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I have IMU angular velocity measurements and trying to constrain the rotational part of a B-Spline trajectory by the measurements as follows(Eq 259 of Diebel's paper https://www.astro.rug.nl/software/kapteyn/_downloads/attitude.pdf).

$$e = \textbf{R}(\textbf{r})\omega- W(\textbf{r})G(\textbf{r})\dot{\textbf{r}}$$

Where $\textbf{r}$ is the rotation vector and $W(\textbf{r})G(\textbf{r})\dot{\textbf{r}}$ a standard rotation vector to angular velocity conversion. $\textbf{R}$ is local to the world rotation matrix to convert local angular velocity to the world one. And the rotation vector is a function of four nearby control points $\textbf{c}$. The problem is how to calculate analytic Jacobian of above. The first term $\textbf{R}\omega$ is easy and will be look like $[[\textbf{R}\omega]_{\times} f_1 \ \ [\textbf{R}\omega]_{\times} f_2 ...]$, where $f_n$ are a spline weight function and it is 3 by 12 matrix. And applying the product rule the second term becomes similar to

$$\frac{\partial W(\textbf{r})}{\partial \textbf{r}}G(\textbf{r})\dot{\textbf{r}}+W(\textbf{r})\frac{\partial G(\textbf{r})}{\partial \textbf{r}}\dot{\textbf{r}}+W(\textbf{r})G(\textbf{r})\frac{\partial \dot{\textbf{r}}}{\partial \textbf{r}}$$

Of course, it is wrong. I realized that when I was trying to evaluate the mid term as

$$W(\textbf{r})\frac{\partial G(\textbf{r})}{\partial \textbf{r}}\dot{\textbf{r}}=W(\textbf{r})\frac{\partial G(\textbf{r})}{\partial \textbf{r}}\frac{\partial \textbf{r}}{\partial \textbf{c}}\dot{\textbf{r}}$$

The matrix dimension does not match.

I guess I am doing wrong with matrix to vector derivative. I have been checking some of online materials for the vector to vector or vector to matrix product rule but it has not been successful.

Can anyone give me a hint?

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I realized that there is a direct conversion method for the rotation vector to angular velocity problem. Their jacobian is easier than the jacobian of the above. The implementation is here: https://github.com/ethz-asl/kalibr

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