Twist (mechanics, $(\mathbf{\omega },\mathbf{v})$)

A rigid-body twist represents an instantaneous rigid body velocity field by a pair of three-dimensional vectors, conventionally ordered $(\mathbf{\omega },\mathbf{v})$. In a stated reference frame, $\mathbf{\omega }$ is angular velocity and $\mathbf{v}$ is the velocity field evaluated at that frame's origin. Every point at position $\mathbf{x}$ has instantaneous velocityThe linear component therefore need not be the center of mass velocity. Both components must use the same coordinate frame; their physical units also differ.^[ref]

The formula follows by differentiating the point position $\mathbf{x}=R(t)\mathbf{X}+\mathbf{p}(t)$, where $R$ is a rotation matrix and $\mathbf{X}$ is fixed in the body. The skew-symmetric matrix $\dot{R}{R}^T$ acts as the cross product with $\mathbf{\omega }$. HenceAt the body's chosen origin $\mathbf{x}=\mathbf{p}$, this correctly returns $\dot{\mathbf{p}}$. Moving the reference origin by a fixed vector $\mathbf{a}$, without changing its axes, gives ${\mathbf{v}}^{\prime }=\mathbf{v}+\mathbf{\omega }\times \mathbf{a}$, with the same angular velocity. The numbers change while the physical velocity field is unchanged.

When $\mathbf{\omega }\ne 0$, define the screw pitch and a point on the instantaneous axis byThe vector triple-product identity gives $\mathbf{u}(\mathbf{q})=h\mathbf{\omega }$. Points on $\mathbf{q}+\lambda \mathbf{\omega }$ share that axial velocity; off-axis points additionally rotate about it. A constant rigid-body twist therefore generates a screw motion. Zero angular velocity instead gives pure rigid-body translation, for which these divisions are not defined.

For an original numerical example take $\mathbf{\omega }=(0,0,2)rad/s$ and $\mathbf{v}=(1,0,0.4)m/s$. At $(0,1,0)\mathrm{m}$ the velocity is $(-1,0,0.4)m/s$. The axis passes through $\mathbf{q}=(0,0.5,0)\mathrm{m}$, with axial speed $0.4m/s$ and screw pitch $h=0.2m/rad$. If the twist stays constant for a revolution, the axial advance is $2\pi h\simeq 1.257\mathrm{m}$.

A twist describes instantaneous motion; it is not generally a six-coordinate finite pose or its ordinary derivative. Its reference-frame convention matters when using a robot's measured joint rates to predict endpoint velocity.

The screw pitch of a rigid-body twist is its signed axial translation per unit rotation angle about the instantaneous motion axis. For angular velocity $\mathbf{\omega }\ne 0$ and linear twist component $\mathbf{v}$ in the same reference frame,The units are length per radian. This mechanics convention differs from using the word pitch for an axial distance per complete revolution.^[ref]

To derive the formula, write the instantaneous velocity field as $\mathbf{u}(\mathbf{x})=\mathbf{v}+\mathbf{\omega }\times \mathbf{x}$. At a point $\mathbf{q}$ on the instantaneous axis, the velocity is parallel to $\mathbf{\omega }$, so $\mathbf{u}(\mathbf{q})=h\mathbf{\omega }$. Taking its scalar product with $\mathbf{\omega }$ removes the cross-product term and gives the displayed result. A suitable closest axis point iswhich verifies $\mathbf{u}(\mathbf{q})=h\mathbf{\omega }$ directly.

Although $\mathbf{v}$ depends on the chosen origin, $h$ does not. Translating that origin by $\mathbf{a}$ replaces $\mathbf{v}$ by $\mathbf{v}+\mathbf{\omega }\times \mathbf{a}$; the added term is perpendicular to $\mathbf{\omega }$. A common rotation of the coordinate axes preserves the scalar product and the norm as well. Multiplying the entire rigid-body twist by a nonzero scalar changes the motion rate but preserves $h$.

For an original mechanism with constant axial advance $2\mathrm{mm}$ per full turn,At five revolutions per second, $\dot{\theta }=10\pi rad/s$ and the axial speed is $h\dot{\theta }=10mm/s$. The advance per revolution is $2\pi h$, which corresponds to screw lead. In a multiple-start screw thread, the spacing between neighbouring threads need not equal that lead, so a thread's engineering pitch must not silently replace the twist parameter.

When $h=0$ and $\mathbf{\omega }\ne 0$, the instantaneous motion is pure rotation about an axis, even if points away from that axis have nonzero linear velocity. When $\mathbf{\omega }=0$ and $\mathbf{v}\ne 0$, the motion is pure rigid-body translation and the displayed quotient is undefined; an infinite-pitch convention can describe that limiting case. A finite nonzero constant pitch produces a screw motion, with off-axis point paths following helices.