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Real Time Kinematic GPS in an Urban Canyon Environment Extended Kalman Filtering (EFK) The EKF describes the evolution of the states, the measurement model relates the state vector to the GPS observations through the design matrix H. Regular updates by the measurement into the state vector is crucial as the system will diverge if there is no measurement provided over a long period of time, driven by the system input noise. The observations for the float filter are double difference carrier phase and code observables. The advantage of using a filter here is that with more measurements available, the estimation of the float solution is more accurate and also reduces the size of the search space. In summary, the process and measurement models for the extended Kalman Filter are given by 
The symbol Wk and Vk denotes variance of the process noise and are zero-mean normal distributed white noise and characterized by covariance matrices Q and R respectively. The essential implementations for the extended Kalman filter are given as: 
In the static model, the state transition matrix Fk can be described as 
For static positioning, the state vector contains only the position, and Wk is set to a small number. The measurement equation can be written as in equation (9) 
R1, R2, R3, R4, are the measurement consists of pseudorange and phase observation and hence the relation between the measurements and the states (position vector) is not linear. Equation (15) can be linearised by approximating with Taylor series expansion about the predicted value of the states at and retaining only the first-order terms (equation 17). The linearised measurement equation in the extended Kalman filter is defined as 
where H is the Jacobian matrix given as follows 
where 
The measurement vector z is expressed as 
In the dynamics model of the filter, three states which will be nine variables that are three linear degrees of freedom (position vector), the correspondence velocity variables (velocity vector) and correspondence acceleration variables (acceleration vector) are considered. In this application, the state model can be written as 
This model can be represented by the equation 
The relation between the previous states and the current states are govern by the transition matrix Fk, 
The H matrix is the same as that of the static section only that it is expanded with zeros considering the velocity and the acceleration vectors. Hence the H matrix in the kinematic state now becomes a 2nx9 matrix. Observation vector is however the same as the static model. In implementing the EKF the initial estimates of the state vector, the corresponding velocity and acceleration components were obtained from the first epoch measurement. In the EKF models the parameters used in initializing the filters are the same. However, there are differences in the size of the matrices which change to reflect the state estimates. It is important to note here that the covariance matrix is actually an estimate of the statistics of the estimation error vector. Wrongly modelling of the system dynamics or of the process or measurement noises can cause the true estimation error uncertainties to be quite different from the covariance matrix computed by the filter. Modelling only a subset of the total set of errors (suboptimal filter) will also cause an inaccurate covariance matrix. When this occurs, the accuracy of the positioning may be substantially degraded. The process whereby the covariance matrix of the mechanized filter is made to closely approximate the true covariance matrix is referred to as Kalman filter tuning. One way in which GPS Kalman filters are often tuned is through the use of adaptive tuning. Specifically, this refers to dynamically setting the process noise Q as a function of vehicle motion. This approach is used in accounting for wrong modelling in the state dynamics model. In this case, the errors are not Gaussian noise, but may be biases in turns. Therefore, the correct Q depends on the vehicle profile. |