Matrix Definitions Phi and Q

Fundamentals
Title	Matrix Definitions Phi and Q
Author(s)	J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level	Advanced
Year of Publication	2011

The determination of the state transition matrix [math]\displaystyle{ {\boldsymbol \Phi} }[/math] and Process Noise matrix [math]\displaystyle{ {\mathbf Q} }[/math] is usually based on physical models describing the estimation problem. For instance, for satellite tracking or orbit determination, they are derived from the orbital motion equations. Elemental formulations, i.e., for the SPP and PPP navigation, are covered by the examples given as follows:

Static positioning

The state vector to determine is given by [math]\displaystyle{ \widehat{\mathbf X}=(dx,dy,dz,\delta t) }[/math] where the coordinates are considered as constants (because the receiver is kept fixed) and the clock offset can be modelled as a white noise with zero mean. Under these conditions matrix [math]\displaystyle{ \Phi }[/math] and [math]\displaystyle{ {\mathbf Q} }[/math] are given by:

[math]\displaystyle{ {\boldsymbol \Phi} (n)=\left( \begin{array}{cccc} 1 & & & \\ & 1 & & \\ & & 1 & \\ & & & 0 \\ \end{array} \right) \qquad {\mathbf Q}(n)=\left( \begin{array}{cccc} 0 & & & \\ & 0 & & \\ & & 0 & \\ & & & \sigma _{\delta t}^2 \\ \end{array} \right) \qquad \mbox{(1)} }[/math]

being [math]\displaystyle{ \sigma _{dt} }[/math] the uncertainty in the clock prediction model (for instance [math]\displaystyle{ \sigma _{dt}=1\,ms=300\,km }[/math] for a unknown clock i.e. 1 leap-millisecond). Notice that the prediction model for the coordinates is exact and, thence, the associated elements in matrix [math]\displaystyle{ {\mathbf Q} }[/math] are null.

Kinematic positioning

• If it is a vehicle running at a high velocity, the coordinates can be modeled as a white noise with zero mean, the same as the clock offset:

[math]\displaystyle{ {\boldsymbol \Phi} (n)=\left( \begin{array}{cccc} 0 & & & \\ & 0 & & \\ & & 0 & \\ & & & 0 \\ \end{array} \right) \qquad {\mathbf Q}(n)=\left( \begin{array}{cccc} \sigma_{dx}^2 & & & \\ & \sigma_{dy}^2& & \\ & & \sigma_{dz}^2 & \\ & & & \sigma _{\delta t}^2 \\ \end{array} \right) \qquad \mbox{(2)} }[/math]

• If it is a vehicle running at a low velocity, the coordinates can be modelled as a random walk process with its uncertainty growing with time:

[math]\displaystyle{ {\boldsymbol \Phi} (n)=\left( \begin{array}{cccc} 1 & & & \\ & 1 & & \\ & & 1 & \\ & & & 0 \\ \end{array} \right) \qquad {\mathbf Q}(n)=\left( \begin{array}{cccc} Q^{\prime}_{dx} \Delta t & & & \\ & Q^{\prime}_{dy} \Delta t & & \\ & & Q^{\prime}_{dz} \Delta t & \\ & & & \sigma _{\delta t}^2 \\ \end{array} \right) \qquad \mbox{(3)} }[/math]