Wind identification along a flight trajectory,part 1: 3D-kinematic approach

This paper deals with the identification of the wind profile along a flight trajectory by means of a three-dimensional kinematic approach. The approach is then applied to a recent aircraft accident, that of Flight Delta 191, which took place at Dallas-Fort Worth International Airport on August 2, 1985.In the 3D-kinematic 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 (DFDR) and air traffic control radar (ATCR). This leads to a least-square problem, which is solved analytically.Key to the precision of the identified wind profile is the correct identification of the accelerometer biases and the impact velocity components. In turn, this depends on 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, stable identification takes place if the integration time is properly chosen.Application of the method developed to the case of Flight Delta 191 shows that the identification problem has a stable solution if the integration time is larger than 180 sec. Numerical computation shows that, for Flight Delta 191, the maximum wind velocity difference determined with the 3D-kinematic approach was W _x =124 fps in the longitudinal direction, W _y =66 fps in the lateral direction, and W _h =71 fps in the vertical direction.Notations a _b measured acceleration, ft/sec² - a _bx ,a _by ,a _bz measured acceleration components, body axes system, ft/sec² - b accelerometer bias, ft/sec² - b _x ,b _y ,b _z accelerometer bias components, body axes system, ft/sec² - g acceleration of gravity, ft/sec² - h altitude above sea level, positive upward, ft - m mass, lb sec²/ft - p air pressure, lb/ft² - R gas constant for air, ft²/sec² °R - S reference wing area, ft² - t running time, sec - T air temperature, °R - V _a airspeed, ft/sec - V _ax ,V _ay ,V _az airspeed components, Earth axes system, ft/sec - V _e inertial velocity, ft/sec - V _ex ,V _ey ,V _ez inertial velocity components, Earth axes system, ft/sec - V _ia indicated airspeed, ft/sec - W mg=weight, lb - W wind velocity, ft/sec - W _x ,W _y ,W _z wind velocity components, Earth axes system, ft/sec - x, y, z Cartesian coordinates, Earth axes system, ft - x longitudinal coordinate, positive northward, Earth axes system, ft - y lateral coordinate, positive eastward, Earth axes system, ft - z vertical coordinate, positive downward, Earth axes system, ft - angle of attack, rad - path inclination, rad - pitch angle, rad - bank angle, rad - air density, lb sec²/ft⁴ - sideslip angle, rad - integration time, sec - roll angle, rad This research was supported by Air Line Pilots Association (ALPA), United States Aviation Underwriters (USAU), and Texas Advanced Technology Program, Grant No. TATP-003604020. This paper is based on Ref. 1.     

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.     

Real-time onboard wind and windshear determination,part 2: Detection

This paper is concerned with windshear detection in connection with real-time wind identification (Ref. 1). It presents a comparative evaluation of two techniques, one based on the shear/downdraft factor and one based on the wind difference index. The comparison is done with reference to a particular microburst, that which caused the 1985 crash of Flight Delta 191 at Dallas-Fort Worth International Airport.The shear/downdraft factor has the merit of combining the effects of the shear and the downdraft into a single entity. However, its effectiveness is hampered by the fact that, in a real situation, the windshear is accompanied by free-stream turbulence, which tends to blur the resulting signal. In turn, this results in undesirable nuisance warnings if the magnitude of the shear factor due to free-stream turbulence is temporarily larger than that due to true windshear. Therefore, proper filtering is necessary prior to using the shear/downdraft factor in detection and guidance. One effective way for achieving this goal is to average the shear/downdraft factor over a specified time interval . The effect of on the average shear/downdraft factor is studied.     

Real-time onboard wind and windshear determination,part 1: Identification

Standard wind identification techniques employed in the analysis of aircraft accidents are post-facto techniques; they are processed after the event has taken place and are based on the complete time histories of the DFDR/ATCR data along the entire trajectory. By contrast, real-time wind identification techniques are processed while the event is taking place; they are based solely on the knowledge of the preceding time histories of the DFDR/ATCR data.In this paper, a real-time wind identification technique is developed. First, a 3D-kinematic approach is employed in connection with the DFDR/ATCR data covering the time interval preceding the present time instant. The aircraft position, inertial velocity, and accelerometer bias are determined by matching the flight trajectory computed from the DFDR data with the flight trajectory available from the ATCR data. This leads to a least-square problem, which is solved analytically every seconds, with / small.With the inertial velocity and accelerometer bias known, an extrapolation process takes place so as to predict the inertial velocity profile over the subsequent -subinterval. At the end of this subinterval, the extrapolated inertial velocity and the newly identified inertial velocity are statistically reconciled and smoothed. Then, the process of identification, extrapolation, reconciliation, and smoothing is repeated. Subsequently, the wind is computed as the difference between the inertial velocity and the airspeed, which is available from the DFDR data. With the wind identified, windshear detection can take place (Ref. 1).As an example, the real-time wind identification technique is applied to Flight Delta 191, which crashed at Dallas-Fort Worth International Airport on August 2, 1985. The numerical results show that the wind obtained via real-time identification is qualitatively and quantitatively close to the wind obtained via standard identification. This being the case, it is felt that real-time wind identification can be useful in windhsear detection and guidance, above all if the shear/downdraft factor signal is replaced by the wind difference signal (Ref. 1).This paper and its companion (Ref. 1) are based on Refs. 2–4.This research was supported by the Aviation Research and Education Foundation and by Texas Advanced Technology Program, Grant No. TATP-003604020.     

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.     

