I emphasize that my question is about sampling, not resampling.
I'm reading the Probabilistic Robotics book by Thrun et al, Chapter 4 on Non-Parametric Filters. The section on Particle filters has an algorithm which states that for each particle index $m$ (see line 4):
sample $x_t^{[m]} \sim p(x_t|u_t,x_{t-1}^{[m]})$
The text's explanation of this step is quoted as:
Line 4. generates a hypothetical state $x_t^{[m]}$ for time t based on the particle $x_{t-1}$ and the control $u_t$. The resulting sample is index by $m$, indicating that it is generated from the $m$-th particle in $\chi_{t-1}$. This step involves sampling from the state transition distribution $p(x_t|u_t,x_{t-1})$. To implement this step, one needs to be able to sample from this distribution. The set of particles obtained after $M$ iterations is the filter's representation of $\bar{bel}(x_t)$.
If I understand correctly, this step says that the m-th sampled particle $x_t^{[m]}$ is obtained by advancing the previous m-th particle with control command $u_t$. I assume that the motion is not deterministic, so the result of this motion is a conditional probability, based on the particle's previous position $u_t$.
However, I'm confused over how exactly to construct this conditional probability $p(x_t|u_t,x_{t-1}^{[m]})$. Is this information usually given? Or is it constructed from the distribution of the other particles?