Equation expansion using Symbolic Toolbox

Hello all,

I would like to evaluate the quaternion expression using Symbolic Toolbox. I used the following convention.

I used the following code and the subsititutions.

syms g b a I J K real


A = cos(a/2)+ sin(a/2)*I;
B = cos(b/2)+ sin(b/2)*J;

C = cos(g/2)+ sin(g/2)*K;

q_rpy(I,J,K) = C*B*A;

F = expand(q_rpy)

F = subs(F,I*J,K)
F = subs(F,J*K,I)
F = subs(F,I*K,-J)
F = subs(F,K*J,-I)

F = subs(F,I^2,-1)

F = subs(F,J^2,-1)

F = subs(F,K^2,-1)

Is there any way to achieve the correct answer as above?

[Paul]

I don't think this can be done easily because the SMT assumes that multiplication commutes (among other things), which is not true for the "mutliplication" of i, j, and k. So it looks like some user-written functions would be needed. I'm curious if anyone comes up with a simple solution.

[Ravindra]

Hi Paul,

you said rightly. Hope someone might have an idea/correct solution.

Answer 1

[James Tursa]

This has been discussed on this forum before, and there is no easy solution to getting the Symbolic Toolbox to deal with non-commutative objects such as quaternions directly. There are somewhat indirect methods as discussed here:

https://www.mathworks.com/matlabcentral/answers/505682-how-to-remove-commutativity-from-symbolic-multiplication?s_tid=srchtitle

Another approach is to store all your quaternions as 4-element vectors and call your own quaternion multiply routines.

Finally, you might also be interested in this related post which discusses simplifying quaternion expressions that assume unit quaternions:

https://www.mathworks.com/matlabcentral/answers/671288-madgwick-filter-quaternion-multiplication?s_tid=srchtitle

[Ravindra]

Thanks James!

I checked the links you have provided. There is no suitable solution. I thought it might be easy to expand the 24 possible combinations using Matlab. Seems like I have to follow the old classical way on paper.

[John D'Errico]

The only real solution is to write your own class. And then it would seem simple.

And of course, the question is is this toolbox is mathematically correct and works:

https://www.mathworks.com/matlabcentral/fileexchange/1176-quaternion-toolbox?s_tid=srchtitle

I've not checked that. I've also not checked to see if it is compatible with the symbolic toolbox. If not, then I would just write a toolbox that was.

[Ravindra]

Thanks John. I will check the toolbox on the link you have provided.