114 may use the statistical representation to compute a long-term average and a short-term average that can be compared to determine if the short-term average is deviating from the long-term average. If the short-term average deviates from (e.g., increases relative to) the long-term average, the ASD module 114 may determine that the INS filter 113 is operating in a disturbed environment (e.g., detecting a temperature, detecting fluctuations (i.e., changes) in the temperature, detecting a rate of change in the temperature, detecting vibrations, detecting fluctuations in the vibrations, detecting a rate of change in the vibrations, detecting a magnetic field, detecting fluctuations in the magnetic field, detecting a rate of change in the magnetic field, or any other type of potential disturbance in the environmental operating conditions that may be detected by a source of information, etc.) for the current epoch.

Accordingly, the ASD module 114 may adjust, e.g., increase, the spectral density for the current epoch such that the model (e.g., error model) is generated or modified for the disturbed environmental operating conditions. Advantageously, the INS filter 113 can adjust the spectral density for a model (e.g., error model) based on the varying environmental conditions in which the INS filter 113 operates. As a result, the INS filter 113 can use the model (e.g., error model) as part of the update procedure to efficiently and accurately determine a solution (e.g., navigation solution) in varying environmental operating conditions.

FIG. 2 is a flow diagram of a sequence of steps for using internal data to adapt a spectral density based on varying environmental operating conditions according to the one or more embodiments as described herein. As used herein, internal data may be any data that is generated, used, or obtained during a filtering process that is implemented by any of a variety of different filters such as, but not limited to, INS filter 113. For example, IMU error correction data that is generated during an update procedure implemented by INS filter 113 may be an example of internal data. However, it is expressly contemplated that the one or more embodiments as described herein are applicable to any of a variety of different types of internal data. As such, the use of IMU error correction data in relation to FIG. 2 is for illustrative purposes only.

Procedure 200 starts at step 205 and continues to step 210. At step 210, the ASD module 114 obtains error correction data for a current epoch (C_k). For example, the error correction may be a gyroscope error correction, e.g., z-gyroscope bias state correction. For this example, let it be assumed that the current epoch is update epoch 5 and the INS filter 113 implements the update procedure to determine the following error correction for the current epoch (C_k):

Update Epoch
C_k(°/s)

5
0.053

Above, °/s may be a unit angular velocity and represent degrees per second. According to the one or more embodiments as described herein, epoch and update epoch may be used interchangeably.

The procedure continues to step 215 and the ASD module 114 creates a rolling history using error corrections from a number of consecutive epochs (e.g., update epochs) that includes the current epoch and one or more previous epochs. In an embodiment, the number, e.g., window size, of consecutive update epochs is 5 and the ASD module 114 creates a rolling history using the error correction values from the 5 consecutive update epochs.

A window size of 5 is for illustrative purposes only, and it is expressly contemplated that the window size may be of any value. In an embodiment, the window size may be any value that is equal to or greater than 2. For this example, let it be assumed that the window size is 5. As such, the ASD module 114 may create a rolling history (History Epoch 1) that includes 4 previous update epochs, e.g., update epochs 1 through 4, and the current update epoch, e.g., update epoch 5. Therefore, History Epoch 1 includes the error corrections from update epochs 1 through 5. For example, the ASD module 114 may create History Epoch 1 as follows:

History Epoch 1

Update Epoch
C_k(°/s)

1
0.099

2
0.036

3
0.024

4
0.038

5
0.053

After the completion of procedure 200 where the current epoch is epoch 5, the next epoch will be the current epoch. For this example, let it be assumed that the next epoch is epoch 6. When epoch 6 is the current update epoch, the procedure 200 repeats for a next iteration, and the ASD module 114 may create a next rolling history (History Epoch 2) at step 215 for the next iteration. History Epoch 2 may be created by removing the oldest epoch (e.g., update epoch 1) from History Epoch 1 and adding the error correction value for update epoch 6 to History Epoch 1. Therefore, History Epoch 2 includes epochs 2, 3, 4, 5, 6 and their respective error correction values.

Accordingly, and on each iteration of procedure 200, the ASD module 114 may create a next rolling history at step 215 for the current update epoch and one or more previous update epochs.

