Effectiveness of Attitude Estimation Processing Approaches in Tolerating Radiation Soft Errors

This article compares the radiation-induced soft-error tolerance effectiveness of five different attitude estimation (AE) processing approaches that are classically used in inertial navigation systems (INSs) of autonomous things. Results of 14-MeV neutron and thermal neutron radiation testing campaigns indicate that all the AE processing approaches investigated can be critically perturbed by single-event upsets (SEUs), yielding inappropriate output data that would place INS unavailable for a while. Moreover, the radiation testing results also suggest the Kalman filter (KF)-based AE approaches presented better effectivenesses in tolerating SEUs than AE approaches based on gradient descent.


Effectiveness of Attitude Estimation Processing Approaches in Tolerating Radiation Soft Errors Tarso Kraemer Sarzi Sartori , Luiz Henrique Laurini , Hassen Fourati , and Rodrigo Possamai Bastos
Abstract-This article compares the radiation-induced softerror tolerance effectiveness of five different attitude estimation (AE) processing approaches that are classically used in inertial navigation systems (INSs) of autonomous things. Results of 14-MeV neutron and thermal neutron radiation testing campaigns indicate that all the AE processing approaches investigated can be critically perturbed by single-event upsets (SEUs), yielding inappropriate output data that would place INS unavailable for a while. Moreover, the radiation testing results also suggest the Kalman filter (KF)-based AE approaches presented better effectivenesses in tolerating SEUs than AE approaches based on gradient descent.

I. INTRODUCTION
T HE attitude estimation (AE) processing is vital for inertial navigation systems (INSs) of objects such as robots, autonomous vehicles, drones [1], and satellites [2] to estimate the object's attitude (spatial orientation) with respect to a reference and hence provide real-time attitude to control movements and maneuvers. Estimating the attitude of objects by processing onboard measurements has a long history, and plenty of optimal AE models have been developed over the last decades [3], [4], [5], [6], [7]. Algorithms based on AE models are usually embedded in INS and involve a two-part process: 1) estimation of the object's orientation based on onboard sensors' measurements and 2) filtering of noisy measurements. The inertial measurement unit (IMU) of INS is composed of sensors, essentially gyroscopes and accelerometers often designed as microelectromechanical systems (MEMSs) that enable the tracking of rotational and translational movements. In addition, for applications demanding 3-D attitude determination, the measurements of a third sensor are normally required to measure the Earth's magnetic field. Hybrid IMUs that incorporate a three-axis magnetometer are also known as magnetic, angular rate, and gravity (MARG) sensors [7]. In addition, INS contains analog-to-digital interface circuits and an onboard processing system, including at least a processor and memories for data and programs (see Fig. 1).
In space orbits [8], [9], airplane altitudes [10], and at ground levels [11], INS components are exposed to radiation-induced soft errors such as single-event effects (SEEs) that can invert memory bits (not permanently), the single-event upsets (SEUs), or even halt the operation (hang or crash), requiring thus a system reboot via either a software reset or a power cycle (off and on), that is, SEU-induced single-event functional interrupts (SEFIs) or single-event latch-up (SEL) effects.
Soft errors during the processing of AE algorithms in INS can interfere with their responses, failing to properly estimate the object's attitude. Only a few related works discuss soft SEEs in INS components, for example, Bazzano et al. [12] evaluated a commercial off-the-shelf (COTS) INS under protons, Stansberry [13] and Oudea et al. [14] analyzed SEEs in COTS accelerometers under protons and heavy ions, and the articles [15], [16], and [17] briefly discuss SEEs in IMUs. Concerning other types of possible errors in INS such as effects of total ionizing dose (TID) and noise sources, which are both not the target herein, Pitt et al. [18] studied COTS accelerometers facing TID and Zhi-Yong et al. [19] investigated IMUs for harsh electromagnetic environments. Finally, our previous work in [20] assessed neutron-induced soft SEEs in AE processing approaches based on the classical AE model known as the novel quaternion Kalman filter (NQKF) [5], essentially underlining three aspects.
