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NASA NTRS1993

Tsien's method for generating non-Keplerian trajectories. Part 2: The question of thrust to orbit a sphere and the restricted three-body problem

Tsien's method is extended to treat the orbital motion of a body undergoing accelerations and decelerations. A generalized solution is discussed for the generalized case where a body undergoes azimuthal and radial thrust and the problem is further simplified for azimuthal thrust alone. Judicious selection of thrust could generate either an elliptic or hyperbolic trajectory. This is unexpected especially when the body has only enough energy for a lower state trajectory. The methodology is extended treating the problem of vehicle thrust for orbiting a sphere and vehicle thrust within the classical restricted three-body problem. Results for the latter situation can produce hyperbolic trajectories through eigen value decomposition. Since eigen values for no-thrust can be imaginary, thrust can generate real eigen values to describe hyperbolic trajectories. Keplerian dynamics appears to represent but a small subset of a much larger non-Keplerian domain especially when thrust effects are considered. The need for high thrust long duration space-based propulsion systems for changing a trajectory's canonical form is clearly demonstrated.

Murad, P. A.↗

NASA NTRS1991

Free return trajectories for Mars missions

Free return trajectories in earth-Mars mission opportunities between 2000 and 2020 are identified and tradeoffs in mission design involving such trajectories are considered. A method of searching for trajectories containing no deterministic maneuvers is developed which is suitable for generating contours. Sample contours are presented for the trajectories and are used to show the variations in trajectory parameters found in the real world.

Wolf, Aron A.↗

NASA NTRS1990

TROJID: A portable software package for upper-stage trajectory optimization

Performance optimization for upper-stage exoatmospheric vehicles often is performed within the framework of a full capability trajectory simulation package requiring either a large mainframe computer or powerful work-station. Since these software packages tend to include capabilities providing for high-fidelity boost and reentry simulations, the programs usually are quite large and not very portable. The program TROJID is an attempt to provide an environment for the optimization of upper-stage trajectories within a small package capable of being run on a standard desktop microcomputer. Utilizing a state-of-the-art nonlinear programming algorithm and a trajectory simulator implementing impulsive burns and an analytic coast phase propagator, TROJID is capable of producing trajectories for optimal multi-burn upper-stage orbit transfers. The package has been designed to allow full generality in definition of both the trajectory simulator and the parameter optimization problem.

Hammes, Steven M.↗

NASA NTRS1990

Numerical derivative techniques for trajectory optimization

The adoption of robust numerical optimization techniques in trajectory simulation programs has resulted in powerful design and analysis tools. These trajectory simulation/optimization programs are widely used, and a representative list includes the GTS system, the POST program, and newer collocation methods such as OTIS and FONPAC. All of these programs rely on optimization algorithms which require objective function and constraint gradient data during the iteration process. However, most trajectory optimization problems lack simple analytical expressions for these derivatives. In the general case a function evaluation involves integrating aerodynamic, propulsive, and gravity forces over multiple trajectory phases with complex control models. With the newer collocation methods, the integration is replaced by defect constraints and cubic approximations for the state. While analytic gradient expressions can sometimes be derived for trajectory optimization problems, the derivation is cumbersome, time consuming, and prone to mistakes. Fortunately, an alternate method exists for the gradient evaluation, namely finite difference approximations. In this paper some finite difference gradient techniques developed for use with the GTS system are presented. These techniques include methods for computing first and second partial derivatives of single and multiple sets of functions. A key feature of these methods is an error control mechanism which automatically adjusts the perturbation size to obtain accurate derivative values.

Hallman, Wayne P.↗

NASA NTRS1994

Compatibility of TOPEX/Poseidon trajectory propagation with JPL and GSFC/FDF operational software

Two independent trajectory software systems are used to perform the TOPEX/Poseidon operational orbit determination and propagation: the Goddard Trajectory Determination System (GTDS) at the GSF/FDF and the Double Precision Trajectory System (DPTRAJ) at JPL. GTDS is used for operational tracking and TDRS-based orbit determination. DPTRAJ is used for ephemeris generation necessary to conduct day-to-day mission operations. This paper describes the DPTRAJ/GTDS trajectory comparison analysis conducted jointly by JPL and GSFC to ensure the compatibility of these two independent trajectory software systems.

