Gimbal lock

Gimbal lock: If the pitch (green) is 90 °, i.e. the nose of the aircraft is pointing upwards, rotations around the roll (blue) and yaw axes (purple) have the same effect. A rotation around the roll axis of the original coordinate system is no longer possible.

Gimbal lock is in the mechanics a critical state of a gimbal bearing ( English Gimbal ). In this state, two of the possible three axes of rotation are parallel to one another and the system therefore has one less degree of freedom. The term is also used for a mathematical problem in three-dimensional simulation , in which rotations with respect to certain axes can no longer be realized with the help of rotation operators for previously defined axes.

Mechanical blockage of a cardan bearing

With a cardanic suspension, the object to be stored is held by three independently movable bearings, each of which enables rotation around an axis. The first of the axes is fixed in space, the second axis is arranged perpendicular to it and can rotate freely around the first axis. The third axis is arranged perpendicular to the second axis and can rotate freely around it. The object to be stored can rotate freely around the third axis.

By suitable rotation of the third axis about the second axis, it can be achieved that the third and the first axis are parallel to one another. A rotation of the object about an axis that is perpendicular to both the first and the second axis is therefore no longer possible. Rotations around this axis are blocked .

With mechanical systems, such as the cardanic bearing, a restriction of the freedom of movement can be observed when approaching this state, since even small rotations around the blocked axis require considerably larger rotations around the free axes. It also happens that inaccurate positioning with respect to one of the two almost parallel axes is erroneously corrected by rotation with respect to the other axis. This leads to inaccuracies in the positioning.

Restricted degrees of freedom in navigation

In the case of an aircraft, the position of the aircraft in space is defined by the roll, pitch and yaw angles , which characterize the turns of the aircraft through rotations around the longitudinal , transverse and vertical axes . Theoretically, rotations around all three axes are possible in any position of the aircraft, so the aircraft has three independent degrees of freedom.

After defining a reference position, for example as "longitudinal and transverse axes parallel to the horizon plane, longitudinal axis in north direction", the position of the aircraft can also be clearly indicated by three independent angles with respect to this reference position:

the direction of flight as the angle between the projection of the longitudinal axis of the aircraft on the horizon plane and a reference direction in this plane
the climb angle as the angle between the longitudinal axis of the aircraft and its projection onto the horizon plane
the roll angle as the angle between the transverse axis of the aircraft and its projection onto the horizon plane

A gimbal lock occurs when the projection of the longitudinal axis on the horizon plane disappears: When flying vertically up or down (climbing angle 90 °), the flight direction cannot be influenced, the roll angle changes due to yaw movements (rotations around the vertical axis). Rolling movements (rotations around the longitudinal axis) do not affect any of the three angles mentioned. Although they change the angle between the transverse axis of the aircraft and a reference direction in the horizon plane, this is not one of the angles mentioned and cannot be derived from them.

root cause

The cause of a gimbal lock is that the position of one axis of rotation depends on the rotations with respect to the other two axes of rotation.

Mathematical description

A rotation in three dimensions around the cardan angle can be represented by the rotation matrix . It can be represented by the matrix product of three matrices, each of which only depends on one of the angles. ${\ displaystyle (\ alpha, \ beta, \ gamma)}$ ${\ displaystyle \ mathbf {R}}$

{\ displaystyle {\ mathbf {R}} = {\ begin {bmatrix} 1 & 0 & 0 \\ 0 & \ cos \ alpha & - \ sin \ alpha \\ 0 & \ sin \ alpha & \ cos \ alpha \ end {bmatrix}} { \ begin {bmatrix} \ cos \ beta & 0 & \ sin \ beta \\ 0 & 1 & 0 \\ - \ sin \ beta & 0 & \ cos \ beta \ end {bmatrix}} {\ begin {bmatrix} \ cos \ gamma & - \ sin \ gamma & 0 \\\ sin \ gamma & \ cos \ gamma & 0 \\ 0 & 0 & 1 \ end {bmatrix}}}

A gimbal lock occurs when and thus and become. Then becomes : ${\ displaystyle \ beta = {\ tfrac {\ pi} {2}}}$ ${\ displaystyle \ cos \ beta = 0}$ ${\ displaystyle \ sin \ beta = 1}$ ${\ displaystyle {\ mathbf {R}}}$

