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NASA NTRS2003

Lunar transfer trajectory design and the four-body problem

The existence of a ballistic trajectory from the Earth to orbit about the Moon was long considered to be impossible based on analysis of the three-body problem. In 1990 a ballistic trajectory from the Earth to lunar orbit was discovered while analyzing a plan to salvage the Muses A (Hiten) spacecraft. This trajectory utilized the Sun's gravity in conjunction with the Earth and Moon's gravity and was thus the first example of a practical four-body trajectory design. This paper presents a review of lunar transfer trajectories that go beyond three-body theory and the Jacobi integral. These include Hiten, Lunar A and the Genesis return trajectory from the vicinity of the Moon to Earth.It is shown that these trajectories may be analyzed by piecing together segments where three-body motion dominates.

Muses↗

NASA NTRS2017

UAV Trajectory Modeling Using Neural Networks

Massive small unmanned aerial vehicles are envisioned to operate in the near future. While there are lots of research problems need to be addressed before dense operations can happen, trajectory modeling remains as one of the keys to understand and develop policies, regulations, and requirements for safe and efficient unmanned aerial vehicle operations. The fidelity requirement of a small unmanned vehicle trajectory model is high because these vehicles are sensitive to winds due to their small size and low operational altitude. Both vehicle control systems and dynamic models are needed for trajectory modeling, which makes the modeling a great challenge, especially considering the fact that manufactures are not willing to share their control systems. This work proposed to use a neural network approach for modelling small unmanned vehicle's trajectory without knowing its control system and bypassing exhaustive efforts for aerodynamic parameter identification. As a proof of concept, instead of collecting data from flight tests, this work used the trajectory data generated by a mathematical vehicle model for training and testing the neural network. The results showed great promise because the trained neural network can predict 4D trajectories accurately, and prediction errors were less than 2:0 meters in both temporal and spatial dimensions.

Neural Networks↗

NASA NTRS2020

Simulation of a Representative Future Trajectory-Based Operations Environment

Trajectory-Based Operations in the National Airspace System is a key aspect of advanced air traffic management research. Trajectory-Based Operations focuses on modernizing the current operating paradigm to increase efficiency, predictability, resilience, and flexibility while migrating toward greater operational autonomy across the airspace. Research conducted at the National Aeronautics and Space Administration supports the transition from current airspace operations to Trajectory-Based Operations targeting a 2035-2045 implementation timeframe. Simulation scenarios that demonstrate a representative Trajectory-Based Operations environment in that timeframe, and enable the evaluation of advanced airborne tools such as strategic airborne trajectory management services, are necessary to support this research effort. This report describes characteristics and assumptions made about the future operating environment that were applied to a scenario development methodology to study a representative 2040 Trajectory-Based Operations environment in a simulation use case. This report also includes descriptions of the study design and analysis approach, discussion of the simulation results, and application of these scenarios to future research activities.

Trajectory Based Operations↗

NASA NTRSDate not supplied

Trajectory Planning for Mission Survivability of Autonomous Vehicles in Moderately to Extremely Uncertain Environments

Trajectory planning is a particularly challenging task for autonomous vehicles when the uncertainties in their operating environment cannot be characterized statistically. In this paper, we propose a receding horizon control strategy with novel trajectory planning policies that enable dynamic updating of the planned trajectories of autonomous vehicles operating in environments where potential conflicts are, from a statistical perspective, either partially known or completely unknown. The proposed policies utilize two metrics: (1) the number of feasible trajectories; and (2) the robustness of the feasible trajectories. We measure the effectiveness of the suggested policies in terms of mission survivability, which is defined as the probability that the primary mission is accomplished or, if that is not possible, the vehicle lands safely at an alternative site. We show that a linear combination of both metrics is an effective objective function when there is a mix of partially known and unknown uncertainties. When the operating environment is dominated by unknown disturbances, maximizing the number of feasible trajectories results in the highest mission survivability. These findings have significant implications for achieving safe aviation. autonomy.

Aviation autonomy↗

NASA NTRS2013

Trajectory Optimization Using Adjoint Method and Chebyshev Polynomial Approximation for Minimizing Fuel Consumption During Climb

This paper describes two methods of trajectory optimization to obtain an optimal trajectory of minimum-fuel- to-climb for an aircraft. The first method is based on the adjoint method, and the second method is based on a direct trajectory optimization method using a Chebyshev polynomial approximation and cubic spine approximation. The approximate optimal trajectory will be compared with the adjoint-based optimal trajectory which is considered as the true optimal solution of the trajectory optimization problem. The adjoint-based optimization problem leads to a singular optimal control solution which results in a bang-singular-bang optimal control.

