Finite Thrust Research Articles

Dynamic modeling and analysis of a large-scale hoop-column antenna using the referenced nodal coordinate formulation

Hoop-column antenna (HCA) is a popular type of deployable antenna in aerospace engineering. Extracting a precise model and performing a comprehensive dynamic analysis is the foundation for ensuring the HCA works properly in complex space environments. However, the multiscale characteristics and strong nonlinearities make most modeling methods invalid because of low accuracy and efficiency. Dynamic modeling and analysis are conducted for HCA constructed by beams and cables using the referenced nodal coordinate formulation in this work. The straight and initially curved beam elements and pre-tensioned cable elements are introduced based on the absolute nodal coordinate formulation. The HCA is described in a non-inertial frame to separate the overall motion and deformations. The nonlinear dynamic equations of the HCA are firstly linearized around its tensioned equilibrium configuration and natural frequencies and mode shapes are analyzed. The symmetric and antisymmetric mode shapes are observed and corresponding characteristics are revealed. The mistuning ratio is introduced to study the mistuning properties. The nonlinear dynamic responses of the HCA for orbital transfer under finite thrust are simulated. The present work provides a general dynamic modeling approach for linear and nonlinear dynamic analysis of the HCA. The numerical results also provide some guidance for the parameter design and vibration control of the HCA.

Pareto-optimal control of relative motion in the orbital maneuvering problem of spacecraft with finite thrust

To solve the time-free rendezvous problem of two spacecraft, the multi-criteria optimization of the relative motion trajectory of the linear motion model in an orbiting cylindrical reference frame is studied. The equations for describing the secular and periodic parameters of the relative motion are obtained. The structure of the nominal control program for the longitudinal motion control variant with finite transversal thrust is investigated in some detail, and its analytical solutions are obtained. An algorithm for solving the Pareto-optimal control program for arbitrary boundary conditions and thrust control acceleration values in the standard time including maneuver time and total time is developed. The algorithm uses the Pareto optimal method to achieve two kinds of multi-objective optimization (total time optimization and fuel optimization). The numerical calculation results on the geostationary planar orbit parameter correction variants are given.

Research for Optimal Programs for Controlling the Relative Motion of a Spacecraft with Limited Thrust

The problem of optimal control of the relative motion of a spacecraft with a finite thrust engine in arbitrary circumcircular orbits using the Pontryagin’s maximum principle is considered. Motion is studied in an orbital cylindrical coordinate system, using variables written in the form of secular and periodic components of relative motion in the orbital plane. The main attention is paid to the analysis of the optimal control structure with a free and transversal orientation of the thrust vector in the presence of passive areas on the trajectory. As a criterion for choosing the optimal control, the motor operating time of the corrective motors was considered. Characteristic control structures for various areas of initial driving conditions have been determined, and estimates of the marginal costs of motor time have been obtained.

An Optimized Plane-Change Solution for Microsatellite Formation Flying

The cross-track baseline requirement in a satellite formation flying mission requires an orbital plane-change maneuver during the acquisition phase. While the optimization of orbital plane-change maneuver has been well studied, the constraints of the propulsion system, such as the power limitation, battery charging cycle, and finite thrust durations, are often neglected. In addition, the self-pressurized chemical propulsion system used in this satellite mission experiences a performance degradation effect over prolonged use in terms of specific impulse and thrust force. To overcome these constraints on the orbital plane-change maneuver process, this article proposes an optimized algorithm by minimizing Edelbaum’s equation using a selective constrained ensemble Kalman filter that includes propulsion’s performance variation and firing duration constraints. Monte Carlo simulations have been conducted to benchmark the proposed method against the theoretical impulsive thrusts in terms of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta V$ </tex-math></inline-formula> , and the commonly adopted natural precession-based plane-change method in terms of the total time taken. Results from the simulation have shown that the required <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta V$ </tex-math></inline-formula> of this proposed method is similar to the theoretical impulsive burn method. In addition, it requires a 75% shorter time than the natural precession method in achieving the desired plane change and is 36% more fuel efficient than the direct firing method. Furthermore, this study has verified that this algorithm can be directly integrated with the existing formation control algorithm.

Indirect optimization for finite thrust orbit transfer and cooperative rendezvous using an initial guess generator

In this paper, the finite thrust nonplanar transfer and cooperative rendezvous are investigated. The indirect method is adopted in the optimization process. The energy-optimal to fuel-optimal continuation is performed by using the homotopy method. An initial guess generator (IGG) is designed based on a simplified dynamical model to overcome the difficulty in providing the initial guess. Some reasonable simplifications on the dynamical model are applied based on the near-circular near-coplanar transfer. The initial guess can be obtained by analytically solving a system of linear equations. Therefore, the IGG has high computational efficiency. Moreover, the terminal time and state are estimated by IGG to deal with the cooperative rendezvous problem. Numerical examples are presented and the validity of the “IGG-Homotopy” combined algorithm is verified.

