Inertial sensors comprise
accelerometers and
gyroscopes, which measure the specific force and inertial angular velocity about a single axis, respectively [
36]. An
inertial measurement unit (IMU) encompasses multiple accelerometers and gyroscopes, usually three of each, obtaining three-dimensional measurements of the specific force and angular rate [
2] viewed in the platform frame
(
Section 2.5). However, the individual accelerometers and gyroscopes are not aligned with the
axes but with those of the non-orthogonal accelerometers
and gyroscopes
frames, which are also defined in
Section 2.5. The output of the inertial sensors must hence first be transformed from the
and
frames to
, as described in
Section 2.6 and
Section 2.7, and then from the
frame to the body frame
as explained in
Section 2.8, where they can be employed by the navigation system. The accelerometers and gyroscopes are assumed to be infinitesimally small and located at the IMU reference point (
Section 2.8), which coincides with the origin of these three frames
.
The IMU is physically attached to the aircraft structure in a strapdown configuration, so both the displacement
and the Euler angles
that describe the relative position and rotation between the body
and platform
frames are constant. Accelerometers can be divided by their underlying technology into pendulous and vibrating beams, while gyroscopes are classified into spinning mass, optical (ring laser or fiber optic), and vibratory [
4]. Current inertial sensor development is mostly focused on
micro machined electromechanical system (MEMS) sensors (there exist both pendulous and vibrating beam MEMS accelerometers, but all MEMS gyroscopes are vibratory), which makes direct use of the chemical etching and batch processing techniques used by the electronics integrated circuit industry to obtain sensors with small size, low weight, rugged construction, low power consumption, low price, high reliability, and low maintenance [
30]. On the negative side, the accuracy of MEMS sensors is still low, although tremendous progress has been achieved in the last two decades, and more is expected in the future.
The different errors that appear in the measurements provided by accelerometers and gyroscopes are described in
Section 2.1.
Section 2.2 presents a model for the measurements of a single inertial sensor, while
Section 2.3 and
Section 2.4 focus on how to obtain the specific values for white noise and bias on which the model relies from the documentation.
Section 2.5 describes the reference systems required to represent the IMU measurements. Additional errors appear when three accelerometers or gyroscopes are employed together, and these are modeled in
Section 2.6 for accelerometers and
Section 2.7 for gyroscopes. The analysis of the inertial sensors concludes with
Section 2.9, which provide a comprehensive error model for the IMU measurements. The final model also depends on the relative position of the IMU with respect to the body frame, which is described in
Section 2.8.
2.2. Single-Axis Inertial Sensor Error Model
As the inertial sensors provide measurements at equispaced discrete times , this section focuses on obtaining a discrete model for the bias and white noise errors of a single-axis inertial sensor. The results obtained here will be employed in the following sections to generate a comprehensive IMU model.
Let us consider a sensor in which the difference between its measurement at any given time
and the real value of the physical magnitude being measured at that same time
can be represented by a zero mean white noise Gaussian process
with spectral density
:
Dividing (
2) by
and integrating results in:
Assuming that the measurement and real value are both constant over the integration interval (note that the stochastic process
cannot be considered constant over any interval) [
34] yields
This expression results in the white noise sensor error
, which is the difference between the sensor measurement
and the true value
. Its mean and variance can be readily computed:
Based on these results, the white noise error can be modeled by a discrete random variable identically distributed to the above continuous white noise error, that is, one that results in the same mean and variance, where
is a standard normal random variable:
Let us now consider a second model in which the measurement error or bias is given by a first-order random walk process or integration of a zero mean white noise Gaussian process
with spectral density
:
Its mean and variance can be quickly computed:
These results indicate that the bias can be modeled by a discrete random variable identically distributed to the continuous random walk above:
where
is a standard normal random variable. Operating with the above expression results in the final expression for the discrete bias as well as its mean and variance:
A comprehensive single-axis sensor error model without a scale factor can hence be constructed by adding together the influence of the system noise provided by (
8) and the bias given by (
13) [
35], while assuming that the standard normal random variables
and
are uncorrelated (note that the expected value and variance of each of the two discrete components of this sensor model coincide with those of their continuous counterparts, but their combined mean and variance provided by expressions (
17) and (
18) differ from that of the combination of the two continuous error models given by (
5) and (
9). This is the case even if considering that the two zero mean white noise Gaussian processes
and
are independent and hence uncorrelated. It is however possible to obtain a discrete model whose discrete bias and white noise components are not only identically distributed to those of their continuous counterparts [
34], even adding the equivalence of covariance between the bias and the sensor error, but this results in a significantly more complex model that behaves similarly to the one above at all but has the shortest time samples after sensor initialization. The authors have decided not to do so in the model described in this article, reducing complexity with little or no loss of realism):
The discrete sensor error or difference between the measurement provided by the sensor at any given discrete time and the real value of the physical variable being measured at that same discrete time is the combination of a bias or first-order random walk and a white noise process, and it depends on three parameters: the bias offset , the bias instability , and the white noise . The contributions of these three different sources to the sensor error as well as to its first and second integrals (gyroscopes measure angular velocity, and their output needs to be integrated once to obtain attitude, while accelerometers measure specific force and are integrated once to obtain velocity and twice to obtain position) are very different and inherent to many of the challenges encountered when employing accelerometers and gyroscopes for inertial navigation, as explained below.
