As the core component of an inertial navigation system, the measurement accuracy of the IMU directly determines the overall performance of the navigation system. Two-dimensional calibration of the IMU primarily involves calibrating the error parameters of the accelerometers and gyroscopes in the horizontal plane (typically a combination of pitch-roll or azimuth-pitch). A dual-axis rate table, with its high-precision angle positioning and attitude control capabilities, is the core equipment for achieving this calibration. This article, based on industry standards and engineering practices, details the entire process of two-dimensional IMU calibration using a two-axis rate table, covering four main stages: pre-calibration preparation, core calibration procedures, data processing and verification, and final steps, ensuring the standardization and repeatability of calibration process and reliability of the calibration results.
I. Preparations before calibration
Pre-calibration preparation is fundamental to ensuring calibration accuracy. It needs to be carried out in four aspects: equipment selection and inspection, environmental condition control, IMU installation and debugging, and software system setup, to ensure that each step meets the calibration requirements.
(Ⅰ) Equipment selection and inspection
1. Dual-axis rate table selection : Based on the IMU's accuracy level and calibration requirements, select a dual-axis rate table that meets the requirements for angular position accuracy, angular rate stability, and axis perpendicularity. For medium-to-high accuracy IMUs (such as navigation-grade IMUs), the rate table's angular position accuracy should be better than 10″, and the axis perpendicularity better than 5″; for consumer-grade IMUs, the rate table accuracy can be appropriately reduced (angular position accuracy ≤ 30″). Simultaneously, the rate table must support static positioning and dynamic rate output modes, and meet the calibration requirements for accelerometer zero bias and scale factor, as well as gyroscope zero bias and scale factor.
2. Auxiliary equipment checks : Prepare a high-precision power supply ( output voltage stability ≤0.1% ) to power the IMU, ensuring that voltage fluctuations do not introduce measurement errors; use a data acquisition card ( sampling rate ≥100Hz, resolution ≥16-bit ) to acquire the acceleration and angular velocity signals output by the IMU, as well as the angular position/angular rate feedback signals of the rate table; check the servo control system with the rate table to ensure smooth axis rotation without step loss or jitter. In addition, tools such as a levelling instrument and torque wrench are required for leveling and fixing the IMU after installation.
3. Equipment Calibration and Verification : Preliminary calibration of the dual-axis rate table is performed to verify its angular position , angular rate accuracy, and axis perpendicularity, among other technical specifications . The actual values and commanded values for each axis of the rate table at different angular positions are measured to ensure the deviations are within acceptable limits. The rate table's horizontal reference plane is checked to ensure its levelness is better than 5 ″. Simultaneously, the IMU is powered on and preheated, its initial output status is recorded, and initial equipment malfunctions are eliminated.
(Ⅱ) Environmental condition control
1. Temperature control : The error parameters of the IMU are significantly affected by temperature. The calibration environment temperature should be controlled at (20±2)℃, and the temperature change rate should be ≤0.5℃/h. This can be achieved through a constant temperature laboratory or a temperature control system to ensure temperature stability during calibration and reduce the impact of temperature drift on the calibration results.
2. Vibration and Interference Control : The calibration environment must be far away from vibration sources (such as machine tools, fans , heavy vehicles, etc. ), and vibration isolation measures should be taken on the ground (such as constructing a vibration isolation foundation or installing vibration isolation pads) to ensure that the environmental vibration acceleration is ≤0.01g. At the same time, avoid strong electromagnetic interference, and ground the rate table, IMU and data acquisition equipment (grounding resistance ≤4Ω) to reduce electromagnetic noise interference to the IMU output signal.
3. Air pressure and humidity control : For IMUs that rely on air pressure for calibration (such as some combined IMUs with barometers), the ambient air pressure should be stabilized at standard atmospheric pressure (101.325kPa±1kPa), and the relative humidity should be controlled at 40%~60% to avoid humidity changes causing the internal circuits of the IMU to become damp or the insulation performance to deteriorate.
(Ⅲ) IMU Installation and Debugging
1. Mechanical Installation : Secure the IMU to the worktable of the dual-axis rate table using a dedicated clamp , ensuring that the IMU's sensing axis is aligned with the rate table's coordinate axes. Typically, the IMU's X-axis should be parallel to the rotation axis of the rate table's inner (or outer) axis, and the Z-axis should be perpendicular to the rate table's worktable plane (i.e., along the direction of gravity). Use a torque wrench to tighten the clamp to the specified torque, avoiding excessive looseness which could cause IMU displacement during calibration, or excessive tightness which could cause IMU structural deformation.
2. Axis Alignment Calibration : The alignment accuracy between the IMU and the rate table is calibrated using a level and laser positioning instrument. First, adjust the rate table to a horizontal position, ensuring that the IMU's Z-axis is parallel to the direction of gravity. Then, by rotating the rate table, verify the parallelism between the IMU's sensing axis and the rate table's rotation axis. The parallelism error should be ≤5″. If the alignment accuracy does not meet the requirements, adjust the fixture position and repeat the calibration until it meets the standard.
