Navigation Systems - Dilution of Precision and Least Squares

This flashcard set covers the vocabulary and key concepts from Lecture 8 on Navigation Systems, focusing on Dilution of Precision (DOP), least squares measurement processing, and positioning methodologies like TOA and TDOA.

Dilution of Precision (DOP)

A factor that expresses the level of degradation of the accuracy of an aspect of the navigation solution due to the relative geometry of the satellites.

GDOP

Geometric Dilution of Precision, which expresses total uncertainty by combining both geometry and time.

PDOP

Position Dilution of Precision, expressing the uncertainty of 3D positioning ($x$, $y$, and $z$).

HDOP

Horizontal Dilution of Precision, expressing the uncertainty in the user's North/South and East/West directions.

VDOP

Vertical Dilution of Precision, expressing the uncertainty in the user's vertical direction.

TDOP

Time Dilution of Precision, expressing the uncertainty in time synchronization.

NDOP

North Dilution of Precision, expressing the specific uncertainty in the North/South direction.

EDOP

East Dilution of Precision, expressing the specific uncertainty in the East/West direction.

User Equivalent Ranging Error (UERE)

The average noise level of a satellite signal, used to estimate positioning uncertainty when multiplied by a DOP value; a common example is $3\,m$.

Idealized Satellite Configuration

A configuration of 4 satellites in the shape of a tetrahedron, with three satellites equally spaced $120^{\circ}$ apart along the horizon and one satellite directly overhead.

Epsilon ($\epsilon$)

The smallest magnitude number a machine (such as MATLAB) can represent, used as a stopping condition for iterations when the change in value is smaller than this threshold.

Interface Control Document (ICD)

The technical document that specifies the GPS system constraints, including the requirement that time deltas never exceed $\pm 1/2$ week in legacy ephemeris.

Topocentric Frame

A local level coordinate frame on the Earth's surface required for correctly calculating HDOP and VDOP.

Observation Matrix ($\mathbf{H}$)

The matrix relating the four state estimates ($x$, $y$, $z$ position and clock error) to the range observations or pseudoranges to each satellite.

Cofactor Matrix

A $4 \times 4$ matrix derived from the observation matrix ($\mathbf{H}$) used to calculate various DOP values.

TOA

Time Of Arrival navigation, also known as circular mode, which requires estimating a clock state in addition to position coordinates.

TDOA

Time Difference of Arrival navigation, also known as hyperbolic mode, which utilizes the range differences between satellites and a master station.

Residuals

The difference between the actual measurements and the ranges that should be measured based on the current estimate of position $x$.

Redundancy

A representation of how much new information is brought by each measurement or the extent to which two measurements should agree.

Circular Error Probable (CEP)

The size of a circle about the mean that will contain $50\%$ of the navigation solutions.

Distance Root Mean Squared (DRMS)

A measure of positioning accuracy, where $2dRMS$ represents the size of the circle or ellipse containing approximately $65\%$ of solutions.

Minimum Detectable Blunder (MDB)

A parameter for blunder detection that identifies the smallest bias in an input measurement that can be detected given the geometric distribution and measurement errors.

Minimum Detectable Error (MDE)

A parameter representing the smallest error in the output states that can be reliably detected by the system.

Carrier to Noise Density ($C/N_0$)

A metric used in GNSS as a proxy for Signal to Noise Ratio (SNR), used to evaluate the uncertainty of individual satellite measurements.

Blunder Detection

A consistency check process used to identify incorrect measurements, especially when redundant observations are available.