{\ displaystyle {\ begin {aligned} {\ mathbf {R}} & = {\ begin {bmatrix} 1 & 0 & 0 \\ 0 & \ cos \ alpha & - \ sin \ alpha \\ 0 & \ sin \ alpha & \ cos \ alpha \ end {bmatrix}} {\ begin {bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ - 1 & 0 & 0 \ end {bmatrix}} {\ begin {bmatrix} \ cos \ gamma & - \ sin \ gamma & 0 \\\ sin \ gamma & \ cos \ gamma & 0 \\ 0 & 0 & 1 \ end {bmatrix}} \\ & = {\ begin {bmatrix} 0 & 0 & 1 \\\ sin \ alpha \ cos \ gamma + \ cos \ alpha \ sin \ gamma & - \ sin \ alpha \ sin \ gamma + \ cos \ alpha \ cos \ gamma & 0 \\ - \ cos \ alpha \ cos \ gamma + \ sin \ alpha \ sin \ gamma & \ cos \ alpha \ sin \ gamma + \ sin \ alpha \ cos \ gamma & 0 \ end {bmatrix}} \\ & = {\ begin {bmatrix} 0 & 0 & 1 \\\ sin (\ alpha + \ gamma) & \ cos (\ alpha + \ gamma) & 0 \\ - \ cos (\ alpha + \ gamma) & \ sin (\ alpha + \ gamma) & 0 \ end {bmatrix}} \ end {aligned}}}

Only the combination appears in this matrix . This means that the two independently chosen angles and lead to a transformation that has only one parameter. The parameters are no longer independent of each other. As a result, one of the degrees of freedom has been lost. ${\ displaystyle \ alpha + \ gamma}$ ${\ displaystyle \ alpha}$ ${\ displaystyle \ gamma}$

Problem areas

In practice, a gimbal lock occurs in the navigation of ships (especially submarines and ROVs ), aviation and especially space travel , as well as in all applications of gimbal-mounted gyros, mainly gyrocompasses and stabilization gyros (in spacecraft ).

It is possible that both axes follow the same orientation or that the axes continue in the orientation of the other axis. Both cases deliver an incorrectly displayed flight attitude and an incorrectly displayed course , which has the consequence that the downstream controls react incorrectly and thus lead to a worsening of the situation. This rocks up to uncontrolled tumbling in space travel. The Gemini program , for example, worked with a gyro compass that had a fourth axis for counter-steering. This was done without the compass for Apollo spaceships , and the fourth axis was reinstalled in the space shuttle .

The Apollo Guidance Computer , which has been significantly improved compared to the Gemini Guidance Computer , had a warning lamp in the display and control unit for an impending gimbal lock if the axis position changed by 70 °.

resolution

With mechanical bearings, the problem can be solved by adding another bearing. In the uncritical state, the freedom of movement with regard to this bearing is redundant to that with regard to the three existing bearings.

Alternatively, the problem can be alleviated by using servomotors to change the position of the axes when the blocked state is approached, thus avoiding the blockage.

Inertial navigation sensors use three independent position sensors (and three rotation rate sensors) so that this problem does not occur. The data will then be billed electronically.

In computer models , such as three-dimensional simulations or some computer games , the problem can be circumvented by using quaternions . Another solution can be to convert the global orientations into local coordinate systems, especially when quaternions are not helpful.

Individual evidence

^ Adrian Popa: Re: What is meant by the term gimbal lock? . June 4, 1998.
↑ Jonathan Strickland: What is a gimbal - and what does it have to do with NASA? . 2008.
↑ Chris Verplaetse: Overview of Pen Design and Navigation Background . 1995. Archived from the original on February 14, 2009.

Web links

NASA site
Gimbal lock
Explanation of the gimbal lock on Youtube (video, English)

[1] Adrian Popa: Re: What is meant by the term gimbal lock? . June 4, 1998.

[2] Jonathan Strickland: What is a gimbal - and what does it have to do with NASA? . 2008.

[3] Chris Verplaetse: Overview of Pen Design and Navigation Background . 1995. Archived from the original on February 14, 2009.