classical mechanics - Understanding terms Twist and Wrench

In kinematics, physics and especially robotics, we often encounter the terms Twist and Wrench. Twist is (LinearVelocity, AngularVelocity) and Wrench is (Force, Torque). The reason I'm confused is I see different definitions and I'm no longer sure if I'm using these terms correctly...

• Per Wikipedia, Twist is defined in context of screw theory and it has linear velocity and angular velocity both along same axis.


• Some robotics literature uses Twist to have linear velocity part as (V + L X W). This allows to computer work more easily.


• In ROS, popular robotics platform, Twist is simply independent linear and angular velocities.

Questions:

1. Is it correct to use the term Twist simply as combination of individual linear and angular velocities for any body?


2. Is there any similar term to describe combined linear and angular accelerations?

Answer

Both twist and wrenches are screws. "Screw" is the general term, and "Twist" is the specific application to motion whereas "Wrench" is the specific application to forces and momentum. All of them combine the linear and angular aspects of the thing they describe in one 6×1 object. The definitions pertain to rigid body mechanics in general and are not specific to robotics.

I hope the following definitions will help you out:

• Ray/Axis A 3D Screw is an object that represents a line in space (direction & location) in addition to a magnitude and a pitch value. A screw has 6 components and they are arranged as a vector $\mathbf{e}$ of direction and a vector $\mathbf{m}$ of moment. There are two possible ways to represent a screw, a) Direction then Moment or b) Moment then Direction $$ {\rm Screw} = \left\{ \begin{array}{cl} \begin{pmatrix} \mathbf{e} \\ \mathbf{m} \end{pmatrix} & \mbox{ray coordinates} \\ \begin{pmatrix} \mathbf{m} \\ \mathbf{e} \end{pmatrix} & \mbox{axis coordinates} \end{array} \right. $$


• Line Composition Consider a line in space with unit direction vector $\hat{\mathbf{e}}$ and any point on the line $\mathbf{r}$. The line object can be represented with the following coordinates $$ {\rm Line} = \left\{ \begin{array}{cl} \begin{pmatrix} \hat{\mathbf{e}} \\ \mathbf{r}\times\hat{\mathbf{e}} \end{pmatrix} & \mbox{ray coordinates} \\ \begin{pmatrix} \mathbf{r}\times\hat{\mathbf{e}} \\ \hat{\mathbf{e}} \end{pmatrix} & \mbox{axis coordinates} \end{array} \right. $$ The direction vector is such because it remains the same throughout 3D space (free vector), whereas the moment vector needs to be transformed if the location of interest changes (line vector). This is evident above where the moment vector is defined as the cross product between location and the direction vector.


• Screw Composition Consider the line above, but add a scalar magnitude $s$ and a scalar pitch $h$. The screw object is similar to the line object but with an additional term parallel to the direction $\hat{\mathbf{e}}$ in the moment vector. $$ {\rm Screw} = \left\{ \begin{array}{cl} s \begin{pmatrix} \hat{\mathbf{e}} \\ \mathbf{r}\times\hat{\mathbf{e}}+ h\, \hat{\mathbf{e}} \end{pmatrix} & \mbox{ray coordinates} \\ s \begin{pmatrix} \mathbf{r}\times\hat{\mathbf{e}}+ h\, \hat{\mathbf{e}} \\ \hat{\mathbf{e}} \end{pmatrix} & \mbox{axis coordinates} \end{array} \right. $$ A pitch represents any components of the moment vector that are parallel to the direction vector as a scalar ratio $h = \frac{\| \mathbf{m}_\parallel \|}{\| \mathbf{e} \|}$


• 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.}


• Twists A twist is a screw representing motion (infinitesimal rotation, velocity and spatial acceleration, joint axis). The direction vector angular part, and the moment vector is the linear part (at a fixed point A). For example velocities are $$\begin{align} & \mbox{Axis Coordinates} & & \mbox{Ray Coordinates} \\ \mathbf{v}_A & = \begin{pmatrix} \mathbf{v}_A \\ {\boldsymbol \omega} \end{pmatrix} & \mathbf{v}_A & = \begin{pmatrix} {\boldsymbol \omega} \\ \mathbf{v}_A \end{pmatrix} \end{align}$$ Axis coordinates are the most common for twists, but not always. A lot of confusion arises from this as people often use twists and axis coordinates interchangingly. Remember, a twist represents some kind of motion and the coordinate representation has to do with the order in which the direction vector and moment vector is represented.



• Wrench A wrench is a screw representing loading (force, momentum, impulse). The direction vector linear part, and the moment vector is the angular part (at a fixed point A). For example forces are $$\begin{align} & \mbox{Axis Coordinates} & & \mbox{Ray Coordinates} \\ \mathbf{f}_A & = \begin{pmatrix} {\boldsymbol \tau}_A \\ \mathbf{F} \end{pmatrix} & \mathbf{f}_A & = \begin{pmatrix} \mathbf{F} \\ {\boldsymbol \tau}_A \end{pmatrix} \end{align}$$ Ray coordinates are the most common for wrenches, but not always.