3. Electrical Connection and Debugging : Connect the IMU to the power supply and data acquisition card, ensuring secure wiring and good contact to avoid signal loss or distortion caused by loose connections. Power on and preheat the IMU; the preheating time depends on the IMU type (navigation-grade IMUs typically require 30-60 minutes, consumer-grade IMUs require 10-20 minutes) to allow the IMU's internal temperature to stabilize. During preheating, monitor the stability of the IMU's output signal. If signal fluctuations, excessive noise, or other abnormalities occur, troubleshoot the wiring or equipment.
(Ⅳ) Software system setup
1. Control software configuration : Install the dual-axis rate table control software and configure the rate table's axis parameters (such as shaft diameter, transmission ratio), control mode (static/dynamic), angular position/angular rate settings , etc. Simultaneously, set the data acquisition trigger conditions to ensure that data acquisition only begins after the rate table's attitude has stabilized, avoiding signal interference during the transition process.
2. Data acquisition software debugging : Debug the data acquisition software, setting parameters such as sampling rate, sampling duration, and data storage format (e.g., CSV, MAT file). Establish a synchronous acquisition mechanism for the IMU output signal and the rate table feedback signal, ensuring that their timestamps are aligned with an error ≤1ms. Verify the integrity and accuracy of data acquisition through simulated acquisition tests, and troubleshoot issues such as data loss and delays.
3. Calibration algorithm deployment : Based on calibration requirements (such as accelerometer bias/scaling factor calibration, gyroscope bias/scaling factor calibration), deploy the corresponding calibration algorithm (such as least squares method, Kalman filter method). Initialize the algorithm parameters, such as the number of iterations and convergence threshold, to ensure that the algorithm can accurately solve for the IMU's error parameters.
II. Core Calibration Process
The core calibration process revolves around the two core components of the IMU: the accelerometer and the gyroscope. Based on the static positioning and dynamic rate control capabilities of the dual-axis rate table, the error parameters in the two dimensions are calibrated step by step. This process takes the "pitch-roll" two-dimensional calibration as an example, covering three key steps: accelerometer static calibration, gyroscope static zero-bias calibration, and gyroscope dynamic rate calibration.
(Ⅰ) Static calibration of accelerometer
The purpose of static calibration of an accelerometer is to solve for its zero bias and scaling factor. It uses the projection of gravitational acceleration under different attitudes as a reference input, and establishes an error model and solves for the parameters by measuring the acceleration signal output by the IMU.
1. Attitude planning for calibration : Based on the pitch and roll two-dimensional directions, six typical static attitudes are planned (ensuring that gravitational acceleration can fully cover the X, Y, and Z sensitive axes of the accelerometer). The specific attitudes are as follows: ① Pitch 0°, Roll 0° (Z-axis positive along the direction of gravity); ② Pitch 0°, Roll 180° (Z-axis negative along the direction of gravity); ③ Pitch 90°, Roll 0° (X-axis positive along the direction of gravity); ④ Pitch 90°, Roll 180° (X-axis negative along the direction of gravity); ⑤ Pitch 0°, Roll 90° (Y-axis positive along the direction of gravity); ⑥ Pitch 0°, Roll 270° (Y-axis negative along the direction of gravity).
2. Attitude Adjustment and Stabilization : Angular position commands for each attitude are sequentially sent via the dual-axis rate table control software. After the rate table drives the IMU to rotate to the target attitude, it remains statically stable. The stabilization time for each attitude is ≥30s, ensuring the stability of the acceleration signal output by the IMU (signal fluctuation amplitude ≤0.001g). During stabilization, the angular position feedback signal of the rate table is monitored in real time. If the attitude deviation exceeds the allowable range (≤5″), the rate table automatically performs compensation adjustments.
3. Data Acquisition and Recording : After each attitude stabilizes, the data acquisition software is activated to acquire the X, Y, and Z axis acceleration signals output by the IMU. The sampling duration is ≥10s, and the sampling rate is ≥100Hz. Simultaneously, the actual angular position of the rate table (pitch angle θ, roll angle φ) is recorded to calculate the projection values of gravitational acceleration on each sensitive axis (reference input). The acquired data is stored according to attitude, clearly labeled with attitude information and timestamps.
4. Error model establishment and parameter solution : The error model of the accelerometer is established, ignoring cross-coupling errors (which can be simplified in two-dimensional calibration). The error model is as follows:
a = K(a + b) (i=X,Y,Z)
Where a is the acceleration of the i-th axis output by the IMU, K is the scale factor of the i-th axis, a is the reference acceleration of the i-th axis (projection of gravitational acceleration), and b is the zero bias of the i-th axis. Based on the reference acceleration a (calculated from θ and φ, such as Z-axis reference acceleration a=g·cosθ·cosφ, X-axis reference acceleration a=g·sinθ, Y-axis reference acceleration a=g·sinφ·cosθ, where g is gravitational acceleration, taken as 9.80665m/s²) and the corresponding a, K and b are solved using the least squares method.
(Ⅱ)Gyroscope static zero bias calibration
The static zero bias of a gyroscope refers to the output deviation of the gyroscope when there is no angular rate input. It needs to be solved by long-term data acquisition while the IMU is stationary.
