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All sourcesNASA NTRSDOE OSTI
NASA NTRS2007

Comparison of a Simple Patched Conic Trajectory Code to Commercially Available Software

Often in spaceflight proposal development, mission designers must eva luate numerous trajectories as different design factors are investiga ted. Although there are numerous commercial software packages availab le to help develop and analyze trajectories, most take a significant amount of time to develop the trajectory itself, which isn't effectiv e when working on proposals. Thus a new code, PatCon, which is both q uick and easy to use, was developed to aid mission designers to condu ct trade studies on launch and arrival times for any given target pla net. The code is able to run quick analyses, due to the incorporation of the patched conic approximation, to determine the trajectory. PatCon provides a simple but accurate approximation of the four body moti on problem that would be needed to solve any planetary trajectory. P atCon has been compared to a patched conic test case for verification, with limited validation or comparison with other COTS software. This paper describes the patched conic technique and its implementation i n PatCon. A description of the results and comparison of PatCon to ot her more evolved codes such as AGI#s Satellite Tool Kit and JAQAR As trodynamics# Swingby Calculator is provided. The results will include percent differences in values such as C3 numbers, and Vinfinity at a rrival, and other more subjective results such as the time it takes to build the simulation, and actual calculation time.

AndersonPark, Brooke M.↗
NASA NTRS2005

Simulation Results of the Huygens Probe Entry and Descent Trajectory Reconstruction Algorithm

Cassini/Huygens is a joint NASA/ESA mission to explore the Saturnian system. The ESA Huygens probe is scheduled to be released from the Cassini spacecraft on December 25, 2004, enter the atmosphere of Titan in January, 2005, and descend to Titan s surface using a sequence of different parachutes. To correctly interpret and correlate results from the probe science experiments and to provide a reference set of data for "ground-truthing" Orbiter remote sensing measurements, it is essential that the probe entry and descent trajectory reconstruction be performed as early as possible in the postflight data analysis phase. The Huygens Descent Trajectory Working Group (DTWG), a subgroup of the Huygens Science Working Team (HSWT), is responsible for developing a methodology and performing the entry and descent trajectory reconstruction. This paper provides an outline of the trajectory reconstruction methodology, preliminary probe trajectory retrieval test results using a simulated synthetic Huygens dataset developed by the Huygens Project Scientist Team at ESA/ESTEC, and a discussion of strategies for recovery from possible instrument failure.

Kazeminejad, B.↗
NASA NTRS2007

SELENE Translunar Trajectory Reconfiguration Plan Provided for the Case of Main Engine Anomaly

In this paper, the reconfiguration of translunar trajectory in case of main engine anomaly is investigated. The objectives of the trajectory design are to reduce the excessive velocity at the Lunar encounter as well as to reduce the total required Delta-v to complete the sequence. 3-impulse Hohmann transfer based trajectory is adopted and possible trajectories are categorized under 2-body approximation. The solutions obtained are applied to more sophisticated models (3-body approximation and 4-body) and yields feasible trajectory.

Kawakatsu, Yasuhiro↗
NASA NTRS2006

Trajectory and Aeroheating Environment Development and Sensitivity Analysis for Capsule-shaped Vehicles

Recently, NASA's Exploration Systems Research and Technology Project funded several tasks that endeavored to develop and evaluate various thermal protection systems and high temperature material concepts for potential use on the crew exploration vehicle. In support of these tasks, NASA Langley's Vehicle Analysis Branch generated trajectory information and associated aeroheating environments for more than 60 unique entry cases. Using the Apollo Command Module as the baseline entry system because of its relevance to the favored crew exploration vehicle design, trajectories for a range of lunar and Mars return, direct and aerocapture Earth-entry scenarios were developed. For direct entry, a matrix of cases was created that reflects reasonably expected minimum and maximum values of vehicle ballistic coefficient, inertial velocity at entry interface, and inertial flight path angle at entry interface. For aerocapture, trajectories were generated for a range of values of initial velocity and ballistic coefficient that, when combined with proper initial flight path angles, resulted in achieving a low Earth orbit either by employing a full lift vector up or full lift vector down attitude. For each trajectory generated, aeroheating environments were generated which were intended to bound the thermal protection system requirements for likely crew exploration vehicle concepts. The trades examined clearly pointed to a range of missions / concepts that will require ablative systems as well as a range for which reusable systems may be feasible. In addition, the results clearly indicated those entry conditions and modes suitable for manned flight, considering vehicle deceleration levels experienced during entry. This paper presents an overview of the analysis performed, including the assumptions, methods, and general approach used, as well as a summary of the trajectory and aerothermal environment information that was generated.

