Inertial navigation temperature drift compensation method based on multi-dimensional modeling

Abstract

This invention discloses an inertial navigation temperature drift compensation method based on multidimensional modeling, relating to the field of inertial navigation. It includes: S1. Collecting temperature, three-axis acceleration, and three-axis angular velocity data within a temperature range of -40℃ to 80℃ using a constant temperature control system, and constructing a time-aligned multivariate feature matrix; S2. After standardization and rotation transformation, constructing a fifth-order nonlinear coupled function set using Chebyshev polynomials, establishing an orthogonal tensor model, and applying L1 constraints; S3. Constructing a drift response curve set based on temperature rise and fall experimental data, obtaining a 6×1 drift sensitivity matrix; S4. Removing outlier samples through Mahalanobis distance analysis and extracting a high-confidence training subset; S5. Constructing a multi-path modeling network containing a hot coding module, convolutional channels, and an attention mechanism for cross-axis modeling; S6. Training the network and generating a steady-state compensation tensor; S7. Inputting new data into the model, sequentially maintaining the positive six-axis output, and completing active temperature drift compensation. Beneficial effects: Improved navigation accuracy and compensation robustness across temperature zones.

Description

Technical Field

[0001] This invention relates to the field of inertial navigation, and more specifically, to an inertial navigation temperature drift compensation method based on multidimensional modeling. Background Technology

[0002] Inertial navigation systems (INS) have wide applications in aerospace, weapon guidance, and high-precision positioning, but their accuracy is severely limited by the drift error of the inertial measurement unit (IMU) under temperature variations. Traditional compensation methods often employ linear fitting or polynomial regression, which, while effective under isothermal conditions, have limited ability to handle nonlinear temperature responses, coupling characteristics, and high-frequency interference. With improved computing power, nonlinear modeling techniques such as neural networks have been gradually introduced, and some studies have begun to use convolutional networks or RNN structures for temperature drift modeling, resulting in improved compensation accuracy.

[0003] However, these methods still have three shortcomings: First, most of them adopt a single-path structure, which cannot dynamically adapt to the characteristics of different temperature zones; second, the correction process often uses the whole set of data for unified correction, ignoring the temperature response asymmetry between each axis channel; and third, the filtering process is insufficient, which can easily introduce high-frequency noise or phase distortion, affecting navigation stability. Summary of the Invention

[0004] The purpose of this invention is to provide an inertial navigation temperature drift compensation method based on multidimensional modeling to solve the problems mentioned in the background art. However, these methods still have three shortcomings: First, most of them adopt a single-path structure, which cannot dynamically adapt to the characteristics of different temperature zones; second, the correction process often uses the whole set of data for unified correction, ignoring the temperature response asymmetry between each axis channel; and third, the filtering process is insufficient, which easily introduces high-frequency noise or phase distortion, affecting navigation stability.

[0005] Technical solution: An inertial navigation temperature drift compensation method based on multidimensional modeling includes the following steps:

[0006] S1. Under the full temperature range of -40℃ to 80℃, a closed-loop constant temperature control system is used to perform fixed-point temperature control on the inertial measurement unit, and temperature data, triaxial acceleration data and triaxial angular velocity data are collected simultaneously to construct a multivariate feature matrix with a unified timestamp and sample alignment mechanism.

[0007] S2. After applying standardization and first-order coordinate system rotation transformation to the multivariable feature matrix, construct a fifth-order nonlinear coupling function group using Chebyshev polynomials. With temperature and first-order derivative as inputs and the triaxial acceleration data and triaxial angular velocity data as outputs, establish an orthogonal tensor coupling model and apply L1 norm constraints to achieve sparse modeling.

[0008] S3. Based on the heating / cooling response experimental data of the inertial measurement unit, construct a set of drift response curves for the triaxial acceleration data and the triaxial angular velocity data, calculate the slope coefficients respectively to obtain a 6×1 dimension drift sensitivity matrix, which is used as the initial directional weight matrix for residual extraction.

[0009] S4. Perform Mahalanobis distance analysis on the residual data obtained under the effect of the drift sensitivity matrix, calculate the normalized distance based on the sample covariance matrix, remove samples with a confidence level of 95%, and construct a high-confidence training subset.

[0010] S5. Construct a multipath modeling network including a temperature thermal encoding embedding module, three layers of normalized convolutional channels, a parallel attention mechanism, and a dynamic path selection structure. Input a shrunken high-confidence training subset into the multipath modeling network to perform cross-axis deep modeling of the temperature-output relationship.

[0011] S6. The multi-path modeling network is trained using an objective function composed of the in-channel L2 constraint loss term and the residual feedback weighting. The steady-state temperature drift compensation tensor is obtained by iterative optimization using a channel-specific, fixed step size, and AdamW optimization strategy.

[0012] S7. Using the temperature data, the triaxial acceleration data, and the triaxial angular velocity data as inputs, feed them into the trained multi-path modeling network, call the temperature drift compensation tensor, and perform stepwise corrections on the six-axis outputs respectively to complete the active compensation for the temperature drift of the inertial measurement unit.

[0013] The multi-path modeling network introduces a five-segment temperature mapping mask activation structure, which dynamically adapts the cross-temperature zone modeling performance based on different paths in the temperature segment label activation and suppression network.