The procedure 200 continues to step 220 and the ASD module 114 computes a statistical value from the error correction values of the rolling history. In this example, the current epoch is update epoch 5. As such, the ASD module 114 may compute a statistical value using the error correction values in History Epoch 1. In an embodiment, the ASD module 114 may use a root mean square (RMS) to compute the statistical value as follows:

$RMS = \sqrt{\frac{\sum_{k = 0}^{n} C_{k}^{2}}{n}},$

- where n is the window size.

Although the example used herein calculates the statistical value using the RMS, it is expressly contemplated that the one or more embodiments as described herein can perform any of a variety of different statistical analyses on a rolling history of internal data according to the one or more embodiments as described herein. For example, other statistical analyses that may be performed include, but are not limited to, calculating a standard deviation value for internal data of a rolling history, calculating an absolute accumulation value for internal data of a rolling history, calculating a slope value for internal data of a rolling history, etc.

Returning to the example, the five error correction values for History Epoch 1 are 0.099, 0.036, 0.024, 0.038, and 0.053. As such, the ASD module 114 uses the RMS to compute the statistical value for History Epoch 1 as 0.056 based on the RMS formula as described above. An RMS value is a measure of the magnitude of a set of numbers. As such, the statistical value may represent a magnitude of the error correction values included in a rolling history (e.g., 5 epochs).

The procedure continues to step 225 and the ASD module 114 utilizes the statistical value to compute a long-term average and a short-term average for the current epoch. As will be described in further detail below, the long-term average and the short-term average of the RMS value can be used to detect and quantify a level of disturbance in the environmental operating conditions for the current epoch. In an embodiment, the ASD module 114 may compute the short-term average for the current epoch (S_k) and long-term average for the current epoch (L_k) as:

$S_{k} = α {RMS}_{k} + (1 - α) S_{k - 1}$

$L_{k} = β {RMS}_{k} - (1 - β) L_{k - 1}$

- where α is a short-term weight and β is a long-term weight. In an embodiment, α and β may be preconfigured or user defined.

Short-term averages respond quickly to changes in values (e.g., RMS value), while long-term averages are slower to react to changes in values. Stated another way, an increasing RMS value for a rolling history that includes the current epoch will have a greater impact on the short-term average than on the long-term average.

According to the one or more embodiments as described herein, the long-term average can be used as a baseline performance indicator, while the short-term average can be used as an indicator for a current time frame (e.g., the rolling history that includes the current update epoch). Therefore, it would be expected that the long-term average and the short-term average will be the same or similar when the RMS value for the rolling history that includes the current epoch is consistent with the historical RMS values that define the long-term average. This would likely be indicative of the INS filter 113 operating in a steady state environment during the current update epoch. Conversely, it would be expected that the short-term average would increase relative to the long-term average when the RMS value indicates that that the magnitude of the error corrections for the rolling history that includes the current epoch is large relative to the historical RMS values. This would likely be indicative of the INS filter 113 operating in a disturbed environment during the current update epoch.

The procedure continues to step 230 and the ASD module 114 computes a ratio value for the current epoch using the long-term average and short-term average. In an embodiment, the long-term average is used as the denominator of the computed ratio and the short-term average is used as the numerator of the computed ratio. In an embodiment, the ASD module 114 calculates the ratio value for the current epoch (Ratio_k) as follows:

${Ratio}_{k} = \frac{S_{k}}{L_{k}}$

Because the long-term average provides an indication regarding the baseline performance and the short-term average provides an indication regarding the recent (i.e., rolling history that is more heavily weighted by the current update epoch) behavior, the ratio value can be used to quantify the behavior of the INS filter 113 during the current update epoch relative to the baseline performance of the INS filter 113. An increasing ratio value, e.g., increasing from a value of 1, may provide an indication that the INS filter 113 is behaving differently from its baseline performance. Therefore, an increasing ratio value can be indicative of the INS filter 113 operating in more disturbed environmental operating conditions during the current update epoch. However, a ratio value of 1 or a ratio value that is close to 1 may provide an indication that the INS filter 113 is behaving similarly to its baseline performance. Therefore, a ratio value of 1 or close to 1 can be indicative of the INS filter 113 operating in a steady state environment during the current update epoch.

