As the core component of an inertial navigation system, the measurement accuracy of an inertial measurement unit (IMU) directly determines the overall performance of the navigation system. The two-dimensional calibration of IMU mainly focuses on calibrating the error parameters of accelerometers and gyroscopes in the horizontal plane (usually pitch roll or azimuth pitch combination). The dual axis turntable, with its high-precision angle positioning and attitude control capabilities, has become the core equipment for achieving this calibration. Based on industry standards and engineering practices, this article elaborates on the entire process of using a dual axis turntable for IMU two-dimensional calibration, covering four major steps: pre calibration preparation, core calibration process, data processing and verification, and final work, ensuring the standardization, repeatability, and reliability of the calibration process and results.

1、 Preparation before calibration

Preparation before calibration is the foundation for ensuring calibration accuracy, which needs to be carried out from four aspects: equipment selection and inspection, environmental condition control, IMU installation and debugging, and software system construction, to ensure that each link meets the calibration requirements.

（1） Equipment selection and inspection

1.  Dual axis turntable selection: Based on the accuracy level and calibration requirements of IMU, select a dual axis turntable that meets the requirements for angular position accuracy, angular velocity stability, and shaft verticality. For medium to high precision IMUs (such as navigation grade), the accuracy of the turntable angle position should be better than 10 inches, and the verticality of the shaft system should be better than 5 inches; For consumer grade IMUs, the turntable accuracy can be appropriately reduced (angular position accuracy ≤ 30 "). At the same time, the turntable needs to support static positioning and dynamic rate output modes to meet the calibration requirements of accelerometer zero bias, scale factor, and gyroscope zero bias and scale factor.

2.  Auxiliary equipment inspection: 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 bits) to collect the acceleration and angular velocity signals output by the IMU, as well as the angular position/velocity feedback signals of the turntable; Check the servo control system of the turntable to ensure smooth rotation of the shaft system without any step loss or shaking. In addition, tools such as level gauges and torque wrenches need to be prepared for horizontal calibration and fixation after IMU installation.

3.  Equipment calibration verification: Conduct preliminary calibration on the dual axis turntable to verify its technical indicators such as angular position, angular velocity accuracy, and shaft verticality. Measure the deviation between the actual values and command values of each axis of the turntable at different angle positions, ensuring that the deviation is within the allowable range; Check the horizontal reference plane of the turntable to ensure that the levelness of the reference plane is better than 5 inches. At the same time, power on and preheat the IMU, record its initial output state, and troubleshoot the initial equipment malfunction.

（2） Environmental condition control

1.  Temperature control: The error parameters of 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 temperature control system to ensure temperature stability during the calibration process and reduce the impact of temperature drift on the calibration results.

2.  Vibration and interference control: The calibration environment should be kept away from vibration sources (such as machine tools, fans, heavy vehicles, etc.), and isolation measures should be taken on the ground (such as building isolation foundations or installing isolation pads) to ensure that the environmental vibration acceleration is ≤ 0.01g. At the same time, strong electromagnetic interference should be avoided by grounding the turntable, IMU, and data acquisition equipment (grounding resistance ≤ 4 Ω) to reduce the interference of electromagnetic noise on the IMU output signal.

3.  Pressure and humidity control: For IMUs that rely on pressure assisted calibration (such as combination IMUs with some barometers), the ambient pressure should be stable at standard atmospheric pressure (101.325kPa ± 1kPa), and the relative humidity should be controlled between 40% and 60% to avoid humidity changes that may cause moisture or insulation performance degradation in the IMU's internal circuits.

（3） IMU installation and debugging

1.  Mechanical installation: Fix the IMU on the worktable of the dual axis turntable with a dedicated fixture, ensuring that the sensitive axis of the IMU is aligned with the coordinate axis of the turntable. It is usually required that the X-axis of the IMU be parallel to the rotation axis of the inner (or outer) axis of the turntable, and the Z-axis be perpendicular to the plane of the turntable worktable (i.e. along the direction of gravity). Use a torque wrench to fix the fixture according to the specified torque to avoid displacement of the IMU during calibration due to loose installation, or deformation of the IMU structure due to excessive tightening.

2.  Axis alignment calibration: Use a spirit level and laser positioning instrument to calibrate the axis alignment accuracy of IMU and turntable. Firstly, adjust the turntable worktable to a horizontal position, ensuring that the Z-axis of the IMU is parallel to the direction of gravity; Then, by rotating the turntable, verify the parallelism between the IMU sensitive axis and the turntable rotation axis, with a parallelism error of ≤ 5 inches. If the alignment accuracy does not meet the requirements, adjust the fixture position and repeat calibration until it meets the standard.

