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 formula_1. 
The linear velocity vector formula_2 at "P" is expressed in terms of the velocity vector formula_3 at "C" as:
formula_4
where formula_1 is the angular velocity vector.
The material acceleration at "P" is:
formula_6
where formula_7 is the angular acceleration vector.
The spatial acceleration formula_8 at "P" is expressed in terms of the spatial acceleration formula_9 at "C" as:
formula_10
which is similar to the velocity transformation above. 
In general the spatial acceleration formula_8 of a particle point "P" that is moving with linear velocity formula_2 is derived from the material acceleration formula_13 at "P" as:
formula_14