Accelerations
One of the most important quantities for you to measure are accelerations. If your two or more-axis accelerometer is mounted along the traditional axes of the aircraft it will be the easiest to code for. The three traditional axes are:
- From the nose through the center of gravity on the line of symmetry, the Y-axis, roll
- From the center of gravity out of the fuselage toward the tip of the right wing, the X-axis, pitch
- From the center of gravity away from the earth, the Z-axis, yaw
Figure of Rigid Aircraft Axes
So, now for some basic Trigonometry, everyone remembers SOH CAH TOA.
In general, the following picture is true for a vehicle moving through space.
Figure of the Direction of the Force of Gravitation on a Body
As you can see from the image, the angle of pitch relative to the surface of the earth is the same angle offset of the weight vector relative to the z-axis in the body frame of reference. If we put our accelerometer so that one of its axes is parallel to the body's z-axis at its center of gravity we are measuring this offset vector. Which is really neat, because it means that we can express the angle of the body in level, non-accelerating motion as ratios of the accelerations
Pitch:
The angle theta between the actual gravity vector and the measured gravity is related to the pitch of the aircraft (pitch = theta + 90°). If we know theta, we know our pitch! Since we know the magnitude of the earth’s gravity, simple calculus gives us our pitch angle:
Woot, we calculated the pitch orientation of our airplane using an accelerometer. Pretty easy, huh?
The real formula that we need to use for the software looks like this:
accelerometer = cos (theta) * gravity
theta = acos (accelerometer / gravity)
And since pitch = theta + 90°pitch = asin (accelerometer / gravity)
The real formula that we need to use for the software looks like this:
pitch = atan2(accelerometer / gravity, z / gravity)
Common piezo-electric accelerometers return in units of, g, 32.17 ft/s^2 or 9.81 m/s^2. We also know some more things about the flight that let us calculate the angles relative to the ground. More on this later, it is a bit more than basic trigonometry to describe. These equations assume non-accelerating flight. You can use a magnetometer to get the relative plane in space with less math, but magnetometers generally take more interface programming in my experience.
Roll:
Roll needs a second accelerometer with an axis perpendicular to the first so that we can figure out the resultant vector between them and then the angle. Essentially the vector between the accelerometers becomes the "gravitational" acceleration and the relative readings lets us calculate the angle with an atan2 function. The second accelerometer will have some other things to manage such as the effects of the distance between them on the accelerations measured. Physics fun and none of the boring class.
Yaw:
Yaw is the hardest of the angles to measure. The only answer is to use a magnetometer or a compass. In many ways, yaw can be solved by dead reckoning. Dead reckoning is all that is important for most of the projects in the DIY garage. They will be covered later.
Next we will discuss gyroscopes and the beauty of rates and integral calculus