The procedure continues to step 235 and the ASD module 114 utilizes the computed ratio value as a scale factor for the spectral density. In an embodiment, the computed ratio value may be used to adjust or maintain the spectral density for an error model. Specifically, and as explained above, P_k⁻=Φ_k−1,kP⁻Φ_k−1,k^T+Q_kof the update procedure implemented by the INS filter 113 may be the error model. Additionally, Q_kmay correspond to the spectral density. Therefore, the ASD module 114 may multiply Q_kby Ratio_k, which serves as a scale factor, to adjust the spectral density. For example, and when the ratio value is large value, the spectral density is increased to account for the disturbed environmental operating conditions. When the ratio value is 1 or close to 1, the spectral density remains at a steady state value for the steady state environment.

Therefore, and in embodiment, the error model may be generated or modified for varying environmental operation conditions at different points in time (e.g., start-up, after start-up, etc.) by adjusting the spectral density using Ratio_kthat is computed according to the one or more embodiments as described herein.

The procedure optionally continues to step 240, and the filter (e.g., INS filter 113) uses the model (e.g., error model) with a spectral density that is scaled using the ratio value. For example, an INS filter 113 may use an error model to account for the IMU errors and compute a solution (e.g., navigation solution). As a result, the INS filter 113 can efficiently and accurately implement the update procedure to compute a solution (e.g., navigation solution) in the varying environmental operating conditions at different points in time (e.g., start-up, after start-up, etc.) based on the adaptable spectral noise density according to the one or more embodiments as described herein. As such, the one or more embodiments as described herein provide an improvement over conventional systems that use static models that cannot be efficiently utilized in varying environmental conditions.

Specifically, such conventional systems may use a steady state static model that will mischaracterize the increased IMU errors in disturbed environments since the process noise is set for operating in a steady state environment. Further, other conventional systems may use a static model with increased noise. However, such static models introduce increased noise that negatively impacts the INS filter 113 during steady state.

In contrast, the one or more embodiments as described herein can adapt/adjust the spectral noise density such that different models (e.g., error models) can be used for varying environmental operating conditions. Advantageously, the INS filter 113 can efficiently and accurately compute a solution (e.g., navigation solution) in the varying environmental operating conditions, at different points in time during operation, based on the adaptable spectral noise density.

The use of the adjusted spectral density (e.g., adjusted based on internal data as described in relation to FIG. 2) in varying environmental operating conditions to compute a solution is a practical application and solution provided by the one or more embodiments as described herein. Specifically, the spectral density that is adjusted based on internal data as described herein is a practical/technical solution that allows a filter (e.g., INS filter 113) to seamlessly, on-the-fly, and in near-real time compute a solution (e.g., navigation solution) efficiently while operating in an environment with changing/varying conditions. This is an improvement when compared to the discussed conventional systems and techniques that can either use a static steady-state model or a model with increased spectral density to compute a solution, both of which are nonoptimal in environments with changing/varying operating conditions. Therefore, the one or more embodiments as described herein provide an improvement in the existing technological fields of solution filtering, navigation solution filtering, Kalman filtering, signal tracking filtering, to name a few.

The procedure continues to step 245 and the ASD module 114 continues to a next epoch. In this example, the next update epoch is update epoch 6. As such, the procedure continues to step 210 for a next iteration of procedure 200 where update epoch 6 is the current update epoch. Thus, the procedure 200 is performed again and iteratively for a current update epoch, e.g., update epoch 6. For example, an error correction value is obtained for update epoch 6, a rolling history (History epoch 2) is created, a value (e.g., RMS) is computed for the rolling history, long-term and short-term averages are computed, a ratio value is computed for update epoch 6, and the ratio value is used as a scale factor for the spectral density.

In addition to using internal data, the one or more embodiments as described herein can adjust the spectral density for varying environmental operating conditions based on data from one or more sensors (e.g., from the one or more environmental sensors 117 and/or one or more sensors of IMU 112 such as an IMU temperature sensor). As used herein, data from the one or more sensors may be referred to as “external data” since the data is not obtained from, i.e., not internal to, the filtering process implemented by INS filter 113 (e.g., Kalman Filter).

FIG. 3 is a flow diagram of a sequence of steps for using external data to adapt a spectral density based on varying environmental operating conditions according to the one or more embodiments as described herein.

