Low-thrust Orbit Transfer Research Articles

Homotopy method for minimum consumption orbit transfer problem

The numerical resolution of the low thrust orbital transfer problem around the Earth with the maximization of the final mass or minimization of the consumption is investigated. This problem is difficult to solve by shooting method because the optimal control is discontinuous and a homotopic method is proposed to deal with these difficulties for which convergence properties are established. For a thrust of 0.1 Newton and a final time 50% greater than the minimum one, we obtain 1786 switching times.

Optimal low-thrust orbital transfers in a central gravity field

The results of studying some problems of optimization of low-thrust transfers between arbitrary elliptic orbits in a Newtonian gravity field are expounded. An approximate solution obtained by the averaging method is presented. Analytical solutions of the averaged equations are given for a wide class of maneuvers. The problem of constructing numerical solutions to the exact equations of motion of a spacecraft between high orbits is discussed

Fuel-Optimal Transfers Between Coplanar Circular Orbits Using Variable-Specific-Impulse Engines

Minimum fuel circular-to-circular low-thrust orbital transfers with prescribed transfer time and with the thrust always directed parallel to the velocity vector are developed for a variable-specific-impulse system. The changeable specific impulse serves as the control variable. The power input is assumed to be constant, and the resulting thrust magnitude is inversely proportional to the chosen specific impulse. With these engine specifications, and under the assumption that the thrust magnitude always remains small compared to the gravitational forces, it is shown that the optimal specific impulse profile increases linearly with time. The associated optimal thrust magnitude is proportional to the vehicle mass; that is, the thrust per unit of vehicle mass remains constant. The mass fraction obtained with variable specific impulse is shown to be greater than or equal to that obtained with constant specific impulse; the advantage can be significant for orbital transfers with large changes in the semimajor axis.

Planar High-Thrust and Low-Thrust Orbital Transfers from Earth to Europa

We review analytical formulae for low-thrust orbital transfers from a circular initial orbit to a circular target orbit and to escape, using constant thrust applied in the same direction as the velocity vector, or in the opposite direction. The formulae give approximations for the maneuver time, final mass fraction, and evolution of the semimajor axis. For comparison, the associated results for high-thrust Hohmann and Oberth transfers are summarized. A simple computer program employing the aforementioned relationships is a useful tool for analysis of planar planetary and interplanetary space missions. This implementation yields fast and reasonably accurate approximations to trajectory performance boundaries. Consequently, the approach can provide trajectory analysis for each spacecraft configuration during the conceptual space mission design phase. As an example, a mission from Low-Earth Orbit (LEO) to Jupiter’s moon Europa is analyzed.

Nonlinear control tools for low thrust orbital transfer

We consider the problem of orbital transfer under propulsion constraints. The solution is arrived at by appealing to several recent nonlinear control design methods including backstepping, forwarding, and a modified form of Jurdjevic-Quinn control.

Accessibility, stabilizability, and feedback control of continuous orbital transfer.

This paper investigates the problem of low-thrust orbital transfer using orbital element feedback from a control-theoretic standpoint, concepts of controllability, feedback stabilizability, and their interaction. The Gauss variational equations (GVEs) are used to model the state-space dynamics. First, the notion of accessibility, a weaker form of controllability, is presented. It is then shown that the GVEs are globally accessible. Based on the accessibility result, a nonlinear feedback controller is derived that asymptotically steers a vehicle from an initial elliptic Keplerian orbit to any given elliptic Keplerian orbit. The performance of the new controller is illustrated by simulating an orbital transfer between two geosynchronous Earth orbits. It is shown that the low-thrust controller requires less fuel than an impulsive maneuver for the same transfer time. Closed-form, analytic expressions for the new orbital transfer controller are given. Finally, it is proved, based on a topological nonlinear stabilizability test, that there does not exist a continuous closed-loop controller that can transfer a spacecraft to a parabolic escape trajectory.

Practical Methods for Optimal Control using Nonlinear Programming

Practical Methods for Optimal Control using Nonlinear Programming

Very low-thrust trajectory optimization using a direct SQP method

The direct transcription or collocation method has demonstrated notable success in the solution of trajectory optimization and optimal control problems. This approach combines a sparse nonlinear programming algorithm with a discretization of the trajectory dynamics. A challenging class of optimization problems occurs when the spacecraft trajectories are characterized by thrust levels that are very low relative to the vehicle weight. Low-thrust trajectories are demanding because realistic forces due to oblateness, aerodynamic drag, and third-body perturbions often dominate the thrust. Furthermore because the thrust is so low, significant changes to the orbits require very long duration trajectories. When a collocation method is applied to a problem of this type, the resulting nonlinear program is very large because the trajectories are long, and very nonlinear because of the perturbing forces. This paper describes the application of the transcription method to the solution of very low-thrust orbit transfers. The vehicle dynamics are defined using a modified set of equinoctial coordinates, and the trajectory modeling is described using these dynamics. A solution is presented for a representative transfer using a spacecraft with a thrust acceleration of approximately 1.25×10 −7 km/s 2 . This transfer requires over 578 revolutions, and leads to a sparse optimization problem with 416 123 variables and 249 674 constraints. Issues related to the numerical conditioning and problem formulation are discussed.