Wave parameter identification problem for ocean test structure data,part 2, discrete formulation

This paper deals with the solution of the wave parameter identification problem for ocean test structure data. A discrete formulation is assumed. An ocean test structure is considered, and wave elevation and velocities are assumed to be measured with a number of sensors. Within the frame of linear wave theory, a Fourier series model is chosen for the wave elevation and velocities. Then, the following problem is posed: Find the amplitudes of the various wave components of specified frequency and direction, so that the assumed model of wave elevation and velocities provides the best fit to the measured data. Here, the term best fit is employed in the least-square sense over a given time interval.At each time instant, the wave representation involves four indexes (frequency, direction, instrument, time); hence, four-dimensional arrays are required. This formal difficulty can be avoided by switching to an alternative representation involving only two indexes (frequency-direction, instrument-time); hence, standard vector-matrix notation can be used. Within this frame, optimality conditions are derived for the amplitudes of the assumed wave model.A characteristic of the wave parameter identification problem is that the condition number of the system matrix can be large. Therefore, the numerical solution is not an easy task and special procedures must be employed. Specifically, Gaussian elimination is avoided and advantageous use is made of the Householder transformation, in the light of the least-square nature of the problem and the discretized approach to the problem.Numerical results are presented. The effect of various system parameters (number of frequencies, number of directions, sampling time, number of sensors, and location of sensors) is investigated in connection with global or strong accuracy, local or weak accuracy, integral accuracy, and condition number of the system matrix.From the numerical experiments, it appears that the wave parameter identification problem has a unique solution if the number of directions is smaller than or equal to the number of sensors; it has an infinite number of solutions otherwise. In the case where a unique solution exists, the condition number of the system matrix increases as the size of the system increases, and this has a detrimental effect on the accuracy. However, the accuracy can be improved by proper selection of the sampling time and by proper choice of the number and location of the sensors.Generally speaking, the computations done for the discrete case exhibit better accuracy than the computations done for the continuous case (Ref. 5). This improved accuracy is a direct consequence of having used advantageously the Householder transformation and is obtained at the expense of increased memory requirements and increased CPU time.This work was supported by Exxon Production Research Company, Houston, Texas. This paper is based partly on Refs. 1–4.     

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.     

Abort landing in the presence of windshear as a minimax optimal control problem,part 2: Multiple shooting and homotopy

In Part 1 of the paper (Ref. 2), we have shown that the necessary conditions for the optimal control problem of the abort landing of a passenger aircraft in the presence of windshear result in a multipoint boundary-value problem. This boundary-value problem is especially well suited for numerical treatment by the multiple shooting method. Since this method is basically a Newton iteration, initial guesses of all variables are needed and assumptions about the switching structure have to be made. These are big obstacles, but both can be overcome by a so-called homotopy strategy where the problem is imbedded into a one-parameter family of subproblems in such a way that (at least) the first problem is simple to solve. The solution data to the first problem may serve as an initial guess for the next problem, thus resulting in a whole chain of problems. This process is to be continued until the objective problem is reached.Techniques are presented here on how to handle the various changes of the switching structure during the homotopy run. The windshear problem, of great interest for safety in aviation, also serves as an excellent benchmark problem: Nearly all features that can arise in optimal control appear when solving this problem. For example, the candidate for an optimal trajectory of the minimax optimal control problem shows subarcs with both bang-bang and singular control functions, boundary arcs and touch points of two state constraints, one being of first order and the other being of third order, etc. Therefore, the results of this paper may also serve as some sort of user's guide for the solution of complicated real-life optimal control problems by multiple shooting.The candidate found for an optimal trajectory is discussed and compared with an approximate solution already known (Refs. 3–4). Besides the known necessary conditions, additional sharp necessary conditions based on sign conditions of certain multipliers are also checked. This is not possible when using direct methods.An extended abstract of this paper was presented at the 8th IFAC Workshop on Control Applications of Nonlinear Programming and Optimization, Paris, France, 1989 (see Ref. 1).This paper is dedicated to Professor Hans J. Stetter on the occasion of his 60th birthday.     

Wave parameter identification problem for ocean test structure data,part 1, continuous formulation

This paper deals with the solution of the wave parameter identification problem for ocean test structure data. A continuous formulation is assumed. An ocean test structure is considered, and wave elevation and velocities are assumed to be measured with a number of sensors. Within the frame of linear wave theory, a Fourier series model is chosen for the wave elevation and velocities. Then, the following problem is posed: Find the amplitudes of the various wave components of specified frequency and direction, so that the assumed model of wave elevation and velocities provides the best fit to the measured data. Here, the term best fit is employed in the least-square sense over a given time interval.At each time instant, the wave representation involves three indexes (frequency, direction, instrument); hence, three-dimensional arrays are required. This formal difficulty can be avoided by switching to an alternative representation involving only two indexes (frequency-direction, instrument); hence, standard vector-matrix notation can be used. Within this frame, optimality conditions are derived for the amplitudes of the assumed wave model.Numerical results are presented. The effect of various system parameters (number of frequencies, number of directions, sampling time, number of sensors, and location of sensors) is investigated in connection with global or strong accuracy, local or weak accuracy, integral accuracy, and condition number of the system matrix.From the numerical experiments, it appears that the identification problem has a unique solution if the number of directions is smaller than or equal to the number of sensors; it has an infinite number of solutions otherwise. In the case where a unique solution exists, the condition number of the system matrix increases as the size of the system increases, and this has a detrimental effect on the accuracy. However, the accuracy can be improved by proper selection of the sampling time and by proper choice of the number and location of the sensors.This work was supported by Exxon Production Research Company, Houston, Texas. This paper is based partly on Ref. 1–4.     

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

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

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.