positioning accuracy of complex array. Therefore, the low-complexity self-calibration method for IMU array proposed by the present application is a stable calibration method which can be well applied to different array types.

Description

Technical Field

[0001] This invention belongs to the technical field of inertial measurement unit (IMU) calibration, specifically relating to a low-complexity self-calibration method for IMU arrays. Background Technology

[0002] Inertial sensors are devices used to measure the acceleration, angular velocity, and changes in orientation of an object. They typically operate based on the principles of inertial force and rotation, providing information about an object's motion state without external reference. Inertial sensors are widely used in many fields, including aerospace, automotive engineering, consumer electronics, and motion tracking. Since rigid bodies in three-dimensional space have six degrees of freedom—three rotational and three translational—inertial measurement units (IMUs) containing three-dimensional gyroscopes and accelerometers are commonly used to measure the motion of objects. Today, with advancements in Micro-Electro-Mechanical Systems (MEMS) technology, the production of miniature accelerometers, gyroscopes, and magnetometers has reached unprecedented scales, while prices have decreased significantly. However, these low-cost IMUs often have poor performance, making reliable positioning difficult in many applications. Leveraging the small size, low cost, and low power consumption of these MEMS-IMUs, arrays of dozens or even hundreds of IMUs can be constructed, and their measurement data can be fused to create highly cost-effective "super sensors."

[0003] The benefits of using IMU arrays extend beyond improved measurement accuracy; they also enhance operational reliability through sensor fault detection and isolation. Since IMU arrays measure force fields, not wave fields, signal models for IMU array measurements can be developed based on classical mechanics results regarding forces in rotating coordinate systems. Therefore, accurate knowledge of the spatial geometric poses between IMUs is crucial, as uncertainties in IMU geometric poses can affect the fusion of measurement results. The literature “H. Carlsson, I. Skog and J. Jaldén, On-the-fly geometric calibration of inertial sensor arrays, 2017 International Conference on Indoor Positioning and Indoor Navigation (IPIN), Sapporo, Japan, 2017, pp. 1-6, doi:10.1109 / IPIN.2017.8115879” only solved one type of uncertainty problem, namely, when all IMU coordinate systems in the IMU array are the same, the problem caused by sensor installation errors and unclear physical positions of sensing elements inside the sensor chip. However, it still has the problems of IMU geometric pose uncertainty and measurement data fusion difficulties in complex IMU arrays. Summary of the Invention

[0004] To address the aforementioned problems, this invention proposes a low-complexity self-calibration method for IMU arrays. This method utilizes the Iterative Closest Point (ICP) algorithm to align the coordinate systems of the IMUs on the array, obtaining information for each IMU. A signal model is then established, and maximum likelihood estimation is used to solve for the geometric distance between each IMU. This innovation effectively overcomes the difficulties in data fusion caused by sensor installation errors and the unclear physical positions of sensing elements within the sensor chip, further improving the applicability and positioning accuracy of complex arrays.

[0005] The technical solution of this invention is:

[0006] A low-complexity self-calibration method for IMU arrays includes the following steps:

[0007] S1. Design an IMU array according to actual needs, determine the IMU deployment method and the number of IMUs used, number the IMUs according to their initial positions, and obtain the initial relative position information sequence of each IMU.

[0008] S2. The following error model is used to compensate for the static parameters:

[0009]

[0010] in, and These represent the acceleration and angular velocity vectors output by the IMU, respectively, where s and ω represent the corresponding true values; b a and b g n represents the zero bias error of the accelerometer and gyroscope, respectively. a and n g These represent the Gaussian noise of the accelerometer and gyroscope, respectively.

