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I have the following Figure and the equations:

Three Axis for measuring tilt

Equations for measuring tilt

The body on the Figures is a tri-axial accelerometer sensor, which measures accelaration in meters/seconds².

The goal is to calculate the tilt of the following angles using acceleration values:

  • ρ: angle of the X-axis relative to the ground (orange line);
  • Φ: angle of the Y-axis relative to the ground (orange line);
  • θ: angle of the Z-axis relative to the gravity (green line).

Could someone explain how to find equations 1,2 and 3 from the figure above?

Source of the equations and figure: https://www.thierry-lequeu.fr/data/AN3461.pdf

There is another similar and more detailed source that uses the same equations, but I also could not understand how to find them: https://www.analog.com/en/app-notes/an-1057.html

I have already implemented them and it is working, I just want help to understand how to obtain the equations. Here is the code:

let pitch = Math.atan(ax / Math.sqrt((Math.pow(ay,2) + Math.pow(az,2))) );
let roll = Math.atan(ay / Math.sqrt((Math.pow(ax,2) + Math.pow(az,2))) );
let theta = Math.atan(Math.sqrt((Math.pow(ax,2) + Math.pow(ay,2))) /az);

Thanks in advance.

Matheus
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1 Answers1

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This is the Pythagorean theorem, finding the 2D distance between 0,0 and a point represented by the two numbers given. If we assign that to a new function it may be clearer:

distance(a, b) { return sqrt((pow(a,2) + pow(b,2))) }

Then angles are calculated by using the inverse tangent function with a distance from that function representing one side of the triangle. For example, the pitch in your question divides the x acceleration by the distance between 0,0 and the acceleration in the YZ plane.

pitch = atan(x / distance(y, z))
aptriangle
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