Quasilinearization Method and Summation of the WKB Series

Solutions obtained by the quasilinearization method (QLM) are compared with the WKB solutions. Expansion of the $p$-th QLM iterate in powers of $\hbar$ reproduces the structure of the WKB series generating an infinite number of the WKB terms with the first $2^p$ terms reproduced exactly. The QLM quantization condition leads to exact energies for the P\"{o}schl-Teller, Hulthen, Hylleraas, Morse, Eckart potentials etc. For other, more complicated potentials the first QLM iterate, given by the closed analytic expression, is extremely accurate. The iterates converge very fast. The sixth iterate of the energy for the anharmonic oscillator and for the two-body Coulomb Dirac equation has an accuracy of 20 significant figures

The Estimation of Precision Pilot Model Parameters Using Inverse Simulation. Internal Report No. 9706

The practice of using mathematical models to simulate pilot behaviour in one-axis stabilisation tasks is a well known conventional simulation problem. In this report a system is developed whereby a mathematical model of a pilot is used as the controller of a rudimentary helicopter model. The main differences between this and other similar scenarios that have been found in the literature are that firstly, inverse simulation is used to provide results that are used as the forcing functions in the model of the pilot/helicopter system, and secondly a constrained optimisation routine is utilised to obtain values for the parameters within the pilot model itself. It will be shown that as the pilot is required to fly different manoeuvres, defined by standards set by the United States Army, or indeed if the severity of the set manoeuvres is varied, the pilot is required to adjust certain human parameters to fly the manoeuvre in a superlative manner. The report considers initially the pilot and helicopter models and subsequently analyses the system as a whole, illustrating how the pilot model can change depending on the circumstances

The Estimation of Helicopter Pilot Workload Using Inverse Simulation: Longitudinal Manoeuvre Analysis. Internal Report No. 9625

In the preceding report the concept of estimating pilot workload using inverse simulation was introduced. The report examined the ADS-33C defined Rapid Side-step Mission Task Element (MTE), and illustrated how various quickness parameters could be obtained from the lateral cyclic pitch and stick displacement time histories. These quickness parameters were plotted on charts and it was shown how the resulting plots could be used to discriminate between two dissimilar helicopter configurations, or identify which manoeuvres were more aggressive and would probably lead to a higher level of workload being placed upon the pilot. The intention of this report is to provide a supplementary study to the previous one by analysing another linear repositioning manoeuvre, the Rapid Acceleration / Deceleration or Quick-hop MTE. The longitudinal cyclic channel will be investigated in terms of pitch and stick displacement and the equivalent quickness parameters calculated and plotted on charts. A final study mirroring the previous one, on control system influence by the introduction of a Stability and Control Augmentation System, (SCAS) and the alteration of the longitudinal cyclic actuator constant will also be carried out

The Estimation of Helicopter Pilot Workload Using Inverse Simulation. Internal Report No. 9624

In the first instance this report describes the means by which inverse simulation can be used as a pilot workload estimation tool. An alternative approach to defining the mathematical model of the ADS-33 Rapid Side-step Mission Task Element (MTE) is presented and is used to drive various inverse simulation runs. Studies are conducted into three varying aggression side-step MTEs and the comparison of two dissimilar helicopter configurations based on the Westland Lynx, simulated using the same side-step. It is shown how the resulting time-histories and quickness charts can be utilised in pilot workload and handling qualities estimation. A third quickness parameter associated with the lateral cyclic stick displacements required to fly the side-step MTEs is introduced and is shown to be capable of discriminating between the pilot workload required for each side-step and vehicle configuration. The latter study in the report presents the preliminary findings on the effects of workload by firstly, introducing a Stability and Control Augmentation System and secondly investigating the effects of altering the value of the lateral cyclic actuator time constant

On the Development of Multiple Manoeuvre Mission Sequences for Inverse Simulation. Internal Report No. 9802

As part of the continuing programme of work and collaboration between the Defence Evaluation and Research Agency (DERA) and Glasgow University (GU), the author was invited to attend the final phase of flight simulation trials entitled ‘TWINS’ at DERA, Bedford; using the Advanced Flight Simulator (AFS) large motion system. The precise nature and details of the five-day trial are given in [1] but the main thrust of the trial was essentially divided into two areas: 1. The simulation of American Design Standard (ADS) Mission Task Elements (MTEs) using a software image database of Coltishall airfield with the appropriate ADS-33 visual cues. 2. The simulation of a mission sequence based on the Haxton Down software image database which comprised fourteen individual tasks. The tasks were either based on ADS MTEs or Nap-of-the-earth (NOE) flight. A full description of the manoeuvre elements is given in Appendix A of [1]. The inverse simulation package HELINV at GU contains a library of manoeuvres based both on ADS MTEs and NOE flight. However, the manoeuvres are separate and individual and until recently it was not possible to run a simulation of combinations of two or more manoeuvres. A request was put forward to develop a method whereby it was possible to choose several elements (MTE or NOE) from the manoeuvre menu and piece them together to form what has been termed a ‘mini-mission sequence’ and then inverse simulate the mission as a whole. This report describes that development and presents the results from several simulated mission runs

