maths.tex

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\DeclarePairedDelimiter{\opair}{\langle}{\rangle}

Let
$\vec{s}_{i}$
be the position, relative to the sensor array orthocenter, of the
$i^{th}$ sensor of the tracked device in the tracked device's coordinate system.
Let:
$$\vec{r}_{i} = \opair{
cos(\theta_{vert,i}) \times cos(\theta_{horiz,i}),
cos(\theta_{vert,i}) \times sin(\theta_{horiz,i}),
sin(\theta_{vert,i})
}$$
be the direction vectors of the rays projected from the lighthouse (chosen, for
simplicity, to be located at $\vec{0}$), in terms of the measured angles
$(\theta_{vert,i},\theta_{horiz,i})$ from the sensor array.
\newline\indent
These rays can be written, equivalently, as the vectors from the lighthouse to
the sensor positions in the world coordinate system,
$\vec{r}_{i} = \mathbf{A} \vec{s}_{i} + \vec{b}$
, where $\mathbf{A}$ is the rotation matrix and $\vec{b}$ is the translation
vector that together take a point in the device coordinate system to the world
coordinate system.
\newline\indent
Estimate the true device position and orientation, $(\mathbf{A},\vec{b})$,
as $(\mathbf{A}',\vec{b}')$, such that:
$$(\mathbf{A}',\vec{b}')=argmin_{(\mathbf{A}',\vec{b}')}\sum_{i} \|proj_{\vec{r}_{i}^{\perp}}(\mathbf{A}' \vec{s}_{i} + \vec{b}')\|$$


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