GPS/UWB tightly coupled system modeling

The system equation is defined as:

where τ is the observation period; X is the state vector; F is the state transition matrix, G is the process noise model, and D is the relative acceleration noise, which obeys a Gaussian distribution law with zero mean and standard deviation of σ along each coordinate axis. Define the covariance matrix of the process noise as Q=σ2GGT, for receiver a and receiver b, with the following equation:

in which In is an identity matrix of n×n.

The observation equation for the relative localization of this experiment can be defined as:

where y is the observation vector, which includes the GPS pseudorange, the Doppler shift, and the actual distance between receivers a and b based on the UWB measurements; h is a nonlinear function, which is derived from Eqs. (4) and (5) as well as the true relative distance between the two vehicles R^ab=rab→Trab→; if the number of visible satellites of receivers a and b is m+1, under this condition, the observation vector y and the measurement noise ζ are expressed as follows:

Assuming that the observations are independent of each other, the measurement noise covariance matrix can be defined as the following equation:

If σρ2,σϑ2,σr2 are the variance of the pseudorange, the Doppler shift, and the UWB measurement error, respectively, it follows that.

where 1 is denoted as a matrix in which all elements are 1.