# Standard Notation/Name for Velocity Transformations

I've been looking to see if there's any standard notation for a matrix to convert an end-effector velocity vector $$\xi_n^0 = \begin{bmatrix} v \\ \omega \end{bmatrix}$$ from one frame of reference to another (in my case, base frame to end-effector frame).

The math works, I just don't quite know how to refer to this velocity (tool) frame transform matrix when talking about it, or if there's a standard notation I could use (e.g., $$J$$ for Jacobian, $$A$$, $$T$$, and $$H$$ for homogenous transforms, etc.).

The matrix (in context) is: $$\begin{equation} \xi^b_n = \begin{bmatrix} {R_b^a}^T & -{R_b^a}^T S(d_b^a)\\ 0_{3\text{x}3} & {R_b^a}^T \end{bmatrix} \xi^a_n \end{equation}$$

Where S is the skew-symmetric: $$\begin{equation} \begin{bmatrix} 0 & -d_3 & d_2 \\ d_3 & 0 & -d_1 \\ -d_2 & d_1 & 0 \end{bmatrix} \end{equation}$$

• In the Modern Robotics textbook by Lynch and Park, the transpose of this matrix is defined as the adjoint representation $$Ad(T_{ab})$$ of the homogeneous transformation matrix $$T_{ab} = (R_{ab}, d_{ab}) \in SE(3)$$.
• In Featherstone's Spatial Notation, this same matrix (as defined in MR) is simply called the spatial transformation matrix $${^a\!X}_b$$ and performs the same operation on spatial velocities and accelerations.