Position, Navigation, and Timing Technologies in the 21st Century. Группа авторов

(38.3)) by solving a weighted nonlinear least‐squares (WNLS) problem. However, in practice, the BTSs’ states are unknown, in which case the mapper/navigator framework can be employed [18, 25].

Consider a mapper with knowledge of its own state vector (by having access to GNSS signals, for example) to be present in the navigator’s environment as depicted in Figure 38.3.

      The mapper’s objective is to estimate the BTSs’ position and clock bias states and share these estimates with the navigator through a central database. For simplicity, assume the position states of the BTSs to be known and stored in a database. In the sequel, it is assumed that the mapper is producing an estimate and an associated estimation error variance for each of the BTSs.

Consider M mappers and N BTSs. Denote the state vector of the j‐th mapper by , the pseudorange measurement by the j‐th mapper on the i‐th BTS by , and the corresponding measurement noise by . Assume to be independent for all i and j with a corresponding variance . Define the set of measurements made by all mappers on the i‐th BTS as

Figure 38.3 Mapper and navigator in a cellular environment (Khalife et al. [18]; Khalife and Kassas [25]).

      Source: Reproduced with permission of IEEE, ION.

      where and . The clock bias is estimated by solving a weighted least‐squares (WLS) problem, resulting in the estimate

      and the associated estimation error variance , where W is the weighting matrix. The true clock bias of the i‐th BTS can now be expressed as , where w_i is a zero‐mean Gaussian random variable with variance .

      Since the navigating receiver is using the estimate of the BTS clock bias, which is produced by the mapping receiver, the pseudorange measurement made by the navigating receiver on the i‐th BTS becomes

      where and η_i ≜ v_i − w_i models the overall uncertainty in the pseudorange measurement. Hence, the vector is a zero‐mean Gaussian random vector with a covariance matrix ∑ = C + R, where is the covariance matrix of and is the covariance of the measurement noise vector . The Jacobian matrix H of the nonlinear measurements with respect to is given by , where

      The navigating receiver’s state can now be estimated by solving a WNLS problem. The WNLS equations are given by

      where l is the iteration number, and denotes the nonlinear measurements evaluated at the current estimate .

      38.4.2 Radio SLAM Framework

      A dynamic estimator, such as an extended Kalman filter (EKF), can be used in the radio SLAM framework for stand‐alone receiver navigation (i.e. without a mapper). Certain a priori knowledge about the BTSs’ and/or receiver’s states must be satisfied to make the radio SLAM estimation problem observable [27, 40–42].

      To demonstrate a particular formulation of the radio SLAM framework, consider the simple case where the BTSs’ positions are known. Also, assume the receiver’s initial state vector to be known (e.g. from a GNSS navigation solution). Using the pseudoranges (Eq. (38.3)), the EKF will estimate the state vector composed of the receiver’s position and velocity as well as the difference between the receiver’s clock bias and each BTS and the difference between the receiver’s clock drift and each BTS, specifically

      where ; δt_r and are the receiver’s and the i‐th BTS clock bias, respectively; and and are the receiver’s and the i‐th BTS clock drift, respectively.

      Assuming the receiver to be moving with velocity random walk dynamics, the system’s dynamics after discretization at a uniform sampling period T can be modeled as

      (38.4)