How Do Quaternions Rotate a Vector to a Specific Orientation?

[Four]

Homework Statement

given z = (0, 0, 0, 1), and v = (0, x, y, z),

and the following properties hold

[tex] v \times u = L [/tex]
[tex] L \times v = u [/tex]
[tex] u \times L = v [/tex]

v, u, l, and z are unit quaternions

In other words; we define forward (v), up (u) and right(L), find a unit quaternion that when you apply it on z, you get v, in the proper orientation.

solve for q = (a, b, c, d)

Homework Equations

(equation to solve for q)
[tex]qzq^{-1} = v[/tex]
Left side applys q on z and right side is v

The Attempt at a Solution

[tex] qz = vq [/tex]
[tex] dk = (xi + yj + zk)(a + bi + cj + dk) [/tex]

after expanding and grouping the terms on the same axis together. In a matrix form I got

[tex] \left(
\begin{array}{cccc}
x & y & z & 0 \\
0 & z & y & x \\
z & 0 & -x & y \\
-y & x & -1 & z
\end{array}
\right)

\left(
\begin{array}{c}
a \\
b \\
c \\
d
\end{array}
\right)
=
\left(
\begin{array} {c}
0 \\
0 \\
0 \\
0
\end{array}
\right)
[/tex]

[a, b, c, d] = [0, 0, 0, 0] ? That doesn't seem right?

Shouldn't I need to use atleast the vector u?

Thank you for your help

 

[mighty2000]

.

Hello,

Thank you for your post. Your attempt at a solution is a good start, but there are a few things that need to be corrected.

Firstly, the equation qzq^(-1)=v is not correct. It should be qzq^(-1)=vq. This is because quaternion multiplication is not commutative, so the order of multiplication matters.

Secondly, the matrix you have written is not correct. The correct matrix should be:

\left(
\begin{array}{cccc}
a & -d & c & -b \\
d & a & -b & -c \\
-c & b & a & -d \\
b & c & d & a
\end{array}
\right)

This can be derived from the definition of quaternion multiplication: (a+bi+cj+dk)(e+fi+gj+hk) = (ae-bf-cg-dh) + (af+be+ch-dg)i + (ag-bh+ce+df)j + (ah+bg-cf+de)k

Thirdly, you are correct in thinking that you need to use the vector u in your solution. In order to fully define the quaternion q, you need to find values for a, b, c, and d that satisfy all three equations given in the problem. This means that you need to use the vector u in addition to the vector v.

I hope this helps. Good luck with your solution!