  We propose a new metric for robot state estimation based on the recently
introduced $\text{SE}_2(3)$ Lie group definition. Our metric is related to
prior metrics for SLAM but explicitly takes into account the linear velocity of
the state estimate, improving over current pose-based trajectory analysis. This
has the benefit of providing a single, quantitative metric to evaluate state
estimation algorithms against, while being compatible with existing tools and
libraries. Since ground truth data generally consists of pose data from motion
capture systems, we also propose an approach to compute the ground truth linear
velocity based on polynomial interpolation. Using Chebyshev interpolation and a
pseudospectral parameterization, we can accurately estimate the ground truth
linear velocity of the trajectory in an optimal fashion with best approximation
error. We demonstrate how this approach performs on multiple robotic platforms
where accurate state estimation is vital, and compare it to alternative
approaches such as finite differences. The pseudospectral parameterization also
provides a means of trajectory data compression as an additional benefit.
Experimental results show our method provides a valid and accurate means of
comparing state estimation systems, which is also easy to interpret and report.
