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At least 181 records · Page 10

Textile Material Lessons Learned During the Design and Qualification of the NASA Orion Capsule Parachute Assembly System

The NASA Orion Capsule Parachute Assembly System (CPAS) development and qualification testing was completed in September 2018. Over the course of the airdrop and ground testing campaign, the team benefited from the ability to design, test, and adjust the design based on observations and inspection results. While the design team used the best knowledge available and attempted to utilize best practices, a number of lessons were learned that should be documented for consideration by future designers. This paper describes these lessons learned including the use of textile reefing loops, the importance of performing joint tests, the impact of using bight sleeves on parachute deployment, and a surprising number of design changes required in the CPAS system after the decision was made to change the suspension line braid to save system mass.

Anderson, Brian P.↗

Aerodynamic Performance of the 2018 InSight Mars Lander

InSight touched down in Elysium Planitia on 26 November 2018, becoming NASA’s eighth successful entry, descent, and landing (EDL) at Mars. InSight was a build-to-print of the successful 2008 Phoenix EDL system, flying a non-spinning, ballistic trajectory with a 70-degree sphere-cone aeroshell (2.65-meter diameter), disk-gap-band parachute, and pulsed terminal descent and landing engines. This work discusses entry aerodynamic performance for InSight up to parachute deployment, including pre-flight aerodynamics predictions and comparisons with post-flight reconstruction, as well as comparisons with the Phoenix reconstruction.

Ashley M Korzun↗

InSight Entry, Descent and Landing Post-Flight Performance Assessment

On November 26, 2018, the Interior Exploration using Seismic Investigations, Geodesy and Heat Transport (InSight) lander successfully touched down on the surface of Mars. NASA Langley Research Center’s (LaRC) Program to Optimize Simulated Trajectories II (POST2) was used during both project development and flight operations to assess the Entry, Descent and Landing (EDL) vehicle performance against related requirements across the expected range of possible environmental and spacecraft conditions. During flight operations, these analyses were used to evaluate the need for updating flight software EDL parameters and the effects of executing a trajectory correction maneuver (TCM). Therefore, the NASA LaRC POST2 simulation had a critical role during the cruise, approach and ultimately EDL phases of the mission. This paper presents results of the final pre- and post- EDL flight performance assessments. A summary of the reconstructed “as-flown” trajectory with a comparison of key EDL metrics to the nominal and three-sigma bound pre-EDL predictions is also provided. Emphasis on the trajectory between entry interface and parachute deploy is provided as this is the period during which deviations from the pre-EDL prediction occurred. The post-flight assessment provides important verification of the POST2 models and analysis techniques used for InSight, providing critical feed-forward information to future lander missions.

Robert W Maddock↗

Mars InSight Entry, Descent, and Landing Trajectory and Atmosphere Reconstruction

The InSight mission landed on the surface of Mars on November 26th, 2018. The InSight system performance met all design requirements, although several performance metrics fell near the boundaries of the predictions. The peak deceleration was high, the overall timeline was short, and the landing site was uprange and crossrange from the target. This paper describes the reconstruction of the entry, descent, and landing trajectory and atmosphere. The approach utilizes a Kalman filter to blend sensor data to obtain the vehicle trajectory. The aerodynamic database is used in combination with the sensed accelerations to obtain estimates of the atmosphere-relative state, which in turn is used to derive the free-stream atmospheric conditions during entry, until the time of parachute deployment. The results indicate that the reconstructed atmosphere was approximately 1σbelow the preflight atmosphere. Analysis of the reconstructed vehicle attitude angles indicate that the aerodynamic lift was oriented downward at entry. The vehicle developed a roll rate during entry, which directed a component of the lift to the north. The low density and aerodynamic lift direction are determined to be the primary causes of the high deceleration, short timeline, and location of the landing site relative to the target.