Chebyshev Polynomial↗

NASA NTRS2018

Trajectory Specification Language for Air Traffic Control

Trajectory Specification is a method of specifying aircraft trajectories with tolerances such that the position at any instant in time is constrained to a precisely defined bounding space. The bounding space at any instant in time is defined by tolerances relative to a reference trajectory that specifies position as a function of time. The tolerances are dynamic and are based on the aircraft navigation capabilities and the traffic situation. This paper proposes a standard Trajectory Specification Language (TSL) based on the Extensible Markup Language (XML) to represent these specifications and to communicate them by datalink. The language can be used to downlink trajectory requests from air to ground and to uplink trajectory assignments from ground to air. The XML format can be converted to binary for operational use, if necessary, using Efficient XML Interchange (EXI) or Abstract Syntax Notation (ASN.1).

XM↗

NASA NTRS1995

Design of High-Accuracy Multiple Flyby Trajectories Using Constrained Optimization

The trajectory optimization technique described in this paper provides several distinct advantages over previous formulations. First, fully numerically integrated trajectory modeling is used. That is, no approximations to the trajectory are made and the inclusion of any level of complicated force models desired is allowed. Second, only trajectory propagation is used so there is no requirement for optimization. This is accomplished by the novel method of splitting the trajectory into independent legs, which are then subjected to constrained optimization. Third, each of the trajectory legs may be specified by any convenient set of parameters particularly useful for that leg. Any of these parameters may then be subject to constraints. Fourth, the nonlinear optimization problem is solved by solving a sequence of linear problems which converges to the optimal nonlinear solution. Fifth, the robustness of this formulation requires little or no user interaction with the optimization once a feasible problem has been posed.

flyby↗

NASA NTRS2022

Trajectory Specification Applied to Terminal Airspace

Despite major efforts to automate air traffic control (ATC), it is still performed by humans today. The complexity and safety-criticality of ATC makes it very difficult to safely automate, but it must be automated to increase airspace capacity (the density of traffic that can be safely managed) and airport throughput (the number of arrivals and departures that an airport can safely handle in a given period of time) beyond what is possible with human controllers. This paper presents the Trajectory Specification (TS) concept, which can help to safely automate ATC. TS is a method of specifying aircraft trajectories such that the position at any given time in flight is restricted to a precisely defined bounding space, removing all ambiguity as to where the flight is allowed to be. The bounding space or volume is determined by tolerances relative to a reference trajectory (position as a function of time). The tolerances are dynamic and are based on the aircraft navigation capabilities and the traffic situation. The tolerances can be a piecewise linear function of time or distance along the route, allowing the tolerances to vary as needed, typically increasing with time for departures and decreasing for arrivals. A Trajectory Specification Language (TSL) is proposed for communicating trajectories from aircraft to ATC as requests and from ATC to aircraft as assignments. The TS concept requires a new generation of airborne Flight Management Systems (FMS) that understand the TSL and can fly the assigned trajectories, but this paper focuses on the ATC functions and the prototype ATC algorithms and software that were developed to test the TS concept. Assuming conformance, TS can guarantee safe separation for an arbitrary length of time even in the event of an ATC system or communication outage. It can help to achieve the high level of safety and reliability needed for ATC automation, and it can also reduce the reliance on ATC backup systems for tactical conflict detection and resolution during normal operation. TS can be applied to any controlled airspace, including enroute, terminal, and urban airspace, but this paper presents algorithms and software for arrival spacing and conflict detection and resolution in the terminal airspace serving a major airport. In a fast-time simulation of a full day of traffic in a major terminal airspace, all conflicts were resolved in near real time, demonstrating the computational feasibility and the preliminary operational feasibility of the TS concept. This paper is a compilation of previous papers, and it adds significant information that was omitted from those papers due to length limitations. It also updates some of the results of those earlier papers due to algorithm refinements and corrections of minor software errors.

air traffic control, trajectory↗

NASA NTRSDate not supplied

Sensitivity of Optimal Midcourse Correction Scheduling for Robust Cislunar Trajectory Design

