2.5 Rigid Body Motion in Three-Dimensional Space

A model that describes the translational and rotational motion of one or more objects must express the motion of those objects with respect to a set of coordinate systems. It must be possible to determine the position, velocity, angular orientation, and rotational rate of each object in the different coordinate systems to perform system simulation. For example, when Newton's Law of motion is applied to determine the translational motion of an object, you must determine the direction of forces and moments acting on the object with respect to inertial (nonrotating and nonaccelerating) space. This may be difficult, because the forces and moments acting on the body are frequently defined in terms of a body-fixed coordinate system that accelerates and rotates with the object.

In an aircraft flight simulation, there are typically two primary coordinate systems [7]. One coordinate system is fixed to the body of the aircraft and moves with it in both translation and rotation. This coordinate system has its origin at the aircraft center of mass and is called the body-fixed coordinate system. The other originates on the surface of the earth at a defined location and is called the earth-fixed coordinate system. The relations between these coordinate systems define the aircraft position, velocity, Euler angles (roll, pitch, and yaw angles), and rotation rates.

In many cases, the effects of the rotation of the earth and its curvature are negligible, which allows the earth to be modeled as flat and...