Usefulness of the Instantaneous Axis of Rotation in Robotics Hello Physics Exchange,
I finally have some decent understanding of the concept of instantaneous axis of rotation in the context of a single rigid body but I am still trying to piece together why it is useful in robotics. So far, I see how kinematics problems involving multiple connected parts can be more easily solved using the instantaneous axis of rotation or the instantaneous center of rotation for 2D problems.
For example, a robotic arm is composed of many components called links that are interconnected via joints. The hand effector is the last component of the robotic arm. In general, the objective is to position the hand effector in a certain position and orientation via a single or sequence of specific movements (translations+rotations). In light of this, how does the concept of ICR help and why is it is so useful? There many different ways to achieve a specific configuration of the hand effector with rotations and translations about different points... I would like to appreciate the usefulness of the ICR (screw theory) in the context of robotics.
 A: Algebraically, screw theory (or spatial algebra) provides for compact notation with conveys the intent of the calculation a lot better than vector notation.
For example, the velocity kinematics of the 2nd body recursively defined from the 1st body in a chain of bodies is
$$ \underbrace{ \boldsymbol{v}_2 }_\text{vel 2} = \underbrace{ \boldsymbol{v}_1  }_\text{vel 1}+ \underbrace{\boldsymbol{s}_2}_\text{axis} \underbrace{ \dot{q}_2 }_\text{  speed}
$$
You read out this equation as adding the joint motion to the body 1 motion to calculate the motion of body 2.
In vector form, the above is one of two forms, depending if the joint is prismatic or revolute
$$ \begin{array}{l}
 \text{prismatic}  \\ \hline
 \vec{v}_2  = \vec{v}_1 + \vec{\omega}_1 \times ( \vec{r}_2 -\vec{r}_1 ) + \vec{z}_2 \dot{q}_2
 \\
 \vec{\omega}_2 = \vec{\omega}_1 \\ \text{revolute} \\ \hline 
\vec{v}_2  = \vec{v}_1 + \vec{\omega}_1 \times ( \vec{r}_2 -\vec{r}_1 ) \\
\vec{\omega}_2 = \vec{\omega}_1 + \vec{z}_2 \dot{q}_2
\end{array}$$
The above becomes even more evident when doing accelerations kinematics
$$ \underbrace{ \boldsymbol{a}_2 }_\text{acc 2} = \underbrace{ \boldsymbol{a}_1  }_\text{acc 1}+ \underbrace{\boldsymbol{s}_2}_\text{axis} \underbrace{ \ddot{q}_2 }_\text{  accel} + \underbrace{ \boldsymbol{v}_2 \times \boldsymbol{s}_2 \dot{q}_2 }_\text{bias}
$$
Compared to the vector form
$$ \begin{array}{l}
 \text{prismatic}  \\ \hline
 \vec{a}_2  = \vec{a}_1 + \vec{\alpha}_1 \times ( \vec{r}_2 -\vec{r}_1 ) + \vec{\omega}_1 \times ( \vec{v}_2 -\vec{v}_1 ) + \vec{z}_2 \ddot{q}_2 + \vec{\omega}_1 \times \vec{z}_2 \dot{q}_2
 \\
 \vec{\alpha}_2 = \vec{\alpha}_1   \\ \text{revolute} \\ \hline 
\vec{a}_2  = \vec{a}_1 + \vec{\alpha}_1 \times ( \vec{r}_2 -\vec{r}_1 )+ \vec{\omega}_1 \times ( \vec{v}_2 -\vec{v}_1 ) \\
\vec{\alpha}_2 = \vec{\alpha}_1 + \vec{z}_2 \ddot{q}_2 + \vec{\omega}_1 \times \vec{z}_2 \dot{q}_2
\end{array}$$
Similar compactness exist for the equations of motion
$$ \boldsymbol{f}_1 - \boldsymbol{f}_2 = \mathbf{I}_1 \boldsymbol{a}_1 + \boldsymbol{v}_1 \times \mathbf{I}_1 \boldsymbol{v}_1 $$
Geometrically, screw theory provides easy visualization of important axes on the mechanism. Each unit twist represents an axis in space (with associated direction and location and pitch value). The same goes for wrenches which represent lines of action in space.
As I explained in this post in a lot more detail, each screw quantity can be geometrically decomposed into the following:


*

*Screw Decomposition For both ray and axis representation the properties of a screw with direction vector (non unit) $\mathbf{e}$ and moment vector $\mathbf{m}$ are found with the following formulas $$\begin{align}
\mbox{Magnitude} & & s & = \| \mathbf{e} \|  \\
\mbox{Unit Direction} & & \hat{\mathbf{e}} &=\frac{\mathbf{e}}{\| \mathbf{e} \|} \\
\mbox{Position Closest To Origin} & & \mathbf{r} & = \frac{\mathbf{e} \times \mathbf{m}}{ \| \mathbf{e} \|^2 }\\
\mbox{Pitch} & & h & = \frac{\mathbf{e} \cdot \mathbf{m}}{ \| \mathbf{e} \|^2 }\\
\end{align}$$ NOTE: $\times$ is the vector cross product, and $\cdot$ the vector dot product.

System modeling is possible with screw theory such as this paper because it allows for manipulation of complex systems using linear algebra.
Synthesis of motion space (kinematics) and reaction space is useful in the design of mechanisms and robot control systems.
For example the total motion space the end effector is the collection of all the joint axis twists in a single matrix
$$ \mathbf{S} = \begin{bmatrix} \boldsymbol{s}_1 & \boldsymbol{s}_2 & \ldots & \boldsymbol{s}_n \end{bmatrix} $$
The null space of the above $\mathbf{R}$ can be estimated such that $\mathbf{S}^\top \mathbf{R} = \mathbf{0}$
Now the space of reaction wrenches defines the lines of action that can react against the end effector without causing any motion.
In my thesis I expand on the usefulness of four subspaces defined in dynamics with the above method. This method has been used in a recent publication to examine the internal forces generated in a human knee when the system of foot and leg kinematics is considered as a twist space $\mathbf{S}$.
