# The Gyroscope and Autopilot

Many of us first encountered the gyroscope as a toy. Invented nearly 170 years ago, it appears to be a simple spinning wheel surrounded by rings that balances on a pedestal. In fact, it is much more. When properly designed, it can actually maintain its orientation in xyz space in a traveling vehicle. This feature started to be exploited in the early 20^{th} century to automatically pilot ships and planes. So how does it really work?

Let’s start with a warm-up example. Assume you have a wheel spinning counterclockwise as shown in the attached slide. The wheel has an angular (spin) velocity of **w _{s}** , measured in radians/second. The wheel has a moment of inertia I, which can be calculated from its mass and radius:

I = mR^{2}

Define the vector **L **to be the spin angular momentum of the wheel. Its magnitude is given by

**L** = I**w _{s} **= mR

^{2}

**w**

_{s}The direction of **L** is given by the famous right-hand rule of physics: holding your right hand up and curling your fingers in the direction of rotation of the wheel, your thumb will point in the direction of the spin angular momentum, as shown in the slide.

Now we apply an external force **F** to the spinning wheel. This exerts a torque **T** on the wheel. The force twists the wheel in the xy plane. The resulting torque, another vector, is just the cross product of the force **F** and the length over which the force is applied:

**T** = **F** x **b**

If you haven’t encountered the cross product before, no worries, here’s what it means. The cross product gives the torque’s magnitude as the product of the magnitudes of **F** and **b** and its direction is in the positive z axis, perpendicular to x and y. You can use the right-hand rule again here. Point your hand in the direction of **b**, curl your fingers in the direction of **F**, and your thumb will point in the direction of **T**. The upward torque causes the spin angular momentum **L **to move towards **T** to the direction to **L’**. This is called precession. If you’re confused, check out this awesome MIT lecture by Professor Walter Lewin: https://www.youtube.com/watch?v=N92FYHHT1qM

What does this example have to do with a gyroscope? A gyroscope is a wheel that can spin inside of its support structure. This structure is made up of 2 or 3 gimbals. Each gimbal allows the wheel to spin around an axis. The 3-gimbaled gyroscope prevents any torque from being exerted on the spinning wheel. Therefore, the direction of the gyroscope’s spin angular momentum will not change, even as the vehicle in which it is traveling pitches and rolls. This is the way that all inertial guidance systems – like autopilots – can use gyroscopes to lock onto the direction of their destination.

Bu wait a minute. Just because the gyroscope locks onto the direction, how does the autopilot get the vehicle to go there? There are sensors (shaft encoders) in the gyroscope frame that sense motion in the gimbals. These readings get fed back into the computerized autopilot system to keep the plane traveling in desired direction.

How does that work? The autopilot uses features on the plane to change its direction. Directions here are not described as x, y, and z, but roll, pitch, and yaw. The roll direction denotes rotation about the plane’s lengthwise axis. Pitch is rotation about its side-to-side axis, and yaw is rotation about its vertical axis. On the tail of the plane, the rudder controls yaw and things called elevators control pitch. The ailerons on the rear edge of each wing control roll. The autopilot uses the feedback from the gyroscope sensors to adjust the rudder, elevators, and ailerons to keep the plane on course.

Now that you understand how a gyroscope can help a plane autopilot itself, you’re probably wondering about automatic landings. This is the subject to which we will will turn next week.