Optimal transfer orbit design of spacecraft with finite thrust based on Legendre pseudospectral method

This article explores the application of the Legendre pseudospectral method to spacecraft orbital transfer with finite thrust optimization problem. Firstly, the model of the orbital transfer optimization control problem was established, while equations of motion were simplified based on some hypotheses. The performance was optimized to minimize the cumulative fuel consumption. The control variable was the thrust attack angle, and terminal state variable constraints included path angle, altitude, and velocity constraints. Then, the optimal control problem was transformed into a nonlinear programming problem (NLP) using the Legendre pseudospectral method. The dynamic optimization problem was transformed into a static parameter optimization problem. The state variables and control variables were selected as the optimal parameters at all collocation nodes. Lastly, the parameter optimization problem was solved using the SNOPT (Sparse Nonlinear Optimizer) software package. The SNOPT software package shows high convergence for a nonlinear programming problem. During the simulation, it was noted that the Legendre pseudospectral method is not sensitive to orbital transfer initial conditions. It was also observed that the optimal solutions of the orbital transfer optimization problem are fairly good in robustness. Therefore, the Legendre pseudospectral method is a viable approach to the spacecraft orbital transfer with a finite thrust optimization problem. The orbit optimization method proposed in this paper can also provide reference and guidance for solving other interplanetary orbital transfer optimization problems.

Reachable Set of Spacecraft With Finite Thrust Based on Grid Method

Reachable sets are sets of position states that the spacecraft can reach from its initial state under mass and time constraints. In this study, a method to solve the reachable set of spacecraft with finite thrust is proposed. Solving reachable sets can be transformed into solving its boundary surface using the distance fields over grids method. A grid is established to represent the boundary surface, and each node on the grid can be determined by an optimal control problem. A modified sequential convex programming with nonlinear dynamic correction is used to solve optimal control problems with a theoretical proof of convergence. Moreover, the symmetry plane of the reachable set is found to reduce the computation in applications. The method is tested with two types of orbits, namely, geosynchronous and highly elliptical orbits, and the properties of the spacecraft’s reachability in two orbits are obtained. Numerical simulations show that the proposed method can succeed in solving reachable sets under mass and time constraints in two types of initial orbits even when a long period is considered.

Delta-V and ground metric investigations for responsive satellite formation flying

This paper studies a responsive mission involving a dual spacecraft flying formation system. A graphical solution based on the selective dependent variables for acquiring the local and global optimum impulsive velocities is proposed to obtain the minimum orbital transfer velocity changes as well as improving its nadir ground coverage. These studies are conducted using finite thrust of single, dual and three-impulse modes under the condition of unperturbed environment. Based on the obtained results, the relationship between the number of impulses applied and the amount of fuel consumed to retain the leader-follower formation separation is inversely proportional. Whereas, in all cases of the ground metric studies, the formation inclination has shown a dominant influence over other parameters for maximizing the total nadir ground coverage and the swath length.

Influence of Steering Angle Profiles on the Orbit Transfer Trajectory

This paper considers a planar orbit transfer trajectory design problem using finite thrust modeling. In this problem, the steering angles associated with the thrust direction are calculated from the predetermined profile format, and the unknown parameters in the profile are directly optimized. Three profile formats that were implemented in previous lunar exploration missions are considered. In addition, a steering angle profile defined in the rotating frame and the optimal steering angle profile are newly studied to compare the performances. To this end, the direct parameter optimization problem and the indirect optimization problem are formulated, and the characteristics of the steering angle profile and its influence on the transfer trajectory are analyzed.

Optimization of finite-thrust trajectories with fixed angular distance

The article presents a new approach for optimizing the trajectories of spacecraft with finite thrust. The recently proposed angular variable – the auxiliary longitude – is considered as an independent variable. The advantages of using this variable to optimize a trajectory with a fixed angular distance and free transfer duration are shown. A complete system of necessary optimality conditions for the considered representation of the optimal control problem is presented. An approach to solving the minimum-fuel problem is considered that is based on the sequential solution of the problems of optimizing the limited power trajectories and the trajectories with a constant exhaust velocity. This approach includes verifying the existence of a solution to the minimum-fuel problem by solving the thrust minimization problem. We present a method for optimization of the finite-thrust trajectories with constant exhaust velocity, based on the application of the maximum principle and the continuation method, which does not require the user to specify an initial guess for the initial values of costate variables. The article presents numerical examples to illustrate the application of such approach, and discusses the properties of the obtained optimal trajectories.

