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.     

Optimal Control of a Ship for Collision Avoidance Maneuvers

We consider a ship subject to kinematic, dynamic, and moment equations and steered via rudder under the assumptions that the rudder angle and rudder angle time rate are subject to upper and lower bounds. We formulate and solve four Chebyshev problems of optimal control, the optimization criterion being the maximization with respect to the state and control history of the minimum value with respect to time of the distance between two identical ships, one maneuvering and one moving in a predetermined way.Problems P1 and P2 deal with collision avoidance maneuvers without cooperation, while Problems P3 and P4 deal with collision avoidance maneuvers with cooperation. In Problems P1 and P3, the maneuvering ship must reach the final point with a given lateral distance, zero yaw angle, and zero yaw angle time rate. In Problems P2 and P4, the additional requirement of quasi-steady state is imposed at the final point.The above Chebyshev problems, transformed into Bolza problems via suitable transformations, are solved via the sequential gradient-restoration algorithm in conjunction with a new singularity avoiding transformation which accounts automatically for the bounds on rudder angle and rudder angle time rate.The optimal control histories involve multiple subarcs along which either the rudder angle is kept at one of the extreme positions or the rudder angle time rate is held at one of the extreme values. In problems where quasi-steady state is imposed at the final point, there is a higher number of subarcs than in problems where quasi-steady state is not imposed; the higher number of subarcs is due to the additional requirement that the lateral velocity and rudder angle vanish at the final point.     

Collision Avoidance by a Ship with a Moving Obstacle: Computation of Feasible Command Strategies

A ship moving from a point A to a point B detects a moving small obstacle at close range. Hence, the ship has to perform a maneuver to avoid collision with the moving obstacle. Using a realistic model of a tanker ship, a method is proposed for computing feasible rudder command strategies for performing the collision-avoidance maneuver.     

Collision Avoidance by a Ship with a Moving Obstacle: Computation of Feasible Command Strategies

Typographical errors on page 57 of Ref. 1 are corrected.     

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.     

Optimal Control of a Ship for a Course-Changing Maneuver

The steering control of a ship during a course-changing maneuver is formulated as a Bolza optimal control problem, which is solved via the sequential gradient-restoration algorithm (SGRA). Nonlinear differential equations describing the yaw dynamics of a steering ship are employed as the differential constraints, and both amplitude and slew rate limits on the rudder are imposed. Two performance indices are minimized: one measures the time integral of the squared course deviation between the actual ship course and a target course; the other measures the time integral of the absolute course deviation. Numerical results indicate that a smooth transition from the initial set course to the target course is achievable, with a trade-off between the speed of response and the amount of course angle overshoot.     

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 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 Control of a Ship for Course Change and Sidestep Maneuvers

We consider a ship subject to kinematic, dynamic, and moment equations and steered via rudder under the assumptions that the rudder angle and rudder angle time rate are subject to upper and lower bounds. We formulate and solve four Mayer problems of optimal control, the optimization criterion being the minimum time.Problems P1 and P2 deal with course change maneuvers. In Problem P1, a ship initially in quasi-steady state must reach the final point with a given yaw angle and zero yaw angle time rate. Problem P2 differs from Problem P1 in that the additional requirement of quasi-steady state is imposed at the final point.Problems P3 and P4 deal with sidestep maneuvers. In Problem P3, a ship initially in quasi-steady state must reach the final point with a given lateral distance, zero yaw angle, and zero yaw angle time rate. Problem P4 differs from Problem P3 in that the additional requirement of quasi-steady state is imposed at the final point.The above Mayer problems are solved via the sequential gradient-restoration algorithm in conjunction with a new singularity avoiding transformation which accounts automatically for the bounds on rudder angle and rudder angle time rate.The optimal control histories involve multiple subarcs along which either the rudder angle is kept at one of the extreme positions or the rudder angle time rate is held at one of the extreme values. In problems where quasi-steady state is imposed at the final point, there is a higher number of subarcs than in problems where quasi-steady state is not imposed; the higher number of subarcs is due to the additional requirement that the lateral velocity and rudder angle vanish at the final point.     

