The attainability domain of a coasting vehicle

In this paper, the attainability domain of a coasting vehicle in horizontal flight is determined. The coasting vehicle is assumed to control perfectly the angle of attack, assumed unlimited. The aerodynamic forces acting on the vehicle are explicitly modelled as nonlinear functions of the angle of attack.     

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 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:

Thrust programs for minimum propellant consumption during the vertical take-off and landing of a rocket

In this paper, the thrust programs for minimum propellant consumption during vertical take-off and landing maneuvers of a rocket in a vacuum are derived for an arbitrary central gravitational field. It is shown that, regardless of the mathematical form of the attracting force, the extremal arc for take-off is composed of a maximum thrust subarc followed by a coasting subarc, and the extremal arc for landing is composed of a coasting subarc followed by a maximum thrust subarc.     

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 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.     

Abort landing in windshear: Optimal control problem with third-order state constraint and varied switching structure

Optimal abort landing trajectories of an aircraft under different windshear-downburst situations are computed and discussed. In order to avoid an airplane crash due to severe winds encountered by the aircraft during the landing approach, the minimum altitude obtained during the abort landing maneuver is to be maximized. This maneuver is mathematically described by a Chebyshev optimal control problem. By a transformation to an optimal control problem of Mayer type, an additional state variable inequality constraint for the altitude has to be taken into account; here, its order is three. Due to this altitude constraint, the optimal trajectories exhibit, depending on the windshear parameters, up to four touch points and also up to one boundary arc at the minimum altitude level. The control variable is the angle of attack time rate which enters the equations of motion linearly; therefore, the Hamiltonian of the problem is nonregular. The switching structures also includes up to three singular subarcs and up to two boundary subarcs of an angle of attack constraint of first order. This structure can be obtained by applying some advanced necessary conditions of optimal control theory in combination with the multiple-shooting method. The optimal solutions exhibit an oscillatory behavior, reaching the minimum altitude level several times. By the optimization, the maximum survival capability can also be determined; this is the maximum wind velocity difference for which recovery from windshear is just possible. The computed optimal trajectories may serve as benchmark trajectories, both for guidance laws that are desirable to approach in actual flight and for optimal trajectories may then serve as benchmark trajectories both for guidance schemes and also for numerical methods for problems of optimal control.This paper is dedicated to Professor George Leitmann on the occasion of his seventieth birthday.     

An exact multi-objective mixed integer nonlinear optimization approach for aircraft conflict resolution

An exact mixed integer nonlinear optimization (MINO) model is presented for tackling the aircraft conflict detection and resolution problem in air traffic management. Given a set of flights and their configurations, the aim of the problem was to provide new configurations such that all conflict situations are avoided, with conflict situation understood to mean an event in which two or more aircraft violate the minimum safety distances that must be maintained in flight. The proposed model solves the problem using both horizontal (velocity and angle turn change) and vertical (altitude level change) maneuvers. As a special case, another model is presented in which only horizontal maneuvers are used. The models proposed are based on a geometric construction which involves trigonometric functions, so the constraint system is included via a large set of trigonometric and nonconvex inequalities. A multicriteria approach is presented to provide useful information to air traffic control officers about the maneuvers to be performed. The main results of an extensive computational experiment are reported in which the performance of the state-of-the-art nonconvex MINO solver Minotaur is studied.     

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.     

On the dynamics of a rigid body carrying a material point

In this paper we consider a system consisting of an outer rigid body (a shell) and an inner body (a material point) which moves according to a given law along a curve rigidly attached to the body. The motion occurs in a uniform field of gravity over a fixed absolutely smooth horizontal plane. During motion the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. We present a derivation of equations describing both the free motion of the system over the plane and the instances where collisions with the plane occur. Several special solutions to the equations of motion are found, and their stability is investigated in some cases. In the case of a dynamically symmetric body and a point moving along the symmetry axis according to an arbitrary law, a general solution to the equations of free motion of the body is found by quadratures. It generalizes the solution corresponding to the classical regular precession in Euler??s case. It is shown that the translational motion of the shell in the free flight regime exists in a general case if the material point moves relative to the body according to the law of areas.     

Maximum survival capability of an aircraft in a severe windshear

This paper is concerned with guidance strategies and piloting techniques which ensure near-optimum performance and maximum survival capability in a severe 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 assumed.First, six particular guidance schemes are considered, namely: constant alpha guidance; maximum alpha guidance; constant velocity guidance; constant absolute path inclination guidance; constant rate of climb guidance; and constant pitch guidance. Among these, it is concluded that the best one is the constant pitch guidance.Next, in an effort to improve over the constant pitch guidance, three additional trajectories are considered: the optimal trajectory, which minimizes the maximum deviation of the absolute path inclination from a reference value, while employing global information on the wind flow field; the gamma guidance trajectory, which is based on the absolute path inclination and which approximates the behavior of the optimal trajectory, while employing local information on the windshear and the downdraft; and the simplified gamma guidance trajectory, which is the limiting case of the gamma guidance trajectory in a severe windshear and which does not require precise information on the windshear and the downdraft.The essence of the simplified gamma guidance trajectory is that it yields a quick transition to horizontal flight. Comparative numerical experiments show that the survival capability of the simplified gamma guidance trajectory is superior to that of the constant pitch trajectory and is close to that of the optimal trajectory.Next, with reference to the simplified gamma guidance trajectory, the effect of the feedback gain coefficient is studied. It is shown that larger values of the gain coefficient improve the survival capability in a severe windshear; however, excessive values of the gain coefficient are undesirable, because they result in larger altitude oscillations and lower average altitude.Finally, with reference to the simplified gamma guidance trajectory, the effect of time delays is studied, more specifically, the time delay ₁ in reacting to windshear onset and the time delay ₂ in reacting to windshear termination. While time delay ₂ has little effect on survival capability, time delay ₁ appears to be critical in the following sense: smaller values of ₁ correspond to better survival capability in a severe windshear, while larger values of ₁ are associated with a worsening of the survival capability in a severe windshear.This research was supported by NASA-Langley Research Center, Grant No. NAG-1-516, and by Boeing Commercial Airplane Company.     

