# How to model projectile motion using dynamical system matrices

I'm trying to figure out how to model projectile motion which can be defined using a simple [quadratic] equation which defines projectile parabola:

$$y = y_0 + v_{0y}t - \dfrac{1}{2} gt^2$$

I came across some slides from Brown University which model the motion as follows:

\begin{align} \text{Matrix form: }& \begin{pmatrix} 0 & 100 & -10\end{pmatrix} \\ \\ \text{state: }& \begin{matrix}y & y' & y'' \\ p & v & a\end{matrix} \\ \\ &\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 2 & 1 \\ \end{pmatrix} \end{align}

But I'm not quite sure how they came up with that transition matrix. Can anyone shed some light how to pick the state variables and how to model the motion?

• Apr 2 '19 at 16:54
• Thanks, for the link. Pretty cool, but I understand the dynamics I just don't quite seem to be able to figure out how to rewrite it into dynamical systems matrix representation Apr 2 '19 at 23:43

He started from $$y = y_0 + v_0 t + \frac 1 2 a t^2$$ but then for some reason best known to himself decided to take $$\frac 1 2 a$$ as $$y''$$ rather than a, (and ignored the fact that his $$y''(t)$$ is no longer equal to $$d y'(t) / dt$$, but hey, Lewis Carroll was also a mathematician, and this guy, like Humpy Dumpty, can make words mean whatever he feels like they ought to mean).
So he gets the approximate equations $$\begin{matrix} y(t) \\ y'(t) \\ y''(t) \end{matrix} \approx \begin{matrix} y(t-1) + y'(t-1) \\ y'(t-1) + 2y''(y-1) \\ y''(t-1) \end{matrix}$$ or $$\begin{bmatrix} y(t) & y'(t) & y''(t) \end{bmatrix} \approx \begin{bmatrix} y(t-1) & y'(t-1) &y''(t-1) \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 2 & 1\\ \end{bmatrix}$$ He can't seem to make up his mind whether $$y''$$ is really a variable, or just the arbitrary constant $$-10$$, (or was that $$+10$$, or $$\pm 20$$ - I've lost the will to care...
It would be more obvious what's going on if you replace the 2's with 1's in his equations, and let $$y''(t)$$ actually BE the second derivative of $$y(t)$$, not something close.