[0011] S3. Acquire dynamic data of the IMU array. Specifically, obtain the dynamic data of all N IMUs through random dynamic rotation and movement. Define the acceleration and angular acceleration output by the k-th IMU in the n-th frame as follows: and The output data of each IMU is obtained as A k :

[0012]

[0013] The acceleration and angular acceleration values ​​of each IMU are recorded to obtain dataset A:

[0014] A = [A1 A2 … A] N ]

[0015] S4, IMU geometric pose calculation, specifically:

[0016] Data between 10% and 90% of the gyroscope range is selected from dataset A as the solution data. The attitude of each IMU in the IMU array is solved using the ICP algorithm, that is, the rotation matrix R between the coordinate systems of each IMU is solved and unified to the Earth coordinate system. Specifically, the gyroscope output data is regarded as three-dimensional point cloud data, and the solution is performed by point cloud registration. The point cloud registration problem is defined as follows:

[0017]

[0018] Among them, P s The source point cloud refers to the gyroscope output value whose coordinate system needs to be calibrated. P t For the target point cloud, this refers to the gyroscope output value in the Earth coordinate system. |P s | This refers to the total amount of gyroscope output data collected during the motion period. The value of the nth point in the target point cloud refers to the gyroscope output value of the nth frame in the Earth coordinate system. The value of the nth point in the source point cloud is denoted as the gyroscope output value of the nth frame whose coordinate system needs to be corrected. R and t are the rotation matrix between coordinate systems and the time error of the gyroscope output data, respectively. and Let R and t be the maximum likelihood estimates.

[0019] Based on the maximum likelihood estimate obtained from the solution Instead of the actual rotation matrix R, let We continue to use the maximum likelihood estimate to solve for the relative position r of each IMU in the array, specifically:

[0020] The signal model of the accelerometer is established as follows:

[0021]

[0022] Among them, s k Let s represent the specific force at a point in the rotating coordinate system, ω represent the specific force at the origin O of the array coordinate system, and r represent the angular velocity. k Let O be the distance from the point to the origin O of the coordinate system. Angular acceleration is represented by ×, and the vector product is represented by ×.

[0023] Introducing Ω a Let Ω be a skew-symmetric matrix representation of a three-dimensional column vector a, where Ω a If b = a × b, then the accelerometer signal model is transformed into:

[0024]

[0025] Among them, S k =R k ×s k W=R k ×ω R k Let be the rotation matrix for correcting the k-th IMU to the Earth coordinate system, where s k ω represents the acceleration information measured by the IMU array, and ω represents the angular velocity information measured by the IMU array. This indicates the corresponding angular acceleration information. Represents a three-dimensional column vector The skew-symmetric matrix representation;

[0026] The physical quantity S k The measurement vector y s The model is as follows:

[0027] y s =h s (W)+H s φ+n s

[0028] in,

[0029]

[0030] ns This indicates the measurement error of the accelerometer.

[0031] The gyroscope signal model is established as follows:

[0032] y ω =h ω (ω)+n ω

[0033] in n ω This indicates the measurement error of the gyroscope.

[0034] By concatenating all the measurements into a single vector, the signal model of the entire array becomes:

[0035] y = h(ω) + Hφ + n

[0036] in,

[0037]

[0038] Assuming the measurement error n has zero mean and follows a Gaussian distribution with a known covariance matrix Q, the log-likelihood function for establishing the array signal model is:

[0039]

[0040] in,

[0041] The maximum likelihood estimator of {W,φ} is given by the following equation:

[0042]

[0043] Since the array signal model is partially linear, the log-likelihood function is lumped by fixing the parameter W and maximizing the likelihood function of the linear parameter φ. For a fixed value of angular velocity W... * The solution is given by the least squares method:

[0044]

[0045] in,

[0046] The least squares solution of φ is calculated. make Thus, relevant information about the translational acceleration s was obtained;

[0047] Substituting the estimated value of φ back into the lumped maximum likelihood estimator:

[0048]

[0049] After obtaining the required variables, an array signal model is established using the initial relative position information sequence of the IMU and data from samples at various time points. The correlation value of the translational acceleration s is calculated, and the relative position r is solved using the least squares method. (k) :

[0050]

[0051] in, in s represents the acceleration information of the k-th IMU at time n, and s represents the translational acceleration information at each time. After each loop, the relative position sequence is updated until it converges, thus completing the calculation of the relative positions of each IMU in the IMU array.

[0052] The beneficial effects of this invention are as follows: the method of this invention is used to determine the geometric pose of each IMU in three-dimensional space within a complex inertial sensor array, thereby improving the subsequent fusion effect. This invention utilizes the measurement values ​​of the inertial measurement unit (IMU) sensor array under arbitrary motion, directly estimating the position and attitude of the IMU within the array without the need for a specialized rotating platform. It does not require reference motion or external devices, and the computational complexity of each iteration is linear, which is a significant advantage. This effectively solves the problems of sensor installation errors and unknown physical positions of sensing elements within the sensor chip, thus improving the applicability and positioning accuracy of complex arrays. Therefore, the method of this invention is a stable calibration method that is well-suited for different array types. Attached Figure Description

[0053] Figure 1 This is a flowchart of an embodiment of the present invention.