Design of fuel-saving lunar captures using finite thrust and gravity-braking

The emergence of microsatellite missions leaving low Earth orbits to pursue the exploration of the solar system is currently leading the development of new techniques of mission analysis. In fact, the small size of these satellites affects the performance of the propulsion systems, in particular because of the limited mass budget available for propellant stowage. This work focuses on the analysis of lunar microsatellite missions, proposing a method to design fuel-saving lunar captures. The solution is aimed at establishing the capture conditions taking advantage of the gravity-braking effect of the Moon, before injecting the spacecraft into a lunar capture orbit. The maneuvers are developed in the form of guidance laws, implemented based on primer vector theory and on the definition of a quasi-integral of motion for the elliptic restricted 4-body problem. The method is verified, by means of numerical analysis, on a 6U CubeSat equipped with a monopropellant thruster and considering deployment conditions compatible with microsatellite lunar missions. The achievement of lunar capture in a few days from the deployment, with propellant usage on the order of tenths of grams, corresponding to ΔV on the order of some meters per second, indicates the suitability of the method to expand the possible profiles of microsatellite missions to the Moon.

A fast computational method for the landing footprints of space-to-ground vehicles

Fast computation of the landing footprint of a space-to- ground vehicle is a basic requirement for the deployment of parking orbits, as well as for enabling decision makers to develop real-time programs of transfer trajectories. In order to address the usually slow computational time for the determination of the landing footprint of a space-to-ground vehicle under finite thrust, this work proposes a method that uses polynomial equations to describe the boundaries of the landing footprint and uses back propagation (BP) neural networks to quickly determine the landing footprint of the space-to-ground vehicle. First, given orbital parameters and a manoeuvre moment, the solution model of the landing footprint of a space-to-ground vehicle under finite thrust is established. Second, given arbitrary orbital parameters and an arbitrary manoeuvre moment, a fast computational model for the landing footprint of a space-to-ground vehicle based on BP neural networks is provided. Finally, the simulation results demonstrate that under the premise of ensuring accuracy, the proposed method can quickly determine the landing footprint of a space-to-ground vehicle with arbitrary orbital parameters and arbitrary manoeuvre moments. The proposed fast computational method for determining a landing footprint lays a foundation for the parking-orbit configuration and supports the design of real-time transfer trajectories.

Trajectory Optimization of a Lunar Direct Landing from Halo-Orbits

This paper illustrates the problem of optimizing the trajectory of a direct landing from a halo orbit to the lunar surface with given coordinates. The study is considered in circular restricted three body problem. The descent operation consists of two stages: change in velocity for descent from orbit (the descent is considered as an instantaneous change in velocity of a spacecraft) and the main stage of braking, which is considered as a task of optimal performance of the spacecraft with finite thrust. The formulation and solution of the optimal control problem is presented, as well as numerical examples illustrating the accessibility of various regions of the lunar surface and examples of landing trajectories.

How Many Impulses Redux

A central problem in orbit transfer optimization is to determine the number, time, direction, and magnitude of velocity impulses that minimize the total impulse. This problem was posed in 1967 by T. N. Edelbaum, and while notable advances have been made, a rigorous means to answer Edelbaum’s question for multiple-revolution maneuvers has remained elusive for over five decades. We revisit Edelbaum’s question by taking a bottom-up approach to generate a minimum-fuel switching surface. Sweeping through time profiles of the minimum-fuel switching function for increasing admissible thrust magnitude, and in the high-thrust limit, we find that the continuous thrust switching surface reveals the N-impulse solution. It is also shown that a fundamental minimum-thrust solution plays a pivotal role in our process to determine the optimal minimum-fuel maneuver for all thrust levels. Remarkably, we find that the answer to Edelbaum’s question is not generally unique, but is frequently a set of equal-Δv extremals. We further find, when Edelbaum’s question is refined to seek the number of finite-duration thrust arcs for a specific rocket engine, that a unique extremal is usually found. Numerical results demonstrate the ideas and their utility for several interplanetary and Earth-bound optimal transfers that consist of up to eleven impulses or, for finite thrust, short thrust arcs. Another significant contribution of the paper can be viewed as a unification in astrodynamics where the connection between impulsive and continuous-thrust trajectories are demonstrated through the notion of optimal switching surfaces.

