For context, I have a target moving in space which I am able to locate using vision sensors. I end up obtaining $M$ noisy samples of the trajectory $[x(t), y(t)]$ for some time instants $\{t_k\}$. Moreover, I have a mobile robot which needs to follow such trajectory (after all samples have been collected). To design a controller for the robot, I require to construct a reference $x_r(t), y_r(t)$ which coincides (as best as possible) with $x(t), y(t)$. Moreover, due to controlability issues, I need the reference $x_r(t), y_r(t)$ to be $n$-times differentiable for, say, $n=3$.
For simplicity we can focus only on the $x$ coordinate. I know that $|x^{(i)}(t)|\leq L_i, \forall t\geq 0, i=0,\dots,n$ and some (known) constants $L_0,\dots,L_n$. In addition, we can assume that the $M$ noisy samples $\tilde{x}_0,\dots,\tilde{x}_M$ at time instants $0<t_1,\dots,t_M$ satisfy $$ \tilde{x}_\mu = x(t_\mu) + \epsilon_\mu $$ with $\epsilon_\mu$ being Gaussian noise with 0 mean and covariance $E_\mu, \mu=1,\dots,M$ and known constants $E_\mu$.
In this context, how can we compute an $n$-times differentiable estimate $\hat{x}(t)$ of $x(t), \forall t\geq 0$ taking into account the samples $\tilde{x}_\mu$ and the knowledge of the constraints given by $L_i, E_\mu$?
Ideally, I would want the estimator to be unbiased in some sense. By this I mean that for given $t\geq 0$ the average of $\hat{x}(t)$ according to the distribution of the noises $\epsilon_1,\dots,\epsilon_M$ be $x(t)$ (How can I formalize this idea?).
Up to now, I am just interpolating the samples, testing polynomials, splines, etc. But I don't think I'm using all the information I have, particularly the knowledge of the constants $L_0,\dots,L_n,E_1,\dots,E_M$ nor taking care of this unbiased-ness property.
I'm sure there must be a more inteligent/elegant way to tackle this problem and I just don't know the name of the method/technique or where to look for.
I'm looking for suggestions/ideas/directions which may help me continue on my own. Thanks!