(Ⅲ)Gyroscope dynamic rate calibration
The purpose of gyroscope dynamic rate calibration is to solve for its scaling factor. Using the known angular rate output by the dual-axis rate table as a reference input, an error model is established and the scaling factor is solved by measuring the output signal of the gyroscope.
1. Calibration attitude selection : Select a horizontal attitude with 0° pitch and 0° roll. At this attitude, the IMU has no angular rate input, and the gyroscope output only contains zero bias and noise. The rate table does not need to rotate in this attitude; simply keep the stage horizontal and stable.
2. Long-term data acquisition : Start the data acquisition software and acquire the output signals of the gyroscope's X, Y, and Z axes. The sampling time should be ≥60 minutes and the sampling rate ≥100Hz. During the acquisition process, continuously monitor the ambient temperature and rate table attitude to ensure temperature stability (fluctuation ≤0.2℃) and no attitude drift (deviation ≤5″) to avoid introducing additional errors from external factors.
3. Zero bias calculation : The acquired gyroscope output data is preprocessed to remove outliers (using the 3σ criterion), and then the average value of the output signal of each axis is calculated. This average value is the static zero bias b of the gyroscope (i=X,Y,Z). At the same time, the standard deviation of the data is calculated to assess the noise level of the gyroscope. If the standard deviation is too large (exceeding the IMU technical specifications), equipment failure or environmental interference needs to be investigated.
4. Rate point planning : Based on the IMU's range and the actual application scenario, plan dynamic rate points in both pitch and roll dimensions. Select 5-7 rate points for each dimension, covering forward and reverse rates (e.g., -100°/s, -50°/s, 0°/s, 50°/s, 100°/s), where the 0°/s rate point is used to verify the consistency of static zero bias. The selection of rate points must ensure that they do not exceed the IMU's range and that the rate table can stably output the rate (rate stability ≤ 0.1°/s).
5. Rate Output and Stabilization : Commands for each rate point are sequentially sent in the pitch and roll dimensions via dual-axis rate table control software. After the rate table drives the IMU to rotate to the target rate, it maintains dynamic stability with a stabilization time ≥20s. During stabilization, the angular rate feedback signal of the rate table is monitored in real time. If the rate deviation exceeds the allowable range (≤0.5°/s), the rate table automatically performs rate compensation.
6. Data Acquisition and Recording : After each rate point stabilizes, start the data acquisition software to acquire the output signal of the corresponding sensitive axis of the gyroscope (e.g., acquire the X-axis gyroscope output when rotating in the pitch dimension, and acquire the Y-axis gyroscope output when rotating in the roll dimension). The sampling time is ≥10s, and the sampling rate is ≥100Hz. At the same time, record the actual angular velocity of the rate table (reference input ω), and store the data according to the rate point and dimension.
7. Error Model Establishment and Parameter Solving : A rate error model for the gyroscope is established, ignoring cross-coupling errors. The model is as follows:
ω = K(ω + b) (i=X,Y)
Where ω is the output angular rate of the i-th axis of the gyroscope, K is the scale factor of the i-th axis, ω is the reference angular rate of the i-th axis (the actual output rate of the rate table), and b is the static zero bias of the i-th axis (already solved in the static calibration). Substitute ω and the corresponding ω at each rate point into the model, and solve for K using the least squares method.
Ⅲ. Data processing and validation
Data processing and verification are key steps to ensure the reliability of calibration results. The raw data collected must be preprocessed, and after solving for error parameters, residual analysis, repeatability verification, and accuracy verification must be performed. If the verification fails, the process must be returned to the core calibration procedure for recalibration.
1. Outlier removal : The 3σ criterion or Grubbs criterion is used to detect and remove outliers from the original data (acceleration, angular rate signals). For the 3σ criterion, the mean μ and standard deviation σ of the data are calculated. Data exceeding the range [μ-3σ, μ+3σ] are identified as outliers and replaced by interpolation of adjacent data or directly removed.
2. Filtering : The preprocessed raw data is low-pass filtered to remove high-frequency noise. A Butterworth low-pass filter is selected, and the cutoff frequency is determined based on the IMU bandwidth (usually 1/5 to 1/3 of the IMU bandwidth) to avoid over-filtering and signal distortion. The filtered data is used for subsequent error parameter calculation.
3. Data synchronization alignment : To address the timestamp discrepancy between the IMU output signal and the rate table feedback signal, linear interpolation is used for synchronization alignment. This ensures that each set of IMU output data corresponds to an accurate rate table attitude or rate state, with a synchronization error ≤1ms.
4. Parameter solution optimization: Substitute the preprocessed data into the error models of the accelerometer and gyroscope, and use the least squares method to solve for error parameters such as zero bias and scaling factor. For complex scenarios, the Kalman filter method can be used to optimize the parameter solution results, improving the accuracy and stability of parameter estimation.
5. Residual analysis : Calculate the residuals between the observed values (IMU output) and the model predictions at each calibrated attitude/rate point. The residuals reflect the fitting accuracy of the error model. If the mean of the residuals is close to 0 and the standard deviation is small (acceleration residual standard deviation ≤ 0.002g, angular rate residual standard deviation ≤ 0.1°/s), it indicates that the model fits well. If the residuals are too large or show a clear trend, the error model (e.g., considering cross-coupling error) or the validity of the calibration data needs to be re-examined.
