Editing Gimbal lock (section)

==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&mdash;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&mdash;limit the motion of the platform close to that point.

===In two dimensions===
[[File:Theodolite vermeer.svg|left|thumb]]
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.<ref>{{cite web|url= http://www.madsci.org/posts/archives/aug98/896993617.Eg.r.html |title= Re: What is meant by the term gimbal lock? |author= Adrian Popa |date= June 4, 1998}}</ref> 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 ''T''<sup>2</sup>&rarr;''S''<sup>2</sup> from the [[torus]] ''T''<sup>2</sup> to the sphere ''S''<sup>2</sup> (given by the point with given azimuth and elevation) is not a [[covering map]] at these points.

===In three dimensions===
[[File:Gimbal 3 axes rotation.gif|thumb|Gimbal with 3 axes of rotation. A set of three gimbals mounted together to allow three degrees of freedom: roll, pitch and yaw. When two gimbals rotate around the same axis, the system loses one degree of freedom.]]
[[Image:no gimbal lock.png|thumb|Normal situation: the three gimbals are independent]]
[[Image:gimbal lock.png|thumb|Gimbal lock: two out of the three gimbals are in the same plane, one degree of freedom is lost]]

Consider a case of a level-sensing platform on an aircraft flying due north with its three gimbal axes mutually perpendicular (i.e., [[Roll (flight)|roll]], [[Pitch (aviation)|pitch]] and [[Yaw angle|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 system]]s, that can be done by mounting the inertial sensors directly to the body of the vehicle (this is called a [[strapdown]] system)<ref>{{cite web|url=http://xenia.media.mit.edu/~verp/projects/smartpen/node8.html#SECTION00322000000000000000|title=Overview of Pen Design and Navigation Background|author=Chris Verplaetse|year=1995|url-status=dead|archiveurl=https://web.archive.org/web/20090214023126/http://xenia.media.mit.edu/~verp/projects/smartpen/node8.html#SECTION00322000000000000000|archivedate=2009-02-14}}</ref> 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.<ref>{{cite web|url=https://patentscope.wipo.int/search/en/detail.jsf?docId=WO2006060611&tab=PCTDESCRIPTION|title=Articulated gas bearing support pads|author=Chappell, Charles, D.|year=2006}}</ref>

===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.<ref>{{cite web|url=http://www.hq.nasa.gov/alsj/e-1344.htm|title=Apollo Guidance and Navigation - Considerations of Apollo IMU Gimbal Lock - MIT Instrumentation Laboratory Document E-1344|author=David Hoag|year=1963|access-date=2006-10-08|archive-date=2021-08-27|archive-url=https://web.archive.org/web/20210827041008/https://www.hq.nasa.gov/alsj/e-1344.htm|url-status=dead}}</ref> Some of the reasoning behind this decision is apparent from the following quote:

{{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.|[[David Hoag]]|''Apollo Lunar Surface Journal''}}

They preferred an alternate solution using an indicator that would be triggered when near to 85 degrees pitch.

{{Quote|Near that point, in a closed stabilization loop, the torque motors could theoretically be commanded to flip the gimbal 180 degrees instantaneously. Instead, in the [[Apollo Lunar Module|LM]], the computer flashed a "gimbal lock" warning at 70 degrees and froze the IMU at 85 degrees|Paul Fjeld|''Apollo Lunar Surface Journal''}}

Rather than try to drive the gimbals faster than they could go, the system simply gave up and froze the platform. From this point, the spacecraft would have to be manually moved away from the gimbal lock position, and the platform would have to be manually realigned using the stars as a reference.<ref>{{cite web|url=http://www.hq.nasa.gov/alsj/gimbals.html|title=Gimbal Angles, Gimbal Lock, and a Fourth Gimbal for Christmas|author1=Eric M. Jones|author2=Paul Fjeld|year=2006|access-date=2006-10-08|archive-date=2009-05-29|archive-url=https://web.archive.org/web/20090529014229/http://www.hq.nasa.gov/alsj/gimbals.html|url-status=dead}}</ref>

After the Lunar Module had landed, [[Michael Collins (astronaut)|Mike Collins]] aboard the Command Module joked "How about sending me a fourth gimbal for Christmas?"

===Robotics===
[[File:Automation of foundry with robot.jpg|thumb|right|Industrial robot operating in a foundry.]]
In robotics, gimbal lock is commonly referred to as "wrist flip", due to the use of a "triple-roll wrist" in [[robotic arm]]s, where three axes of the wrist, controlling yaw, pitch, and roll, all pass through a common point.

An example of a wrist flip, also called a wrist singularity, is when the path through which the robot is traveling causes the first and third axes of the robot's wrist to line up. The second wrist axis then attempts to spin 180° in zero time to maintain the orientation of the end effector. The result of a singularity can be quite dramatic and can have adverse effects on the robot arm, the end effector, and the process.

The importance of avoiding singularities in robotics has led the American National Standard for Industrial Robots and Robot Systems – Safety Requirements to define it as "a condition caused by the collinear alignment of two or more robot axes resulting in unpredictable robot motion and velocities".<ref>ANSI/RIA R15.06-1999</ref>