3.  Electrical connection and debugging: Connect the IMU to the power supply and data acquisition card to ensure secure wiring and good contact, avoiding signal loss or distortion caused by virtual connections. Power on and preheat the IMU, with the preheating time determined according to the IMU type (navigation grade IMUs typically require 30-60 minutes of preheating, while consumer grade IMUs require 10-20 minutes of preheating), to achieve a stable internal temperature of the IMU. During the preheating process, monitor the stability of the IMU output signal. If there are abnormal situations such as signal jumps or excessive noise, it is necessary to troubleshoot the wiring or equipment.

（4） Software system construction

1.  Control software configuration: Install the dual axis turntable control software, configure the axis parameters of the turntable (such as shaft diameter, transmission ratio), control mode (static/dynamic), angular position/angular velocity settings, etc. At the same time, set the triggering conditions for data collection to ensure that the turntable posture is stable before starting data collection, avoiding signal interference during the transition process.

2.  Debugging of data collection software: Debug the data collection software, set parameters such as sampling rate, sampling duration, and data storage format (such as CSV, MAT files). Establish a synchronous acquisition mechanism for IMU output signals and turntable feedback signals, ensuring that their timestamps are aligned with an error of ≤ 1ms. Through simulation acquisition testing, verify the integrity and accuracy of data acquisition, and troubleshoot issues such as data loss and delay.

3.  Calibration algorithm deployment: Based on calibration requirements (such as accelerometer zero bias/scale factor calibration, gyroscope zero bias/scale factor calibration), deploy corresponding calibration algorithms (such as least squares method, Kalman filter method). Initialize the algorithm parameters, such as iteration count, convergence threshold, etc., to ensure that the algorithm can accurately solve the error parameters of IMU.

2、 Core calibration process

The core calibration process revolves around the two core components of IMU, the accelerometer and gyroscope. Based on the static positioning and dynamic rate control capabilities of the dual axis turntable, the error parameter calibration in two directions is completed step by step. This process takes the two-dimensional calibration of "pitch roll" as an example, covering three key steps: static calibration of accelerometers, static zero bias calibration of gyroscopes, and dynamic rate calibration of gyroscopes.

（1） Static calibration of accelerometer

The purpose of static calibration of an accelerometer is to solve its zero bias and scale factor, using the projection of gravity acceleration at different poses as a reference input. By measuring the acceleration signal output by the IMU, an error model is established and parameters are solved.

1.  Calibration attitude planning: Based on the pitch roll two-dimensional directions, plan 6 typical static attitudes (ensuring that the gravity acceleration can fully cover the X, Y, and Z sensitive axis directions of the accelerometer), with specific attitudes as follows: ① pitch 0 °, roll 0 ° (Z-axis forward along the gravity direction); ② Pitch 0 °, roll 180 ° (Z-axis reverse 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 reverse 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 reverse along the direction of gravity).

2.  Attitude adjustment and stability: The dual axis turntable control software sequentially sends angular position commands for each posture, and the turntable drives the IMU to rotate to the target posture, maintaining static stability. The stabilization time for each posture is ≥ 30s, ensuring that the acceleration signal output by the IMU is stable (signal fluctuation amplitude ≤ 0.001g). During the stabilization process, real-time monitoring of the angular position feedback signal of the turntable. If the attitude deviation exceeds the allowable range (≤ 5 "), the turntable will automatically compensate and adjust.

3.  Data acquisition and recording: After each posture is stabilized, start the data acquisition software to collect the X, Y, and Z axis acceleration signals output by the IMU. The sampling time is ≥ 10s and the sampling rate is ≥ 100Hz. At the same time, record the actual angular position of the turntable (pitch angle θ, roll angle φ), which is used to calculate the projection values of gravity acceleration on each sensitive axis (reference input). Classify and store the collected data according to posture, and label clear posture information and timestamps.

4.  Error model establishment and parameter solving: Establish an error model for the accelerometer, 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）

Among them, 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 (gravity acceleration projection) of the i-th axis, and b is the zero offset of the i-th axis. Based on the reference acceleration a (calculated from θ and φ, such as Z-axis reference acceleration a=g · cos θ · cos φ, X-axis reference acceleration a=g · sin θ, Y-axis reference acceleration a=g · sin φ · cos θ, where g is the gravitational acceleration, taken as 9.80665m/s2) and the corresponding a, the least squares method is used to solve K and b.

（2） Static zero bias calibration of gyroscope

The static zero bias of a gyroscope refers to the output deviation of the gyroscope when there is no angular velocity input, which needs to be solved through long-term data acquisition in the stationary state of the IMU.

（3） Dynamic rate calibration of gyroscope

The purpose of dynamic rate calibration of a gyroscope is to solve its scaling factor. Using the known angular rate output by a dual axis turntable 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 of pitch 0 ° and roll 0 °. At this time, the IMU has no angular rate input, and the gyroscope output only contains zero bias and noise. In this posture, the turntable does not need to rotate, just keep the stage level and stable.

