Reference Line¶

Fem(Finite Element Method)pos Smooth¶

The optimization variables are points(x, y) position. The object function can be wrriten as:

C o s t_{a l l} = C o s t_{s m o o t h} + C o s t_{l e n g t h} + C o s t_{d e v i a t i o n}

As we can see,

\vec{v_{2}} = \vec{v_{0}} + \vec{v_{1}}

and the shorter $| \vec{v_{2}} |$ is, the smoother $p_{0} \to p_{1} \to p_{2}$ become. In vector format:

$\begin{aligned} \vec{v_{0}} & = P_{0} - P_{1} \vec{v_{1}} & = P_{2} - P_{1} \vec{v_{2}} & = \vec{v_{0}} + \vec{v_{1}} \end{aligned}$ So,

C o s t_{s m o o t h} = \sum_{k = 1}^{n - 2} | | 2 P_{k} - P_{k - 1} + P_{k + 1} | |_{2}^{2}

This cost can also be descriped as:

As $θ$ decreases, $c o s θ$ decreases too. So,

C o s t_{s m o o t h} = \sum_{k = 1}^{n - 2} \frac{\vec{v_{0}} \cdot \vec{v_{1}}}{| \vec{v_{0}} | \cdot | \vec{v_{1}} |}

As we can see, if we use $θ$ for smooth, the cost will not be a qp problem, we should solve it with a nonlinear solver.

C o s t_{l e n g t h} = \sum_{k = 0}^{n - 2} | | P_{k + 1} - P_{k} | |_{2}^{2}

C o s t_{d e v i a t i o n} = \sum_{k = 0}^{n - 1} | | P_{k} - P_{k_{r e f}} | |_{2}^{2}

Assumption:

$| P_{0} P_{1} | \approx | P_{1} P_{2} |$ , each piece of segment has almost the same length;
$θ$ is small, so $s i n θ \approx c o s θ$ ;
$d s \approx a r c P_{1} \to P_{2}$ , distance between two points is approximately equal to the arc of them.

As we can see in above figure, $| P_{0} P_{1} | = | P_{1} P_{2} |$ , $| P_{0} A | = | P_{0} P_{1} |$ :

\frac{L}{d s} = \frac{d s}{R}

L = | \vec{P_{0} P_{1}} + \vec{P_{1} P_{2}} | = \frac{| \vec{P_{1} P_{2}} |^{2}}{R}