Numerical Computation of Optimal Trajectories for Coplanar, Aeroassisted Orbital Transfer

This paper is concerned with the problem of the optimal coplanaraeroassisted orbital transfer of a spacecraft from a high Earth orbitto a low Earth orbit. It is assumed that the initial and final orbits arecircular and that the gravitational field is central and is governed by theinverse square law. The whole trajectory is assumed to consist of twoimpulsive velocity changes at the begin and end of one interior atmosphericsubarc, where the vehicle is controlled via the lift coefficient.The problem is reduced to the atmospheric part of the trajectory, thusarriving at an optimal control problem with free final time and liftcoefficient as the only (bounded) control variable. For this problem,the necessary conditions of optimal control theory are derived. Applyingmultiple shooting techniques, two trajectories with different controlstructures are computed. The first trajectory is characterized by a liftcoefficient at its minimum value during the whole atmospheric pass. For thesecond trajectory, an optimal control history with a boundary subarcfollowed by a free subarc is chosen. It turns out, that this secondtrajectory satisfies the minimum principle, whereas the first one fails tosatisfy this necessary condition; nevertheless, the characteristicvelocities of the two trajectories differ only in the sixth significantdigit.In the second part of the paper, the assumption of impulsive velocitychanges is dropped. Instead, a more realistic modeling with twofinite-thrust subarcs in the nonatmospheric part of the trajectory isconsidered. The resulting optimal control problem now describes the wholemaneuver including the nonatmospheric parts. It contains as controlvariables the thrust, thrust angle, and lift coefficient. Further,the mass of the vehicle is treated as an additional state variable. For thisoptimal control problem, numerical solutions are presented. They are comparedwith the solutions of the impulsive model.     

Optimal Interplanetary Orbital Transfers via Electrical Engines

The Hohmann transfer theory, developed in the 19th century, is the kernel of orbital transfer with minimum propellant mass by means of chemical engines. The success of the Deep Space 1 spacecraft has paved the way toward using advanced electrical engines in space. While chemical engines are characterized by high thrust and low specific impulse, electrical engines are characterized by low thrust and hight specific impulse. In this paper, we focus on four issues of optimal interplanetary transfer for a spacecraft powered by an electrical engine controlled via the thrust direction and thrust setting: (a) trajectories of compromise between transfer time and propellant mass, (b) trajectories of minimum time, (c) trajectories of minimum propellant mass, and (d) relations with the Hohmann transfer trajectory. The resulting fundamental properties are as follows:

Optimal Trajectories for Earth-to-Mars Flight

This paper deals with the optimal transfer of a spacecraft from a low Earth orbit (LEO) to a low Mars orbit (LMO). The transfer problem is formulated via a restricted four-body model in that the spacecraft is considered subject to the gravitational fields of Earth, Mars, and Sun along the entire trajectory. This is done to achieve increased accuracy with respect to the method of patched conics.The optimal transfer problem is solved via the sequential gradient-restoration algorithm employed in conjunction with a variable-stepsize integration technique to overcome numerical difficulties due to large changes in the gravitational field near Earth and near Mars. The optimization criterion is the total characteristic velocity, namely, the sum of the velocity impulses at LEO and LMO. The major parameters are four: velocity impulse at launch, spacecraft vs. Earth phase angle at launch, planetary Mars/Earth phase angle difference at launch, and transfer time. These parameters must be determined so that V is minimized subject to tangential departure from circular velocity at LEO and tangential arrival to circular velocity at LMO.For given LEO and LMO radii, a departure window can be generated by changing the planetary Mars/Earth phase angle difference at launch, hence changing the departure date, and then reoptimizing the transfer. This results in a one-parameter family of suboptimal transfers, characterized by large variations of the spacecraft vs. Earth phase angle at launch, but relatively small variations in transfer time and total characteristic velocity.For given LEO radius, an arrival window can be generated by changing the LMO radius and then recomputing the optimal transfer. This leads to a one-parameter family of optimal transfers, characterized by small variations of launch conditions, transfer time, and total characteristic velocity, a result which has important guidance implications. Among the members of the above one-parameter family, there is an optimum–optimorum trajectory with the smallest characteristic velocity. This occurs when the radius of the Mars orbit is such that the associated period is slightly less than one-half Mars day.     

