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 (x0, y0) as Δx = x – x0 and Δy = y – y0, respectively, which can be understood as a displacement vector (Δx, Δy). 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 (xk, yk) and the unknown TOT of this transmitter by TOT k. The TOA measurements at the reference point and a subsequent time, denoted by
where c is the speed of light, and
Similar to Eq. 40.1, we have TOT k = nTfield +
where
By single difference, Eq. 40.8c eliminates
(40.9a)
where
(40.9c)
With Eq. 40.9b from at least three non‐co‐located transmitters, an instantaneous solution for (Δx, Δy, Δt) can be obtained via, say, the iterative least squares method. With a kinematic model for the displacement and clock state, a sequential solution for (Δx, Δy, Δt) 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 Tfield is 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.
In the present setting of dead reckoning from a known location, the initial stationary TOA measurements can be accumulated into a “deterministic” reference. The time difference with respect to such a reference at a fixed time point as in Eq. (40.8) is akin to applying a well‐calibrated bias to all subsequent measurements, which can then be treated as approximately “uncorrelated.” This initial measurement error, if not calibrated out to an insignificant level, would make subsequent time‐difference measurements time‐correlated. Because it is constant, albeit random, it can be viewed as an unknown bias and modeled as an extra state to the positioning Kalman filter.
Strictly speaking, no matter how well a parameter is calibrated, the subsequent measurements can be time‐correlated via the residual calibration errors. Furthermore, time differencing can also be applied continuously to consecutive measurements, that is, between t+1 and t, rather than between t and t0 = 0 as in Eq. (40.8). Consecutive time differences between times (t+1 and t) and (t and t−1) are correlated through the common epoch t. The standard Kalman filter cannot be applied to consecutive time‐difference measurements because the measurements are correlated. In [92], a fixed‐lag smoother was used where the previous position is introduced as an extra state. With both the current and previous positions in the filter, the position difference becomes directly observable by the time differences of carrier phase measured at the current and previous epochs [92]. A similar formulation can be used for SOOP when processing consecutive time differences of phase and/or range measurements.
In the field test environment depicted in Figure
