Yi Yi GXij

Notice that if the original system of differential equations X = F(X, t) is linear, the second and higher order partial derivatives of F(X,t) are zero (i.e., fxP = 0, i > 2). The same statements apply to G(Xi,ti) in Eq. (4.2.5). Hence, for a linear system there is no need to deal with a state or observational deviation vector or a reference solution. However, for the orbit determination problem, F(X,t) and G(Xi,ti) will always be nonlinear in X(t), thus requiring that we deal with deviation vectors and a reference trajectory in order to linearize the system.

Generally in this text, uppercase X and Y will represent the state and the observation vectors and lowercase x and y will represent the state and observation deviation vectors as defined by Eq. (4.2.3). However, this notation will not always be adhered to and sometimes x and y will be referred to as the state and observation vectors, respectively.

Example 4.2.1

Compute the A matrix and the H matrix for a satellite in a plane under the inflience of only a central force. Assume that the satellite is being tracked with range observations, p, from a single ground station. Assume that the station coordinates, (XS, is), and the gravitational parameter are unknown. Then, the state vector, X, is given by

0 0

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