Many of the early attempts at powered flight failed because, among other reasons, basic stability and control of the vehicle was inadequate. Central to the astounding progress in this field (initial powered human flight to lunar exploration in 66 years) has been the development and application of the stability and control technologies.

In modern aerospace vehicle development, a central feature of any project is the engineering simulation model of the flight dynamics of the vehicle. This model is a mathematical representation of the steady-state performance and dynamic response that is expected of the proposed vehicle. This math model allows the stability and control engineer to develop control laws to allow a human pilot (and increasingly an autopilot) to maneuver the vehicle to launch, perform its mission, and (if not an expendable weapon or probe) recover safely for re-use. The math model also forms the basis for simulators used to train a pilot in the skills needed to operate the vehicle safely and effectively.

This brief article is intended to explain to a novice student how these math models are created and used.

To be able to predict the resulting motion of any vehicle (whether
automobile, boat, skateboard, or Space Shuttle) it is necessary
to understand the physics of how forces cause objects to move.
This field of study is called *dynamics* and is a college-level
course in most engineering curricula. (Dynamics follows a similar
course, *statics*, that examines how objects, especially
structures, transmit loads and forces through structural elements.)

Using basic equations from dynamics, mathematical equations are written that describe how the vehicle will move in response to forces that are applied to the vehicle. For example, it is pretty easy to describe how a rocket will accelerate when a constant thrust is provided by the rocket's engine. More difficult is to describe and predict how the sloshing of fuel in the rocket's fuel tank will cause the rocket's structure to vibrate or throw the rocket off-course. Another type of modeling problem would be to understand and predict, in a mathematical equation, how an aircraft will respond to hitting an updraft in the atmosphere, or how the aircraft will respond to the deflection of various control surfaces at different airspeeds.

These equations are usually in the form of *differential*
equations, in which the rate of change of some quantity is described
as being dependent upon other quantities and their rates of change.

The set of mathematical equations that describe these motions
are collectively called a *math model* or *simulation model*
of the vehicle. These equations can range in complexity from a
single equation on an engineer's workpad to a complex set of software
routines. The more complex vehicle models are, for convenience,
broken down into *subsystem models* that deal with different
sets of forces:

An atmospheric subsystem model, which is usually reused by
many different vehicle models, describes how the Earth's (or other
planet's) atmosphere *behaves*. The atmosphere has several
properties that change with altitude such as pressure, temperature,
and how fast sound travels through the air (speed of sound); these
in turn have a quite important effect on how a vehicle traveling
through the atmosphere will behave.

An aerodynamic subsystem model describes how the vehicle will respond to forces caused by motion of the vehicle through the atmosphere, and predicts the effects of each different control surface (such as the flaps, rudders, ailerons, etc.) upon the motion of the vehicle. While spacecraft operate primarily in the absence of an atmosphere, they usually have to fly through the atmosphere of the Earth (or another planet) during launch or landing (if the mission of the spacecraft includes landing). Thus, most aerospace vehicle math models include an aerodynamic subsystem model.

A propulsion subsystem model describes how any motors or engines will behave and what forces will act on the vehicle to which they are attached.

A landing gear subsystem model is required when the vehicle is in contact with the ground in order to model how the ground reaction forces are created and how they affect the motion of the vehicle.

An inertial properties subsystem model provides details about how the mass and inertia of the vehicle might change with time. For simple vehicles (such as skateboards) the mass properties (weight, moments of inertia) don't change, but any fuel-carrying vehicle will have varying mass properties. If the circumstances require, the 'inertia model' will include such dynamic elements as fuel slosh, landing gear movement, and in a highly-detailed model, take into account the motion of control surfaces.

Some vehicles are relatively flexible, requiring the development of a flexible structure subsystem model to account for motion of various parts of the vehicle in relation to other parts. Airships and other long, slender vehicles made of lightweight materials will require some attention to this aspect of modeling.

Finally, any electrical, mechanical, or electronic system that assists the pilot in moving the control surfaces has to be described mathematically. This includes the springiness of control cables in a light airplane, the hydraulic actuators of older commercial transports, and the 'fly-by-wire' flight control systems of tactical aircraft and newer transports (and all spacecraft).

The modeler bases the subsystem models on various pieces of information. Aerodynamic models are often the most complex models and are usually based on wind-tunnel data, in which the forces and moments (moments are rotary forces) exerted on a scaled model are measured at various speeds, flow angles, and with combinations of control surface deflections, until enough data is available to predict the forces and moments that will act on the full-scale vehicle. Increasingly more data is being added by using a technology called Computational Fluid Mechanics (CFD) in which the same forces and moments are predicted in a computer program, using the geometry (shape) of the vehicle in a virtual wind tunnel.

The resulting aerodynamic subsystem model will predict what the forces and moments would be as a result of any combination of control surface deflections, thrust settings, and flight conditions.