Optimal take-off trajectories in the presence of windshear

This paper is concerned with optimal flight trajectories in the presence of windshear. With particular reference to take-off, eight fundamental optimization problems [Problems (P1)–(P8)] are formulated under the assumptions that the power setting is held at the maximum value and that the airplane is controlled through the angle of attack.Problems (P1)–(P3) are least-square problems of the Bolza type. Problems (P4)–(P8) are minimax problems of the Chebyshev type, which can be converted into Bolza problems through suitable transformations. These problems are solved employing the dual sequential gradient-restoration algorithm (DSGRA) for optimal control problems.Numerical results are obtained for a large number of combinations of performance indexes, boundary conditions, windshear models, and windshear intensities. However, for the sake of brevity, the presentation of this paper is restricted to Problem (P6), minimax h, and Problem (P7), minimax . Inequality constraints are imposed on the angle of attack and the time derivative of the angle of attack.The following conclusions are reached: (i) optimal trajectories are considerably superior to constant-angle-of-attack trajectories; (ii) optimal trajectories achieve minimum velocity at about the time when the windshear ends; (iii) optimal trajectories can be found which transfer an aircraft from a quasi-steady condition to a quasi-steady condition through a windshear; (iv) as the boundary conditions are relaxed, a higher final altitude can be achieved, albeit at the expense of a considerable velocity loss; (v) among the optimal trajectories investigated, those solving Problem (P7) are to be preferred, because the altitude distribution exhibits a monotonic behavior; in addition, for boundary conditions BC2 and BC3, the peak angle of attack is below the maximum permissible value; (vi) moderate windshears and relatively severe windshears are survivable employing an optimized flight strategy; however, extremely severe windshears are not survivable, even employing an optimized flight strategy; and (vii) the sequential gradient-restoration algorithm (SGRA), employed in its dual form (DSGRA), has proven to be a powerful algorithm for solving the problem of the optimal flight trajectories in a windshear.Portions of this paper were presented at the AIAA Atmospheric Flight Mechanics Conference, Snowmass, Colorado, August 19–21, 1985. The authors are indebted to Boeing Commercial Aircraft Company, Seattle, Washington and to Pratt and Whitney Aircraft, East Hartford, Connecticut for supplying some of the technical data pertaining to this study.This research was supported by NASA-Langley Research Center, Grant No. NAG-1-516. The authors are indebted to Dr. R. L. Bowles, NASA-Langley Research Center, Hampton, Virginia, for helpful discussions.This paper is based in part on Refs. 1–5.     

Optimal abort landing trajectories in the presence of windshear

This paper is concerned with optimal flight trajectories in the presence of windshear. The abort landing problem is considered with reference to flight in a vertical plane. It is assumed that, upon sensing that the airplane is in a windshear, the pilot increases the power setting at a constant time rate until maximum power setting is reached; afterward, the power setting is held constant. Hence, the only control is the angle of attack. Inequality constraints are imposed on both the angle of attack and its time derivative.The performance index being minimized is the peak value of the altitude drop. The resulting optimization problem is a minimax problem or Chebyshev problem of optimal control, which can be converted into a Bolza problem through suitable transformations. The Bolza problem is then solved employing the dual sequential gradient-restoration algorithm (DSGRA) for optimal control problems. Numerical results are obtained for several combinations of windshear intensities, initial altitudes, and power setting rates.For strong-to-severe windshears, the following conclusions are reached: (i) the optimal trajectory includes three branches: a descending flight branch, followed by a nearly horizontal flight branch, followed by an ascending flight branch after the aircraft has passed through the shear region; (ii) along an optimal trajectory, the point of minimum velocity is reached at about the time when the shear ends; (iii) the peak altitude drop depends on the windshear intensity, the initial altitude, and the power setting rate; it increases as the windshear intensity increases and the initial altitude increases; and it decreases as the power setting rate increases; (iv) the peak altitude drop of the optimal abort landing trajectory is less than the peak altitude drop of comparison trajectories, for example, the constant pitch guidance trajectory and the maximum angle of attack guidance trajectory; (v) the survival capability of the optimal abort landing trajectory in a severe windshear is superior to that of comparison trajectories, for example, the constant pitch guidance trajectory and the maximum angle of attack guidance trajectory.Portions of this paper were presented at the IFAC 10th World Congress, Munich, Germany, July 27–31, 1987 (Paper No. IFAC-87-9221).This research was supported by NASA Langley Research Center, Grant No. NAG-1-516, by Boeing Commercial Airplane Company (BCAC), and by Air Line Pilots Association (ALPA). Discussions with Dr. R. L. Bowles (NASA-LRC) and Mr. C. R. Higgins (BCAC) are acknowledged.     