Figure 2 and
Figure 3 represent the performance of a fictitious sensor of
,
, and
working at a frequency of
, and they are intended to showcase the different behavior and relative influence on the total error of each of its three components. The figures show the theoretical variation with time of the sensor error mean (
Figure 2) and standard deviation (
Figure 3) given by (
17) and (
18) together with the average of fifty different runs. In addition,
Figure 2 also includes ten of those runs to showcase the variability in results implicit to the random variables (although the data are generated at
, for visibility purposes, the figure only employs 1 out of every 1000 points, so it appears far less noisy than the real data), while
Figure 3 shows the theoretical contribution to the standard deviation of each of the three components. In addition to the near equivalence between the theory and the average of several runs, the figures show that the bias instability is the commanding long-term factor in the deviation between the sensor measurement and its zero mean (the standard deviation of the bias instability grows with the square root of time while the other two components are constant). As discussed in
Section 2.1, the bias drift or bias instability is indeed the most important quality parameter of an inertial sensor. This is also the case when the sensor output is integrated, as discussed below.
Let us integrate the sensor error over a timespan
to evaluate the growth with time of both its expected value and its variance (as the interest lies primarily in
, a simple integration method such as the rectangular rule is employed):
Figure 4 and
Figure 5 follow the same pattern as
Figure 2 and
Figure 3 but applied to the error integral instead of to the error itself. They would represent the attitude error resulting from integrating the gyroscope output or the velocity error expected when integrating the specific force measured by an accelerometer. The conclusions are the same as before but significantly more accentuated. Not only is the expected value of the error constant instead of zero (
has been employed in the experiment), but the growth in the standard deviation (over a nonzero mean) is much quicker than before. The bias instability continues to be the dominating factor but now increases with a power of
, while the bias offset and white noise contributions also increase with time, although with powers of
and
, respectively. Let us continue the process and integrate the error a second time:
Figure 6 and
Figure 7 show the same type of figures but applied to the second integral of the error (
has been employed in the experiment). In this case, the degradation of the results with time is even more intense to the point where the measurements are useless after a very short period of time. Unless corrected by the navigation system, this is equivalent to the error in position obtained by double integrating the output of the accelerometers.
Let us summarize the main points of the single-axis inertial sensor discrete error model developed in this section, which includes the influence of the bias and the system error but not that of the scale factor and cross-coupling errors included in the three-dimensional error model of
Section 2.9. The error
, which applies to specific force for accelerometers and inertial angular velocity in the case of gyroscopes, depends on three factors: bias offset
, bias drift
, and white noise
. Its mean is always zero, but the error standard deviation grows with time (
) due to the bias drift with constant contributions from the bias offset and the white noise. When integrating the error to obtain
, which is equivalent to ground velocity for accelerometers and attitude for gyroscopes, the initial speed error or initial attitude error
becomes the fourth contributor, and an important one indeed, as it becomes the mean of the first integral error at any time. The standard deviation, which measures the spread over the nonzero mean, increases very quickly with time because of the bias instability (
), with contributions from the offset (
) and the white noise (
). If integrating a second time to obtain
, or position in case of the accelerometer, the initial position error
turns into the fifth contributor. The expected value of the position error grows linearly with time due to the initial velocity error with an additional constant contribution from the initial position error, while the position standard deviation (measuring spread over a growing average value) grows extremely quick due mostly to the bias instability (
) but also because of the bias offset (
) and the white noise (
).