Christopher D Karlgaard↗

Mars InSight Entry, Descent, and Landing Trajectory and Atmosphere Reconstruction

The InSight mission landed on the surface of Mars on November 26th, 2018. The InSight system performance met all design requirements, although several performance metrics fell near the boundaries of the predictions. The peak deceleration was high, the overall timeline was short, and the landing site was uprange and crossrange from the target. This paper describes the reconstruction of the entry, descent, and landing trajectory and atmosphere. The approach utilizes a Kalman filter to blend sensor data to obtain the vehicle trajectory. The aerodynamic database is used in combination with the sensed accelerations to obtain estimates of the atmosphere-relative state, which in turn is used to derive the free-stream atmospheric conditions during entry, until the time of parachute deployment. The results indicate that the reconstructed atmosphere was approximately 1σbelow the preflight atmosphere. Analysis of the reconstructed vehicle attitude angles indicate that the aerodynamic lift was oriented downward at entry. The vehicle developed a roll rate during entry, which directed a component of the lift to the north. The low density and aerodynamic lift direction are determined to be the primary causes of the high deceleration, short timeline, and location of the landing site relative to the target.

Christopher D Karlgaard↗

Mars Sample Return, Sample Retrieval Lander, Reaction Control System Jet Interaction Supersonic Wind Tunnel Test Overview with CFD Predictions

NASA's Mars Sample Return campaign will be launching several missions over the next decade that will work together to return rock samples from Mars back to Earth. The Sample Retrieval Lander (SRL) will deliver the Mars Ascent Vehicle and fetch rover to the surface of Mars in 2006. Rock samples collected by the Mars 2020 Perseverance rover, landing in early 2021, will be loaded on the the ascent vehicle to be launched into Mars orbit for retrieval by yet another spacecraft. The Sample Retrieval Lander will be a blunt entry capsule similar to past Mars entry vehicles like Mars Science Laboratory and Viking. The vehicle will fly a guided entry, using a small lift vector produced by a non-zero trim angle of attack to eliminate downrange and crossrange position errors at the point of parachute deploy. This energy and heading management is achieved with a reaction control system (RCS) that directs the bank angle of the vehicle and also minimizes unwanted capsule dynamics. The reaction control system and control design is based on the Mars Science Laboratory and Mars 2020 RCS systems. However, due to packaging constraints, the backshell of this new entry vehicle has a different geometry than those earlier designs. To certify the RCS system for flight the project must characterize the jet plume interactions with the capsule backshell that could impair or significantly augment the RCS control authority. This characterization will be done through a combination of computational fluid dynamics (CFD) analysis and wind tunnel test. Two candidate arrangements of the RCS jets have been identified for the SRL vehicle and are currently under evaluation before final selection. The aero/RCS plume interactions of these candidate configurations have been measured in a supersonic wind tunnel test in NASA Langley's Unitary Plan Wind Tunnel. The test was conducted in the fall of 2020 and data is currently being reduced. An overview of the candidate RCS configurations are presented here with an overview of the wind tunnel model design, jet scaling and scaled nozzle design, and the test matrix. Preliminary CFD runs are presented with an assessment of the predicted plumes and their interaction with the wake flow of the vehicle. The predicted effects of the model sting is provided as well. This high fidelity wind tunnel test is being conducted much earlier in the SRL project than would normally be done. The test was funded as part of a CFD evaluation task funded by NASA's Aerosciences Evaluation and Test Capabilities Project. The objective of the evaluation task was to compare the ability of CFD to predict complex flows with data that can be measured in the Langley Unitary Plan Wind Tunnel. RCS Jet interactions were selected as a type of complex flow that is important to NASA missions. In addition to providing useful data to the SRL project, there was added emphasis on quantifying the accuracy of the CFD predictions and wind tunnel test data. An overview of the uncertainty quantification methodologies for computational and experimental portions of this test is presented.

blunt body↗

Interpretation of Vehicle Tumbling Predictions from 6-DOF Entry and Descent Simulation

Blunt body entry vehicles are subject to dynamic instability during terminal descent. This often manifests as limit cycle oscillations in total angle of attack, but can diverge into tumbling behavior under certain conditions. For the Mars Sample Return Earth Entry Vehicle (MSR EEV), there is a constraint on the orientation of the sample tubes so backward impact is impermissible. Past missions have chosen to deploy parachutes to preclude tumbling, but active events after release of MSR EEV have been ground-ruled out with the intent to maximize system reliability. Prevention of tumbling during subsonic descent is a design driver for MSR EEV. During preliminary design of the MSR EEV, six degree-of-freedom numerical simulations indicated an unacceptably high probability of tumbling for a 60degree sphere-cone forebody geometry, which necessitated a design change. Decreasing the forebody angle was expected to improve dynamic stability, but would also adversely impact mass, aeroheating, and manufacturing risk. Hence there was strong motivation to understand the physical drivers for the onset of tumbling, and to determine: (a) whether the causes of tumbling are representative of physically realizable vehicle entry configurations and (b) what changes can be made to existing design and analysis practices to ensure a stable vehicle.