The procedure 300 starts at step 305 and continues to step 310 where the ASD module 114 obtains sensor data for a current epoch. In an embodiment, the sensor data may be obtained from one or more different sensors. For example, the sensors may include, but are not limited to, IMU sensors, optional environmental sensors 117 that are coupled to body of interest 102, etc.

As an illustrative example, let it be assumed that the ASD module 114 obtains temperature data (i.e., external data) from a temperature sensor and vibration sensor data from a vibration sensor, both of which are optional environmental sensors 117. Although the illustrative example used herein describes data being obtained from optional environmental sensors 117, it is expressly contemplated that the one or more embodiments as described herein are applicable to data from IMU sensors, a combination of optional environmental sensors 117 and IMU sensors, or any other sensors that are, for example, external to the operation of a filtering process.

The procedure continues to step 315 and the ASD module 114 compares each of the obtained sensor data with one or more corresponding threshold value. As an illustrative example, let it be assumed that there is a single temperature threshold value t. Further, let it be assumed there are two vibration threshold values (e.g., a lower vibration threshold value and a higher vibration threshold value).

Although the example as described herein uses threshold, it is expressly contemplated that the ASD module 114 may utilize different sensor patterns, profiles created for sensors, a combination of sensor patterns and profiles, etc. to determine when certain conditions, indicative of a disturbed environment, are met to adjust the spectral density.

For this example, the temperature threshold value may be used to define an operating environment disturbed by temperature. Specifically, if the temperature sensor data is less than or equal to the temperature threshold value, the ASD module 114 may determine that the INS filter 113 is not operating in an environment that is disturbed by temperature (e.g., the INS filter is operating in a steady state environment) for the current epoch. However, if the temperature sensor data is greater than the temperature threshold value, the ASD module 114 may determine that the INS filter 113 is operating in an environment that is disturbed by temperature for the current epoch.

The vibration threshold values may be used to define an operating environment disturbed by vibrations. Specifically, if the vibration sensor data is less than or equal to the lower vibration threshold value, the ASD module 114 may determine that the INS filter 113 is not operating in an environment that is disturbed by vibrations (e.g., the INS filter is operating in a steady state environment) for the current epoch. If the vibration sensor data is greater than the lower vibration threshold value but less than or equal to the higher vibration threshold value, the ASD module 114 may determine that the INS filter 113 is operating in an environment that is mildly disturbed by vibrations for the current epoch. If the vibration sensor data is greater than the higher vibration threshold value, the ASD module 114 may determine that the INS filter 113 is operating in an environment that is greatly disturbed by vibrations for the current epoch.

The procedure continues to step 320 and the ASD module 114 determines if at least one threshold value is exceeded. For example, the ASD module 114 may determine if the temperature sensor data exceeds the temperature threshold value to determine if the environmental operating conditions are disturbed by temperatures. Additionally, the ASD module 114 may determine if the vibration sensor data exceeds either of the two vibration threshold values to determine if the environmental operation conditions are mildly or greatly disturbed by vibrations.

If at least one threshold value is exceeded, the procedure continues from step 320 to step 325. At step 325, the ASD module 114 determines that the filter (e.g., INS filter 113) is operating in a disturbed environment. As such, the ASD module 114 adjusts the spectral density to an increased value or maintains the spectral density at an existing increased value. The determination of whether the ASD module 114 adjusts the spectral density to an increased value or maintains it at an existing increased value may be based on the value of the spectral density for the previous epoch. For example, if the spectral density value for the previous epoch is at a steady state value, the ASD module 114 adjusts the spectral density value to an increased value to account for the disturbed environmental operating conditions. However, if the spectral density value for the previous epoch is at the increased value because of the disturbed environment operating conditions during the previous epoch, the ASD module 114 maintains the spectral density value at the increased value. Because the increased spectral density value is input to a model (e.g., error model), the IMU corrections are more heavily weighted such that the IMU corrections have a larger influence/impact when used by the error model to compute the solution (e.g., navigation solution) in the disturbed environmental operating conditions.

