Uncertainty analysis for accurate ground truth trajectories with robotic total stations

TL;DR

This work proposes a method to compute the six-DOF uncertainty from the fusion of three RTSs based on Monte Carlo (MC) methods, which relies on point-to-point minimization to propagate the noise of R TSs on the pose of the robotic platform.

Abstract

In the context of robotics, accurate ground truth positioning is essential for the development of Simultaneous Localization and Mapping (SLAM) and control algorithms. Robotic Total Stations (RTSs) provide accurate and precise reference positions in different types of outdoor environments, especially when compared to the limited accuracy of Global Navigation Satellite System (GNSS) in cluttered areas. Three RTSs give the possibility to obtain the six-Degrees Of Freedom (DOF) reference pose of a robotic platform. However, the uncertainty of every pose is rarely computed for trajectory evaluation. As evaluation algorithms are getting increasingly precise, it becomes crucial to take into account this uncertainty. We propose a method to compute this six-DOF uncertainty from the fusion of three RTSs based on Monte Carlo (MC) methods. This solution relies on point-to-point minimization to propagate the noise of RTSs on the pose of the robotic platform. Five main noise sources are identified to model this uncertainty: noise inherent to the instrument, tilt noise, atmospheric factors, time synchronization noise, and extrinsic calibration noise. Based on extensive experimental work, we compare the impact of each noise source on the prism uncertainty and the final estimated pose. Tested on more than 50 km of trajectories, our comparison highlighted the importance of the calibration noise and the measurement distance, which should be ideally under 75 m. Moreover, it has been noted that the uncertainty on the pose of the robot is not prominently affected by one particular noise source, compared to the others.

Figures (5)

• Figure 1: Setup used to record a reference trajectory during a snowstorm. Three RTS are each tracking a specific active prism, all mounted on a Clearpath Warthog robotic platform.
• Figure 2: Visualization of error propagation of the MC method applied with a point-to-point minimization. Each triplet of interpolated prism measurements $\widehat{\bm{q}}^i_j$ of trajectories $\widehat{\mathcal{Q}^i}$ in the world frame $\mathcal{F}^W$ are denoted by crosses enclosed within ellipses representing their corresponding covariances $\widehat{\bm{\Sigma}}^i_j$. The points $\bm{r}_1$, $\bm{r}_2$, and $\bm{r}_3$ are the reference prism positions taken in laboratories, along with their covariance. A point-to-point minimization minimizes the distance between samples of the same color, in both kinds of distributions. The result of this minimization is the estimated vehicle pose $\bm{\xi}_j$ along with its covariance $\bm{\Lambda}_j$.
• Figure 3: Influence of the noise sources from \ref{['tab:budget']}, in relation with the measured distance between an RTS and its assigned prism. The square root of the Frobenius norm is used to estimate the similarity between covariance matrices.
• Figure 4: Top view of a reference trajectory generated from a deployment in the Montmorency forest. Interpolated prism paths $\widehat{\mathcal{Q}}^i$ are represented by dots in red (prism 1), blue (prism 2), and green (prism 3). The RTS-estimated pose $\bm{\xi}$ is displayed by black points, along with a GNSS-estimated position represented in orange. Both covariances $\widehat{\bm{\Sigma}}^i_j$ and $\bm{\Lambda}_j$ are shown with shaded ellipsoids. RTS positions are indicated by an indigo cross, along with the start and stop positions indicated by red stars.
• Figure 5: Impact of noise sources on the final pose uncertainty, for both the translation and the orientation, respectively in blue and red.