7B-4-7 
i 
where p m is the smoothed and corrected 
pseudoranges, R' is the geometric range 
without any error, and S tu is the GPS 
receiver clock offset. The measurement noise 
vector V is considered to be a zero-mean 
white process. Let 
h(x)=R‘+ô u (12) 
Equation (11) can be linearized by using a 
truncated Taylor series expansion about the 
predicted state vector X(k / k -1) at time k. 
Hence, (11) can be approximated as 
Z i (t) = H i (t)-x + V i (13) 
where 
\ dx ) x(k/k-\) 
The resulting measurement equation of the 
integrated DGPS/DR system can be written 
in component form as: 
~H\t) 
V" 
Z\t) 
H\t) 
V 2 
z‘(t) 
= 
H'(t) 
X + 
V‘ 
Z"(0 
H n (t) 
_v n _ 
where n is the number of satellites. It 
should be noted that (14) can also be 
used for the case where n is less than 
3. 
3.2 Computational Effective EKF 
The discrete-time forms of (9) and (14) 
can be written as 
X{k +1) = <&(*) • X(k) + T(k) • W{k) (15) 
Z(k) = H(k)-X(k)+V(k) (16) 
where 0(&) is the state transition matrix, 
T(k) is the coefficient matrix, H(k) is the 
measurement matrix, and d>(/:), V(k) and 
H(k) are sparse matrices. w(k) and v(k) are 
white noise with associated covariance 
matrices Q(k) and R(k), respectively. 
The EKF algorithm is given in Table 1. For 
the EKF, most of the computational loads are 
due to the covariance matrix updating laws 
(Lewis, 1986). Since the state transition 
matrix <f>(&) is a sparse matrix (about 82 
percent of the elements in <X>(/Q are zero). If 
only multiplications with the non-zero 
elements are needed for implementation the 
structure of the sparse matrices should be 
exploited when implementing the EKF 
algorithm. This will reduce the number of