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2016-02-02 18:04:41 +0200 | asked a question | Is matrix calculus possible within sagemath? For example, is it possible to compute the following partial derivative in Sagemath: $$ A_{ij} = \frac{\partial e_{ij}(x)}{\partial x_i} = \begin{pmatrix} -R_{ij}^T R_i^T & R_{ij}^T \frac{\partial R_i^T}{\partial \theta_i} (t_j - t_i) \\ 0^T & -1 \end{pmatrix} $$ for $$ e_{ij}(x) = \begin{pmatrix} R_{ij}^T (R_i^T (t_j - t_i) - t_{ij}) \\ \theta_j - \theta_i - \theta_{ij} \end{pmatrix} $$ and $$ x_i^T = (t_i^T, \theta_i) $$ $$ z_{ij}^T = (t_{ij}^T, \theta_{ij}) $$ The background shouldn't be important, but just for completeness: this example is an excerpt from Robotics, especially the problem of Pose Graph Simultaneous Mapping and Localization. $x_i$ defines a pose in $\mathbb{R}^2$ with translation $t_i$ and rotation angle $\theta_i$, $z_{ij}$ defines a transform between two poses and $e_{ij}(x)$ defines the error between two poses, i.e. a transform between them. |

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