6. Repeatability verification : Under the same environmental conditions and calibration procedures, perform three complete calibration experiments and determine the error parameters for each calibration. Calculate the coefficient of variation (the ratio of standard deviation to mean) of the three parameters. If the coefficient of variation is ≤1%, the calibration results have good repeatability; if the coefficient of variation is too large, issues such as equipment stability and environmental interference need to be investigated, and recalibration should be performed.
7. Accuracy Verification : Select attitude/velocity points not involved in calibration as verification points. Substitute the calibrated error parameters into the error model to compensate the IMU output, and calculate the error between the compensated IMU output and the reference input. If the compensated error meets the IMU technical specifications (e.g., acceleration measurement error ≤ 0.01g, angular rate measurement error ≤ 0.5°/s), the calibration accuracy is satisfactory. If the error does not meet the requirements, the calibration process needs to be optimized again (e.g., add more attitude/velocity points for calibration, adjust the error model), and calibration should be performed again.
8. Temperature stability verification (optional) : If the IMU needs to operate over a wide temperature range, calibration experiments can be repeated at different temperature points (e.g., -10℃, 0℃, 20℃, 40℃, 60℃) to verify the variation of error parameters with temperature. A temperature compensation model for the error parameters can be established to improve the measurement accuracy of the IMU under different temperature conditions.
9. Data classification storage : Preprocessed raw data, error parameter solution results, residual analysis reports, verification results, etc., are categorized and stored according to calibration date, IMU number, and calibration environment conditions. Data storage formats adopt common formats (such as CSV, MAT, PDF) to ensure data readability and traceability.
10. Data backup : Perform multiple backups of archived data (such as local hard drives and cloud storage) to prevent data loss. Backup data must have clearly labeled filenames and explanatory documents, clearly defining the corresponding target, process, and conditions.
Ⅳ. Finishing work
The final steps mainly include archiving calibration data, restoring and maintaining equipment, and preparing a calibration report to ensure the traceability of the calibration process and provide a basis for the subsequent use and maintenance of the IMU. The calibration report is a summary of the calibration work and must comprehensively and accurately record the calibration process and results, mainly including the following:
1. Equipment shutdown and disassembly : After calibration, turn off the power to the dual-axis rate table, IMU, and data acquisition equipment. Disconnect the IMU from the fixture in sequence and remove the IMU. Avoid collisions and vibrations during disassembly to protect the sensitive components of the IMU.
2. Equipment cleaning and maintenance : Clean the dual-axis rate table , shaft system, and fixtures to remove dust and debris; perform a visual inspection of the IMU to ensure it is undamaged and that the wiring ports are clean. Record the equipment's usage status and maintenance details to provide a basis for periodic equipment calibration.
3. Equipment parameter restoration : Restore the parameters of the dual-axis rate table and data acquisition equipment to their default states, close the control software and acquisition software, and ensure that the equipment is in a safe standby state.
4. The calibration report includes the following:
(1) Calibration object information: IMU model, serial number, manufacturer, and technical specifications;
(2) Calibration equipment information: dual-axis rate table model and accuracy class, data acquisition equipment model and sampling parameters, and auxiliary equipment list;
(3) Calibrating environmental conditions: temperature, humidity, air pressure, vibration;
(4) Calibration process description: Calibration attitude/velocity point planning, data acquisition parameters, error model, and solution algorithm;
(5) Calibration results: accelerometer zero bias and scale factor, gyroscope zero bias and scale factor, residual analysis results, repeatability verification results, and accuracy verification results;
(6) Conclusions and recommendations: Whether the calibration results meet the standards, recommendations for IMU usage (such as temperature compensation, periodic recalibration cycle), and recommendations for equipment maintenance.
Ⅴ. Precautions
In summary, the standard procedure for IMU two-dimensional calibration using a dual-axis rate table must strictly follow the logical sequence of "pre-calibration preparation - core calibration - data processing and verification - finishing work," focusing on key aspects such as equipment accuracy, environmental control, axis alignment, and data synchronization. Through a standardized calibration procedure and rigorous verification methods, the IMU's error parameters can be accurately determined, significantly improving its measurement accuracy and ensuring the reliable operation of the inertial navigation system.
1. If rate table attitude drift or abnormal IMU output signal occurs during calibration, calibration must be stopped immediately, the fault must be investigated, and the calibration must be restarted to avoid generating invalid calibration data.
2. The preheating time of the IMU must strictly adhere to the technical requirements. Insufficient preheating will lead to unstable error parameters and affect calibration accuracy.
3. The alignment accuracy of the axis system of a dual-axis rate table directly affects the calibration results. The rate table needs to be calibrated regularly to ensure that the axis system accuracy meets the requirements.
4. The temperature, vibration, electromagnetic interference and other factors of the calibration environment have a significant impact on the IMU output. Environmental conditions must be strictly controlled and isolation and shielding measures should be taken when necessary.
5. The calibration report must be reviewed by professionals to ensure the accuracy and standardization of the report content, and it should be archived and stored after the review is approved.