Table 4 shows the standard units of the different sources of error for both accelerometers and gyroscopes.
2.3. Obtainment of System Noise Values
This section focuses on the significance of system or white noise error
and how to obtain it from sensor specifications, which often refer to the integral of the output instead of the output itself. As the integral of white noise is a random walk process, the angle random walk of a gyroscope is equivalent to white noise in the angular rate output, while velocity random walk refers to the specific force white noise in accelerometers [
37]. The discussion that follows applies to gyroscopes but is fully applicable to accelerometers if replacing the angular rate by specific force and attitude or angle by ground velocity.
Angle random walk, measured in (
), (
), or equivalent, describes the average deviation or error that occurs when the sensor output signal is integrated due to system noise exclusively, without considering other error sources such as bias or scale factor [
41]. If integrating multiple times and obtaining a distribution of end points at a given final time
, the standard deviation of this distribution, containing the final angles at the final time, scales linearly with the white noise level
, the square root of the integration step size
, and the square root of the number of steps
, as noted by the last term of (
21). This means that an angle random walk of 1
translates into a standard deviation for the error of 1
after 1
, 10
after 100
, and
after 1000
.
Manufacturers often provide this information as the power spectral density PSD of the white noise process in () or equivalent, where it is necessary to take its square root to obtain , or as the root PSD in () that is equivalent to . Sometimes, it is even provided as the PSD of the random walk process, not the white noise, in units () or equivalent. It is then necessary to multiply this number by the square root of the sampling interval or divide it by the square root of the sampling frequency to obtain the desired value.
2.4. Obtainment of Bias Drift Values
This section describes the meaning of bias instability
(also known as bias stability or bias drift) and how to obtain it from sensor specifications. As in the previous section, the discussion is centered on gyroscopes, but it is fully applicable to accelerometers as well. Bias instability can be defined as the potential of the sensor error to stay within a certain range for a certain time [
42]. A small number of manufacturers directly provide sensor output changes over time, which directly relates with the bias instability (also known as in-run bias variation, bias drift, or rate random walk) per the second term of (
18). If provided with an angular rate change of x (
) (
) in t (
), then
can be obtained as follows [
34,
43]:
As the bias drift is responsible for the growth of sensor error with time (
Figure 2 and
Figure 3), manufacturers more commonly provide bias stability measurements that describe how the bias of a device may change over a specified period of time [
35], typically around 100
. Bias stability is usually specified as a
value with units (
) or (
), which can be interpreted as follows according to (
16)—(
18). If the sensor error (or bias) is known at a given time t, then a
bias stability of 0.01
over 100
means that the bias at time
is a random variable with the mean error at time t and standard deviation 0.01
, and expression (
25) can be used to obtain
. As the bias behaves as a random walk over time whose standard deviation grows proportionally to the square root of time, the bias stability is sometimes referred as a bias random walk.
In reality, bias fluctuations do not really behave as a random walk. If they did, the uncertainty in the bias of a device would grow without bound as the timespan increased, which is not the case. In practice, the bias is constrained to be within some range, and therefore, the random walk model is only a good approximation to the true process for short periods of time [
35].