Optimal penetration landing trajectories in the presence of windshear

This paper is concerned with optimal flight trajectories in the presence of windshear. The penetration landing problem is considered with reference to flight in a vertical plane, governed by either one control (the angle of attack, if the power setting is predetermined) or two controls (the angle of attack and the power setting). Inequality constraints are imposed on the angle of attack, the power setting, and their time derivatives.The performance index being minimized measures the deviation of the flight trajectory from a nominal trajectory. In turn, the nominal trajectory includes two parts: the approach part, in which the slope is constant; and the flare part, in which the slope is a linear function of the horizontal distance. In the optimization process, the time is free; the absolute path inclination at touchdown is specified; the touchdown velocity is subject to upper and lower bounds; and the touchdown distance is subject to upper and lower bounds.Three power setting schemes are investigated: (S1) maximum power setting; (S2) constant power setting; and (S3) control power setting. In Scheme (S1), it is assumed that, immediately after the windshear onset, the power setting is increased at a constant time rate until maximum power setting is reached; afterward, the power setting is held constant; in this scheme, the only control is the angle of attack. In Scheme (S2), it is assumed that the power setting is held at a constant value, equal to the prewindshear value; in this scheme, the only control is the angle of attack. In Scheme (S3), the power setting is regarded as a control, just as the angle of attack.Under the above conditions, the optimal control problem is solved by means of the primal sequential gradient-restoration algorithm (PSGRA). Numerical results are obtained for several combinations of windshear intensities and initial altitudes. The main conclusions are given below with reference to strong-to-severe windshears.In Scheme (S1), the touchdown requirements can be satisfied for relatively low initial altitudes, while they cannot be satisfied for relatively high initial altitudes; the major inconvenient is excess of velocity at touchdown. In Scheme (S2), the touchdown requirements cannot be satisfied, regardless of the initial altitude; the major inconvenient is defect of horizontal distance at touchdown.In Scheme (S3), the touchdown requirements can be satisfied, and the optimal trajectories exhibit the following characteristics: (i) the angle of attack has an initial decrease, which is followed by a gradual, sustained increase; the largest value of the angle of attack is attained near the end of the shear; in the aftershear region, the angle of attack decreases gradually; (ii) initially, the power setting increases rapidly until maximum power setting is reached; then, maximum power setting is maintained in the shear region; in the aftershear region, the power setting decreases gradually; (iii) the relative velocity decreases in the shear region and increases in the aftershear region; the point of minimum velocity occurs at the end of the shear; and (iv) depending on the windshear intensity and the initial altitude, the deviations of the flight trajectory from the nominal trajectory can be considerable in the shear region; however, these deviations become small in the aftershear region, and the optimal flight trajectory recovers the nominal trajectory.A comparison is shown between the optimal trajectories of Scheme (S3) and the trajectories arising from alternative guidance schemes, such as fixed controls (fixed angle of attack, coupled with fixed power setting) and autoland (angle of attack controlled via path inclination signals, coupled with power setting controlled via velocity signals). The superiority of the optimal trajectories of Scheme (S3) is shown in terms of the ability to meet the path inclination, velocity, and distance requirements at touchdown. Therefore, it is felt that guidance schemes based on the properties of the optimal trajectories of Scheme (S3) should prove to be superior to alternative guidance schemes, such as the fixed control guidance scheme and the autoland guidance scheme.Portions of this paper were presented at the AIAA 26th Aerospace Sciences Meeting, Reno, Nevada, January 11–14, 1988 (Paper No. AIAA-88-0580).This research was supported by NASA-Langley Research Center, Grant No. NAG-1-516, by Boeing Commercial Airplane Company (BCAC), and by Air Line Pilots Association (ALPA).The authors are indebted to Dr. R. L. Bowles, NASA-Langley Research Center, and to Dr. G. R. Hennig, Boeing Commercial Airplane Company, for helpful discussions.     