For example, and as previously explained, P_k⁻=Φ_k−1,kP⁻Φ_k−1,k^T+Q_kof the update procedure implemented by the INS filter 113 may be the error model. Additionally, Q_kmay correspond to the spectral density. Therefore, the ASD module 114 may multiply Q_kby an increased scale factor to increase the spectral density. As such, the error model may be generated or modified by increasing the spectral density to account for the disturbed environmental operating conditions.

The procedure continues from step 325 to step 330. At optional step 330, the filter (e.g., INS filter 113) uses the model (e.g., error model), which includes an increase to the allowable system noise based on the increased spectral density, to efficiently and accurately compute a solution in the disturbed environmental operating conditions. The procedure continues to step 335, and the ASD module continues to a next epoch. The procedure then repeats to determine if the spectral density should be at an increased value or a steady state value for the next epoch.

If at least one threshold value is not exceeded at step 320, the procedure continues from step 320 to step 340. At step 340, the ASD module 114 determines that the filter (e.g., INS filter 113), is not operating in disturbed environment. As such, the ASD module 114 adjusts the spectral density to a steady state value or maintains the spectral density at an existing steady state value. For example, if the spectral density value for the previous epoch is at a steady state value, the ASD module 114 maintains the spectral density value at the steady state value to account for the steady state environment. However, if the spectral density value for the previous epoch is at the increased value because of the disturbed environment operating conditions during the previous epoch, the ASD module 114 may adjust the spectral density value to the steady state value.

As previously explained, P_k⁻=Φ_k−1,kP⁻Φ_k−1,k^T+Q_kof the update procedure implemented by the INS filter 113 may be the error model. Additionally, Q_kmay correspond to the spectral density. Therefore, the ASD module 114 may multiply Q_kby a steady state scale factor. As such, the error model may be generated or modified by adjusting the spectral density to account for the steady state environment.

The procedure continues from step 340 to step 345. At optional step 345, the filter (e.g., INS filter 113) uses the steady state model (e.g., steady state error model) to efficiently and accurately compute a solution (e.g., navigation solution) in the steady state environment. The procedure continues to step 350 and the ASD module continues to a next epoch. The procedure then repeats to determine if the spectral density should be at an increased value or a steady state value for the next epoch.

Accordingly, the environmental conditions in which the filter (e.g., INS filter) 113 is operating may be evaluated to adapt the spectral noise density at different points in time (e.g., at start-up, after start-up). This in turn updates/modifies the error model to account for varying environmental conditions at the different points in time. Advantageously, the (e.g., INS filter 113) can efficiently and accurately determine a solution (e.g., navigation solution) in varying environmental operating conditions at the different points in time. As such, the one or more embodiments as described herein provide an improvement in the existing technological field of solution filtering (e.g., navigation solution filtering, Kalman filtering, signal tracking filtering, etc.) when compared to conventional systems that may use static error models.

The use of the adjusted spectral density (e.g., adjusted based on external data as described in relation to FIG. 3) in varying environmental operating conditions is a practical application and solution provided by the one or more embodiments as described herein. Specifically, the spectral density that is adjusted based on external data (e.g., sensor data) as described herein is a practical/technical solution that allows a filter (e.g., INS filter 113) to seamlessly, on-the-fly, and in near-real time compute a solution (e.g., navigation solution) efficiently while operating in an environment with changing/varying conditions. This is an improvement when compared to the discussed conventional systems and techniques that can either use a static steady-state model or a model with increased spectral density to compute a solution, both of which are nonoptimal in environments with changing/varying operating conditions.

Although not explicitly included in FIG. 3, it is expressly contemplated that a rolling history may be created for the obtained sensor data and sensor data from one or more previous epochs as described above in relation to FIG. 2. The creation of the rolling history may be performed between steps 310 and 315 of FIG. 3. One or more statistical analyses may be performed on the sensor data of the rolling history to obtain a statistical representation for the current epoch from the rolling history, as described above in relation to FIG. 2. The statistical analyses may be performed after creating the rolling history and before performing step 320 of FIG. 3. The statistical representation may be compared to the threshold value at step 320 and the procedure 300 may be performed as described above to determine if the filter is operating in a steady state or disturbed environment.

Instead of comparing the statistical representation to a threshold at step 320, it is expressly contemplated that a long-term average and short-term average, as described above, may be computed for the statistical representation from the sensors. The long-term and short-term averages may then be compared to adjust the spectral density, as described above in relation to FIG. 2.

