alt="images"/> (compensated accelerometer output) and (temperature) in this example.
Figure 3.6 Gyro 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.
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,
(3.1)
(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 3.7 Directions of modeled sensor cluster errors.
3.3.4.1 ISA Calibration Parameters
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 3.1 to 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 misalignments can 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.
Compensation
In this case, calibration amounts to estimating the values of and , given input–output pairs , where is known from controlled calibration conditions and is recorded under these conditions. For accelerometers, controlled conditions may include the direction and magnitude of gravity, conditions on a shake table, or those on a centrifuge. For gyroscopes, controlled conditions may include the relative direction of the rotation axis of Earth (e.g. with sensors mounted on a two‐axis indexed rotary table), or controlled conditions on a rate table.
The full set of input–output pairs under sets of calibration conditions yields a system of linear equations
(3.5)
in the unknown parameters (the elements of the matrix ) and 3 unknown parameters (rows of the 3‐vector ), which will be overdetermined for . In that case, the system of linear equations may be solvable for the calibration parameters by using the method of least‐squares,
(3.6)
provided that the matrix is nonsingular.
The values of and determined in this way are called calibration parameters.
Estimation of the calibration parameters can also be done using Kalman filtering, a by‐product of which would be the covariance matrix of calibration parameter uncertainty. This covariance matrix is also useful in modeling system‐level
