Optimization problem with many parameters and different functions

Hello dear MATLAB community.

I have an optimization problem. I want to calculate a time optimal trajectory with the fmincon function. I have N supporting points for the trajectory which are described through a circle (see the picture), so I have the middle points of the circle (xmk and ymk) and two parameters: the radius rk < 2 (just an example) and phi < 2*pi So I have for each points the function:

xk = xmk + rk *cos(phi)
yk = ymk + rk *sin(phi)

My criteria to optimize the trajectoy takes the curvature and the length of the trajectory which is described through the N supporting points as mentioned before. So actually my optimization problem calculates the minimum of

kappa*s

My question is, how can I minimize the problem kappa*s, which is directly dependent on xk and yk, and they are again directly dependent on rk and phi. So I need N different values for rk and phi, that I get the N optimal supporting points for the trajectory.

I hope someone can help me with this problem.

Thank you in advance

David

[Matt J]

But what are the unknown parameters that you are trying to choose? You say you "have" xmk, ymk, rk, phik as known and given input data, so what don't you have?