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

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 performance.

      3.3.4.2 Calibration Parameter Drift

      INS calibration parameters may not be exactly constant over time. Their values may change significantly over the operational life of the INS. Specifications for calibration stability generally divide these calibration parameter variations into two categories:

      Turn‐on to turn‐on changes that occur between a system shut‐down and the next start‐up. They may be caused by temperature transients or power turn‐on effects during shut‐downs and turn‐ons, and may represent stress relief mechanisms within materials and between assembled parts. They are generally considered to be independent from turn‐on to turn‐on, so the model for the covariance of calibration errors for the th turn‐on would be of the form

(3.7)

where is the covariance of turn‐on‐to‐turn‐on parameter changes. The initial value at the end of calibration is usually determinable from error covariance analysis of the calibration process. Note that this is the covariance model for a random walk, the covariance of which grows without bound.

Long‐term drift, sometimes called “aging,” which has been attributed to such long‐term phenomena as material migration within solids or ion diffusion within crystalline materials. Its calibration parameter uncertainty covariance equation has the same form as Eq. (3.7), but with now representing the calibration parameter drift in the time interval between successive discrete times within an operational period.

       Predicting Incipient System or Sensor

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