1) The high contribution of SEFI/SEL on the total number of neutron-induced soft errors that can lead an INS to critical failures. 2) The influence of applying lower load averages on the AE processing system to reduce the number of failures. 3) Case-study AE algorithm's results concerning SEU-induced mismatches presented no significant impact as the AE processing could very rapidly recover itself to provide correct responses in all related scenarios tested. Unlike the aforementioned related works, this article focuses on comparing the effectiveness of different AE processing approaches based on four classical AE models facing neutron-induced soft SEEs: 1) quaternion-based extended Kalman filter (EKF) [4]; 2) NQKF [5]; 3) quaternion-based indirect Kalman filter (IKF) [6]; and 4) quaternion-based gradient descent (Gradient) [7]. More specifically, regarding our previous work [20] in which we investigated only the AE model NQKF [5], this article analyzes AE processing approaches based on three other classical AE models (EKF [4], IKF [6], and Gradient [7]). In addition, we also present herein new results comparing the computation times of the different AE processing approaches investigated, correlating them with the rates of neutron-induced failures to suggest better options taking into account both aspects.
This article is organized as follows. Section II introduces the AE approaches addressed in this work. Section III explains the test setup used to assess the AE approaches under neutron radiation effects, the hypotheses of the radiation-induced failures on the AE, the computing strategies implemented for executing the AE approach, and finally the summary and evaluation of the results observed in the radiation campaigns. Section IV shows the conclusions of the work.

II. CASE-STUDY AE APPROACHES A. Case-Study AE Models
An AE algorithm is used to estimate the attitude or orientation of an object with respect to a known reference, based on the onboard sensors' measurements. The following equation shows a simplified model for the sensors: These three equations describe, respectively, the gyroscope, accelerometer, and magnetometer measurements. In fact, if the initial orientation is given, only the gyroscope information should be enough to estimate the attitude. However, due to the gyroscope bias, b ω added to the real angular velocity ω 0 , and integration errors [6], the estimation error increases over time. To compensate for these errors, other sensors are used.
The attitude of an object can be represented through different forms. Euler angles φ, θ, and ψ are the most simple representation, having an intuitive physical interpretation commonly referred to as roll, pitch, and yaw, respectively [21] (see Fig. 1). Nevertheless, the Euler angles are susceptible to singularity problems [22], demanding more computational efforts if dynamically used on the AE. Another well-known AE representation, not susceptible to singularity problems, is the attitude quaternion q that is a normalized hypercomplex vector with four components [5] q = q 1 q 2 q 3 q 4 T .
The rotation matrix C B N [q] in (1) is a nonlinear function of the attitude quaternion (cf., [20]). The rotation matrix is capable of transforming a vector defined in the navigation frame (N) (such as the gravity g and the Earth's magnetic field h) to the body frame (B). Other important parameters to take into account are the sensors' measurement noises, represented by v ω , v a , and v m in (1), inherent to each sensor. To filter such noises and combine the sensor's data to estimate the attitude, many different approaches have been proposed. One wellknown approach is the Kalman filter (KF), which is an optimal recursive algorithm for state estimation of linear systems. The KF algorithm assumes the sensor's noises follow a Gaussian normal distribution with zero mean and a determined standard deviation, depending on the sensors. The filter is first initialized with a guess of the state of the system q 0/0 , that in the case of this work is just the quaternion, and an error covariance matrix P 0/0 , which is an estimate of the error in the filter estimation. Furthermore, the filter makes a prediction (q k/k−1 , P k/k−1 ) in the iteration k using the physical model of the system, which in the case of AE can be based on the angular velocity ω provided by the gyroscopes. Finally, the predictions are updated using another set of sensors' measurements, such as acceleration a from accelerometers and magnetic field m from magnetometers, generating the estimates q k/k and P k/k that will feed the algorithm in the next iteration.
The four AE algorithms assessed in this work are as follows.
• EKF: The EKF is one of the most applied algorithms for real-time spacecraft AE [3] as they are already wellestablished [4]. Since the KF is designed for linear systems, it is necessary to linearize the measurement equations (accelerometer and magnetometer in 1) to properly use it for the attitude quaternion estimation.
