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:
- Is it correct to use the term Twist simply as combination of individual linear and angular velocities for any body?
- Is there any similar term to describe combined linear and angular accelerations?
Best 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:
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
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.