Abban's notes :: Racing paths

$G_{i}$ represents the frame of $i$ -th gate of the track.

Local $+z$ -axis marks the side where a the drone should enter from. Bezier anchor points are placed in order, where $d$ is a constant:

$d\cdot\hat{Z}_{G_{i}}$ for entering point
$\begin{bmatrix}0 & 0 & 0\end{bmatrix}^{T}$ in the frame $G_{i}$
$-d\cdot\hat{Z}_{G_{i}}$ for exiting point

Aim is to minimise the constant $d$ to reduce time spent in straight line travelling through the gate, while ensuring no collisions happen.

Points in the path are stored in the following structure for an $n$ gate track, where $A_{j}$ is the anchor point, and $C_{j}$ is the control point for the $j$ -th point, and there are three of these for each7 $i$ -th gate:

0 & 1 & 2 & 3 & 4 & \dots & i & i & i & \dots & n-1 & n \\ A_{0} & C_{0} & C_{1} & A_{1} & C_{1} & \dots & C_{i} & A_{i} & C_{i} & \dots & C_{n} & A_{n} \end{bmatrix}$$ The total number of anchor points is given by $3n-2$

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