[0014] Preferably, the construction of the multivariate feature matrix in S1 includes the following steps:

[0015] S1-1. In the configuration of sampling points in the full temperature range of -40℃ to 80℃, the constant temperature points are set as one sampling point every 2.5℃ in the range of -40℃ to 80℃, for a total of 49 points. The closed-loop constant temperature control system is used to ensure that the stability error of each point is less than ±0.2℃, the data acquisition time for each temperature point is not less than 30 minutes, and the sampling frequency is set to 1000Hz.

[0016] S1-2. The collected temperature data, triaxial acceleration data, and triaxial angular velocity data are aligned by timestamps to form an N-row, 7-column floating-point matrix. The column vectors are, in order, timestamp, temperature value, X / Y / Z acceleration value, and X / Y / Z angular velocity value. Zero-mean, unit variance standardization is used.

[0017] Preferably, the generation of the S1-2 feature matrix data structure further includes the following steps:

[0018] S1-2-1. Using the three-dimensional Euclidean norm of the triaxial acceleration data and the triaxial angular velocity data, samples with a first derivative rate of change exceeding three times the standard deviation of the mean are removed.

[0019] S1-2-2. Initially fix the inertial measurement unit in a known orientation coordinate system, record the reference three-axis direction vectors through manual calibration, and perform direction cosine matrix remapping processing on the collected temperature data, the three-axis acceleration data, and the three-axis angular velocity data.

[0020] Preferably, the calculation of the drift sensitivity matrix in S3 includes the following steps:

[0021] S3-1. Perform four rounds of temperature excitation on the inertial measurement unit, each round including heating and cooling stages, with the temperature change rate set to ±2℃ / min, record the temperature change response corresponding to the triaxial acceleration data and the triaxial angular velocity data, and set the sampling frequency to 2000Hz;

[0022] S3-2. The least squares method is used to fit the output curve of each round of experimental data with temperature change, extract the slope value of the linear segment, and calculate the average value of the four rounds as the drift response coefficient, which corresponds to the six-axis data respectively, and combine them to form a 6×1 dimension drift sensitivity matrix.

[0023] Preferably, the S3-2 slope response vector extraction further includes the following steps:

[0024] S3-2-1. Only segments with residual mean square error less than 0.002 are retained in the fitting results, and the corresponding temperature range must cover at least 10℃. Those that do not meet this requirement are automatically discarded.

[0025] S3-2-2. Perform correlation analysis on the temperature drift coefficients of the triaxial acceleration data and the triaxial angular velocity data. Channel data with a Pearson correlation coefficient lower than 0.4 are not included in the final matrix to ensure accurate residual extraction direction.

[0026] Preferably, the multi-path modeling network in S5 further includes the following structure:

[0027] S5-1. Normalize the temperature value to the interval -1,1, input the piecewise hyperbolic tangent mapping function, output a 32-dimensional encoded tensor to guide the convolution path selection mask, and map the normalized temperature value to the corresponding temperature segment label index to activate the matching path and block the other paths.

[0028] S5-2. Combining the encoding tensor, guide a set of three-layer feature extraction sub-networks to model temperature region features. This sub-network consists of three one-dimensional convolutional layers. Each layer contains a 1D convolutional operation with a kernel size of 5. The number of convolutional channels in each layer are 64, 128, and 256, respectively. Each layer is followed by LayerNorm and GELU activation functions to prevent gradient explosion.

[0029] S5-3. The output of each convolutional layer is fed into a 4-head attention mechanism module. Each head uses a scaled dot product attention function to keep the output dimension consistent and construct an axial feature context.

[0030] S5-4. Based on the temperature zone label index output by S5-1, activate the corresponding path from five fixed pre-trained paths, and mask the output of the remaining paths to make the modeling of each temperature zone independent.

[0031] Preferably, the S5-4 dynamic pathway activation structure further includes the following steps:

[0032] S5-4-1. Divide the temperature values ​​into five segments: extremely low temperature -40℃ to -20℃, low temperature -20℃ to 0℃, medium temperature 0℃ to 30℃, high temperature 30℃ to 55℃, and extremely high temperature 55℃ to 80℃, and establish pathways for each segment.

[0033] S5-4-2. Configure a fixed mask encoding matrix for each temperature range, such as 1, 0, 0, 0, 0. Select the modeling path in the network corresponding to the current temperature range through matrix multiplication and output the compensation tensor.

[0034] Preferably, during the training of the multi-path modeling network in S6, the AdamW optimizer is used, with an initial learning rate of 0.001 and a weight decay coefficient of 0.01. The learning rate is adaptively adjusted to 80% of its original value after every 100 training rounds. Each training batch includes 200 sample frames, and the total number of training rounds is 300. The loss function consists of a linear weighted sum of the in-channel L2 residual term and the full-channel drift balance term, with a ratio of 4:1.

[0035] Preferably, the maintenance correction process in S7 includes the following steps:

[0036] S7-1. Select the compensation tensor channel group using the current temperature value, and perform element-wise weighting of the six-axis data with the corresponding compensation tensor channel group. The compensation value range is limited to ±0.01g and ±0.01° / s.

[0037] S7-2. All compensated triaxial acceleration and triaxial angular velocity data are processed by a third-order Butterworth low-pass filter with the cutoff frequency set to 0.2 times the original sampling frequency to prevent high-frequency disturbances from being generated during compensation.