Salama, Ahmed↗

NASA NTRS1993

Trajectory design for the Deep Space Program Science Experiment (DSPSE) mission

In 1994, the Deep Space Program Science Experiment (DSPSE) spacecraft will become the first spacecraft to perform, in succession, both a lunar orbiting mission and a deep-space asteroid encounter mission. The primary mission objective is to perform a long-duration flight-test of various new-technology lightweight components, such as sensors, in a deep-space environment. The mission has two secondary science objectives: to provide high-resolution imaging of the entire lunar surface for mapping purposes and flyby imaging of the asteroid 1620 Geographos. The DSPSE mission is sponsored by the Strategic Defense Initiative Organization (SDIO). As prime contractor, the Naval Research Laboratory (NRL) is building the spacecraft and will conduct mission operations. The Goddard Space Flight Center's (GSFC) Flight Dynamics Division is supporting NRL in the areas of The Deep Space Network (DSN) will provide tracking support. The DSPSE mission will begin with a launch from the Western Test Range in late January 1994. Following a minimum 1.5-day stay in a low-Earth parking orbit, a solid kick motor burn will boost DSPSE into an 18-day, 2.5-revolution phasing orbit transfer trajectory to the Moon. Two burns to insert DSPSE into a lunar polar orbit suitable for the mapping mission will be followed by mapping orbit maintenance and adjustment operations over a period of 2 sidereal months. In May 1994, a lunar orbit departure maneuver, in conjunction with a lunar swingby 26 days later, will propel DSPSE onto a heliocentric transfer that will intercept Geographos on September 1, 1994. This paper presents the characteristics, deterministic delta-Vs, and design details of each trajectory phase of this unique mission, together with the requirements, constraints, and design considerations to which each phase is subject. Numerous trajectory plots and tables of significant trajectory events are included. Following a discussion of the results of a preliminary launch window analysis, a summary of the deterministic impulsive delta-V budget required to establish the baseline mission trajectory design is presented.

Carrington, D.↗

NASA NTRS1996

Attitude and Trajectory Estimation Using Earth Magnetic Field Data

The magnetometer has long been a reliable, inexpensive sensor used in spacecraft momentum management and attitude estimation. Recent studies show an increased accuracy potential for magnetometer-only attitude estimation systems. Since the Earth's magnetic field is a function of time and position, and since time is known quite precisely, the differences between the computer and measured magnetic field components, as measured by the magnetometers throughout the entire spacecraft orbit, are a function of both the spacecraft trajectory and attitude errors. Therefore, these errors can be used to estimate both trajectory and attitude. Traditionally, satellite attitude and trajectory have been estimated with completely separate system, using different measurement data. Recently, trajectory estimation for low earth orbit satellites was successfully demonstrated in ground software using only magnetometer data. This work proposes a single augmented extended Kalman Filter to simultaneously and autonomously estimate both spacecraft trajectory and attitude with data from a magnetometer and either dynamically determined rates or gyro-measured body rates.

Deutschmann, Julie↗

NASA NTRS1997

Estimating Attitude, Trajectory, and Gyro Biases in an Extended Kalman Filter using Earth Magnetic Field Data from the Rossi X-Ray Timing Explorer

Traditionally satellite attitude and trajectory have been estimated with completely separate systems, using different measurement data. The estimation of both trajectory and attitude for low earth orbit satellites has been successfully demonstrated in ground software using magnetometer and gyroscope data. Since the earth's magnetic field is a function of time and position, and since time is known quite precisely, the differences between the computed and measured magnetic field components, as measured by the magnetometers throughout the entire spacecraft orbit, are a function of both the spacecraft trajectory and attitude errors. Therefore, these errors can be used to estimate both trajectory and attitude. This work further tests the single augmented Extended Kalman Filter (EKF) which simultaneously and autonomously estimates spacecraft trajectory and attitude with data from the Rossi X-Ray Timing Explorer (RXTE) magnetometer and gyro-measured body rates. In addition, gyro biases are added to the state and the filter's ability to estimate them is presented.

