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 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.     

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 Planetary Orbital Transfers via Chemical Engines and Electrical Engines

Because the orbital periods for planetary orbital transfers are of order hour, the primary objective of an optimal trajectory is to minimize the propellant consumption. In this paper, we present a systematic investigation of optimal trajectories for planetary orbital transfer. Major results on thrust control, propellant consumption, and flight time are presented with particular reference to LEO-to-GEO transfer. The following results were obtained with the sequential gradient-restoration algorithm in either single-subarc form or multiple-subarc form:

Guidance Trajectories for Spacecraft Rendezvous

In a previous paper of Miele et al. (J. Optim. Theory Appl. 132(1), 2007), we employed the single-subarc sequential gradient-restoration algorithm to optimize the three-dimensional rendezvous between a target spacecraft in a planar circular orbit and a chaser spacecraft with an initial separation distance and separation velocity. The achieved continuous solutions are characterized by two, three, or four subarcs depending on the performance index (time, fuel) and the constraints. In this paper, based on the solutions in Miele et al. (J. Optim. Theory Appl. 132(1), 2007), we employ the multiple-subarc sequential gradient-restoration algorithm to produce pieced guidance trajectories implementable in real time via constant control components. In other words, in this investigation, we force the controls to behave as parameters in each subarc. With the above understanding, we investigate four problems: (P1) minimum time, fuel free; (P2) minimum fuel, time free; (P3) minimum time, fuel given; (P4) minimum fuel, time given. Problem P1 results in a two-subarc solution, each subarc with constant controls: a max-thrust accelerating subarc followed by a max-thrust braking subarc. Problem P2 results in a four-subarc solution, each subarc with constant controls: an initial coasting subarc, followed by a max-thrust braking subarc, followed by another coasting subarc, followed by another max-thrust braking subarc. Problems P3 and P4 result in two, three, or four-subarc solutions depending on the performance index and the constraints, albeit with constant controls in each subarc. For each of the problems studied, the performance index of the multiple-subarc pieced guidance trajectory approximates well the performance index of the single-subarc continuous optimal trajectory of Miele et al. (J. Optim. Theory Appl. 132(1), 2007) as well as the performance index of the multiple-subarc pieced optimal trajectory: the pairwise relative differences in performance index are less than 1/100. This research was supported by NSF under Grant CMS-0218878.     

Multiple-Subarc Gradient-Restoration Algorithm, Part 2: Application to a Multistage Launch Vehicle Design

In Part 1 (see Ref. 2), a multiple-subarc gradient-restoration algorithm (MSGRA) was developed with the intent of enhancing the robustness of gradient-restoration algorithms and also enlarging the field of applications. Indeed, MSGRA can be applied to optimal control problems involving multiple subsystems as well as discontinuities in the state and control variables at the interface between contiguous subsystems.In Part 2 (this paper), MSGRA is applied to compute the optimal trajectory for a multistage launch vehicle design, specifically, a rocket-powered spacecraft ascending from the Earth surface to a low Earth orbit (LEO). Single-stage, double-stage, and triple-stage configurations are considered. For multistage configurations, discontinuities in the mass occur at the interfaces between consecutive stages.The numerical results show that, given the current levels of the engine specific impulse and spacecraft structural factor, the single-stage version is not feasible at this time, while the double-stage and triple-stage versions are feasible. Further increases in the specific impulse and decreases in the structural factor are needed if the single-stage configuration has to become feasible.Also, the numerical results show that the optimal trajectory requires initially maximum thrust, followed by modulated thrust so as to satisfy the maximum acceleration constraint, followed by nearly zero thrust for coasting flight, followed by a final burst with moderate thrust so as to increase the spacecraft velocity to the circular velocity needed for LEO insertion. The above properties of the optimal thrust time history are useful for developing the guidance scheme approximating in real time the optimal trajectory for a launch vehicle design.     

An optimal policy for a fish harvest

A dynamical model for harvesting a fish population system is proposed by introducing control into the known Verhulst-Pearl model. An optimal control problem including some parameters is stated, and the usual necessary conditions are applied. For specific parameter values, the candidate control policy is deduced, and optimality is verified by applying a sufficiency theorem. The optimal trajectories may contain maximum and minimum control arcs as well as a singular subarc. The significance of the singular arc is interpreted in terms of the system dynamics.     

