GPS with High Rate Sensors

Stochastic Processes (GPS)

A stochastic process is a family of random variables indexed by a parameter for continuous-time stochastic processes orfor discrete-time stochastic sequences. For a deterministic signal such asthe value of x is determined by the value of t. Once t is known, the value of x(t) is known. There is no element of chance. Alternatively, signals […]

Linear Systems with Random Inputs (GPS)

Therefore, this section considers the statistical properties of the output of a linear system when the system model is known and the input is white. The results of this section are often used to model error components with significant time correlation via the state augmentation approach. In such situations, linear stochastic systems forced by Gaussian […]

State Models for Stochastic Processes (GPS) Part 1

As motivation for the study of stochastic systems, consider the error dynamics for a single channel of a tangent plane INS, as depicted in Figure 4.4 and the following equation whererepresents error in the navigation state. The error model is driven by two sources of error: accelerometer error ea and gyro error eg. These errors […]

State Models for Stochastic Processes (GPS) Part 2

Scalar Gauss-Markov Process The scalar Gauss-Markov process refers to the special case of eqn. (4.79) where the state, input, and output are each scalar variables: where w is Gaussian white noise with PSD denoted by a2. In eqn. (4.90), the parameter t is the correlation time. This process has already been used in Examples 4.14 […]

Discrete-time Equivalent Models (GPS)

When the dynamics of the system of interest evolve in continuous time, but analysis and implementation are more convenient in discrete-time, we will require a means for determining a discrete-time model in the form of eqn. (4.65) which is equivalent to eqn. (4.57) at the discrete-time instants Specification of the equivalent discrete-time model requires computation […]

Linear State Estimation (GPS)

For deterministic systems, Section 3.6 discussed the problem of state estimation. This section considers state estimation for stochastic, linear, discrete-time state space systems where: is a known signal. The standard notation and assumptions stated in Section 4.6.1 apply. The state estimate is computed according to whereis called the measurement residual andis a designer specified gain […]

Detailed Examples (GPS) Part 1

This section presents a few detailed examples intended to demonstrate the utility of random variables and stochastic processes in relation to navigation applications. Sections 4.9.1 and 4.9.2 focus on frequently asked questions. Sections 4.9.3 and 4.9.4 discuss navigation examples that are relatively simple compared to those in Part II of the topic, but follow the […]

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 […]

Detailed Examples (GPS) Part 3

One Dimensional Position Aided INS Next, consider the same system as in the previous example, but with a position measurement available at T =1 second intervals. The measurement noiseis white and Gaussian with variance During each intervalthe INS integrates the INS mechanization equations as described in eqn. (4.143). At timethe INS position estimateis subtracted from […]

Complementary Filtering (GPS)

The approach described in Section 4.9.4 and depicted in Figure 4.8 is an example of a feed-forward complementary filter implementation. For this example, the kinematics are given by eqn. (4.141),represents the acceleration measurement in eqn. (4.142), the output prediction is ;and the error estimator is defined in eqns. (4.145) anddesigned by the choice of L.superscript […]

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