Deutschmann, Julie↗

NASA NTRS1996

Capture Conditions for Merging Trajectory Segments to Model Realistic Aircraft Descents

A typical commercial aircraft trajectory consists of a series of flight segments. An aircraft switches from one segment to another when certain specified variables reach their desired values. Trajectory synthesis for air traffic control automation must be consistent with practical pilot procedures. We examine capture conditions for merging trajectory segments to model commercial aircraft descent in trajectory synthesis. These conditions translate into bounds on measurements of atmospheric wind, pressure, and temperature. They also define ranges of thrust and drag feasible for a descent trajectory. Capture conditions are derived for the Center-TRACON Automation System developed at NASA Ames Research Center for automated air traffic control. Various uses of capture conditions are discussed. A Boeing 727-200 aircraft is used to provide numerical examples of capture conditions.

Zhao, Yiyuan↗

NASA NTRS1999

Optimization of Insertion Cost for Transfer Trajectories to Libration Point Orbits

The objective of this work is the development of efficient techniques to optimize the cost associated with transfer trajectories to libration point orbits in the Sun-Earth-Moon four body problem, that may include lunar gravity assists. Initially, dynamical systems theory is used to determine invariant manifolds associated with the desired libration point orbit. These manifolds are employed to produce an initial approximation to the transfer trajectory. Specific trajectory requirements such as, transfer injection constraints, inclusion of phasing loops, and targeting of a specified state on the manifold are then incorporated into the design of the transfer trajectory. A two level differential corrections process is used to produce a fully continuous trajectory that satisfies the design constraints, and includes appropriate lunar and solar gravitational models. Based on this methodology, and using the manifold structure from dynamical systems theory, a technique is presented to optimize the cost associated with insertion onto a specified libration point orbit.

Howell, K. C.↗

NASA NTRS2002

Trajectories for High Specific Impulse High Specific Power Deep Space Exploration

Flight times and deliverable masses for electric and fusion propulsion systems are difficult to approximate. Numerical integration is required for these continuous thrust systems. Many scientists are not equipped with the tools and expertise to conduct interplanetary and interstellar trajectory analysis for their concepts. Several charts plotting the results of well-known trajectory simulation codes were developed and are contained in this paper. These charts illustrate the dependence of time of flight and payload ratio on jet power, initial mass, specific impulse and specific power. These charts are intended to be a tool by which people in the propulsion community can explore the possibilities of their propulsion system concepts. Trajectories were simulated using the tools VARITOP and IPOST. VARITOP is a well known trajectory optimization code that involves numerical integration based on calculus of variations. IPOST has several methods of trajectory simulation; the one used in this paper is Cowell's method for full integration of the equations of motion. An analytical method derived in the companion paper was also evaluated. The accuracy of this method is discussed in the paper.

Polsgrove, Tara↗

NASA NTRS2002

Simulation of Trajectories for High Specific Impulse Deep Space Exploration

Difficulties in approximating flight times and deliverable masses for continuous thrust propulsion systems have complicated comparison and evaluation of proposed propulsion concepts. These continuous thrust propulsion systems are of interest to many groups, not the least of which are the electric propulsion and fusion communities. Several charts plotting the results of well-known trajectory simulation codes were developed and are contained in this paper. These charts illustrate the dependence of time of flight and payload ratio on jet power, initial mass, specific impulse and specific power. These charts are intended to be a tool by which people in the propulsion community can explore the possibilities of their propulsion system concepts. Trajectories were simulated using the tools VARITOP and IPOST. VARITOP is a well known trajectory optimization code that involves numerical integration based on calculus of variations. IPOST has several methods of trajectory simulation; the one used in this paper is Cowell's method for full integration of the equations of motion. The analytical method derived in the companion paper was also used to simulate the trajectory. The accuracy of this method is discussed in the paper.

