Gimbal lock

See also: http://en.wikipedia.org/wiki/Gimbal_lock

Euler angles have a major deficiency, and that is, that it is possible, in some rotation sequences, to reach a situation where two of the three Euler angles cause rotation around the same axis of the object. In the case below, rotation around the x axis becomes indistinguishable in its effect from rotation around the z axis, so the z and x axis angles collapse into one transformation, and the rotation reduces from three degrees of freedom to two.

Imagine that we are using the Euler angle convention of starting with a rotation around the x axis, followed by the y axis, followed by the z axis.

Here we see a Spitfire aircraft, flying across the screen. The x axis is left to right (tail to nose), the y axis is from the left wing tip to the right wing tip (going away from the screen), and the z axis is from bottom to top:

images/spitfire_0.png

Imagine we wanted to do a slight roll with the left wing tilting down (rotation about x) like this:

images/spitfire_x.png

followed by a violent pitch so we are pointing straight up (rotation around y axis):

images/spitfire_y.png

Now we’d like to do a turn of the nose towards the viewer (and the tail away from the viewer):

images/spitfire_hoped.png

But, wait, let’s go back over that again. Look at the result of the rotation around the y axis. Notice that the x axis, as was, is now aligned with the z axis, as it is now. Rotating around the z axis will have exactly the same effect as adding an extra rotation around the x axis at the beginning. That means that, when there is a y axis rotation that rotates the x axis onto the z axis (a rotation of \pm\pi/2 around the y axis) - the x and y axes are “locked” together.

Mathematics of gimbal lock

We see gimbal lock for this type of Euler axis convention, when \cos(\beta) = 0, where \beta is the angle of rotation around the y axis. By “this type of convention” we mean using rotation around all 3 of the x, y and z axes, rather than using the same axis twice - e.g. the physics convention of z followed by x followed by z axis rotation (the physics convention has different properties to its gimbal lock).

We can show how gimbal lock works by creating a rotation matrix for the three component rotations. Recall that, for a rotation of \alpha radians around x, followed by a rotation \beta around y, followed by rotation \gamma around z, the rotation matrix R is:

R = \left(\begin{smallmatrix}\operatorname{cos}\left(\beta\right) \operatorname{cos}\left(\gamma\right) & - \operatorname{cos}\left(\alpha\right) \operatorname{sin}\left(\gamma\right) + \operatorname{cos}\left(\gamma\right) \operatorname{sin}\left(\alpha\right) \operatorname{sin}\left(\beta\right) & \operatorname{sin}\left(\alpha\right) \operatorname{sin}\left(\gamma\right) + \operatorname{cos}\left(\alpha\right) \operatorname{cos}\left(\gamma\right) \operatorname{sin}\left(\beta\right)\\\operatorname{cos}\left(\beta\right) \operatorname{sin}\left(\gamma\right) & \operatorname{cos}\left(\alpha\right) \operatorname{cos}\left(\gamma\right) + \operatorname{sin}\left(\alpha\right) \operatorname{sin}\left(\beta\right) \operatorname{sin}\left(\gamma\right) &- \operatorname{cos}\left(\gamma\right) \operatorname{sin}\left(\alpha\right) + \operatorname{cos}\left(\alpha\right) \operatorname{sin}\left(\beta\right) \operatorname{sin}\left(\gamma\right)\\- \operatorname{sin}\left(\beta\right) & \operatorname{cos}\left(\beta\right) \operatorname{sin}\left(\alpha\right) & \operatorname{cos}\left(\alpha\right) \operatorname{cos}\left(\beta\right)\end{smallmatrix}\right)

When \cos(\beta) = 0, \sin(\beta) = \pm1 and R simplifies to:

R = \left(\begin{smallmatrix}0 & - \operatorname{cos}\left(\alpha\right) \operatorname{sin}\left(\gamma\right) + \pm{1} \operatorname{cos}\left(\gamma\right) \operatorname{sin}\left(\alpha\right) & \operatorname{sin}\left(\alpha\right) \operatorname{sin}\left(\gamma\right) + \pm{1} \operatorname{cos}\left(\alpha\right) \operatorname{cos}\left(\gamma\right)\\0 & \operatorname{cos}\left(\alpha\right) \operatorname{cos}\left(\gamma\right) + \pm{1} \operatorname{sin}\left(\alpha\right) \operatorname{sin}\left(\gamma\right) & - \operatorname{cos}\left(\gamma\right) \operatorname{sin}\left(\alpha\right) + \pm{1} \operatorname{cos}\left(\alpha\right) \operatorname{sin}\left(\gamma\right)\\- \pm{1} & 0 & 0\end{smallmatrix}\right)

When \sin(\beta) = 1:

R = \left(\begin{smallmatrix}0 & \operatorname{cos}\left(\gamma\right) \operatorname{sin}\left(\alpha\right) - \operatorname{cos}\left(\alpha\right) \operatorname{sin}\left(\gamma\right) & \operatorname{cos}\left(\alpha\right) \operatorname{cos}\left(\gamma\right) + \operatorname{sin}\left(\alpha\right) \operatorname{sin}\left(\gamma\right)\\0 & \operatorname{cos}\left(\alpha\right) \operatorname{cos}\left(\gamma\right) + \operatorname{sin}\left(\alpha\right) \operatorname{sin}\left(\gamma\right) & \operatorname{cos}\left(\alpha\right) \operatorname{sin}\left(\gamma\right) - \operatorname{cos}\left(\gamma\right) \operatorname{sin}\left(\alpha\right)\\-1 & 0 & 0\end{smallmatrix}\right)

From the angle sum and difference identities (see also geometric proof, Mathworld treatment) we remind ourselves that, for any two angles \alpha and \beta:

\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \, \cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta

We can rewrite R as:

R = \left(\begin{smallmatrix}0 & V_{1} & V_{2}\\0 & V_{2} & - V_{1}\\-1 & 0 & 0\end{smallmatrix}\right)

where:

V_1 = \operatorname{cos}\left(\gamma\right) \operatorname{sin}\left(\alpha\right) - \operatorname{cos}\left(\alpha\right) \operatorname{sin}\left(\gamma\right) = \sin(\alpha - \gamma) \, V_2 = \operatorname{cos}\left(\alpha\right) \operatorname{cos}\left(\gamma\right) + \operatorname{sin}\left(\alpha\right) \operatorname{sin}\left(\gamma\right) = \cos(\alpha - \gamma)

We immediately see that \alpha and \gamma are going to lead the same transformation - the mathematical expression of the observation on the spitfire above, that rotation around the x axis is equivalent to rotation about the z axis.

It’s easy to do the same set of reductions, with the same conclusion, for the case where \sin(\beta) = -1 - see http://www.gregslabaugh.name/publications/euler.pdf.