Spatial Acceleration

In physics, the study of rigid body motion allows for several ways to define the acceleration of a body. The usual definition of acceleration entails following a single particle/point of a rigid body and observing its changes in velocity. Spatial acceleration entails looking at a fixed (unmoving) point in space and observing the change in velocity of the particles that pass through that point. This is similar to the definition of acceleration in fluid dynamics, where typically one measures velocity and/or acceleration at a fixed point inside a testing apparatus.

Definition

Consider a moving rigid body and the velocity of a point ''P'' on the body being a function of the position and velocity of a center-point ''C'' and the angular velocity $\vec \omega$.

The linear velocity vector $\vec v_P$ at ''P'' is expressed in terms of the velocity vector $\vec v_C$ at ''C'' as:

$\vec v_P = \vec v_C + \vec \omega \times (\vec r_P-\vec r_C)$

where $\vec \omega$ is the angular velocity vector.

The material acceleration at ''P'' is:

$\vec a_P = \frac$

$\vec a_P = \vec a_C + \vec \alpha \times (\vec r_P-\vec r_C) + \vec \omega \times (\vec v_P-\vec v_C)$

where $\vec \alpha$ is the angular acceleration vector.

The spatial acceleration $\vec \psi_P$ at ''P'' is expressed in terms of the spatial acceleration $\vec \psi_C$ at ''C'' as:

$\vec \psi_P = \frac$

$\vec_ = \vec_+\vec\times(\vec_-\vec_)$

which is similar to the velocity transformation above.

In general the spatial acceleration $\vec \psi_P$ of a particle point ''P'' that is moving with linear velocity $\vec v_P$ is derived from the material acceleration $\vec a_P$ at ''P'' as:

$\vec_=\vec_-\vec\times\vec_$

References

*.
*. This reference effectively combines screw theory with rigid body dynamics for robotic applications. The author also chooses to use spatial accelerations extensively in place of material accelerations as they simplify the equations and allows for compact notation. Se
online presentation, page 23
also from same author.
*JPL DARTS page has a section on spatial operator algebra (link

as well as an extensive list of references (link

.
*{{cite book, title=Springer Handbook of Robotics, author=Bruno Siciliano, Oussama Khatib, publisher=Springer, year=2008. Page 41 (link: Google Book

defines spatial accelerations for use in rigid body mechanics.

Rigid bodies
Acceleration