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All sourcesNASA NTRSDOE OSTI
NASA NTRS2017

Distributed Spatiotemporal Motion Planning for Spacecraft Swarms in Cluttered Environments

This paper focuses on trajectory planning for spacecraft swarms in cluttered environments, like debris fields or the asteroid belt. Our objective is to reconfigure the spacecraft swarm to a desired formation in a distributed manner while minimizing fuel and avoiding collisions among themselves and with the obstacles. In our prior work we proposed a novel distributed guidance algorithm for spacecraft swarms in static environments.1 In this paper, we present the Multi-Agent Moving-Obstacles Spherical Expansion and Sequential Convex Programming (MAMO SE–SCP) algorithm that extends our prior work to include spatiotemporal constraints such as time-varying, moving obstacles and desired time-varying terminal positions. In the MAMO SE–SCP algorithm, each agent uses a spherical-expansion-based sampling algorithm to cooperatively explore the time-varying environment, a distributed assignment algorithm to agree on the terminal position for each agent, and a sequential-convex-programming-based optimization step to compute the locally-optimal trajectories from the current location to the assigned time-varying terminal position while avoiding collision with other agent and the moving obstacles. Simulations results demonstrate that the proposed distributed algorithm can be used by a spacecraft swarm to achieve a time-varying, desired formation around an object of interest in a dynamic environment with many moving and tumbling obstacles.

Hadaegh, Fred Y.↗
NASA NTRSDate not supplied

Dynamic Spectrum Allocation in Urban Air Transportation System via Deep Reinforcement Learning

The emerging concepts of Urban Air Mobility (UAM) and Advanced Air Mobility (AAM) open a new paradigm for urban air transportation. A big challenge is that these new aerial vehicles will quickly saturate the already crowded aviation spectrum, which is an essential resource to ensure reliable communications for safe operations. In this paper, we consider an air transportation system where multiple aerial vehicles are operated to transport passengers or cargo from different sources to destinations along their pre-defined paths. During the flight, the minimum communication Quality of Service (QoS) requirement must be achieved to ensure flight safety. Our objective is to minimize the average mission completion time by jointly optimizing the velocity selection and spectrum allocation for all aerial vehicles. We formulate the optimization problem as a multi-stage Markov Decision Process (MDP) where the optimization variables are coupled together. A multi-agent Deep Reinforcement Learning (DRL) based solution is proposed where Value Decomposition Networks (VDN) algorithm is utilized to take discrete actions. Additionally, we propose a heuristic greedy algorithm as a baseline solution. Simulation results show that our learning based solution outperforms the heuristic greedy algorithm and another Orthogonal Multiple Access (OMA) solution in minimizing the mission completion time.

Ruixuan Han↗
NASA NTRSDate not supplied

Dynamic Spectrum Allocation in Urban Air Transportation System via Deep Reinforcement Learning

The emerging concepts of Urban Air Mobility (UAM) and Advanced Air Mobility (AAM) open a new paradigm for urban air transportation. A big challenge is that these new aerial vehicles will quickly saturate the already crowded aviation spectrum, which is an essential resource to ensure reliable communications for safe operations. In this paper, we consider an air transportation system where multiple aerial vehicles are operated to transport passengers or cargo from different sources to destinations along their pre-defined paths. During the flight, the minimum communication Quality of Service (QoS) requirement must be achieved to ensure flight safety. Our objective is to minimize the average mission completion time by jointly optimizing the velocity selection and spectrum allocation for all aerial vehicles. We formulate the optimization problem as a multi-stage Markov Decision Process (MDP) where the optimization variables are coupled together. A multi-agent Deep Reinforcement Learning (DRL) based solution is proposed where Value Decomposition Networks (VDN) algorithm is utilized to take discrete actions. Additionally, we propose a heuristic greedy algorithm as a baseline solution. Simulation results show that our learning based solution outperforms the heuristic greedy algorithm and another Orthogonal Multiple Access (OMA) solution in minimizing the mission completion time.