A new approach to optimal trajectory design is the determination of optimal trajectories that are robust to initial trajectory dispersions, navigation errors, maneuver execution errors, and environment modeling errors. This paper investigates the sensitivity of cislunar robust optimal trajectory design to launch date, duration of navigation measurement passes, and navigation measurement frequency. For a given cislunar trajectory from translunar injection (TLI) to lunar orbit insertion (LOI), the optimal locations of midcourse corrections, also known as trajectory correction maneuvers (TCM) are determined by minimizing the final 3-σ ∆v subject to a final 3-σ position dispersion constraint for a given launch date, specified measurement pass duration prior to each maneuver, and measurement frequency. Optical navigation (OpNav) is assumed, and OpNav field-of-view (FOV) and lighting constraints are employed. These constraints turn out to be important elements of the problem. The sensitivity of the optimal TCM locations are then investigated by varying the launch date, duration of OpNav measurement passes, and the OpNav measurement frequency, and then re-optimizing the locations of the TCMs. Given the problem parameters provided herein, results show that while the optimal TCM locations with respect to TLI vary greatly from one launch date to another, their locations with respect to LOI are nearly invariant over a 2-month launch window. Results also show that in all cases the optimal location of the last TCM is found to be at the point where the OpNav lunar FOV constraint is first violated. For all other TCMs, OpNav measurement pass duration and measurement frequency can have a moderate to large affect on the optimal TCM locations.

Linear Covariance Analysis↗

NASA NTRSDate not supplied

Artemis I Off-Nominal-Trajectory Design and Optimization

In order to achieve the Artemis I mission objectives to human-rate the Orion spacecraft even under off-nominal conditions, trajectories were developed to handle contingency scenarios: alternate missions for contingencies before or during the Trans-Lunar Injection burn, recoveries to return Orion to the nominal trajectory after a delayed or partial burn, and abort trajectories to return the spacecraft to Earth after a critical failure. In addition, disposal trajectories were generated for situations where recovery was not an option. Details are given on the trajectory characteristics, solution families, the generation of the trajectories, and operations considerations.

Artemis I↗

NASA NTRS2022

Trajectory Generation for Flexible-Joint Space Manipulators

Space manipulator arms often exhibit significant joint flexibility and limited motor torque. Future space missions, including satellite servicing and large structure assembly, may involve the manipulation of massive objects, which will accentuate these limitations. Currently, astronauts use visual feedback on-orbit to mitigate oscillations and trajectory following issues. Large time delays between orbit and Earth make ground teleoperation difficult in these conditions, so more autonomous operations must be considered to remove the astronaut resource requirement and expand robotic capabilities in space. Trajectory planning for autonomous systems must therefore be considered to prevent poor trajectory tracking performance. We provide a model-based trajectory generation methodology that incorporates constraints on joint speed, motor torque, and base actuation for flexible-joint space manipulators while minimizing total trajectory time. Full spatial computer simulation results, as well as physical experiment results with a single-joint robot on an air bearing table, show the efficacy of our methodology.

Space robotics↗

NASA NTRS1972

Estimating trajectory correction requirements for multiple outer planet missions.

General approach to the problem of estimating trajectory correction requirements for multiple outer planet flyby missions when the navigation system uses onboard optical measurements made during approach to each target planet to complement the ground-based radio measurements. The accuracy and reliability of the onboard measurement system plays a critical role in sizing the trajectory correction capability required. An illustration of the combined use of radio and optical measurements is provided for the particular case of a Jupiter-Uranus-Neptune mission. Use of the statistical technique developed for computing the trajectory correction margin required to account for uncertainties in subsystem performance, permits trajectory correction savings of 100 to 20 m/sec over 'worst case' designs. This represents weight savings of about 50% of the science payload. For the example case trajectory correction requirements are estimated for two candidate optical systems and the radio alone case. The use of onboard measurements allows a trajectory correction savings of approximately 140 m/sec.

Friedman, L. D.↗

NASA NTRS1974

Space Trajectory Error Analysis Program (STEAP) for halo orbit missions. Volume 1: Analytic and user's manual

Development, test, conversion, and documentation of computer software for the mission analysis of missions to halo orbits about libration points in the earth-sun system is reported. The software consisting of two programs called NOMNAL and ERRAN is part of the Space Trajectories Error Analysis Programs (STEAP). The program NOMNAL targets a transfer trajectory from Earth on a given launch date to a specified halo orbit on a required arrival date. Either impulsive or finite thrust insertion maneuvers into halo orbit are permitted by the program. The transfer trajectory is consistent with a realistic launch profile input by the user. The second program ERRAN conducts error analyses of the targeted transfer trajectory. Measurements including range, doppler, star-planet angles, and apparent planet diameter are processed in a Kalman-Schmidt filter to determine the trajectory knowledge uncertainty. Execution errors at injection, midcourse correction and orbit insertion maneuvers are analyzed along with the navigation uncertainty to determine trajectory control uncertainties and fuel-sizing requirements. The program is also capable of generalized covariance analyses.