• NQKF: The linearization necessary for the EKF algorithm can provoke undesirable effects, such as sensitivity to initial conditions and an increase in the computational load. For dealing with them, Choukroun et al. [5] developed a novel algorithm, presenting a pseudo-measurement linear equation to be used with the KF, eliminating the linearization procedure and being less sensitive to initial AE errors.
• IKF: Suh [6] proposed an adaptive KF approach to compensate for external accelerations (other than gravity). Instead of estimating the quaternions, they estimate the attitude quaternion error, function of the gyroscope bias and noise, and then convert them into quaternions.
• Gradient: Madgwick et al. [7] proposed a novel AE algorithm using the gradient descent, an iterative optimization algorithm, to minimize a cost function in terms of the attitude quaternion. The proposed approach is recursive as well as the traditional KF, but is less computationally costly, achieving similar degrees of precision. The KF-based algorithms as well as the Gradient algorithm possess some parameters that need to be adjusted according to the respective sensors' noises. For the KF, it is necessary to set three covariance matrices, one for each sensor being used (gyroscope, accelerometer, and magnetometer), based on the standard deviation of the sensors' measurement noise. For the Gradient algorithm, a parameter called β (see [7]), representing the magnitude of the gyroscope measurement error, needs also to be tuned before operation.

B. Input and Output Datasets
Two different input datasets were generated to test the case-study AE approaches processing under radiation. The first input dataset is composed of 1000 input vectors, and the second of 333. Each input vector contains nine components, which represent the sensor's measurements stacked for a specific moment in time. Each sensor (accelerometer, magnetometer, and gyroscope) provides a 3-D physical measurement (acceleration, magnetic field, and angular velocity), totaling nine components when stacked in an input vector. The sample time of the 333-input dataset was 0.1 s, whereas, for the 1000-input dataset, it was 0.01 s. More details about the generation and characteristics of the sensor's measurements can be found in [20]. The case-study AE algorithm initially executes the input dataset's first input vector, generating a corresponding output vector (i.e., a case-study AE algorithm's four components estimated quaternion q k/k or simply q), and so on for the next input vectors. Therefore, as the input datasets contain 1000 and 333 input vectors, output datasets of 1000 and 333 output vectors are generated, respectively. The case-study AE processing approaches were named according to the AE algorithm's acronyms, that is, EKF, IKF, Gradient, and NQKF that executed input datasets of 333 input vectors, and 1000_NQKF that executed input dataset of 1000 input vectors.

A. Neutron Radiation Test Setups
To assess the effectiveness of the case-study AE processing approaches in tolerating soft errors, the processing system abstracted in Fig. 1 was implemented in a Raspberry Pi 4 Model B board, herein referred to as the system under test (SUT). The operating system (OS) installed on the SUT was the 32-bit Raspberry Pi OS Lite. The AE algorithms were implemented in C/C++ language and stored along with the input datasets in a secure digital (SD) card. The SUT running the case-study AE approaches was exposed to neutrons in four different radiation testing campaigns. The first and second campaigns were performed in July and August 2021 at the Institute Laue-Langevin (ILL) in Grenoble (France) by using the thermal and epi-thermal neutron irradiation station (TENIS) [23], which generates a neutron beam in a wide spectrum of energy, but with a large component in the thermal region (approximately 99%, corresponding to the range from 20 meV to 1 eV). In the August campaign, the SUT was moved outside the main neutron beam window due to the high number of SEFIs/SELs initially observed, hence being exposed to residual thermal neutron flux, and not to the maximum flux. The third and fourth campaigns were performed in February and July 2022 through the 14-MeV neutron generator GENEPI2 at the LPSC's GENESIS facility in Grenoble (France) [24]. Fig. 2 shows the SUT (4 GB of SDRAM) inside the neutron generator chamber. Fig. 3 depicts the test setup used in the neutron radiation campaigns. Another Raspberry Pi, wherein the control computer (CC) outside the neutron chamber, was applied to send commands, receive data, and monitor the experiment. In all campaigns, the algorithms and datasets were stored in SD cards placed outside the radiation chamber, hence not exposed to radiation effects. The input dataset was loaded once-from the SD card to a first-in-first-out (FIFO) buffer on the data memory (SDRAM)-before the beginning of the case-study AE algorithm's execution. Then, in the ILL campaigns, for each input vector, the case-study AE algorithm produced an output vector that was immediately written into the program memory. Otherwise, in the LPSC campaigns, right after each output vector was calculated, it was stored in another FIFO buffer on the data memory. Only after the entire execution of the input dataset, the output dataset stored in the FIFO buffer was thus sent to the CC's SD card.

