Maths - Conversion Euler To Matrix

Definition of terms:

This depends on what conventions are used for the Euler Angles. The following are derived on the euler angle page, the first assumes NASA Standard Airplane:

[R] =
c θ *c φ -c θ *s φ s θ
cψ*sφ + s ψ* s θ*c φ cψ*cφ -s ψ*s θ*s φ -s ψ*c θ
sψ*sφ - c ψ* s θ*c φ sψ*cφ + c ψ*s θ*s φ c ψ*c θ

 

Or, from the same page, this uses NASA Standard Airplane taking angles in the reverse order:

[R] =
c ψ*c θ sψ*cφ + c ψ*s θ*s φ sψ*sφ - c ψ* s θ*c φ
-s ψ*c θ cψ*cφ -s ψ*s θ*s φ cψ*sφ + s ψ* s θ*c φ
s θ -c θ *s φ c θ *c φ

 

Java code to do conversion:

/** this conversion uses NASA standard aeroplane conventions as described on page:
*   https://www.euclideanspace.com/maths/geometry/rotations/euler/index.htm
*   Coordinate System: right hand
*   Positive angle: right hand
*   Order of euler angles: [R3][R2][R1] = [about z][about y][about x] = [bank][attitude][heading]
*   matrix row column ordering:
*   [m00 m01 m02]
*   [m10 m11 m12]
*   [m20 m21 m22]*/
public final void rotate(double heading, double attitude, double bank) {
// Assuming the angles are in radians.
double c1 = Math.cos(heading);
double s1 = Math.sin(heading);
double c2 = Math.cos(attitude);
double s2 = Math.sin(attitude);
double c3 = Math.cos(bank);
double s3 = Math.sin(bank);
m00 = c1 * c2;
m01 = -s1 * c2;
m02 = s2;
m10 = s1 * c3+(c1 * s2 * s3);
m11 = (c1*c3) - (s1 * s2 * s3);
m12 = -c2 * s3;
m20 = (s1 * s3) - (c1 * s2 * c3);
m21 = (c1 * s3) + (s1 * s2 * c3);
m22 = c2*c3;
}

Example

we take the 90 degree rotation from this: rightUp to this: rightForward

As shown here the axis angle for this rotation is:

heading = 0 degrees
bank = 90 degrees
attitude = 0 degrees

so substituteing this in the above formula gives:

[R] =
c θ *c φ -c θ *s φ s θ
cψ*sφ + s ψ* s θ*c φ cψ*cφ -s ψ*s θ*s φ -s ψ*c θ
sψ*sφ - c ψ* s θ*c φ sψ*cφ + c ψ*s θ*s φ c ψ*c θ
[R] =
1 0 0
0 0 -1
0 1 0

This agrees with the matix rotations here.

other examples in 90 degree steps are shown here.


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