quaternion kinematic formulation in attitude dynamics block does not seem to be complete.

In the attitude dynamics block help files the kinematic equation for quaternion is given as

Infact there should be a (-1/2) coefficient as a multiplier. This I believe is a typo in the help file, because the actual block seems to be calculating using this factor. Do you have any comments ?

[Umar]

Hi @Gökhan,

After reviewing the documentation provided in the link below

https://www.mathworks.com/help/aeroblks/attitudedynamics.html#mw_79668ddf-be10-4a93-aab5-7903fde3b29c

and articles online about what you mentioned clearly signifies the presence of the 0.5 coefficient in the quaternion kinematic equation; it is a critical component. However, the equation provided in the documentation is technically relevant to rotational dynamics, my interpretation is that both frameworks relate to rotational motion, they serve different purposes. Rotational dynamics provides a physical basis rooted in forces and torques, while quaternion kinematics offers a computational method for managing rotations without singularities. They are not considered the same; rather, they complement each other within the broader context of rotational motion analysis.

Based on these comments, if you still doubt or suspect a typo in the documentation for community of people have years of technical experience then by all means consider reaching out to the support team for clarification.

https://supportcases.mathworks.com/mwsupport/s/casecreation?language=en_US&c__caseParameter=Report_a_Bug&s_tid=srl22_br

Hope this helps.

[James Tursa]

@Gökhan This is definitely a typo. The -1/2 (or +1/2 in other quaternion conventions) needs to be there. Good catch. You should submit a bug report with the documentation error.

[Gökhan]

Many thanks for your responses @William Rose , @James Tursa and @Umar. I did some tests to find out if MATLAB is internally using this equation or is it just a typo in the help file itself. It seems, internally it is using a (-1/2) factor to implement the equation. I manually generated a MATLAB functon block to code the same equation and compare the output quaternion with the output of the block. When I used (-1/2) factor, there is a very small discrepancy (in the order of 1e-5 to 1e-7) which tells me that MATLAB indeed is implementing with the (-1/2) factor. However what also bothers me, I am using the same integration methods and etc but still where does this discrepancy come from I wonder. Because in the long run it seems to be accumulating more and more. Now thinking may be I am doing something wrong, I decided to do another test. I compare MATLAB with MATLAB itself, i.e. I compared the output quaternion of built in "Attitude Dynamics" and built in "Spacecraft Dynamics" outputs as seen below.

So you see I am not doing anything extra. I just make sure that the same initial conditions and same orbital parameters are used in both settings. Then I plotted the difference between the quaternions generated. The output is as follows.

the difference order is small (1e-7) however this is due to the initial w=[0,0.1,0] deg/s. If I give lareger w or more complex initial conditions like non-zero components for [wx,wy,wz], the delta increases.

My understanding is that "Space Dynamics" block is just the combination of "Orbit Propagator" block and "Attitude Dynamics" block. As such, one would expect the same output hence a delta of zero (as is the case for comparing output w for which the delta is zero). What is with quaternion integration I do not understand? Any ideas ?

Answer 1

[William Rose]

@Gökhan,

I think you are correct. A factor of (-1/2) should multiply the right hand side matrices in the equation here for the time derivative of the quaternion.

See Phillips et al 2001, “Review of attitude representation used for aircraft kinematics”, Journal of Aircraft 38 (4): 718-737, equation 110.

Also see Suh YS 2010, “Orientation estimation using a quaternion-based indirect Kalman filter”, IEEE Trans Instrum Measure 59(12): 3296-3305, equation 18.

[William Rose]

I used the quaternion update equation when analyzing head acceleration during soccer heading. I made animations using the quaternion update equation (with the factor of -1/2 above) and the angular velocities measured with an accelerometer. The animations were consistent with motion capture data. Therefore I have confidence in the equation.