Euler Angles

Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors

  • Euler angles are a way to describe the orientation of a rigid body in three-dimensional space.
  • They are defined by three successive rotations around different axes.
  • There are two main types of Euler angles: extrinsic and intrinsic.
  • There exists twelve possible sequences of rotation axes, divided in two groups.
    • Proper Euler Angles (zxz, xyx, yzy, zyz, xzx, yxy)
    • Tait-Briyan angles (xyz, yzx, zxy, xzy, zyx, yxz)
  • For first and last angle, the valid range could be [-\(\pi\), \(\pi\)]
  • Second angle covers \(\pi\) radians, it could be [\(0\), \(\pi\)] or [-\(\pi/2\), \(\pi/2\)]
    • Proper Euler Angles has gimbal lock on \(n\pi\)
    • Tait-Briyan angles has gimbal lock on \(\pi/2 + n\pi\)

Intrinsic Euler Angles

Intrinsic Euler angles describe rotations about the axes of a moving coordinate system (body frame). Here's how they work:

  1. First Rotation (α): Rotate the object around its own axis.
  2. Second Rotation (β): Rotate the object around its own axis, which has moved with the first rotation.
  3. Third Rotation (γ): Rotate the object around its own axis, which has moved with the first two rotations.

Build ACTIVE Rotation matrix from Intrinsic Euler Angle

\(R = R_{I \rightarrow \alpha \rightarrow \beta \rightarrow \gamma} = R_{\alpha}R_{\beta}R_{\gamma}\)

Build PASSIVE Rotation matrix from Intrinsic Euler Angle

\(R = R_{\gamma \leftarrow \beta \leftarrow \alpha \leftarrow I} = R_{\gamma}R_{\beta}R_{\alpha}\)

(Be aware that \(R_{?}\) is differ depends on ACTIVE or PASSIVE)

Extrinsic Euler Angles

Extrinsic Euler angles describe rotations about the axes of a fixed coordinate system (global frame). Here's how they work:

  1. First Rotation (α): Rotate the object around the fixed axis.
  2. Second Rotation (β): Rotate the object around the fixed axis.
  3. Third Rotation (γ): Rotate the object around the fixed axis.

Build ACTIVE Rotation matrix from Extrinsic Euler Angle

\(R = R_{\gamma \leftarrow \beta \leftarrow \alpha \leftarrow I} = R_{\gamma}R_{\beta}R_{\alpha}\)

Build PASSIVE Rotation matrix from Extrinsic Euler Angle

\(R = R_{I \rightarrow \alpha \rightarrow \beta \rightarrow \gamma} = R_{\alpha}R_{\beta}R_{\gamma}\)

(Be aware that \(R_{?}\) is differ depends on ACTIVE or PASSIVE)

See Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors for detailed derivation

  • This document describes about conversion between PASSIVE Rotation matrix and Intrinsic Euler Angles.
  • This document uses \(\psi\) as first angle of rotation, \(\theta\) as second angle of rotation and \(\phi\) as third angle of rotation.