Analysis of the IMU Calibration Model

Why does IMU Calibration matter?

IMU measurements can easily drift very far without proper calibration which results in large errors in the position and orientation estimation. These errors are different and may be caused by manufacturing defects, temperature variations, sensor noise among others.

• Packaging alignment error:

The angle between the accelerometer detection axis (sensitive axis) and the packaging reference axis (as shown in Figure 2). “Input axis alignment” is another term for this error. The unit of packaging alignment error is “degrees”. Packaging technology usually controls the alignment accuracy of the bare chip and the packaging within approximately 1°.

• Orthogonal/alignment error:

The angle between multiple-axis devices and the ideal angle displacement (usually 90°)

• Cross-axis sensitivity:

Measures the output generated on another axis when an acceleration is applied to one axis, usually expressed as a percentage. The coupling between two axes is caused by alignment error, etching inaccuracy, and circuit crosstalk.

• Package alignment error α and sensor alignment error θ.
• α represents the angle between the sensor axis and the package axis.
• θ represents the deviation of the sensor axis from the orthogonal axis, that is, 90 ° – ∠ysensor – xsensor.

• 0g bias level:

Refers to the output level when there is no acceleration (0 input). Analog sensors are usually represented as V or mV, while digital sensors are denoted by code number (LSB). The 0g bias is specified by a specific power supply voltage and is usually proportional to it (in most cases, the nominal value of the 0g bias is half of the power supply voltage).

0g bias is often defined from multiple aspects:

1. 0g voltage (V), specifying the expected possible voltage range for output at 0g acceleration.

2. The deviation of the output from the ideal value, also known as the initial bias error, is specified at 25°C and is expressed as the acceleration error (g) or the output signal (mV for analog sensors and LSB for digital sensors).

3. The relationship between 0g offset and temperature, or the offset temperature coefficient (mg/°C), describes the variation in output for every 1 °C change in temperature.

4. Bias voltage sensitivity, describing the relationship between the change of “0 bias level” and the change of power supply. The unit of this parameter is usually mv/V, mg/V or LSB/V.

5. 0g total error, including all errors.

• Accelerometer noise density:

Total noise: The random deviation relative to the ideal output, equal to the product of the noise density and the square root of the noise bandwidth. The unit of this parameter is usually mg-RMS.

• Output data rate:

In digital output accelerometers, it defines the sampling rate of the data. Bandwidth refers to the highest frequency signal that can be sampled without aliasing at the specified output data rate. According to the Nyquist sampling criterion, the bandwidth is half of the output data rate.

In analog output accelerometers, the bandwidth is defined as the signal frequency at which the response drops to -3dB of the DC (or low-frequency) acceleration response.

Main characteristics of the gyroscope

• Measurement Range:

The maximum value of the angular rate input in both positive and negative directions by the gyroscope represents the measurement range of the gyroscope. The larger this value is, the stronger the gyroscope’s ability to sense the rate is. The unit is: °/sec

• Sensitivity:

Refers to the minimum increment of the input angular rate that can be detected under the specified input angular rate. The smaller the value, the better. When choosing, it is necessary to select an appropriate measurement range for different applications. As the measurement range becomes larger, the sensitivity will correspondingly decrease.

• Packaging error:

The angle error between the diagonal of the bare chip and the diagonal of the package. The smaller, the better.

• Non-linearity:

Scale factor: The ratio of the gyroscope output to the input angular rate. This ratio is obtained by fitting a straight line using the input/output data measured within the entire range of input angular rate through the least squares method.

The residual of the scale factor fitting determines the credibility of the fitting data and characterizes the deviation degree of the actual input/output data of the gyroscope. A non-linearity of 0.1% of the full scale means the non-linearity of the scale factor.

• Initial zero bias error

Zero bias refers to the output of the gyroscope in the zero input state. It is equivalent to representing the average value of the output over a long period of time as the input angular rate.

• Zero bias stability

The long-term steady-state output in the zero input state is a stable random process, that is, the steady-state output fluctuates and oscillates around the mean (zero bias). It is conventionally represented by the mean square deviation, which is defined as the zero bias stability.

The initial zero bias error of 3°/sec can be regarded as a static error, which does not fluctuate over time and thus is easy to be corrected through software. The size of the 0.007°/sec zero drift value indicates the degree of dispersion of the observed values around the zero bias, and it is difficult to be corrected through software.

Note: For micro-mechanical gyroscopes, due to the significant influence of temperature on their structural materials, the zero bias stability is provided at 25°C.

• Output noise:

When the gyroscope is in a zero-input state, its output signal is the superposition of white noise and slowly varying random functions. The slowly varying random function can be used to determine the zero bias or zero bias stability index. White noise is defined as the standard deviation of the equivalent rotational angular rate under the square root of the unit detection bandwidth, in units of (°/sec)/√Hz or (°/hr)/√Hz. This white noise can also be expressed by the angle random walk coefficient in units of °/√hr, which refers to the random time accumulation of the gyroscope output error coefficient generated by the white noise. When the external conditions remain basically unchanged, it can be considered that the main statistical characteristics of the various noises analyzed above do not change with the passage of time.

Accelerometer/Gyroscope abstract model

Based on the previous characteristics, accelerometers and gyroscopes are calibrated in three aspects: zero bias, scale factor, and cross-coupling (geometric correction).
1. Zero bias is related to temperature.
2. Scale factor is related to temperature.
3. The geometric correction matrix is fixed.

It can be understood in a more simplified way as three sets of rulers in three directions. The rulers measure the values (angular velocity/angular velocity) in three directions. The zero mark of the ruler needs to be corrected.

1. x’, y’, z’ represent the original output axes (sensitive axes) of the IMU chip.
2. x, y, z are the actual output axes after the module (P-Box) is installed.
Note: The IMU chip is first installed in the module and then in the P-box. This can be regarded as a single coordinate transformation.

Conclusion

Calibration of IMU is not a possible step but is a basic necessity of every application that requires precise motion tracking. Since IMUs are still in development, calibration is getting more advanced so that it is possible to achieve high levels of precision in ever smaller form factors.

With the ongoing developments in sensor technology, the methods of calibration will further develop as well, with the next-generation technology being able to track motion even more accurately in regard to robotics and augmented reality and autonomous systems.