Mohinder S. Grewal - Global Navigation Satellite Systems, Inertial Navigation, and Integration

Theoretically, one can recover the sensor input from the sensor output so long as the input–output relationship is known and invertible. Lock‐in (or “dead zone”) errors and quantization errors are the only ones shown with this problem. The cumulative effects of both types (lock‐in and quantization) often benefit from zero‐mean input noise or dithering. Also, not all digitization methods have equal cumulative effects. Cumulative quantization errors for sensors with frequency outputs are bounded by one‐half least significant bit (LSB) of the digitized output, but the variance of cumulative errors from independent sample‐to‐sample A/D conversion errors can grow linearly with time.

In inertial navigation, integration turns white noise into random walks.

The accuracy demands on sensors used in inertial navigation cannot always be met within the tolerance limits of manufacturing, but can often be met by calibrating those errors after manufacture and using the results to compensate them during operation. Calibration is the process of characterizing the sensor output, given its input. Sensor error compensation is the process of determining the sensor input, given its output. Sensor design is all about making that process easier. Another problem is that any apparatus using physical phenomena that might be used to sense rotation or acceleration may also be sensitive to other phenomena, as well. Many sensors also function as thermometers, for example.

Figure 3.6is a schematic of such an error compensation procedure, using the example of a gyroscope that is also sensitive to acceleration and temperature (not an unusual situation). The first problem is to determine the input–output function

where the ellipsis “ ” allows for the effects of more variables to be compensated. The functional characterization is usually done using a set of controlled input values and measured output values. The next problem is to determine its inverse,

and use it with independently sensed values for the variables involved – (sensor output), (compensated accelerometer output) and (temperature) in this example.

Figure 3.6Gyro error compensation example.

If the input–output function is common to all sensors of the same design, then this only has to be done once. Otherwise, it can become expensive.

There are also methods using nonlinear Kalman filtering and auxiliary sensor aiding for tracking and updating compensation parameters that may drift over time.

The individual sensor input axes within an inertial sensor assembly (ISA) must be aligned to a common reference frame, and this can be combined with sensor‐level calibration of all sensor compensation parameters, as illustrated in Figure 3.5. Figure 3.7illustrates how input axis misalignments and scale factors at the ISA level affect sensor outputs, in terms of how they are related to the linear input–output model,

(3.1)

(3.2)

where is a vector representing the inputs (accelerations or rotation rates) to three inertial sensors with nominally orthogonal input axes, is a vector representing the corresponding outputs, is a vector of sensor output biases, and the corresponding elements of are labeled in Figure 3.7.

Figure 3.7Directions of modeled sensor cluster errors.

The parameters and of this model can be estimated from observations of sensor outputs when the inputs are known, the process called calibration.

The purpose of calibration is sensor compensation, which is essentially inverting the input‐output of Equation 3.1to obtain

(3.3)

the sensor inputs compensated for scale factor, misalignment, and bias errors.

This result can be generalized for a cluster of gyroscopes or accelerometers, the effects of individual biases, scale factors, and input axis misalignmentscan be modeled by an equation of the form

(3.4)

where is the Moore–Penrose pseudoinverse of the corresponding , which can be determined by calibration.