3D Rigid Body Dynamics

The difficulty of describing the positions of the body-fixed axis of a rotating body is approached through the use of Euler angles: spin ψ̇, nutation θ and precession φ shown below in Figure 1. In this case we surmount the difficulty of keeping track of the principal axes fixed to the body by making their orientation the unknowns in our equations of motion; then the angular velocities and angular accelerations which appear in Euler’s equations are expressed in terms of these fundamental unknowns, the positions of the principal axes expressed as angular deviations from some initial positions. Euler angles are particularly useful to describe the motion of a body that rotates about a fixed point, such as a gyroscope or a top or a body that rotates about its center of mass, such as an aircraft or spacecraft. Unfortunately, there is no standard formulation nor standard notation for Euler angles. We choose to follow one typically used in physics textbooks. However, for aircraft and spacecraft motion a slightly different one is used; the primary difference is in the definition of the ”pitch” angle. For aircraft motion, we usually refer the motion to a horizontal rather than to a vertical axis. In a description of aircraft motion, ψ would be the ”roll” angle; φ the ”yaw” angle; and θ the ”pitch” angle. The pitch angle θ would be measured from the horizontal rather than from the vertical, as is customary and useful to describe a spinning top.