Ruixuan Han↗
NASA NTRS2022

Contribution of Spicules to Solar Coronal Emission

Recent high-resolution imaging and spectroscopic observations have generated renewed interest in spicules' role in explaining the hot corona. Some studies suggest that some spicules, often classified as type II, may provide significant mass and energy to the corona. Here we use numerical simulations to investigate whether such spicules can produce the observed coronal emission without any additional coronal heating agent. Model spicules consisting of a cold body and hot tip are injected into the base of a warm (0.5 MK) equilibrium loop with different tip temperatures and injection velocities. Both piston- and pressure-driven shocks are produced. We find that the hot tip cools rapidly and disappears from coronal emission lines such as Fe xii 195 and Fe xiv 274. Prolonged hot emission is produced by preexisting loop material heated by the shock and by thermal conduction from the shock. However, the shapes and Doppler shifts of synthetic line profiles show significant discrepancies with observations. Furthermore, spatially and temporally averaged intensities are extremely low, suggesting that if the observed intensities from the quiet Sun and active regions were solely due to type II spicules, one to several orders of magnitude more spicules would be required than have been reported in the literature. This conclusion applies strictly to the ejected spicular material. We make no claims about emissions connected with waves or coronal currents that may be generated during the ejection process and heat the surrounding area.

Shanwlee Sow Mondal↗
NASA NTRS2010

Route Generation for a Synthetic Character (BOT) Using a Partial or Incomplete Knowledge Route Generation Algorithm in UT2004 Virtual Environment

This paper presents a new Route Generation Algorithm that accurately and realistically represents human route planning and navigation for Military Operations in Urban Terrain (MOUT). The accuracy of this algorithm in representing human behavior is measured using the Unreal Tournament(Trademark) 2004 (UT2004) Game Engine to provide the simulation environment in which the differences between the routes taken by the human player and those of a Synthetic Agent (BOT) executing the A-star algorithm and the new Route Generation Algorithm can be compared. The new Route Generation Algorithm computes the BOT route based on partial or incomplete knowledge received from the UT2004 game engine during game play. To allow BOT navigation to occur continuously throughout the game play with incomplete knowledge of the terrain, a spatial network model of the UT2004 MOUT terrain is captured and stored in an Oracle 11 9 Spatial Data Object (SOO). The SOO allows a partial data query to be executed to generate continuous route updates based on the terrain knowledge, and stored dynamic BOT, Player and environmental parameters returned by the query. The partial data query permits the dynamic adjustment of the planned routes by the Route Generation Algorithm based on the current state of the environment during a simulation. The dynamic nature of this algorithm more accurately allows the BOT to mimic the routes taken by the human executing under the same conditions thereby improving the realism of the BOT in a MOUT simulation environment.

Hanold, Gregg T.↗
NASA NTRSDate not supplied

Bootstrapping Multi-Agent Unmanned Aerial Vehicle (UAV) System Integration Using Ground-Based Assets: Lessons Learned

The highly dynamic nature of UAVs imposes significant challenges when conducting initial testing ranging from safety risks posed by high-capacity lithium batteries and spinning propellers to rigorous timing demands on controllers and the consequences of failures mid-air. Flight testing of a single vehicle is time and labor intensive due to these challenges and more, and the complexity increases exponentially with the number of vehicles. While simulations and hardware-in-the-loop bench testing can provide adequate environments for preliminary validation, differences in system deployment architecture, software interfaces, and hardware infrastructure between simulation and a fleet of real UAVs create a sizable gap that must be navigated carefully during system integration. In support of the Autonomy Teaming and TRAjectories for Complex Trusted Operational Reliability (ATTRACTOR) project, which had the goal of establishing a basis of certification of trust and trustworthiness in multi-agent autonomous systems, this gap was tackled from two directions. First, a novel mixed-reality simulation environment was engineered to blur the transition from simulation to flight hardware. Second, a fleet of Unmanned Surface Vehicles (USVs) was developed as a test and evaluation platform that more closely represented the final aerial fleet while eliminating many of the risks associated with air vehicles. This paper delves into the second element, analyzing the efficacy of the USV platform in performing system integration testing for the UAV system. In this paper we present the USV fleet and its role in reducing the aforementioned gaps in deployment architecture, software interfaces, and hardware infrastructure when moving from simulation to flight. An overview of the hardware and software onboard the vehicles will be provided along with supporting infrastructure. The system integration process will be documented including results in supporting both the overarching design reference mission (DRM) of ATTRACTOR and individual research efforts conducted during the project. Finally, we will discuss some of the practical lessons learned regarding the testing, deployment, and operation of multi-agent autonomous systems.

