Computer mechanization of six-degree of freedom flight equations

Fogarty, L. E.; Howe, R. M.

doi:10.1177/003754976801100407

Cited by 12 publications

(10 citation statements)

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“…The equations of motion, which underpin the flight model of the simulated aircraft (7,8) , are the focal point of the simulator. The forces and moments which define aircraft motion are solved as a set of nonlinear differential equations to compute the aircraft motion and trajectory (9) . These equations are solved by a computer program (typically executed on a PC) with sufficient accuracy that the simulator model faithfully replicates the aircraft motion (linear and angular) throughout the flight envelope.…”

Section: Organisation Of a Flight Simulatormentioning

confidence: 99%

The impact of flight simulation in aerospace

Allerton¹

2010

Aeronaut. j.

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Section: Organisation Of a Flight Simulatormentioning

confidence: 99%

The impact of flight simulation in aerospace

Allerton¹

2010

Aeronaut. j.

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“…The scalar rotational flight equations are invariably written with respect to aircraft body axes. However, the translational equations can be written with respect to either body or flight path axes [6]. Here we will use flight path axes, since they seem to be somewhat more suitable for the modified-Euler algotithm.…”

Section: Examplesolutions Of Flight Eauation~mentioning

confidence: 99%

An improved numerical integration method for flight simulation

Howe¹

1989

Flight Simulation Technologies Conference and Exhibit

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In this paper a modified form of Euler integration is described which, when applied to the six-degree of freedom flight equations, retains and enhances many of the advantages of AB-2 integration and at the same time eliminates the disadvantages. The scheme is based on the Euler integration formula, but with the state-variable derivative represented at the midpoint of each integration step. In this case the conventional first-order Euler method actually becomes second order, with a very small accompanying error coefficient. To apply this method to the six-degree-of-freedom flight equations it is necessary to define velocity states at half-integer frame times and position states at integer frame times. It is shown through dynamic error analysis that the modified Euler method has an error coefficient which is one-tenth that associated with AB-2. The method also exhibits minimal output delay in response to transient inputs. The modified Euler method may also be useful in the integration of state and costate equations in real-time mechanization of Kalrnan filters for navigation and control systems. The ever increasing complexity of the math models used as a basis for real time flight simulation has continued to apply pressure on digital processor speed requirements for such simulations. More effective numerical integration algorithms can help relieve some of this pressure. The most popular method currently in use for flight simulation is the Adams-Bashforth second-order predictor method, usually referred to as AB-2. Its advantages include second-order accuracy with respec1 to integration step size, only one required pass through the state equations per integration step, and compatibility with real-time inputs. Disadvantages include stability problems associated with extraneous roots and response delays of one or two frames following transient inputs. In the next section we consider AB-2 along with two other second-order integration methods suitable for real-time simulation of dynamic systems. The first is a two-pass realtime predictor-corrector algorithm and the second is a singlepass version of the same. We then introduce the modified-Euler method and describe its application to the flight equations. The accuracy of the various methods is compared by means of time-history plots of the dynamic error in simulating aircraft response to a control-surface input. 2. Some Real-time Second-order Inteeration Aleorithm~ Consider first the AB-2 predictor integration method applied to the state equation given by Professor of Aerospace Engineering Associate Fellow, AIAA Copyright 0 American Institute of Aeronauticd wd Astronautics, Inc., 1989. AU rights reserved.

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“…It has been noted 19 that the use of the flight path or §"Due to the absence of a digital computer system large enough to solve the complex equations developed using the structural and aerodynamic mathematical models that describe the physics of the elastic aircraft." iThis simulation language is now available at over 250 installations around the world.…”

Section: Equation Developmentmentioning

confidence: 99%

“…Therefore, using Eqs. (18)(19)(20), the uncoupled equations for arbitrarily shaped bodies are (21a) (21b) (21c)…”

Section: Table 1 Collected Equations Of Motionmentioning

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Aeroservoelasticity in the time domain

Cutchihs

Purvist

BuntonJ³

1983

Journal of Aircraft

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The objective of this investigation was to develop and evaluate a nonlinear technique for the stability assessment of high-performance aircraft in the time domain. The technique employed is compatible with popular digital flutter and/or vibration mathematical models and is adaptable to various aircraft control systems arid/or modal information. The aircraft may possess servocontrol systems of high complexity and may dictate the use of aerodynamic schemes of the user's choice. The method of approach involves the use of the Advanced Continuous Simulation Language (ACSL), chosen because of its hybrid computer-like features and its conversational language. All modules of the simulation are solved in the time domain as the vehicle is subjected to disturbances of the user's choice. The vehicle motion rates are then output in the time domain for the engineer's concluding assessment of stability. Instabilities experienced by an early version of the F-16 aircraft are used as check cases. The stability boundary obtained is unconservative in its agreement with that determined in flight test, whereas the frequencies obtained are in excellent agreement. Conclusions include the following: 1) the simultaneous solution of large systems of aerodynamic, inertial, elastic, and servocontrol equations can be obtained using advanced simulation languages with mnemonic features that aid in their implementation and use; and 2) some improvements, when compared to frequency domain results, can be obtained using this nonlinear time domain simulation approach, especially when studying the behavior of phenomena with nonlinear effects or response. NomenclatureA = tensor or matrix of inertia terms; see Eq. (20) a = acceleration B = direction cosines b = reference length for lateral equations C = stability derivatives, conventional subscripts c = mean chord D = collection of 7 terms; see Eq. (20) F =force / = function to be searched (G) = generalized force vector g = gravitational constant H = angular momentum h = altitude 7 = inertia terms J = integer used in search algorithm [K] = stiffness matrix L,M,N = body moments analogous to the three components of the vector M i _[M] = mass matrix M -Mach number m = mass [P] = load vector p,q,r = roll, pitch, and yaw rates; see Fig. 1 Q = intermediate vector; see Eq. (14) q = dynamic pressure, VipV 2 [q] = normal coordinates R = position vector S = reference area, wing planform s = Laplace transform variable T = thrust t = time u, v, w = velocities along x,y, z axes x,y,h = positional coordinates X,Y,Z = force components x ' = variable for aerodynamic functions V = velocity => = stands for, or implies a = angle of attack |8 = sideslip angle 5 f j = Kronecker delta d = control surface deflection = permutation symbol = roll angle = mode shape matrix = yaw angle = pitch angle = rotation rate vector = circular natural frequency Subscripts a = aileron B = body-fixed axes c.m. = center of mass E = Euler (inertia!) axes F = flaperon HT = horizontal tail ij t k = tensor notation indices L,R -left, right N = nor...

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Computer mechanization of six-degree of freedom flight equations

Abstract: The authors were introduced last month when we published their article Trajectory optimization by a direct descent process. Preparation of this paper was supported under NASA contract NASr 54(06).

Cited by 12 publications

References 1 publication

The impact of flight simulation in aerospace

The impact of flight simulation in aerospace

An improved numerical integration method for flight simulation

Aeroservoelasticity in the time domain

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