B. Hypotheses of Radiation-Induced Failures in the AE
Before the radiation campaigns, the golden reference data/results containing the case-study AE approaches' output vectors were generated via simulation without any radiation effects. Otherwise, the radiation testing data/results are the AE quaternions calculated in the SUT by the AE approaches facing the neutron flux effects. Every AE approach's execution is defined herein as a run, that is, the complete execution of either 1000 input vectors for the first case-study input dataset or 333 for the second one.
For the sake of making the results of the radiation testing campaigns more intuitive, the radiation testing data were converted into Euler angles afterward, adopting a rotation sequence ψ, θ, and φ (i.e., Yaw-Pitch-Roll). The Euler angles' mean absolute error (MAE) between the radiation testing data and the golden reference data were also calculated after the campaigns through the following equation: in which n is the number of input vectors for a specific run, y i is the respective golden reference data, and y i rad is the Euler angle calculated considering the radiation testing data. We adopted 1 • as the Euler angle's MAE threshold to classify different hypotheses of radiation-induced failures in the AE processing system.
• No Failure: The result of the AE algorithm does not differ from the golden reference and the run is complete, for example, the AE algorithm correctly computes the entire input dataset.
• Tolerable Failure: The run is complete but an SEU-induced mismatch between the radiation testing data/results and the golden reference is observed and the Euler angles' MAEs are lesser than the threshold stipulated.
• Critical Failure: a) Mismatch and Complete Run: An SEU causes a mismatch between radiation testing data/results and the golden reference, being the Euler angles' MAEs greater than the threshold stipulated, however, the input dataset is fully processed. b) Incomplete Run (Hang or Crash): An SEFI/SEL stops the AE algorithm during the input dataset processing, however a mismatch between the radiation testing data/results and the golden reference is not observed before the interruption. For example, for the input dataset with 1000 input vectors, the AE algorithm was computing the 500th input vector when an SEU interfered with an essential function of the SUT, halting its operation and requiring a software reboot (relaunch of the AE approach). The 500 computed radiation testing data could be stored in the program memory and they match with their respective golden reference data, however, the output dataset was not complete. c) Mismatch and Incomplete Run: An SEFI/SEL stops the AE algorithm during the input dataset processing and a mismatch between the radiation testing data/results and the golden reference is observed before the interruption. For example, for the input dataset with 333 input vectors, the AE algorithm was computing the 100th input vector when an SEU interfered with an essential function of the SUT, halting its operation and requiring an SUT's reboot. The 100 computed radiation testing data could be stored in the program memory, however, mismatches in relation to their golden reference data were observed.
• Processing Failure: It is either a tolerable failure or a critical failure resulting in a mismatch and complete run-herein referred to as tolerable processing failures and critical processing failures, respectively-however, the SEU occurs in the operations that make the processing of the AE algorithm. Hence, tolerable and critical failures that are induced by SEUs in the FIFO buffers (used to store the input and output vectors) are not considered processing failures.

C. Computing Strategies Applied to Assess AE Approaches
Three different computing strategies for the case-study AE algorithms were assessed during the radiation testing campaigns. Fig. 4 abstracts these computing strategies that were defined considering the four cores of the SUT.
• Strategy 1: Run three independent and redundant processes of the case-study AE algorithm in three different cores at the same time, that is, the same case-study AE algorithm was separately executed at the same time by core 1-3, providing thus three output datasets that are equal if no failure was induced by the radiation.