Byrnes, D. V.↗

NASA NTRS1973

Four-body trajectory optimization

A collection of typical three-body trajectories from the L1 libration point on the sun-earth line to the earth is presented. These trajectories in the sun-earth system are grouped into four distinct families which differ in transfer time and delta V requirements. Curves showing the variations of delta V with respect to transfer time, and typical two and three-impulse primer vector histories, are included. The development of a four-body trajectory optimization program to compute fuel optimal trajectories between the earth and a point in the sun-earth-moon system are also discussed. Methods for generating fuel optimal two-impulse trajectories which originate at the earth or a point in space, and fuel optimal three-impulse trajectories between two points in space, are presented. A brief qualitative comparison of these methods is given. An example of a four-body two-impulse transfer from the Li libration point to the earth is included.

Pu, C. L.↗

NASA NTRS1975

ASTP (SA-210) Launch vehicle operational flight trajectory. Part 3: Final documentation

Trajectory data are presented for a nominal and two launch window trajectory simulations. These trajectories are designed to insert a manned Apollo spacecraft into a 150/167 km. (81/90 n. mi.) earth orbit inclined at 51.78 degrees for rendezvous with a Soyuz spacecraft, which will be orbiting at approximately 225 km. (121.5 n. mi.). The launch window allocation defined for this launch is 500 pounds of S-IVB stage propellant. The launch window opening trajectory simulation depicts the earliest launch time deviation from a planar flight launch which conforms to this constraint. The launch window closing trajectory simulation was developed for the more stringent Air Force Eastern Test Range (AFETR) flight azimuth restriction of 37.4 degrees east-of-north. These trajectories enclose a 12.09 minute launch window, pertinent features of which are provided in a tabulation. Planar flight data are included for mid-window reference.

Carter, A. B.↗

NASA NTRS1973

Apollo/Saturn 5 postflight trajectory: AS-512

The postflight trajectory is presented for the Apollo/Saturn V AS-512 flight. An analysis is included of the orbital and powered flight trajectories of the launch vehicle and the free flight trajectories of the expended S-IC and S-II stages. Trajectory dependent parameters are provided in earth-fixed launch site, launch vehicle navigation, and geographic polar coordinate systems. The time history of the trajectory parameters for the launch vehicle is presented from guidance reference release to Command Service Module (CSM) separation. Tables of significant parameters at engine cutoff, stage separation, parking orbit insertion, and translunar injection are included. Figures of such parameters as altitude, surface and cross range, and the magnitude of total velocity and acceleration as a function of range time for the powered flight trajectories are presented.

Pinson, G. T.↗

NASA NTRS1979

Shuttle entry trajectory reconstruction using inflight accelerometer and gyro measurements

An error analysis has been made of a Shuttle postflight entry trajectory reconstruction process to obtain trajectory state estimation errors and to assess the impact of these errors on Shuttle aerodynamic force coefficient extraction. In this analysis, the entry trajectory is assumed to be reconstructed via numerical integration of onboard accelerometer and gyro measurements and constrained to satisfy ground-based radio tracking. The trajectory state estimation errors are calculated using a Kalman-Schmidt sequential filter assuming various measurement error models and combinations of ground-based tracking. The resultant trajectory estimation errors are analyzed in a simplified perturbation process to establish the accuracy to which postflight aerodynamic force coefficients can be determined. Results are presented which show that the principal error sources affecting the trajectory reconstruction and thus the force coefficient extraction, assuming perfect atmospheric density knowledge, are the accelerometer and gyro resolution, acceleration-sensitive gyro drifts, and the alignment uncertainties associated with integration on the Shuttle.

Compton, H. R.↗

NASA NTRS1981

Interplanetary trajectory optimization

A procedure for minimizing total impulsive Delta-V for constrained multiple-flyby trajectories, which was originally developed for application to satellite tours, has been modified for application to interplanetary trajectories. The modification includes adding to the cost function the Delta-V required to escape from a parking orbit about the launch planet and the Delta-V required for insertion into orbit about the arrival planet. The hyperbolic excess velocity vector with respect to the launch planet and the launch date have been added to the set of independent variables for the optimization. Each trajectory originates at departure from the parking orbit rather than at a fixed position in space, as is the case for the satellite tour application. The multi-conic trajectory propagation techniques and the Newton optimization algorithm of the original method have been retained. Examples of the application of this new method are given for several types of Galileo interplanetary trajectory options, including Mars powered flyby, broken plane, VEGA, and Delta VEGA trajectories.

Damario, L. A.↗