Block 6 Flashcards

Click a question to reveal the answer.

1. Where in space is the GPS constellation, and how often does each satellite orbit?

Medium Earth Orbit (MEO) at about 20,200 km altitude with a 12-hour orbital period. The constellation is sized so at least four satellites are in view from anywhere on Earth at any time.

2. What is GNSS, and what constellations does it include?

Global Navigation Satellite System: the family of satellite navigation systems including GPS (USA), GLONASS (Russia), Galileo (EU), and BeiDou (China). Modern receivers can mix satellites from multiple constellations, which improves availability and geometry. The math is the same; only the satellite-position lookup table changes.

3. What does a GPS satellite continuously broadcast?

Its precise orbital position (broadcast ephemeris), an atomic timestamp, and a unique pseudo-random noise (PRN) code that lets the receiver identify which satellite the signal came from.

4. Define the pseudorange. Why is it called "pseudo"?

\(\tilde{\rho}_i = \lVert \mathbf{r}_u - \mathbf{r}^{(i)} \rVert + c\,\delta t_u + \epsilon_i\): true geometric range plus an unknown receiver-clock bias \(\delta t_u\) times the speed of light, plus noise. The "pseudo" refers to the clock-bias term, which corrupts every measurement equally because the receiver clock is not synchronized with the satellite atomic clocks.

5. How many satellites are needed to solve for 3D position and clock bias?

At least four. The unknowns are \([x, y, z, \delta t_u]\): three position components plus one clock bias. Each pseudorange contributes one equation, so four equations are required for a unique solution. With more than four, the system is over-determined and you solve it as nonlinear least squares.

6. Why is GPS positioning a nonlinear problem?

Because the pseudorange equation contains \(\lVert \mathbf{r}_u - \mathbf{r}^{(i)} \rVert\), a Euclidean norm in user position. There is no closed-form solution; you have to iterate. Standard methods are Newton-Raphson, Levenberg-Marquardt, or general nonlinear least-squares routines (MATLAB fitnlm, scipy least_squares).

7. State the GPS positioning least-squares problem.

\(\hat{\mathbf{x}} = \arg\min_{\mathbf{x}}\sum_i \big(\tilde{\rho}_i - h_i(\mathbf{x})\big)^2\), where \(h_i(\mathbf{x}) = \lVert \mathbf{r}_u - \mathbf{r}^{(i)} \rVert + c\,\delta t_u\) is the predicted pseudorange to satellite \(i\). Minimizing the sum of squared residuals gives the maximum-likelihood estimate under independent Gaussian noise on each pseudorange.

8. What is Dilution of Precision (DOP)?

A scalar that quantifies how satellite geometry amplifies pseudorange noise into position error: \(\sigma_{\text{position}} = \mathrm{DOP} \times \sigma_\rho\). DOP depends only on geometry, not on signal strength or receiver hardware.

9. List the four common DOP variants and what they measure.

HDOP (horizontal position uncertainty per unit pseudorange noise), VDOP (vertical), PDOP (combined 3D position), TDOP (clock bias). Each is a scalar derived from the position-covariance matrix \(\mathbf{P}_{\text{pos}} = \sigma_\rho^2 (H^\top H)^{-1}\) at the solution.

10. Typical good and bad HDOP values?

HDOP \(\approx 1\) to \(2\) is good; HDOP \(> 4\) is degraded and often flagged by receivers. HDOP \(\gg 4\) is pathological — position is essentially unconstrained in some direction. The Block 6 demo includes a clustered-satellite case where HDOP exceeds 30.

11. Why is VDOP almost always worse than HDOP?

GPS satellites only sit at or above the horizon — the receiver never sees one "below" it. The vertical component of the geometry is always partially under-constrained relative to horizontal, so vertical position uncertainty is larger for the same pseudorange noise. This is why aviation altimetry typically uses pressure or radar altimeters as the primary vertical reference, not GPS.

12. What are the four dominant GPS error sources and their typical magnitudes?

Ionospheric delay (1 to 5 m, mitigated by dual-frequency receivers or broadcast model corrections); tropospheric delay (0.5 to 2 m, mitigated by the Saastamoinen model); multipath (0.5 to 3 m, mitigated by antenna placement and signal processing); receiver noise (0.1 to 0.3 m, set by hardware quality). Lumped: \(\sigma_\rho \approx 3\) to \(5\) m for L1 SPS.

13. Quick numeric: \(\sigma_\rho = 3\) m, HDOP = 1.5. What is \(\sigma_{\text{horiz}}\)? What if a solar storm doubles \(\sigma_\rho\)?

Baseline: \(\sigma_{\text{horiz}} = 1.5 \times 3 = 4.5\) m. Solar storm: \(\sigma_{\text{horiz}} = 1.5 \times 6 = 9\) m. With good geometry, doubling pseudorange noise just doubles position error. With a bad geometry, the same doubling lands you in a much worse place because DOP multiplies the increase.

14. Why is GPS easy to jam, and what is spoofing?

The GPS signal at Earth's surface is about \(-130\) dBm — roughly a billionth of a typical Wi-Fi signal. A small-power transmitter on the right frequency overwhelms it locally and denies positioning to every receiver in the area. Spoofing is the next level: counterfeit signals that mimic real GPS broadcasts and trick receivers into computing a wrong position. Both threats motivate fault detection (Block 8) and alternative-navigation backups.

15. How does GPS feed into a Kalman filter? What is \(\mathbf{R}\) set to?

Two paths exist (Block 7): loose coupling treats the GPS solver's output position as a linear measurement, with \(\mathbf{R} \approx (\mathrm{HDOP}\times\sigma_\rho)^2\,\mathbf{I}\). Tight coupling feeds raw pseudoranges into the filter directly, requiring an Extended Kalman Filter to handle the nonlinearity. Either way, the value of \(\sigma_\rho\) from the error budget above sets the diagonal of \(\mathbf{R}\).

16. What does the Block 6 MATLAB demo (GPSPositioningDemo.m) demonstrate?

2D GPS positioning from pseudoranges using nonlinear least-squares (fitnlm). Two scenarios with the same \(\sigma_\rho\) and the same number of satellites — one with good (well-spread) geometry, one with bad (clustered) geometry — produce dramatically different position uncertainties. The demo computes HDOP from the Jacobian-based covariance and shows that \(\sigma_{\text{position}} = \mathrm{HDOP} \times \sigma_\rho\) matches the empirical position error.