Measuring Motion

We've studied a number of inertial sensors. The first sensor that we looked at was the accelerometer and that measures the total acceleration of a body. That acceleration might be due to forces acting on the body or the acceleration might be due to gravity acting on the body. If the only force acting on a body is gravity, then we can use the acceleration measured on the body to determine the roll and pitch angles with respect to the gravity vector or the downward direction. We looked at gyroscopes which are able to measure components of the angular velocity of the body. And, we looked at magnetometers which can sense components of the Earth’s magnetic field with respect to the coordinate frame attached to the body of the object. We can combine information from accelerometers and magnetometers to determine the orientation of the body with respect to magnetic north.

The earliest types of inertial navigation systems were referred to as gimballed and they are very large instruments that I showed very early on in this lecture. It contains a free floating gyro-stabilized platform referred to as the stable element and it’s shown here by the coordinate frame I. That platform is connected to the vehicle by a gimbal mechanism. As the orientation of the vehicle changes, the body frame which is attached to the vehicle changes, but the inertial coordinate frame I remains fixed in space. We can, therefore, determine the orientation of a vehicle by measuring the angles of the gimbals between the inertial frame which is fixed in space and the body frame which is attached to the vehicle.

The accelerometers are mounted on the gyro-stabilized platform, so they measure acceleration of the vehicle in the inertial frame. But if we want to, and we know the orientation of the vehicle with respect to the inertial frame, we can rotate the acceleration vector in to the vehicles body frame. This is a schematic of the gimballed inertial measurement unit used on the Apollo spacecraft. The stable element is shown here and it contains three spinning gyroscopes and they maintain the orientation of that platform constant with respect to the stars. That platform also contains three accelerometers indicated here, XA, YA and ZA.

The gimbal mechanism effectively allows the spacecraft to rotate about the stable platform and the gimbal angles are measured by rather old fashioned sensors by today’s standard called resolvers. Today, almost all inertial measurement units use what’s called the strap-down configuration. We have a body frame B attached to the vehicle and the accelerometers and gyroscopes measure acceleration along each of the body frame axis and the gyroscopes measure the angular velocity components about each of the body frame coordinate axes. As the vehicle orientation changes, so does the body frame B, we measure the angular velocity in the coordinate frame B and we integrate that over time to determine the orientation of the vehicle. Once we know the orientation of the vehicle in the body frame, we can rotate it back to the inertial frame to determine our acceleration with respect to the stars.