Application of Sparse Identification of Nonlinear Dynamics for Physics-Informed Learning

Advances in machine learning and deep neural networks has enabled complex engineering tasks like image recognition, anomaly detection, regression, and multi-objective optimization, to name but a few. The complexity of the algorithm architecture, e.g., the number of hidden layers in a deep neural network, typically grows with the complexity of the problems they are required to solve, leaving little room for interpreting (or explaining) the path that results in a specific solution. This drawback is particularly relevant for autonomous aerospace and aviation systems, where certifications require a complete understanding of the algorithm behavior in all possible scenarios. Including physics knowledge in such data-driven tools may improve the interpretability of the algorithms, thus enhancing model validation against events with low probability but relevant for system certification. Such events include, for example, spacecraft or aircraft sub-system failures, for which data may not be available in the training phase. This paper investigates a recent physics-informed learning algorithm for identification of system dynamics, and shows how the governing equations of a system can be extracted from data using sparse regression. The learned relationships can be utilized as a surrogate model which, unlike typical data-driven surrogate models, relies on the learned underlying dynamics of the system rather than large number of fitting parameters. The work shows that the algorithm can reconstruct the differential equations underlying the observed dynamics using a single trajectory when no uncertainty is involved. However, the training set size must increase when dealing with stochastic systems, e.g., nonlinear dynamics with random initial conditions

In-Time UAV Flight-Trajectory Estimation and Tracking Using Bayesian Filters

Rapid increase of UAV operation in the next decade in areas of on-demand delivery, medical transportation services, law enforcement, traffic surveillance and several others pose potential risks to the low altitude airspace above densely populated areas. Safety assessment of airspace demands the need for a novel UAV traffic management (UTM) framework for regulation and tracking of the vehicles. Particularly for low-altitude UAV operations, quality of GPS measurements feeding into the UAV is often compromised by loss of communication link caused by presence of trees or tall buildings in proximity to the UAV flight path. Inaccurate GPS locations may yield to unreliable monitoring and inaccurate prognosis of remaining battery life and other safety metrics which rely on future expected trajectory of the UAV. This work therefore proposes a generalized monitoring and prediction methodology for autonomous UAVs using in-time GPS measurements. Firstly, a typical 4D smooth trajectory generation technique from a series of waypoint locations with associated expected times-of-arrival based on B-spline curves is presented. Initial uncertainty in the vehicle's expected cruise velocity is quantified to compute confidence intervals along the entire flight trajectory using error interval propagation approach. Further, the generated planned trajectory is considered as the prior knowledge which is updated during its flight with incoming GPS measurements in order to estimate its current location and corresponding kinematic profiles. Estimation of position is denoted in dicrete state-space representation such that position at a future time step is derived from position and velocity at current time step and expected velocity at the future time step. A linear Bayesian filtering algorithm is employed to efficiently refine position estimation from noisy GPS measurements and update the confidence intervals. Further, a dynamic re-planning strategy is implemented to incorporate unexpected detour or delay scenarios. Finally, critical challenges related to uncertainty quantification in trajectory prognosis for autonomous vehicles are identified, and potential solutions are discussed at the end of the paper. The entire monitoring framework is demonstrated on real UAV flight experiments conducted at the NASA Langley Research Center

A Fast Monte Carlo Method for Model-Based Prognostics Based on Stochastic Calculus

This work proposes a fast Monte Carlo method to solve differential equations utilized in model-based prognostics. The methodology is derived from the theory of stochastic calculus, and the goal of such a method is to speed up the estimation of the probability density functions describing the independent variable evolution over time. In the prognostic scenarios presented in this paper, the stochastic differential equations describe variables directly or indirectly related to the degradation of a monitored system. The method allows the estimation of the probability density functions by solving the deterministic equation and approximating the stochastic integrals using samples of the model noise. By so doing, the prognostic problem is solved without the Monte Carlo simulation based on Euler's forward method, which is typically the most time consuming task of the prediction stage. Three different prognostic scenarios are presented as proof of concept: (i) life prediction of electrolytic capacitors, (ii) remaining time to discharge of Lithium-ion batteries, and (iii) prognostic of cracked structures under fatigue loading. The paper shows how the method produces probability density functions that are statistically indistinguishable from the distributions estimated with Euler's forward Monte Carlo simulation. However, the proposed solution is orders of magnitude faster when computing the time-to-failure distribution of the monitored system. The approach may enable complex real-time prognostics and health management solutions with limited computing power

