Trajectory Optimization Techniques Research Articles

Multiple Method 2-D Trajectory Optimization Satisfying Waypoints and No-Fly Zone Constraints

Minimum time to target is one of the primary goals of a global strike mission. The Hypersonic Cruise Vehicle and the Common Aero Vehicle are currently being investigated for mission effectiveness. Additional mission requirements include passage through intermediate waypoints and avoidance of no-fly zones. Thus, a real-time or near real-time autonomous trajectory generation technique is desired to minimize the flight time, satisfy terminal and multiple intermediate state constraints, and remain within specified control limitations. The research herein presents a baseline technique, an analytical geometric trajectory optimization technique, and a dynamic optimization technique. Numerical examples for constant speed trajectories as well as decelerating flight are used to demonstrate and compare the presented techniques. These results show the significant time savings achievable through optimization, the accuracy and computation efficiency of the geometric solution, and the robustness and application of the dynamic optimization technique.

A new method for smooth trajectory planning of robot manipulators

A new method for smooth trajectory planning of robot manipulators is described in this paper. In order to ensure that the resulting trajectory is smooth enough, an objective function containing a term proportional to the integral of the squared jerk (defined as the derivative of the acceleration) along the trajectory is considered. Moreover, a second term, proportional to the total execution time, is added to the expression of the objective function. In this way it is not necessary to define the total execution time before running the algorithm. Fifth-order B-splines are then used to compose the overall trajectory. With respect to other trajectory optimization techniques, the proposed method enables one to set kinematic constraints on the robot motion, expressed as upper bounds on the absolute values of velocity, acceleration and jerk. The algorithm has been tested in simulation yielding good results, which have also been compared with those provided by another important trajectory planning technique.

Trajectory Optimization via Simulation Using Incremented Genetic Algorithms

In this paper, a new method for the numerical calculation of optimal, or nearly optimal, solutions to trajectory problems is presented. This method, which is applicable to a large category of problems, is based on a modified genetic algorithm that is an optimum-seeking method that attempts to apply the rules of biological evolution, mutation, and selection to technical optimization problems. The modification is achieved by repeatedly applying a global optimization genetic algorithm method while incrementing the number of design variables (i.e., increasing the size of the chromosomes). The trajectory data used for the optimization process are produced via simulation of the equations of motion using numerical integration techniques. The trajectory optimization technique is tested on a supersonic missile moving in the vertical plane but it is equally applicable to three-dimensional trajectories with minimum formulation time and effort, and ideally suited for parallel computing. The results show the successful application of the incremented genetic algorithm to trajectory problems and its superiority over the fixed chromosome-size genetic algorithm technique.

Genetic Algorithm and Calculus of Variations-Based Trajectory Optimization Technique

A genetic algorithm is used cooperatively with the Davidon‐Fletcher‐Powell penalty function method and the calculus of variations to optimize low-thrust, Mars-to-Earth trajectories for the Mars Sample Return Mission. The return trajectory is chosen thrust-coast-thrust a priori, has a fixed time of flight, and is subject to initial and final position and velocity equality constraints. The global search properties of the genetic algorithm combine with the local search capabilities of the calculus of variations to produce solutions that are superior to those generated with the calculus of variations alone, and these solutions are obtained more quickly and require less user interaction than previously possible. The genetic algorithm is not hampered by ill-behaved gradients and is relatively insensitive to problems with a small radius of convergence, allowing it to optimize trajectories for which solutions had not yet been obtained. The use of the calculus of variations within the genetic algorithm optimization routine increased the precision of the final solutions to levels uncommon for a genetic algorithm.

Optimal Guidance for the Space Shuttle Transition

A guidance method for the space shuttle's transition from hypersonic entry to subsonic cruising flight is presented. The method evolves from a numerical trajectory optimization technique in which kinetic energy and total energy (per unit weight) replace velocity and time in the dynamic equations. This allows the open end-time problem to be transformed to one of fixed terminal energy. In its ultimate form, 'E-Guidance' obtains energy balance (including dynamic-pressure-rate damping) and path length control by angle-of-attack modulation and cross-range control by roll angle modulation. The guidance functions also form the basis for a pilot display of instantaneous maneuver limits and destination. Numerical results illustrate the E-Guidance concept and the optimal trajectories on which it is based.

