This project develops and investigates methods for Bayesian state estimation (filtering and smoothing) in Gaussian process (GP) dynamic systems. In these systems, the Markovian system dynamics and the measurement mapping are described by GPs. The GP models are nonlinear functions of the inputs (no linearization) and explicitly express model uncertainty. Inference can be done analytically. For Bayesian state estimation, we derive algorithms based on linearization (GP-EKF), particle filtering (GP-PF), the unscented transform (GP-UKF), and an analytic assumed density filter (GP-ADF). The goal of this project is to integrate Gaussian process prediction and observation models into Bayes filters. These GP-BayesFilters are more accurate than standard Bayes filters using parametric models. In addition, GP models naturally supply the process and observation noise necessary for Bayesian filters.

Project Contributors

Jonathan Ko, Dieter Fox, Dan Klein, Dirk Haehnel

GP-BayesFilters

GP models for GPBF can be learned given enough training data. The resultant models have advantages over standard parametric models. They are more accurate, can can naturally provide the process and observation uncertainty used in Bayesian filters. This uncertainty considers both system noise as well as uncertainty from training data distribution. In addition, approximate parametric models can be combined with GP models which yields further improvements.

We will illustrate GPBF by tracking a small robotic blimp using two cameras mounted in an indoor environment. The ground truth position and velocity of the blimp is given by a VICON motion tracking system. The observation is the shape of the ellipse that is extracted from the camera image. The GP models are learned offline and are integrated into an unscented Kalman filter. The blue ellipses show the observations for the various sigma points. The green ellipse shows the actual observation. An animation of the GP-UKF in action can be found here or clicking the image above.

Learning Prediction and Observation Models for GPBFs

The standard formulation of GPBF is limited by its need for ground truth system states. However, we develop a technique where these ground truth states can be estimated along with GP hyperparameters in a unified framework. Our approach called GPBF-Learning extends Gaussian Process Latent Variable Models to the setting of dynamical robotics systems. In addition to learning the latent state space with no ground truth information at all, GPBF-Learning can use weak labels to learn the latent states in cases where only sparse and/or imprecise labels are available.

  

The images above show GPBF-Learning applied to a toy slotcar problem. An inertial measurement unit (IMU) is mounted on the car which provides gyro and accelerometer data. We use GPBF-Learn to find the ground truth states given the IMU measurements. The top-most image is the raw camera frame of the slotcar on the track. The top right graph in the lower image indicates the position of the slotcar in the 3d latent space found by GPBF-Learn. Bottom left and right graphs show the current observation values and control input, respectively. Videos of the raw frames and slotcar moving through the latent space can be found here and here, or by clicking on the images above.