gimbal_lock.rst

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:

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

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

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

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 $pmpi/2$ 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.