[0038] Compared with the prior art, the advantages of this invention are:

[0039] (1) The traditional single-axis or independent axis modeling is extended to a multi-channel joint modeling temperature-inertial response coupling structure, which significantly improves the accuracy of temperature drift modeling and the synergy between dimensions.

[0040] (2) Compared with the existing first-order / second-order fitting models, a 9th-order nested structure is used and Bayesian regularization is introduced to improve the stability and generalizability of the high-order model in the boundary temperature region.

[0041] (3) Instead of relying on traditional mean / variance elimination, a residual-driven dynamic 3σ window mechanism is used to eliminate irreversible drift data and improve the stability of training samples.

[0042] (4) Based on the Transformer structure, the nonlinear relationship between temperature and inertial data is reconstructed. Compared with conventional convolutional networks, it has a stronger feature capture capability and is particularly suitable for complex thermal drift characteristics.

[0043] (5) Instead of using static weight interpolation, a dynamic path activation mechanism driven by temperature embedding vector is introduced, which activates different modeling subnetworks according to the current temperature zone and adaptively compensates for weight selection.

[0044] (6) Compared with the traditional whole attitude or velocity vector correction method, we propose a dimension-by-dimensional (each axis) independent correction strategy, which can avoid the cumulative error caused by thermal drift asymmetry between dimensions.

[0045] (7) Unlike directly outputting the correction result, this method introduces a third-order Butterworth filter for zero-phase-delay post-processing, which effectively suppresses high-frequency noise in the model and improves the availability of the compensation signal. Attached Figure Description

[0046] Figure 1 This is a schematic diagram of the overall process of an inertial navigation temperature drift compensation method based on multidimensional modeling according to the present invention. Detailed Implementation

[0047] For examples, please refer to Figure 1 A method for compensating for temperature drift in inertial navigation based on multidimensional modeling includes the following steps:

[0048] S1. Under the full temperature range of −40℃ to 80℃, a closed-loop constant temperature control system is used to perform fixed-point temperature control on the inertial measurement unit, while simultaneously collecting temperature data, triaxial acceleration data and triaxial angular velocity data, and constructing a multivariate feature matrix with a unified timestamp and sample alignment mechanism;

[0049] Specifically, the "ring-type isothermal control system for the inertial measurement unit" refers to the process during the temperature drift modeling data acquisition phase. The inertial measurement unit (IMU) is placed in a closed thermally controlled cavity with stable temperature control capabilities, and its operating temperature is controlled using a ring-type heating or cooling method to ensure a stable distribution across the entire set range (e.g., -40℃ to 85℃). This ring-type isothermal system typically uses an outer PID thermal control film heating element in conjunction with an internal fan convection system to ensure temperature uniformity and time controllability during the temperature control process. This guarantees the stability of the IMU output signal at different temperature zones, thus providing controllable variable inputs for subsequent drift modeling.

[0050] S2. After applying standardization and first-order rotation transformation of the coordinate system to the multivariable feature matrix, a fifth-order nonlinear coupling function group is constructed using Chebyshev polynomials. With temperature and first-order derivative as inputs and triaxial acceleration data and triaxial angular velocity data as outputs, an orthogonal tensor coupling model is established, and L1 norm constraints are applied to achieve sparse modeling.

[0051] Specifically, the steps of applying standardization and first-order coordinate system rotation transformation to the multivariable feature matrix refer to performing the following two processing operations during the preprocessing of the multivariable feature matrix composed of data collected by the inertial measurement unit (including temperature, triaxial acceleration, and triaxial angular velocity):

[0052] All channel data (per column) are processed to have zero mean and unit variance:

[0053] ,

[0054] Where μ is the sample mean and σ is the sample standard deviation.

[0055] A first-order coordinate system rotation transformation maps the acquired raw triaxial data (inertia + temperature) to an approximately orthogonal reference system that minimizes assembly errors. This process involves rotating the system after estimating the IMU installation deviation matrix R.

[0056] ,

[0057] Where R is a three-dimensional rotation matrix, which is generally derived from calibration experiments or initially estimated and then fine-tuned through principal component analysis.

[0058] The acquired raw triaxial data (inertia + temperature) is mapped to an approximately orthogonal reference frame that minimizes assembly error. This process involves estimating the IMU installation deviation matrix R and then rotating it.

[0059] Specifically, “constructing a fifth-order nonlinear coupled function set” refers to constructing a set of functions containing nonlinear terms of order five or lower in order to more accurately express the nonlinear response of inertial output to temperature changes during the multidimensional temperature-output modeling process. Each function is a high-order modeling expression of an output channel.

[0060] Function group form:

[0061] ,

[0062] in Let be the predicted drift value for the i-th axis, and ak be the fitting coefficient. For temperature, This is the offset constant.

[0063] Coupling characteristics description: In order to capture the temperature correlation across channels, this function group achieves coupled modeling by jointly optimizing the parameters of each axis and sharing the temperature vector input, thereby overcoming the problem that traditional independent modeling methods cannot capture the consistency of multi-axis drift.

[0064] Specifically, "establishing an orthogonal tensor coupling model" refers to organizing the output data of all sensor channels (usually six channels in total, including three-axis acceleration and three-axis angular velocity) into a high-order tensor structure under different temperature zones during the in-depth modeling of the temperature drift of the inertial measurement unit, and performing orthogonal decomposition on each dimension to extract its potential coupling relationship.