Guidance strategies for near-optimum take-off performance in a windshear

This paper is concerned with guidance strategies for near-optimum performance in a windshear. This is a wind characterized by sharp change in intensity and direction over a relatively small region of space. The take-off problem is considered with reference to flight in a vertical plane.First, trajectories for optimum performance in a windshear are determined for different windshear models and different windshear intensities. Use is made of the methods of optimal control theory in conjunction with the dual sequential gradient-restoration algorithm (DSGRA) for optimal control problems. In this approach, global information on the wind flow field is needed.Then, guidance strategies for near-optimum performance in a wind-shear are developed, starting from the optimal trajectories. Specifically, three guidance schemes are presented: (A) gamma guidance, based on the relative path inclination; (B) theta guidance, based on the pitch attitude angle; and (C) acceleration guidance, based on the relative acceleration. In this approach, local information on the wind flow field is needed.Next, several alternative schemes are investigated for the sake of completeness, more specifically: (D) constant alpha guidance; (E) constant velocity guidance; (F) constant theta guidance; (G) constant relative path inclination guidance; (H) constant absolute path inclination guidance; and (I) linear altitude distribution guidance.Numerical experiments show that guidance schemes (A)–(C) produce trajectories which are quite close to the optimum trajectories. In addition, the near-optimum trajectories associated with guidance schemes (A)–(C) are considerably superior to the trajectories arising from the alternative guidance schemes (D)–(I).An important characteristic of guidance schemes (A)–(C) is their simplicity. Indeed, these guidance schemes are implementable using available instrumentation and/or modification of available instrumentation.Portions of this were presented at the AIAA 24th Aerospace Sciences Meeting, Reno, Nevada, January 6–9, 1986. The authors are indebted to Boeing Commercial Aircraft Company, Seattle, Washington and to Pratt and Whittney Aircraft, East Hartford, Connecticut for supplying some of the technical data pertaining to this study.The authors are indebted to Dr. R. L. Bowles, NASA-Langley Research Center, Hampton, Virginia for helpful discussions. They are also indebted to Mr. Z. G. Zhao, Aero-Astronautics Group, Rice University, Houston, Texas for analytical and computational assistance.This research was supported by NASA-Langley Research Center, Grant No. NAG-1-516. This paper, a continuation of Ref.1, is based in part on Refs. 2–3.     

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.     

Quasi-steady flight to quasi-steady flight transition in a windshear: Trajectory optimization and guidance

