# UKF for a serie of observations with covariance

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 $$Z_t$$?

• Thanks for your answer, you're actually right, I should see those as control inputs. I know the covariance will grow, but this is a reduced problem that fits into a bigger architecture. So from my understanding now the covariance matrix $Z_t$ of my measurements is actually the process noise (often called $Q_t$), correct? Sep 9, 2021 at 12:48