[0065] Technical implementation:

[0066] Data Structures:

[0067] Constructing a third-order tensor Where N: number of samples; 6: six inertial channels (Ax, Ay, Az, Gx, Gy, Gz); Z: number of temperature zones or temperature segmentation dimension.

[0068] Orthogonal tensor decomposition: Using Tucker decomposition or Higher-order Singular Value Decomposition (HOSVD), the tensor is deconstructed into a kernel tensor. Three orthogonal factor matrices , , :

[0069] ,

[0070] in This represents the nth modular multiplication of the tensor. Orthogonality ensures that the information between factor matrices does not interfere with each other, which is beneficial for extracting the coupling features between the dominant mode and the channel in temperature drift.

[0071] Specifically, "applying L1 norm constraints" in the optimization process of establishing a tensor coupling model involves applying an L1 norm regularization term to the kernel tensor G or the key parameter matrix in the network, which takes the form:

[0072] ,

[0073] in Let λ represent the L1 norm of the tensor (i.e., the sum of the absolute values ​​of all elements), and λ be the weighting coefficient.

[0074] S3. Based on the heating / cooling response experimental data of the inertial measurement unit, a set of drift response curves for triaxial acceleration data and triaxial angular velocity data is constructed. After calculating the slope coefficients, a 6×1 dimension drift sensitivity matrix is ​​obtained, which is used as the initial directional weight matrix for residual extraction.

[0075] Specifically, constructing a set of drift response curves for triaxial acceleration and triaxial angular velocity data refers to acquiring IMU data from each channel under multiple temperature zones, subtracting the calibrated output value at the standard temperature, and obtaining a set of response curves showing the thermal drift error as a function of temperature. Each response curve reflects the offset trend of the data on one axis as a function of temperature.

[0076] Specific steps: Record the IMU six-axis output at multiple temperature setpoints; calculate the offset value using 25℃ or a certain constant temperature reference point as a reference; fit six independent response curves, each corresponding to one channel (Ax, Ay, Az, Gx, Gy, Gz).

[0077] Curve form: The response curve can be fitted using a high-order polynomial or a neural network, and its value is used for subsequent model supervised learning training or as an evaluation metric baseline.

[0078] S4. Perform Mahalanobis distance analysis on the residual data obtained under the influence of the drift sensitivity matrix, calculate the normalized distance based on the sample covariance matrix, remove samples with a confidence level of 95%, and construct a high-confidence training subset.

[0079] Specifically, “constructing a high-confidence training subset” refers to selecting a subset of samples from the complete modeling sample set that meets the following conditions to supervise model training and reduce the impact of overfitting or abnormal perturbations: Kalman filter residuals are less than the dynamic threshold (e.g., 3σ); the sampling temperature range is uniformly covered and there is no concentrated accumulation; the drift trend has good continuity and no abrupt changes; the temperature change rate (dT / dt) is less than the preset threshold (e.g., 1℃ / min).

[0080] S5. Construct a multi-path modeling network that includes a temperature thermal encoding embedding module, three layers of normalized convolutional channels, a parallel attention mechanism, and a dynamic path selection structure. Input the shrinking high-confidence training subset into the multi-path modeling network to perform cross-axis deep modeling of the temperature-output relationship.

[0081] Specifically, "cross-axis deep modeling of temperature-output relationship" refers to the design of neural network models that, instead of independently modeling the temperature drift curves of each axis sensor, all six channels are treated as a whole input structure, and cross-dimensional interaction modeling is performed in deep networks (especially attention or residual coupling layers) to capture potential cross-axis thermal drift coordination or coupling patterns.

[0082] Introduce cross-fusion structures at deeper levels (≥3rd layer), such as multi-channel residual fusion modules;

[0083] A multi-head attention mechanism is used to allocate the contribution of inter-axis drift in different temperature zones.

[0084] During training, the six-channel output is uniformly optimized to maintain consistency in error directionality and amplitude.

[0085] S6. The multi-path modeling network is trained using an objective function composed of the in-channel L2 constraint loss term and the residual feedback weighting. The steady-state temperature drift compensation tensor is obtained by iterative optimization using a channel-specific, fixed step size, and AdamW optimization strategy.

[0086] Specifically, the "objective function composed of the weighted sum of the loss term and the residual feedback" refers to the optimization objective function used when training a multi-path inertial drift modeling network, which consists of two explicitly defined parts:

[0087] LossTerm: Typically the mean square error (MSE) between the predicted output and the actual drift value.

[0088] Residual Feedback Weighted Term: This refers to the inverse weighting metric introduced from the residual magnitude of each temperature zone in the previous training iteration. This metric is used to improve the model's attention to learning edge temperature zones and small sample temperature zones.

[0089] The objective function takes the following form:

[0090] ,

[0091] in Let i be the residual of the i-th temperature region. The weights are obtained by inversely proportional to the mean residual of the most recent iteration (i.e., the larger the residual, the higher the weight), thereby achieving adaptive optimization and focusing of the temperature distribution during model training.

[0092] S7. Using temperature data, triaxial acceleration data, and triaxial angular velocity data as input, feed them into the trained multi-path modeling network, call the temperature drift compensation tensor, and perform step-by-step correction on the six-axis output rows to complete the active compensation for temperature drift of the inertial measurement unit.

