Gimbal lock is the loss of one degree of freedom in a multi-dimensional mechanism at certain alignments of the axes. In a three-dimensional three-gimbal mechanism, gimbal lock occurs when the axes of two of the gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate two-dimensional space. The term can be misleading in the sense that none of the individual gimbals is actually restrained. All three gimbals can still rotate freely about their respective axes of suspension. Nevertheless, because of the parallel orientation of two of the gimbals' axes, there is no gimbal available to accommodate rotation about one axis, leaving the suspended object effectively locked (i.e. unable to rotate) around that axis. The problem can be generalized to other contexts, where a coordinate system loses definition of one of its variables at certain values of the other variables.
Gimbals
A gimbal is a ring that is suspended so it can rotate about an axis. Gimbals are typically nested one within another to accommodate rotation about multiple axes. They appear in gyroscopes and in inertial measurement units to allow the inner gimbal's orientation to remain fixed while the outer gimbal suspension assumes any orientation. In compasses and flywheel energy storage mechanisms they allow objects to remain upright. They are used to orient thrusters on rockets. Some coordinate systems in mathematics behave as if they were real gimbals used to measure the angles, notably Euler angles. For cases of three or fewer nested gimbals, gimbal lock inevitably occurs at some point in the system due to properties of covering spaces.
In engineering While only two specific orientations produce exact gimbal lock, practical mechanical gimbals encounter difficulties near those orientations. When a set of gimbals is close to the locked configuration, small rotations of the gimbal platform require large motions of the surrounding gimbals. Although the ratio is infinite only at the point of gimbal lock, the practical speed and acceleration limits of the gimbals—due to inertia (resulting from the mass of each gimbal ring), bearing friction, the flow resistance of air or other fluid surrounding the gimbals (if they are not in a vacuum), and other physical and engineering factors—limit the motion of the platform close to that point.
In two dimensions
Gimbal lock can occur in gimbal systems with two degrees of freedom such as a theodolite with rotations about an azimuth (horizontal angle) and elevation (vertical angle). These two-dimensional systems can gimbal lock at zenith and nadir, because at those points azimuth is not well-defined, and rotation in the azimuth direction does not change the direction the theodolite is pointing. Consider tracking a helicopter flying towards the theodolite from the horizon. The theodolite is a telescope mounted on a tripod so that it can move in azimuth and elevation to track the helicopter. The helicopter flies towards the theodolite and is tracked by the telescope in elevation and azimuth. The helicopter flies immediately above the tripod (i.e. it is at zenith) when it changes direction and flies at 90 degrees to its previous course. The telescope cannot track this maneuver without a discontinuous jump in one or both of the gimbal orientations. There is no continuous motion that allows it to follow the target. It is in gimbal lock. So there is an infinity of directions around zenith for which the telescope cannot continuously track all movements of a target. Note that even if the helicopter does not pass through zenith, but only near zenith, so that gimbal lock does not occur, the system must still move exceptionally rapidly to track it, as it rapidly passes from one bearing to the other. The closer to zenith the nearest point is, the faster this must be done, and if it actually goes through zenith, the limit of these "increasingly rapid" movements becomes infinitely fast, namely discontinuous. To recover from gimbal lock the user has to go around the zenith – explicitly: reduce the elevation, change the azimuth to match the azimuth of the target, then change the elevation to match the target. Mathematically, this corresponds to the fact that spherical coordinates do not define a coordinate chart on the sphere at zenith and nadir. Alternatively, the corresponding map T2→S2 from the torus T2 to the sphere S2 (given by the point with given azimuth and elevation) is not a covering map at these points.
In three dimensions
Consider a case of a level-sensing platform on an aircraft flying due north with its three gimbal axes mutually perpendicular (i.e., roll, pitch and yaw angles each zero). If the aircraft pitches up 90 degrees, the aircraft and platform's yaw axis gimbal becomes parallel to the roll axis gimbal, and changes about yaw can no longer be compensated for.
Solutions This problem may be overcome by use of a fourth gimbal, actively driven by a motor so as to maintain a large angle between roll and yaw gimbal axes. Another solution is to rotate one or more of the gimbals to an arbitrary position when gimbal lock is detected and thus reset the device. Modern practice is to avoid the use of gimbals entirely. In the context of inertial navigation systems, that can be done by mounting the inertial sensors directly to the body of the vehicle (this is called a strapdown system) and integrating sensed rotation and acceleration digitally using quaternion methods to derive vehicle orientation and velocity. Another way to replace gimbals is to use fluid bearings or a flotation chamber.
On Apollo 11 A well-known gimbal lock incident happened in the Apollo 11 Moon mission. On this spacecraft, a set of gimbals was used on an inertial measurement unit (IMU). The engineers were aware of the gimbal lock problem but had declined to use a fourth gimbal. Some of the reasoning behind this decision is apparent from the following quote:
The advantages of the redundant gimbal seem to be outweighed by the equipment simplicity, size advantages, and corresponding implied reliability of the direct three degree of freedom unit. They preferred an alternate solution using an indicator that would be triggered when near to 85 degrees pitch.
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