Modeling of Flight Dynamics

Flight dynamics is the study of vehicle motions through air or space. Unlike cars and trains, these motions are in three dimensions, unconstrained by road or rail. Flight dynamics is rooted in classical mechanics. Newton’s and Euler’s laws are quite adequate to calculate their motions. Relativistic effects are relegated to miniscule perturbations.

An aerospace vehicle experiences six degrees of freedom. Three translational degrees describe the motion of the center of mass (c. m.), also called the trajectory, and three attitude degrees orient the vehicle. If the c. m. of the vehicle is used as
reference point, the translational and attitude motions can be described separately. Tracking a missile means recording the position coordinates of its c. m. Maintaining attitude of an aircraft requires the pilot to watch carefully the attitude indicator without reference to the aircraft’s position.

Newton’s second law governs the translational degrees of freedom and Euler’s law controls the attitude dynamics. Both must be referenced to an inertial reference frame, which includes not just the linear and angular momenta but also their time derivatives. As long as the coordinate system is inertial, the equations are simple, but if body coordinates are introduced additional terms appear that make the adjustments for the time-dependent coordinate transformations.

My goal is to model flight dynamics in a form that is invariant under time – dependent coordinate transformations. To that end, these additional terms must be suppressed. A time operator, the rotational time derivative, will accomplish this feat. With it we can formulate the equations of motion in an invariant tensor form, independent of coordinate systems.

To clarify that approach, let me use Newton’s second law as presented in any physics book. With p the linear momentum vector and/ the external force vector, the time rate of change of the linear momentum equals the external force

Implied is that the time derivative is taken with respect to the inertial reference frame I. If we want to change the reference frame to the vehicle’s body frame B, Newton’s law must be written

dp

d t

with u> the angular velocity of the body relative to the inertial frame. For pro­gramming, we have to coordinate the two equations. Because of the time deriva­tives, we express the first equation in inertial coordinates and the second one in body coordinates. Brackets and superscripts I or В indicate the coordinated vectors

-dpi1

d t

where [£2]B is the skew-symmetric form of ш, expressed in body coordinates. The time derivative is not a tensor concept because it changes its form as the inertial coordinates are replaced by the body coordinates. It is not invariant under the transformation matrix (T]B7 of the body coordinates with respect to the inertial coordinates, i. e., the right and left sides of the transformation are dissimilar:

If we introduce the rotational time derivative D1 relative to frame /, Newton’s law has the same form in both coordinate systems,

[.D>pY = uv

[D’p]B = [f]B

and the rotational time derivative transforms like a first-order tensor:

[D’p]B = [Tf^D’p]1

With [T]BI representing any, even time-dependent, coordinate transformations, Newton’s law can be expressed in the invariant tensor form

D’p = f (1.2)

valid in any coordinate system. This tensorial formulation is the key to the invariant modeling of flight dynamics. It will allow us to derive the mathematical model first without consideration of coordinate systems. After having made desired changes, we pick the appropriate coordinate systems and code the component form.

The motto “from tensor modeling to matrix coding” will guide us through kinematics and dynamics to the simulation of aerospace vehicles. This approach has served me well over 30 years. I hope that you will also benefit from it by the diligent study of the following chapters.

The second chapter, “Mathematical Concepts in Modeling,” lays the foundation through classical mechanics, a branch of physics. The axioms of mechanics and the principle of material indifference provide the sure footing for the modeling tasks.

With the hypothesis that points and frames are sufficient to model dynamic prob­lems, I build a nomenclature that is self-defining. For instance, the displacement of missile M from the tracking radar R is modeled by the displacement vector smr of the two points, whereas the angular velocity of body frame В with respect to the Earth E is given by the angular velocity vector u>BE. You will encounter other symbols that use points and frames, like linear velocity, angular momentum, moment of inertia, etc.

I permit only physical variables that are invariant under time-dependent coordi­nate transformations, that is, true tensor concepts. A construct like a radius vector has no place in our toolbox. Coordinate systems are abstract entities relating the components of a vector to Euclidean space. They have measure and direction, but no common origin. With these provisos we build our models with Cartesian tensors, as physical concepts, independent of coordinate systems.

With these tools we assail geometrical problems, like the near collision of two airplanes, both flying along straight lines; the miss distance of a missile impacting a plane; the imaging of an object on a focal plane array; and others. Problems at the end of the chapter invite you to practice your skills.

The third chapter, “Frames and Coordinate Systems,” distinguishes carefully between the two concepts. Frames are models of physical objects consisting of mutually fixed points, but coordinate systems have no physical reality. They are, as already characterized, mathematical abstracts. We make use of the nice properties of the transformation matrices between Cartesian coordinate systems. They are orthogonal, and therefore their inverse is the transpose. As the direction cosine matrix, they play an important part in flight mechanics.

No engineering discipline other than flight mechanics has to deal with so many coordinate systems. We will work with most of them: heliocentric, inertial, Earth, geographic, body, wind, and flight-path coordinate systems. We distinguish be­tween round rotating Earth and flat Earth. In Chapter 10,1 shall also introduce the oblate Earth and the geodetic coordinate system.