As the core component of an inertial navigation system, the measurement accuracy of the IMU directly determines the overall performance of the navigation system. Two-dimensional calibration of the IMU primarily involves calibrating the error parameters of the accelerometers and gyroscopes in the horizontal plane (typically a combination of pitch-roll or azimuth-pitch). A dual-axis rate table, with its high-precision angle positioning and attitude control capabilities, is the core equipment for achieving this calibration. This article, based on industry standards and engineering practices, details the entire process of two-dimensional IMU calibration using a two-axis rate table, covering four main stages: pre-calibration preparation, core calibration procedures, data processing and verification, and final steps, ensuring the standardization and repeatability of calibration process and reliability of the calibration results.
I. Preparations before calibration
Pre-calibration preparation is fundamental to ensuring calibration accuracy. It needs to be carried out in four aspects: equipment selection and inspection, environmental condition control, IMU installation and debugging, and software system setup, to ensure that each step meets the calibration requirements.
(Ⅰ) Equipment selection and inspection
1. Dual-axis rate table selection : Based on the IMU's accuracy level and calibration requirements, select a dual-axis rate table that meets the requirements for angular position accuracy, angular rate stability, and axis perpendicularity. For medium-to-high accuracy IMUs (such as navigation-grade IMUs), the rate table's angular position accuracy should be better than 10″, and the axis perpendicularity better than 5″; for consumer-grade IMUs, the rate table accuracy can be appropriately reduced (angular position accuracy ≤ 30″). Simultaneously, the rate table must support static positioning and dynamic rate output modes, and meet the calibration requirements for accelerometer zero bias and scale factor, as well as gyroscope zero bias and scale factor.
2. Auxiliary equipment checks : Prepare a high-precision power supply ( output voltage stability ≤0.1% ) to power the IMU, ensuring that voltage fluctuations do not introduce measurement errors; use a data acquisition card ( sampling rate ≥100Hz, resolution ≥16-bit ) to acquire the acceleration and angular velocity signals output by the IMU, as well as the angular position/angular rate feedback signals of the rate table; check the servo control system with the rate table to ensure smooth axis rotation without step loss or jitter. In addition, tools such as a levelling instrument and torque wrench are required for leveling and fixing the IMU after installation.
3. Equipment Calibration and Verification : Preliminary calibration of the dual-axis rate table is performed to verify its angular position , angular rate accuracy, and axis perpendicularity, among other technical specifications . The actual values and commanded values for each axis of the rate table at different angular positions are measured to ensure the deviations are within acceptable limits. The rate table's horizontal reference plane is checked to ensure its levelness is better than 5 ″. Simultaneously, the IMU is powered on and preheated, its initial output status is recorded, and initial equipment malfunctions are eliminated.
(Ⅱ) Environmental condition control
1. Temperature control : The error parameters of the IMU are significantly affected by temperature. The calibration environment temperature should be controlled at (20±2)℃, and the temperature change rate should be ≤0.5℃/h. This can be achieved through a constant temperature laboratory or a temperature control system to ensure temperature stability during calibration and reduce the impact of temperature drift on the calibration results.
2. Vibration and Interference Control : The calibration environment must be far away from vibration sources (such as machine tools, fans , heavy vehicles, etc. ), and vibration isolation measures should be taken on the ground (such as constructing a vibration isolation foundation or installing vibration isolation pads) to ensure that the environmental vibration acceleration is ≤0.01g. At the same time, avoid strong electromagnetic interference, and ground the rate table, IMU and data acquisition equipment (grounding resistance ≤4Ω) to reduce electromagnetic noise interference to the IMU output signal.
3. Air pressure and humidity control : For IMUs that rely on air pressure for calibration (such as some combined IMUs with barometers), the ambient air pressure should be stabilized at standard atmospheric pressure (101.325kPa±1kPa), and the relative humidity should be controlled at 40%~60% to avoid humidity changes causing the internal circuits of the IMU to become damp or the insulation performance to deteriorate.
(Ⅲ) IMU Installation and Debugging
1. Mechanical Installation : Secure the IMU to the worktable of the dual-axis rate table using a dedicated clamp , ensuring that the IMU's sensing axis is aligned with the rate table's coordinate axes. Typically, the IMU's X-axis should be parallel to the rotation axis of the rate table's inner (or outer) axis, and the Z-axis should be perpendicular to the rate table's worktable plane (i.e., along the direction of gravity). Use a torque wrench to tighten the clamp to the specified torque, avoiding excessive looseness which could cause IMU displacement during calibration, or excessive tightness which could cause IMU structural deformation.
2. Axis Alignment Calibration : The alignment accuracy between the IMU and the rate table is calibrated using a level and laser positioning instrument. First, adjust the rate table to a horizontal position, ensuring that the IMU's Z-axis is parallel to the direction of gravity. Then, by rotating the rate table, verify the parallelism between the IMU's sensing axis and the rate table's rotation axis. The parallelism error should be ≤5″. If the alignment accuracy does not meet the requirements, adjust the fixture position and repeat the calibration until it meets the standard.
