Re: [Jsbsim-devel] Euler Angles - Gimbal Lock - Quaternions

[Marius N. <ma...@u-...>]

julie barbic wrote:

> I would like to know if anyone could explain me what is exactly the 
> gimbal lock. I've read a lot of papers about that issue, but I still 
> don't see what loss of degree of freedom appears. 

Mathematically, the "gimbal lock" can be described as a discontinuity in
the Euler angles. For example, consider the case of an aircraft which is
pitching up, wings level. As the pitch angle reaches 90 deg (nose up)
the roll angle of the aircraft switches from a value of zero to a value
of 180 deg. At exactly 90 deg pitch the roll angle is basically unknown.
The quaternion attitude representation, on the other hand, does not
exhibit such discontinuities.

Quaternions are a little more abstract concepts, in other words
difficult to "visualize". They are related to another attitude
representation - the "Euler Axis Formulation" by a simple change of
variable, so let me explain what this Euler Axis is: The simple idea
behind it is that the arbitrary attitude of an aircraft can be
completely defined by a single rotation about a particular axis (called
the Euler axis). Thus to represent the attitude of the aircraft you need
4 parameters - the rotation angle and the 3 components of the Euler axis
unit vector.

There are several reasons why people prefer to work in quaternions even
though it is less intuitive. Besides the gimbal lock problem, the
quaternion mathematics have polynomials instead of trigonometric
functions thus you avoid the loss of precision induced by the Taylor
series approximations. Also, the kinematic equations which integrate the
angular rates to get the attitude are linear for the quaternion
formulation, which in case of an Inertial Navigation System, allows the
use of linear Kalman filters.

A concise paper which I found very useful is AIAA 2000-4302 "Review of
Attitude Representations Used for Aircraft Kinematics" by Phillips and
Hailey. Search for this in the AIAA Journal of Aircraft, Vol. 38, no. 4.
Or order it online from www.aiaa.org

Cheers,

-- 
Marius Niculescu
Unmanned Dynamics LLC
(541) 308-0894
http://www.u-dynamics.com