2.6. Accelerometer Triad Sensor Error Model
An IMU is equipped with an accelerometer triad composed by three individual accelerometers, each of which measures the projection of the specific force over its sensing axis as described in
Section 2.2 while incurring in an error
that can be modeled as a combination of bias offset, bias drift, and white noise (
16). The three accelerometers can be considered infinitesimally small and located at the IMU
reference point, which is defined as the intersection between the sensing axes of the three accelerometers. As the accelerometer frame
is centered at the IMU reference point and its three non-orthogonal axes coincide with the accelerometers’ sensing axes, (
30) joins together the measurements of the three individual accelerometers:
where
is the specific force viewed in the accelerometer frame
,
represents its measurement also viewed in
,
is the error introduced by each accelerometer (
16), and
is a square diagonal matrix containing the scale factor errors
for each accelerometer (
Section 2.1). It is however preferred to obtain an expression in which the specific forces are viewed in the orthogonal platform frame
instead of the accelerometer frame
. As both share the same origin,
where
and
, defined by (
26) and (
27), contain the cross-coupling errors
generated by the misalignment of the accelerometer sensing axes. The scale factor and cross-coupling errors contain fixed and temperature-dependent error contributions (refer to
Section 2.1) that can be modeled as normal random variables:
where
and
can be obtained from the sensor specifications. Equation (
31) can be transformed to make it more useful by defining the accelerometer scale and cross-coupling error matrix
:
Considering that the scale and cross-coupling errors are uncorrelated and very small, and taking into account the expressions for the mean and variance of the sum and product of two random variables [
44], the different components
of
can be obtained as follows
:
Let us also define the accelerometer error transformation matrix
as
A process similar to that employed above leads to:
Taking into account the expressions for the mean and variance of the sum and product of two random variables [
44], and knowing that the cross-coupling errors are very small
, it can be proven that the bias and white noise errors viewed in the platform frame
respond to a expression similar to (
16):
where each
and
is a random vector composed by three independent standard normal random variables
. Note that as the bias drift is mostly a warm-up process that stabilizes itself after a few minutes of operation, the random walk within (
42) is not allowed to vary freely but is restricted to within a band of width
. The final model for the accelerometer measurements viewed in
results in
where
is described in (
34) through (
37) and
is provided by (
42). This model relies on inputs for the bias offset
, bias drift
, white noise
, scale factor error
, and cross-coupling error
.
Section 6.2 provides an example on how to obtain these values from the data sheet provided by the accelerometer manufacturer.
2.7. Gyroscopes Triad Sensor Error Model
The IMU is also equipped with a triad of gyroscopes, each of which measures the projection of the inertial angular velocity over its sensing axis as described in
Section 2.2. The obtainment of the gyroscope triad model is fully analogous to that of the accelerometers in the previous section, with the added difficulty that the transformation between the gyroscope frame
and platform frame
relies on six small angles instead of three. The starting point hence is:
where
is the inertial angular velocity viewed in the platform frame
,
represents its measurement also viewed in
,
is the error introduced by each gyroscope (
16),
is a square diagonal matrix containing the scale factor errors
, and
and
, defined by (
28) and (
29), contain the cross-coupling errors
generated by the misalignment of the gyroscope sensing axes.
Operating in the same way as in
Section 2.6 leads to:
where each
and
is a random vector composed by three independent standard normal random variables
. As in the case of the accelerometers, the random walk within (
45) representing the bias drift is not allowed to vary freely but is restricted to within a band of width
. This model relies on inputs for the bias offset
, bias drift
, white noise
, scale factor error
, and cross-coupling error
, which can be obtained from the gyroscope specifications. An example of this process is included in
Section 6.2. The gyroscope scale and cross-coupling error matrix
responds to:
2.8. Mounting of Inertial Sensors
Equations (
43) and (
46) contain the relationships between the specific force
and inertial angular velocity
and their measurements
when evaluated and viewed in the platform frame
. However, from the point of view of the navigation system, both magnitudes need to be evaluated and viewed in the body frame
instead of
. These equations thus need to be modified so they relate
with
as well as
with
, respectively, as described in
Section 2.9 below. To do that, it is necessary to define the relative pose (position plus attitude) between the
and
frames, and to distinguish between the true position
and attitude
, and their estimations by the IMU processor (
and
). Note that the IMU, represented by the platform frame
, should be mounted in the aircraft as close as possible to the center of gravity (this reduces errors, as described in
Section 2.9), and it is loosely aligned with the aircraft body axes.
