Chapter 9.3.2: INERTIAL SYSTEMS TECHNOLOGIES: Systematic Errors

9.3.2 Systematic Errors

These are errors that can be calibrated and compensated.

9.3.2.1 Sensor-Level Models These are sensor output errors in addition to additive zero-mean white noise and time-correlated noise considered above. The same models apply to accelerometers and gyroscopes. Some of the more common types of sensor errors are illustrated in Fig. 9.13:

(a) Bias, which is any nonzero sensor output when the input is zero
(b) Scale factor error, often resulting from aging or manufacturing tolerances

(c) Nonlinearity, which is present in most sensors to some degree
(d) Scale factor sign asymmetry (often from mismatched push-pull amplifiers)
(e) A dead zone, usually due to mechanical stiction or lockin [for a ring laser gyroscope (RLG)]
(f ) Quantization error, inherent in all digitized systems. It may not be zeromean when the input is held constant, as it could be under calibration conditions.

We can recover the sensor input from the sensor output so long as the input/output relationship is known and invertible. Dead-zone errors and quantization errors are the only ones shown with this problem. The cumulative effects of both types (dead zone 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 0.5 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.

9.3.2.2 ISA-Level Models For a cluster of N ≥ 3 gyroscopes or accelerometers, the effects of individual biases, scale factors, and input axis misalignments can be modeled by an equation of the form

where the components of the vector b_z are three aggregate biases, the components of the z_input and z_output vectors are the sensed values (accelerations or angular rates) and output values from the sensors, respectively, and the elements m_ijof the "scale factor and misalignment matrix" M represent the individual scale factor deviations and input axis misalignments as illustrated in Fig. 9.14 for N = 3 orthogonal input axes. The larger arrows in the figure represent the nominal input axis directions (labeled #1, #2, and #3) and the smaller arrows (labeled m_ij ) represent the directions of scale factor deviations (i = j) and input axis misalignments (i ≠ j).

where † represents the Moore-Penrose matrix inverse.

The compensation model of Eq. 9.13 is the one used in system implementation for compensating sensor outputs using a single constant matrix M and vector b_z.

9.3.2.3 Calibrating Sensor Biases, Scale Factors, and Misalignments Calibration is the process of taking sensor data to characterize sensor inputs as functions of sensor outputs. It amounts to estimating the values of M and b_z, given input-output pairs [z_{input, k}, z_{output, k}], where z_{input, k} is known from controlled calibration conditions (e.g., for accelerometers, from the direction and magnitude of gravity, or from conditions on a shake table or centrifuge) and z_{output, k} is measured under these conditions.

Calibration Data Processing The full set of data under K sets of calibration conditions yields a system of 3K linear equations

in the 3N unknown parameters m_{i, j}(the elements of the matrix M) and 3 unknown parameters b_{i, z} (rows of the 3-vector b_z), which will be overdetermined for K > N + 1. In that case, the system of linear equations may be solvable for the 3(N + 1) calibration parameters by using the method of least-squares

provided that the matrix Z^TZ is nonsingular.

Calibration Parameters The values of M and b_z 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 matrix is also useful in modeling system-level performance.

< Previous Excerpt Next Excerpt >

Purchase This Book

TABLE OF CONTENTS

Chapter 9.3.2: INERTIAL SYSTEMS TECHNOLOGIES: Systematic Errors

Contact Preferences

This is embarrasing...

Customize Your GlobalSpec Experience

Select Your Free Newsletters

Industry Newsletters

Select Your Free Product Alerts

This is embarrasing...