Robinson, Jeffrey S.↗
NASA NTRS2009

Validation of Multibody Program to Optimize Simulated Trajectories II Parachute Simulation with Interacting Forces

A capability to simulate trajectories of multiple interacting rigid bodies has been developed, tested and validated. This capability uses the Program to Optimize Simulated Trajectories II (POST 2). The standard version of POST 2 allows trajectory simulation of multiple bodies without force interaction. In the current implementation, the force interaction between the parachute and the suspended bodies has been modeled using flexible lines, allowing accurate trajectory simulation of the individual bodies in flight. The POST 2 multibody capability is intended to be general purpose and applicable to any parachute entry trajectory simulation. This research paper explains the motivation for multibody parachute simulation, discusses implementation methods, and presents validation of this capability.

Raiszadeh, Behzad↗
NASA NTRS2007

Efficient Optimization of Low-Thrust Spacecraft Trajectories

A paper describes a computationally efficient method of optimizing trajectories of spacecraft driven by propulsion systems that generate low thrusts and, hence, must be operated for long times. A common goal in trajectory-optimization problems is to find minimum-time, minimum-fuel, or Pareto-optimal trajectories (here, Pareto-optimality signifies that no other solutions are superior with respect to both flight time and fuel consumption). The present method utilizes genetic and simulated-annealing algorithms to search for globally Pareto-optimal solutions. These algorithms are implemented in parallel form to reduce computation time. These algorithms are coupled with either of two traditional trajectory- design approaches called "direct" and "indirect." In the direct approach, thrust control is discretized in either arc time or arc length, and the resulting discrete thrust vectors are optimized. The indirect approach involves the primer-vector theory (introduced in 1963), in which the thrust control problem is transformed into a co-state control problem and the initial values of the co-state vector are optimized. In application to two example orbit-transfer problems, this method was found to generate solutions comparable to those of other state-of-the-art trajectory-optimization methods while requiring much less computation time.

Lee, Seungwon↗
NASA NTRS2011

Rapid Calculation of Spacecraft Trajectories Using Efficient Taylor Series Integration

A variable-order, variable-step Taylor series integration algorithm was implemented in NASA Glenn's SNAP (Spacecraft N-body Analysis Program) code. SNAP is a high-fidelity trajectory propagation program that can propagate the trajectory of a spacecraft about virtually any body in the solar system. The Taylor series algorithm's very high order accuracy and excellent stability properties lead to large reductions in computer time relative to the code's existing 8th order Runge-Kutta scheme. Head-to-head comparison on near-Earth, lunar, Mars, and Europa missions showed that Taylor series integration is 15.8 times faster than Runge- Kutta on average, and is more accurate. These speedups were obtained for calculations involving central body, other body, thrust, and drag forces. Similar speedups have been obtained for calculations that include J2 spherical harmonic for central body gravitation. The algorithm includes a step size selection method that directly calculates the step size and never requires a repeat step. High-order Taylor series integration algorithms have been shown to provide major reductions in computer time over conventional integration methods in numerous scientific applications. The objective here was to directly implement Taylor series integration in an existing trajectory analysis code and demonstrate that large reductions in computer time (order of magnitude) could be achieved while simultaneously maintaining high accuracy. This software greatly accelerates the calculation of spacecraft trajectories. At each time level, the spacecraft position, velocity, and mass are expanded in a high-order Taylor series whose coefficients are obtained through efficient differentiation arithmetic. This makes it possible to take very large time steps at minimal cost, resulting in large savings in computer time. The Taylor series algorithm is implemented primarily through three subroutines: (1) a driver routine that automatically introduces auxiliary variables and sets up initial conditions and integrates; (2) a routine that calculates system reduced derivatives using recurrence relations for quotients and products; and (3) a routine that determines the step size and sums the series. The order of accuracy used in a trajectory calculation is arbitrary and can be set by the user. The algorithm directly calculates the motion of other planetary bodies and does not require ephemeris files (except to start the calculation). The code also runs with Taylor series and Runge-Kutta used interchangeably for different phases of a mission.