On Exactness Of The Supersymmetric WKB Approximation Scheme

Exactness of the lowest order supersymmetric WKB (SWKB) quantization condition $\int^{x_2}_{x_1} \sqrt{E-\omega^2(x)} dx = n \hbar \pi$, for certain potentials, is examined, using complex integration technique. Comparison of the above scheme with a similar, but {\it exact} quantization condition, $\oint_c p(x,E) dx = 2\pi n \hbar$, originating from the quantum Hamilton-Jacobi formalism reveals that, the locations and the residues of the poles that contribute to these integrals match identically, for both of these cases. As these poles completely determine the eigenvalues in these two cases, the exactness of the SWKB for these potentials is accounted for. Three non-exact cases are also analysed; the origin of this non-exactness is shown to be due the presence of additional singularities in $\sqrt{E-\omega^2(x)}$, like branch cuts in the $x-$plane.Comment: 11 pages, latex, 1 figure available on reques

The Estimation of Helicopter Pilot Workload Using Inverse Simulation. Internal Report No. 9624

In the first instance this report describes the means by which inverse simulation can be used as a pilot workload estimation tool. An alternative approach to defining the mathematical model of the ADS-33 Rapid Side-step Mission Task Element (MTE) is presented and is used to drive various inverse simulation runs. Studies are conducted into three varying aggression side-step MTEs and the comparison of two dissimilar helicopter configurations based on the Westland Lynx, simulated using the same side-step. It is shown how the resulting time-histories and quickness charts can be utilised in pilot workload and handling qualities estimation. A third quickness parameter associated with the lateral cyclic stick displacements required to fly the side-step MTEs is introduced and is shown to be capable of discriminating between the pilot workload required for each side-step and vehicle configuration. The latter study in the report presents the preliminary findings on the effects of workload by firstly, introducing a Stability and Control Augmentation System and secondly investigating the effects of altering the value of the lateral cyclic actuator time constant

Mathematical Models of Three Slalom Types for Inverse Simulation. Internal Report No. 9716

In the absence of adequate data from flight trials, a request was put forward for the development of mathematical models of various configurations of slalom manoeuvres. The models were to be based on the existing manoeuvres as flown by United States Army, specified in ADS-33D, the Defence Evaluation and Research Agency (DERA), Bedford and the German Aerospace Research Establishment (DLR), the definitions of which have been stated previously by the DERA. This report describes the development of the manoeuvres and their utilisation within the inverse simulation package HELINV, at Glasgow University. It will be shown that data acquired from the inverse simulations of these manoeuvres can be used in workload calculations using software developed by Glasgow Caledonian University, and this process in itself will verify or disprove the validity of the manoeuvres that have been developed. Although there is potentially no real substitute for genuine data obtained from actual flight trials, it is hoped that these slalom simulations will prove to be a useful tool when used in conjunction with the inverse simulation, workload estimation metrics and handling qualities software

Helicopter inverse simulation for workload and handling qualities estimation

Helicopter handling qualities are investigated using inverse simulation as the method of providing state and control information for the appropriate quantitative metrics. The main aim of the work was to develop a more comprehensive and versatile method of quantifying handling qualities levels using the available inverse algorithm "Helin v". Subsequently, the assessment of the helicopter model inherent in Helinv, "Helicopter Generic Simulation", (HGS) for its suitability to handling qualities studies was paramount. Since the Helinv inverse algorithm operates by initially defining a mathematical flight test manoeuvre for the vehicle to "fly", considerable time was given to modelling suitable handling qualities assessment manoeuvres. So-called "attitude quickness" values were then calculated thus providing an initial objective insight into handling qualities level of the vehicle under test. Validation of the tasks formed an integral part of successfully fulfilling the flight test manoeuvre development objective. The influence of the human is captured by the inclusion of a pilot model and the development of a novel method of parameter estimation, supplements the overall objective of modifying Helinv results to achieve potentially more realistic responses and thus correspondingly more realistic handling qualities. A comparative study of two helicopters, one based on the Westland Lynx battlefield/utility type and the other, a hypothetically superior configuration effectively demonstrates the capability of inverse simulation to deliver results adequate for initial handling qualities studies. Several examples are used to illustrate the point. Helinv has been shown to be versatile and efficient and can be used in initial handling qualities studies. The advantages of such a technique are clear when it is seen that actual flight testing, ground based or airborne is extremely costly, as the flight test manoeuvres must be representative of real life, reproducible and of course, as risk free as possible. Many inverse simulation runs and handling qualities calculations have been carried out for different helicopter configurations and manoeuvres thus illustrating the advantages of the technique and fulfilling all the aims mentioned above

Periodic Quasi - Exactly Solvable Models

Various quasi-exact solvability conditions, involving the parameters of the periodic associated Lam{\'e} potential, are shown to emerge naturally in the quantum Hamilton-Jacobi approach. It is found that, the intrinsic nonlinearity of the Riccati type quantum Hamilton-Jacobi equation is primarily responsible for the surprisingly large number of allowed solvability conditions in the associated Lam{\'e} case. We also study the singularity structure of the quantum momentum function, which yields the band edge eigenvalues and eigenfunctions.Comment: 11 pages, 5 table