Overview of Inertial Measurement Unit
IMU Sensors
The main components of the inertial measurement unit include accelerometer and gyroscope, IMU processor, calibration parameter memory, temperature sensor and related power system.
A device with a gyroscope and accelerometer but no other components is sometimes referred to as an inertial sensor assembly (ISA). Most IMU sensors contain 3 accelerometers and 3 single-freedom gyroscopes mounted on 3 quadrature sensitive axes. However, some IMUs have redundant inertial sensors in the oblique axis to prevent one sensor from failing.
IMU Working Principle
The IMU processor converts the output of the inertial sensor dimensionally, compensates for known error terms, and checks the range of output to detect whether the sensor is normal. The IMU processor may also provide closed-loop force feedback or rebalancing loop control to the accelerometer and/or gyroscope. Dimensional conversion refers to the conversion of the direct output of an inertial sensor, such as voltage difference, current or pulse signal, into specific force or angular rate. Many IMUs are obtained by comparing the force and angular rate integration within the sampling period 𝜏𝑖
\( \upsilon_{ib}^b\left(\,t\,\right)=\int_{t-\tau_i}^tf_{ib}^b\left(\,t^{\prime}\,\right)\mathrm{d}t^{\prime}\,,\quad\alpha_{ib}^b\left(\,t\,\right)=\int_{t-\tau_i}^t\omega_{ib}^b\left(\,t^{\prime}\,\right)\mathrm{d}t^{\prime}\, \)
They are respectively delta – \(\upsilon\) and delta – \(\theta\). delta – \(\theta\) is just \(\alpha_{ib}^b\), the attitude increment. However, delta – \(\upsilon\)(\(\upsilon_{ib}^b\)) is not a velocity increment. IMIU outputs specific force and angular rate, or their integrals. Typical output frequencies range from 100 to 1000Hz.
Some IMUs sample the sensor at a higher frequency than the output frequency of the IMU. The results of multiple samples at the sensor end can be simply added up and sent out in a data output cycle. Or combined processing output to reduce the cone error and paddle error.
Calibration of Inertial Measurement Unit
IMU calibrations
The constant error of the inertial sensor is calibrated in the internal field and stored in the memory. In actual operation, the IMU processor corrects the sensor output according to the pre-calibrated error results. Calibration parameters generally include zero deviation of accelerometer and gyroscope, scale factor and cross-coupling error, and deviation error of gyroscope related to g. These error terms will vary with temperature, so it is necessary to perform calibration in a certain temperature range, and also install a temperature sensor in the IMU. However, the temperature inside each inertial sensor does not necessarily match the ambient temperature of the IMU, so some high-precision IMUs take temperature control measures instead of IMU temperature compensation.
Calibration by Kalman filter
Applying one set of IMU calibration coefficients to all IMUs produced in the same batch can minimize calibration costs. The determination of cross-coupling errors requires calibration at the IMU level, so each inertial measurement unit or sensor needs to be calibrated separately in order to obtain high accuracy. The Kalman filter is used to identify the calibration coefficient from the measurement data. The calibration process described here refers to the internal field calibration, which is different from the field calibration.
A deeper accelerometer error that can be compensated for by the IMU processor is the size effect error. In order to calculate navigation parameters at a point in space, the angular rate and specific force of the inertial measurement unit must be measured based on a reference point, sometimes called the impact center.
In practical applications, the structural layout of the inertial sensor results in a distance of several centimeters between the measurement points of the individual sensors (for MEMS IMU modules and solutions, the distance is smaller). For gyroscope measurements, there is no problem. However, the rotation of the accelerometer around the reference point causes it to be sensitive to a centripetal force that is not present at the reference point, and the angular acceleration causes it to measure the Euler force. The pseudo-acceleration of the accelerometer with respect to the reference point can be given by the following formula.
\( (\alpha_{bx}^{bP}\, \, \, \alpha_{by}^{bP}\, \, \, \alpha_{bz}^{bP})=-(Ω_{ib}^{b}Ω_{ib}^{b} + Ω_{ib}^{b}) (r_{bx}^{b} \: \: \:r_{by}^{b} \: \: \: r_{bz}^{b}) \)
\(r_{bx}^{b}\), \(r_{by}^{b}\) and \(r_{bz}^{b}\) represent the displacement of the X-axis, Y-axis, and Z-axis accelerometers to the IMU reference point respectively, and these quantities are known constants.