Genetic Algorithm and Calculus of Variations-Based Trajectory Optimization Technique

Genetic Algorithm and Calculus of Variations-Based Trajectory Optimization Technique

Trajectory Optimization for Air-to-Surface Missiles with Imaging Radars

The use of trajectory optimization techniques is presented for the terminal guidance of an air-to-surface missile using a Doppler beam sharpening (DBS) radar seeker. The terminal guidance problem is characterized by a stealthy terrain-following phase that is followed by a climb and dive onto the target (a bunt trajectory). The imaging properties of DBS radars impose additional azimuth plane constraints on the trajectory that have to be incorporated into the optimization process. The various mission phases are interrelated, and the performance objectives come into conflict with the hardware constraints. The trajectory optimizer is used to generate offline open-loop controls that satisfy the various mission requirements. Numerical examples are used to illustrate the method and its efficacy.

Optimal, Low-Thrust Earth-Orbit Transfers Using Higher-Order Collocation Methods

A trajectory optimization technique based upon higher-order collocation is used to solve optimal, low-thrust, Earth-orbit transfer problems. The optimal control problem solved is defined, and the solution method for solving this problem is described. For several example cases analyzed, a spacecraft is transferred from low Earth orbit to a variety of final mission orbits. A range of thrust accelerations from approximately 1 to 10 - 3 g was used. A comparison is made between the optimal transfers found in this work and the transfers found by using analytical blended control methods. Finally, conclusions drawn from this work are discussed.

Windshear recovery using thrust vectoring

This study presents a preliminary evaluation of the application of Thrust Vectoring (TV) technology for improving windshear recovery of civil aircraft in microburst windshear encounters. A numerical trajectory optimization technique based on the method of multiple shooting is applied to achieve a microburst escape with minimal loss of altitude for a Boeing 727 type aircraft. Although the application of TV does result in a slight improvement in recovery altitude, there is also a price to be paid in the sense that TV utilization significantly hampers the ability to gain specific energy during recovery. Maintaining an energy buffer during a microburst encounter is highly desirable to achieve robustness with respect to uncertainness in windshear size and strength. TV aided windshear recovery has also been compared with an alternative approach to improve the windshear survivability capability, namely, the application of lateral maneuvering. The comparison shows that lateral maneuvering is not only a far more effective means to improve the recovery altitude, it also vastly improves energy preservation. In a view of the results established in this study, we feel that TV does not hold out great promise as a useful means to truly improve windshear recovery.

211 Trajectory optimization techniques and software implementation

211 Trajectory optimization techniques and software implementation

Trajectory Optimization Techniques and Software Implementation

In the past years many in-depth studies on new launch vehicles with airbreathing engines have been conducted worldwide. The extreme performance requirements and the small design margins for such vehicles have in particular renewed the interest in trajectory optimization. The paper reports on recent progress in the state of the art of numerical techniques and their implementation. Questions of robustness, ease of use, efficiency are addressed; user-and software requirements as well as a particular architecture are described. The prototype of an advanced user-interface is presented. The optimal ascent of an advanced launch vehicle is discussed as an example.

Collisions in space: A retrospective overview of ISAS studies

A chronological review of studies in ISAS concerning collisions in space is presented. The collision probability in space with artificial orbiting bodies was estimated, and a Space Traffic Control System was proposed, in 1971. The design of a space station for safety against collision hazards was discussed in 1972. A trajectory optimization technique for low-thrust multiple rendezvous mission in order ti sweep space debris around the earth was developed in 1977. In 1984, the collision probability was reestimated using space bedris data accumulated for more than a decade. Several experimental projects in ISAS, such as hypervelocity impact experiments using a railgun system, sampling and measuring of alumina particles in exhaust plume of solid-propellant propellant rocket motors, and a result of analysis on the behavior of such alumina particles in orbit are also introduced.

Decarburization of stainless steel: Part II. A mathematical model and a process optimization for industrial scale systems

A mathematical model is presented for describing the bath temperature and composition trajectories for the decarburization of stainless steel for processes operating on an indus-trial scale. The model is based on a set of component balances, written with the aid of driving force expressions, the appropriate equilibrium relationships and the heat balance. The resultant set of ordinary differential equations were solved numerically. The predic-tions based on the model were compared with experimentally measured bath composition and temperature paths obtained for the operation of a 40 ton electric furnace. The predic-tions and measurements showed very good agreement. The mathematical model was then combined with a trajectory optimization technique to compute the optimal blowing programi .e., oxygen-argon content of the gas supplied) such that the total cost of the operation is minimized. This calculation was repeated for a series of cost factors.