To increase the realism, this article assumes that the real displacement between the two frames is deterministic, while the relative rotation is stochastic. In this way, each simulation run exhibits a slightly different IMU platform attitude with respect to the aircraft body:
As the IMU reference point is fixed with respect to the structure but the aircraft center of mass slowly moves as the fuel load diminishes, it is possible to establish a linear model that provides the displacement between the origins of both frames according to the aircraft mass (the aircraft masses
and
) when the fuel tank is fully loaded or empty as inputs, as are the displacements between the IMU reference point and the aircraft center of mass
and
.:
The platform Euler angles respond to the stochastic model provided by (
51), in which each specific Euler angle is obtained as the product of the user-selected standard deviations (
,
,
) by a single realization of a standard normal random variable
(
,
, and
).
Once the real pose between the
and
frames is established, it is necessary to specify its estimation employed by the IMU processor in the comprehensive model introduced in
Section 2.9, which is discussed in
Section 5.2. Stochastic models are employed for both the translation
and rotation
, changing their values from one execution to the next:
As in the previous case, the model relies on two user-selected standard deviations (
and
), as well as six realizations of a standard normal random variable
, which are denoted as
,
,
,
,
, and
.
Section 6.2 suggests values for the five required settings (
), although they can be adjusted by the user.
can be considered quasi-stationary as it slowly varies based on the aircraft mass, and the relative position of their axes
remains constant because the IMU is rigidly attached to the aircraft structure. Although Euler angles have been employed in this section, from this point on, it is more practical to employ the rotation matrix
to represent the rotation between two different frames [
45]. The time derivatives of
and
are hence zero:
2.9. Comprehensive Inertial Sensor Error Model
Two considerations are required to establish the measurement equations for the inertial sensors viewed in the body frame
. First, let us apply the composition rules of
Appendix A.6 considering
as
,
as
, and
as
, which results in:
Second, it is also necessary to consider that as
is a rotation matrix in which all rows and columns are unitary vectors, the projection of the
frame bias and white noise errors
and
onto the
frame does not change their stochastic properties:
As the inertial angular velocity does not change when evaluated in the
and
frames (
55), its measurement in the body frame can be derived from (
46) by first projecting it from
to
based on the real rotation matrix
and then projecting back the measurement into
based on the estimated rotation matrix
. The bias and white noise error is also projected according to (
60):
The expression for the specific force measurement is significantly more complex because the back and forth transformations of the specific force between the
and
frames need to consider the influence of the lever arm
, as indicated in (
58). The additional terms introduce errors in the measurements, so as indicated in
Section 2.8, it is desirable to locate the IMU as close as possible to the aircraft center of mass.
Note that this expression cannot be directly evaluated as the estimated values for the inertial angular velocity and acceleration (
) are unknown by the IMU until obtained by the navigation filter. The IMU can however rely on the gyroscope readings, directly replacing
with
and computing
based on the difference between the present and previous
readings, resulting in:
Table 5 lists the error sources contained in the comprehensive inertial sensor error model represented by (
61), (
63). The first two columns list the different error sources, while the third column specifies their origin according to the criterion established in the first paragraph of
Section 2.1. The section where each error is described appears on the fourth column, which is followed by the seeds (refer to
Section 6 for the meaning of the terms
and
) employed to ensure the results variability for different aircraft (
) as well as different flights (
).
Note that all the required error sources (
column) need to be specified by the user. As an example,
Section 6.2 suggests values appropriate for a low SWaP aircraft. It is worth pointing out that all errors are modeled as stochastic variables or processes (with the exception of the
displacement between the body and platform frames, which is deterministic), as expressions (
61), (
63) rely on the errors provided by (
59), (
60), the scale and cross-coupling matrices given by (
34), (
47), and the transformations given by (
50), (
51), (
52), (
53).
In the case of the accelerometer triad, the stochastic nature of the fixed and run-to-run error contributions to the model relies on three realizations of normal distributions for the bias offset, three for the scale factor errors, three for the cross-coupling errors, and nine for the mounting errors, while the in-run error contributions require three realizations each for the bias drift and system noise at every discrete sensor measurement. The gyroscope triad is similar but requires six realizations to model the cross-coupling errors instead of three while using the same six realizations as the accelerometer triad to model the true and estimated rotation between the and frames.
Expressions (
61), (
63) can be rewritten to show the measurements as functions of the full errors (
), which represent all the errors introduced by the inertial sensors with the exception of white noise.