Optimal Starting Conditions for the Rendezvous Maneuver, Part 1: Optimal Control Approach

We consider the three-dimensional rendezvous between two spacecraft: a target spacecraft on a circular orbit around the Earth and a chaser spacecraft initially on some elliptical orbit yet to be determined. The chaser spacecraft has variable mass, limited thrust, and its trajectory is governed by three controls, one determining the thrust magnitude and two determining the thrust direction. We seek the time history of the controls in such a way that the propellant mass required to execute the rendezvous maneuver is minimized. Two cases are considered: (i) time-to-rendezvous free and (ii) time-to-rendezvous given, respectively equivalent to (i) free angular travel and (ii) fixed angular travel for the target spacecraft. The above problem has been studied by several authors under the assumption that the initial separation coordinates and the initial separation velocities are given, hence known initial conditions for the chaser spacecraft. In this paper, it is assumed that both the initial separation coordinates and initial separation velocities are free except for the requirement that the initial chaser-to-target distance is given so as to prevent the occurrence of trivial solutions. Analyses performed with the multiple-subarc sequential gradient-restoration algorithm for optimal control problems show that the fuel-optimal trajectory is zero-bang, namely it is characterized by two subarcs: a long coasting zero-thrust subarc followed by a short powered max-thrust braking subarc. While the thrust direction of the powered subarc is continuously variable for the optimal trajectory, its replacement with a constant (yet optimized) thrust direction produces a very efficient guidance trajectory: Indeed, for all values of the initial distance, the fuel required by the guidance trajectory is within less than one percent of the fuel required by the optimal trajectory. For the guidance trajectory, because of the replacement of the variable thrust direction of the powered subarc with a constant thrust direction, the optimal control problem degenerates into a mathematical programming problem with a relatively small number of degrees of freedom, more precisely: three for case (i) time-to-rendezvous free and two for case (ii) time-to-rendezvous given. In particular, we consider the rendezvous between the Space Shuttle (chaser) and the International Space Station (target). Once a given initial distance SS-to-ISS is preselected, the present work supplies not only the best initial conditions for the rendezvous trajectory, but simultaneously the corresponding final conditions for the ascent trajectory.     

3D Geosynchronous Transfer of a Satellite: Continuation on the Thrust

The minimum-time transfer of a satellite from a low and eccentric initial orbit toward a high geostationary orbit is considered. This study is preliminary to the analysis of similar transfer cases with more complicated performance indexes (maximization of payload, for instance). The orbital inclination of the spacecraft is taken into account (3D model), and the thrust available is assumed to be very small (e.g. 0.3 Newton for an initial mass of 1500 kg). For this reason, many revolutions are required to achieve the transfer and the problem becomes very oscillatory. In order to solve it numerically, an optimal control model is investigated and a homotopic procedure is introduced, namely continuation on the maximum modulus of the thrust: the solution for a given thrust is used to initiate the solution for a lower thrust. Continuous dependence of the value function on the essential bound of the control is first studied. Then, in the framework of parametric optimal control, the question of differentiability of the transfer time with respect to the thrust is addressed: under specific assumptions, the derivative of the value function is given in closed form as a first step toward a better understanding of the relation between the minimum transfer time and the maximum thrust. Numerical results obtained by coupling the continuation technique with a single–shooting procedure are detailed.     