Scott, James R.↗
NASA NTRS2010

High-Speed Solution of Spacecraft Trajectory Problems Using Taylor Series Integration

It has been known for some time that Taylor series (TS) integration is among the most efficient and accurate numerical methods in solving differential equations. However, the full benefit of the method has yet to be realized in calculating spacecraft trajectories, for two main reasons. First, most applications of Taylor series to trajectory propagation have focused on relatively simple problems of orbital motion or on specific problems and have not provided general applicability. Second, applications that have been more general have required use of a preprocessor, which inevitably imposes constraints on computational efficiency. The latter approach includes the work of Berryman et al., who solved the planetary n-body problem with relativistic effects. Their work specifically noted the computational inefficiencies arising from use of a preprocessor and pointed out the potential benefit of manually coding derivative routines. In this Engineering Note, we report on a systematic effort to directly implement Taylor series integration in an operational trajectory propagation code: the Spacecraft N-Body Analysis Program (SNAP). The present Taylor series implementation is unique in that it applies to spacecraft virtually anywhere in the solar system and can be used interchangeably with another integration method. SNAP is a high-fidelity trajectory propagator that includes force models for central body gravitation with N X N harmonics, other body gravitation with N X N harmonics, solar radiation pressure, atmospheric drag (for Earth orbits), and spacecraft thrusting (including shadowing). The governing equations are solved using an eighth-order Runge-Kutta Fehlberg (RKF) single-step method with variable step size control. In the present effort, TS is implemented by way of highly integrated subroutines that can be used interchangeably with RKF. This makes it possible to turn TS on or off during various phases of a mission. Current TS force models include central body gravitation with the J2 spherical harmonic, other body gravitation, thrust, constant atmospheric drag from Earth's atmosphere, and solar radiation pressure for a sphere under constant illumination. The purpose of this Engineering Note is to demonstrate the performance of TS integration in an operational trajectory analysis code and to compare it with a standard method, eighth-order RKF. Results show that TS is 16.6 times faster on average and is more accurate in 87.5% of the cases presented.

Scott, James R.↗
NASA NTRS2012

Direct Method Transcription for a Human-Class Translunar Injection Trajectory Optimization

This paper presents a new trajectory optimization software package developed in the framework of a low-to-high fidelity 3 degrees-of-freedom (DOF)/6-DOF vehicle simulation program named Mission Analysis Simulation Tool in Fortran (MASTIF) and its application to a translunar trajectory optimization problem. The functionality of the developed optimization package is implemented as a new "mode" in generalized settings to make it applicable for a general trajectory optimization problem. In doing so, a direct optimization method using collocation is employed for solving the problem. Trajectory optimization problems in MASTIF are transcribed to a constrained nonlinear programming (NLP) problem and solved with SNOPT, a commercially available NLP solver. A detailed description of the optimization software developed is provided as well as the transcription specifics for the translunar injection (TLI) problem. The analysis includes a 3-DOF trajectory TLI optimization and a 3-DOF vehicle TLI simulation using closed-loop guidance.

