Continuous-time batch trajectory estimation using temporal basis functions

Abstract

Roboticists often formulate estimation problems in discrete time for the practical reason of keeping the state size tractable; however, the discrete-time approach does not scale well for use with high-rate sensors, such as inertial measurement units, rolling-shutter cameras, or sweeping laser imaging sensors. The difficulty lies in the fact that a pose variable is typically included for every time at which a measurement is acquired, rendering the dimension of the state impractically large for large numbers of measurements. This issue is exacerbated for the simultaneous localization and mapping problem, which further augments the state to include landmark variables. To address this tractability issue, we propose to move the full Maximum-a-Posteriori estimation problem into continuous time and use temporal basis functions to keep the state size manageable. We present a full probabilistic derivation of the continuous-time estimation problem, derive an estimator based on the assumption that the densities and processes involved are Gaussian and show how the coefficients of a relatively small number of basis functions can form the state to be estimated, making the solution efficient. Our derivation is presented in steps of increasingly specific assumptions, opening the door to the development of other novel continuoustime estimation algorithms through the application of different assumptions at any point. We use the simultaneous localization and mapping problem as our motivation throughout the paper, although the approach is not specific to this application. Results from two experiments are provided to validate the approach: (i) self-calibration involving a camera and a high-rate inertial measurement unit, and (ii) perspective localization with a rolling-shutter camera.

Trajectory representation and landmark projection for continuous-time structure from motion

“…No previous publication has compared these choices, instead each paper makes a hard commitment to one of the methods. In Furgale et al (2015) some of the choices are discussed, but a comparison is left for future work.…”

Direct semi-dense SLAM for rolling shutter cameras

“…Although each of these changes is foreshadowed in previous work, it remains a significant challenge to bring these components together to produce a full working system. We demonstrate that our system is capable of building accurate maps and performing accurate camera tracking without recourse to other data such as that provided by inertial sensors in [15], known scene structure in [6] or direct depth as in [9]. To the best of our knowledge, our method proposed in this paper is the first full solution to monocular, direct (feature-less) SLAM that can handle RS cameras.…”

Spline Error Weighting for Robust Visual-Inertial Fusion

“…This makes it brittle on rolling-shutter cameras, unless the camera motion can be well represented under the chosen knot spacing (such as when the knot spacing is equal to the frame distance, as in [13]). A hint in this direction is also provided in the rolling-shutter camera calibration work of Oth et al [19,10], where successive re-parametrization of the spline (by adding knots in intervals with large residuals) was required to attain an accurate calibration. In this paper we amend the SplineFusion approach [17,21] by making the trajectory approximation error explicit.…”