Angular Velocity Estimation From the Angular Acceleration Matrix

Computing the angular velocity $ω$ from the angular acceleration matrix is a nonlinear problem that arises when one wants to estimate the three-dimensional angular velocity of a rigid-body from point-acceleration measurements. In this paper, two new methods are proposed, which compute estimates of the angular velocity from the symmetric part $WS$ of the angular acceleration matrix. The first method uses a change of coordinate frame of $WS$ prior to performing the square-root operations. The new coordinate frame is an optimal representation of $WS$ with respect to the overall error amplification. In the second method, the eigenvector spanning the null space of $WS$ is estimated. As $ω$ lies in this space, and because its magnitude is proportional to the absolute value of the trace of $WS$⁠, it is a simple matter to obtain $ω$⁠. A simulation shows that, for this example, the proposed methods are more accurate than those existing methods that use only centripetal acceleration measurements. Moreover, their errors are comparable to other existing methods that combine tangential and centripetal acceleration measurements. In addition, errors of 2.15% in the accelerometer measurements result in errors of approximately 3% in the angular-velocity estimates. This shows that accelerometers are competitive with angular-rate sensors for motions of the type of the simulated example, provided that position and orientation errors of the accelerometers are accounted for.