Matthew P Vaughan↗
NASA NTRS1998

Integration of the Remote Agent for the NASA Deep Space One Autonomy Experiment

This paper describes the integration of the Remote Agent (RA), a spacecraft autonomy system which is scheduled to control the Deep Space 1 spacecraft during a flight experiment in 1999. The RA is a reusable, model-based autonomy system that is quite different from software typically used to control an aerospace system. We describe the integration challenges we faced, how we addressed them, and the lessons learned. We focus on those aspects of integrating the RA that were either easier or more difficult than integrating a more traditional large software application because the RA is a model-based autonomous system. A number of characteristics of the RA made integration process easier. One example is the model-based nature of RA. Since the RA is model-based, most of its behavior is not hard coded into procedural program code. Instead, engineers specify high level models of the spacecraft's components from which the Remote Agent automatically derives correct system-wide behavior on the fly. This high level, modular, and declarative software description allowed some interfaces between RA components and between RA and the flight software to be automatically generated and tested for completeness against the Remote Agent's models. In addition, the Remote Agent's model-based diagnosis system automatically diagnoses when the RA models are not consistent with the behavior of the spacecraft. In flight, this feature is used to diagnose failures in the spacecraft hardware. During integration, it proved valuable in finding problems in the spacecraft simulator or flight software. In addition, when modifications are made to the spacecraft hardware or flight software, the RA models are easily changed because they only capture a description of the spacecraft. one does not have to maintain procedural code that implements the correct behavior for every expected situation. On the other hand, several features of the RA made it more difficult to integrate than typical flight software. For example, the definition of correct behavior is more difficult to specify for a system that is expected to reason about and flexibly react to its environment than for a traditional flight software system. Consequently, whenever a change is made to the RA it is more time consuming to determine if the resulting behavior is correct. We conclude the paper with a discussion of future work on the Remote Agent as well as recommendations to ease integration of similar autonomy projects.

Dorais, Gregory A.↗
NASA NTRS2022

Multi-Agent Control Approach to the Stability of Linear Deployable Systems

Collisions and undesired dynamic responses may occur during the expansion phase of deployable elements on a spacecraft. This problem is more pronounced in large and complex deployable elements where characterization of transient dynamics using testing and simulation can be challenging. This paper introduces a multi-agent control theory approach to address the problem of collisions and transient behavior in deployable lattice systems. We introduce the graph theoretic and relative error vector-based state-space structures to represent the geometry of a lattice-based deploy system. We show that under the assumption of almost-strict dissipativity, the closed-loop feedback gains of the relative error dynamics are the spring and damper constants in the structure. Moreover, we introduce the control perspective of shaping transient dynamics using eigenstructure assignment and balanced coordinate transformation.

Pablos, Juan L. de↗
NASA NTRS2004

Interaction and Communication of Agents in Networks and Language Complexity Estimates

Knowledge acquisition and sharing are arguably the most critical activities of communicating agents. We report about our on-going project featuring knowledge acquisition and sharing among communicating agents embedded in a network. The applications we target range from hardware robots to virtual entities such as internet agents. Agent experiments can be simulated using a convenient simulation language. We analyzed the complexity of communicating agent simulations using Java and Easel. Scenarios we have studied are listed below. The communication among agents can range from declarative queries to sub-natural language queries. 1) A set of agents monitoring an object are asked to build activity profiles based on exchanging elementary observations; 2) A set of car drivers form a line, where every car is following its predecessor. An unsafe distance cm create a strong wave in the line. Individual agents are asked to incorporate and apply directions how to avoid the wave. 3) A set of micro-vehicles form a grid and are asked to propagate information and concepts to a central server.

Smid, Jan↗
NASA NTRS2003

NASA Virtual Glovebox (VBX): Emerging Simulation Technology for Space Station Experiment Design, Development, Training and Troubleshooting