3. Electrical Connection and Debugging : Connect the IMU to the power supply and data acquisition card, ensuring secure wiring and good contact to avoid signal loss or distortion caused by loose connections. Power on and preheat the IMU; the preheating time depends on the IMU type (navigation-grade IMUs typically require 30-60 minutes, consumer-grade IMUs require 10-20 minutes) to allow the IMU's internal temperature to stabilize. During preheating, monitor the stability of the IMU's output signal. If signal fluctuations, excessive noise, or other abnormalities occur, troubleshoot the wiring or equipment.
(Ⅳ) Software system setup
1. Control software configuration : Install the dual-axis rate table control software and configure the rate table's axis parameters (such as shaft diameter, transmission ratio), control mode (static/dynamic), angular position/angular rate settings , etc. Simultaneously, set the data acquisition trigger conditions to ensure that data acquisition only begins after the rate table's attitude has stabilized, avoiding signal interference during the transition process.
2. Data acquisition software debugging : Debug the data acquisition software, setting parameters such as sampling rate, sampling duration, and data storage format (e.g., CSV, MAT file). Establish a synchronous acquisition mechanism for the IMU output signal and the rate table feedback signal, ensuring that their timestamps are aligned with an error ≤1ms. Verify the integrity and accuracy of data acquisition through simulated acquisition tests, and troubleshoot issues such as data loss and delays.
3. Calibration algorithm deployment : Based on calibration requirements (such as accelerometer bias/scaling factor calibration, gyroscope bias/scaling factor calibration), deploy the corresponding calibration algorithm (such as least squares method, Kalman filter method). Initialize the algorithm parameters, such as the number of iterations and convergence threshold, to ensure that the algorithm can accurately solve for the IMU's error parameters.
II. Core Calibration Process
The core calibration process revolves around the two core components of the IMU: the accelerometer and the gyroscope. Based on the static positioning and dynamic rate control capabilities of the dual-axis rate table, the error parameters in the two dimensions are calibrated step by step. This process takes the "pitch-roll" two-dimensional calibration as an example, covering three key steps: accelerometer static calibration, gyroscope static zero-bias calibration, and gyroscope dynamic rate calibration.
(Ⅰ) Static calibration of accelerometer
The purpose of static calibration of an accelerometer is to solve for its zero bias and scaling factor. It uses the projection of gravitational acceleration under different attitudes as a reference input, and establishes an error model and solves for the parameters by measuring the acceleration signal output by the IMU.
1. Attitude planning for calibration : Based on the pitch and roll two-dimensional directions, six typical static attitudes are planned (ensuring that gravitational acceleration can fully cover the X, Y, and Z sensitive axes of the accelerometer). The specific attitudes are as follows: ① Pitch 0°, Roll 0° (Z-axis positive along the direction of gravity); ② Pitch 0°, Roll 180° (Z-axis negative along the direction of gravity); ③ Pitch 90°, Roll 0° (X-axis positive along the direction of gravity); ④ Pitch 90°, Roll 180° (X-axis negative along the direction of gravity); ⑤ Pitch 0°, Roll 90° (Y-axis positive along the direction of gravity); ⑥ Pitch 0°, Roll 270° (Y-axis negative along the direction of gravity).
2. Attitude Adjustment and Stabilization : Angular position commands for each attitude are sequentially sent via the dual-axis rate table control software. After the rate table drives the IMU to rotate to the target attitude, it remains statically stable. The stabilization time for each attitude is ≥30s, ensuring the stability of the acceleration signal output by the IMU (signal fluctuation amplitude ≤0.001g). During stabilization, the angular position feedback signal of the rate table is monitored in real time. If the attitude deviation exceeds the allowable range (≤5″), the rate table automatically performs compensation adjustments.
3. Data Acquisition and Recording : After each attitude stabilizes, the data acquisition software is activated to acquire the X, Y, and Z axis acceleration signals output by the IMU. The sampling duration is ≥10s, and the sampling rate is ≥100Hz. Simultaneously, the actual angular position of the rate table (pitch angle θ, roll angle φ) is recorded to calculate the projection values of gravitational acceleration on each sensitive axis (reference input). The acquired data is stored according to attitude, clearly labeled with attitude information and timestamps.
4. Error model establishment and parameter solution : The error model of the accelerometer is established, ignoring cross-coupling errors (which can be simplified in two-dimensional calibration). The error model is as follows:
a = K(a + b) (i=X,Y,Z)
Where a is the acceleration of the i-th axis output by the IMU, K is the scale factor of the i-th axis, a is the reference acceleration of the i-th axis (projection of gravitational acceleration), and b is the zero bias of the i-th axis. Based on the reference acceleration a (calculated from θ and φ, such as Z-axis reference acceleration a=g·cosθ·cosφ, X-axis reference acceleration a=g·sinθ, Y-axis reference acceleration a=g·sinφ·cosθ, where g is gravitational acceleration, taken as 9.80665m/s²) and the corresponding a, K and b are solved using the least squares method.
(Ⅱ)Gyroscope static zero bias calibration
The static zero bias of a gyroscope refers to the output deviation of the gyroscope when there is no angular rate input. It needs to be solved by long-term data acquisition while the IMU is stationary.
