Position, Navigation, and Timing Technologies in the 21st Century

Figure 40.22 Geometry and coverage on estimation error and receiver complexity.

A radio receiver can relatively easily measure the TOA of a variety of SOOP such as those DTV signals described in Section 40.2and AF/FM signals [86–89] and cellular signals [46, 90]. But it needs a means of determining the TOT in order to generate range measurements. Once the ranges to signal sources at known locations are available, the receiver location can be determined. In Section 40.3, we describe a calibration method that uses the initial position information, either known a priori as is the case for many navigation systems or from an aiding source (thus cooperative), to determine the TOT and the clock drift as well. As long as the operations of the signal sources and receiver are not interrupted, the one‐time calibration remains valid for subsequent relative positioning. The aiding source for this method can be a digital map, a visual determination at a known road intersection, or a cooperative navigator (either remote or co‐located).

Instead of the absolute position ( x , y ), a receiver may calculate its position relative to a reference point ( x 0, y 0) as Δ x = x – x 0and Δ y = y – y 0, respectively, which can be understood as a displacement vector (Δ x , Δ y ). Adding successive displacements onto the initial position yields a continuous navigation solution [91], thus making radio dead reckoning. Like a self‐contained inertial navigation solution, the accuracy of a radio dead‐reckoning solution cannot be better than the initial condition. However, unlike the inertial solution, whose errors keep grow due to time integration of the accelerometer bias and gyro drift, the radio dead‐reckoning solution errors may stay bounded due to direct displacement estimation.

Denote the location of the k ‐th transmitter by ( x k, y k) and the unknown TOT of this transmitter by TOT k. The TOA measurements at the reference point and a subsequent time, denoted by (with ) and TOA k(with TOT k), respectively, are given by

(40.8a)

(40.8b)

where c is the speed of light, and and w kare uncorrelated measurement errors assumed to be zero‐mean Gaussian with variances and ( σ k) 2, respectively. This assumption becomes invalid in the presence of NLOS signals, as further discussed in Section 40.4.3.

Similar to Eq. 40.1, we have TOT k= nT field+ + c Δ t , with the last term accounting for the unknown clock offset between the transmitter and receiver to estimate. Taking the difference between Eqs. 40.8aand 40.8bgives

(40.8c)

where is the combined measurement error, a Gaussian with zero mean and variance + .

By single difference, Eq. 40.8celiminates but requires estimation of Δ t instead. The equation can be further linearized around ( x 0, y 0) as

(40.9a)

(40.9b)

where

(40.9c)

With Eq. 40.9bfrom at least three non‐co‐located transmitters, an instantaneous solution for (Δ x , Δ y , Δ t ) can be obtained via, say, the iterative least squares method. With a kinematic model for the displacement and clock state, a sequential solution for (Δ x , Δ y , Δ t ) can be obtained via the extended Kalman filter as formulated in [23]. In other words, from the relative ranges, we can derive displacement vectors as a form of relative positioning. The concept of time‐differencing measurements to a transmitter in order to remove the unknown time of transmission is similar to time differencing of GNSS carrier phase measurements in order to remove ambiguity. It has been applied to GNSS‐only positioning [92] and to GNSS/INS positioning [28, 93].

In the above formulation, a constant nominal period T fieldis used. However, a practical radio transmitter is subject to a certain frequency error. Experimental data in Figure 40.20(f) show such a clock drift. While omitted for short data sets, the frequency error can be accounted for in the time‐differencing formulation ( Eq. 40.8c) by introducing a slow‐varying drift term per transmitter to be then estimated jointly.