So, in this question I'd be grateful for hints and further information if I am correct or no.
To calculate the position upon range-measurments to fixed anchors (like GPS) you need to solve the trilateration problem, for example: non-linear least squares, geometrical algorithms or the particle filter, which also is able to solve the trilateration-problem as such.
Due to noise/errors the result might be a jagged line -> you can use the Kalman-Filter to smooth it. So far: Particle - calculation, Kalman - smoothing. Now:
Is it possible to use a Kalman-Filter NOT to smoothen an already existing result, BUT to solve the trilateration as such?
Regarding the particle filter: How to use the particle filter NOT to solve trilateration, BUT to smoothen an already existing result (e.g. calculated with NLLS)?
Best and thank you for any hints, papers, videos, solutions etc.!
The Kalman filter is an optimal solver for linear Gaussian problems. It is often used to solve the trilateration problem (Question 1). To use it in this problem the Jacobian (partial derivative of the range measurement with respect to the position) is linearized at the current position estimate. That process, linearization of the Jacobian, defines the Kalman filter as an Extended Kalman Filter, or EKF in the literature. That works well for GPS because the range to the transmitter is so great that the error in the Jacobian estimate due to position error is small enough to be negligible if the Kalman filter is crudely initialized, for example within 100 km. It breaks down when the 'fixed anchors' are closer to the user. The closer the anchor, the more quickly the line-of-sight vector to the anchor is changing with the position estimate. In these cases Unscented Kalman Filters (UKF) or Particle Filters (PF) are sometimes used instead of an EKF.
The best introduction to the KF and EKF in my view is Applied Optimal Estimation by Gelb. That book has been in print since 1974, and there is a reason why. A discussion of the breakdown of the EKF when the anchor is close can be found in the paper "The Scaled Unscented Transformation" by Julier, which can be found here.
For question 2, the answer is yes, certainly a PF could be used to smooth a solution that is created, for example, by replacing the range measurements with an epoch-by-epoch result from a least-squares solver for the position. I would not recommend the approach. The power of the PF, and the reason we pay the price of computing everything for each particle, is that it handles the non-linearities. To 'pre-linearize' the problem before handing it to the PF defeats its purpose.