[0054] Figure 2 This is an IMU array designed independently in this invention.

[0055] Figure 3 This is a simulation diagram of the IMU array results calculated in MATLAB according to the present invention.

[0056] Figure 4 This is a comparison chart of the trajectory calculations of the IMU array after calibration by the algorithm of this invention and the IMU array after precise calibration. Detailed Implementation

[0057] The technical solution of the present invention will now be described in detail with reference to the accompanying drawings and embodiments:

[0058] Example

[0059] like Figure 1 As shown, this example includes the following execution steps:

[0060] 1. Design a circular IMU array, such as Figure 2As shown, the IMU array in this example consists of 16 MEMS-IMUs. On the front of the array, eight MEMS-IMUs are arranged in a circle with a radius of 1 cm, with each adjacent IMU increasing in angle by 45° clockwise. Meanwhile, the back of the array is symmetrically arranged with the front, further improving the array's sensing capabilities and data collection efficiency.

[0061] The IMU used in this example is the BMI270, which consists of a 16-bit digital triaxial accelerometer with a range of ±8g and a 16-bit digital triaxial gyroscope with a range of ±500dps and an operating frequency of 100Hz.

[0062] 2. IMU internal static parameter calibration:

[0063] Equation (1) is the basic error model for gyroscopes and accelerometers:

[0064]

[0065] in and These represent the specific force and angular velocity vectors output by the IMU, respectively, with s and ω representing the corresponding true values. A a and A g b represents the multiplicative error of the accelerometer and gyroscope, respectively. a and b g n represents the additive error of the accelerometer and the gyroscope, respectively. a and n g These represent the Gaussian noise of the accelerometer and gyroscope, respectively.

[0066] In this example, the IMU was calibrated using a dedicated turntable, which facilitates subsequent algorithm verification and compensates for static errors such as sensitivity axis misalignment, unknown scaling factors, and deviations. The general error model after calibration is shown in Equation (2).

[0067]

[0068] In the formula and These represent the acceleration and angular velocity vectors output by the IMU, respectively, with s and ω representing the corresponding true values. a and b g n represents the zero bias error of the accelerometer and gyroscope, respectively. a and n g These represent the Gaussian noise of the accelerometer and gyroscope, respectively.

[0069] 3. Collect and store dynamic data from the IMU array:

[0070] 3-1. After initial calibration of the IMU, connect it to the computer for power. Initially, hold the IMU stationary for 5-10 seconds to allow for subsequent coarse zero-bias calibration. Then, perform random dynamic rotation and movement of the IMU array, and acquire dynamic data from N=16 IMUs using the corresponding driver. Let the acceleration and angular acceleration output by the k-th IMU in the n-th frame be respectively... and Then the output data of each IMU is A k .

[0071]

[0072] 3-2. Storing the acceleration and angular acceleration values ​​of each IMU obtained in 3-1 to a computer via a wired connection. Recording the dataset A collected by the IMU array.

[0073] A = [A1 A2 … A] 16 (4)

[0074] 4. IMU geometric pose calculation:

[0075] 4-1. Data Preprocessing. Since the IMU operates at 100Hz, excessively large datasets during the data acquisition process may slow down the computation. Furthermore, the gyroscope's measurement range is relatively small, and fluctuations beyond this range can significantly interfere with the matching algorithm. Additionally, smaller datasets from the gyroscope result in higher levels of random noise, which can affect the subsequent two algorithms, both of which require larger amounts of IMU data to obtain accurate results.

[0076] To address the aforementioned issues, this example employs several optimization measures to improve algorithm accuracy while reducing the amount of data. Specifically, data ranging from 10% to 90% of the gyroscope's range is selected. This helps reduce the amount of data while avoiding jitter issues caused by gyroscope data exceeding its range.

[0077] 4-2. Use the ICP algorithm to solve for the initial attitude of each IMU in the IMU array, that is, solve for the rotation matrix R between the coordinate systems of each IMU, and unify them to the Earth coordinate system.