Model Based Diagnostics and Prognostics Framework for Systems Health Management

In order to tackle and solve the system health prediction problem, it is essential to have awareness of the current state and health of the system, especially since it is necessary to perform condition-based predictions. To be able to accurately predict the future state of any system, it is required to possess knowledge of its current and future operations. Given models of the current and future system behavior, the general approach of model-based prognostics can be employed as a solution to the prior stated prediction problem. In case of electric aircrafts, computing remaining flying time is safety-critical, since an aircraft that runs out of power (battery charge) while in the air will eventually lose control leading to catastrophe. In order to tackle and solve the prediction problem, it is essential to have awareness of the current state and health of the system, especially since it is necessary to perform condition-based predictions. To be able to predict the future state of the system, it is also required to possess knowledge of the current and future operations of the vehicle. This presentation will cover a physics based-modeling approach implemented for case-studies in battery and composite structures for prognostics. Given models of the current and future system behavior, a general approach of model-based prognostics can be employed as a solution to the prediction problem and further for decision making

Real-Time UAV Trajectory Prediction for Safety Monitoring in Low-Altitude Airspace

The rising number of small unmanned aerial vehicles (UAVs) expected in the next decade will enable a new series of commercial, service, and military operations in low altitude airspace as well as above densely populated areas. These operations may include on-demand delivery, medical transportation services, law enforcement operations, traffic surveillance and many more. Such unprecedented scenarios create the need for robust, efficient ways to monitor the UAV state in time to guarantee safety and mitigate contingencies throughout the operations. This work proposes a generalized monitoring and prediction methodology that utilizes realtime measurements of an autonomous UAV following a series of way-points. Two different methods, based on sinusoidal acceleration profiles and high-order splines, are utilized to generate the predicted path. The monitoring approach includes dynamic trajectory re-planning in the event of unexpected detour or hovering of the UAV during flight. It can be further extended to different vehicle types, to quantify uncertainty affecting the state variables, e.g., aerodynamic and other environmental effects, and can also be implemented to prognosticate safety-critical metrics which depend on the estimated flight path and required thrust. The proposed framework is implemented on a simplified, scalable UAV modeling and control system traversing 3D trajectories. Results presented include examples of real-time predictions of the UAV trajectories during flight and a critical analysis of the proposed scenarios under uncertainty constraints

An Uncertainty Quantification Framework for Autonomous System Tracking and Health Monitoring

This work proposes a perspective towards establishing a framework for uncertainty quantification of autonomous system tracking and health monitoring. The approach leverages the use of a predictive process structure, which maps uncertainty sources and their interaction according to the quantity of interest and the goal of the predictive estimation. It is systematic and uses basic elements that are system agnostic, and therefore needs to be tailored according to the specificity of the application. This work is motivated by the interest in low-altitude unmanned aerial vehicle operations, where awareness of vehicle and airspace state becomes more relevant as the density of autonomous operations grows rapidly. Predicted scenarios in the area of small vehicle operations and urban air mobility have no precedent, and holistic frameworks to perform prognostics and health management (PHM) at the system- and airspace-level are missing formal approaches to account for uncertainty. At the end of the paper, two case studies demonstrate implementation framework of trajectory tracking and health diagnosis for a small unmanned aerial vehicle

Architecture and Information Requirements to Assess and Predict Flight Safety Risks During Highly Autonomous Urban Flight Operations

As aviation adopts new and increasingly complex operational paradigms, vehicle types, and technologies to broaden airspace capability and efficiency, maintaining a safe system will require recognition and timely mitigation of new safety issues as they emerge and before significant consequences occur. A shift toward a more predictive risk mitigation capability becomes critical to meet this challenge. In-time safety assurance comprises monitoring, assessment, and mitigation functions that proactively reduce risk in complex operational environments where the interplay of hazards may not be known (and therefore not accounted for) during design. These functions can also help to understand and predict emergent effects caused by the increased use of automation or autonomous functions that may exhibit unexpected non-deterministic behaviors. The envisioned monitoring and assessment functions can look for precursors, anomalies, and trends (PATs) by applying model-based and data-driven methods. Outputs would then drive downstream mitigation(s) if needed to reduce risk. These mitigations may be accomplished using traditional design revision processes or via operational (and sometimes automated) mechanisms. The latter refers to the in-time aspect of the system concept. This report comprises architecture and information requirements and considerations toward enabling such a capability within the domain of low altitude highly autonomous urban flight operations. This domain may span, for example, public-use surveillance missions flown by small unmanned aircraft (e.g., infrastructure inspection, facility management, emergency response, law enforcement, and/or security) to transportation missions flown by larger aircraft that may carry passengers or deliver products. Caveat: Any stated requirements in this report should be considered initial requirements that are intended to drive research and development (R&D). These initial requirements are likely to evolve based on R&D findings, refinement of operational concepts, industry advances, and new industry or regulatory policies or standards related to safety assurance