[0093] Specifically, the "temperature drift compensation tensor" refers to a multidimensional array structure of six-axis sensor error predictions obtained under different temperature input conditions through a trained multi-path inertial drift modeling network. This structure can be directly used for subsequent real-time data compensation.

[0094] The tensor dimension is defined as [Z, 6], where Z is the number of temperature zones (e.g., five zones), and 6 represents six channels for triaxial acceleration and triaxial angular velocity. Each element represents the prediction drift error of a specific channel within a given temperature zone. This tensor serves as a static buffer for the model output and is deployed in the computational unit of the navigation system to perform compensation mapping. Unlike traditional coefficients, this tensor is a nonlinear dynamic response expression learned by the network structure, which can be dynamically adjusted for accurate temperature drift correction.

[0095] The multi-path modeling network introduces a five-segment temperature mapping mask activation structure, which dynamically adapts the cross-temperature zone modeling performance based on different paths in the temperature segment label activation and inhibition network.

[0096] Specifically, the five-segment temperature mapping mask activation structure refers to a sparse activation path structure consisting of five temperature ranges (such as extremely low, low, medium, high, and extremely high) mapped to the network using temperature value input. Each temperature range controls the opening and closing of specific network sub-paths through a mask mechanism, thereby achieving temperature range adaptive modeling.

[0097] It consists of the following components: a temperature zone discriminator, which determines which of the five zones the current input temperature zone belongs to based on the heat encoding vector;

[0098] The mask generation module generates a set of 5-bit binary masks in the form of [1,0,0,0,0].

[0099] Mask control activation uses a mask to activate or mask the pathways of five independent modeling branches, activating only the branch structure corresponding to the matching temperature zone.

[0100] Specifically, the inertial measurement unit uses a MEMS-grade six-axis sensor module with a temperature response range of -40℃ to 85℃; the data sampling frequency is set to 200Hz. In step S2, the method for removing drifting and unstable samples employs three-axis fusion residual analysis, estimating the difference between the state value and the original observation using Kalman filtering and dynamically removing samples based on a three-standard-deviation principle. In step S3, the constructed temperature response model is a ninth-order polynomial. Model training uses Bayesian regularization to control the variance of higher-order term coefficients and prevent overfitting. The modeling network in step S4 uses a three-layer residual convolution module followed by a four-head attention mechanism module, each head containing two 128-dimensional fully connected layers and Softmax normalization processing. In step S5, the hot coding module linearly normalizes the temperature to the [-1,1] interval and then maps it to a 32-dimensional temperature region embedding vector using a tanh family activation function for path activation. In step S6, the extracted drift tensor is divided into 10 groups according to temperature region, each group containing 6 compensation channels. S7 uses a channel-by-channel weighted correction method, and the output is smoothed by a Butterworth low-pass filter.

[0101] The construction of the multivariate feature matrix in S1 includes the following steps:

[0102] S1-1. Sampling points are configured in the full temperature range of -40℃ to 80℃. The constant temperature points are set as one sampling point every 2.5℃ in the range of -40℃ to 80℃, for a total of 49 points. A closed-loop constant temperature control system is used to ensure that the stability error of each point is less than ±0.2℃. The data acquisition time for each temperature point is not less than 30 minutes, and the sampling frequency is set to 1000Hz.

[0103] S1-2. The collected temperature data, triaxial acceleration data, and triaxial angular velocity data are aligned by timestamps and formed into an N-row, 7-column floating-point matrix. The column vectors are, in order, timestamp, temperature value, X / Y / Z acceleration value, and X / Y / Z angular velocity value. Zero-mean, unit variance standardization is used.

[0104] Specifically, the Kalman filter employs a first-order state model, with its state transition matrix and observation matrix being identity matrices. The noise covariance matrix is ​​set as a diagonal matrix (σ²=0.002) based on prior experience. The residual is calculated by subtracting the filtered estimate from the observed data. A 3σ dynamic window uses the statistical characteristics of the residual distribution that vary with the temperature range to adjust the boundary threshold in real time, thereby improving the accuracy of sample rejection.

[0105] The generation of the S1-2 feature matrix data structure also includes the following steps:

[0106] S1-2-1. Using the three-dimensional Euclidean norm of triaxial acceleration and triaxial angular velocity data, samples with the rate of change of the first derivative exceeding three times the standard deviation of the mean are removed.

[0107] Specifically, the "three-dimensional Euclidean norm" refers to the magnitude or L2 norm of a three-dimensional vector (such as triaxial acceleration or triaxial angular velocity) in Euclidean space.

[0108] S1-2-2. Initially fix the inertial measurement unit in a known orientation coordinate system, record the reference three-axis orientation vectors through manual calibration, and perform orientation cosine matrix remapping processing on the collected temperature data, three-axis acceleration data, and three-axis angular velocity data.

[0109] Specifically, the ninth-order polynomial function is in the form of:

[0110] ,

[0111] Where T is the temperature input and a_i are the coefficient parameters. Maximum a posteriori estimation is based on the Laplace prior to penalize higher-order terms. During training, the EM algorithm is used to estimate the coefficients and noise variance to ensure model stability in the edge temperature range.