Witzberger, Kevin E.↗
NASA NTRS2011

Particle Trajectory and Icing Analysis of the E(sup 3) Turbofan Engine Using LEWICE3D Version 3

Particle trajectory and ice shape calculations were made for the Energy Efficient Engine (E(sup 3)) using the LEWICE3D Version 3 software. The particle trajectory and icing computations were performed using the new "block-to-block" collection efficiency method which has been incorporated into the LEWICE3D Version 3 software. The E(sup 3) was developed by NASA and GE in the early 1980 s as a technology demonstrator and is representative of a modern high bypass turbofan engine. The E(sup 3) flow field was calculated using the NASA Glenn ADPAC turbomachinery flow solver. Computations were performed for the low pressure compressor of the E(sup 3) for a Mach 0.8 cruise condition at 11,887 m assuming a standard warm day for three drop sizes and two drop distributions typically used in aircraft design and certification. Particle trajectory computations were made for water drop sizes of 5, 20, and 100 microns. Particle trajectory and ice shape predictions were made for a 20 micron Langmuir-D distribution and for a 92 mm Super-cooled Large Droplet (SLD) distribution with and without splashing effects for a Liquid Water Content (LWC) of 0.3 g/cu m and an icing time of 30 min. The E3 fan and spinner combination proved to be an effective ice removal mechanism as they removed greater than 36 percent of the mass entering the inlet for the icing cases. The maximum free stream catch fraction for the fan and spinner combination was 0.60 while that on the elements downstream of the fan was 0.03. The non-splashing trajectory and collection efficiency results showed that as drop size increased impingement rates increased on the spinner and fan leaving less mass to impinge on downstream components. The SLD splashing case yielded more mass downstream of the fan than the SLD non-splashing case due to mass being splashed from the upstream inlet lip, spinner and fan components. The ice shapes generated downstream of the fan were either small or nonexistent due to the small available mass and evaporation except for the 92 m SLD splashing case. Relatively large ice shapes were predicted for internal guide vane #1 and rotor #1 for the 92 m SLD splashing case due to re-impingement of splashed mass.

Bidwell, Colin S.↗
NASA NTRS2012

Creation of an Upper Stage Trajectory Capability Boundary to Enable Booster System Trade Space Exploration

The problem of trajectory optimization is important in all space missions. The solution of this problem enables one to specify the optimum thrust steering program which should be followed to achieve a specified mission objective, simultaneously satisfying the constraints.1 It is well known that whether or not the ascent trajectory is optimal can have a significant impact on propellant usage for a given payload, or on payload weight for the same gross vehicle weight.2 Consequently, ascent guidance commands are usually optimized in some fashion. Multi-stage vehicles add complexity to this analysis process as changes in vehicle properties in one stage propagate to the other stages through gear ratios and changes in the optimal trajectory. These effects can cause an increase in analysis time as more variables are added and convergence of the optimizer to system closure requires more analysis iterations. In this paper, an approach to simplifying this multi-stage problem through the creation of an upper stage capability boundary is presented. This work was completed as part of a larger study focused on trade space exploration for the advanced booster system that will eventually form a part of NASA s new Space Launch System.3 The approach developed leverages Design of Experiments and Surrogate Modeling4 techniques to create a predictive model of the SLS upper stage performance. The design of the SLS core stages is considered fixed for the purposes of this study, which results in trajectory parameters such as staging conditions being the only variables relevant to the upper stage. Through the creation of a surrogate model, which takes staging conditions as inputs and predicts the payload mass delivered by the SLS upper stage to a reference orbit as the response, it is possible to identify a "surface" of staging conditions which all satisfy the SLS requirement of placing 130 metric tons into low-Earth orbit (LEO).3 This identified surface represents the 130 metric ton capability boundary for the upper stage, such that if the combined first stage and boosters can achieve any one staging point on that surface, then the design is identified as feasible. With the surrogate model created, design and analysis of advanced booster concepts is streamlined, as optimization of the upper stage trajectory is no longer required in every design loop.

Walsh, Ptrick↗
NASA NTRS2012

Comparison of Trajectory Models in Calculations of N2-broadened Half-widths and N2-induced Line Shifts for the Rotational Band of H2O-16 and Comparison with Measurements

In this work, Complex Robert-Bonamy calculations of half-widths and line shifts were done for N2-broadening of water for 1639 transitions in the rotational band using two models for the trajectories. The first is a model correct to second order in time, the Robert-Bonamy parabolic approximation. The second is the solution of Hamilton's equations. Both models use the isotropic part of the atom-atom potential to determine the trajectories. The present calculations used an intermolecular potential expanded to 20th order to assure the convergence of the half-widths and line shifts. The aim of the study is to assess if the difference in the half-widths and line shifts determined from the two trajectory models is greater than the accuracy requirements of the spectroscopic and remote sensing communities. The results of the calculations are compared with measurements of the half-widths and line shifts. It is shown that the effects of the trajectory model greatly exceed the needs of current remote sensing measurements and that line shape parameters calculated using trajectories determined by solving Hamilton's equations agree better with measurement.