The International Space Station demonstrates the greatest capabilities of human ingenuity, international cooperation and technology development. The complexity of this space structure is unprecedented; and training astronaut crews to maintain all its systems, as well as perform a multitude of research experiments, requires the most advanced training tools and techniques. Computer simulation and virtual environments are currently used by astronauts to train for robotic arm manipulations and extravehicular activities; but now, with the latest computer technologies and recent successes in areas of medical simulation, the capability exists to train astronauts for more hands-on research tasks using immersive virtual environments. We have developed a new technology, the Virtual Glovebox (VGX), for simulation of experimental tasks that astronauts will perform aboard the Space Station. The VGX may also be used by crew support teams for design of experiments, testing equipment integration capability and optimizing the procedures astronauts will use. This is done through the 3D, desk-top sized, reach-in virtual environment that can simulate the microgravity environment in space. Additional features of the VGX allow for networking multiple users over the internet and operation of tele-robotic devices through an intuitive user interface. Although the system was developed for astronaut training and assisting support crews, Earth-bound applications, many emphasizing homeland security, have also been identified. Examples include training experts to handle hazardous biological and/or chemical agents in a safe simulation, operation of tele-robotic systems for assessing and diffusing threats such as bombs, and providing remote medical assistance to field personnel through a collaborative virtual environment. Thus, the emerging VGX simulation technology, while developed for space- based applications, can serve a dual use facilitating homeland security here on Earth.

Smith, Jeffrey D.↗
NASA NTRSDate not supplied

Data Augmentation for Intelligent Contingency Management Using Generative Adversarial Neural Networks

Artificial intelligence (AI)-based techniques for intelligent contingency management (ICM) require that intelligent agents learn various aspects of system dynamics to create and execute contingencies. For high assurance contingency management, agents achieve the most compelling results through supervised or semi-supervised machine learning, for which agents require large datasets to learn the dynamics of the system. Unfortunately, data collection in aerospace applications can be costly, due to both time and resources. Presented work describes a framework for data augmentation of ICM databases containing training data for machine learning models. This framework populates the database with the outputs of generative adversarial network (GAN) models that were trained on flight data. Methods for evaluating the suitability of these models based on the equations of motion, as well as other physical constraints, are discussed. The paper demonstrates the utility of this database for training intelligent agents on the NASA T2 generic transport aircraft model and experimental vertical takeoff and landing (VTOL) simulation model.

Generative Machine Learning↗
NASA NTRSDate not supplied

Data Augmentation for Intelligent Contingency Management Using Generative Adversarial Neural Networks

Artificial intelligence (AI)-based techniques for intelligent contingency management (ICM) require that intelligent agents learn various aspects of system dynamics to create and execute contingencies. For high assurance contingency management, agents achieve the most compelling results through supervised or semi-supervised machine learning, for which agents require large datasets to learn the dynamics of the system. Unfortunately, data collection in aerospace applications can be costly, due to both time and resources. Presented work describes a framework for data augmentation of ICM databases containing training data for machine learning models. This framework populates the database with the outputs of generative adversarial network (GAN) models that were trained on flight data. Methods for evaluating the suitability of these models based on the equations of motion, as well as other physical constraints, are discussed. The paper demonstrates the utility of this database for training intelligent agents on the NASA T2 generic transport aircraft model and experimental vertical takeoff and landing (VTOL) simulation model.

Generative Machine Learning↗
NASA NTRS2019

Efficacy of Climate Forcings in PDRMIP Models

Quantifying the efficacy of different climate forcings is important for understanding the real‐world climate sensitivity. This study presents a systematic multimodel analysis of different climate driver efficacies using simulations from the Precipitation Driver and Response Model Intercomparison Project (PDRMIP). Efficacies calculated from instantaneous radiative forcing deviate considerably from unity across forcing agents and models. Effective radiative forcing (ERF) is a better predictor of global mean near‐surface air temperature (GSAT) change. Efficacies are closest to one when ERF is computed using fixed sea surface temperature experiments and adjusted for land surface temperature changes using radiative kernels. Multimodel mean efficacies based on ERF are close to one for global perturbations of methane, sulfate, black carbon, and insolation, but there is notable intermodel spread. We do not find robust evidence that the geographic location of sulfate aerosol affects its efficacy. GSAT is found to respond more slowly to aerosol forcing than CO2 in the early stages of simulations. Despite these differences, we find that there is no evidence for an efficacy effect on historical GSAT trend estimates based on simulations with an impulse response model, nor on the resulting estimates of climate sensitivity derived from the historical period. However, the considerable intermodel spread in the computed efficacies means that we cannot rule out an efficacy‐induced bias of ±0.4 K in equilibrium climate sensitivity to CO2 doubling when estimated using the historical GSAT trend.