2.  Long term data collection: Start the data collection software to collect the output signals of the gyroscope's X, Y, and Z axes, with a sampling time of ≥ 60 minutes and a sampling rate of ≥ 100Hz. During the collection process, continuously monitor the ambient temperature and turntable posture to ensure temperature stability (fluctuation ≤ 0.2 ℃) and no drift in posture (deviation ≤ 5 "), avoiding additional errors introduced by external factors.

3.  Zero bias calculation: Preprocess the collected gyroscope output data, remove outliers (using the 3 σ criterion), and then calculate the average value of the output signals of each axis, which is the static zero bias b (i=X, Y, Z) of the gyroscope. Simultaneously calculate the standard deviation of the data and evaluate the noise level of the gyroscope. If the standard deviation is too large (exceeding the IMU technical specifications), it is necessary to troubleshoot the equipment or environmental interference.

4.  Rate point planning: Based on the range of IMU and actual application scenarios, plan dynamic rate points in pitch and roll dimensions. Select 5-7 velocity points for each dimension, covering forward and reverse velocities (such as -100 °/s, -50 °/s, 0 °/s, 50 °/s, 100 °/s), with the 0 °/s velocity point used to verify the consistency of static zero bias. The selection of the rate point should ensure that it does not exceed the range of the IMU, and the turntable can stably output the rate (rate stability ≤ 0.1 °/s).

5.  Rate output and stability: Through the dual axis turntable control software, commands for each rate point are sent sequentially in the pitch and roll dimensions. After the turntable drives the IMU to rotate to the target speed, it maintains dynamic stability for at least 20 seconds. During the stabilization process, the real-time monitoring of the angular velocity feedback signal of the turntable is carried out. If the velocity deviation exceeds the allowable range (≤ 0.5 °/s), the turntable automatically compensates for the velocity.

6.  Data collection and recording: After each rate point stabilizes, start the data collection software to collect the output signals of the gyroscope corresponding to the sensitive axis (such as collecting the X-axis gyroscope output when the pitch dimension rotates, and collecting the Y-axis gyroscope output when the roll dimension rotates), with a sampling time of ≥ 10s and a sampling rate of ≥ 100Hz. At the same time, record the actual angular velocity of the turntable (reference input ω), and store the data classified by rate point and dimension.

7.  Error model establishment and parameter solving: Establish a rate error model for the gyroscope, ignoring cross coupling errors. The model is as follows:

ω = K(ω + b) （i=X,Y）

Among them, ω is the output angular velocity of the i-th axis of the gyroscope, K is the scale factor of the i-th axis, ω is the reference angular velocity of the i-th axis (the actual output velocity of the turntable), and b is the static zero bias of the i-th axis (which has been solved in static calibration). Substitute the ω and corresponding ω of each rate point into the model, and use the least squares method to solve K.

3、 Data processing and validation

Data processing and validation are key steps in ensuring the reliability of calibration results. It is necessary to preprocess the collected raw data, solve the error parameters, and perform residual analysis, repeatability validation, and accuracy validation. If the validation fails, it is necessary to return to the core calibration process for recalibration.

1.  Outlier removal: Use the 3 σ criterion or Grubbs criterion to detect and remove outliers in the raw data (acceleration, angular velocity signals). For the 3 σ criterion, calculate the mean μ and standard deviation σ of the data, and identify data that exceeds the range of [μ -3 σ, μ+3 σ] as outliers, replacing or directly removing them with interpolation of adjacent data.

2.  Filtering processing: Low pass filtering is applied to the preprocessed raw data to remove high-frequency noise. Choose  Butterworth  Low pass filter, with a cut-off frequency determined by the bandwidth of the IMU (usually 1/5~1/3 of the IMU bandwidth), to avoid signal distortion caused by excessive filtering. The filtered data is used for subsequent error parameter solving.

3.  Data synchronization alignment: Linear interpolation method is used for synchronization alignment to address the timestamp deviation between IMU output signal and turntable feedback signal. Ensure that each IMU output data corresponds to an accurate turntable attitude or velocity state, with a synchronization error of ≤ 1ms.

4.  Parameter optimization: Substitute the preprocessed data into the error models of the accelerometer and gyroscope, and use the least squares method to solve error parameters such as zero bias and scale 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 residual between the observed values (IMU output) at each calibration attitude/velocity point and the predicted values of the model, which reflects 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 velocity residual standard deviation ≤ 0.1 °/s), it indicates that the model fitting effect is good; If the residual is too large or there is a clear trend, it is necessary to re-examine the error model (such as considering cross coupling errors) or calibrate the validity of the data.

