A Smooth trajectory generation

Reference

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This post is some idea about the paper “Minimum Snap Trajectory Generation and Control for Quadrotors” by Daniel Mellinger and Vijay Kumar. Thanks for thier work, I got a deeper understanding of Trajectory Generation and Optimisation problem with Quadratic programming.

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\[ \begin{aligned} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z + xy \end{aligned} \]

$c^\intercal= \begin{bmatrix} c_1 & c_2 & \dots&c_{12} \\ \end{bmatrix}$ $A=\begin{bmatrix} -1 & 0 & 0 & \dots & 0 \\ 0 & -1 & 0 & \dots & 0 \\ 0 & 0 & -1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & -1\\ 1 & 0 & 0 & \dots & 0 \\ 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \end{bmatrix}\quad b=\begin{bmatrix} 0\\ 0\\ 0\\ \vdots\\ 0\\ P_{1,max}\\ P_{2,max}\\ P_{3,max}\\ \vdots\\ P_{12,max} \end{bmatrix}$ $G=\begin{bmatrix} 1 &1 &\dots&1 \\ \end{bmatrix} \quad h=P_{demand}$