The maximum Perron roots of digraphs with some given parameters

We characterize the extremal digraphs which attain the maximum Perron root of digraphs with given arc connectivity and number of vertices. We also characterize the extremal digraphs which attain the maximum Perron root of digraphs given diameter and number of vertices.     

Minimum-fuel rocket trajectories involving intermediate-thrust arcs

The optimal trajectories in the neighborhood of an optimal intermediate-thrust arc are investigated for the minimumfuel orbit rendezvous problem with fixed specific impulse. Since such an arc is singular, the thrust acceleration magnitude being the singular control component, a second-variation analysis leads to the identification of a field of neighboring, singular arcs in a state space of dimension four rather than six, provided that a suitable Jacobi condition is met. A given neighboring initial six-dimensional state vector does not generally lie on a neighboring singular arc, and junction onto the appropriate singular arc must be accomplished by a short period of strong variations in the acceleration. This contributes an addition to the fuel expenditure which is of order 5/2 rather than 2 in the initial state displacement. The minimization of this higher-order cost, in the case of bounded acceleration, leads to an unsymmetrical version of Fuller's problem, whose solution requires an infinite number of switches between maximum and zero thrust during the short period. For unbounded thrust, the junction simplifies to either coast-impulse-singular trajectories or impulse-coast-impulse-singular trajectories. The neighboring singular arc meets the final condition in 4 dimensions, rather than 6 dimensions, and rendezvous must be completed by another, terminal short period of strong variations in the acceleration. Implications for midcourse guidance near a singular arc are discussed.     

Junction phenomena for optimal control with state-variable inequality constraints of third order

It is known that extremal arcs governed by inequality constraints of third order (constraint relations that must be differentiated three times to generate a control equation) cannot join an unconstrained arc, except in special cases. But a control problem is exhibited, for which every extremal includes a constrained arc of third order. The constrained arc joins the end of an infinite sequence of consecutive unconstrained arcs of finite total duration. Evidence (but not proof) is given that this phenomenon is typical, rather than exceptional. An analogous phenomenon is well known for optimal control problems with singular arcs of second order.     

Relations among whitney sets,self-similar arcs and quasi-arcs

We study in this paper some relations among self-similar arcs, Whitney sets and quasi-arcs: we prove that any self-similar arc of dimension greater than 1 is a Whitney set; give a geometric sufficient condition for a self-similar arc to be a quasi-arc, and provide an example of a self-similar arc such that any subarc of it fails to be at-quasi-arc for anyt ≥ 1, which answers an open question on Whitney sets. We also show that self-similar arcs with the same Hausdorff dimension need not be Lipschitz equivalent. Supported by Special Funds for Major State Basic Research Projects of China, Morningside Center of Mathematics, NSFC (No. 10241003) and ZJNFS (No. 101026).     

Wind identification along a flight trajectory,part 3: 2D-dynamic approach

This paper deals with the identification of the wind profile along a flight trajectory by means of a two-dimensional dynamic approach. In this approach, the wind velocity components are computed as the difference between the inertial velocity components and the airspeed components. The airspeed profile as well as the nominal thrust, drag, and lift profiles are obtained from the available DFDR measurements. The actual values of the thrust, drag, and lift are assumed to be proportional to the respective nominal values via multiplicative parameters, called the thrust, drag, and lift factors. The thrust, drag, and lift factors plus the inertial velocity components at impact are determined by matching the flight trajectory computed from DFDR data with the flight trajectory available from ATCR data. This leads to a least-square problem which is solved analytically under the additional requirement of closeness of the multiplicative factors to unity. Application of the 2D-dynamic approach to the case of Flight Delta 191 shows that, with reference to the last 180 sec before impact, the values of the multiplicative factors were 1.09, 0.84, and 0.89; this implies that the actual values of the thrust, drag, and lift were 9% above, 16% below, and 11% below their respective nominal values. For the last 60 sec before impact, the aircraft was subject to severe windshear, characterized by a horizontal wind velocity difference of 123 fps and a vertical wind velocity difference of 80 fps. The 2D-dynamic approach is applicable to the analysis of windshear accidents in take-off or landing, especially for the case of older-generation, shorter-range aircraft which do not carry the extensive instrumentation of newer-generation, longer-range aircraft. The same methodology can be extended to the investigation of aircraft accidents originating from causes other than windshear (e.g., icing, incorrect flap position, engine malfunction), above all if its precision is further increased by combining the 2D-dynamic approach and the 2D-kinematic approach.     

