ExponentialProfile inflection point instability

[willtoth]

Describe the bug
I plotted the values returned by the calculateInflectionPoint function added to the sample code. I noticed the profile looks fine (velocity & position) but the inflection point velocity and position oscillates between the goal and setpoint after the actual inflection point. This seems to depend on both the kV value and the setpoint/setpoint delta. e.g. the default values in the template are fine, but tweaking slightly generates this type of issue.

To Reproduce
Drive the sim elevator up/down to the points referenced in the code

• Link to code: https://github.com/FRC3005/exponential-profile

Expected behavior

This inflection point is important to the profile generation. The question I have is, how is it stable if the inflection point is wrong, and if the inflection point is wrong, should this internal state not be a public API?

Screenshots

From sim in the project

From robot code (similar structure)

Desktop (please complete the following information):

• OS: Windows 11
• Project Information:

Additional context

[calcmogul]

From Discord:

Jordan // 3538M // CSA — Today at 12:01 PM
[This issue] is a numerical instability issue. The inflection point represents when the profile should switch from +V to -V, which if the state is numerically (scale of 1e-9) slightly behind or slightly ahead of the followed profile, the profile will choose one of the following:

1. we are fractionally behind the profile, output +V for a few microseconds then apply -V to reach the goal
2. we are fractionally ahead of the profile, output -V until we are near the goal then apply +V for a few microseconds

Thus, the inflection point oscillates according to numerical stability issues once in the braking stage of the exponential profile.

So the inflection point should either not be part of the public API, or we should document that it's unstable if it's an actually useful quantity.