One pseudorange constrains the receiver to a surface.
GPS Pseudorange Geometry Workbench
GPS Pseudorange Geometry Workbench
Build a four-unknown GPS fix from range spheres, receiver-clock bias, and satellite geometry
animation
simulation
gps
gnss
positioning
geometry
Deterministic GPS pseudorange geometry workbench with draggable satellites, receiver-clock bias, signed measurement noise, Gauss-Newton solving, rank gating, GDOP, and PDOP.
GPS Pseudorange Geometry Workbench
Each satellite contributes one measured range containing the same receiver-clock error. Build the constraints in stages, then solve position and clock together.
Unknownsx, y, z, clock bias
Enabled equations4 pseudoranges
Rank gaterank 4 — fix available
GeometryPDOP 1.87
Try
See why a receiver clock adds a fourth unknown to three position coordinates.
Observe
Choose “Three only”, then change only Enabled measurements from 3 to 4.
Explain
Rank changes from 3 to 4 and the hidden clock term becomes solvable.
Boundary
Every pseudorange contains geometric distance plus one shared receiver-clock bias.
More satellites intersect the candidate geometry.
The shared clock term shifts every pseudorange.
QR solves one Gauss-Newton update without forming an inverse.
Rank, residuals, PDOP, and convergence qualify the fix.
Range-constraint scene
Start with one measured sphere. Drag a satellite or use its keyboard arrow keys to change the teaching geometry.
Solver statusSolved
Four independent measurements reveal all four unknowns.
Receiver estimate0.000, 0.000, 0.000 km
Converged in 3 iterations.
Clock bias300.000 m · 1.00 us
One shared term is estimated with position.
Rank4 of 4
The Jacobian has four independent columns.
GDOP / PDOP2.00 / 1.87
Geometry amplifies range uncertainty.
Predicted position RMS0.00 m
sigma range × PDOP.
Controls
Play follows the same five deterministic stages as Step.
Receiver and measurements
Selected satellite
Deterministic measurement noise
One state, one calculation
The scene, residual ledger, solution, and geometry diagnostics all consume the same pure calculation result.
rho_i = ||r_true - s_i|| + b_clock + noise_i
residual_i = rho_i - (||r - s_i|| + b)
delta = QR_least_squares(J, residual)
[r,b]_(k+1) = [r,b]_k + delta
predicted position RMS = sigma_range × PDOP
residual_i = rho_i - (||r - s_i|| + b)
delta = QR_least_squares(J, residual)
[r,b]_(k+1) = [r,b]_k + delta
predicted position RMS = sigma_range × PDOP
Four independent measurements solve four unknowns.The receiver position and shared clock bias are published together.
Pseudorange and residual ledger
All distances below use the scale-neutral local teaching coordinates stored internally in metres.
| Satellite | Pseudorange | Noise | Residual | Used |
|---|
Deterministic fixture checks
These source-contract fixtures run in the page at 1e-6 relative tolerance. Rounded GDOP/PDOP gates are compared at their contracted two-decimal display precision.
| Fixture | Computed | Expected | Result |
|---|
Technical boundaries. This is a deterministic pseudorange geometry solver with known satellite coordinates and one shared receiver-clock term. It does not acquire RF signals or model ephemeris/orbit error, satellite clock correction, ionosphere, troposphere, relativity, Earth rotation, multipath/NLOS, integer ambiguity, constellation time offsets, RAIM, Kalman filtering, map matching, or receiver power. Local fixture coordinates are scale-neutral solver checks, not satellite orbits. Real GNSS fixes require corrected ephemerides, calibrated measurements, quality checks, and more than the minimum satellites.
Model ownership: this workbench owns multi-satellite pseudorange equations, receiver-clock bias, solver iteration, and geometry amplification. UWB TWR retains two-device timing exchange and NLOS ranging; Location Selector retains technology and deployment choice.