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Six-DoF Equations of Motion
Before we embark on our journey, it would be to your advantage to stop by at Chapter 5 and review Newton’s law and Chapter 6 on Euler’s law. Otherwise, starting from these general principles the derivation here is self-contained.
As always, I shall formulate the equations in an invariant tensor form first, followed by the all-important decision of coordinate systems. The form of the MOI tensor will lead us into two different directions. For missiles with tetragonal symmetry, the equations are so simple that we write them in scalar form. For aircraft with planar symmetry however, we have to live with an inverted moment of inertia matrix.
The selection of the inertial reference frame is always an important decision that you have to make before you begin your modeling task. For near-Earth aerospace vehicles, including low-Earth orbiting satellites, the Earth-centered, ecliptic-oriented frame serves as inertial reference (J2000). It can be combined with a spherical or elliptical Earth shape. However, frequently the trajectory of a vehicle is so slow compared to orbital speed and so close to the Earth that the centrifugal and Coriolis accelerations can be neglected. In these cases we choose the Earth as the inertial frame and unwrap the Earth’s surface into a plane tangential to the launch point. We speak now of the flat-Earth approach.
The choice is yours to make. I will first discuss the easier flat-Earth case, applicable to tactical missiles and aircraft and add two special cases: spinning missile and Magnus rotor. Then I will lure you over to the much more difficult modeling task of the elliptical Earth. We will make excursions into geodesy and study the shape and gravitational field of the Earth before we derive the equations of motions. The inertial frame is the Earth-centered, ecliptic-oriented frame. To provide you with the full spectrum of formulations, I will not only use the inertial frame as a reference frame but also the Earth frame. You will encounter both options in six-DoF simulations.