[0112] Specifically, establish the direction cosine matrix. Each element Let the cosine value of the j-th axis of b in the i-th axis direction of the n-th system satisfy:

[0113] ,

[0114] Remapping acceleration and angular velocity data:

[0115] The original triaxial data is remapped to data in the inertial coordinate system using this matrix:

[0116]

[0117] Temperature data processing method: If the temperature data is collected by an independent sensor for each axis (such as a 3-point thermistor), it is also spatially mapped according to the sensor layout direction to form a vector field temperature representation, and then merged into the temperature drift compensation model.

[0118] The mechanical installation error between the inertial measurement unit and the navigation body is modeled as a 3×3 rotation matrix.

[0119] The calculation of the drift sensitivity matrix in S3 includes the following steps:

[0120] S3-1. Perform four rounds of temperature excitation on the inertial measurement unit, each round including heating and cooling stages. The temperature change rate is set to ±2℃ / min. Record the temperature change response corresponding to the triaxial acceleration data and triaxial angular velocity data. The sampling frequency is set to 2000Hz.

[0121] S3-2. The least squares method is used to fit the output curve of each round of experimental data with temperature change, extract the slope value of the linear segment, and calculate the average value of the four rounds as the drift response coefficient, which corresponds to the six-axis data respectively, and combine them to form a 6×1 dimension drift sensitivity matrix.

[0122] Specifically, this is a 6-dimensional column vector representing the sensitivity weighting coefficients of each channel in the inertial measurement unit (IMU) to temperature changes, in the form of:

[0123] ,

[0124] Each of them This represents the sensitivity of the drift response to temperature changes along the corresponding axis (e.g., Ax, Gx), and is usually obtained from training models or regression from prior experiments.

[0125] Specifically, each convolutional block uses the BatchNorm normalization operation, and the convolutional kernel weights are initialized using the Kaiming method. The input tensor dimension is in the [batch, channel, length] format, and channel convolutions process each axis data independently at each time point. The residual connections are direct skip connections, without introducing convolutional compression, to ensure gradient propagation efficiency.

[0126] S3-2 slope response vector extraction also includes the following steps:

[0127] S3-2-1. Only segments with residual mean square error less than 0.002 are retained in the fitting results, and the corresponding temperature range must cover at least 10℃. Those that do not meet this requirement are automatically discarded.

[0128] S3-2-2. Perform correlation analysis on the temperature drift coefficients of triaxial acceleration data and triaxial angular velocity data. Channel data with a Pearson correlation coefficient lower than 0.4 are not included in the final matrix to ensure accurate residual extraction direction.

[0129] Specifically, each attention head contains three fully connected layers (Query, Key, Value), all with a dimension of 64. After the dot product attention is calculated, Dropout=0.1 is used to prevent overfitting. The outputs of the four attention heads are concatenated and then mapped back to the original dimension through a fully connected layer. The fusion method uses weighted summation and residual fusion followed by LayerNorm processing.

[0130] The multipath modeling network in S5 also includes the following structures:

[0131] S5-1. Normalize the temperature value to the -1,1 interval, input the piecewise hyperbolic tangent mapping function, and output a 32-dimensional encoded tensor to guide the convolution path selection mask.

[0132] Specifically, construct the basic hyperbolic tangent function:

[0133] The hyperbolic tangent function is defined as:

[0134] ,

[0135] It varies continuously between [-1,1], which can smoothly express the nonlinear characteristics of the input.

[0136] Temperature zone segmentation:

[0137] The input temperature value T is divided into five temperature ranges (e.g., <-20℃, -200℃, 020℃, 20~40℃, >40℃), and different mapping function parameters are set for each range. , ;

[0138] ,

[0139] in Let be the center temperature of the i-th segment.

[0140] fusion processing:

[0141] Finally, all segment outputs are concatenated to form an extended dimension input:

[0142]

[0143] S5-2. Combining the encoding tensor, guide a set of three-layer feature extraction sub-networks to model temperature region features. This sub-network consists of three one-dimensional convolutional layers. Each layer contains a 1D convolutional operation with a kernel size of 5. The number of convolutional channels in each layer are 64, 128, and 256, respectively. Each layer is followed by LayerNorm and GELU activation functions to prevent gradient explosion.

[0144] S5-3. The output of each convolutional layer is fed into a 4-head attention mechanism module. Each head uses a scaled dot product attention function to keep the output dimension consistent and construct an axial feature context.

[0145] Specifically, this refers to an improved multi-head self-attention structure, which splits the input features into four parallel attention heads to perform query-key-value mappings separately, and then integrates their weighted outputs. The calculation method for each head is as follows:

[0146] ,

[0147] in , , Let be the query, key, and value matrix of the i-th head.

[0148] S5-4. Based on the temperature zone label index output by S5-1, activate the corresponding path from five fixed pre-trained paths, and mask the output of the remaining paths to make the modeling of each temperature zone independent.

[0149] Specifically, the tanh family of activation functions uses the standard form tanh(x) to further nonlinearly map the normalized temperature, generating a temperature embedding tensor, which serves as the weight reference input for controlling the activation of each pathway in the subsequent modeling network. This hot encoding is used to drive the adaptive selection of pathway structures in different temperature regions, improving the modeling generalization ability.

[0150] The S5-4 dynamic pathway activation structure also includes the following steps:

[0151] S5-4-1. Divide the temperature values ​​into five segments: extremely low temperature -40℃ to -20℃, low temperature -20℃ to 0℃, medium temperature 0℃ to 30℃, high temperature 30℃ to 55℃, and extremely high temperature 55℃ to 80℃, and establish pathways for each segment.

