which the prediction is required r ue8a0 t f r ue8a0 t 0 ue89e u03b4 2 ue89e ue89e u03c0 ue8a0 u03b8 r ue8a0 t f u03b8 r ue8a0 t 0 ue89e where r t i is the tape pack radius at time t i and u03b8 r t i is the angular position of the take up reel at time t i thus r ue8a0 t f r ue8a0 t 0 u03b4 2 ue89e ue89e u03c0 ue8a0 u03b8 r ue8a0 t n u03b8 r ue8a0 t 0 eqn ue89e 5 inaccuracies in this model are accounted for in the kalman filter by an estimate of u201c noise u201d applied to the system u2014 that is by quantifying as applied noise essentially unmeasurable attributes of the system dynamics the applied noise is one of the factors that is used in determining the kalman filter gain k as discussed below with reference to fig2 and 3 to determine the filter gain the system also requires an estimate of the measurement noise or error variance the system implicitly assumes that the errors in the various position measurements can be represented by independent zero mean random variables u03b5 i which each have u201c small u201d variances we know that for a smooth differentiable function g that maps u201c n u201d real numbers to one real number we have var g x 1 u03b5 1 x n u03b5 n u2248 0033 u2202 g u2202 x 1 ue89e x.