Jacobian of $R^{-1} v$ with respect to $R \in SO(3)$

Question

According to "A micro Lie theory for state estimation in robotics", the Jacobian $J_R^{R\cdot{}\mathbf{v}} = -\mathbf{R}[\mathbf{v}]_\times$. Here non-bolded R is an element of SO(3) or an element of the pure quaternions $\mathbb{H}_p$ acting on a vector $\mathbf{v}$, and bold $\mathbf{R}$ is the corresponding rotation matrix. $[\cdot]_\times$ is the skew-symmetric matrix, i.e., $[\mathbf{a}]_\times{}\mathbf{b}=\mathbf{a}\times{}\mathbf{b}.$

I want to find $J_R^{ {R}^{-1} \cdot \mathbf{v}}$, and according to the same paper, it seems as simple as $$J_R^{ {R}^{-1} \cdot \mathbf{v}} = J_{R^{-1}}^{ {R}^{-1} \cdot \mathbf{v}} J_R^{R^{-1}} = (-\mathbf{R}^{-1}[\mathbf{v}]_\times)(-\mathbf{R}) = \mathbf{R}^T[\mathbf{v}]_\times \mathbf{R}.$$

I get the same result if I do the limit by hand, as shown in eq. (123) in the referenced paper below.

However, this seems wrong. I am checking derivations in some source code, and it seems they have used $$ J_R^{ {R}^{-1} \cdot \mathbf{v}} = [\mathbf{R}^T \mathbf{v}]_\times $$ which just seems impossible. What is going on?

A micro Lie theory for state estimation in robotics: http://www.iri.upc.edu/files/scidoc/2089-A-micro-Lie-theory-for-state-estimation-in-robotics.pdf

Answer 1

Maybe I am misunderstanding what you mean by "pure quaternion", but the quaternion $R$ is not pure imaginary; $R = \cos\theta/2 + (\sin\theta/2)n$ where $\theta$ is the angle of rotation and $n$ is the unit normal of the plane of rotation expressed as an imaginary quaternion. Then $R\cdot v = Rv\bar R$ where $\bar R$ is the conjugate, and $R\bar R = \bar RR = 1$.

As for your question, they are equivalent expressions. Consider a vector $w$ and let $v, w$ be represented by imaginary quaternions. Then $$ \mathbf R^T[v]_\times\mathbf Rw = \bar R(v\times(Rw\bar R))R = (\bar RvR)\times(\bar RRw\bar RR) = (\bar RvR)\times w = [\mathbf R^Tv]_\times w, $$ so $\mathbf R^T[v]_\times\mathbf R = [\mathbf R^Tv]_\times$.

Answer 2

The expressions $\mathbf{R}[\mathbf{v}]_{\times}\mathbf{R}^{-1}$ and $[\mathbf{R}\mathbf{v}]_\times$ are equivalent.

The map $f\colon\mathbb{R}^3\to\mathfrak{so}(3)$; $f(\mathbf{v})=[\mathbf{v}]_{\times}$ is an isomorphism of Lie algebra representations from the standard representation to the adjoint representation, $$\mathrm{ad}_X(f(\mathbf{v}))=f(X\mathbf{v})\tag{$\ast$}$$ for all $\mathbf{v}\in\mathbb{R}^3$ and $X\in\mathfrak{so}(3)$. Writing $\mathbf{R}=e^X$ and using $\mathrm{Ad}_{(e^X)}=e^{(\mathrm{ad}_X)}$ and $(\ast)$ we have $$ \mathbf{R}[\mathbf{v}]_{\times}\mathbf{R}^{-1} =\mathrm{Ad}_{(e^X)}f(\mathbf{v}) =e^{(\mathrm{ad}_X)}f(\mathbf{v}) =f(e^X\mathbf{v}) =[\mathbf{R}\mathbf{v}]_\times .$$