AXY:Element Coordination/Multi Dimensional Tracking Algorithm (Position)/Theoretical Background

Stopping (or any other) path from the given initial conditions

Given

Initial position, velocity and acceleration $(p_{0},{\dot {p}}_{0},{\ddot {p}}_{0})$

Path

$p=c+s\cdot cos(\alpha (t))+n\cdot sin(\alpha (t))$ where $\alpha (0)=0$

with:
$|s|=|n|$

$s\cdot n=0$

$p_{0}=c+s$

assuming $|s|=|n|=R$ and $R=1$

we get:
${\dot {p}}=(-s\cdot sin(\alpha (t))+n\cdot cos(\alpha (t)))\cdot {\dot {\alpha }}(t)$

${\dot {p}}_{0}=n\cdot {\dot {\alpha }}(t)$

assuming (positive alpha velocity):
${\dot {\alpha }}(0)=|{\dot {p}}|$

and

$n={\frac {\dot {p}}{|{\dot {p}}|}}$

${\ddot {p}}=(-s\cdot sin(\alpha (t))+n\cdot cos(\alpha (t)))\cdot {\ddot {\alpha }}(t)-(s\cdot cos(\alpha (t))+n\cdot sin(\alpha (t)))\cdot {\dot {\alpha }}^{2}(t)$

${\ddot {p}}_{0}=n\cdot {\ddot {\alpha }}(0)-s\cdot {\dot {\alpha }}^{2}(0)$

multiplying above by n:
${\ddot {p}}_{0}\cdot n={\ddot {\alpha }}(t)$

and
$s={\frac {{\ddot {p}}_{0}-n\cdot {\ddot {\alpha }}(0)}{{\dot {\alpha }}^{2}(0)}}$