Avoiding gimbal lock.(Viewpoint: CG)

To place an object in 3D space, its transformation properties need to be specified: position, scale, and orientation. The first two attributes are easily definable by three numbers for each. The meaning of the x, y, and z positions and scale parameters are easy to understand, visualize, manipulate, and animate for artists. However, that is not the case for orientation.

Using an xyz triplet (three angular values) to manipulate an object's orientation may become impossible--for instance, during some configurations of the three angles, such as gimbal lock--and lead to major problems when animating these values. Gimbal lock is a phenomenon known for a long time, and it has caused severe problems long before computer graphics emerged. According to NASA documents on the Apollo space program, pilots had to keep a close eye on the Gimbal Lock warning light while maneuvering the spacecraft in order to avoid unwanted and dangerous malfunctioning in the guidance and control systems.

The orientation, or angular position, of a rigid object has three degrees of freedom. By holding a camera in our hands, for example, we can pan left and right, tilt it up and down, or roll it without changing the point of interest. The most common and intuitive way to define these attributes is the use of Euler angles: The orientation is represented by three consecutive rotations around the main axes of a reference frame. However, the order of the rotational axes is something the industry has never agreed on, so it is essential to supply this information if we transfer animation data using Euler angles. Each major 3D application has a way to change the order of rotations.

Using three gimbals, it is possible to construct a physical device, a gimbal system, based on the principle of Euler angles. A gimbal is a pivoted device, most often a ring, which rotates around a single axis. By mounting a gimbal inside another one, the inner ring rotates around an additional axis, increasing the degrees of freedom by one. Defining the orientation of an object with Euler angles is like putting it inside a virtual three-gimbal device, and then rotating each ring by the corresponding angle. (Once again, the order of rotations and the coordinate frame axes must be agreed upon.) The outer ring can represent the tilt, the middle ring the pan, and the innermost ring the roll. However, most of the publications on Euler angles refer to the three attributes as yaw, pitch, and roll, as used in aerospace applications.

[ILLUSTRATION OMITTED]

Having full control over the three degrees of freedom, one could conclude that Euler angles (or gimbal systems, in general) are the perfect way to describe orientation. Unfortunately, there are configurations wherein we lose one degree of freedom: the gimbal lock. In this state, one of the gimbal rings is rotated such that it aligns perfectly with another. In this situation, the entire range of rotations is unreachable, and we may need to first re-orient the locked gimbal in order to rotate the ring arbitrarily. If the angles are near the gimbal-lock state, the gimbal system becomes unstable, as even small rotations (or round-off errors of the numerical representation) may yield unexpected results.

[ILLUSTRATION OMITTED]