• Strategy 2: Run a single process of the case-study AE algorithm in a core x determined by the OS, besides running other four processes of four similar algorithms at the same time to increase the amount of computational work that the case-study AE algorithm's processing system (SUT) performs (i.e., increase the computing system's load average). For example, the case-study AE algorithm was executed in core 1 for a while, after in core 3 or another as a function of the OS's process management.
• Strategy 3: Likewise Strategy 2, run a single process of the case-study AE algorithm in a core y determined by the OS, however, no other processes of similar algorithms are running at the same time.

D. Summary of the Radiation Campaigns Results
Table I summarizes the results for the four radiation campaigns performed. The cross section for each AE approach was obtained by dividing the number of critical failures by the fluence (irradiation time × average flux). As the radiation testing campaigns produce much higher fluxes than the natural environment, results are transposed to a real scenario in which the AE processing approaches could be applied  I   RADIATION TESTING RESULTS FOR THE THREE CAMPAIGNS CARRIED OUT AT THE ILL (THERMAL NEUTRONS)  to have an estimate of the failure in time (FIT). Hence, the flux at commercial airplanes' altitudes was taken as a reference because the neutron effects have been identified as primarily responsible for inducing SEUs in avionics systems [25]. The FIT values in Table I for LPSC campaigns were calculated by multiplying the cross section by the 14-MeV neutron flux at commercial airplane altitude of 40 000 ft (around 91.8 neutrons/cm 2 /h [26]). For example, the EKF approach in the February 2021 campaign was approximately 20 failures every billion hours of operation. For the ILL campaigns, the FIT was calculated by multiplying the cross section by the neutron flux for thermal neutrons at the same altitude, approximately 300 times the neutron flux for energies above 10 MeV at sea level (around 13 neutrons/cm 2 /h for New York city [27]).
Table I also highlights that only a few number of runs presented processing failures, mostly noticed in the LPSC campaigns. The total number of failures was higher in the ILL tests, essentially due to SEFIs/SELs (see Table I at columns Number of incomplete runs). As much higher fluxes were applied in the ILL campaigns (100 and 120 · 10 5 neutrons/cm 2 /h against 4.14 and 4.27 · 10 5 neutrons/cm 2 /h at LPSC), the SUT was halted several times possibly due to the perturbation of essential functions of the OS, requiring software/hardware reboot and preventing the AE approaches to fully compute the input dataset. Moreover, the methods used for storing the radiation testing data (see Section III-A) have an influence on the number of incomplete runs. Other factors can also explain the low number of processing failures such as the error detection and correction codes integrated on the Arm Cortex-A72 of the SUT and its cache memories. Actually, whenever an error is detected and cannot be corrected, the cache line is evicted, the error is reported in a register and, in the case of the L1 data cache, causes a "data abort" [28]. Moreover, the Arm Cortex-A72 also has error correction codes for the data coming from the SDRAM, further mitigating the number of failures that could be observed as it would correct at least a single bit of word that was affected by the radiation.  Table I), and therefore they are predominant in Fig. 5. Furthermore, Fig. 5 shows the proportion of tolerable processing failures and critical processing failures for each case-study AE processing approach. Note that the processing failures observed in the Gradient approach were all critical processing failures, unlike the other AE approaches that report also tolerable processing failures. The IKF approach presented the highest proportion of tolerable processing failures, which indicate that the operations in the processing of the IKF algorithm were disturbed by SEUs, however the disturbance was either not enough to make the IKF algorithm diverge or the SEU effect was rapidly mitigated, that is, the IKF algorithm could quickly recover itself, having Euler angles' MAEs below the threshold of 1 • specified (see Section III-B). Fig. 6. Radiation testing results of Euler angles being significantly perturbed by a critical processing failure (SEU-induced mismatch and complete run) during the 1000_NQKF approach processing in the February 2022 campaign. The three left plots show the golden testing and reference Euler angles in degrees for each input vector processed. The three right plots show the MAE and the absolute error between the golden testing and reference Euler angles in degrees for each input vector processed.