The role of the right temporoparietal junction in perceptual conflict: detection or resolution?

The right temporoparietal junction (rTPJ) is a polysensory cortical area that plays a key role in perception and awareness. Neuroimaging evidence shows activation of rTPJ in intersensory and sensorimotor conflict situations, but it remains unclear whether this activity reflects detection or resolution of such conflicts. To address this question, we manipulated the relationship between touch and vision using the so-called mirror-box illusion. Participants' hands lay on either side of a mirror, which occluded their left hand and reflected their right hand, but created the illusion that they were looking directly at their left hand. The experimenter simultaneously touched either the middle (D3) or the ring finger (D4) of each hand. Participants judged, which finger was touched on their occluded left hand. The visual stimulus corresponding to the touch on the right hand was therefore either congruent (same finger as touch) or incongruent (different finger from touch) with the task-relevant touch on the left hand. Single-pulse transcranial magnetic stimulation (TMS) was delivered to the rTPJ immediately after touch. Accuracy in localizing the left touch was worse for D4 than for D3, particularly when visual stimulation was incongruent. However, following TMS, accuracy improved selectively for D4 in incongruent trials, suggesting that the effects of the conflicting visual information were reduced. These findings suggest a role of rTPJ in detecting, rather than resolving, intersensory conflict

Effects of Multimodal Load on Spatial Monitoring as Revealed by ERPs

While the role of selective attention in filtering out irrelevant information has been extensively studied, its characteristics and neural underpinnings when multiple environmental stimuli have to be processed in parallel are much less known. Building upon a dual-task paradigm that induced spatial awareness deficits for contralesional hemispace in right hemisphere-damaged patients, we investigated the electrophysiological correlates of multimodal load during spatial monitoring in healthy participants. The position of appearance of briefly presented, lateralized targets had to be reported either in isolation (single task) or together with a concurrent task, visual or auditory, which recruited additional attentional resources (dual-task). This top-down manipulation of attentional load, without any change of the sensory stimulation, modulated the amplitude of the first positive ERP response (P1) and shifted its neural generators, with a suppression of the signal in the early visual areas during both visual and auditory dual tasks. Furthermore, later N2 contralateral components elicited by left targets were particularly influenced by the concurrent visual task and were related to increased activation of the supramarginal gyrus. These results suggest that the right hemisphere is particularly affected by load manipulations, and confirm its crucial role in subtending automatic orienting of spatial attention and in monitoring both hemispaces

Pupil dilation reveals top–down attentional load during spatial monitoring

It has long been known that the diameter of human pupil enlarges with increasing effort during the execution of a task. This has been observed not only for purely mechanical effort but also for mental effort, as for example the computation of arithmetic problems with different levels of difficulty. Here we show that pupil dilation reflects changes in visuospatial awareness induced by attentional load during multi-tasking. In the single-task condition, participants had to report the position of lateralized, briefly presented, masked visual targets ("right", "left", or "both" sides). In the multitasking conditions, participants also performed additional tasks, either visual or auditory, to increase the attentional load. Sensory stimulation was kept constant across all conditions to rule out the influence of low-level factors. Results show that event-related pupil dilation strikingly increased with task demands, mirroring a concurrent decrease in visuospatial awareness. Importantly, pupil dilation significantly differed between two dual-task conditions that required to process the same number of stimuli but yielded differed levels of accuracy (difficulty). In contrast, pupil dilation did not differ between two conditions which were equally challenging but differed both in the modality of the dual task (auditory vs. visual) and in the number of stimuli to be attended. We conclude that pupil dilation genuinely reflects the top-down allocation of supramodal attentional resources