In addition to utilizing internal data and/or external data to adjust the spectral density as described above in relation to FIGS. 2 and 3, the one or more embodiments as described herein may use misclosure values to switch between filters that use different error models. Specifically, and as will be described in further detail below, a primary filter may use a steady state error model and a secondary/backup filter may use an error model (i.e., disturbed error model) that utilizes more heavily weighted IMU corrections such that the IMU corrections have a larger impact/influence when the solution (e.g., navigation solution) is computed in disturbed environmental conditions. The misclosure values from the filters may be compared to determine which of the primary filter and secondary filter should be used.

FIG. 4 is a flow diagram of a sequence of steps for using misclosure values to switch between filters that use different models according to the one or more embodiments as described herein. Although the examples as described in relation to FIG. 4 use two different filters (e.g., INS filters), it is expressly contemplated that any number of a plurality of filters (e.g., 2, 3, 4, 5, 6, etc.) may be used according to the one or more embodiments as described herein. As such, the discussion of two different filters in relation to FIG. 4 is for illustrative purposes only. In an embodiment, steps 415 through 450 of FIG. 4 may be performed in the solution domain. The solution domain may be a point in time during or after the INS 110 and/or receiver 104 compute a navigation solution and/or update a navigation solution. For example, the INS 104 computes the GNSS/INS solution in the solution domain. In an embodiment, the INS 110 and/or receiver 104 operate in an observation domain at a point in time prior to the INS 110 and/or receiver 104 computing a navigation solution and/or updating a navigation solution. For example, the IMU measurements, the GNSS raw measurements, and the GNSS observables may be obtained in the observation domain.

The procedure 400 starts at step 405 and continues to step 410 where the processor instantiates each of at least two filters (e.g., INS filters) using a different configuration. As an illustrative example, let it be assumed that at least two INS filters include a first INS filter, e.g., filter 1, and a second INS filter, e.g., filter 2. Filter 1 and filter 2 may compute, in parallel, their respective GNSS/INS solutions.

In this example, let it be assumed that filter 1 is the primary INS filter and filter 2 is the secondary/backup filter. As such, filter 1 with a steady state error model may be used to compute a GNSS/INS solution until it is determined that the environmental operating conditions have changed due to a disturbance, (e.g., detecting a temperature, detecting fluctuations (i.e., changes) in the temperature, detecting a rate of change in the temperature, detecting vibrations, detecting fluctuations in the vibrations, detecting a rate of change in the vibrations, detecting a magnetic field, detecting fluctuations in the magnetic field, detecting a rate of change in the magnetic field, or any other type of potential disturbance in the environmental operating conditions that may be detected by a source of information, etc.). When a disturbance is detected, the INS filter 2 with a disturbed error model may be used to compute a GNSS/INS solution in the disturbed environment. When it is determined that the environment has returned to steady state, i.e., without a disturbance, filter 1 may be used again to compute a GNSS/INS solution.

According to the one or more embodiments as described herein, filter 1 and filter 2 are configured differently by using different error models. Specifically, filter 1 may utilize an error model that is based on a spectral density value that corresponds to a steady state environment. For example, the spectral density value may be based on the steady state statistics provided by a manufacturer of the INS filter. Filter 2 may utilize an error model that is based on an increased spectral density value for disturbed environmental operating conditions.

Because filters 1 and 2 are configured differently, filters 1 and 2 operate differently to compute misclosure values and/or output solutions (e.g., GNSS/INS solution) using the same data (e.g., IMU data). As will be described in further detail below, values computed from the misclosure values of filters 1 and 2 can be analyzed in the solution domain to determine if an environmental disturbance has occurred such that, for example, one or more actions (e.g., switchover from filter 1 to filter 2) can be taken.

The procedure 400 continues to step 415 and each of the at least two filters, for a current epoch, computes a misclosure value for each update type that is used to obtain a solution (e.g., GNSS/INS solution). For this example, let it be assumed that there is a single update type and that the update type is a position update. Based on the different configurations (e.g., use of different error models), filters 1 and 2 can each use a received position update to compute a position misclosure value. Accordingly, and for this example, each of filters 1 and 2 computes a position misclosure value.

