Eliminating the Euler Angle singularity without quaternions?

[James_Frogan]

Hi all,

I've formulated using Lagrangian formalism the equations of motion for a spinning top. I know about the gimbal lock/singularity that occurs at theta=0 and I was wondering if there was any other way to do it without dwelving into quaternions.

Yogi published a paper "A Motion of Top by Numerical Calculation" suggesting a replacement: $\dot{\beta} = \dot{\varphi} cos \vartheta$ and $\dot{\alpha} = \dot{\varphi} sin \vartheta$, but this hasn't worked for me (I find myself getting $\ddot{\alpha} = 0$, which isn't true)

I've had a look at quaternions but I'm not inclined to understand it or be able to simply 'convert' my equations over into quaternion calculus, neither (I believe) can I use Lagrangian mechanics on quaternions.

 

[Mbert]

One way to avoid the problem without using quaternions is to lock the possible values of theta to a given range that doesn't include the singularities. For example, theta cannot be smaller than 1E-6 degrees and larger than 179.99999 degrees. But the simplest way is using quaternions.

M.

 

[James_Frogan]

Thanks Mbert, unfortunately the case I am looking at causes the top to rise to the steady position.. which unfortunately is the theta 0 position.

Looks like I'll have to dwelve into quaternions. Is it possible to 'convert' my equations in Euler angles into quaternions if I have the equations in Euler's (rigid body) equations? Or must everything be redefined again?

 

[Mbert]

To convert from Euler angles to quaternions take a look at this website:

http://www.flipcode.com/documents/matrfaq.html#Q60"

Especially the part:

Q60. How do I convert Euler rotation angles to a quaternion?
-------------------------------------------------------------------

Converting Euler rotation angles to quaterions can be achieved through
the use of quaternion multiplication. Each rotation angle is converted
to an axis-angle pair, with the axis corresponding to one of the
Euclidean axii. The axis-angle pairs are converted to quaternions and
multiplied together. The final quaternion is the desired result.

The following code segment demonstrates this:
---------------------------------------------

quaternion_from_euler( QUATERNION *q, VFLOAT ax, VFLOAT ay, VFLOAT az )
{
VECTOR3 vx = { 1, 0, 0 }, vy = { 0, 1, 0 }, vz = { 0, 0, 1 };
QUATERNION qx, qy, qz, qt;

quaternion_from_axisangle( qx, &vx, rx );
quaternion_from_axisangle( qy, &vy, ry );
quaternion_from_axisangle( qz, &vz, rz );

quaternion_multiply( &qt, &qx, &qy );
quaternion_multiply( &q, &qt, &qz );
}
---------------------------------------------

M.

 

[James_Frogan]

Thanks Mbert, I came across several articles on how to convert the angles themselves to quaternion, however the equations of motion are in the forms of: $\ddot{\phi},\ddot{\psi},\ddot{\vartheta} = f(\phi,\psi,\vartheta,\dot{\phi},\dot{\psi},\dot{\\theta})$, so I cannot apply the conversions in that manner.

Does Lagrange formalism work on quaternions?

 

[Mbert]

Then, you should have a look at:

http://www.fho-emden.de/~hoffmann/quater12012002.pdf

M.