Lamouroux, J.↗
NASA NTRS2012

Adaptive Trajectory Prediction Algorithm for Climbing Flights

Aircraft climb trajectories are difficult to predict, and large errors in these predictions reduce the potential operational benefits of some advanced features for NextGen. The algorithm described in this paper improves climb trajectory prediction accuracy by adjusting trajectory predictions based on observed track data. It utilizes rate-of-climb and airspeed measurements derived from position data to dynamically adjust the aircraft weight modeled for trajectory predictions. In simulations with weight uncertainty, the algorithm is able to adapt to within 3 percent of the actual gross weight within two minutes of the initial adaptation. The root-mean-square of altitude errors for five-minute predictions was reduced by 73 percent. Conflict detection performance also improved, with a 15 percent reduction in missed alerts and a 10 percent reduction in false alerts. In a simulation with climb speed capture intent and weight uncertainty, the algorithm improved climb trajectory prediction accuracy by up to 30 percent and conflict detection performance, reducing missed and false alerts by up to 10 percent.

Seperation Assurance↗
NASA NTRS2015

Lunar Cube Transfer Trajectory Options

Numerous Earth-Moon trajectory and lunar orbit options are available for Cubesat missions. Given the limited Cubesat injection infrastructure, transfer trajectories are contingent upon the modification of an initial condition of the injected or deployed orbit. Additionally, these transfers can be restricted by the selection or designs of Cubesat subsystems such as propulsion or communication. Nonetheless, many trajectory options can be considered which have a wide range of transfer durations, fuel requirements, and final destinations. Our investigation of potential trajectories highlights several options including deployment from low Earth orbit (LEO), geostationary transfer orbits (GTO), and higher energy direct lunar transfers and the use of longer duration Earth-Moon dynamical systems. For missions with an intended lunar orbit, much of the design process is spent optimizing a ballistic capture while other science locations such as Sun-Earth libration or heliocentric orbits may simply require a reduced Delta-V imparted at a convenient location along the trajectory.

Libration Orbits↗
NASA NTRS2015

Lunar Cube Transfer Trajectory Options

Numerous Earth-Moon trajectory and lunar orbit options are available for Cubesat missions. Given the limited Cubesat injection infrastructure, transfer trajectories are contingent upon the modification of an initial condition of the injected or deployed orbit. Additionally, these transfers can be restricted by the selection or designs of Cubesat subsystems such as propulsion or communication. Nonetheless, many trajectory options can b e considered which have a wide range of transfer duration, fuel requirements, and final destinations. Our investigation of potential trajectories highlights several options including deployment from low Earth orbit (LEO) geostationary transfer orbits (GTO) and higher energy direct lunar transfer and the use of longer duration Earth-Moon dynamical systems. For missions with an intended lunar orbit, much of the design process is spent optimizing a ballistic capture while other science locations such as Sun-Earth libration or heliocentric orbits may simply require a reduced Delta-V imparted at a convenient location along the trajectory.

Libration Orbits↗
NASA NTRS2017

Point-Mass Aircraft Trajectory Prediction Using a Hierarchical, Highly-Adaptable Software Design

A highly adaptable and extensible method for predicting four-dimensional trajectories of civil aircraft has been developed. This method, Behavior-Based Trajectory Prediction, is based on taxonomic concepts developed for the description and comparison of trajectory prediction software. A hierarchical approach to the "behavioral" layer of a point-mass model of aircraft flight, a clear separation between the "behavioral" and "mathematical" layers of the model, and an abstraction of the methods of integrating differential equations in the "mathematical" layer have been demonstrated to support aircraft models of different types (in particular, turbojet vs. turboprop aircraft) using performance models at different levels of detail and in different formats, and promise to be easily extensible to other aircraft types and sources of data. The resulting trajectories predict location, altitude, lateral and vertical speeds, and fuel consumption along the flight path of the subject aircraft accurately and quickly, accounting for local conditions of wind and outside air temperature. The Behavior-Based Trajectory Prediction concept was implemented in NASA's Traffic Aware Planner (TAP) flight-optimizing cockpit software application.