Thermal control of Huygens mock up probe in Earth atmosphere: Thermo‐fluid dynamic simulation and mission data

A mock up of Huygens probe with 76 channels acquisition has been developed at University of Padova and successfully flown with a stratospheric space balloon from Italian Space Agency Base in Trapani on May 30th 2002. Temperature sensors have monitored temperature profiles in critical points of electronics, batteries and structure during raise at 40 km altitude, floating and parachuted descent. A thermal model of the probe has been implemented taking into consideration incoming external fluxes (solar direct, albedo and radiative heat fluxes), internal heat fluxes generation and convective heat transfer with atmosphere as a function of probe altitude. For the evaluation of convective fluxes and probe spinning rates an algebraic turbulent model, developed by the present authors, has been employed. This model is suitable to predict effects such as flow curvature and separation and solid boundary rotations. Simulation results have been utilized during project phase to optimize thermal paths in order to keep critical components in the allowable temperature range and for post flight analysis of mission data. These data show that passive thermal control of the probe has performed as expected contributing to a extremely successful scientific flight.     

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.     

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

This paper deals with the identification of the wind profile along a flight trajectory by means of a two-dimensional kinematic 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 is obtained from flight measurements. The inertial velocity profile is obtained by integration of the measured inertial acceleration. The accelerometer biases and the impact values of the inertial velocity components are determined by matching the computed flight trajectory with the measured flight trajectory, available from the digital flight data recorder and air traffic control radar. This leads to a least-square problem, which is solved analytically for both the continuous formulation and the discrete formulation. Key to the precision of the identification process is the proper selection of the integration time. Because the measured data are noise-corrupted, unstable identification occurs if the integration time is too short. On the other hand, if the integration time is too long, the hypothesis of two-dimensional motion (flight trajectory nearly contained in a vertical plane) breaks down. Application of the 2D-kinematic approach to the case of Flight Delta 191 shows that stable identification takes place for integration times in the range τ = 120 to 180 sec before impact. The results of the 2D-kinematic approach are close to those of the 3D-kinematic approach (Ref. 1), particularly in terms of the inertial velocity components at impact (within 1 fps) and the maximum wind velocity differences (within 2 fps). The 2D-kinematic approach is applicable to the analysis of wind-shear 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.     

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:

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.     

The optimum rectilinear motion of a two-mass system

The forward rectilinear motion of a system of two rigid bodies along a horizontal plane is considered. Forces of dry friction act between the bodies and the plane, and the motion is controlled by internal forces of interaction between the bodies. A periodic motion in which the system moves along a straight line is constructed. The optimum parameters of the system and a control law are found corresponding to the maximum mean velocity of motion of the system as a whole.     

The steady rolling of a disc on a rough plane

The well-known problem of the rolling without slipping of a heavy circular disc along a horizontal plane is considered. The steady motions of a disc for which the angle between the plane of the disc and the supporting plane (the angle of nutation) is constant are investigated. The problem of the range of variation of the angle of nutation within which the given motions are stable, irrespective of the values of the constants of the two linear first integrals or, in other variables, irrespective of the angular velocities of precession and proper rotation, is investigated. It is shown that this range is wider than was established earlier in [1].     

高超声速飞行器纵向动力系统平衡态集的拓扑结构研究

以高超声速飞行器纵向运动的空气动力学模型和结构动力学模型为依据,采用数值分析的方法,研究了高超声速飞行器动力系统平衡点集的拓扑结构.首先根据高超声速飞行器在巡航阶段的飞行边界的限制条件得到在给定的飞行高度和马赫数下的平衡点集,由此近似估算出了高超声速飞行器的飞行包线.然后根据得到的平衡点集,分别研究了高超声速飞行器的迎角、升降舵偏角和发动机的燃料当量比与飞行马赫数和高度的关系,并进行了数值拟合,在此基础上分别描绘了以上三个拟合关系式的曲面关系图。     

Location of rectilinear center trajectories

A rectilinear center trajectory is a polygonal line consisting only of horizontal and vertical segments which minimizes the maximum distance tom given points in the plane. In this paper a polynomial time geometric procedure, to find a center trajectory subject to the number of bends, is presented. When the polygonal is constrained on the extreme segments a modified algorithm is designed.