Polsgrove, Tara↗

NASA NTRS2003

On-Board Entry Trajectory Planning Expanded to Sub-orbital Flight

A methodology for on-board planning of sub-orbital entry trajectories is developed. The algorithm is able to generate in a time frame consistent with on-board environment a three-degree-of-freedom (3DOF) feasible entry trajectory, given the boundary conditions and vehicle modeling. This trajectory is then tracked by feedback guidance laws which issue guidance commands. The current trajectory planning algorithm complements the recently developed method for on-board 3DOF entry trajectory generation for orbital missions, and provides full-envelope autonomous adaptive entry guidance capability. The algorithm is validated and verified by extensive high fidelity simulations using a sub-orbital reusable launch vehicle model and difficult mission scenarios including failures and aborts.

Lu, Ping↗

NASA NTRS2002

Monocular motion adaptation affects the perceived trajectory of stereomotion

Perceived stereomotion trajectory was measured before and after adaptation to lateral motion in the dominant or nondominant eye to assess the relative contributions of 2 cues: changing disparity and interocular velocity difference. Perceived speed for monocular lateral motion and perceived binocular visual direction (BVD) was also assessed. Unlike stereomotion trajectory perception, the BVD of static targets showed an ocular dominance bias, even without adaptation. Adaptation caused equivalent biases in perceived trajectory and monocular motion speed, without significantly affecting perceived BVD. Predictions from monocular motion data closely match trajectory perception data, unlike those from BVD sources. The results suggest that the interocular velocity differences make a significant contribution to stereomotion trajectory perception.

Depth Perception/physiology↗

NASA NTRS2000

Quantifying ataxia: ideal trajectory analysis--a technical note

We describe a quantitative method to assess repeated stair stepping stability. In both the mediolateral (ML) and anterioposterior (AP) directions, the trajectory of the subject's center of mass (COM) was compared to an ideal sinusoid. The two identified sinusoids were unique in each direction but coupled. Two dimensionless numbers-the mediolateral instability index (IML) and AP instability index (IAP)-were calculated using the COM trajectory and ideal sinusoids for each subject with larger index values resulting from less stable performance. The COM trajectories of nine nonimpaired controls and six patients diagnosed with unilateral or bilateral vestibular labyrinth hypofunction were analyzed. The average IML and IAP values of labyrinth disorder patients were respectively 127% and 119% greater than those of controls (p<0.014 and 0.006, respectively), indicating that the ideal trajectory analysis distinguishes persons with labyrinth disorder from those without. The COM trajectories also identify movement inefficiencies attributable to vestibulopathy.

Non-NASA Center↗

NASA NTRS2004

Trajectory Specification for High-Capacity Air Traffic Control

In the current air traffic management system, the fundamental limitation on airspace capacity is the cognitive ability of human air traffic controllers to maintain safe separation with high reliability. The doubling or tripling of airspace capacity that will be needed over the next couple of decades will require that tactical separation be at least partially automated. Standardized conflict-free four-dimensional trajectory assignment will be needed to accomplish that objective. A trajectory specification format based on the Extensible Markup Language is proposed for that purpose. This format can be used to downlink a trajectory request, which can then be checked on the ground for conflicts and approved or modified, if necessary, then uplinked as the assigned trajectory. The horizontal path is specified as a series of geodetic waypoints connected by great circles, and the great-circle segments are connected by turns of specified radius. Vertical profiles for climb and descent are specified as low-order polynomial functions of along-track position, which is itself specified as a function of time. Flight technical error tolerances in the along-track, cross-track, and vertical axes define a bounding space around the reference trajectory, and conformance will guarantee the required separation for a period of time known as the conflict time horizon. An important safety benefit of this regimen is that the traffic will be able to fly free of conflicts for at least several minutes even if all ground systems and the entire communication infrastructure fail. Periodic updates in the along-track axis will adjust for errors in the predicted along-track winds.