Simple Guidance Scheme for Low-Thrust Orbit Transfers

3Pao, L. Y., “Minimum-Time Control Characteristics of Flexible Structures,” Journal of Guidance, Control, and Dynamics, Vol. 19, No. 1, 1996, pp. 123–129. 4Singh,T., andVadali, S.R., “RobustTime-Optimal Control:AFrequency Domain Approach,” Journal of Guidance, Control, and Dynamics, Vol. 17, No. 2, 1994, pp. 346–353. 5Singer, N. C., and Seering, W. P., “Preshaping Command Inputs to Reduce System Vibration,” Journal of Dynamic Systems, Measurement and Control, Vol. 12, March 1990, pp. 76–82. 6Banerjee, A. K., “Dynamics and Control of the Shuttle-Antennae System,” Journal of the Astronautical Sciences, Vol. 41, Jan.–March, 1993, pp. 73–90. 7Singhose, W., Banerjee, A. K., and Seering, W. E., “Slewing Flexible Spacecraft with De ection Limiting Input Shaping,” Journal of Guidance, Control, and Dynamics, Vol. 20, No. 2, 1997, pp. 291–298. 8Singhose,W., and Singer, N., “Effects of Input Shaping on Two-Dimensional Trajectory Following,” IEEE Transactions on Robotics and Automation, Vol. 12, No. 6, 1996, pp. 881–887. 9Spong,M.W., and Vidyasagar, M., RobotDynamics andControl, Wiley, New York, 1989, pp. 24, 259–283. 10Banerjee, A. K., and Singhose, W. E., “Input Shaping with Nonlinear Tracking Control of a Two-Link Flexible Robot,”AAS/AIAA Astrodynamics Specialist Conf., AAS Paper 97-121, Sun Valley, ID, Aug. 1997.

Optimization of Many-Burn Orbital Transfers

Optimization of medium and low-thrust orbital transfers is important for future upper stages. Previous research has generally either considered a fairly small number of burns with medium thrust or has assumed continuous thrusting for low-thrust transfers. The acceleration level considered in this paper is too high to use orbit averaging and too low (in most cases) to use only a few burns. The optimization scheme used is an optimal control solution for each burn, and a parameter optimization for the ensemble of burns. The solution for each burn is available in open-loop and closed-loop form, so that the method also provides a guidance scheme for flying the transfers. The optimization includes oblateness terms, complicating many-burn transfers with plane change. Results compare favorably to several published cases. This paper also includes additional transfer cases and parametric results. These results are easy to obtain in comparison with many-burn (non-averaged) multi-point boundary value problem solutions, which require the determination of a number of sensitive parameters. Cases examined include transfer to high-Earth orbit, transfer to geostationary orbit, transfer to the Global Positioning System (GPS) orbit, escape from orbit, and transfer to the lunar sphere of influence.

Near-optimal low-thrust orbit transfers generated by a genetic algorithm

The effectiveness of a genetic algorithm to design near-optimal low-thrust trajectories is investigated. The orbit transfer is broken into segments and a thrust direction is encoded over each of the segments. A thrust/no-thrust variable allows the trajectory to exhibit coast arcs. Constant and variable thrust propulsion systems are modeled. The thrust profile for a minimum-time Earth-to-Mars constant thrust, and fixed-time Earth-to-Mercury variable thrust trajectories are supplied.

Time-critical low-thrust orbit transfer optimization

Time-critical low-thrust orbit transfer optimization

Optimum burn scheduling for low-thrust orbital transfers

This paper describes a method for optimizing the point at which thrust should be initiated on each passage of a low-thrust, multiple finite-duration perigee-burn ascent trajectory, with the thrust vector always oriented perpendicular to nadir. The quantity being minimized is the rotation of the line of apsides. The semianalytic method developed here requires only a simple root-finding routine for the numerical portion of the solution. A perturbation solution (two-variable expansion) is used to model the flight path during the burn, yielding a burn-on position with a maximum error of approximately 2%.

Gain indexing schemes for low-thrust perturbation guidance

Two perturbation guidance schemes, time-to-go guidance and minimum-distance guidance, are re-examined in the context of a low-thrust orbit transfer problem. The two schemes, which use different techniques for indexing feedback gains, are shown to be comparable in performance. Both schemes are found to produce terminal state errors that are orders of magnitude smaller than those obtained in several previous studies. Various small modifications or enhancements of the algorithms are thought to account for a portion of the dramatic improvement in results. The problem investigated is a hypothetical Earth to Mars orbit transfer with six state variables, two control variables, and six terminal state constraints.