Gimbal lock

From Wikipedia, the free encyclopedia

Jump to: navigation, search
 This article needs additional citations for verification.
Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (February 2009)
Animation: a set of three gimbals mounted together to allow three degrees of freedom.

Gimbal lock is the loss of one degree of freedom that occurs when the axes of two of the three gimbals needed to apply or compensate for rotations in three dimensional space are driven to the same direction.

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, flywheel energy storage mechanisms, or more commonly drink holders, they allow objects to remain upright. They are used to orient thrusters on rockets[1]. Some coordinate systems in mathematics behave as if there were real gimbals used to measure the angles. For cases of at least three nested gimbals mounted in a certain way, gimbal lock can occur.

Contents

[edit] Gimbal lock in mechanical engineering

[edit] Example

Normal situation: the three gimbals are independent
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, pitch and yaw angles each zero). If the aircraft pitches up 90 degrees, the aircraft and platform's roll axis gimbal becomes parallel to the yaw axis gimbal, and changes about yaw can no longer be compensated for (see illustration).

The word lock is misleading: no gimbal is restrained, all three gimbals can still rotate freely about their respective axis of suspension. Nevertheless, because of the parallel orientation of both the yaw and pitch gimbal axes, there is no axis available to accommodate yaw rotation.

[edit] Solutions

This problem may be overcome by use of a fourth gimbal, intelligently 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 strapdown system[2] 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[3].

[edit] Gimbal lock on Apollo 11

A well-known gimbal lock anecdote 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.[4] 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."

—David Hoag, Apollo Lunar Surface Journal

They preferred an alternate solution using a device that would be triggered when near to 85 degrees pitch. The device failed to function [5]:

"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 LM, the computer flashed a 'gimbal lock' warning at 70 degrees and froze the IMU at 85 degrees"

—Paul Fjeld, Apollo Lunar Surface Journal

After the Apollo had landed, Mike Collins joked "How about sending me a fourth gimbal for Christmas?"

[edit] Gimbal lock in applied mathematics

The problem of the gimbal lock appears when one uses the Euler angles in an application of mathematics, for example in a computer program (3D modeling, embedded navigation systems, 3D video games, metaverses, ...).

Euler angles provide a means for giving a numerical description of any rotation in three dimensional space using three numbers.

To make a comparison, all the translations can be described using three numbers x, y, and z, as the succession of three consecutive linear movements along three perpendicular axes X, Y and Z axes. That's the same for rotations, all the rotations can be described using three numbers α, β, and γ, as the succession of three rotational movements around three axes that are perpendicular one to the next. This similarity between linear coordinates and angular coordinates makes Euler angles very intuitive, but unfortunately they suffer from the gimbal lock problem.

[edit] Loss of a degree of freedom with Euler angles

A rotation in 3D space can be represented numerically with matrices in several ways. One of these representations is:

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

with α and γ constrained in the interval [ − π,π], and β constrained in the interval [0,π].

Let's examine for example what happens when β = 0. Knowing that \cos\, 0 = 1 and \sin \,0 = 0, the above expression becomes equal to:

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

The second matrix is the identity matrix and has no effect on the product. Carrying out matrix multiplication of first and third matrices:

\begin{align} R &= \begin{bmatrix} \cos \alpha \cos \gamma -\sin \alpha \sin \gamma & -\cos \alpha \sin \gamma - \sin \alpha \cos \gamma & 0 \\ \sin \alpha \cos \gamma + \cos \alpha \sin \gamma & -\sin \alpha \sin \gamma + \cos \alpha \cos \gamma & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{align}

And finally using the trigonometry formulas:

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

Changing α's and γ's values in the above matrix has the same effects: the rotation's angle α + γ changes, but the rotation's axis remains in the Z direction. The last column and the last line in the matrix won't change: one degree of freedom has been lost.

The only solution for α and γ to recover different roles is to get β away from the 0 value.

A similar problem appears when β = π.

One can choose another convention for representing a rotation with a matrix using Euler angles than the Z-X-Z convention above, and also choose other variation intervals for the angles, but at the end there is always at least one value for which a degree of freedom is lost.

Note that the gimbal lock problem does not make Euler angles "wrong" (they always play at least their role of a well-defined coordinates system), but it makes them unsuited for some practical applications.

[edit] The quaternion solution

Another representation for rotations in 3D space is the quaternions.

A quaternion is a tuple made of 4 numbers (s,\ x,\ y,\ z) that represents a geometrical similarity in the general case. If the relation s2 + x2 + y2 + z2 = 1 is verified, then the quaternion can be used to represent a rotation.

From a practical point of view, a rotation of an angle \mathbf{\theta} around an axis directed by an normalized vector \vec{N} = (x_0, y_0, z_0) is represented by the quaternion (\cos (\theta / 2),\ x_0 \cdot \sin (\theta / 2),\ y_0 \cdot \sin (\theta / 2),\ z_0 \cdot \sin (\theta / 2) ).

There is no problem similar to the gimbal lock with quaternions. This can be explained intuitively by the fact that a quaternion describes a rotation in one single move ("please turn \mathbf{\theta} radians around the axis driven by vector \vec{N}"), while the Euler angles are made of three successive rotations.

Besides that, quaternions also have other advantages over Euler angles.

[edit] References

  1. ^ Jonathan Strickland (2008). "What is a gimbal -- and what does it have to do with NASA?". http://science.howstuffworks.com/gimbal.htm. 
  2. ^ Chris Verplaetse (1995). "Overview of Pen Design and Navigation Background". http://xenia.media.mit.edu/~verp/projects/smartpen/node8.html#SECTION00322000000000000000. 
  3. ^ Chappell, Charles, D. (2006). "Articulated gas bearing support pads". http://www.wipo.int/pctdb/en/wo.jsp?IA=US2005043537&DISPLAY=DESC. 
  4. ^ David Hoag (1963). "Apollo Guidance and Navigation - Considerations of Apollo IMU Gimbal Lock - MIT Instrumentation Laboratory Document E-1344". http://www.hq.nasa.gov/alsj/e-1344.htm. 
  5. ^ Eric M. Jones and Paul Fjeld (2006). "Gimbal Angles, Gimbal Lock, and a Fourth Gimbal for Christmas". http://www.hq.nasa.gov/alsj/gimbals.html. 

[edit] See also

[edit] External links

Retrieved from "http://en.wikipedia.org/wiki/Gimbal_lock"
Personal tools
Languages