Context: Positioning technology accuracy and cost trade-offs.
GPS Accuracy: Standard GPS time-of-flight ranging measures satellite signal travel time \(t = d/c\) where \(c = 3 \times 10^8\) m/s. With 4 satellites, multilateration solves for \((x, y, z)\) position with typical error 5-10m. Worked example: DGPS uses ground reference station to measure error vector \(\vec{e}\) and transmit corrections, reducing error to 1-3m. RTK GPS measures carrier phase (wavelength \(\lambda \approx 19\) cm for L1 signal) for centimeter accuracy: \(\Delta d = \lambda \cdot \Delta \phi / 2\pi\) where \(\Delta \phi\) is phase difference.
BLE Beacon Range Estimation: Received Signal Strength Indicator (RSSI) follows the log-distance path loss model \(\text{RSSI} = A - 10n\log_{10}(d/d_0)\) where \(A\) is the measured RSSI at reference distance \(d_0\) (typically 1m) and \(n\) is the path loss exponent (typically 2-4 indoors). Worked example: With \(A = -40\) dBm (measured at 1m), \(n=2.5\), measured RSSI = -65 dBm at unknown distance \(d\): solving \(-65 = -40 - 10(2.5)\log_{10}(d/1\text{m})\) gives \(-25 = -25\log_{10}(d)\), so \(d = 10^{1.0} = 10\) m. In practice, accuracy degrades to approximately 3m due to multipath reflections, body absorption, and environmental variability.
UWB Time-of-Flight: UWB measures two-way time-of-flight with nanosecond precision. Signal travels distance \(d = c \cdot t / 2\) (round-trip). Worked example: With 1 ns timing resolution, distance resolution is \(c \cdot 10^{-9} / 2 = 0.15\) m. UWB’s short pulses (~2 ns) resolve multipath: direct signal at time \(t_1\), reflected signal at \(t_2 > t_1 + 2\) ns can be distinguished, enabling 10-30cm accuracy indoors.
Cost-Accuracy Trade-off: Standard GPS receiver: $0, 5-10m accuracy. DGPS correction service: +$500/year, 1-3m accuracy (5x improvement). RTK base station: +$5,000, 2cm accuracy (250x improvement over standard GPS, but $5,000 upfront for the last meter of improvement). Worked example: For delivery tracking (zone-level), GPS’s 5-10m is sufficient. For warehouse robot navigation (must align to pallet), UWB’s 10-30cm is mandatory despite 100x cost premium.