Optimal Trajectories for Spacecraft Rendezvous

The efficient execution of a rendezvous maneuver is an essential component of various types of space missions. This work describes the formulation and numerical investigation of the thrust function required to minimize the time or fuel required for the terminal phase of the rendezvous of two spacecraft. The particular rendezvous studied concerns a target spacecraft in a circular orbit and a chaser spacecraft with an initial separation distance and separation velocity in all three dimensions. First, the time-optimal rendezvous is investigated followed by the fuel-optimal rendezvous for three values of the max-thrust acceleration via the sequential gradient-restoration algorithm. Then, the time-optimal rendezvous for given fuel and the fuel-optimal rendezvous for given time are investigated. There are three controls, one determining the thrust magnitude and two determining the thrust direction in space. The time-optimal case results in a two-subarc solution: a max-thrust accelerating subarc followed by a max-thrust braking subarc. The fuel-optimal case results in a four-subarc solution: an initial coasting subarc, followed by a max-thrust braking subarc, followed by another coasting subarc, followed by another max-thrust braking subarc. The time-optimal case with fuel given and the fuel-optimal case with time given result in two, three, or four-subarc solutions depending on the performance index and the constraints. Regardless of the number of subarcs, the optimal thrust distribution requires the thrust magnitude to be at either the maximum value or zero. The coasting periods are finite in duration and their length increases as the time to rendezvous increases and/or as the max allowable thrust increases. Another finding is that, for the fuel-optimal rendezvous with the time unconstrained, the minimum fuel required is nearly constant and independent of the max available thrust. Yet another finding is that, depending on the performance index, constraints, and initial conditions, sometime the initial application of thrust must be delayed, resulting in an optimal rendezvous trajectory which starts with a coasting subarc. This research has been supported by NSF under Grant CMS-0218878.     

Optimal power-limited rendezvous of a spacecraft with bounded thrust and general linear equations of motion

The solution of the fixed-time optimal power-limited rendezvous with a general linear system of ordinary differential equations and a bound on the magnitude of the applied thrust is presented. Necessary and sufficient conditions for thrust saturation in an optimal solution are included.Because of the generality of the linear system of equations of motion, controllability considerations are required for a complete solution of this problem. It is shown that the condition of controllability can be defined completely in terms of a class of primer vectors associated with this problem. Moreover, it is shown that two distinct versions of the primer vector appear in this problem. Therefore, there is not a unique primer vector associated with every rendezvous problem.The work is applied to the problem of the rendezvous of a spacecraft near a satellite in circular orbit. The optimal rendezvous trajectory is determined by the interaction of a primer vector and the bound on the thrust magnitude. The results of computer simulations are presented graphically.     

Optimal apogee burn time for low thrust spinning satellite in low altitude

In this study, the optimal burn time for low-thrust impulsive propulsion systems is investigated to raise the perigee altitude of a low-Earth orbit. The maneuver is done using spin-stabilized attitude control and impulsive thrusting system for a time interval centered about apogee point. On the one hand, the low value of the thrust level causes more burn time needed to accomplish the transfer. This, in turn, will cause more thrust loss due to the deviation between the thrust axis (spin axis) and the velocity vector of the satellite. On the other hand, for small thrust duration, the transfer needs more revolutions around the Earth and more travel in lower altitudes with dense atmosphere and more drag loss. To transfer the satellite with minimum propellant mass, a compromise between velocity losses due to both drag and thrust deviation angle should be made. An analytical approximate correlation between average thruster burn time and total required propellant mass is formulated in this study and an analytical optimal solution for burn time is found. Nonlinear programming is used to find optimal burn time history. Comparing the analytical and numerical results shows a very good match.     