In an embodiment, a misclosure value may represent or indicate a difference (i.e., divergence) between two solutions that are utilized to determine a combined solution at a filter for an update type. For example, a misclosure value may represent or indicate a difference between a GNSS solution and an INS solution for an update type at filter 1. An increasing misclosure value may be indicative of the INS solution and GNSS solution becoming more different. Conversely, a decreasing misclosure value or a misclosure value nearing zero may be indicative of the INS solution and the GNSS solution becoming more equal/consistent. As an example, let it be assumed that the current update epoch is update epoch 5 and that filter 1 and filter 2, using two different error models, compute the following misclosure values for position:

Update
INS Filter 1
INS Filter 2

Epoch
Position Misclosure (m)
Position Misclosure (m)

5
10.665
0.913

In the table above, m may be a unit and represent meters. Further, the table above indicates that the INS solution and GNSS solution are more different at filter 1 than they are at filter 2.

Procedure 400 continues to step 420 and the ASD module 114 creates, for the at least two filters, a rolling history using misclosure values from a number of consecutive epochs (e.g., update epochs) that includes the current epoch and one or more previous epochs. In an embodiment, the number, e.g., window size, of consecutive update epochs is 5 and the ASD module 114 creates a rolling history using the misclosure values from the 5 consecutive update epochs.

A window size of 5 is for illustrative purposes only, and it is expressly contemplated that the window size may be of any value. In an embodiment, the window size may be any value that is equal to or greater than 2. For this example, let it be assumed that the window size is 5. As such, the ASD module 114 may create a rolling history (History Epoch 1) that includes 4 previous update epochs, e.g., update epochs 1 through 4, and the current update epoch, e.g., update epoch 5. Therefore, History Epoch 1 includes the misclosure values from update epochs 1 through 5. For example, the ASD module 114 may create History Epoch 1 as follows:

History Epoch 1

Update
Filter 1
Filter 2

Epoch
Position Misclosure (m)
Position Misclosure (m)

1
1.958
0.615

2
4.849
0.338

3
6.593
0.338

4
8.891
0.695

5
10.665
0.913

After the completion of procedure 400 where the current epoch is epoch 5, the next epoch will be the current epoch. For this example, let it be assumed that the next epoch is epoch 6. When epoch 6 is the current update epoch, the procedure 400 repeats for a next iteration, and the ASD module 114 may create a next rolling history (History Epoch 2) at step 420 for the next iteration. History Epoch 2 may be created by removing the oldest epoch (e.g., update epoch 1) from History Epoch 1 and adding the error correction value for update epoch 6 to History Epoch 1. Therefore, History Epoch 2 includes epochs 2, 3, 4, 5, and 6 and their respective misclosure values.

The procedure continues to step 425 and the ASD module 114 computes, for each filter, a residual value for the update type. In an embodiment, the ASD module 114 computes a residual value for the update type utilizing the misclosure values from the rolling history. The ASD module 114 may use an RMS to compute the position residual value in a similar manner as described above with relation to step 220 of FIG. 2. As such, the ASD module 114 uses the RMS to compute the position residual value for filter 1 as 7.261. The ASD module 114 uses the RMS to compute the position residual value for filter 2 as 0.620.

An RMS value is a measure of the magnitude of a set of numbers. Since a misclosure value represents the difference between the GNSS solution and the INS solution, an increasing RMS value indicates an increasing magnitude of difference between the GNSS solution and the INS solution over the rolling history. As such, a residual value may represent a magnitude of the difference between the GNSS solution and the INS solution at a filter for an update type over the rolling history.

Procedure 400 continues to step 430 and the ASD module 114 computes, for the current epoch, a misclosure ratio value for the update type. For example, the position residual values for filters 1 and 2 that correspond to update epoch 5 are 7.261 and 0.620, respectively. As such, the ASD module 114 may compute the position misclosure ratio (PMR) value at update epoch 5 as follows:

$PMR = \frac{7.261}{0.6 2 0} = 1 1.7 1 1$

In this example, filter 1 corresponds to the numerator and filter 2 corresponds to the denominator. According to the one or more embodiments as described herein, a ratio value increasing from