5.5: Inertia-based Sensors

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A moving mass does not loose its kinetic energy, but for friction. Likewise, a resting mass will resist acceleration. Both effects are due to “inertia” and can be exploited to measure acceleration and speed.

5.5.1. Accelerometer

An accelerometer can be thought of as a mass on a dampened spring. Considering a vertical spring with a mass hanging down from it, we can measure the acting force F = kx (Hooke’s law) by measuring the displacement x that the mass has stretched the spring. Using the relationship F = am, we can now calculate the acceleration a on the mass m. On earth, this acceleration is roughly 9.81 kgm/s². In practice, these spring/mass systems are realized using microelectromechanical systems (MEMS), such as a cantilevered beam whose displacement can be measured, e.g., using a capacitive sensor. Accelerometers measure up to three axes of translational accelerations. Infering a position from this requires integration twice, thereby amplifying any noise, making position estimates using accelerometers alone infeasible. As gravity provides a constant acceleration vector, accelerometers are very good at estimating the pose of an object with respect to gravity.

5.5.2. Gyroscopes

A gyroscope is an electro-mechanical device that can measure rotational orientation. It is complementary to the accelerometer that measures translational acceleration. Classically, a gyroscope consists of a rotating disc that could freely rotate in a system of pivots and gimbals. When moving the system, the inertial momentum keeps the original orientation of the disc, allowing to measure the orientation of the system relative to where the system was started. A variation of the gyroscope is the rate gyro, which measures rotational speed.

What a rate gyro measures can most intuitively be illustrated by considering the implementation of an optical rate gyro. In an optical gyro, a laser beam is split into two beams and send around a circular path in two opposite directions. If this setup is rotated against the direction of one of these laser beams, one laser will have to travel slightly longer than the other, leading to a measurable phase-shift at the receptor. This phase shift is proportional to the rotational speed of the setup. As light with the same frequency and phase will add, and lights with the same frequency but opposite phases will cancel each other, light at the detector will be darker for high rotational velocities. As small-scale optical rate gyros are not practical for multiple reasons, MEMS rate gyros rely on a mass suspended by springs. The mass is actively vibrating, making it subject to Coriolis forces, when the sensor is rotated. Coriolis forces can be best understood by moving orthogonally to the direction of rotation on a vinyl disk player. In order to move in a straight line, you will not only need to move forwards, but also sideways. The necessary acceleration to change the speed of this sideway motion is counteracting the Coriolis force, which is both proportional to the lateral speed (the vibration of the mass in a MEMS sensor) and the rotational velocity, which the device wishes to measure. Note that the MEMS gyro would only be able to measure acceleration if it were not vibrating.

Gyroscopes can measure the rotational speed around three axes, which can be integrated to obtain absolute orientation. As an accelerometer measures along three axes of translation, the combination of both sensors can provide information on motion in all six degrees of freedom. Together with a magnetometer (compass), which provides absolute orientation, this combination is also known as Inertial Measurement Unit (IMU).