Piecewise-Linear Pathways to the Optimal Solution Set in Linear Programming

An optimal control problem with four linear controls describing a sophisticated concern model is investigated. The numerical solution of this problem by combination of a direct collocation and an indirect multiple shooting method is presented and discussed. The approximation provided by the direct method is used to estimate the switching structure caused by the four controls occurring linearly. The optimal controls have bang-bang subarcs as well as constrained and singular subarcs. The derivation of necessary conditions from optimal control theory is aimed at the subsequent application of an indirect multiple shooting method but is also interesting from a mathematical point of view. Due to the linear occurrence of the controls, the minimum principle leads to a linear programming problem. Therefore, the Karush–Kuhn–Tucker conditions can be used for an optimality check of the solution obtained by the indirect method.     

Reflections on the Hohmann Transfer

Walter Hohmann was a civil engineer who studied orbital maneuvers in his spare time. In 1925, he published an important book (Ref. 1) containing his main result, namely, that the most economical transfer from a circular orbit to another circular orbit is achieved via an elliptical trajectory bitangent to the terminal orbits. With the advent of the space program some three decades later, the Hohmann transfer maneuver became the most fundamental maneuver in space.In this work, we present a complete study of the Hohmann transfer maneuver. After revisiting its known properties, we present a number of supplementary properties which are essential to the qualitative understanding of the maneuver. Also, we present a simple analytical proof of the optimality of the Hohmann transfer and complement it with a numerical study via the sequential gradient-restoration algorithm. Finally, as an application, we present a numerical study of the transfer of a spacecraft from the Earth orbit around the Sun to another planetary orbit around the Sun for both the case of an ascending transfer (orbits of Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto) and the case of a descending transfer (orbits of Mercury and Venus).     

On harvesting two competing populations

The problem of harvesting two competing populations is formulated in an optimal control setting. The maximum sustained rent (MSR) solution is introduced and is shown to be not only totally singular, but also to play a central role in solutions to the harvesting problem. It is further shown that nonsingular extremal subarcs must in general approach and leave the MSR along partially singular curves. A numerical example is introduced to demonstrate this phenomenon. In the case where the populations are driven onto the MSR in minimum time, however, the optimal control is shown to be bang-bang with at most one switch.The author is indebted to Professor D. H. Jacobson and Dr. D. H. Martin for helpful discussions during the preparation of this paper.     

Optimum constrained disorbit by multiple impulses

The solution of the problem of optimally disorbiting a satellite initially in an elliptical orbit about a planet using many impulses has not been discussed in the open literature. This paper presents the analytical solution of the problem in closed form. It is assumed that the reentry trajectory is such that it intersects the atmosphere at a given angle ₄. This constraint is imposed by the safe recovery of the satellite. The problem is formulated as an optimal control problem. In the configuration space, optimum subarcs are segments of straight lines. The switching curve is obtained by integration of a Riccati equation. Criteria for the selection of the optimum trajectory are derived. Variations of the minimum characteristic velocity are discussed.The authors thank the National Aeronautics and Space Administration, Applied Mathematics Division, for the Grant No. NGR-06-003-033 under which this work was carried out.     

SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control

Parametric nonlinear optimal control problems subject to control and state constraints are studied. Two discretization methods are discussed that transcribe optimal control problems into nonlinear programming problems for which SQP-methods provide efficient solution methods. It is shown that SQP-methods can be used also for a check of second-order sufficient conditions and for a postoptimal calculation of adjoint variables. In addition, SQP-methods lead to a robust computation of sensitivity differentials of optimal solutions with respect to perturbation parameters. Numerical sensitivity analysis is the basis for real-time control approximations of perturbed solutions which are obtained by evaluating a first-order Taylor expansion with respect to the parameter. The proposed numerical methods are illustrated by the optimal control of a low-thrust satellite transfer to geosynchronous orbit and a complex control problem from aquanautics. The examples illustrate the robustness, accuracy and efficiency of the proposed numerical algorithms.     

