Detailed Examples (GPS) Part 2

Scalar Analysis

Let x represent one component of the position error, the component-wise error density is:

Statistic	RMS	drms	2drms	CEP	R95
Radius
Probability	.393	.632	.982	.500	.950

 

Table 4.1: Summary of various horizontal error statistics in w coordinates and their relationships to the error standard deviation a assuming a circular error distribution

Given a set of samplesof the random variable x, the RMS error is

For large values of N, the RMS value is expected to converge to

From distribution tables for Gaussian random variables (see p. 48 in [107]),

Therefore, if the altitude h is estimated to be 100m withthen the probability thatis 68%, thatis 95%, etc.

Summary Comparison

Table 4.1 summarizes the various horizontal error statistics defined above in relation to the one dimensional error \ standard deviationThe table is useful for converting between error statistics.

Example 4.25 If an analyst is interested in finding the R95 error statistic that is equivalent to a stated drms error statistic of 10.0 m, then from Table 4.1:

Therefore, the equivalent R95 statistic is 17.3 m.

Repeating the process of Example 4.25 for each of the other possible combinations of accuracy statistics in two dimensions results in Table 4.2.

		drms	2drms	CEP	R95
	1.0	1.4	2.8	1.2	2.4
drms	0.7	1.0	2.0	0.8	1.7
2drms	0.4	0.5	1.0	0.4	0.9
CEP	0.8	1.2	2.4	1.0	2.1
R95	0.4	0.6	1.2	0.5	1.0

Table 4.2: Summary of conversion factors from the statistic listed in the in the leftmost column to the statistic indicated in the top row, assuming a circular error distribution

To use this table, find the row corresponding to the given statistic. Multiply this row by the numeric value of that statistic to obtain all the other statistics as indicated in the (top) header row. For example, if the R95 statistic is 1.0 m, them the other statistics are

Instrument Specifications

Inertial instrument specifications quantify various aspects of sensor performance: range, bandwidth, linearity, random walk, rate random walk, etc. A subset of these parameters specify the expected behavioral characteristics of the stochastic sensor errors. The purpose of this example is to illustrate the relationship between these parameters, the sensor data, and the stochastic error model. The section has two goals:

• to illustrate how the parameters of the instrument error models are determined from a set of instrument data; and

• to clarify how the data from an instrument specification data sheet relates to the type of Gauss-Markov model discussed in Section 4.6.

The example will focus on a gyro error model using a simplified version of the error model described in [9, 10] after setting all the flicker noise components to zero. The methodology is similar for accelerometers [11]. This section contains a very basic description of the error model and model parameter estimation method. Many details of the technical procedures have been excluded.

Assume that the output y of a gyro attached to a non-rotating platform is modeled as

The initial conditions are. The stochastic process noiseand measurement noiseare independent, white, and Gaussian withThe nomenclature and units related to the various symbols are shown in Table 4.3.

Symbol	Interpretation		Units
	Random bias, bias, fixed drift
K	Rate random walk
N	Angular random walk
	Rate ramp, Random ramp

Table 4.3: Parameter definitions for the gyro error model of eqns. (4.1384.139). The PSDis defined in eqn. (4.140).

Prior to estimating the model parameters K and N, the effect of the initial conditionsmust first be removed. For each sequence of available datathe first step is to remove the trend defined by

This can be performed by least squares curve fitting as discussed in Section 5.3.2. Representing the data in Y as a column vector, we have that where

Because the sampling times are unique, the matrix A has full row rank andis nonsingular. Therefore, the least squares estimate is computed as (see eqn. (5.26) on p. 176)

Given a large set of instruments, withestimated for each, the distribution ofcould be estimated and the variance parameterscan be determined.

Figure 4.6: Power spectral density of eqn. (4.140).

Givenand the data setthe sequence of residual measurementscan be formed. This process is referred to as detrending and Y is the detrended data. The covariance sequenceis computed fromunder the ergodic assumption, and the PSD of _ is computed as the Fourier transform ofUsing the results of Section 4.5, forthe transfer function model for the system of eqns. (4.138-4.139) is

Given thatandare independent, the PSD of y is

Therefore, a curve fit in the frequency domain, of the theoretical model for Sy from eqn. (4.140) to the PSD data computed from Y provides the estimates of N and K.

Example 4.26 Figure 4.6 displays a graph of eqn. (4.140) along with its component partsFrom the graph,is the approximately0.0045and __ is approximately

The instrument specification parameters are sometimes referred to as Allan variance parameters. This name refers to the Allan variance method that is an alternative approach from the PSD for processing the data set Y to estimate the parameters of the stochastic error model. The Allan variance method was originally defined as a means to quantify clock (or oscillator) stability.

One Dimensional INS

This example analyzes the growth of uncertainty as measured by error variance in a simple unaided inertial navigation system. The example will illustrate that the analysis methods of this topic allow quantitative analysis of each contribution to the system error to be considered in isolation.

Consider a one dimensional single accelerometer inertial navigation system (INS) implemented in an inertial frame. The actual system equations are

An accelerometer is available that provides the measurement

where the manufacturer specifies that:

•is Gaussian white noise with PSD equal to

• b is the random walk process

whereis Gaussian and white with PSD equal to

• the distribution of the initial bias is

The plant state vector is x = [p, v, b].

The implemented navigation equations are

whereis the best estimate of the accelerometer bias b that is available at time t. These equations are integrated forward in time starting fromthe expected initial conditions where we assume that

The navigation system state

vector iswhere the differential equation forbecause the mean ofis zero.

Therefore, the dynamic equations for the error states

and using the definition of Q in eqn. (4.59) in Section 4.6.1

For any time t, due to the fact that in this example F and r are time invariant, the state error covariance can be found in closed form as

Verification that eqn. (4.144) is the solution of eqn. (4.100) is considered in Exercise 4.18. In this example,

and the two terms in eqn. (4.144) are

and

where we have used the fact thatTherefore, the error variance of each of the three states is described by

The growth of the error variance has two components. The first component is due to the initial uncertainty in each error state. This component dominates initially. The second component is due to the driving process noise componentsThe driving noise component dominates the growth for large time intervals. The driving noise is determined by the quality of the inertial instruments. Better instruments will yield slower growth of the INS error. However, without some form of aiding measurement, the covariance of the state will increase without bound.