Standard Notation/Name for Velocity Transformations

Question

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}

Answer 1

This matrix has different names based on what version of screw theory, spatial notation, etc. in which you're using it. In general, when working with these various formulations, the 6-element velocity vector has the angular velocity in the top half and the linear velocity in the bottom half (and it generally represents a vector field rather than a vector).

The closest examples of a similar functionally-equivalent matrix are the following:

• 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.