(Ⅲ)Gyroscope dynamic rate calibration
The purpose of gyroscope dynamic rate calibration is to solve for its scaling factor. Using the known angular rate output by the dual-axis rate table as a reference input, an error model is established and the scaling factor is solved by measuring the output signal of the gyroscope.
1. Calibration attitude selection : Select a horizontal attitude with 0° pitch and 0° roll. At this attitude, the IMU has no angular rate input, and the gyroscope output only contains zero bias and noise. The rate table does not need to rotate in this attitude; simply keep the stage horizontal and stable.
2. Long-term data acquisition : Start the data acquisition software and acquire the output signals of the gyroscope's X, Y, and Z axes. The sampling time should be ≥60 minutes and the sampling rate ≥100Hz. During the acquisition process, continuously monitor the ambient temperature and rate table attitude to ensure temperature stability (fluctuation ≤0.2℃) and no attitude drift (deviation ≤5″) to avoid introducing additional errors from external factors.
3. Zero bias calculation : The acquired gyroscope output data is preprocessed to remove outliers (using the 3σ criterion), and then the average value of the output signal of each axis is calculated. This average value is the static zero bias b of the gyroscope (i=X,Y,Z). At the same time, the standard deviation of the data is calculated to assess the noise level of the gyroscope. If the standard deviation is too large (exceeding the IMU technical specifications), equipment failure or environmental interference needs to be investigated.
4. Rate point planning : Based on the IMU's range and the actual application scenario, plan dynamic rate points in both pitch and roll dimensions. Select 5-7 rate points for each dimension, covering forward and reverse rates (e.g., -100°/s, -50°/s, 0°/s, 50°/s, 100°/s), where the 0°/s rate point is used to verify the consistency of static zero bias. The selection of rate points must ensure that they do not exceed the IMU's range and that the rate table can stably output the rate (rate stability ≤ 0.1°/s).
5. Rate Output and Stabilization : Commands for each rate point are sequentially sent in the pitch and roll dimensions via dual-axis rate table control software. After the rate table drives the IMU to rotate to the target rate, it maintains dynamic stability with a stabilization time ≥20s. During stabilization, the angular rate feedback signal of the rate table is monitored in real time. If the rate deviation exceeds the allowable range (≤0.5°/s), the rate table automatically performs rate compensation.
6. Data Acquisition and Recording : After each rate point stabilizes, start the data acquisition software to acquire the output signal of the corresponding sensitive axis of the gyroscope (e.g., acquire the X-axis gyroscope output when rotating in the pitch dimension, and acquire the Y-axis gyroscope output when rotating in the roll dimension). The sampling time is ≥10s, and the sampling rate is ≥100Hz. At the same time, record the actual angular velocity of the rate table (reference input ω), and store the data according to the rate point and dimension.
7. Error Model Establishment and Parameter Solving : A rate error model for the gyroscope is established, ignoring cross-coupling errors. The model is as follows:
ω = K(ω + b) (i=X,Y)
Where ω is the output angular rate of the i-th axis of the gyroscope, K is the scale factor of the i-th axis, ω is the reference angular rate of the i-th axis (the actual output rate of the rate table), and b is the static zero bias of the i-th axis (already solved in the static calibration). Substitute ω and the corresponding ω at each rate point into the model, and solve for K using the least squares method.
Ⅲ. Data processing and validation
Data processing and verification are key steps to ensure the reliability of calibration results. The raw data collected must be preprocessed, and after solving for error parameters, residual analysis, repeatability verification, and accuracy verification must be performed. If the verification fails, the process must be returned to the core calibration procedure for recalibration.
1. Outlier removal : The 3σ criterion or Grubbs criterion is used to detect and remove outliers from the original data (acceleration, angular rate signals). For the 3σ criterion, the mean μ and standard deviation σ of the data are calculated. Data exceeding the range [μ-3σ, μ+3σ] are identified as outliers and replaced by interpolation of adjacent data or directly removed.
2. Filtering : The preprocessed raw data is low-pass filtered to remove high-frequency noise. A Butterworth low-pass filter is selected, and the cutoff frequency is determined based on the IMU bandwidth (usually 1/5 to 1/3 of the IMU bandwidth) to avoid over-filtering and signal distortion. The filtered data is used for subsequent error parameter calculation.
3. Data synchronization alignment : To address the timestamp discrepancy between the IMU output signal and the rate table feedback signal, linear interpolation is used for synchronization alignment. This ensures that each set of IMU output data corresponds to an accurate rate table attitude or rate state, with a synchronization error ≤1ms.
4. Parameter solution optimization: Substitute the preprocessed data into the error models of the accelerometer and gyroscope, and use the least squares method to solve for error parameters such as zero bias and scaling factor. For complex scenarios, the Kalman filter method can be used to optimize the parameter solution results, improving the accuracy and stability of parameter estimation.
5. Residual analysis : Calculate the residuals between the observed values (IMU output) and the model predictions at each calibrated attitude/rate point. The residuals reflect the fitting accuracy of the error model. If the mean of the residuals is close to 0 and the standard deviation is small (acceleration residual standard deviation ≤ 0.002g, angular rate residual standard deviation ≤ 0.1°/s), it indicates that the model fits well. If the residuals are too large or show a clear trend, the error model (e.g., considering cross-coupling error) or the validity of the calibration data needs to be re-examined.