On extremal decomposition problems

The capacity approach and symmetrization method arc adapted to some extremal decomposition problems on the unit disk or an annulus. The problems on the maximum product of the interior radii of pairwise nonoverlapping domains and the maximum product of the Robin radii of such domains are considered. New invariants with respect to the M?bius transformations of the Riemann sphere are introduced. In particular, for these invariants problems on extremal decomposition with free poles on the unit circle are investigated. Bibliography: 19 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 54–74.     

Numerical Calculation of Acoustic Sources for the Landing Gear of Aeroplane during Take-off and Landing

In this paper we present Orthogonal Sub-grid scale stabilized method with static subscales applied on turbulent flow around landing gear of aircraft during take-off and landing. Numerical results of proposed method are compared with famous LES Smagorinsky method for turbulent flow and it is shown how obtained results affect allocation of acoustic sources and then propagation of acoustic waves in computational domain. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimum coasting flight in a horizontal plane

Busemann's concept of optimum altitude for rectilinear coasting flight has been extended to flight in a horizontal plane. The reachable domain of a vehicle in coasting flight in a horizontal plane, starting from an initial velocity, is obtained. It is shown that the area covered first increases with the altitude and then decreases to zero when the ceiling is reached. In particular, there exists an altitude for maximum longitudinal range and another altitude for maximum lateral range. Optimum variations in the lift coefficient and the bank angle to reach the boundary of the footprint are discussed.     

Maximum Divert for Planetary Landing Using Convex Optimization

This paper presents a real-time solution method of the maximum divert trajectory optimization problem for planetary landing. In mid-course, the vehicle is to abort and retarget to a landing site as far from the nominal as physically possible. The divert trajectory must satisfy velocity constraints in the range and cross range directions and a total speed constraint. The thrust magnitude is bounded above and below so that once on, the engine cannot be turned off. Because this constraint is not convex, it is relaxed to a convex constraint and lossless convexification is proved. A transformation of variables is introduced in the nonlinear dynamics and an approximation is made so that the problem becomes a second-order cone problem, which can be solved to global optimality in polynomial time whenever a feasible solution exists. A number of examples are solved to illustrate the effectiveness and efficiency of the solution method.     

Airport runway scheduling

Airport runway optimization is an ongoing challenge for air traffic controllers. Since demand for air-transportation is predicted to increase, there is a need to realize additional take-off and landing slots through better runway scheduling. In this paper, we review the techniques and tools of operational research and management science that are used for scheduling aircraft landings and take-offs. The main solution techniques include dynamic programming, branch and bound, heuristics and meta-heuristics.     

Airport runway scheduling

Airport runway optimization is an ongoing challenge for air traffic controllers. Since demand for air-transportation is predicted to increase, there is a need to realize additional take-off and landing slots through better runway scheduling. In this paper, we review the techniques and tools of operational research and management science that are used for scheduling aircraft landings and take-offs. The main solution techniques include dynamic programming, branch and bound, heuristics and meta-heuristics.     

Experimental investigation of the effect of aircraft trailing vortices on roofs

Roofs of buildings in the vicinity of airports can be damaged by trailing vortices of aircraft during take-off and landing. Preliminary experimental investigations have been conducted in a water towing tank in order to examine a discrepancy between damage probability assumptions and actual roof damage frequency. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

有等待席位限制的平行多跑道航班起降排队模式研究

根据实际平行多跑道机场终端区飞行和控制程序,利用排队理论讨论2种有等待席位限制的航班起降排队模式:①分列独立、②联合协作模型.求出了这2种排队模式下相关指标.通过比较2种模式下航班队列服务质量和系统工作指标,发现②比①支持更多的航班起降架次并有较高的服务质量.从而,为优化航班起降提出一些建议.