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We address the problem of uncertainty propagation and Bayesian fusion on unimodular Lie groups. Starting from a stochastic differential equation (SDE) defined on Lie groups via Mckean-Gangolli injection, we first convert it to a parametric SDE in exponential coordinates. The coefficient transform method for the conversion is stated for both Ito's and Stratonovich's interpretation of the SDE. Then we derive a mean and covariance fitting formula for probability distributions on Lie groups defined by a concentrated distribution on the exponential coordinate. It is used to derive the mean and covariance propagation equations for the SDE defined by injection, which coincides with the result derived from a Fokker-Planck equation in previous work. We also propose a simple modification to the update step of Kalman filters using the fitting formula, which improves the fusion accuracy with moderate computation time.

This paper addresses uncertainty propagation on unimodular matrix Lie groups that have a surjective exponential map. We derive the exact formula for the propagation of mean and covariance in a continuous-time setting from the governing Fokker-Planck equation. Two approximate propagation methods are discussed based on the exact formula. One uses numerical quadrature and another utilizes the expansion of moments. A closed-form second-order propagation formula is derived. We apply the general theory to the joint attitude and angular momentum uncertainty propagation problem and numerical experiments demonstrate two approximation methods. These results show that our new methods have high accuracy while being computationally efficient., Comment: 37 pages

Dynamic state estimation, as opposed to kinematic state estimation, seeks to estimate not only the orientation of a rigid body but also its angular velocity, through Euler's equations of rotational motion. This paper demonstrates that the dynamic state estimation problem can be reformulated as estimating a probability distribution on a Lie group defined on phase space (the product space of rotation and angular momentum). The propagation equations are derived non-parametrically for the mean and covariance of the distribution. It is also shown that the equations can be approximately solved by ignoring the third and higher moments of the probability distribution. Numerical experiments show that the distribution constructed from the propagated mean and covariance fits the sample data better than an extended Kalman filter., Comment: 8 pages, 4 figures

This article reports the modeling of inertial rotational Brownian motion as an Ornstein-Uhlenbeck process evolving on the cotangent bundle of the rotation group, SO(3). The benefit of this approach and the use of a different parameterization of rotations allows the handling of particles with arbitrary shapes, without requiring any simplifying assumptions on the shape or the structure of the viscosity tensors. The resultant Fokker-Planck equation for the joint orientation and angular momentum probability distribution can be solved approximately using an `ansatz' Gaussian distribution in exponential coordinates., Comment: 10 pages, 5 figures

Rearrangement tasks have been identified as a crucial challenge for intelligent robotic manipulation, but few methods allow for precise construction of unseen structures. We propose a visual foresight model for pick-and-place rearrangement manipulation which is able to learn efficiently. In addition, we develop a multi-modal action proposal module which builds on the Goal-Conditioned Transporter Network, a state-of-the-art imitation learning method. Our image-based task planning method, Transporters with Visual Foresight, is able to learn from only a handful of data and generalize to multiple unseen tasks in a zero-shot manner. TVF is able to improve the performance of a state-of-the-art imitation learning method on unseen tasks in simulation and real robot experiments. In particular, the average success rate on unseen tasks improves from 55.4% to 78.5% in simulation experiments and from 30% to 63.3% in real robot experiments when given only tens of expert demonstrations. Video and code are available on our project website: https://chirikjianlab.github.io/tvf/, Comment: IEEE IROS 2022