I have some doubts about how to implement a UKF-like algorithm when I only have motion observations and no control inputs.
Assume I have a robot with state $s_t = (x_t, y_t, \theta_t)$ and the corresponding 3x3 covariance matrix $P_t$. I obtain a serie of $k$ measurements $z_i = (\Delta x_i, \Delta y_i, \Delta \theta_i)$ for $i = t \: ... \: t+k-1$, each with covariance $Z_i$. I need to compute the covariance of the resulting state $s_{t+k}$.
Since the transformation from $s_t$ to $s_{t+k}$ is non-linear, I need to approximate the posterior distribution of $s_{t+k}$ with a Gaussian and I am trying to use a UKF-like approach.
My problem is that I do not exactly understand how I should use both the covariance of the state $s_t$ and that of the measurements $z_t \: ... \: z_{t+k-1}$. I already have the nonlinear function $g$ such that $s_{t+1} = g(s_t, z_t)$ and I compute all the intermediate states $s_t \: ... \: s_{t+k}$. But I do not have a control input (generally called $u_t$) so is the prediction step just to compute sigma points from $P_t$? And in the update step how do I take into consideration the measurement covariance $Z_t$?