Karr, David A.↗
NASA NTRS2017

High-Fidelity Multi-Rotor Unmanned Aircraft System Simulation Development for Trajectory Prediction Under Off-Nominal Flight Dynamics

The NASA Unmanned Aircraft System (UAS) Traffic Management (UTM) project is conducting research to enable civilian low-altitude airspace and UAS operations. A goal of this project is to develop probabilistic methods to quantify risk during failures and off nominal flight conditions. An important part of this effort is the reliable prediction of feasible trajectories during off-nominal events such as control failure, atmospheric upsets, or navigation anomalies that can cause large deviations from the intended flight path or extreme vehicle upsets beyond the normal flight envelope. Few examples of high-fidelity modeling and prediction of off-nominal behavior for small UAS (sUAS) vehicles exist, and modeling requirements for accurately predicting flight dynamics for out-of-envelope or failure conditions are essentially undefined. In addition, the broad range of sUAS aircraft configurations already being fielded presents a significant modeling challenge, as these vehicles are often very different from one another and are likely to possess dramatically different flight dynamics and resultant trajectories and may require different modeling approaches to capture off-nominal behavior. NASA has undertaken an extensive research effort to define sUAS flight dynamics modeling requirements and develop preliminary high fidelity six degree-of-freedom (6-DOF) simulations capable of more closely predicting off-nominal flight dynamics and trajectories. This research has included a literature review of existing sUAS modeling and simulation work as well as development of experimental testing methods to measure and model key components of propulsion, airframe and control characteristics. The ultimate objective of these efforts is to develop tools to support UTM risk analyses and for the real-time prediction of off-nominal trajectories for use in the UTM Risk Assessment Framework (URAF). This paper focuses on modeling and simulation efforts for a generic quad-rotor configuration typical of many commercial vehicles in use today. An overview of relevant off-nominal multi-rotor behaviors will be presented to define modeling goals and to identify the prediction capability lacking in simplified models of multi-rotor performance. A description of recent NASA wind tunnel testing of multi-rotor propulsion and airframe components will be presented illustrating important experimental and data acquisition methods, and a description of preliminary propulsion and airframe models will be presented. Lastly, examples of predicted off-nominal flight dynamics and trajectories from the simulation will be presented.

Foster, John V.↗
NASA NTRS2015

Integrated Vehicle and Trajectory Design of Small Spacecraft with Electric Propulsion for Earth and Interplanetary Missions

This paper investigates the feasibility of Earth-transfer and interplanetary mission architectures for miniaturized spacecraft using emerging small solar electric propulsion technologies. Emerging small SEP thrusters offer significant advantages relative to existing technologies and will enable U-class systems to perform trajectory maneuvers with significant Delta V requirements. The approach in this paper is unique because it integrates trajectory design with vehicle sizing and accounts for the system and operational constraints of small U-class missions. The modeling framework includes integrated propulsion, orbit, energy, and external environment dynamics and systems-level power, energy, mass, and volume constraints. The trajectory simulation environment models orbit boosts in Earth orbit and flyby and capture trajectories to interplanetary destinations. A family of small spacecraft mission architectures are studied, including altitude and inclination transfers in Earth orbit and trajectories that escape Earth orbit and travel to interplanetary destinations such as Mercury, Venus, and Mars. Results are presented visually to show the trade-offs between competing performance objectives such as maximizing available mass and volume for payloads and minimizing transfer time. The results demonstrate the feasibility of using small spacecraft to perform significant Earth and interplanetary orbit transfers in less than one year with reasonable U-class mass, power, volume, and mission durations.

Spangelo, Sara↗