Hello, I am implementing quaternions for a flight dynamic model in order to avoid the singularity problem when using Euler angles (at 90deg pitch). For a bit of background (if necessary), I am performing the following steps: compute quaternion e compute e-dot derive direction cosines compute Euler angles based on the above

My problem is the following: I have stumbled across two different definitions of the quaternion rate (e-dot) calculations. Which one is correct?

I am more confident with this having followed the derivation of it: e0-dot = 0.5 * (-e1*vp - e2*vq - e3*vr) + Lambda * e0; e1-dot = 0.5 * ( e0*vp - e3*vq + e2*vr) + Lambda * e1; e2-dot = 0.5 * ( e3*vp + e0*vq - e1*vr) + Lambda * e2; e3-dot = 0.5 * (-e2*vp + e1*vq + e0*vr) + Lambda * e3;

Note: the lambda is a corretion factor for residual integration errors that serves to maintain the relationship e0^2+e1^2+e2^2+e3^2=1 (Ref: Cooke, Zyda et al.)

!!!! BUT, I have seen this one implemented and published on at least two other sites: E1D =0.5 * (- E4*P - E3*Q - E2*VR) + LAM*E1 E2D =0.5 * (- E3*P + E4*Q + E1*VR) + LAM*E2 E3D =0.5 * ( E2*P + E1*Q - E4*VR) + LAM*E3 E4D =0.5 * ( E1*P - E2*Q + E3*VR) + LAM*E4

They look completely different! Are several interpretations? Are both correct? I require enlightening!

Thank you all in advance,

Mike

-- mike theo (mike.theo@euro-ergon.com), May 30, 2003

Answers

Hello Mike:

There is no universally agreed apon convention for Euler angles. In fact, it is difficult to find two books that define them the same way. The signature (numbers of +'s and -'s or each e-dot is the same (a good sign). The middle two exactly correspond. It is the outer two that are out of whack, but I bet that is entirely due to different conventions for the Euler angles.

Quaternions can be defined a bunch of ways, but Hamilton's left-handed system is the standard.

doug

-- Doug Sweetser (sweetser@alum.mit.edu), May 30, 2003.