Spatial acceleration

In physics the study of rigid body motion provides for several ways of defining the acceleration state of a rigid body. The classical definition of acceleration entails following a single particle/point along the rigid body and observing its changes of velocity. In this article the notion of spatial acceleration is explored, which entails looking at a fixed (unmoving) point in space and observing the changes of velocity of whatever particle/point happens to coincide with the observation point. This is similar to the acceleration definition fluid dynamics where typically one can measure velocity and/or accelerations on a fixed locate inside a testing apparatus.

Definition

Consider a moving rigid body and the velocity of an particle/point P along the body being a function of time and position.

${\vec {v}}_{P}={\vec {v}}(t,{\vec {r}}_{P})$

The spatial acceleration vector of point P is defined as:

${\vec {\psi }}_{P}=\lim _{dt\to \infty }({\vec {v}}(t+dt,r_{P})-{\vec {v}}(t,r_{P}))/dt$

The material acceleration vector of particle P is defined as:

${\vec {a}}_{P}=\lim _{dt\to \infty }({\vec {v}}(t+dt,r_{P}+{\vec {v}}_{P}\cdot dt)-{\vec {v}}(t,r_{P}))/dt$

(to be written)

References

• Frank M. White (2003). Fluid Mechanics. McGraw-Hill Professional. ISBN 0072402172..
• Roy Featherstone (1987). Robot Dynamics Algorithms. Springer. ISBN 0898382300.. 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.
• JPL DARTS page has a section on spatial operator algebra (link: [1]) as well as an extensive list of references (link: [2]).