T B Richardson↗
NASA NTRS2003

Experimental And Numerical Evaluation Of Gaseous Agents For Suppressing Cup-Burner Flames In Low Gravity

Longer duration missions to the moon, to Mars, and on the International Space Station (ISS) increase the likelihood of accidental fires. NASA's fire safety program for human-crewed space flight is based largely on removing ignition sources and controlling the flammability of the material on-board. There is ongoing research to improve the flammability characterization of materials in low gravity; however, very little research has been conducted on fire suppression in the low-gravity environment. Although the existing suppression systems aboard the Space Shuttle (halon 1301, CF3Br) and the ISS (CO2 or water-based form) may continue to be used, alternative effective agents or techniques are desirable for long-duration missions. The goal of the present investigation is to: (1) understand the physical and chemical processes of fire suppression in various gravity and O2 levels simulating spacecraft, Mars, and moon missions; (2) provide rigorous testing of analytical models, which include detailed combustion-suppression chemistry and radiation sub-models, so that the model can be used to interpret (and predict) the suppression behavior in low gravity; and (3) provide basic research results useful for advances in space fire safety technology, including new fire-extinguishing agents and approaches.

Takahashi, Fumiaki↗
NASA NTRS2024

An On-Board Off-Board Framework for Online Replanning: Applied to UAVs in Urban Environments

Autonomous systems are being used in a multitude of areas at an increasing rate and require a high level of adaptivity and intelligence to operate safely, especially under faulty conditions. This paper introduces a novel genetic algorithm tailored for UAV trajectory replanning, with an improved execution time via search space reduction based on the operating conditions of the UAV and its remaining mission. A unique characteristic of the replanning agent is its fast-start and adaptive properties, pre-seeding candidates with partial solutions and dynamically tuning elitism, crossover, and mutation rates in correspondence to the average fitness and diversity of the population. A population restart mechanism and early stopping mechanism are evaluated as well to assess their effect on solution quality and runtime. Previous work on genetic algorithms for UAV replanning were conducted with short trajectories in a small state space. Our UAV operates in a 56,000 square meter simulated urban environment, with static obstacles and a total of 53 possible waypoints. The agent increases the safety and reliability of UAV autonomy when operating under faulty conditions and when replanning is required.

Machine Learning↗
NASA NTRS2005

NASA Tech Briefs, September 2005

Topivs include: Diamond-Coated Carbon Nanotubes for Efficient Field Emission; Improved Anode Coatings for Direct Methanol Fuel Cells; Advanced Ablative Insulators and Methods of Making Them; PETIs as High-Temperature Resin-Transfer-Molding Materials; Stable Polyimides for Terrestrial and Space Uses; Low-Density, Aerogel-Filled Thermal-Insulation Tiles; High-Performance Polymers Having Low Melt Viscosities; Nonflammable, Hydrophobic Aerogel Composites for Insulation; Front-Side Microstrip Line Feeding a Raised Antenna Patch; Medium-Frequency Pseudonoise Georadar; Facilitating Navigation Through Large Archives; Program for Weibull Analysis of Fatigue Data; Comprehensive Micromechanics-Analysis Code - Version 4.0; Component-Based Visualization System; Software for Engineering Simulations of a Spacecraft; LabVIEW Interface for PCI-SpaceWire Interface Card; Path Following with Slip Compensation for a Mars Rover; International Space Station Electric Power System Performance Code-SPACE; Software for Automation of Real-Time Agents, Version 2; Software for Optimizing Plans Involving Interdependent Goals; Computing Gravitational Fields of Finite-Sized Bodies; Custom Sky-Image Mosaics from NASA's Information Power Grid; ANTLR Tree Grammar Generator and Extensions; Generic Kalman Filter Software; Alignment Stage for a Cryogenic Dilatometer; Rugged Iris Mechanism; Treatments To Produce Stabilized Aluminum Mirrors for Cryogenic Uses; Making AlNx Tunnel Barriers Using a Low-Energy Nitrogen-Ion Beam; Making Wide-IF SIS Mixers with Suspended Metal-Beam Leads; Sol-Gel Glass Holographic Light-Shaping Diffusers; Automated Counting of Particles To Quantify Cleanliness; Phase Correction for GPS Antenna with Nonunique Phase Center; Compact Infrasonic Windscreen; Broadband External-Cavity Diode Laser; High-Efficiency Solar Cells Using Photonic-Bandgap Materials; Generating Solid Models from Topographical Data; Computationally Lightweight Air-Traffic-Control Simulation; Spool Valve for Switching Air Flows Between Two Beds; Partial Model of Insulator/ Insulator Contact Charging; Asymmetric Electrostatic Radiation Shielding for Spacecraft; and Reusable Hybrid Propellant Modules for Outer-Space Transport.

