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Why are there square brackets around [V_b] when deriving it, but not when using it in the update?

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The vertical pipes denote the magnitude. It looks like the magnitude of the velocity vector must be above a threshold value (epsilon) for the robot to be considered in motion. The square brackets denote vector/matrix. Then, in the equation denoted with 1., you are multiplying a matrix with a vector, so no brackets or pipes are necessary.

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Your question lacks sufficient information to provide a precise answer. It seems that $\mathcal{V}_b$ is a vector in $\mathbb{R}^3$ and is used in combination with the Jacobian of an "end-effector" $b$.

I believe that the braked notation around $\mathcal{V}_b$ means: "Take the components of the right-hand side and assign them to the components of the left-hand side." The author likely uses this notation because the logarithm of an orthogonal 3x3 matrix results in a skew-symmetric 3x3 matrix, which has only 3 degrees of freedom.

The slides you mentioned present a less common formulation of inverse kinematics, acknowledging the manifold structure of $SO(3)$ and utilizing the quasi-Newton algorithm on the tangent space. The brackets are likely used to represent elements of the tangent space of $SO(3)$ in $\mathbb{R}^3$ (as they are isomorphic).

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