[0078] The ICP algorithm is currently the most widely used point cloud registration algorithm. Point cloud registration refers to the process of registering two point clouds P... s and P t Output a transformation T such that T(P) s ) and P tThe degree of overlap is maximized, and the transformation T includes a rotation matrix R and a translation vector t. The innovation of this invention lies in treating the gyroscope output data as a 3D point cloud, while the theoretical output values ​​of the gyroscope on a rigid body should be identical. In transformation T, R and t represent the rotation matrix between coordinate systems and the time error of the gyroscope output data, respectively. That is, let... as close as possible The point cloud registration problem can then be described by equation (5).

[0079]

[0080] Among them, P s The source point cloud refers to the gyroscope output value whose coordinate system needs to be calibrated. P t For the target point cloud, the gyroscope output value in the Earth coordinate system. |P s | This refers to the total amount of gyroscope output data collected during the motion period. The value of the nth point in the target point cloud refers to the gyroscope output value of the nth frame in the Earth coordinate system. The value of the nth point in the source point cloud is denoted as the gyroscope output value of the nth frame whose coordinate system needs to be corrected. R and t are the rotation matrix between coordinate systems and the time error of the gyroscope output data, respectively. and The maximum likelihood estimates of R and t

[0081] Since the initial pose of each IMU in the IMU array can be roughly estimated, i.e., a roughly accurate parameter estimate already exists, this algorithm can quickly converge to the desired registration result. If the initial pose is unknown, it can be approximated using this algorithm. The outputs of the accelerometer and gyroscope are the acceleration and angular velocity values ​​in the current frame, and the time difference between the data from each IMU in the IMU array is very small. Therefore, in this example, only rotation is considered, and the effect of translation is ignored. Let... Let represent the centroids of the source and target point clouds, respectively. The centroids are calculated by adding the coordinates of all points and dividing by the number of points. Let Transform both the source and target point clouds to centroid coordinates to obtain centroid-free coordinates, which speeds up the matching algorithm. The point cloud registration problem can be simplified as Equation (6).

[0082]

[0083] Next, the covariance matrix is ​​calculated, which is the product of the centroid-free coordinate matrix of the source point cloud and the transpose of the centroid-free coordinate matrix of the target point cloud. Then, perform Singular Value Decomposition (SVD) on the covariance matrix H to obtain H = UΣV TThen the optimal rotation matrix for the point cloud registration problem is shown in equation (7).

[0084]

[0085] Step 4-3. Using the R obtained in Step 4-2, continue to use maximum likelihood estimation to solve for the relative position r of each IMU in the IMU array.

[0086] To determine the relative positions of the IMUs in the array, it is necessary to establish corresponding signal models for the accelerometers and gyroscopes respectively for calculating their relative positions. The first step is to establish the accelerometer signal model. Basic kinematics stipulates that the specific force at a point in a rotating coordinate system (specific force, simply put, refers to the non-gravitational external force acting on a unit mass, usually represented by a vector) can be decomposed into the specific force, eccentric term, and Euler term at another point. Based on this theory, the formula can be written as follows:

[0087]

[0088] Among them, s k Let s represent the specific force at a point in the rotating coordinate system, where s is the specific force (also called translational acceleration) at the origin O of the array coordinate system. This force arises from the inertia of the rigid body during rotation and is related to the motion state of the rigid body and the direction of the rotation axis. This is a key variable for subsequent calculations of relative position. The middle part is the centrifugal term, which arises from the centrifugal force of the rigid body during rotation. Its magnitude is related to the distance r from point A to the center of rotation O. k And it is related to the angular velocity ω; the last term is the Euler term, which arises from the change in angular velocity of the rigid body during rotation, and is related to the angular acceleration of the rigid body and the distance r from point A to the axis of rotation. k and angular acceleration Related, × represents the vector product.

[0089] Therefore, according to the above theory, the acceleration at point A can be decomposed into the vector sum of three parts: the translational acceleration at the origin O, the centrifugal term, and the Euler term. Since the coordinate systems of each IMU in a complex IMU array are not the same, it is necessary to add the relevant information of the rotation matrix R to transform all IMU coordinate systems to the Earth coordinate system.

