Maths - Quaternion to Euler - Example

In order to check we are getting the correct result this page starts with a euler angle, converts it to a quaternion, then converts it back to to euler angle. When we have done the complete rond trip we end up with the same angles which gives confidence that the method is correct.

So lets start with the following euler angle:

heading = 10 degrees 
attitude = 20 degrees 
bank = 30 degrees

We then convert to quaternion using the first method on this page. Note that most maths libraries use radians so you will probably need to convert degrees to radians by multiplying by PI/180 = 0.01745.

c1 = cos(5 deg) = 0.9962  
c2 = cos(10 deg) = 0.9848  
c3 = cos(15 deg) = 0.9659  
s1 = sin(5 deg) = 0.0872  
s2 = sin(10 deg) = 0.1736  
s3 = sin(15 deg) = 0.2588  

So using the formular gives the following result:
 
qw = 0.9437  
qx = 0.2685  
qy = 0.1277  
qz = 0.1448

We now convert back to euler angles using the method on this page.

First we work out the heading part: this involves using the atan2 function. Implemetations of this function may vary, so be careful about the order of the operands, this page describes the issues.
 
tan(heading y part): usually the first operand= 0.1633  
tan(heading x part): usually the second operand= 0.9254

I used atan(y/x) instead of atan2(y,x) as I was looking up the values in 4 figure tables instead of using a computer program. However, because the angles are all in the first quadrant, I believe these are equivalent since in fist quadrant: 
tan(a) = sin(a)/cos(a) = opposite/adjacent = y/x 
and 
atan2(y,x) = atan2(opposite,adjacent)

tan(heading) = 0.1764

heading = 10 degrees

 

Calculating the attitude:
 
sin(attitude) = 0.3419

attitude = 20 degrees

 

Calculating the bank:
 
tan(bank y part) = 0.4698  
tan(bank x part) = 0.8138  
tan(bank) = 0.5773

bank = 30 degrees

 

An OpenOffice.org with these calulations is here


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