Optimal Guidance for the Space Shuttle Transition

A guidance method for the space shuttle's transition from hypersonic entry to subsonic cruising flight is presented. The method evolves from a numerical trajectory optimization technique in which kinetic energy and total energy (per unit weight) replace velocity and time in the dynamic equations. This allows the open end-time problem to be transformed to one of fixed terminal energy. In its ultimate form, E-Guidance obtains energy balance (including dynamic-pressure-rate damping) and path length control by angle-of-attack modulation and cross-range control by roll angle modulation. The guidance functions also form the basis for a pilot display of instantaneous maneuver limits and destination. Numerical results illustrate the E-Guidance concept and the optimal trajectories on which it is based.

Optimal Trajectories for Multidimensional Nonlinear Processes by Iterated Dynamic Programming

A trajectory optimization technique for multidimensional nonlinear processes is presented. Problems which are cast in a discrete-time mold are considered. The method is based on dynamic programming and employs a combination of the technique of functional approximation and the method of region-limiting strategies. The cost function at each stage is approximated by a quadratic polynomial in a region which is restricted to be of a size judged appropriate to reduce the error in the approximation. Minimal costs are evaluated at a set of points, called base points. A new control trajectory and an improved state trajectory are then generated within an extrapolation region. The iterative application of this procedure yields an optimal trajectory. Contained in the algorithm is a simple procedure which eliminates matrix inversion to determine the coefficients of the approximating polynomial. The present algorithm is applicable to problems with one bounded control action. It accounts for inequality constraints on state variables in a straightforward manner. The algorithm is applied to solve a number of trajectory optimization problems.

Trajectory optimization techniques in chemical reaction engineering. II. Comparison of the methods

AIChE JournalVolume 18, Issue 4 p. 867-869 Communication to the Editor Trajectory optimization techniques in chemical reaction engineering. II. Comparison of the methods R. K. Jaspan, R. K. Jaspan Department of Chemical and Petroleum Engineering University of Pittsburgh, Pittsburgh, Pennsylvania 15213Search for more papers by this authorJ. Coull, Corresponding Author J. Coull Department of Chemical and Petroleum Engineering University of Pittsburgh, Pittsburgh, Pennsylvania 15213Department of Chemical and Petroleum Engineering University of Pittsburgh, Pittsburgh, Pennsylvania 15213Search for more papers by this author R. K. Jaspan, R. K. Jaspan Department of Chemical and Petroleum Engineering University of Pittsburgh, Pittsburgh, Pennsylvania 15213Search for more papers by this authorJ. Coull, Corresponding Author J. Coull Department of Chemical and Petroleum Engineering University of Pittsburgh, Pittsburgh, Pennsylvania 15213Department of Chemical and Petroleum Engineering University of Pittsburgh, Pittsburgh, Pennsylvania 15213Search for more papers by this author First published: July 1972 https://doi.org/10.1002/aic.690180440Citations: 7AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume18, Issue4July 1972Pages 867-869 RelatedInformation

Trajectory Optimization for the Nonlinear Combined Estimation and Control Problem

A trajectory optimization technique for the nonlinear combined estimation and control problem with quadratic cost is presented. Rather than arbitrarily invoking a separation principle, which is not generally valid for nonlinear systems, controller and estimator parameters are chosen simultaneously with the system nominal control and nominal trajectory to minimize a quadratic performance index of the Bolza form. Initially, a linear filter is used in the feedback loop. Subsequently, a nonlinear extended Kalman filter is utilized with this improved nominal trajectory. It is shown in the numerical example considered that the nonlinear system performance is much more sensitive to the choice of the nominal trajectory than to the selection of a more accurate nonlinear estimator.

Trajectory optimization techniques in chemical reaction engineering: I. Boundary condition iteration methods

AbstractA new generalization of boundary condition iteration (BCI) methods is developed, based on a suggestion of Denn and Aris. This simplifies to Horn's equation for up to two state equations and to previous boundary iteration methods when is explicitly soluble for as a function of the constants, the state and the adjoint variables. The new general method is also applicable when this latter function is unobtainable.Distinct improvement in the convergence rates of the existing BCI methods has been obtained through the introduction of a correction specific to each state variable.A convergence procedure for use with Horn's equation is proposed and the resultant algorithm has desirable properties.