Entry Vehicle↗

Predicting Unreinforced Fabric Mechanical Behavior with Recurrent Neural Networks

Unreinforced woven fabrics are widely employed in various high-performance applications, including parachute deployment systems, airbags, and ballistic armor. The analysis of such materials is inherently complex due to the multiscale structure of these materials, and the dependence of macroscale behavior on changes that occur at lower scales. Previously, NASA’s Multiscale Analysis Tool (NASMAT) showed its capability in predicting unreinforced fabric behavior at the macroscale by capturing finite rotations that occur at the mesoscale. Though effective, the tool can face high computational cost for large, complex problems, motivating the need for the development of a surrogate model that can capture the same behavior. A recurrent neural network (RNN) was developed and trained on virtual NASMAT data to mimic the physics-based solutions while improving the computational runtime. The architecture of the RNN to best simulate the fabric behavior was carefully crafted based on heuristic knowledge of predicting physics-based temporal data, manual hyperparameter case studies, and Hyperband optimization.. The resultant model was able to predict a variety of stress-strain curves for fabrics with different mesoscale geometries, and was further validated by comparing to experimental data for the K706 style Kevlar plain-weave fabric, demonstrating the ability of the model to effectively capture the geometric changes in the fabric without explicitly calculating them, as is done in NASMAT. Furthermore, the tool showed its ability to improve on the runtime by a factor of 10 for fabric solutions compared to the multiscale tool, which would further enable the simulation of complex loading scenarios on unreinforced fabrics.

Fabric↗

Mars 2020 Perseverance Entry Controller Design and Flight Reconstruction

On February 18th, 2021, NASA landed Perseverance on the Jezero crater (on Mars). An entry Guidance, Navigation, and Control (GNC) system delivered the vehicle to the desired landing ellipse. The navigation filter propagated position and attitude states initialized from cruise using Inertial Measurement Unit (IMU) measurements. The entry guidance modulated the lift vector through bank commands to reach the parachute deploy conditions. The entry controller commanded the propulsive Reaction Control System (RCS) to track the bank commands while doing rate damping on angle-of-attack and sideslip. This paper describes the design and the as-flown performance of the entry controller.

Way, David W.↗

Mars 2020 Entry, Descent, and Landing as Observed by Radio Science Techniques at UHF and X-band Frequencies

On February 18, 2021, NASA’s Mars 2020 Rover “Perseverance” conducted the Entry, Descent, and Landing (EDL) sequence to land safely on the surface of Mars at Jezero Crater. Due to the criticality of the EDL sequence, redundant telecommunications strategies were utilized to observe the event. The rover transmitted an X-band (8.4 GHz) signal encoded with Multiple Frequency Shift Keying tones direct-to-Earth and a UHF (401.5 MHz) signal with 8 kbps encoded telemetry to orbiters, which relayed the signal back to Earth at X-band. NASA’s Deep Space Network complexes at Goldstone, California and Madrid, Spain received the direct-to-Earth X-band signal. Two of the largest steerable radio telescopes in the world, the 100-meter Green Bank Telescope in West Virginia and the Effelsberg 100-meter Radio Telescope in Germany also received the UHF carrier signal on Earth. During the EDL event, full spectrum recordings of the UHF and X-band signals were captured at these observatories using open-loop receivers. Real-time and post-processing signal analysis to determine power level, received frequency, and Doppler shift allowed determination of the spacecraft’s health and safety state during critical sequences, including event detection such as atmospheric entry, plasma blackouts, and parachute deployment. Simultaneous observations of the UHF and X-band radio links allow for redundancy in the event of a fault and greatly improve confidence in Doppler event detection.

