A parametrized virtual point reduced estimation framework for determining an unknown force’s magnitude, direction and position of action

This article challenges the question of whether a force applied to a rigid interface can be estimated jointly with its position and direction of action utilizing a model which does not take such a force explicitly into account. Firstly, we discuss linear model-based estimation as a possibility to determine the amplitude, direction and position of a given unknown force acting on a rigid interface. For this purpose, the estimation results of two linear augmented Kalman filters are considered. One configuration delivers estimates for a set of equivalent forces applied to a rigid interface, which are subsequently reduced to a virtual point. The other configuration considers inputs reduced into the virtual point of interest, estimating directly the forces and moments acting on it. After the estimation process, a nonlinear system of equations is solved to characterize an arbitrary unknown acting force in amplitude, position, and direction. Thereafter, this article proposes two novel approaches based on a nonlinear augmented state–space model defined by using the Virtual Point Transformation (VPT) method that incorporates the explicit position or the lever arms of the unknown applied force as well as its direction of action with respect to the considered virtual point as parameters to be estimated. An observability analysis for these nonlinear systems is provided utilizing recent results to determine the observability of nonlinear systems with direct feedthrough and based on Lie derivatives. It was found that only the latter formulation is fully observable and hence an Augmented Extended Kalman Filter (AEKF) parametrized via the lever arms between the unknown force and the considered virtual point is chosen. The proposed methodology is then experimentally validated on an automotive-grade rubber mount component. The results show a superior performance of the nonlinear estimation framework in the time- and frequency-domain over the linear counterparts. Moreover, the unknown position and direction at which the force is applied can be estimated with higher precision and without further processing using the AEKF framework.