I'm enquiring you about a very hard problem (for me of course!). I'm starting to manipulate rotations using quaternion and I'm facing the following difficulty:
I have a combination of rotation such:
$Q1' \oplus Q' \oplus Q2 \oplus Q = Qr$
where Q1, Q2, Q and Qr are unitary quaternion not pure (with real part), Q' is conjugate of Q, $\oplus$ is quaternion multiplication
I known Q1, Q2, and Qr, and I tried to find Q. The basic response will be to develop quaternion multiplication and solve the equation, but it's very tedious !
Exist a way to find Q more easier ?
Thanks
You have an equation of form $abcx=d$ where, as you specify, you know the quaternions $a,b,c,d$ and seek to find $x$.
Now for any quaternion of form $p+qi+rj+sk$ its "conjugate" is $p-qi-rj-sk$ and the product of that quaternion with its conjugate is the real number $p^2+q^2+r^2+s^2.$ If $p,q,r,s$ are not all zero, i.e. if you didn't start with the zero quaternion, then the inverse of the quaternion is the conjugate divided by the above sum of four squares.
So to solve $abcx=d$ for $x$ one starts by multiplying sides on the left by the inverse of $a$ [denoted $a^{-1}$] and gets $a^{-1}abcx=a^{-1}d.$ Since $a^{-1}a=1$ this stage is $bcx=a^{-1}d.$ As you can see we only need to continue this using $b^{-1}$ and then $c^{-1}$ and end up with $x=c^{-1}b^{-1}a^{-1}d.$ Hope this helps.