6. Repeatability verification : Under the same environmental conditions and calibration procedures, perform three complete calibration experiments and determine the error parameters for each calibration. Calculate the coefficient of variation (the ratio of standard deviation to mean) of the three parameters. If the coefficient of variation is ≤1%, the calibration results have good repeatability; if the coefficient of variation is too large, issues such as equipment stability and environmental interference need to be investigated, and recalibration should be performed.
7. Accuracy Verification : Select attitude/velocity points not involved in calibration as verification points. Substitute the calibrated error parameters into the error model to compensate the IMU output, and calculate the error between the compensated IMU output and the reference input. If the compensated error meets the IMU technical specifications (e.g., acceleration measurement error ≤ 0.01g, angular rate measurement error ≤ 0.5°/s), the calibration accuracy is satisfactory. If the error does not meet the requirements, the calibration process needs to be optimized again (e.g., add more attitude/velocity points for calibration, adjust the error model), and calibration should be performed again.
8. Temperature stability verification (optional) : If the IMU needs to operate over a wide temperature range, calibration experiments can be repeated at different temperature points (e.g., -10℃, 0℃, 20℃, 40℃, 60℃) to verify the variation of error parameters with temperature. A temperature compensation model for the error parameters can be established to improve the measurement accuracy of the IMU under different temperature conditions.
9. Data classification storage : Preprocessed raw data, error parameter solution results, residual analysis reports, verification results, etc., are categorized and stored according to calibration date, IMU number, and calibration environment conditions. Data storage formats adopt common formats (such as CSV, MAT, PDF) to ensure data readability and traceability.
10. Data backup : Perform multiple backups of archived data (such as local hard drives and cloud storage) to prevent data loss. Backup data must have clearly labeled filenames and explanatory documents, clearly defining the corresponding target, process, and conditions.
Ⅳ. Finishing work
The final steps mainly include archiving calibration data, restoring and maintaining equipment, and preparing a calibration report to ensure the traceability of the calibration process and provide a basis for the subsequent use and maintenance of the IMU. The calibration report is a summary of the calibration work and must comprehensively and accurately record the calibration process and results, mainly including the following:
1. Equipment shutdown and disassembly : After calibration, turn off the power to the dual-axis rate table, IMU, and data acquisition equipment. Disconnect the IMU from the fixture in sequence and remove the IMU. Avoid collisions and vibrations during disassembly to protect the sensitive components of the IMU.
2. Equipment cleaning and maintenance : Clean the dual-axis rate table , shaft system, and fixtures to remove dust and debris; perform a visual inspection of the IMU to ensure it is undamaged and that the wiring ports are clean. Record the equipment's usage status and maintenance details to provide a basis for periodic equipment calibration.
3. Equipment parameter restoration : Restore the parameters of the dual-axis rate table and data acquisition equipment to their default states, close the control software and acquisition software, and ensure that the equipment is in a safe standby state.
4. The calibration report includes the following:
(1) Calibration object information: IMU model, serial number, manufacturer, and technical specifications;
(2) Calibration equipment information: dual-axis rate table model and accuracy class, data acquisition equipment model and sampling parameters, and auxiliary equipment list;
(3) Calibrating environmental conditions: temperature, humidity, air pressure, vibration;
(4) Calibration process description: Calibration attitude/velocity point planning, data acquisition parameters, error model, and solution algorithm;
(5) Calibration results: accelerometer zero bias and scale factor, gyroscope zero bias and scale factor, residual analysis results, repeatability verification results, and accuracy verification results;
(6) Conclusions and recommendations: Whether the calibration results meet the standards, recommendations for IMU usage (such as temperature compensation, periodic recalibration cycle), and recommendations for equipment maintenance.
Ⅴ. Precautions
In summary, the standard procedure for IMU two-dimensional calibration using a dual-axis rate table must strictly follow the logical sequence of "pre-calibration preparation - core calibration - data processing and verification - finishing work," focusing on key aspects such as equipment accuracy, environmental control, axis alignment, and data synchronization. Through a standardized calibration procedure and rigorous verification methods, the IMU's error parameters can be accurately determined, significantly improving its measurement accuracy and ensuring the reliable operation of the inertial navigation system.
1. If rate table attitude drift or abnormal IMU output signal occurs during calibration, calibration must be stopped immediately, the fault must be investigated, and the calibration must be restarted to avoid generating invalid calibration data.
2. The preheating time of the IMU must strictly adhere to the technical requirements. Insufficient preheating will lead to unstable error parameters and affect calibration accuracy.
3. The alignment accuracy of the axis system of a dual-axis rate table directly affects the calibration results. The rate table needs to be calibrated regularly to ensure that the axis system accuracy meets the requirements.
4. The temperature, vibration, electromagnetic interference and other factors of the calibration environment have a significant impact on the IMU output. Environmental conditions must be strictly controlled and isolation and shielding measures should be taken when necessary.
5. The calibration report must be reviewed by professionals to ensure the accuracy and standardization of the report content, and it should be archived and stored after the review is approved.
