Constrained Trajectories Research Articles

A Methodology for the Computation of Constrained Orbits and its Application to the Design of Solar System Trajectories

The purpose of this paper is the development of a procedure for the determination of constrained trajectories of spacecraft in the Solar System and its application to the computation of orbits, such as heteroclinic connections in the Sun-Earth-Moon system, that cannot be computed using only a two or three-body problem approach. Starting from a nominal trajectory computed in a simplified force model, the aim is to compute a new one, close to the original, but satisfying more realistic equations of motion. The formulation adopted allows for the existence of discontinuities in velocity and specific constraints for the states of the orbit, including boundary conditions. The continuity requirement of the orbit is accomplished in position by a differential correction process. The discontinuities in velocity are minimized using an optimization procedure. Some significant applications are presented, mainly regarding libration point orbit missions. It will be shown how using the initial input, the constraints and weights, an orbit, initially determined in the Circular Restricted Three–Body Problem, can be refined in order to fulfill the equations of motion of the Restricted n–Body Problem and specific scopes. In all the cases, an accurate initial guess is a key point.

Shape-Based Approximation of Constrained Low-Thrust Space Trajectories Using Fourier Series

Shape-Based Approximation of Constrained Low-Thrust Space Trajectories Using Fourier Series

Shape Based Approximation of Constrained Low-Thrust Space Trajectories using Fourier Series

Covers advancements in spacecraft and tactical and strategic missile systems, including subsystem design and application, mission design and analysis, materials and structures, developments in space sciences, space processing and manufacturing, space operations, and applications of space technologies to other fields.

Optimization of Impulsive Trajectories from a Circular Orbit to an Excess Velocity Vector

Feasible transfer trajectories are constructed to serve as initial guesses for determining constrained optimal impulsive escape trajectories, from a circular orbit to a target hyperbolic excess velocity vector. The proximity of these feasible solutions to their corresponding optima are quantified. The objective of this work is to improve the feasible solutions, thereby enabling general anytime escape trajectories to be targeted that result in substantially reduced fuel expenditure. The procedure is currently restricted to one- and three-impulse escape trajectories from a circular parking orbit. Two separate three-impulse methods are presented, each specific to whether the time of flight is free or fixed. Numerical results, obtained using nonlinear programming algorithms, are presented to validate the robustness of the method. Analysis and results are nondimensional, and therefore are applicable to departures from a circular orbit about any celestial body.

Control of constraint forces and trajectories in a rich sensory and actuation environment

A simple control strategy is proposed and applied to a class of non-linear systems that have abundant sensory and actuation channels as in living systems. The main objective is the independent control of constrained trajectories of motion, and control of the corresponding constraint forces. The peripheral controller is a proportional, derivative and integral (PID) controller. A central controller produces, via pattern generators, reference signals that are the desired constrained position and velocity trajectories, and the desired constraint forces. The basic tenet of the this hybrid control strategy is the use of two mechanisms: 1. linear state and force feedback, and 2. non-linear constraint velocity feedback - sliding mode feedback. The first mechanism can be envisioned as a high gain feedback systems. The high gain attribute imitates the agonist–antagonist co-activation in natural systems. The strategy is applied to the control of the force and trajectory of a two-segment thigh-leg planar biped leg with a mass-less foot cranking a pedal that is analogous to a bicycle pedal. Five computational experiments are presented to show the effectiveness of the strategy and the performance of the controller. The findings of this paper are applicable to the design of orthoses and prostheses to supplement functional electrical stimulation for support purposes in the spinally injured cases.

