2D Rotation matrices (4): velocity and acceleration fields in planar rotational motion

Antonio Sala, UPV

Difficulty: ** ,       Relevance: ,      Duration: 33:13

Summary:

This video analyses rotations in a “differential way” i.e., as a “motion”, understanding the kinematics of planar rotation changing with time $x_w(t)\; =\; R(t)x_b$.

Such a study of kinematics will be addressed via a discussion of the following three topics:

• Linearization (Jacobians) of small angle deviations. Linearization in body frame (skew-symmetric matrix $J\; =\; [0-\; 1;\; 10]$ so $R(𝜃)\; \approx \; I\; +\; J𝜃$ for small $𝜃$).

• Velocity field: a changing rotation can be expressed as $\frac{d{x}_w}{dt}=\mathrm{\Omega }{x}_w$ where $\mathrm{\Omega }=\; wJ$, is an angular velocity skew-symmetric matrix, with $J$ being above skew-symmetric matrix, infinitesimal generator of motions. The velocity field is orthogonal to position.

• Acceleration field: carrying out further derivatives, it is proved that acceleration in world frame is $(\alpha J\; -{\mathrm{\Omega }}^2\cdot \; I)x_b$, where $\alpha J$ is responsible for the “tangential acceleration” component and $-{\mathrm{\Omega }}^2\cdot \; I$ is the “normal acceleration” component.

*Link to my [whole collection] of videos in English. Link to larger [Colección completa] in Spanish.