F. Critical Processing Failures in AE Approaches
Figs. 6-8 show valuable experimental findings regarding runs that presented critical processing failures, that is, significant differences between the radiation-testing data/results and the golden reference. Each figure presents, for its respective AE approach, in the three left plots the golden reference data in orange (response outside the radiation), and in blue the radiation testing data (response under radiation) in the form of Euler angles-in degrees-for each respective processed input vector. The three right plots show the absolute error between the golden reference data and the radiation testing data, and the MAE for each angle-in green and red, respectively-for each respective input vector. Fig. 6 shows the results of the 1000_NQKF approach in the February 2022 campaign. The most probable hypothesis for the significant error observed between the responses is that an SEU occurred during the computation of the 408th input vector of the input dataset, considerably disturbing the algorithm. The MAE for the roll (φ), pitch (θ ), and yaw (ψ) angles are 95 • , 9.4 • , and 15.6 • s, respectively, and the absolute errors reached about 300 • , 95 • , and 225 • for the respective three angles, considering the peaks. It is important to note that an error of 300 • is equivalent to an error of −60 • , considering the range of 360 • of the trigonometric circle. However, the metric used in this work is the Euler angle's MAE (see Section III-B) that purely considers the absolute difference between the golden Euler angles and the AE approach responses under radiation. The input dataset used in the 1000_NQKF approach considers sensors with a sampling time of 0.01 s, that is, the sensors provide 100 input vectors per second. Therefore, considering a shorter computation time than the sensors' sample time, the SEU effect would have remained for about 3 s as it persisted for around 300 input vectors. Even though the AE errors in the 1000_NQKF approach response were high, the 1000_NQKF algorithm could rapidly follow the correct response using the next input vectors. Fig. 7 shows a critical processing failure obtained in the July 2022 campaign for the IKF approach. Probably, an SEU occurred during the computation of the 108th input vector Fig. 7. Radiation testing results of Euler angles being significantly perturbed by a critical processing failure (SEU-induced mismatch and complete run) during the IKF approach processing in the July 2022 campaign. The three left plots show the golden testing and reference Euler angles in degrees for each input vector processed. The three right plots show the MAE and the absolute error between the golden testing and reference Euler angles in degrees for each input vector processed. reflecting a high peak in Euler angle ψ, and remaining until the 135th input vector. The Euler angles' MAEs were 0.2 • , 0.075 • , and 7.4 • for φ, θ, and ψ, respectively. As this input dataset assumes a sampling time of 0.1 s for the sensors, making the same assumptions as in the previous analysis, the SEU effect would have remained about 2.7 s, presenting a high absolute error peak of around 270 • in ψ.
Finally, Fig. 8 presents a critical processing failure observed on the Gradient approach in the July 2022 campaign. Probably an SEU disturbed the processing of the 168th input vector, and the Gradient algorithm could not recover itself until the end of the input dataset, showing a delayed behavior in relation to the golden reference data. On the other hand, the response presents a tendency of recovery as the absolute error tends slowly to zero. The Euler angles' MAEs in this case were 37.5 • , 19 • , and 22.4 • for φ, θ, and ψ, respectively. In general, regarding all KF-based AE processing approaches tested, when the input dataset is completely computed and a processing failure occurred, the KF-based AE algorithm rapidly recovers itself to its ideal responses. This is probably due to the recursive and adaptive nature of KF algorithms. Besides the estimation of the attitude based on the previous responses, the KF algorithm also relies on information from multiple sensors. Assuming the measurements are continuously available, the KF adapts its internal gains based on the confidence level of past estimates and current measurements, improving the AE response over time. Otherwise, the processing failures obtained in the Gradient approach were all critical, and some of them, such as the run in Fig. 8, could not completely recover themselves until the end of the input dataset. However, we underline that both the KF-based and Gradient AE processing approaches need some tuned parameters according to the characteristics of the sensors (see Section II-A). When an SEU disturbs the AE approach processing, if the sensors' measurements are reliable and continuously available, the calibration of these AE approaches' parameters is fundamental for improving the algorithms' convergence.