Highly Constrained Optimal Launch Ascent Guidance

This paper is concerned with theoretical and algorithmic development for fast generation of optimal launch ascent trajectories through the atmosphere and guidance, particularly in highly constrained missions. The theoretical investigation oers an insight into what may be called atmospheric primer vector theory, in which the critical roles of the primer vector in the determination of optimal endo-atmospheric ascent trajectories are explored in some detail. The algorithmic development focuses on ascent missions tightly constrained by aerodynamic bending moment limit. The expansion of algorithm capability aims at rapid and automated generation of such highly constrained optimal ascent trajectories. When carried on-board, this process realizes optimal closed-loop ascent guidance. An in-depth application of the theoretical and algorithmic developments in this paper is conducted using the vehicle model and mission profile of the Ares I Crew Launch Vehicle. Extensive simulations are conducted to compare optimal closed-loop with conventional open-loop ascent guidance approaches. The tests oer some rich and revealing comparisons on how each approach performs on inequality constraint observance and injected mass in the presence of winds and dispersions.

Cascade Prediction Filters With Adaptive Zeros to Track the Time-Varying Resonances of the Vocal Tract

In this paper, a simple and reliable technique is proposed to track vocal tract resonances in continuous speech. The approach is based on the use of predictor filters with adaptive zeros whose constrained trajectories guarantee the successful tracking of the frequency and the damping of each resonance. The zeros are adapted using a gradient-based algorithm to minimize an instantaneous prediction residual according to the principle of minimal disturbance yielding an adaptive structure capable of tracking fast-changing resonance parameters.

Optimal Trajectory Reconfiguration and Retargeting for Reusable Launch Vehicles

Reconfigurable inner-loop control laws improve the fault tolerance of a vehicle to control effector failures. The control effectors can produce significant perturbations to the nominal forces and can also affect the flight conditions over which the vehicle can be controlled. Three-degree-of-freedom dynamical models typically include wing-body aerodynamic force effects but ignore the aerodynamic forces produced by the control surfaces. These trim effects are a component of 6-degree-of-freedom models; however, such models are cumbersome for the purposes of trajectory optimization. In this work, a method for including the trim effects as well as control-induced trajectory constraints in a 3-degree-of-freedom model is presented. The method uses nonlinear control allocation to determine the control effector positions, required to rotationally balance the vehicle, and uses that information to compute force perturbations and new control-induced trajectory constraints. The contributions of the control effectors to the vehicle lift and drag are computed, as well as constraints that define regions of the flight envelope where the vehicle cannot be rotationally balanced. An example is presented that includes control failure effects in a maximum downrange trajectory optimization problem. Constrained trajectories are planned by solving an optimal control problem using a Legendre pseudospectral method and state-dependent constraints, resulting from a control failure, are included. Results show that these methods hold the potential to enhance safety margins even when control effectors have failed.

Generalized inverse approach for static behaviour of constrained structures

The problem of structure subjected to certain external causes is established by a mathematical system consisting of equilibrium equations and prescribed constraint conditions. The satisfaction of constrained trajectories depends on an exact calculation of constraint forces required for satisfying the constraints. Most work related to this subject calculate the constraint forces using a numerical approach including undetermined Lagrange multipliers. However, the Lagrange multipliers method has difficulty in handling systems subjected to several tens of constraints. Starting from the difference in potential energy of structures with and without constraints, this study derives the equilibrium equation of constrained structures. It is shown that the derived equations coincide with the results obtained from previous work. The effectiveness and simplicity of the proposed method are illustrated through the static control of continuous structure and the reanalysis of a simple truss structure with a floating substructure. Copyright © 2005 John Wiley & Sons, Ltd.

Analytical method for constrained mechanical and structural systems

The objective of this study is to present an accurate and simple method to describe the motion of constrained mechanical or structural systems. The proposed method is an elimination method to require less effort in computing Moore-Penrose inverse matrix than the generalized inverse method provided by Udwadia and Kalaba. Considering that the results by numerical integration of the derived second-order differential equation to describe constrained motion veer away the constrained trajectories, this study presents a numerical integration scheme to obtain more accurate results. Applications of holonomically or nonholonomically constrained systems illustrate the validity and effectiveness of the proposed method.

