BAYES ALGORITHM WITH ITERATIVE DIFFERENTIAL CORRECTION FOR NONLINEAR SYSTEMS DETERMINATION OF UPDATED ESTIMATES We are now in a position to obtain the updated estimate for the nonlinear observation and target dynamic model cases [5, pp. 424–443]. We shall use the example of the ballistic projectile traveling through the atmosphere for definiteness in our discussion. Assume that the past measurements have " permitted us to obtain the state vector estimate Xðt À Þ at the time t À , the last time observations were made on the target. As done at the end of Section , it is convenient to designate. | Tracking and Kalman Filtering Made Easy. Eli Brookner Copyright 1998 John Wiley Sons Inc. ISBNs 0-471-18407-1 Hardback 0-471-22419-7 Electronic 17 BAYES ALGORITHM WITH ITERATIVE DIFFERENTIAL CORRECTION FOR NONLINEAR SYSTEMS DETERMINATION OF UPDATED ESTIMATES We are now in a position to obtain the updated estimate for the nonlinear observation and target dynamic model cases 5 pp. 424-443 . We shall use the example of the ballistic projectile traveling through the atmosphere for definiteness in our discussion. Assume that the past measurements have permitted us to obtain the state vector estimate X t at the time t the last time observations were made on the target. As done at the end of Section it is convenient to designate this past time t Q with the index k and write X t as X k. Assume that the measurements are being made in polar coordinates while the projectile is being tracked in rectangular coordinates that is the state-vector is given in rectangular coordinates as done in Section . Although previously the projectile trajectory plane was assumed to contain the radar we will no longer make this assumption. We will implicitly assume that the radar is located outside the plane of the trajectory. It is left to the reader to extend and to this case. This is rather straightforward and we shall refer to these equations in the following discussions as if this generalization has been made. This extension is given elsewhere 5 pp. 106110 . By numerically integrating the differential equation given by starting with X t at time t we can determine X t . As before we now find it convenient to also refer to X t Xn as X k it being the estimate of the predicted state vector at time t or n based on the measurement at time t or k n . We can compute the transition matrix n k by numerical integration of the differential equation given by with A given by . In turn n k can be used to determine the covariance matrix of X t 367 368 .
