slack variables vs. differentiable approximations

[Peter230655]

As per my quick tests the latest version of #476 converges much better than the previous one.
Yet, I think the differentiable approximation still converges more reliably. (E.g. I tried to let opty decide the time t_m, and this seemed to converge better)

Hence my question:
In a practical application, e.g. the skateboard jump or the examples in Posa et al 2013, would the differentiable approximation lead to inacceptable results?

Of course I think, it is very good that opty can handle slack variables!

Update
I simply compared the two methods, the results seem close.

• optimal h: (h_slack - h_differentiable) / h_differentiable = - 7 * 10^(-4), that is h_slack a faster
• only visible difference is in F(t) at the very beginning.
• With random initial guesses, the differentiable method is better.
• increasing N from 40 to 80 seems to make the differentiable method a bit faster
• increasing N to 120 the slack variable method does not find a solution even with 20,000 iterations

Overall in this simulation the results look similar to me, with the differentiable method more "forgiving",

[moorepants]

The question is always whether a model simulates reality sufficiently for your purposes and then there are other needs like computational efficiency. Your question as posed isn't really answerable because modeling decisions reflect the problem you are solving and the physical thing you are simulating. Using smoothed discontinuities in the dynamics may be a justifiable modeling choice in certain cases and in some cases it isn't.

[Peter230655]

Makes sense, so in a way the end justifies the means.
Maybe I will find more examples.