Enter the elements of the rotation matrix into the calculator to determine the Euler angles.

Euler Angle Formula

The following formula is used to calculate the Euler angles from a rotation matrix. Assuming the rotation sequence is ZYX, the Euler angles can be calculated as follows:ψ = atan2(R21, R11)θ = atan2(-R31, sqrt(R11^2 + R21^2))φ = atan2(R32, R33)Variables:

  • ψ, θ, φ are the Euler angles (yaw, pitch, roll respectively) Rij represents the elements of the rotation matrix atan2(y, x) is the arctangent function with two arguments

To calculate the Euler angles, use the atan2 function with the appropriate elements of the rotation matrix as arguments. For ψ (yaw), use R21 and R11. For θ (pitch), use -R31 and the square root of the sum of the squares of R11 and R21. For φ (roll), use R32 and R33.

What is an Euler Angle?

Euler angles are a set of three angles that describe the orientation of a rigid body in a three-dimensional space. Named after the Swiss mathematician Leonhard Euler, these angles provide a way to represent the rotation of the body from a set initial orientation to its current orientation. The three angles are typically denoted as φ (phi), θ (theta), and ψ (psi), and each corresponds to a rotation about one of the three principal axes (X, Y, Z) of the body. The specific sequence of these rotations can vary, leading to different possible conventions for representing Euler angles.