# 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?

Edit:

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

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

## Related Questions

### Why $C_p$ is zero for photon gas?

1  Asked on February 21, 2021

### Problems understanding some cosmological models

0  Asked on February 21, 2021

### Why does falling water become less and less focused as it falls down?

2  Asked on February 21, 2021

### What fraction of the mass of all baryonic matter are black holes?

1  Asked on February 21, 2021 by arman-armenpress

### What is the relationship between lattice field theory and grid-based numerical methods?

1  Asked on February 21, 2021

### Transparent Boundary Conditions 1D Klein Gordon

0  Asked on February 21, 2021

### How to approach this problem? (electrostatics, capacitors)

0  Asked on February 21, 2021 by wolfcho

### Can a system of cubic anharmonic oscillators have multiple stable equilibria?

1  Asked on February 21, 2021

### Voyager distance

1  Asked on February 21, 2021 by relayman357

### Electric and magnetic fields are two aspects of the same entity

2  Asked on February 21, 2021 by satya

### Why does Gauss’s law apply to any shape of a closed surface?

2  Asked on February 21, 2021 by jelle-3-0

### Navier Stokes: $(u⋅∇)u$ vs $u⋅∇u$

3  Asked on February 21, 2021 by travisg

### Does viscosity arise from the increase in intertia created by the movement of molecules?

0  Asked on February 21, 2021

### How is the temporal profile of a wave signal obtained via its overlapping with a pulse signal?

0  Asked on February 21, 2021

### Can a particle have multiple spin states at the same time?

1  Asked on February 21, 2021

### Time evolution of a state on adding a constant to the Hamiltonian of the Schrodinger equation

2  Asked on February 21, 2021 by a_point_particle

### What determines whether a water-cooled nuclear reactor has a positive or negative void coefficient?

2  Asked on February 21, 2021

### Difference between relativistic and non-relativistic classical equations of electron

1  Asked on February 21, 2021

### Do angular quantities (such as torque) depend upon the distance from the axis or distance from the origin?

2  Asked on February 21, 2021 by thepixelgenome

### Einstein’s light speed postulate

2  Asked on February 21, 2021 by nik