Let $\boldsymbol{f}$ be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space H, and let $\boldsymbol{g}$ be an associated square-integrable, zero-mean, random vector with realizations which are not observable in a Hilbert space K. We seek an optimal filter in the form of a closed linear operator X acting on the observable realizations of a proximate vector $\boldsymbol{f}_{\epsilon } \approx \boldsymbol{f}$ that provides the best estimate $\widehat{\boldsymbol{g}}_{\epsilon} = X \boldsymbol{f}_{\epsilon}$ of the vector $\boldsymbol{g}$ . We assume the required covariance operators are known. The results are illustrated with a typical example.

Let Xt be an n-dimensional diffusion process and S(t) be a set-valued function. Suppose Xt is invisible when it is hidden by S(t), but we can see the process exactly otherwise. In this paper, we derive the optimal estimator E[f(X1) | Xs1Xs∉S(s), 0 ≤ s ≤ 1] for a bounded Borel function f. We illustrate some computations for Gauss-Markov processes.

This paper addresses the problem of identifying the uncertainties present in a robotic contact situation. These uncertainties are errors and misalignments of an object with respect to its ideal position. The paper describes how to solve for the errors caused during grasping and errors present when coming into contact with the environment. A force sensor is used together with Kalman Filters to solve for all the uncertainties. The straightforward use of a force sensor and the Kalman Filters is found to be effective in finding only some of the uncertainties in robotic contact. The other uncertainties form dependencies that cannot be estimated in this manner. This dependency brings about the problem of observability. To make the unobservable uncertainties observable a sequence of contacts are used. The error covariance matrix of the Kalman Filter (KF) is used to obtain new contacts that are required to solve for all the uncertainties completely. There is complete freedom in choosing which unobservable quantity to be excited in forming the next contact. The paper describes how these new contacts can be randomly executed. A two dimensional contact situation will be used to demonstrate the effectiveness of the method. Experimental data are also presented to prove the validity of the procedure. Due to the non-linear relationship between the uncertainties and the forces, an Extended Kalman Filter (EKF) has been used.