[0090] To simplify the calculation, Ω is introduced. a Let Ω be a skew-symmetric matrix representation of a three-dimensional column vector a, where Ω a If b = a × b, then the above accelerometer signal model can be transformed into:

[0091]

[0092] Among them, S k =R k ×s kW=R k ×ω R k Let be the rotation matrix for correcting the k-th IMU to the Earth coordinate system, where s k ω represents the acceleration information measured by the IMU array, and ω represents the angular velocity information measured by the IMU array. This indicates the corresponding angular acceleration information. Represents a three-dimensional column vector The skew-symmetric matrix representation;

[0093] The physical quantity S k The measurement vector y s The model is as follows:

[0094] y s =h s (W)+H s φ+n s (10)

[0095] in:

[0096]

[0097] n s This indicates the measurement error of the accelerometer.

[0098] Secondly, the gyroscope signal model is established. This model does not involve force decomposition and is based on the error n between the actual and measured values. ω The relationship can be modeled as follows:

[0099] y ω =h ω (ω)+n ω (11)

[0100] in n ω This indicates the measurement error of the gyroscope.

[0101] Then, by connecting all the measurements into a single vector, the signal model of the entire array becomes:

[0102] y=h(ω)+Hφ+n (12)

[0103] in:

[0104]

[0105] As can be seen from this formula, the signal model of the inertial sensor array consists of a nonlinear part h(W) that depends only on W and a nonlinear part s that depends only on the translational acceleration s. The linear part Hφ is composed of y, and it can be seen intuitively that y is determined by two uncertain variables W and φ. Therefore, it is necessary to estimate the most suitable values ​​of W and φ in order to minimize the impact of noise on the system.

[0106] Assuming the measurement error n has zero mean and follows a Gaussian distribution with a known covariance matrix Q, the log-likelihood function of the array signal model is given by the following equation:

[0107]

[0108] in

[0109] Therefore, the maximum likelihood estimator of {W,φ} is given by the following equation:

[0110]

[0111] Since the array signal model is partially linear, we can centralize the log-likelihood function by fixing the parameter W and maximizing the likelihood function of the linear parameter φ, then substitute the result into the likelihood function. For a fixed angular velocity W, the solution is given by the least squares method.

[0112]

[0113] in

[0114] The least squares solution of φ is calculated. By replacing φ with it, we obtain relevant information about the translational acceleration s.

[0115] Substituting the estimated value of φ back into the equation yields the lumped maximum likelihood estimator:

[0116]

[0117] At this point, all the necessary variables have been calculated, and the relative position information can be calculated. The position calculation process is as follows:

[0118] First, provide reference position information (which can be a specific IMU or the geometric center of the array). Based on this reference position, write out the initial IMU relative position information sequence. Then, given data from samples at various times, establish the array signal model and calculate the correlation value of the translational acceleration s. For formula (8), calculate the relative position r using the least squares method. (k) The relevant formulas are as follows:

[0119]

[0120] in:

[0121] in This represents the acceleration information of the k-th IMU at time n. s represents the translational acceleration information at each time step. The relative position sequence is updated after each iteration until convergence, thus completing the calculation of the relative positions of each IMU in the IMU array.

[0122] This invention designs two sets of experiments to demonstrate the superiority of the algorithm. Since the actual coordinate system of the gyroscope and accelerometer is difficult to measure precisely, a rotating platform is used to accurately compensate the angle between each adjacent IMU in the IMU array to 45°, ensuring that the IMUs in the array can accurately reflect the true value of the geometric attitude. In the first set of experiments, MATLAB is used for data verification. By calculating the difference between the set value and the actual measured value, the error term is obtained, and the error terms of the 16 IMUs are summed and averaged. The final results show that the average angle error of the X-axis is 0.0324°, the average angle error of the Y-axis is 0.1260°, and the average angle error of the Z-axis is 0.0233°. See the detailed simulation results. Figure 3 The second set of experiments used the algorithm proposed in this invention to measure geometric pose and compared the movement trajectory with that of a precisely calibrated IMU array. In the experiment, the operator moved the handheld IMU array, forming a rectangular trajectory, approximately 6m x 2.8m. The trajectory obtained by the IMU array calibrated using the method of this invention is very close to the trajectory of the precisely calibrated IMU array; see details below. Figure 4 As can be seen, for complex IMU arrays, the method proposed in this invention can estimate the geometric pose of each IMU very well, overcoming the problems of IMU geometric pose uncertainty and measurement data fusion difficulties in complex IMU arrays. Experimental results prove that this invention provides excellent estimation of IMU geometric pose uncertainty in complex IMU arrays, without the need for an external rotating platform, and is a low-complexity self-calibration method for IMU arrays.

