The axes of random infinitesimal rotations and the propagation of orientation uncertainty

• Off-diagonal elements of the covariance of a rigid body orientation are meaningful.
• Cross-correlation terms relate with the distribution of axes of random rotations.
• Angular uncertainty of points on object’s surface depends on distribution of axes.
• Benefits: smart use of orientation data in robotic operations, e.g. path planning.

Perception systems can measure the orientation of a solid 3D object; however, their measurements will contain some uncertainties. In many robotic applications, it is important to propagate the orientation uncertainties of a rigid object onto the uncertainties of specific points on its surface. The orientation uncertainty can be reported as a 3 × 3 covariance matrix. We show that the off-diagonal elements of this matrix provide important clues about the angular uncertainties of points on the object’s surface. Specifically, large off-diagonal elements correspond to a highly concentrated distribution of axes of random infinitesimal rotations which causes large variability in the angular uncertainties of surface points. In particular, experimental data indicate that the ratio of maximum to minimum angular uncertainties can exceed three. In contrast, small off-diagonal elements correspond to a uniform distribution of axes which causes the angular uncertainty of all points on the object’s surface to be almost constant.