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.

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. \mathbf v_P = \mathbf v_C + \boldsymbol \omega \times (\mathbf r_P - \mathbf r_C) where \boldsymbol \omega is the angular velocity vector. The material acceleration at P is: \mathbf a_P = \frac{d \mathbf v_P}{dt} = \mathbf a_C + \boldsymbol \alpha \times (\mathbf r_P - \mathbf r_C) + \boldsymbol \omega \times (\mathbf v_P - \mathbf v_C) where \boldsymbol \alpha is the angular acceleration vector. The spatial acceleration \boldsymbol \psi_P at P is expressed in terms of the spatial acceleration \boldsymbol \psi_C at C as: \begin{align} \boldsymbol \psi_P &= \frac{\partial \mathbf v_P}{\partial t} \\[1ex] &= \boldsymbol \psi_{C} + \boldsymbol \alpha \times (\mathbf{r}_{P} - \mathbf{r}_{C}) \end{align} which is similar to the velocity transformation above. In general the spatial acceleration \boldsymbol \psi_P of a particle point P that is moving with linear velocity \mathbf v_P is derived from the material acceleration \mathbf a_P at P as: \boldsymbol{\psi}_{P} = \mathbf{a}_{P} - \boldsymbol{\omega} \times \mathbf{v}_{P}

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. See 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: [https://web.archive.org/web/20040609052639/http://dshell.jpl.nasa.gov/References/index.php). This reference defines spatial accelerations for use in rigid body mechanics.

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