Claims

1. A low-complexity self-calibration method for IMU arrays, characterized in that, Includes the following steps: S1. Design an IMU array according to actual needs, determine the IMU deployment method and the number of IMUs used, number the IMUs according to their initial positions, and obtain the initial relative position information sequence of each IMU. S2. The following error model is used to compensate for the static parameters: ， in, and These represent the specific force and angular velocity vectors output by the IMU, respectively. and These represent the corresponding actual values; and These represent the zero bias errors of the accelerometer and gyroscope, respectively. and These represent the Gaussian noise of the accelerometer and gyroscope, respectively. S3. Acquire dynamic data of the IMU array, specifically by randomly rotating and moving the array to obtain dynamic data from all N IMUs. Define the... The first frame The acceleration and angular acceleration output by each IMU are respectively and The output data of each IMU is obtained as follows : ， The acceleration and angular acceleration values ​​of each IMU are recorded to obtain the dataset. : ； S4, IMU geometric pose calculation, specifically: From the dataset Data between 10% and 90% of the gyroscope's range was selected as the solution data. The ICP algorithm was used to solve for the attitude of each IMU in the IMU array, that is, to solve for the rotation matrix of the coordinate system between each IMU. This is done by unifying the data onto the Earth coordinate system, specifically by treating the gyroscope output data as 3D point cloud data and solving it using point cloud registration. The point cloud registration problem is defined as follows: ， in, For the target point cloud Point value, referring to the first point in the Earth coordinate system. Frame gyroscope output value The data output from the gyroscope in the IMU that needs to be calibrated. The first point in the source cloud Point value, referring to the point value of the coordinate system that needs to be corrected. Frame gyroscope output value; and These are the rotation matrix between coordinate systems and the time error of the gyroscope output data, respectively. According to the solution We continue to use maximum likelihood estimation to solve for the relative positions of the IMUs in the array. Specifically: The signal model of the accelerometer is established as follows: ， in, The specific force at the origin O of the array coordinate system. Angular velocity, Let O be the distance from the point to the origin O of the coordinate system. Angular acceleration, It is a vector product; Introduction Represents a three-dimensional column vector The skew-symmetric matrix representation, where The accelerometer signal model is then transformed into: ， in, ,in For the first A rotation matrix for IMU correction to Earth coordinate system. This indicates the acceleration information measured by the IMU array; Represents a three-dimensional column vector The skew-symmetric matrix representation, This represents the angular velocity information measured by the IMU array, while This indicates the corresponding angular acceleration information; Will be determined by physical quantities The measurement vector formed The model is as follows: ， in, ， This indicates the measurement error of the accelerometer; The gyroscope signal model is established as follows: ， in ; By concatenating all the measurements into a single vector, the signal model of the entire array becomes: ， in, ， Assuming measurement error It has zero mean and a known covariance matrix. If the signal is Gaussian distributed, then the log-likelihood function for establishing the array signal model is: ， in, ; The maximum likelihood estimator is given by the following formula: ， Since the array signal model is partially linear, it can be solved by fixing parameters. And maximize the linear parameter The likelihood function is used to condense the log-likelihood function for a fixed value of angular velocity. The solution is given by the least squares method: ， in, ; Thus, the translational acceleration was obtained. Related information; Will Substituting the estimated values ​​back into the ensemble maximum likelihood estimator: ， After obtaining the required variables, the array signal model is established using the initial relative position information sequence of the IMU and the data from samples at various time points, and the translational acceleration is calculated. The relevant values ​​are used to calculate the relative positions using the least squares method. : ， in, , ,in Indicates the first One IMU in Information on acceleration at any given moment. It contains the translational acceleration information at each moment; after each loop, the relative position sequence is updated until it converges, thus completing the calculation of the relative positions of each IMU in the IMU array.

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