6.  Repeatability verification: Under the same environmental conditions and calibration process, repeat 3 complete calibration experiments and solve the error parameters for each calibration. Calculate the coefficient of variation (the ratio of standard deviation to mean) of the parameters three times. If the coefficient of variation is ≤ 1%, it indicates that the calibration results have good repeatability; If the coefficient of variation is too large, it is necessary to investigate issues such as equipment stability and environmental interference, and recalibrate.

7.  Accuracy verification: Select attitude/velocity points that have not participated in calibration as verification points, substitute the calibrated error parameters into the error model, compensate for 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 (such as acceleration measurement error ≤ 0.01g, angular velocity measurement error ≤ 0.5 °/s), it indicates that the calibration accuracy meets the standard; If the error does not meet the standard, it is necessary to re optimize the calibration process (such as adding calibration attitude/velocity points, adjusting the error model) and perform calibration again.

8.  Temperature stability verification (optional): If the IMU needs to operate within a wide temperature range, calibration experiments can be repeated at different temperature points (such as -10 ℃, 0 ℃, 20 ℃, 40 ℃, 60 ℃) to verify the variation of error parameters with temperature. Establish a temperature compensation model for error parameters to improve the measurement accuracy of IMU in different temperature environments.

9.  Data classification storage: Store preprocessed raw data, error parameter solution results, residual analysis reports, validation results, etc. by calibration date, IMU number, and calibration environment conditions. The data storage format adopts universal formats (such as CSV, MAT, PDF) to ensure the readability and traceability of the data.

10.  Data backup: Perform multiple backups of archived data (such as local hard drives, cloud storage) to avoid data loss. Backup data should be clearly labeled with file names and documentation, specifying the calibration objects, processes, and conditions corresponding to the data.

4、 Closing work

The final work mainly includes archiving calibration data, restoring and maintaining equipment, preparing calibration reports, ensuring the traceability of the calibration process, and providing a basis for the subsequent use and maintenance of IMU. The calibration report is a summary of the calibration work, which needs to comprehensively and accurately record the calibration process and results, mainly including the following contents:

1.  Equipment shutdown and disassembly: After calibration is completed, turn off the dual axis turntable IMU、 Disassemble the power supply of the data acquisition device and disconnect the IMU from the fixture in order, then remove the IMU. During the disassembly process, avoid collisions and vibrations to protect the sensitive components of the IMU.

2.  Equipment cleaning and maintenance: Clean the worktable, shaft system, and fixtures of the dual axis turntable, remove dust and debris; Conduct a visual inspection of the IMU to ensure no damage and clean wiring ports. Record the usage and maintenance status of the equipment, providing a basis for regular calibration of the equipment.

3.  Equipment parameter recovery: Restore the parameters of the dual axis turntable and data acquisition equipment to their default state, 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 contents:

(1)  Calibration object information: IMU model, number, manufacturer, technical specifications;

(2)  Calibration equipment information: model and accuracy level of dual axis turntable, model and sampling parameters of data acquisition equipment, list of auxiliary equipment;

(3)  Calibration environmental conditions: temperature, humidity, air pressure, vibration situation;

(4)  Calibration process description: calibration attitude/velocity point planning, data acquisition parameters, error model, solving algorithm;

(5)  Calibration results: accelerometer zero bias, scale factor, gyroscope zero bias, scale factor, residual analysis results, repeatability verification results, accuracy verification results;

(6)  Conclusion and Suggestions: Whether the calibration results meet the standards, recommendations for the use of IMU (such as temperature compensation, regular recalibration cycles), and equipment maintenance recommendations.

5、 Precautions

In summary, the standard process for using a dual axis turntable for IMU two-dimensional calibration must strictly follow the logical sequence of "pre calibration preparation core calibration data processing and verification final work", with a focus on key aspects such as equipment accuracy, environmental control, axis alignment, and data synchronization. Through standardized calibration procedures and rigorous verification methods, the error parameters of IMU can be accurately solved, significantly improving its measurement accuracy and providing guarantees for the reliable operation of inertial navigation systems.

1.  During the calibration process, if there is any drift in the turntable attitude or abnormal IMU output signal, the calibration should be stopped immediately. After troubleshooting, restart to avoid the generation of invalid calibration data;

2.  The preheating time of IMU must strictly comply with technical requirements. Insufficient preheating can lead to unstable error parameters and affect calibration accuracy;

3.  The alignment accuracy of the axis system of the dual axis turntable directly affects the calibration results, and it is necessary to calibrate the turntable regularly to ensure that the axis system accuracy meets the requirements;

4.  The temperature, vibration, electromagnetic interference and other factors in the calibration environment have a significant impact on the output of IMU. It is necessary to strictly control the environmental conditions and take isolation and shielding measures if necessary;

5.  The calibration report needs to be reviewed by professionals to ensure the accuracy and standardization of the report content. After passing the review, it should be archived and saved.