Chapter 9.3.3: INERTIAL SYSTEMS TECHNOLOGIES: Other Calibration Parameters

9.3.3 Other Calibration Parameters

9.3.3.1 Nonlinearities Sensor input/output nonlinearities are generally modeled by polynomials:

where the first two parameters a₀ = bias and a₁ = scale factor. The polynomial input output model of Eq. 9.17 is linear in the calibration parameters, so they can still be calibrated using a system of linear equations as was used for scale factor and bias.

The generalization of Eq. 9.17 to vector-valued inputs and outputs includes all the cross-power terms between different sensors, but it also includes multidimensional data structures in place of the scalar parameters a_i . Such a model would, for example, include the acceleration sensitivities of gyroscopes and the rotation rate sensitivities of accelerometers.

9.3.3.2 Sensitivities to Other Measurable Conditions Most inertial sensors are also thermometers, and part of the art of sensor design is to minimize their temperature sensitivities. Other bothersome sensitivities include acceleration sensitivity of gyroscopes and rotation rate sensitivities of accelerometers (already mentioned above).

Compensating for temperature sensitivity requires adding one or more thermometers to the sensors and taking calibration data over the expected operational temperature range, but the other sensitivities can be "cross-compensated" by using the outputs of the other inertial sensors. The accelerometer outputs can be used in compensating for acceleration sensitivities of gyroscopes, and the gyro outputs can be used in compensating for angular rate sensitivities of accelerometers.

9.3.3.3 Other Accelerometer Models

Centrifugal Acceleration Effects Accelerometers have input axes defining the component(s) of acceleration that they measure. There is a not-uncommon superstition that these axes must intersect at a point to avoid some unspecified error source. That is seldom the case, but there can be some differential sensitivity to centrifugal accelerations due to high rotation rates and relative displacements between accelerometers. The effect is rather weak, but not always negligible. It is modeled by the equation

where ω is the rotation rate and r_i is the displacement component along the input axis from the axis of rotation to the effective center of the accelerometer. Even manned vehicles can rotate at ω ≈ 3 rad/s, which creates centrifugal accelerations of about 1 g at r_i = 1 m and 0.001 g at 1 mm. The problem is less significant, if not insignificant, for MEMS-scale accelerometers that can be mounted within millimeters of one another.

Center of Percussion Because ω can be measured, sensed centrifugal accelerations can be compensated, if necessary. This requires designating some reference point within the instrument cluster and measuring the radial distances and directions to the accelerometers from that reference point. The point within the accelerometer required for this calculation is sometimes called its "center of percussion." It is effectively the point such that rotations about all axes through the point produce no sensible centrifugal accelerations, and that point can be located by testing the accelerometer at differential reference locations on a rate table.

Angular Acceleration Sensitivities Pendulous accelerometers are sensitive to angular acceleration about their hinge lines, with errors equal to ωhinge, where ω is the angular acceleration in radians per second squared and Δ_hinge is the displacement of the accelerometer proof mass (at its center of mass) from the hinge line. This effect can reach the 1-g level for Δ_hinge ≈ 1 cm and ω ≈ 10³ rad/s², but these extreme conditions are rarely persistent enough to matter in most applications.

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Chapter 9.3.3: INERTIAL SYSTEMS TECHNOLOGIES: Other Calibration Parameters

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