Collision of a solar wind shock wave with the Earth’s bow shock. Wave flow pattern

The wave pattern of the flow developed when a solar wind shock wave propagates along the surface of the Earth’s bow shock is studied. The investigation is carried out in the three-dimensional non-plane-polarized formulation within the framework of the ideal magnetohydrodynamic model in which the medium is assumed to be inviscid and non-heat-conducting and to have the infinite conductivity. The global three-dimensional pattern of the interaction which is a function of the latitude and longitude of elements on the surface of the bow shock is constructed as a mosaic of solutions to the problem of breakdown of a discontinuity developed between the states behind the impinging and bow shocks on the moving curve of intersection of their fronts. The investigation is carried out for typical solar wind parameters and interplanetary magnetic field strength in the Earth’s orbit and for several Mach numbers of the interplanetary shock wave, which makes it possible to trace the evolution of the flow developed as a function of the intensity of the shock perturbation of the solar wind. The solution obtained is necessary for interpreting measurements carried out by spacecraft located in the neighborhood of the Lagrange point and the Earth’s magnetosphere.     

Abort landing in the presence of windshear as a minimax optimal control problem,part 1: Necessary conditions

The landing of a passenger aircraft in the presence of windshear is a threat to aviation safety. The present paper is concerned with the abort landing of an aircraft in such a serious situation. Mathematically, the flight maneuver can be described by a minimax optimal control problem. By transforming this minimax problem into an optimal control problem of standard form, a state constraint has to be taken into account which is of order three. Moreover, two additional constraints, a first-order state constraint and a control variable constraint, are imposed upon the model. Since the only control variable appears linearly, the Hamiltonian is not regular. Thus, well-known existence theorems about the occurrence of boundary arcs and boundary points cannot be applied. Numerically, this optimal control problem is solved by means of the multiple shooting method in connection with an appropriate homotopy strategy. The solution obtained here satisfies all the sharp necessary conditions including those depending on the sign of certain multipliers. The trajectory consists of bang-bang and singular subarcs, as well as boundary subarcs induced by the two state constraints. The occurrence of boundary arcs is known to be impossible for regular Hamiltonians and odd-ordered state constraints if the order exceeds two. Additionally, a boundary point also occurs where the third-order state constraint is active. Such a situation is known to be the only possibility for odd-ordered state constraints to be active if the order exceeds two and if the Hamiltonian is regular. Because of the complexity of the optimal control, this single problem combines many of the features that make this kind of optimal control problems extremely hard to solve. Moreover, the problem contains nonsmooth data arising from the approximations of the aerodynamic forces and the distribution of the wind velocity components. Therefore, the paper can serve as some sort of user's guide to solve inequality constrained real-life optimal control problems by multiple shooting.An extended abstract of this paper was presented at the 8th IFAC Workshop on Control Applications of Nonlinear Programming and Optimization, Paris, France, 1989 (see Ref. 1).This paper is dedicated to Professor Hans J. Stetter on the occasion of his 60th birthday.     

Maximin Approach to the Ship Collision Avoidance Problem via Multiple-Subarc Sequential Gradient-Restoration Algorithm

The ideal strategy for ship collision avoidance under emergency conditions is to maximize wrt the controls the timewise minimum distance between a host ship and an intruder ship. This is a maximin problem or Chebyshev problem of optimal control in which the performance index being maximinimized is the distance between the two ships. Based on the multiple-subarc sequential gradient-restoration algorithm, a new method for solving the maximin problem is developed.Key to the new method is the observation that, at the maximin point, the time derivative of the performance index must vanish. With the zero derivative condition being treated as an inner boundary condition, the maximin problem can be converted into a Bolza problem in which the performance index, evaluated at the inner boundary, is being maximized wrt the controls. In turn, the Bolza problem with an added inner boundary condition can be solved via the multiple-subarc sequential gradient-restoration algorithm (SGRA).The new method is applied to two cases of the collision avoidance problem: collision avoidance between two ships moving along the same rectilinear course and collision avoidance between two ships moving along orthogonal courses. For both cases, we are basically in the presence of a two-subarc problem, the first subarc corresponding to the avoidance phase of the maneuver and the second subarc corresponding to the recovery phase. For stiff systems, the robustness of the multiple-subarc SGRA can be enhanced via increase in the number of subarcs. For the ship collision avoidance problem, a modest increase in the number of subarcs from two to three (one subarc in the avoidance phase, two subarcs in the recovery phase) helps containing error propagation and achieving better convergence results.     

