Non-parametric Estimation of Stochastic Differential Equations with Sparse Gaussian Processes

The application of Stochastic Differential Equations (SDEs) to the analysis of temporal data has attracted increasing attention, due to their ability to describe complex dynamics with physically interpretable equations. In this paper, we introduce a non-parametric method for estimating the drift and diffusion terms of SDEs from a densely observed discrete time series. The use of Gaussian processes as priors permits working directly in a function-space view and thus the inference takes place directly in this space. To cope with the computational complexity that requires the use of Gaussian processes, a sparse Gaussian process approximation is provided. This approximation permits the efficient computation of predictions for the drift and diffusion terms by using a distribution over a small subset of pseudo-samples. The proposed method has been validated using both simulated data and real data from economy and paleoclimatology. The application of the method to real data demonstrates its ability to capture the behaviour of complex systems

Robust 3D IMU-LIDAR Calibration and Multi Sensor Probabilistic State Estimation

Autonomous robots are highly complex systems. In order to operate in dynamic environments, adaptability in their decision-making algorithms is a must. Thus, the internal and external information that robots obtain from sensors is critical to re-evaluate their decisions in real time. Accuracy is key in this endeavor, both from the hardware side and the modeling point of view. In order to guarantee the highest performance, sensors need to be correctly calibrated. To this end, some parameters are tuned so that the particular realization of a sensor best matches a generalized mathematical model. This step grows in complexity with the integration of multiple sensors, which is generally a requirement in order to cope with the dynamic nature of real world applications. This project aims to deal with the calibration of an inertial measurement unit, or IMU, and a Light Detection and Ranging device, or LiDAR. An offline batch optimization procedure is proposed to optimally estimate the intrinsic and extrinsic parameters of the model. Then, an online state estimation module that makes use of the aforementioned parameters and the fusion of LiDAR-inertial data for local navigation is proposed. Additionally, it incorporates real time corrections to account for the time-varying nature of the model, essential to deal with exposure to continued operation and wear and tear. Keywords: sensor fusion, multi-sensor calibration, factor graphs, batch optimization, Gaussian Processes, state estimation, LiDAR-inertial odometry, Error State Kalman Filter, Normal Distributions Transform

Rethinking Trajectory Evaluation for SLAM: a Probabilistic, Continuous-Time Approach

Despite the existence of different error metrics for trajectory evaluation in SLAM, their theoretical justifications and connections are rarely studied, and few methods handle temporal association properly. In this work, we propose to formulate the trajectory evaluation problem in a probabilistic, continuous-time framework. By modeling the groundtruth as random variables, the concepts of absolute and relative error are generalized to be likelihood. Moreover, the groundtruth is represented as a piecewise Gaussian Process in continuous-time. Within this framework, we are able to establish theoretical connections between relative and absolute error metrics and handle temporal association in a principled manner