This chapter wraps up the modeling of geometrical problems. Do not under­estimate their importance. In a typical aerospace simulation you may find that one-third to one-half of the effort is expended to get the geometry right. The next chapter leads us to the kinematics of flight vehicles.

The fourth chapter, “Kinematics of Translation and Rotation,” introduces time and models the motions of vehicles without consideration of forces. We describe the translation of bodies by the displacement vector and their attitude by the ro­tation tensor. Their time derivatives are linear and angular velocities. It is here that I introduce the rotational time derivative, both for vectors and tensors. As already emphasized before, the rotational time derivative enables us to model flight dynamics by equations that are invariant under time-dependent coordinate transformations.

To shift reference frames, from inertial to Earth for instance, Euler’s transforma­tion is introduced. It is the generalization of the familiar form, shown in Eq. (1.1). Many derivations rely on it, particularly the formulation of the translational and attitude equations of motion. Shifting from the inertial to the Earth frame incurs such apparent forces as the Coriolis and centrifugal forces.

Finally in this chapter we solve the fundamental kinematic problem of flight dynamics, namely, given the body rates of the vehicle, determine the attitude angles. We take three approaches. The Euler method integrates the Euler angles directly with the penalty of singularities in the differential equations. Avoiding this disadvantage, the direction cosine and quaternion methods both solve linear differential equations. They are the preferred approach today because their higher computational load is no detraction any longer.

The fifth chapter, “Translational Dynamics,” introduces Newton’s second law for modeling the translational dynamics of aerospace vehicles. It is, together with Chapter 6, the heart of flight dynamics. Starting with the linear momentum, I formulate Newton’s second law first for particles and then for rigid bodies. The earlier teaser on the invariancy of Newton’s law will be fully developed. With Euler’s transformation I derive the Coriolis and Grubin transformations for shifts in reference frames and reference points, respectively. You will also get the first taste of simulations from the derivation of the translational equations for three-, five-, and six-degree-of-freedom (DoF) models.

The sixth chapter, “Attitude Dynamics,” formulates the attitude equations of motions based on Euler’s law. Conventional wisdom says that the attitude equations are a consequence of Newton’s law, but I will give evidence that Leonhard Euler developed them independently.

This chapter will challenge your mechanistic mind more than the rest of the book. I introduce the moment of inertia tensor with its axial and cross product of inertia. The moment of inertia ellipsoid gives a geometrical picture of the principal axes. As the linear momentum is at the center of Newton’s law, so is the angular momentum the heartbeat of Euler’s law. I start with particles and then expand the angular momentum to rigid bodies and eventually to clustered bodies. Euler’s law states that the inertial time rate of change of the angular momentum equals the externally applied moments. Again, we use the rotational time derivative to present Euler’s equation in tensor form, invariant under time-dependent coordinate transformations.

Now we are in a position to formulate the equations of motion of an aerospace vehicle and of a conventional spinning top. Of course, our emphasis is on free flight and on the significance of the c. m. of the vehicle. If the c. m. is used as reference point, Euler’s equation simplifies greatly and becomes dynamically uncoupled from the translational equation. With l as the angular momentum and m the externally applied moment, we can formulate Euler’s equation and combine it with Newton’s Eq. (1.2) for the fundamental equations of flight dynamics:

D’p = f, D’l — m (1.3)

All modeling in flight dynamics begins with these equations. They are the backbone of six-DoF simulations.

The ultimate challenge is the formulation of the dynamics of clustered bodies. With the theorems and proofs you should be able to derive the equations of motion of a shuttle releasing a satellite, the swiveling nozzle of a missile, or an aircraft with rotating propellers.

Finally, I will introduce you to the mysterious world of gyrodynamics. The unexpected response of gyroscopes, their precession and nutation modes can easily be explained by Euler’s law. With the energy theorem we derive two integrals of motion, the conservation of energy and angular momentum, which are pivotal for satellite dynamics.

The seventh chapter, “Perturbation Equations,” completes the assortment of modeling techniques. Although perturbation equations are rarely used for full – up simulations, they are important for stability investigations and control system design. Even here I emphasize the invariant formulation of perturbations, which leads to component perturbations and the general perturbation equations of flight vehicles for unsteady reference flight.

The perturbations of aerodynamic forces and moments are given close atten­tion. Taking advantage of the configurational symmetry of airplanes and missiles, vanishing derivatives of the Taylor series are sifted out and techniques presented for including higher-order derivatives.

As applications, we derive the roll, pitch, and yaw transfer functions for the autopilot designs of Chapter 10. More sophisticated examples are the perturbation equations of aircraft during pull-up, and of missiles executing high g maneuvers. These are illustrations of perturbation equations of unsteady reference flight, in­cluding nonlinear aerodynamic coupling effects.

Part 1 concludes here. It is a comprehensive treatment of Newtonian dynamics, sufficient for any modeling task in flight dynamics. The physical nature of the phenomena is emphasized by the invariant tensor formulation. Yet eventually, we have to feed our computers with instructions and numbers. That practical step is the subject of Part 2.