10.8: What Does an Inertial Measurement Unit Measure?

A common in-the-box assembly of components today is a perpendicular triad of accelerometers (strain-guage typically), along with a triad of angular rate gyros. The six measurements of this inertial measurement unit (IMU) have generally obviated inclinometers, which are functionally equivalent to a pendulum whose angle (following gravity) relative to the housing is measured via a potentiometer.

This combination of sensors within an IMU brings up a fundamental user parameter. First, the accelerometers on a non-accelerating frame will point down (gravity); they can be used to estimate pitch and roll, and hence replace inclinometers. When the platform actually does accelerate, however, the measured acceleration vector is the vector sum of the true acceleration and the gravity effect. So the pitch and roll of an IMU during accelerations is critical if we are to separate out the gravity effect from the measured accelerometer signals. The rate gyros possess a different characteristic: they are completely insensitive to linear acceleration (and gravity), but suffer a bias, so that the integration of a measured rate to deduce angle will drift. A typical drift rate for a fiber optic gyro is \(72\)°/hour, certainly not good enough for a long-term pitch or roll measurement. In the short term, gyros are quite accurate.

The accelerometers and rate gyros are typically taken together to derive a best estimate of pitch and roll. Specifically, the low-frequency components of the accelerometer signals are used to eliminate the drift in the angle estimates; the assumption is that a controlled body generally has only short periods of significant linear acceleration. Conversely, the high-frequency portion of the the rate gyros’ signals are integrated to give a short-term view of attitude. The interesting user parameter is, then, deciding whether what time frame applies to the accelerometer signals, and what time frame applies to the rate gyro signals.

The three accelerometers measure the total derivative of velocity, in the body frame, plus the projection of gravity onto the sensor axes. Using the above notation, assuming the sensor \([x, \, y, \, z]\) is aligned with the body \([x, \, y, \, z]\), and assuming that the sensor is located at the vector \(\vec{r}_S\), this is

\begin{align} \text{acc}_x \,\, &= \,\, \dfrac{\partial u}{\partial t} + qw - rv + \dfrac{dq}{dt} z_S - \dfrac{dr}{dt} y_S + (q y_S + r z_S)p - (q^2 + r^2)x_S - \sin \theta g \\[4pt] \text{acc}_y \,\, &= \,\, \dfrac{\partial v}{\partial t} + ru - pw + \dfrac{dr}{dt} x_S - \dfrac{dp}{dt} z_S + (r z_S + p x_S)q - (r^2 + p^2) y_S + \sin \psi \cos \theta g \\[4pt] \text{acc}_z \,\, &= \,\, \dfrac{\partial w}{\partial t} + pv - qu + \dfrac{dp}{dt} y_S - \dfrac{dq}{dt} x_S + (p x_S + q y_S)r - (p^2 + q^2) z_S + \cos \psi \cos \theta g \end{align}

Here \(g = 9.81 \ m/s^2\), and \( [\phi, \, \theta, \, \psi] \) are the three Euler angle rotations. The accelerations have some intuitive elements. The first term on the right-hand side captures actual, honest-to-goodness linear acceleration. The second and third terms capture centripetal acceleration - e.g., in the \(y\)-channel, an acceleration \(ru\) is reported, the product of the forward velocity \(u\) and the leftward turning rate \(r\). The fourth and fifth terms account for the linear effect of placing the sensor away from the body origin; later terms capture the nonlinear effects. Gravity comes in most naturally in the acceleration in the \(z\)-channel: if the roll and pitch Euler angles are zero, then the sensor thinks the vehicle is accelerating upward at one \(g\).

The rate gyros are considerably easier!

\begin{align} \text{rate}_x \, &= \, p \\[4pt] \text{rate}_y \, &= \, q\\[4pt] \text{rate}_z \, &= \, r. \end{align}

The rate gyros measure the body-referenced rotation rates.