what is the definition of quaternion rates?greenspun.com : LUSENET : quaternions : One Thread
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 theo (firstname.lastname@example.org), May 30, 2003
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 Sweetser (email@example.com), May 30, 2003.