Paielli, Russell A.↗

NASA NTRS2004

Trajectory Analysis

The Systems Analysis Branch performs trajectory and systems analysis for next-generation launch vehicle and space transportation technologies. Currently the branch is supporting the Project Prometheus with analysis of nuclear electric and nuclear thermal propulsion missions to a variety of destinations. Within Project Prometheus a proposed mission to Jupiter and three of the planet sized icy moons that orbit it is developing. The Jupiter Icy Moons Orbiter (JIMO) as the project is being called will enable detailed scientific investigation of Jupiter's moons Callisto, Ganymede, and Europa. These moons were choose to orbit for intensive study in particular because they are each believed to have water, energy, and organic material. The JIMO mission will utilize nuclear fission power and electric propulsion in order to allow the spacecraft to orbit the moons at close range for long durations of time. My assignment this summer was to assist in developing a trajectory analysis for the spacecraft system the Jupiter Icy Moons Orbiter by rewriting an inefficient Excel file into a more efficient FORTRAN program. This program has been created for use planning the trajectory of the Jupiter Icy Moons Orbiter Mission. The program uses a database of thousands of data points representing flight time, burn time, thrust, mass of the propellant used, final mass of the spacecraft, ratio of finals mass to initial mass, and change in velocity that a spacecraft experiences during each phase in the Jupiter Icy Moons Orbiter Mission. The trajectory program is capable of taking a specific user entered specific impulse (isp), final mass fraction, and thrust (P(sub J)/m0) and through the use of cubic splines to fit specific data curves, the program can locate the exact flight time linked to the user specified values of specific impulse and final over initial mass fraction. Currently, the database used by the program to calculate flight times for a given thrust is only for isps in the range of 2000 to 9000 seconds. Although the program is specific to the Jupiter Icy Moons Orbiter mission, it can be easily modified to fit any mission with a propulsion system expected isp range between 2000 and 9000 seconds. This program performs the same tasks as the old Excel file but in a more timely fashion. Finally, this FORTRAN program will replace the Excel file as the tool used to calculate trajectory properties in missions and will provide valuable data to mission analysts for years to come.

Burke, Laura M.↗

NASA NTRS2005

Handling Trajectory Uncertainties for Airborne Conflict Management

Airborne conflict management is an enabling capability for NASA's Distributed Air-Ground Traffic Management (DAG-TM) concept. DAGTM has the goal of significantly increasing capacity within the National Airspace System, while maintaining or improving safety. Under DAG-TM, autonomous aircraft maintain separation from each other and from managed aircraft unequipped for autonomous flight. NASA Langley Research Center has developed the Autonomous Operations Planner (AOP), an onboard decision support system that provides airborne conflict management (ACM) and strategic flight planning support for autonomous aircraft pilots. The AOP performs conflict detection, prevention, and resolution from nearby traffic aircraft and area hazards. Traffic trajectory information is assumed to be provided by Automatic Dependent Surveillance Broadcast (ADS-B). Reliable trajectory prediction is a key capability for providing effective ACM functions. Trajectory uncertainties due to environmental effects, differences in aircraft systems and performance, and unknown intent information lead to prediction errors that can adversely affect AOP performance. To accommodate these uncertainties, the AOP has been enhanced to create cross-track, vertical, and along-track buffers along the predicted trajectories of both ownship and traffic aircraft. These buffers will be structured based on prediction errors noted from previous simulations such as a recent Joint Experiment between NASA Ames and Langley Research Centers and from other outside studies. Currently defined ADS-B parameters related to navigation capability, trajectory type, and path conformance will be used to support the algorithms that generate the buffers.

Barhydt, Richard↗