Onboard Generation of Three-Dimensional Constrained Entry Trajectories

A methodology for very fast design of three-degree-of-freedom (3DOF) entry trajectories subject to all common inequality and equality constraints is developed. The approach makes novel use of the quasi-equilibrium glide phenomenon in lifting entry as a centerpiece for effective and efficient enforcement of the inequality constraints. The highly constrained nonlinear trajectory planning problem is decomposed into two sequential one-parameter search problems. The algorithm is able to generate a complete and feasible 3DOF entry trajectory of 25-min flight time in about 2-3 s on a desktop computer, given the entry conditions, values of constraint parameters, and final conditions. High-fidelity numerical simulations with a reusable launch vehicle model for various entry missions and trajectory planning using the space shuttle model are presented to demonstrate the capability and effectiveness of the algorithm.

Freezing degrees of freedom under stress: Kinematic evidence of constrained movement strategies

The present study investigated the effect of psychological stress imposed on movement kinematics in a computer-simulated batting task involving a backward and forward swing of the forearm. The psychological stress was imposed by a mild electric stimulus following poor performance. Fourteen participants hit a moving ball with a horizontal lever and aimed at a distant target with as much accuracy as possible. The kinematic characteristics appearing under stress were delay of movement initiation, small amplitude of movement and low variability of spatial kinematic events between trials. These features were also found in previous studies in which the experimental task required high accuracy. The characteristic kinematics evident in the present study suggested that the movement strategies adopted by the stressed participants were similar to those that appear under high accuracy demand. Moreover, a correlation analysis between the onset times of kinematic events revealed that temporally consistent movements were reproduced under stress. Taken together, the present findings demonstrated that, under psychological stress, movement strategies tend to shift toward the production of more constrained trajectories, as is seen under conditions of high accuracy demand, even though the difficulty of the task itself does not change.

SUPERVISION OF HETEROGENEOUS CONTROLLERS FOR A MOBILE ROBOT

Operating in dynamic environments requires the use of multi-objective (optimization) controllers. This work presents a methodology for supervising a set of controllers designed according to a set of different specifications, in a hierarchical control structure. In this paper, unlike the theory of switching control, a similar structure which focus on choosing the best controller at a given moment is proposed. This methodology is applied to an autonomous mobile robot that follows constrained trajectories. Due to the complex non-linear behavior of the robot, all possible controllers had to be modeled off-line. Then, the on-line supervisor selects an optimal controller to perform the requested task. This method is demonstrated on the soccer robots of RoGiTeam in RoboCup annual competition.

Visuomotor processing as reflected in the directional discharge of premotor and primary motor cortex neurons.

Premotor and primary motor cortical neuronal firing was studied in two monkeys during an instructed delay, pursuit tracking task. The task included a premovement "cue period," during which the target was presented at the periphery of the workspace and moved to the center of the workspace along one of eight directions at one of four constant speeds. The "track period" consisted of a visually guided, error-constrained arm movement during which the animal tracked the target as it moved from the central start box along a line to the opposite periphery of the workspace. Behaviorally, the animals tracked the required directions and speeds with highly constrained trajectories. The eye movements consisted of saccades to the target at the onset of the cue period, followed by smooth pursuit intermingled with saccades throughout the cue and track periods. Initially, an analysis of variance (ANOVA) was used to test for direction and period effects in the firing. Subsequently, a linear regression analysis was used to fit the average firing from the cue and track periods to a cosine model. Directional tuning as determined by a significant fit to the cosine model was a prominent feature of the discharge during both the cue and track periods. However, the directional tuning of the firing of a single cell was not always constant across the cue and track periods. Approximately one-half of the neurons had differences in their preferred directions (PDs) of >45 degrees between cue and track periods. The PD in the cue or track period was not dependent on the target speed. A second linear regression analysis based on calculation of the preferred direction in 20-ms bins (i.e., the PD trajectory) was used to examine on a finer time scale the temporal evolution of this change in directional tuning. The PD trajectories in the cue period were not straight but instead rotated over the workspace to align with the track period PD. Both clockwise and counterclockwise rotations occurred. The PD trajectories were relatively straight during most of the track period. The rotation and eventual convergence of the PD trajectories in the cue period to the preferred direction of the track period may reflect the transformation of visual information into motor commands. The widely dispersed PD trajectories in the cue period would allow targets to be detected over a wide spatial aperture. The convergence of the PD trajectories occurring at the cue-track transition may serve as a "Go" signal to move that was not explicitly supplied by the paradigm. Furthermore, the rotation and convergence of the PD trajectories may provide a mechanism for nonstandard mapping. Standard mapping refers to a sensorimotor transformation in which the stimulus is the object of the reach. Nonstandard mapping is the mapping of an arbitrary stimulus into an arbitrary movement. The shifts in the PD may allow relevant visual information from any direction to be transformed into an appropriate movement direction, providing a neural substrate for nonstandard stimulus-response mappings.

