Global Positioning Systems, Inertial Navigation, and Integration

By Mohinder S. Grewal, Lawrence R. Weill, and Angus P. Andres

Chapter 9.5.3: INERTIAL SYSTEMS TECHNOLOGIES: Sensor Error Propagation

9.5.3 Sensor Error Propagation

Errors made in compensating for inertial sensor errors will cause navigation errors. Here, we derive some approximating formulas for how errors in individual compensation parameters propagate into velocity and position errors.

9.5.3.1 Scale Factors, Biases, and Misalignments The models for bias, scale factor and input axis misalignment compensation are the same for gyroscopes and accelerometers. The compensated sensor output

for z = ω (for gyroscopes) or z = a (for accelerometers). The sensitivity of the compensated output to bias is then

the 3x3 identity matrix, and the sensitivity to the elements m_{j, k} of M are

If we put these 12 calibration parameters in vector form as

then the matrix of partial derivatives

For analytical purposes, this matrix of partial derivatives would be evaluated under "nominal" conditions, which could be for M = I and b_z = 0. In that case, z_output = z_input, the bias sensitivities will be unitless (e.g., g/g), the scale factor sensitivities will be in units of parts-per-part and the misalignment sensitivities will be in output units per radian.

This matrix can be augmented with additional calibration parameters, such as acceleration-sensitivities for gyroscopes or temperature sensitivities. It only requires taking the necessary partial derivatives.

9.5.3.2 Accelerometer Compensation Error Propagation Acceleration errors due to accelerometer compensation errors in sensor-fixed coordinates and navigation coordinates will then be

where δp_acc.comp is the vector of accelerometer compensation parameter errors, the partial derivative matrix is the one in Eq. 9.93 with z = a, and is the coordinate transformation matrix from sensor-fixed coordinates to navigation coordinates. For example

For gimbaled systems aligned to locally level coordinates with sensor axes pointing north and east

For α carouseled gimbal system aligned to locally level coordinates

where a is the carousel angle.

For a strapdown system aligned to body coordinates of the host vehicle

For a carouseled strapdown system rotating about the body-fixed yaw axis

Velocity and Position Errors. The velocity error sensitivities to each of the compensation parameters will be the integral over time of the acceleration sensitivities, and the position error sensitivities to each of the compensation parameters will be the integral over time of the velocity sensitivities. However, the accelerations must be transformed into navigation coordinates before integration:

where δx_nav is the navigation position error due to compensation parameter errors. The GNSS navigation solution will not include δx_nav, and it is the difference between the INS and GNSS solutions that can be used to estimate the compensation parameter errors.

9.5.3.3 Gyroscope Compensation Error Propagation The principal effect of gyroscope compensation errors on inertial navigation position errors is from the

miscalculation of gravitational acceleration due to the resulting tilt errors, as illustrated in Fig. 9.31, where

and δρΝ is a small rotation about the north-pointing coordinate axis and δρΕ is the corresponding east tilt component.

Tilt Errors Small tilt errors due to calibration errors can be approximated as the horizontal north and east components of a rotation vector δρ:

The east and north tilt components can then be substituted into Eqs. 9.106 and 9.107 to obtain the equations for position error due to tilts. Schuler oscillations are excited when these position errors, in turn, cause tilts.

Velocity Errors For the tilt error angles δρ_Ν, δρ_Ε in radians and g ≈ 9.8 m/s², the corresponding velocity errors will propagate as

Effect of Heading Errors Navigation error sensitivity to rotational error ρυ about the local vertical (i.e., heading error) is usually smaller than the sensitivities to tilt-related errors. The time rate-of-change of position errors due to heading error are

where δx_Ε and δx_Ν are the navigation error components due to heading error δρυ (measured counterclockwise in radians) and ν_Ε and ν_Ν are the vehicle velocity components in the east and north directions, respectively.

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Chapter 9.5.3: INERTIAL SYSTEMS TECHNOLOGIES: Sensor Error Propagation

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