Hyper-Dual Twist
It is an enlarged twist that carries the information of both velocity and acceleration using Hyper-Dual Vectors. Or a movement and its derivative. It uses the Automatic Differentiation property of Dual Numbers/Hyper-Dual Numbers.
If we have an velocity twist: \[ \hat{\nu} = \vec{\omega} + \epsilon \, \vec{v} = (\omega + \epsilon \, v) \hat{s} = |\hat\nu| \, \hat{s} \]
The hyper-dual velocity twist will be:
k\[ \check{\nu} = [\hat{\omega} + \epsilon \, \hat{v} + \check{\epsilon} ( \dot{\hat{\omega}} + \epsilon \, \dot{\hat{v}})] = [\omega + \epsilon \, v + \check{\epsilon}(\dot{\omega} + \epsilon \, \dot{s})] ( \hat{s} + \check{\epsilon} \, \dot{\hat{s}} ) \]
\[ \check{\nu} = (|\nu| + \check{\epsilon} \, |\dot{\nu}|) ( \hat{s} + \check{\epsilon} \, \dot{\hat{s}} ) = \hat{\nu} + \check{\epsilon} \, \dot{\hat{\nu}} = |\check{\nu}| \, \check{s}\]