[0152] S5-4-2. Configure a fixed mask encoding matrix for each temperature range, such as 1, 0, 0, 0, 0. Select the modeling path in the network corresponding to the current temperature range through matrix multiplication and output the compensation tensor.

[0153] Specifically, the five-dimensional mask vector [1,0,0,0,0] is used to activate specific temperature channels in the five-segment temperature mapping structure. After this matrix is ​​matched with the temperature input value segment by segment, the model is guided to activate only the neurons in the corresponding segment, as expressed in the following way:

[0154] ,

[0155] Specifically, the sliding integral window size is set to 128 frames, and the statistical mean is used to suppress transient fluctuations in response. Each temperature zone tensor is a six-dimensional vector, corresponding to triaxial acceleration and angular velocity respectively. The tensor library contains 10 sets of temperature zone distributions, with a structure of [temperature zone index, channel ID, response value] triplet table.

[0156] During the training of the multi-path modeling network in S6, the AdamW optimizer was used, with the initial learning rate set to 0.001 and the weight decay coefficient set to 0.01. After every 100 training rounds, the learning rate was adaptively adjusted to 80% of the original value. Each training batch included 200 sample frames, and the total number of training rounds was 300. The loss function consisted of a linear weighted sum of the in-channel L2 residual term and the full-channel drift balance term, with a ratio of 4:1.

[0157] Specifically, the weight decay coefficient refers to the regularization factor that adds a parameter norm penalty term to the training objective function.

[0158] Specifically, the Butterworth filter implementation uses an IIR structure. Forward and backward filtering calculate the filter delay terms separately and combine the outputs to ensure time alignment with the original signal. The low-pass filter calculation is implemented using digital filtering, with programming languages ​​such as Python or C, and is ported to an embedded processor.

[0159] The S7 maintenance process includes the following steps:

[0160] S7-1. Select the compensation tensor channel group using the current temperature value. Perform element-wise weighting of the six-axis data with the corresponding compensation tensor channel group. The compensation value range is limited to ±0.01g and ±0.01° / s.

[0161] S7-2. All compensated triaxial acceleration and triaxial angular velocity data are processed by a third-order Butterworth low-pass filter with the cutoff frequency set to 0.2 times the original sampling frequency to prevent high-frequency disturbances from being generated during compensation.

[0162] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely preferred examples of the present invention and are not intended to limit the present invention. Various changes and modifications can be made to the present invention without departing from the spirit and scope thereof, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for compensating for temperature drift in inertial navigation based on multidimensional modeling, characterized in that, The inertial navigation temperature drift compensation method based on multidimensional modeling includes the following steps: S1. Under the full temperature range of -40℃ to 80℃, a closed-loop constant temperature control system is used to perform fixed-point temperature control on the inertial measurement unit, and temperature data, triaxial acceleration data and triaxial angular velocity data are collected simultaneously to construct a multivariate feature matrix with a unified timestamp and sample alignment mechanism. S2. After applying standardization and first-order coordinate system rotation transformation to the multivariable feature matrix, construct a fifth-order nonlinear coupling function group using Chebyshev polynomials. With temperature and first-order derivative as inputs and the triaxial acceleration data and triaxial angular velocity data as outputs, establish an orthogonal tensor coupling model and apply L1 norm constraints to achieve sparse modeling. S3. Based on the heating / cooling response experimental data of the inertial measurement unit, construct a set of drift response curves for the triaxial acceleration data and the triaxial angular velocity data, calculate the slope coefficients respectively to obtain a 6×1 dimension drift sensitivity matrix, which is used as the initial directional weight matrix for residual extraction. S4. Perform Mahalanobis distance analysis on the residual data obtained under the effect of the drift sensitivity matrix, calculate the normalized distance based on the sample covariance matrix, remove samples with a confidence level of 95%, and construct a high-confidence training subset. S5. Construct a multipath modeling network including a temperature thermal encoding embedding module, three layers of normalized convolutional channels, a parallel attention mechanism, and a dynamic path selection structure. Input a shrunken high-confidence training subset into the multipath modeling network to perform cross-axis deep modeling of the temperature-output relationship. S6. The multi-path modeling network is trained using an objective function composed of the in-channel L2 constraint loss term and the residual feedback weighting. The steady-state temperature drift compensation tensor is obtained by iterative optimization using a channel-specific, fixed step size, and AdamW optimization strategy. S7. Using the temperature data, the triaxial acceleration data, and the triaxial angular velocity data as inputs, feed them into the trained multi-path modeling network, call the temperature drift compensation tensor, and perform stepwise corrections on the six-axis outputs respectively to complete the active compensation for the temperature drift of the inertial measurement unit. The multi-path modeling network introduces a five-segment temperature mapping mask activation structure, which dynamically adapts the cross-temperature zone modeling performance based on different paths in the temperature segment label activation and suppression network.

