target="_blank" rel="nofollow" href="#ulink_171b8bb3-c8f7-5c69-9495-5c4993146766">Figure 3.5 Common input–output error types.
3.3.2.4 Electrostatic
The surface‐to‐mass ratios of MESG‐scale devices are such that surface electrostatic forces eventually come to dominate acceleration‐induced forces. Electrical signals can then be used to keep a thin proof mass centered in its enclosure during accelerations.
3.3.3 Sensor Errors
3.3.3.1 Additive Output Noise
Sensor noise is most commonly modeled as zero‐mean additive random noise. As a rule, sensor calibration removes all but the zero‐mean noise component. Models and methods for dealing with various forms of zero‐mean random additive noise using Kalman filtering are discussed in Chapter 10.
3.3.3.2 Input–output Errors
The ideal sensor input–output function for rotation and acceleration sensors is linear and unbiased, meaning that the sensor output is zero when the sensor input is zero.
These are repeatable sensor output errors, unlike the zero‐mean random noise considered earlier. The same types of models apply to accelerometers and gyroscopes. Some of the more common types of sensor input–output errors are illustrated in Figure 3.5. These are listed for the specific panels:
1 bias, which is any nonzero sensor output when the input is zero;
2 scale factor error, usually due to manufacturing tolerances;
3 nonlinearity, which is present in most sensors to some degree;
4 scale factor sign asymmetry (often from mismatched push–pull amplifiers);
5 lock‐in, often due to mechanical stiction or (for ring laser gyroscopes) mirror backscatter; and
6 quantization error, inherent in all digitized systems.
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
In inertial navigation, integration turns white noise into random walks.
3.3.3.3 Error Compensation
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 3.6 is 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 “
and use it with independently sensed values for the variables involved –
Figure 3.6 Gyro error compensation example.
If the input–output function
There are also methods using nonlinear Kalman filtering and auxiliary sensor aiding for tracking and updating compensation parameters that may drift over time.
3.3.4 Inertial Sensor Assembly (ISA) Calibration
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 3.7 illustrates 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,
(3.2)
where