Gladden, Roy↗

AERACEPT (Aerosol Rapid Analysis Combined Entry Probe/sonde Technology): Enabling Technology for Missions to the Venus Clouds

AERACEPT (AErosol Rapid Analysis Combined Entry Probe/sonde Technology) is an early-stage technology allowing a single aeroshell body to act as both an entry vehicle and aerosol-sampling passive descent sonde. AERACEPT does not require heat shield separation, deployable parachutes, or descent control, thus reducing the mass, volume, and complexity of planetary aerosol sampling. AERACEPT is particularly well suited for a Venus mission, where the particles of greatest interest are within the subsonic descent regime. AERACEPT uses the aeroshell’s own velocity to drive aerosol capture and separation through a series of embedded inlets. It takes advantage of recently developed thermal protection materials (3D-CC and HEEET) in combination with heritage aerosol sampling technologies from both planetary and airborne science (high-speed inlets and particle separation). The trade space for a given descent trajectory includes the particle capture efficiency for a given size, the bias introduced in the sampled particle size and concentration distributions, and the thermal alteration experienced by the particles during their brief exposure to the internal flow environment. AERACEPT is included in the Nephele mission concept study for a small spacecraft targeting the Venus middle and lower cloud layers. Nephele complements larger missions targeting Venus atmospheric gas analysis, such as DAVINCI and Venera-D, by specifically targeting cloud and haze particles. Because of the short lifetime of the probe in the lower atmosphere, Nephele requires a fast cadence of analysis of the captured particles, and includes the VOLTR dual optical spectrometer (SERS/LIBS) as part of its notional payload. Preliminary modeling based on the Nephele trajectory at 63 km to 39 km indicates AERACEPT can limit sample heating to 30-60 K above ambient. A modified particle tracking model has been implemented to estimate capture efficiency of particles larger than 0.1 µm and total sample volume as part of an inlet and interal flow path geometry trade study. Further modeling and empirical testing is underway to improve these estimates.

AERACEPT↗

Low-Earth Orbit Flight Test of an Inflatable Decelerator (LOFTID) Mission Overview and Science Return

The Low-Earth Orbit (LEO) Flight Test of an Inflatable Decelerator (LOFTID) mission was the culmination of two decades of research and development led by NASA Langley Research Center for Hypersonic Inflatable Aerodynamic Decelerator (HIAD) technology. A HIAD aeroshell can be hard packed into a small volume for launch and then deployed prior to atmospheric entry for stable aerodynamic deceleration through the atmosphere. Much larger than traditional fixed diameter aeroshells that are constrained by the size of launch vehicle shrouds, inflatable decelerators create more drag and start the deceleration process in the upper reaches of the atmosphere with greater efficiency and stability. Large deployable heatshields enable spacecraft to carry bigger, heavier payloads, including scientific instruments and human support systems for planetary landing and exploration. The success of the LOFTID mission could enable new NASA missions to Mars (including access to higher altitudes than currently possible), Venus, and most solar system destinations with atmospheres, as well as cost-effective payload returns to Earth. With its unique 6m diameter inflatable heatshield, LOFTID was the first-of-a-kind orbital reentry flight, and the largest blunt body atmospheric entry of any kind. On November 10, 2022, the LOFTID aeroshell endured the harsh environments of atmospheric reentry while exhibiting stable aerodynamics through the entire spectrum of hypersonic, supersonic, transonic, and subsonic flight. The demonstration confirmed the HIAD technology structural and thermal performance as the aeroshell protected the 1100 kg Reentry Vehicle (RV) entering Earth’s atmosphere at 8 km/s, reaching Mach 30, and experiencing 9.5 g deceleration before deploying parachutes and gently splashing down in the Pacific Ocean, where it was recovered in excellent condition. The LOFTID aeroshell was exposed to an aeroheating environment representative of many Mars and LEO HIAD applications, and successfully demonstrated the ability of the heat-affected inflatable structure to withstand aerodynamic forces that exceeded those expected at Mars. This mission further demonstrated the viability of HIAD technology to deliver large payloads safely and accurately through an atmosphere via a controlled entry, descent, and landing. Whereas LOFTID was indeed a first-of-a-kind flight for an inflatable heatshield, its remarkable performance assured that it will not be the last of its kind.