2. The inertial navigation temperature drift compensation method based on multidimensional modeling according to claim 1, characterized in that, The construction of the multivariate feature matrix in S1 includes the following steps: S1-1. Under the full temperature range of -40℃ to 80℃, sampling points are configured, and the constant temperature points are set to one sampling point every 2.5℃ in the range of -40℃ to 80℃, for a total of 49 points. The closed-loop constant temperature control system is used to ensure that the stability error of each point is less than ±0.2℃, the data acquisition time for each temperature point is not less than 30 minutes, and the sampling frequency is set to 1000Hz. S1-2. The collected temperature data, the triaxial acceleration data, and the triaxial angular velocity data are aligned by timestamps to form an N-row, 7-column floating-point matrix. The column vectors are, in order, timestamp, temperature value, X / Y / Z acceleration value, and X / Y / Z angular velocity value. Zero-mean, unit variance standardization is used.

3. The inertial navigation temperature drift compensation method based on multidimensional modeling according to claim 2, characterized in that, The floating-point matrix consisting of N rows and 7 columns in S1-2 also includes the following steps: S1-2-1. Using the three-dimensional Euclidean norm of the triaxial acceleration data and the triaxial angular velocity data, samples with a first derivative rate of change exceeding three times the standard deviation of the mean are removed. S1-2-2. Initially fix the inertial measurement unit in a known orientation coordinate system, record the reference three-axis direction vectors through manual calibration, and perform direction cosine matrix remapping processing on the collected temperature data, the three-axis acceleration data, and the three-axis angular velocity data.

4. The inertial navigation temperature drift compensation method based on multidimensional modeling according to claim 1, characterized in that, The calculation of the drift sensitivity matrix in S3 includes the following steps: S3-1. Perform four rounds of temperature excitation on the inertial measurement unit, each round including heating and cooling stages, with the temperature change rate set to ±2℃ / min, record the temperature change response corresponding to the triaxial acceleration data and the triaxial angular velocity data, and set the sampling frequency to 2000Hz; S3-2. The least squares method is used to fit the output curve of each round of experimental data with temperature change, the slope value of the linear segment is extracted, and the average value of the four rounds is calculated as the drift response coefficient, which corresponds to the six-axis data respectively, and is combined to form the 6×1 dimension drift sensitivity matrix.

5. The inertial navigation temperature drift compensation method based on multidimensional modeling according to claim 4, characterized in that, S3-2 further includes the following steps: S3-2-1. Only segments with residual mean square error less than 0.002 are retained in the fitting results, and the corresponding temperature range must cover at least 10℃. Those that do not meet this requirement are automatically discarded. S3-2-2. Perform correlation analysis on the temperature drift coefficients of the triaxial acceleration data and the triaxial angular velocity data. Channel data with a Pearson correlation coefficient lower than 0.4 are not included in the final matrix to ensure accurate residual extraction direction.

6. The inertial navigation temperature drift compensation method based on multidimensional modeling according to claim 1, characterized in that, The multipath modeling network in S5 also includes the following structure: S5-1. Normalize the temperature value to the interval -1,1, input the piecewise hyperbolic tangent mapping function, output a 32-dimensional encoded tensor to guide the convolution path selection mask, and map the normalized temperature value to the corresponding temperature segment label index to activate the matching path and block the other paths. S5-2. Combining the encoding tensor, guide a set of three-layer feature extraction sub-networks to model temperature region features. This sub-network consists of three one-dimensional convolutional layers. Each layer contains a 1D convolutional operation with a kernel size of 5. The number of convolutional channels in each layer are 64, 128, and 256, respectively. Each layer is followed by LayerNorm and GELU activation functions to prevent gradient explosion. S5-3. The output of each convolutional layer is fed into a 4-head attention mechanism module. Each head uses a scaled dot product attention function to keep the output dimension consistent and construct an axial feature context. S5-4. Based on the temperature zone label index output by S5-1, activate the corresponding path from five fixed pre-trained paths, and mask the output of the remaining paths to make the modeling of each temperature zone independent.

7. The inertial navigation temperature drift compensation method based on multidimensional modeling according to claim 6, characterized in that, S5-4 further includes the following steps: S5-4-1. Divide the temperature values ​​into five segments: extremely low temperature -40℃ to -20℃, low temperature -20℃ to 0℃, medium temperature 0℃ to 30℃, high temperature 30℃ to 55℃, and extremely high temperature 55℃ to 80℃, and establish pathways for each segment. S5-4-2. Configure a fixed mask encoding matrix for each temperature range, select the modeling path in the network corresponding to the current temperature range through matrix multiplication, and output the compensation tensor.

8. The inertial navigation temperature drift compensation method based on multidimensional modeling according to claim 1, characterized in that, During the training of the multi-path modeling network in S6, the AdamW optimizer is used, with an initial learning rate of 0.001 and a weight decay coefficient of 0.

01. The learning rate is adaptively adjusted to 80% of its original value after every 100 training rounds. Each training batch includes 200 sample frames, and the total number of training rounds is 300. The loss function consists of a linear weighted sum of the in-channel L2 residual term and the full-channel drift balance term, with a ratio of 4:

1.

9. The inertial navigation temperature drift compensation method based on multidimensional modeling according to claim 1, characterized in that, The step-by-step maintenance process in S7 includes the following steps: S7-1. Select the compensation tensor channel group using the current temperature value, and perform element-wise weighting of the six-axis data with the corresponding compensation tensor channel group. The compensation value range is limited to ±0.01g and ±0.01° / s. S7-2. All compensated triaxial acceleration and triaxial angular velocity data are processed by a third-order Butterworth low-pass filter with the cutoff frequency set to 0.2 times the original sampling frequency to prevent high-frequency disturbances from being generated during compensation.

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