This paper is concerned with the near-optimum guidance of an aircraft from quasi-steady flight to quasi-steady flight in a windshear. The take-off problem is considered with reference to flight in a vertical plane. In addition to the horizontal shear, the presence of a downdraft is considered. It is assumed that the power setting is held at the maximum value and that the aircraft is controlled through the angle of attack. Inequality constraints are imposed on both the angle of attack and its time derivative.First, trajectory optimization is considered. The optimal transition problem is formulated as a Chebyshev problem of optimal control: the performance index being minimized is the peak value of the modulus of the difference between the absolute path inclination and a reference value, assumed constant. Two types of optimal trajectories are studied: type 1 is concerned with gamma recovery (recovery of the initial value of the relative path inclination); and type 2 is concerned with quasisteady flight recovery (recovery of the initial values of the relative velocity, the relative path inclination, and the relative angle of attack). The numerical results show that the type 1 trajectory and the type 2 trajectory are nearly the same in the shear portion, while they diverge to a considerable degree in the aftershear portion of the optimal trajectory.Next, trajectory guidance is considered. A guidance scheme is developed so as to achieve near-optimum quasi-steady flight recovery in a windshear. The guidance scheme for quasi-steady flight recovery includes three parts in sequence. The first part refers to the shear portion of the trajectory and is based on the result that this portion of the trajectory depends only mildly on the boundary conditions; therefore, any of the guidance schemes already developed for type 1 trajectories can be employed (for instance, variable gamma guidance). The second part (constant gamma guidance) refers to the initial aftershear portion of the trajectory and is designed to achieve almost velocity recovery. The third part (constant rate of climb guidance) refers to the final aftershear portion of the trajectory and is designed to achieve almost complete restoration of the initial quasi-steady state.While the shear guidance and the initial aftershear guidance employ constant gain coefficients, the final aftershear guidance employs a variable gain coefficient. This is done in order to obtain accuracy and prompt response, while avoiding oscillations and overshoots. The numerical results show that the guidance scheme for quasi-steady flight recovery yields a transition from quasi-steady flight to quasi-steady flight which is close to that of the optimal trajectory, ensures the restoration of the initial quasi-steady state, and has good stability properties.This paper is based on Refs. 1 and 2.This research was supported by NASA-Langley Research Center, Grant No. NAG-1-516, and by Boeing Commercial Aircraft Company. The authors are indebted to Dr. R. L. Bowles, NASA-Langley Research Center, for helpful discussions.     

Quasi-steady flight to quasi-steady flight transition for abort landing in a windshear: Trajectory optimization and guidance

This paper is concerned with the optimal transition and the near-optimum guidance of an aircraft from quasi-steady flight to quasi-steady flight in a windshear. The abort landing problem is considered with reference to flight in a vertical plane. In addition to the horizontal shear, the presence of a downdraft is considered.It is assumed that a transition from descending flight to ascending flight is desired; that the initial state corresponds to quasi-steady flight with absolute path inclination of –3.0 deg; and that the final path inclination corresponds to quasi-steady steepest climb. Also, it is assumed that, as soon as the shear is detected, the power setting is increased at a constant time rate until maximum power setting is reached; afterward, the power setting is held constant. Hence, the only control is the angle of attack. Inequality constraints are imposed on both the angle of attack and its time derivative.First, trajectory optimization is considered. The optimal transition problem is formulated as a Chebyshev problem of optimal control: the performance index being minimized is the peak value of the modulus of the difference between the instantaneous altitude and a reference value, assumed constant. By suitable transformations, the Chebyshev problem is converted into a Bolza problem. Then, the Bolza problem is solved employing the dual sequential gradient-restoration algorithm (DSGRA) for optimal control problems.Two types of optimal trajectories are studied, depending on the conditions desired at the final point. Type 1 is concerned with gamma recovery (recovery of the value of the relative path inclination corresponding to quasi-steady steepest climb). Type 2 is concerned with quasi-steady flight recovery (recovery of the values of the relative path inclination, the relative velocity, and the relative angle of attack corresponding to quasi-steady steepest climb). Both the Type 1 trajectory and the Type 2 trajectory include three branches: descending flight, nearly horizontal flight, and ascending flight. Also, for both the Type 1 trajectory and the Type 2 trajectory, descending flight takes place in the shear portion of the trajectory; horizontal flight takes place partly in the shear portion and partly in the aftershear portion of the trajectory; and ascending flight takes place in the aftershear portion of the trajectory. While the Type 1 trajectory and the Type 2 trajectory are nearly the same in the shear portion, they diverge to a considerable degree in the aftershear portion of the trajectory.Next, trajectory guidance is considered. Two guidance schemes are developed so as to achieve near-optimum transition from quasi-steady descending flight to quasi-steady ascending flight: acceleration guidance (based on the relative acceleration) and gamma guidance (based on the absolute path inclination).The guidance schemes for quasi-steady flight recovery in abort landing include two parts in sequence: shear guidance and aftershear guidance. The shear guidance is based on the result that the shear portion of the trajectory depends only mildly on the boundary conditions. Therefore, any of the guidance schemes already developed for Type 1 trajectories can be employed for Type 2 trajectories (descent guidance followed by recovery guidance). The aftershear guidance is based on the result that the aftershear portion of the trajectory depends strongly on the boundary conditions; therefore, the guidance schemes developed for Type 1 trajectories cannot be employed for Type 2 trajectories. For Type 2 trajectories, the aftershear guidance includes level flight guidance followed by ascent guidance. The level flight guidance is designed to achieve almost complete velocity recovery; the ascent guidance is designed to achieve the desired final quasi-steady state.The numerical results show that the guidance schemes for quasi-steady flight recovery yield a transition from quasi-steady flight to quasi-steady flight which is close to that of the optimal trajectory, allows the aircraft to achieve the final quasi-steady state, and has good stability properties.This research was supported by NASA Langley Research Center, Grant No. NAG-1-516, by Boeing Commercial Airplane Company, and by Air Line Pilots Association.The authors are indebted to Dr. R. L. Bowles (NASA-LRC) and Dr. G. R. Hennig (BCAC) for helpful discussions.     

