Integrals in the Equations of Motion

[Peter230655]

I tried an eom of the form $W = \int_0^{tf} F((x(t), \dfrac{dx}{dt}, params) dt$, but I got this error:

Did I do something wrong, or does opty not accept such eoms?

Thanks for any hint!

[moorepants]

opty would not know how to handle an integral in a equation of motion. opty works with daes and odes. I'm not aware of a form that would have an integral in the equation of motion.

[Peter230655]

Thanks! I just wanted to make sure I would not do anything wrong.
Reason:
I looked at my old sailboat simulation, and I want it to move around the 'bouy'. That is the 'bouy' should be an internal point of the closed path the boat takes.
My idea was to use the winding number which should be 1 or -1, if the buoy is inside the curve. This winding number is an integral.

I will try some other ideas now.

[moorepants]

This sounds like something you need to implement as a constraint, not as an element in the ODE. Does this thing simulate with the integral?

I can think of a PID controller that is similar. You have the I term which is the integral of the state variable up until the current time. This can be part of the ODE and a quantity that is calculated but you'd make a new equation with the integrand to have it integrate as time passes. This will have to be done the same way for opty, there should be no explicit integrals in the EoMs but you can add more equations that will be integrated.

[Peter230655]

Let me try this, if I understood correctly: I will declare a state variable W, then I add W.diff(t) = integrand of winding integral to the eoms.
Then I add an instance constraint W(tf) = 1.
I will try and report.

[moorepants]

If you have these equations:

$$\begin{align} \frac{dx}{dt} = x \\\ \frac{dv}{dt} = \frac{F}{m} \\\ \frac{dF}{dt} = u(t) \end{align}$$

then $F =\int_{t_0}^{t_c}u(t)dt$ at any time $t_c$.

[Peter230655]

I assume, you meant $\dfrac{dx}{dt} = v$. not x. (else the only solution would be $x(t) = e^t$)
I am playing around with it right now. It seems, it might work, but at the moment convergence is very slow.
I keep on playing around and will report.

[moorepants]

yes, typo

[Peter230655]

yes, typo

Just amazing what opty can do!

[Peter230655]

It works, in that opty runs without issues.
However, either my understanding of 'winding number' is flawed - very likely true, or it is not a good numerical tool for what I want to do.
Also this could be true: As I understand it, winding number = +1 or -1 if the point is inside the curve, and 0 it it is not.