It is worth noting that the pendulum accelerometer measures the acceleration of the standard detected mass block, not the acceleration at the hinge. The measurement error of reference point specific force caused by size effect is as following formula,
\( {\delta}f_{ib,size{}}^b=\left( \begin{array}{llllllllll} {{a_{ix,x}^b-a_{ib,x}^b}} \\ {{a_{iy,y}^b-a_{ib,y}^b}} \\ {{a_{iz,z}^b-a_{ib,z}^b}} \end{array}\right)=-\left( \begin{array}{llllllllll} {{a_{bx,x}^{bp}}} \\ {{a_{by,y}^{bp}}} \\ {{a_{bz,z}^{bp}}} \end{array}\right) \)
\( = \begin{bmatrix} -\left({\omega_{ib,y}^b}^{2}+{\omega_{ib,z}^b}^{2}\right)x_{bx}^{b}+\left(\omega_{ib,x}^{b}\omega_{ib,y}^{b}-\ddot{\omega}_{ib,z}^{b}\right)y_{bx}^{b}+\left(\omega_{ib,x}^{b}\omega_{ib,z}^{b}+\ddot{\omega}_{ib,y}^{b}\right)z_{bx}^{b} \\-\left({\omega_{ib,z}^b}^{2}+{\omega_{ib,x}^b}^{2}\right)y_{by}^{b}+\left(\omega_{ib,x}^{b}\omega_{ib,y}^{b}+\ddot{\omega}_{ib,z}^{b}\right)x_{by}^{b}+\left(\omega_{ib,y}^{b}\omega_{ib,z}^{b}-\ddot{\omega}_{ib,x}^{b}\right)z_{by}^{b} \\-\left({\omega_{ib,x}^b}^{2}+{\omega_{ib,y}^b}^{2}\right)z_{bz}^{b}+\left(\omega_{ib,x}^{b}\omega_{ib,z}^{b}-\ddot{\omega}_{ib,y}^{b}\right)x_{bz}^{b}+\left(\omega_{ib,y}^{b}\omega_{ib,z}^{b}+\ddot{\omega}_{ib,x}^{b}\right)y_{bz}^{b} \end{bmatrix} \)
The reference point is the intersection of the three accelerometer sensitive axes,\(y_{b x}^{b}=z_{b x}^{b}=x_{b y}^{b}=z_{b y}^{b}=x_{b z}^{b}=y_{b z}^{b}=0\). The size effect error is simplified as:
\( \delta f_{ib, size}^b = – \left[ \begin{array}{c} \left( {\omega_{ib,y}^{b}}^2 + {\omega_{ib,z}^{b}}^{2}\right)x_{bx}^{b} \\ \left( {\omega_{ib,z}^{b}}^2 + {\omega_{ib,x}^{b}}^{2}\right)y_{by}^{b} \\ \left( {\omega_{ib,x}^{b}}^2 + {\omega_{ib,y}^{b}}^{2}\right)z_{bz}^{b} \end{array} \right] \)
In IMU processors, the size effect correction term is simplified as \(\delta f_{ib, size}^b\). It is important to note that not all IMU outputs are calibrated, nor do they necessarily correct for size effect errors. This also includes some inertial measurement units with temperature sensors.
Inertial sensors are sensitive to vibrations (for example, from the propulsion system). Both vibration caused by machinery and vibration caused by sound waves. In the process of transmitting vibration to the sensor, the intensity of vibration transmission is not only related to the packaging and bonding mode inside the IMU, but also related to the installation of the IMU itself. The transmission of vibration also varies with the frequency and direction of vibration.
Therefore, many IMUs are equipped with shock absorbers, which can also protect the components from impact. When designing a shock absorber, it is necessary to limit the vibration transfer near the mechanical resonance frequency of the sensor (and its resonance frequency), but also to limit the vibration transfer near the IMU solution update frequency (and its resonance frequency).