J M Dinonno↗

Low-Earth Orbit Flight Test of an Inflatable Decelerator (LOFTID) Mission Overview and Science Return

The Low-Earth Orbit (LEO) Flight Test of an Inflatable Decelerator (LOFTID) mission was the culmination of two decades of research and development led by NASA Langley Research Center for Hypersonic Inflatable Aerodynamic Decelerator (HIAD) technology. A HIAD aeroshell can be hard packed into a small volume for launch and then deployed prior to atmospheric entry for stable aerodynamic deceleration through the atmosphere. Much larger than traditional fixed diameter aeroshells that are constrained by the size of launch vehicle shrouds, inflatable decelerators create more drag and start the deceleration process in the upper reaches of the atmosphere with greater efficiency and stability. Large deployable heatshields enable spacecraft to carry bigger, heavier payloads, including scientific instruments and human support systems for planetary landing and exploration. The success of the LOFTID mission could enable new NASA missions to Mars (including access to higher altitudes than currently possible), Venus, and most solar system destinations with atmospheres, as well as cost-effective payload returns to Earth. With its unique 6m diameter inflatable heatshield, LOFTID was the first-of-a-kind orbital reentry flight, and the largest blunt body atmospheric entry of any kind. On November 10, 2022, the LOFTID aeroshell endured the harsh environments of atmospheric reentry while exhibiting stable aerodynamics through the entire spectrum of hypersonic, supersonic, transonic, and subsonic flight. The demonstration confirmed the HIAD technology structural and thermal performance as the aeroshell protected the 1100 kg Reentry Vehicle (RV) entering Earth’s atmosphere at 8 km/s, reaching Mach 30, and experiencing 9.5 g deceleration before deploying parachutes and gently splashing down in the Pacific Ocean, where it was recovered in excellent condition. The LOFTID aeroshell was exposed to an aeroheating environment representative of many Mars and LEO HIAD applications, and successfully demonstrated the ability of the heat-affected inflatable structure to withstand aerodynamic forces that exceeded those expected at Mars. This mission further demonstrated the viability of HIAD technology to deliver large payloads safely and accurately through an atmosphere via a controlled entry, descent, and landing. Whereas LOFTID was indeed a first-of-a-kind flight for an inflatable heatshield, its remarkable performance assured that it will not be the last of its kind.

John M DiNonno↗

Development of an Ejectable Data Recorder Ejection Mechanism for the Low Earth Orbit Flight Test of an Inflatable Decelerator

On November 10, 2022, the 1100kg (2,425 lbs.) LOFTID Reentry Vehicle (RV) was launched on a United Launch Alliance Atlas V as a secondary payload with the Joint Polar Surveyor System-2. The 6-meter diameter (~20 ft.) aeroshell (a type of heat shield) entered the atmosphere at 8 kilometers per second (18,000 miles per hour), and flew nominally, enduring the intended heat pulse that saw temperatures exceeding 1371˚C (2500˚F) on the front side while the payload skin remained only about 38˚C (100˚F). The RV exceeded Mach 30 and the heat-affected aeroshell withstood a pressure pulse that exerted 9g’s deceleration maintaining stable flight through the hypersonic, supersonic, transonic, and subsonic regimes to the parachute deployment. As part of the Agency’s strategic goal “to extend human presence deeper into space and to the moon for sustainable long-term exploration and utilization”, the LOFTID inflatable aerodynamic decelerator or aeroshell technology could one day help land humans on Mars. As with any flight test, data collection is of utmost importance. Without a data downlink and a possibility of the RV sinking before the recovery crew got to it, a secondary data collection method was introduced. The RV would eject a data recorder, which would have a duplicate copy of the on-board flight date, before splashdown and be retrieved separately. This paper discusses the development of the ejection mechanism used to eject the data recorder from the RV during the test flight. The development includes discussions of design constraints, a design overview, the testing program, and lessons learned throughout the process all the way through successful data recorder recovery.

mechanism↗

Analysis of a Landing System for Planetary Payloads Utilizing Passive Energy Absorbing Composite Structure