Multiple Space Debris Collecting Mission—Debris Selection and Trajectory Optimization

This paper investigates the cost requirement for a space debris collecting mission aimed at removing heavy debris from low Earth orbits. The problem mixes combinatorial optimization to select the debris among a list of candidates and functional optimization to define the orbital manoeuvres. The solving methodology proceeds in two steps: Firstly, a specific transfer strategy with impulsive manoeuvres is defined so that the problem becomes of finite dimension; secondly the problem is linearized around an initial reference solution. A Branch and Bound algorithm is then applied iteratively to optimize simultaneously the debris selection and the orbital manoeuvres, yielding a new reference solution. The optimal solutions found are close to the initial guess despite a very complicated design space. The method is exemplified on a representative application case.     

Optimal trajectories between Earth and Mars in their true planetary orbits

The optimal transfers from Earth to Mars and from Mars to Earth, considering the actual planetary orbits, are presented as functions of the corresponding idealized Hohmann transfers. The numerically exact two-impulse optimal trajectories are given in graphical form for all possible Hohmann windows. The two-impulse transfers which are absolute optimals and those for which a third impulse provides the absolute optimal are delineated. These data are designed to provide all the information necessary for quick orbit calculations for preliminary Martian mission analysis. In this form, they are as easy to use as the standard Hohmann transfer approximations and provide much greater accuracy.This research was supported in part by the National Aeronautics and Space Administration through Grant No. NGR-06-003-033.     

Optimal trajectories for aeroassisted,coplanar orbital transfer

This paper considers both classical and minimax problems of optimal control which arise in the study of aeroassisted, coplanar orbital transfer. The maneuver considered involves the coplanar transfer from a high planetary orbit to a low planetary orbit. An example is the HEO-to-LEO transfer of a spacecraft, where HEO denotes high Earth orbit and LEO denotes low Earth orbit. In particular, HEO can be GEO, a geosynchronous Earth orbit.The basic idea is to employ the hybrid combination of propulsive maneuvers in space and aerodynamic maneuvers in the sensible atmosphere. Hence, this type of flight is also called synergetic space flight. With reference to the atmospheric part of the maneuver, trajectory control is achieved by means of lift modulation. The presence of upper and lower bounds on the lift coefficient is considered.Within the framework of classical optimal control, the following problems are studied: (P1) minimize the energy required for orbital transfer; (P2) minimize the time integral of the heating rate; (P3) minimize the time of flight during the atmospheric portion of the trajectory; (P4) maximize the time of flight during the atmospheric portion of the trajectory; (P5) minimize the time integral of the square of the path inclination; and (P6) minimize the sum of the squares of the entry and exit path inclinations. Problems (P1) through (P6) are Bolza problems of optimal control.Within the framework of minimax optimal control, the following problems are studied: (Q1) minimize the peak heating rate; (Q2) minimize the peak dynamic pressure; and (Q3) minimize the peak altitude drop. Problems (Q1) through (Q3) are Chebyshev problems of optimal control, which can be converted into Bolza problems by suitable transformations.Numerical solutions for Problems (P1)–(P6) and Problems (Q1)–(Q3) are obtained by means of the sequential gradient-restoration algorithm for optimal control problems. The engineering implications of these solutions are discussed. In particular, the merits of nearly-grazing trajectories are considered.This research was supported by the Jet Propulsion Laboratory, Contract No. 956415. The authors are indebted to Dr. K. D. Mease, Jet Propulsion Laboratory, for helpful discussions. This paper is a condensation of the investigation reported in Ref. 1.