A Heuristic Study of Relegation of Control in Constrained Robotic Systems

A major control problem in the robotic field is the simultaneous and independent control of constrained trajectories and forces of constraint. The trajectories and the forces are related through the mechanical structure of the system. The task of the controller is to influence the mechanical coupling and allow separate control of the trajectories and the forces. A feasible control strategy is by relegation of control to the state or to the input. Relegation by inputs implies assigning the control of trajectories and forces to independent groups of inputs. In this paper, exact and approximate input relegation strategies are investigated. The effectiveness of the input relegation strategy is tested by digital computer simulation of a three link planar robot in a periodic rubbing maneuver.

Interplanetary Trajectory Optimization with Application to Galileo

A procedure for minimizing total impulsive AK 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 A V required to escape from a parking orbit about the launch plant and the A 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 multiconic 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, Venus-Earth gravity assist, and A V Earth gravity assist trajectories.

Constrained Optimum Trajectories with Specified Range

The characteristics of optimum fixed-range trajectories whose structure is constrained to climb, steady cruise, and descent segments are derived by application of optimal control theory. The performance function consists of the sum of fuel and time costs, referred to as direct operating costs (DOC). The state variable is range-to-go and the independent variable is energy. In this formulation a cruise segment always occurs at the optimum cruise energy for sufficiently large range. At short ranges (500 n. mi. and less) a cruise segment may also occur below the optimum cruise energy. The existence of such a cruise segment depends primarily on the fuel flow vs thrust characteristics and on thrust constraints. If thrust is a free control variable along with airspeed, it is shown that such cruise segments will not generally occur. If thrust is constrained to some maximum value in climb and to some minimum in descent, such cruise segments generally will occur. The performance difference between free thrust and constrained thrust trajectories has been determined in computer calculations for an example transport aircraft.

A time-optimal problem with recoverability constraint

This paper introduces the concept of recoverability as a design consideration. A simple time-optimal regulator problem is modified to explore the feasibility of imposing a recoverability constraint as an explicit restriction on the system state during control synthesis. The results are summarized as two theorems. Comparative numerical data for both unconstrained and constrained trajectories are presented.

Solution of the Free-Fall Guidance and Control Problem in N-Body Space

The solution of the free-fall guidance and control problem in an inverse-square central force field is treated as an introduction. Circular, elliptical, parabolic and hyperbolic trajectories are covered. Consideration of the control problem (steering and thrust cutoff signals) is applied to the case of a rocket where only the direction of the thrust vector and thrust cutoff are controllable. Consideration is given to the energy aspects, and to launch and atmospheric conditions as they affect guidance and control. These techniques are then extended to the solution of the free-fall guidance problem in N-body space. In this latter category, Bonnet's Theorem, as extended by Egorov, is covered to show its application to the solution of the N-body problem by superimposing solutions of the inverse-square central force field problem. The theorem is applied to treat the problem of finding possible conic trajectories in two-body space. The problem of finding possible conic and nonconic trajectories in three-body space is also covered and a solution to the guidance problem for these trajectories is included. The libration points of the N-body problem are treated as special cases of the solutions discussed. The equations discussed are valid for a vehicle of negligible mass acted upon by forces due only to N inverse-square central force fields. No attempt has been made to cover the effects of noninverse-square central force fields, although Bonnet's Theorem can be applied to constrained trajectories such as those that will be made possible with a continuous low thrust vehicle.