Delivery of a payload from space to a planetary surface currently requires the development of an application specific landing system to protect the payload from forces imparted during impact with the planet surface. Often, active energy attenuating systems such as retro-rockets, deployable parachutes, and airbags are utilized within these landing systems to reduce landing impact energy. Unfortunately, these active systems come at a cost; active energy attenuating systems are susceptible to system faults which may limit or completely negate their energy attenuating capability. Additionally, components needing to be stowed such as fuel, parachutes, and airbags increase design complexity, cost, and weight. To overcome these limitations, this study examines the potential of passive energy attenuation through energy absorbing structural design and composite materials to mitigate landing loads for small payload planetary delivery. Researchers at the National Aeronautics and Space Administration (NASA) Langley Research Center (LaRC) have conducted extensive research into developing energy absorbing structures and components for the attenuation of impact energy under various loading conditions including aircraft crash and spacecraft impact. The current study leverages this research to design a lightweight planetary delivery system which utilizes unique outer mold line (OML) geometry and passive energy absorbing structural design to limit landing loads across potential planetary surface environments. The OML geometry is designed to control impact orientation and provide self-righting capabilities for slopped impact surfaces. The internal structure is composed of composite material structures arranged to provide energy absorption which is robust to impact angle and impact velocity. The developed planetary delivery design concept will be evaluated using finite element (FE) model analysis. Simulations of landing impacts with representative soil surface environments will be used to characterize the energy absorbing capabilities of the landing system. Sensitivity of predicted impact force to landing environment, impact angle, and impact velocity will be assessed to identify capabilities and limitations of the initial structural design. Results will be used to determine the feasibility of a lightweight composite structure to passively absorb landing energy for robust planetary payload delivery.

Crashworthiness↗

AERACEPT (Aerosol Rapid Analysis Combined Entry Probe/sonde Technology): Enabling Technology for Planetary Atmospheric Science Probes

Aerosols –clouds, hazes, and dusts –are a key part of planetary mass and energy balance, but difficult to study remotely. Current technology limits in situ measurements to once-in-a-lifetime flagship missions, which is not enough to characterize such highly dynamic systems. AERACEPT (AErosol Rapid Analysis Combined Entry Probe/sonde Technology) is an early-stage technology allowing asingle aeroshell body toact as both anentry vehicle and aerosol-samplingpassivedescent sonde, using the aeroshell’sown velocityto drive aerosol capture and separationthrough a series of embedded inlets.It takes advantage ofrecently developed thermal protection materials (3D-CC and 3MDCP)in combination withheritage aerosol sampling technologies fromboth planetary and airborne science (high-speed inlets and particle separation). By eliminating the need for heat shield separation, deployable parachutes, ordescent control, AERACEPT reducesthe mass, volume, and complexity ofplanetary aerosol sampling. Verifying AERACEPT’s performance involves modeling interdependencies between the size and geometry of the sampling inlets, the material response of the hypersonic phase of entry, the thermal conditions throughout the probe’s descent, the subsonic flow and particle sampling efficiency, the needed amount of sample, andthe cadence at which the sample can be analyzed. AERACEPT’s current predictions, using the Nephele Venus cloud mission concept (sampling between 63km to 39km, all subsonic) indicates AERACEPT can obtain >20 μL of particle material (roughly 10x the limit of detection) from droplets larger than 0.2 μm, with sample heating limited to <15 K above ambient. Validation testing for the thermal material response model has recently been performed at the UIUC Plasmatron facility, and for the flow and particle capture models is planned for this fall at the Ames Fluid Mechanics Laboratory.

AERACEPT↗

Entry Guidance Design and Post-Flight Performance of the Mars 2020 Mission

Like its predecessor Mars Science Laboratory, the Mars 2020 mission successfully utilized a derivative of the Apollo Entry Terminal Point Controller guidance algorithm, whereby bank angle controls range flown along a trajectory. The flight performance of this algorithm in conjunction with a range-trigger for parachute deploy delivered the Perseverance rover to 1.7 km from the expected touchdown location within an ellipse of 7.5 x 5.2 km. This miss distance is largely attributed to atmospheric and aerodynamic modeling uncertainties between the design and as-flown trajectory. This algorithm for guided entry continues to provide a solid basis for Mars missions to improve upon.

Guidance↗

Guidance Performance of Bank-Angle and Alpha-Beta Steering for A Large Robotic Mars Entry Vehicle

Entry performance using Apollo-based entry guidance is assessed with a future entry system for a large robotic payload. Both a traditional bank-angle steering system and an innovative alpha-beta steering concept are considered to compare flight performance. Results indicate alpha-beta steering provides an order of magnitude improvement in crossrange error and mean altitudes up to 1 km higher than bank-angle steering. Results also show alpha-beta steering provides downrange errors with a spread half of that of bank-angle steering. Lastly, alpha-beta steering is found to provide a narrower spread of terminal altitude, Mach number, and dynamic pressure, indicating safer parachute deploy conditions.

Daniel L Engel↗