G. Assessment of AE Approaches in Computing Strategies
To compare the AE processing approaches running in different computing strategies, we considered the classical metrics of central processing unit (CPU) usage, load average, and computation time. Furthermore, we focused on radiation testing data/results obtained in the LPSC campaigns for Strategies 1 and 2 as they presented most processing failures and they were exposed to similar average neutron fluxes (see Table I). Table II summarizes the results considering the different computing strategies tested in the LPSC campaigns.
The CPU usage is defined as the ratio between the time spent by the processor on one or more processes and the time interval measured. The value ranges from 0%, when the CPU time is not used by the processes and N × 100% when the CPU is fully utilized, with N being the number of processing cores (SUT's CPU has four cores). Observing in Table II that the average CPU usages were similar among the processes related to the case-study AE approaches within the same strategy, however significantly different from the other strategy. For example, the Gradient approach in Strategy 1 had an average CPU usage of 10.05%, similar to the EKF in the same strategy but very different from the same approach in Strategy 2 (41.16%). The normalized load average in Table II was obtained by averaging all system processes running or waiting for the processor availability during 15 min of measurements and normalized by the SUT's number of cores. Strategy 1 executes three times more processes in parallel when compared with Strategy 2, consequently, the average CPU usage for each process was reduced, whereas the system load average was increased. Although the number of processes in parallel in Strategy 1 was higher, the number of runs was lower due to the overload in comparison with Strategy 2. Hence, higher numbers of failures induced by SEUs in the FIFO buffer (used for storing output vectors) and of processing failures were observed (see column "Total" in Table II). Fig. 9 shows the average computation time of each input vector and the rates of processing failures and critical processing failures for each one of the AE processing approaches in Strategies 1 and 2 in the July 2022 campaign. The rate of critical processing failures and the rate of processing failures were obtained, respectively, by dividing the number of critical processing failures and the number of processing failures (critical + tolerable) by the irradiation time of each AE approach for Strategies 1 and 2 tested in July 2022. The computation time was calculated by dividing the exposition time of the AE approach by the number of input vectors processed (number of runs × 333) in each strategy for the same campaign. The EKF approach presented the highest computation time in both computing strategies, whereas the NQKF was the lowest one. The highest rates of critical processing failures were observed in the NQKF and Gradient approaches in Strategy 2. Actually, according to Fig. 5, in all campaigns, when an SEU disturbed the processing of the Gradient approach, it provoked a criti-cal processing failure. The KF-based approaches, in general, showed better results considering critical processing failures, and more specifically, the IKF presented a better balance regarding computation time and rates of critical processing failures in both strategies. In Strategy 1, the IKF provided the highest rate of processing failures. Nevertheless, all processing failures were tolerable and the rate of critical processing failures was zero. In Strategy 2, the IKF has one of the lowest rates of processing failures. These observations agree with the data in Fig. 5, showing that the IKF approach has the highest percentage of tolerable processing failures in relation to the other AE approaches.

IV. CONCLUSION
This work implemented, tested, and compared five AE processing approaches through four different radiation campaigns (thermal and 14-MeV neutrons). Three different computing processing strategies were also used for assessing the effectiveness of the AE approaches in tolerating SEU effects, essentially varying the SUT's CPU usage and load average. Strategy 2 (see Section III-C) proved to be more suitable for testing the AE approaches, producing a higher number of runs and failures within a shorter irradiation time. The different levels of soft error tolerance observed among the different strategies are explained, for instance, by the time the AE algorithm takes to execute an input dataset, the number of voluntary and involuntary context switches (able to cause more cache invalidation), and the time spent executing kernel threads. Compared with the AE approach based on gradient descent, KF-based AE approaches showed better results regarding tolerable and critical processing failures, being able, in all cases, to recover themselves after a few seconds. This is probably caused by the KF's adaptive nature, which relies on multiple sensors' measurements, adjusting its internal gains based on its past estimate and current measurements. All processing failures observed on the Gradient approach were critical processing failures, and, in some examples, it could not recover itself until the end of the input dataset. On the other hand, the effectiveness of AE processing approaches can still be improved by optimally tuning the AE algorithms' parameters based on the sensors' noises. Among the KFbased approaches, the IKF presented the lowest rates of critical processing failures and a better balance between computation time and critical processing failures.