• Interpretation Both twists and wrenches represent an object at distance. For example a force $\mathbf{F}$ though a point A has torque ${\boldsymbol \tau}_A = \mathbf{r}_A \times \mathbf{F}$. And the velocity of a body rotating about a point A is $\mathbf{v}_A = \mathbf{r}_A \times \mathbf{\omega}$. Both are the moment vectors of the corresponding screws. In the most common notation these are $$ \begin{align} \mathbf{v}_A & = \begin{pmatrix} \mathbf{r}_A \times {\boldsymbol \omega} \\ {\boldsymbol \omega} \end{pmatrix} & \mbox{twist in (linear,angular)=axis coordinates} \\ \mathbf{f}_A & = \begin{pmatrix} \mathbf{F}\\ \mathbf{r}_A \times \mathbf{F} \end{pmatrix} & \mbox{wrench in (linear,angular)=ray coordinates} \end{align} $$ You can see that these are identical to the line compositions.


• Twist Example A body in motion has angular velocity $\mathbf{\omega} = (1,0,5)$ and linear velocity a point A $\mathbf{v}_A = (-2,4,1)$. Show the motion as a twist in axis coordinates and decompose it into its properties
  
  
  
  • Twist in axis coordinates (6×1 quantity) $$\mathbf{v}_A = \begin{pmatrix} \mbox{moment} \\ \mbox{direction} \end{pmatrix} = \begin{pmatrix} \mathbf{v}_A \\ {\boldsymbol \omega} \end{pmatrix} = \begin{pmatrix} \begin{vmatrix} -2 \\ 4 \\ 1 \end{vmatrix} \\ \begin{vmatrix} 1 \\ 0 \\ 5 \end{vmatrix} \end{pmatrix} $$
  
  
  • Magnitude: $\| \begin{vmatrix} 1 \\ 0 \\ 5 \end{vmatrix} \| = \sqrt{26}$
  
  
  • Direction: $ \frac{ \begin{vmatrix} 1 \\ 0 \\ 5 \end{vmatrix} }{\sqrt{26}} = \begin{vmatrix} \frac{1}{\sqrt{26}} \\ 0 \\ \frac{5}{\sqrt{26}} \end{vmatrix}$
  
  
  • Position: $ \frac{\begin{vmatrix} 1 \\ 0 \\ 5 \end{vmatrix} \times \begin{vmatrix} -2 \\ 4 \\ 1 \end{vmatrix}}{\sqrt{26}^2} = \begin{vmatrix} -\frac{10}{13} \\ -\frac{11}{26} \\ \frac{2}{13} \end{vmatrix}$
  
  
  • Pitch: $\frac{\begin{vmatrix} 1 \\ 0 \\ 5 \end{vmatrix} \cdot \begin{vmatrix} -2 \\ 4 \\ 1 \end{vmatrix}}{\sqrt{26}^2}= \frac{3}{26} $
  
  
  • Parallel velocity vector: $\mbox{(pitch)} {\boldsymbol \omega} = \frac{3}{26} \begin{vmatrix} 1 \\ 0 \\ 5 \end{vmatrix} = \begin{vmatrix} \frac{3}{26} \\ 0 \\ \frac{15}{26} \end{vmatrix}$
  
  
  
  
  

The above represents the geometry of the motion in all the detail that is available from the two pieces of information, the linear and angular velocity at one point.

Similarly for wrenches. The 6 components that define them are decomposed into magnitude, direction, position and pitch.

_{Related posts. Forces as Screws, Motion Screw and Instant Rotation Axis}

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For your second question, linear and angular acceleration does not form a twist (motion screw) because they contain centrifugal terms that do not transform like normal screws. This is because regular acceleration tracks a specific particle, and the screw quantities have a point of measurement fixed in space.

You can however construct an acceleration twist, if instead of using the regular (material) acceleration, you use spatial accelerations. At any point A the spatial acceleration vector ${\boldsymbol \psi}_A$ is the material acceleration $\mathbf{a}_A$ minus the centrifugal terms. $$ {\boldsymbol \psi}_A = \mathbf{a}_A - {\boldsymbol \omega} \times \mathbf{v}_A$$

Then the acceleration twist in axis coordinate is defined as:

$${\boldsymbol \psi}_A = \begin{pmatrix} \mbox{moment} \\ \mbox{direction} \end{pmatrix} = \begin{pmatrix} \mathbf{a}_A - {\boldsymbol \omega} \times \mathbf{v}_A \\ {\boldsymbol \alpha} \end{pmatrix} $$

The above is used the 6×6 equations of motion

$$ \mathbf{f}_A = \mathrm{I}_A {\boldsymbol \psi}_A + \mathbf{v}_A \times \mathrm{I}_A \mathbf{v}_A $$

But that is a subject of another question, as the derivation of the spatial equations of motion is rather involved at this stage.