Source record↗
NASA NTRS2024

Aera-Mip: Emission Pathways, Remaining Budgets, and Carbon Cycle Dynamics Compatible With 1.5 and 2 °C Global Warming Stabilization

While international climate policies now focus on limiting global warming to well below 2 °C or pursuing a 1.5 °C level of global warming, the climate modelling community has not provided an experimental design in which all Earth system models (ESMs) converge and stabilize at the same prescribed global warming levels. This gap hampers accurate estimations based on comprehensive ESMs of the carbon emission pathways and budgets needed to meet such agreed warming levels and of the associated climate impacts under temperature stabilization. Here, we apply the Adaptive Emission Reduction Approach (AERA) with ESMs to provide such simulations in which all models converge at 1.5 and 2.0 °C warming levels by adjusting their emissions over time. These emission-driven simulations provide a wide range of emission pathways and resulting atmospheric CO 2 projections for a given warming level, uncovering uncertainty ranges that were previously missing in the traditional Coupled Model Intercomparison Project (CMIP) scenarios with prescribed greenhouse gas concentration pathways. Meeting the 1.5 °C warming level requires a 40 % (full model range: 7 % to 76 %) reduction in multi-model mean CO 2 -forcing-equivalent (CO 2 -fe) emissions from 2025 to 2030, a 98 % (57 % to 127 %) reduction from 2025 to 2050, and a stabilization at 1.0 (−1.7 to 2.9) PgC yr−1 from 2100 onward after the 1.5 °C global warming level is reached. Meeting the 2.0 °C warming level requires a 47 % (8 % to 92 %) reduction in multi-model mean CO 2 -fe emissions until 2050 and a stabilization at 1.7 (−1.5 to 2.7) PgC yr−1 from 2100 onward. The on-average positive emissions under stabilized global temperatures are the result of a decreasing transient climate response to cumulative CO 2 -fe emissions over time under stabilized global warming. This evolution is consistent with a slightly negative zero emissions commitment – initially assumed to be zero – and leads to an increase in the post-2025 CO 2 -fe emission budget by a factor of 2.2 (−0.8 to 6.9) by 2150 for the 1.5 °C warming level and a factor of 1.4 (0.9 to 2.4) for the 2.0 °C warming level compared to its first estimate in 2025. The median CO 2 -only carbon budget by 2150, relative to 2020, is 800 GtCO 2 for the 1.5 °C warming level and 2250 GtCO 2 for the 2.0 °C warming level. These median values exceed the median IPCC AR6 estimates by 60 % for the 1.5 °C warming level and 67 % for 2.0 °C. Some of the differences may be explained by the choice of the mitigation scenario for non-CO 2 radiative agents. Our simulations highlight shifts in carbon uptake dynamics under stabilized temperature, such as a cessation of the carbon sinks in the North Atlantic and in tropical forests. On the other hand, the Southern Ocean remains a carbon sink centuries after temperatures stabilize. Overall, this new type of warming-level-based emission-driven simulation offers a more coherent assessment across climate models and opens up a wide range of possibilities for studying both the carbon cycle and climate impacts, such as extreme events, under climate stabilization.

Adaptive Emission Reduction Approach↗
NASA NTRS2013

Autonomous Formations of Multi-Agent Systems

Autonomous formation control of multi-agent dynamic systems has a number of applications that include ground-based and aerial robots and satellite formations. For air vehicles, formation flight ("flocking") has the potential to significantly increase airspace utilization as well as fuel efficiency. This presentation addresses two main problems in multi-agent formations: optimal role assignment to minimize the total cost (e.g., combined distance traveled by all agents); and maintaining formation geometry during flock motion. The Kuhn-Munkres ("Hungarian") algorithm is used for optimal assignment, and consensus-based leader-follower type control architecture is used to maintain formation shape despite the leader s independent movements. The methods are demonstrated by animated simulations.

Dhali, Sanjana↗