A Fast Computational Method for the Minimum Duration Transfer Trajectories of Space-to-Ground Vehicles

On the conditions that the spacecraft engine is in finite thrust mode and the maneuver time is given, it takes a long time to compute the minimum duration transfer trajectories of space-to-ground vehicles, which is mainly because the initial values of the adjoint variables involved in the optimization model have no definite physical meanings and the model is sensitive to them. In order to develop space-to-ground transfer trajectory programmes in real time in an uncertain environment for the decision makers, we propose a fast method for computing the minimum duration transfer trajectories of space-to-ground vehicles with the given position of the landing point and the arbitrary maneuver point. First, the optimization model based on the hybrid method is established to compute the minimum duration transfer trajectory. Then, the region composed of maneuverable points is gridded and the initial values of the adjoint variables and the values of partial state variables of the minimum duration transfer trajectories at all gridded points are computed and saved to a database. Finally, the predicted values of the initial values of the adjoint variables and the values of partial state variables at any maneuver point within the region composed of maneuverable points are computed by using a binary cubic interpolation method. Finally, the minimum duration transfer trajectory is obtained by the hybrid method which takes the neighborhood of the predicted values as the search ranges of the initial values of the adjoint variables and the values of partial state variables. Simulation results demonstrate that the proposed method, which requires only 2.93% of the computational time of the hybrid method, can improve substantially the computational time of the minimum duration transfer trajectory of a space-to-ground vehicle under the guarantee of ensuring accuracy. The methodology of converting the time domain into the space domain is well applied in this paper.

Hybrid QPSO and SQP Algorithm with Homotopy Method for Optimal Control of Rapid Cooperative Rendezvous

AbstractThe fuel-optimal rapid cooperative rendezvous problem between two spacecraft under finite thrust, based on the indirect method, was investigated and converted into a two-point boundary valu...

Optimal inspector satellite guidance to quasi-hover via relative teardrop trajectories

Metaheuristic methods and direct methods are used to find minimum time and minimum fuel solutions to inject an inspector satellite into and maintain a relative teardrop trajectory with respect to a resident space object in geosynchronous orbit. The optimal control problem is transcribed to a parameter optimization problem and solved with a particle swarm optimization algorithm, which is then fed as an initial guess for two different nonlinear programming problem solvers, one of which is the Radau pseudospectral method. Two angles are used to represent the direction of the finite thrust for the two burns in three dimensions for both the minimum time and minimum fuel formulations, and an analytic expression for the final six Hill-Clohessy-Wiltshire states after the two finite burns is employed to find the optimal burn directions and burn times. Thus, multiple optimization methods are used to find both minimum time and minimum fuel solutions, providing mission planners with multiple tools to enable optimal quasi-hovering in a prescribed three-dimensional teardrop in order to inspect and characterize a resident space object.

Finite element thrust line analysis of cracked axisymmetric masonry domes reinforced with tension rings

Masonry domes under self-weight are subjected to hoop tensile forces in their lower portion, which lead to vertical cracks appearing along the dome's meridian planes. Almost always, a close inspection reveals such hoop tension cracks. At the final extent of cracking, the dome stands as a series of arches with a common key stone. Tension rings put around the perimeter close to the base of a dome are commonly used to arrest these cracks. However, the behaviour of domes with tension rings is not well understood. This paper presents an enhancement of the finite element thrust line analysis in order to analyse masonry domes with tension rings. The development of a new element to include the effect of both hoop and meridian tension cracks, utilising orthotropic material properties and a penalty approach, is discussed in detail. The case study for a hemispherical cupola with fibre reinforced polymer (FRP) hoop rings shows that the proposed method can accurately estimate the thrust line of the dome.

Minimum time multiple-burn optimization of an upper stage with a finite thrust for satellite injection into geostationary orbit

In this paper, the problem of minimum time multiple-burn optimization of an upper stage with a limited thrust, and engine restart capability for satellite injection into geostationary orbit are considered. The goals are to find thrust vector angle, times of the engine firings, and optimal duration of active phases of the upper stage to minimize fuel consumption and meet the desired boundary conditions. Various flight sequences with multiple burns, from two burns up to six burns, are considered. Also, the optimal trajectory for each sequence is derived. To solve the multi-point boundary value problem, an improved indirect shooting method with high performance is presented and used for an optimal solution. All in all, this novel method presented for multi-burn problem, not only with a very good accuracy, but also with a very fast convergence to the desired end conditions.

Rapid Generation of Time-Optimal Trajectories for Asteroid Landing via Convex Optimization

This paper presents a new method for rapid generation of time-optimal trajectories for asteroid landing via convex optimization. To overcome the nonconvex difficulty due to the free-flight time, a minimum-landing-error problem that can be convexified is proposed to serve as a gateway for solving the time-optimal control problem. The proposed method solves the reduced minimum-landing-error problem via convex optimization first and then searches for the minimum flight time through a combination of extrapolation and bisection methods. The effectiveness of the proposed methodology for rapidly generating time-optimal trajectories is validated through simulations of spacecraft landing on two asteroids with both finite thrust and low thrust.