Optimal, quasi-optimal and superlinear band-Toeplitz preconditioners for asymptotically ill-conditioned positive definite Toeplitz systems

In this paper we are concerned with the solution of Hermitian Toeplitz systems with nonnegative generating functions . The preconditioned conjugate gradient (PCG) method with the well-known circulant preconditioners fails in the case where has zeros. In this paper we consider as preconditioners band-Toeplitz matrices generated by trigonometric polynomials of fixed degree . We use different strategies of approximation of to devise a polynomial which has some analytical properties of , is easily computable and is such that the corresponding preconditioned system has a condition number bounded by a constant independent of . For each strategy we analyze the cost per iteration and the number of iterations required for the convergence within a preassigned accuracy. We obtain different estimates of for which the total cost of the proposed PCG methods is optimal and the related rates of convergence are superlinear. Finally, for the most economical strategy, we perform various numerical experiments which fully confirm the effectiveness of approximation theory tools in the solution of this kind of linear algebra problems.

     

Decomposition technique and optimal trajectories for the aeroassisted flight experiment

The aeroassisted flight experiment (AFE) refers to a spacecraft to be launched and then recovered by the space shuttle in 1994. It simulates a transfer from a geosynchronous Earth orbit (GEO) to a low Earth orbit (LEO). Specifically, the AFE spacecraft is released from the space shuttle and is accelerated by means of a solid rocket motor toward Earth, so as to achieve atmospheric entry conditions close to those of a spacecraft returning from GEO. Following the atmospheric pass, the AFE spacecraft ascends to the specified LEO via an intermediate parking Earth orbit (PEO). The final maneuver includes the rendezvous with and the capture by the space shuttle. The entry and exit orbital planes of the AFE spacecraft are identical with the orbital plane of the space shuttle. In this paper, with reference to the AFE spacecraft, an actual GEO-to-LEO transfer is considered and optimal trajectories are determined by minimizing the total characteristic velocity. The optimization is performed with respect to the time history of the controls (angle of attack and angle of bank), the entry path inclination and the flight time being free. Two transfer maneuvers are considered: (DA) direct ascent to LEO; (IA) indirect ascent to LEO via PEO. While the motion of the AFE spacecraft in a 3D-space is described by a system of six ODEs, substantial simplifications are possible if one exploits these facts: (i) the instantaneous orbital plane is nearly identical with the initial orbital plane; (ii) the bank angle is small; and (iii) the Earth's angular velocity is relatively small. Under these assumptions, the complete system can be decoupled into two subsystems, one describing the longitudinal motion and one describing the lateral motion. The angle of attack history, the entry path inclination, and the flight time are determined via the longitudinal motion subsystem; in this subsystem, the total characteristic velocity is minimized subject to the specified LEO requirement. The angle of bank history is determined via the lateral motion subsystem; in this subsystem, the difference between the instantaneous bank angle and a constant bank angle is minimized in the least square sense subject to the specified orbital inclination requirement. It is shown that both the angle of attack and the angle of bank are constant. This result has considerable importance in the design of nominal trajectories to be used in the guidance of AFE and AOT vehicles.