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I found this formula in some notes but I would like to have a reference (book, paper, etc.) to understand where it comes from. I know that it works only for small angles.

$ \begin{bmatrix} \phi_e\\ \theta_e\\ \psi_e \end{bmatrix} = sgn(q_e^0) \cdot \begin{bmatrix} 2q_e^1\\ 2q_e^2\\ 2q_e^3 \end{bmatrix} $

  • $q_c$ is the quaternion setpoint
  • $q$ is the current quaternion (the one we are "measuring")
  • $q_e=q_c \otimes q^{-1}$ is the error in the orientation
  • While $\begin{bmatrix}\phi_e\\ \theta_e\\ \psi_e \end{bmatrix}$ is a vector of error in roll, pitch, and yaw angles.

Thanks in advance for your help.

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