Kinematics of Translation and Rotation

After spending two chapters in the three-dimensional world of geometry, we are ready to launch into the fourth dimension, time. We will study kinematics, the branch of mechanics that deals with the motion of bodies without reference to force or mass. Later, in Chapters 5 and 6 we will add mass and force, apply Newton’s and Euler’s laws, and study the dynamics of aerospace vehicles.

If you watch the space shuttle take off at the Cape and track its altitude gain in time, you study its launch kinematics. However, if you are in Mission Control, responsible for ascent and orbit insertion, you concern yourself with the effect of mass, drag, and thrust and therefore are accountable for the dynamics of the space shuttle. Dynamics builds on kinematics. Hence we begin with kinematics.

I first introduce the rotation tensor, which defines the mutual orientation of two frames. It is the physical equivalent of the abstract coordinate transformation. Then I go right to the essence of the coordinate-independent formulation of kinematics and introduce the rotational time derivative. It will enable us in Chapters 5 and 6 to formulate Newton’s and Euler’s laws in an invariant form, valid in any allow­able coordinate system. Afterward you are ready for the discussion of linear and angular motions of aircraft, missile, and spacecraft in greater detail. Finally, we wrap up the chapter with the fundamental problem in kinematics of flight, namely, how to calculate the attitude angles from the body’s angular velocity.

Throughout these minutiae we shall remain true to our principle “from in­variant modeling to matrix simulations.” All of the forthcoming kinematic con­cepts are valid in any allowable coordinate system and thus are true tensor con­cepts. So welcome aboard, bring your tool chest, and I will fill it up with more goodies.