I would like to model a plane trajectory, here are my assumptions and conclusions so far:
Assumptions:
- we denote the plane's speed with $v(t)$, its trajectory $x(t)$, the total trajectory time will be $\tau$
- we have start and end times of flights as well as distance, so we have $\tau$ and $\int_0^\tau v(t)dt$
- the plane has an acceleration phase, let's call it $p_1$
- the plane has a plateau phase, $p_2$, where it reaches its cruise speed, $v_c$
- the plane has a deceleration phase, $p_3$.
The plane speed over time is $v(t) = \begin{cases} v_1(t) & t \in p_1 \\ v_c & t \in p_2\\ v_2(t) & t \in p_3 \\ 0 & \text{else} \end{cases}$
Let $p_1 = [0, t_1], p_2 = [t_1, t_2]$ and $p_3 = [t_2, \tau]$, we assume $t_1 = \tau - t_2$ (essentially, same amount of time accelerate as to decelerate) and we rewrite $p_1 = [0, t_1], p_2 = [t_1, \tau- t_1]$ and $p_3 = [\tau-t_1, \tau]$.
We 'll now further assume $v_2(t) = v_1(t - \tau)$ (the deceleration shape is the inverted acceleration shape).
So at this point we are left with determining $v_1(t)$. The model has $m$ parameters for $v_1$ and $t_1$, $v_c$ (note that $t_1$ and $v_c$ can be included as a parameter in $v_1$).
My questions are:
- Are assumptions realistic, make sense, anything missing?
- How would you go about modelling $v_1(t)$
Possible answer 1:
Make the assumption that $v_1(t)$ is linear: $v_1(t) = \frac{v_c}{t_1}t$.
The overall model has two parameters
Possible answer 2:
Go with a sigmoid-like function for $v_1$ but with a bounded input, I thought $\sin$ with the right phase would be good for this since it also has derivatives of zero at $\frac{dv_1(0)}{dt} = \frac{dv_1(t_1)}{dt} = 0$, this leads to: $v_1(t) = v_c\frac{1-\cos{(\omega t)}}{2}$.
$\omega = 2\pi/t_1$
Again two parameters.
Since I don't want to have $v_c$ as a parameter/constant, I thought of introducing $\alpha = \frac{\omega}{v_c}$ thus
$v_1(t) = \frac{\omega}{2\alpha}(1-\cos{(\omega t)})$