Other subsystem models can be equally complex, depending on the type of vehicle and the fidelity required of the model. Propulsion models can be simple thrust and moment estimates or include the calculation of many internal engine states (such as flow rates, pressures, and temperatures in several stages of a turbine engine or rocket fuel turbopumps). Rotorcraft models include models of the rotor dynamics; propeller models are needed for propeller-driven aircraft. Landing gear models often use data obtained from landing gear drop tests and friction measurements from specialized testing equipment. Inertial models are usually generated from adding up individual component weights and locations; only in rare cases do full-scale aircraft get moments of inertia (rotational inertia) measured directly.

Models of flight control systems are usually based on specifications for the hardware and software components involved in the control system; often the flight control software itself is included in the control subsystem model.

Aerospace vehicle mathematical models are written in a variety of software languages; early models were actually made from analog electronics (thus the 'analog computer') that contained no software; the electronic components such as resistors, capacitors and amplifiers represented the differential equations that describe the motion of the vehicle.

Since the 1950s, FORTRAN (for FORmula TRANslater) has been the principal programming language used to write flight dynamic models. Its almost universal application for science applications has been gradually reduced by more modern software languages; however, the bulk of vehicle simulations are still written in FORTRAN [1].

In addition to general purpose languages, such as FORTRAN, BASIC, C, C++, and Ada, there exist specialized languages for differential equation solution. Also used are specialized software environments for describing mathematical equations (Mathematica, MAPLE, MATLAB, xMath, and others) as well as allowing construction of block diagrams of dynamic systems (xMath's SystemBuild and MATLAB's Simulink).

The choice of programming language is dictated by the purpose of the math model and familiarity with the programming tool.

An aerospace vehicle flight dynamics model can take many forms depending on the purpose of the model.

For design work, for example in developing an control law for the vehicle, often the model is in the form of a 'linearized model'. In this form, the dynamics of the vehicle (technically, the perturbations of the vehicle's motion from a specified steady-state condition) can be represented by a set of matrices or Laplace-domain transfer functions.

Since these linear models can be generated from the complete, non-linear math model (but not vice-versa) often the complete model is created first and then linearized for investigation of specific flight conditions. Usually these high-fidelity, complex simulation models are intended to be run in 'batch' mode, or not in real time where the model maneuvers at the same rate it would in actual flight. Real-time models are used exclusively for simulation studies involving human or flight hardware 'in-the-loop.'

A specialized form of a complete non-linear model is usually required if the model is to serve as a real-time simulation model (by 'real-time' it is implied that the simulation will be piloted). [Note - with the increasing computational speed of newer computers, less specialization is required to solve the differential equations of the model fast enough to serve as a piloted simulation model.] Traditionally, each research organization that makes use of an in-house flight simulator facility has developed customized simulation source code in a variety of languages; these are functionally similar but surprisingly non-compatible requiring modifications to a mathematical model when transferred from one organization to another.

It is traditional to generate a set of check cases, composed of 'trim' values and dynamic time history recordings of key parameters, to guarantee proper reimplementation if a simulation model is moved from one type of computer to another.

The development of a mathematical description of the dynamics of a flight vehicle is central to any development project. Development of flight dynamic models require information from many engineering disciplines and often specialized programming languages.

[1] Jackson, E. Bruce: Results of a Flight Simulation Software Methods Survey, AIAA Paper No. 95-3414, August 1995. Presented at the AIAA Flight Simulation Technologies Conference, Baltimore, MD, August 7-9, 1995.

General approach to simulation modeling, with a non-linear F-16 aircraft example:

*Stevens, Brian L. and Lewis, Frank L.: Aircraft Simulation
and Control. John Wiley & Sons, 1992. ISBN 0-417-61397-5*

Modeling of dynamic systems:

*Cannon, Robert H.: Dynamics of Physical Systems. McGraw-Hill,
1967. ISBM 07-009754-2*

Modeling of aircraft dynamics:

*McRuer, Duane; Ashkenas, Irving; and Graham, Dunstan: Aircraft
Dynamics and Automatic Control. Princeton University Press, 1973.
ISBN 0-691-08083-6*

Equations of motion:

*Gainer, Thomas G. and Hoffman, Sherwood: Summary of Transformation
Equations and Equations of Motion used in Free-Flight and Wind-Tunnel
Data Reduction and Analysis. NASA SP-3070, 1972*

*Etkin, Bernard: Dynamics of Atmospheric Flight. John Wiley
& Sons, 1972. ISBN 0-471-24620-4*

*Martin R. Waszak, Carey S. Buttrill and David K. Schmidt:Modeling
and Model Simplification of Aeroelastic Vehicles: An Overview,
NASA TM-107691, September 1992*

Aerospace simulation software:

*Jackson, E. Bruce: Manual
for a Workstation-Based Generic Flight Simulation Program (LaRCsim)
Version 1.4, NASA TM-110164, May 1995*

*Flightgear, a collaborative freeware product, is available
at www.flightgear.org*

Flight vehicle models:

*Jackson, E. Bruce; Cruz, Christopher I.; and Ragsdale, W.
A.: Real-Time Simulation Model of the HL-20 Lifting Body, NASA
TM-107580, July 1992.*

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