Kinematic GNSS Positioning and DOP Calculation Based on a Behavioral Framework

In this paper, we present a behavioral approach to solve kinematic GNSS positioning problems as well as a way to represent the DOP (dilution of precision) of a moving object. The concept of behavioral modeling was first proposed by Jan C. Willems in a series of papers (J.C. Willems, "From time series to linear system - part I, II, and III," Automatica Vol. 22, 1986.) as a generic tool for mathematical modeling of dynamical systems. In the proposed approach, a GNSS positioning problem is firstly described by a kernel representation and then the problem can be solved by a structured total least-squares (STLS) algorithm. STLS algorithm is a modified version of the traditional total least-squares (TLS) method. It can be shown that the STLS algorithm is able to provide better performance than the TLS algorithm for the problems that possess a particular structure. In the case of the present paper, it is shown that the GNSS positioning problem has a Hankel structure (i.e., the geometric matrix of the pseudorange equation is Hankel), therefore the problem can be solved by an STLS algorithm subject to a Hankel structure. On the other hand, a formula for calculating DOP of a dynamic positioning problem is also provided. The proposed method is able to calculate the DOP value for multiple time epochs, in compared with the conventional DOP formula which can only be derived at a single time epoch, so as to reflect the inter-dependency between consecutive time epochs.