Can change in position due to acceleration be expressed using dual quaternions?

Dual quaternions seem like an appealing way to model 6DOF motion since they linearize rotation.

I’ve reviewed what literature I can find on then, and found expressions for translation and change in position for constant velocity, but not for accelerating bodies.

Can change in position over time due to acceleration be expressed using dual quaternions or does the lack of a second derivative (over the dual numbers) make this impossible?

I’ve seen references to a more general Clifford ‘Motor Algebra’? Does it solve this problem?


I am primarily working from the paper: “3D kinematics using dual quaternions: theory and applications in neuroscience” which contains a tutorial covering screw translation and velocity using dual quaternions.

Physics Asked by gibbss on November 11, 2021

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One Answer

It's certainly possible. However, the displacement, velocity, and acceleration cannot all be represented in the same dual quaternion.

Dual numbers are often used to represent a quantity along with its derivative. This usage does not allow for a second derivative to be stored alongside it. The primary paper decided to store velocity quaternions along with their displacement quaternions in dual quaternions. However, one can simply store the displacement, velocity, and acceleration all in different variables. This is how the paper treats line velocities since the line itself took up too many degrees of freedom to store the velocity with the line in the same dual quaternion.

All of that said, acceleration isn't a particularly useful quantity when modeling 6 degree of freedom inertial movement. This is because forces and torques are proportional to the derivative of momentum not velocity. This distinction is only relevant for objects that have non-isotropic inertia tensors, and thus is easily forgotten. This distinction, means one should track momentum, and directly calculate velocity from that, rather than calculating acceleration.

Answered by Rick on November 11, 2021

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