The art and science of air vehicle navigation concerns itself with two basic questions: how is the position of an air vehicle determined?; and how must it be guided to reach its destination? The answers to these questions are rooted in classical mechanics and feedback control theory. Practical solutions to the problems posed by these questions depend on sophisticated devices for measuring vehicular motion and on computing equipment capable of rapidly performing intricate mathematical computations. The design of computer programs capable of solving these problems are shaped by the mathematical models that describe the motion of air vehicles acting under the influence of gravitational, aerodynamic (e.g., lift and drag), and power system forces. These mathematical models are essential to the design and construction of on-board navigation and guidance systems, as well as flight simulators. The development of one such mathematical model in the subject of this series of articles:
In general terms, a mathematical model is a description of some system or process expressed in the language of mathematics. The mathematical model of flight dynamics developed here describes both a physical system (i.e., an aircraft and its surroundings), and a process (i.e., the control of the aircraft along some desired path). The equations of the model define relationships between system variables. The model includes mathematical descriptions of flight dynamics, the coordinate frames against which values of system variables take meaning, and the manner in which control over aircraft motion is effected. Equations of the model are used to convert motion sensor inputs into measurements of aircraft state, compute guidance parameters, derive steering and throttle commands, and perform coordinate frame transformations.
The hypothetical aircraft to which the mathematical model applies does not depend on the type (e.g., rotor driven or jet propelled, large or small, commercial or military) or the number and types of motion and position measuring devices installed on the aircraft. The model only requires that there be some means for estimating aircraft state (i.e., position, velocity, acceleration, and attitude) and and guidance parameters (e.g., track angle error, commanded airspeed). Large commercial and military aircraft are, typically, equipped with devices that provide independent and redundant measurements of acceleration, speed, altitude, attitude, and position by a variety of means. Inertial Reference Systems (IRSs), a collection of accelerometers and gyroscopes, can be used to measure acceleration and attitude. Integrating accelerations over time yields aircraft velocity. A Global Positioning System (GPS), when available, provides accurate measures of position and velocity, among other things. Devices for measuring air pressure (e.g., Pitot tube, altimeter) provide estimates of altitude and airspeed. Doppler radar and ground-based systems, like the Tactical Air Navigation and VHF Omni-directional Ranging Systems (TACAN/VOR), are examples of other means for estimating state-related measures such as velocity, range, and bearing.
Our goal is to derive a sophisticated mathematical model of aircraft dynamics providing the means for (a) predicting aircraft state using motion sensor inputs, and (b) estimating the steering and throttle commands needed to guide the aircraft along a selected track at some desired speed. Using this model, software developers will be able to design navigation and guidance systems capable of guiding an aircraft along a desired path. In this model, feedback is applied to dynamic variables in a way that forces aircraft state to match a desired state.
The derivation of the model is presented in steps. In Part 2: Reference Frames and Coordinate Transformations, a mathematical description of the several frames of reference that serve as a backdrop for the mathematical development is given. Then, in Part 3: Equations of Motion, a dynamic model of flight is developed. The result of this development is a set of differential equations (i.e., the equations of motion) that describe the dynamic behavior of the system. In Part 4: Closed-Loop, Feedback Control Model, equations used to compute steering and throttle commands are derived, which, when applied to the aircraft’s control surface actuators and throttle, result in a desired change in aircraft state. The control laws presented in this section determine the manner in which control compensations are computed and applied.
In Part 5: An Atmospheric Model, equations describing the variations in air temperature, pressure and density with altitude are given. These are needed to convert estimates of airspeed estimate to true airspeed and Mach number. The atmospheric model is also useful when simulating a Pitot-type airspeed indicator.
The mass (or weight, loosely speaking) of our hypothetical aircraft varies over time. The mass of the aircraft directly affects its flight performance, and, consequently, we must account for changes in its mass resulting from fuel loading and fuel consumption. The method whereby aircraft mass is predicted is presented in Part 6: A Mass Model.
In Part 7: Computational View of Guidance and Control, a method for computing aircraft state, guidance parameters and throttle and steering commands is outlined. The contents of this section is useful when applying the mathematical model to the design and construction of either operational or simulated flight systems.
Definitions for the symbols and the constants used in the development of the mathematical model are listed in Appendix A: Symbols Index and Appendix B: Constants Index. The vector notation used follows conventions in common practice. Vector quantities are represented using bolded characters (e.g., ). The dot or inner vector product is indicated using the notation, , while the cross or outer vector product is indicated using the notation . Unless explicitly stated otherwise, all vectors which appear in this report are three-dimensional, with components expressed relative to one of the four frames of reference (i.e., ECEF, NOLL, HOLL or BODY frames) described in Part 2: Reference Frames and Coordinate Transformations. The frame of reference relative to which the vector is expressed is indicated using the superscripts, E, N, H or B. The vector, , for example, is a three-dimensional vector relative to the HOLL frame with components, .
English units are the units of choice. Unless a statement to the contrary appears, it may be assumed that (a) distances are given in nautical miles; (b) force is expressed in pounds; (c) time intervals are measured in seconds; (d) velocities are expressed as knots; and (e) units associated with acceleration are ft/sec2; (f) temperature is given in degrees Fahrenheit; and (g) pressure is expressed in units of the standard atmosphere. Angular displacements are, unless otherwise stated, given in radians.
Unit conversion factors, wherever they appear, are of the form, . The subscript, from, and the superscript, to, indicate the initial and final units of measure for a quantity undergoing conversion. For example, in the equation, <, the distance, y, expressed in units of feet, is converted to an equivalent distance, x, expressed in units of nautical miles.