I think I have an interesting problem that is clearly outside my mathematical/physical comfort zone. Imagine a real ascending pendulum, i.e. a helium balloon with a string and object at the end. I am able to track this object (3D positions with timestamps), and I need to know where the object at the end of the string is in the future. I'm talking about the next few seconds, and it's a small setup, so air density, for example, should be negligible.
I found that there are equations for a "spherical pendulum" and for a "free falling body" that should probably be used here. I probably also need some sort of damped horizontal term and a global position offset.
The way I imagine the problem is that I need to combine these equations somehow. After I measure a series of positions, I run a parameter estimation algorithm to find all the initial parameters at $t0$. Then I need to solve this equation numerically to know where the object will be in the future. First of all, I would be interested to know if people think that this is broadly the way to go?
Now to the mathematical part: I've been trying to simplify the problem - a simple pendulum in 3D space - to see if I can solve it, and I'm really not sure if what I'm doing makes sense - one major problem is that all the literature I've found on pendulums talks about angles and their derivatives, but I'm measuring XYZ positions with time. I thought maybe I could take the pendulum equation and replace the $\theta$ angle and derivatives with position equivalents:
$$\ddot\theta(t) = -µ * \ddot\theta(t) - \frac{g}{l} \sin (\theta(t))$$ $\theta$ to Position: $$x = l * \sin(\theta(t))\\ y = -l * \cos(\theta(t))$$ I was thinking about creating 2 equations for each axis, e.g. X: $$\theta(t) = \arcsin(\frac{x}{l})\\ \dot\theta(t) = \frac{1}{l*\sqrt{1-(\frac{x}{l})^2}}\\ \ddot\theta(t) = \frac{x}{l^3*(1-\frac{x^2}{l^2})^{\frac{3}{2}}}\\ $$
Now substitute that into the first equation? It already seems overly complicated and I haven't even added the x offset and some sort of rotation of the whole system to make this simple pendulum with real 3D measured coordinates work. I haven't looked closely into parameter estimation, from what I've read the python library Gekko would be a good option. Also not sure if these equations would even work with at.
Any help would be greatly appreciated.