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I'm working on an attitude control system in terms of quaternions, and i'm basing my approach on the book Space Vehicle Dynamics and Control (2nd ed.). If we consider a proportionate controller, i.e.

$$\mathbf{u} = -\mathbf{K}_p \mathbf{q}_e $$

it would seem logical to me to define the error quaternions to be the first three elements of

$$\begin{bmatrix} q_{1e} \\ q_{2e} \\ q_{3e} \\ q_{4e} \\ \end{bmatrix} = \begin{bmatrix} q_{1} \\ q_{2} \\ q_{3} \\ q_{4} \\ \end{bmatrix} - \begin{bmatrix} q_{1c} \\ q_{2c} \\ q_{3c} \\ q_{4c} \\ \end{bmatrix}$$

Where $q_i$ are the actual quaternions and $q_c$ are the desired (commanded) quaternions. Yet in the book, they instead define

$$\begin{bmatrix} q_{1e} \\ q_{2e} \\ q_{3e} \\ q_{4e} \\ \end{bmatrix} = \begin{bmatrix} q_{4c} & q_{3c} & -q_{2c} & -q_{1c} \\ -q_{3c} & q_{4c} & q_{1c} & -q_{2c} \\ q_{2c} & -q_{1c} & q_{4c} & -q_{3c} \\ q_{1c} & q_{2c} & q_{3c} & q_{4c}\\ \end{bmatrix}\begin{bmatrix} q_{1} \\ q_{2} \\ q_{3} \\ q_{4} \\ \end{bmatrix}$$

To me it seems like we do this because there are multiple possible quaternions that correspond to the same orientation, but I have not yet found a clear explanation for this. Can anyone enlighten me?

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  • $\begingroup$ Page back in the book a bit to check the math, or do some cyphering. As used in navigation, quaternions define rotations, and rotation is carried out by multiplying them together. So I would expect that $q_e$ would be the solution to either $q_i q_e$ = $q_c$, or $q_e q_i$ = $q_c$. If so, the matrix math is just how the actual numerical solution is calculated. $\endgroup$
    – TimWescott
    Apr 5, 2021 at 15:45
  • $\begingroup$ One reason : With Quaternions you can never get in gimbal lock. $\endgroup$
    – Ben
    Apr 9, 2021 at 14:18

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A possible explanation is that a set of 4 quaternions is not independent, so by converting them to error quaternions, we get 3 independent values that we can use to calculate 3 independent control actions (one for each axis).

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