We introduce Tempered Geodesic Markov Chain Monte Carlo (TG-MCMC) algorithm for initializing pose graph optimization problems, arising in various scenarios such as SFM (structure from motion) or SLAM (simultaneous localization and mapping). TG-MCMC is first of its kind as it unites global non-convex optimization on the spherical manifold of quaternions with posterior sampling, in order to provide both reliable initial poses and uncertainty estimates that are informative about the quality of solutions. We devise theoretical convergence guarantees and extensively evaluate our method on synthetic and real benchmarks. Besides its elegance in formulation and theory, we show that our method is robust to missing data, noise and the estimated uncertainties capture intuitive properties of the data.

-- Extrinsic calibration is essential for multi-sensor fusion, existing methods rely on structured targets or fully-excited data, limiting real-world applicability. Online calibration further suffers from weak excitation, leading to unreliable estimates. T o address these limitations, we propose a reinforcement learning (RL)-based extrinsic calibration framework that formulates extrinsic calibration as a decision-making problem, directly optimizes SE (3) extrinsics to enhance odometry accuracy. Our approach leverages a probabilistic Bingham distribution to model 3D rotations, ensuring stable optimization while inherently retaining quaternion symmetry. A trajectory alignment reward mechanism enables robust calibration without structured targets by quantitatively evaluating estimated tightly-coupled trajectory against a reference trajectory. Additionally, an automated data selection module filters uninformative samples, significantly improving efficiency and scalability for large-scale datasets. Extensive experiments on UA Vs, UGVs, and handheld platforms demonstrate that our method outperforms traditional optimization-based approaches, achieving high-precision calibration even under weak excitation conditions. The code is available at https://github.com/

The need for statistical models of orientations arises in many applications in engineering and computer science. Orientational data appear as sets of angles, unit vectors, rotation matrices or quaternions. In the field of directional statistics, a lot of advances have been made in modelling such types of data. However, only a few of these tools are used in engineering and computer science applications. Hence, this paper aims to serve as a cheat sheet for those probability distributions of orientations. Models for 1-DOF, 2-DOF and 3-DOF orientations are discussed. For each of them, expressions for the density function, fitting to data, and sampling are presented. The paper is written with a compromise between engineering and statistics in terms of notation and terminology. A Python library with functions for some of these models is provided. Using this library, two examples of applications to real data are presented.

Planar-symmetric hands, such as parallel grippers, are widely adopted in both research and industrial fields. Their symmetry, however, introduces ambiguity and discontinuity in the SO(3) representation, which hinders both the training and inference of neural-network-based grasp detectors. We propose a novel SO(3) representation that can parametrize a pair of planar-symmetric poses with a single parameter set by leveraging the 2D Bingham distribution. We also detail a grasp detector based on our representation, which provides a more consistent rotation output. An intensive evaluation with multiple grippers and objects in both the simulation and the real world quantitatively shows our approach's contribution.

This paper presents a stochastic gradient Monte Carlo approach defined on a Cartesian product of SE(3), a domain commonly used to characterize problems in structure-from-motion (SFM) among other areas. The algorithm is parameterized by an inverse temperature such that when the value goes to inifinity, the algorithm is implicitly operating on a delta function with it's peak at the maximum of the base distribution. The proposed algorithm is formulated as a SDE and a splitting scheme is proposed to integrate it. A theoretical analysis on the SDE and its discretization is explored, showing that 1) the resulting Markov process has the appropriate invariant distribution and 2) the sampler will draw samples close to the maximum of the posterior (in terms of expectation of the unnormalized log posterior). Along with the algorithm, a model is defined using the Bingham distribution to characterize typical SFM posteriors which is then used to perform experiments with the algorithm.

In this paper, we propose a method for online estimation of the robot's posture. Our method uses von Mises and Bingham distributions as probability distributions of joint angles and 3D orientation, which are used in directional statistics. We constructed a particle filter using these distributions and configured a system to estimate the robot's posture from various sensor information (e.g., joint encoders, IMU sensors, and cameras). Furthermore, unlike tangent space approximations, these distributions can handle global features and represent sensor characteristics as observation noises. As an application, we show that the yaw drift of a 6-axis IMU sensor can be represented probabilistically to prevent adverse effects on attitude estimation. For the estimation, we used an approximate model that assumes the actual robot posture can be reproduced by correcting the joint angles of a rigid body model. In the experiment part, we tested the estimator's effectiveness by examining that the joint angles generated with the approximate model can be estimated using the link pose of the same model. We then applied the estimator to the actual robot and confirmed that the gripper position could be estimated, thereby verifying the validity of the approximate model in our situation.

In recent years, a deep learning framework has been widely used for object pose estimation. While quaternion is a common choice for rotation representation of 6D pose, it cannot represent an uncertainty of the observation. In order to handle the uncertainty, Bingham distribution is one promising solution because this has suitable features, such as a smooth representation over SO(3), in addition to the ambiguity representation. However, it requires the complex computation of the normalizing constants. This is the bottleneck of loss computation in training neural networks based on Bingham representation. As such, we propose a fast-computable and easy-to-implement loss function for Bingham distribution. We also show not only to examine the parametrization of Bingham distribution but also an application based on our loss function.

We propose a new policy parameterization for representing 3D rotations during reinforcement learning. Today in the continuous control reinforcement learning literature, many stochastic policy parameterizations are Gaussian. We argue that universally applying a Gaussian policy parameterization is not always desirable for all environments. One such case in particular where this is true are tasks that involve predicting a 3D rotation output, either in isolation, or coupled with translation as part of a full 6D pose output. Our proposed Bingham Policy Parameterization (BPP) models the Bingham distribution and allows for better rotation (quaternion) prediction over a Gaussian policy parameterization in a range of reinforcement learning tasks. We evaluate BPP on the rotation Wahba problem task, as well as a set of vision-based next-best pose robot manipulation tasks from RLBench. We hope that this paper encourages more research into developing other policy parameterization that are more suited for particular environments, rather than always assuming Gaussian.

The algorithm is based on rejection sampling, where the proposal distribution is a polynomial approximation of the pdf, and can be sampled from by explicitly evaluating integrals of polynomials over the sphere. Our algorithm gives exact samples, assuming exact computation of an inverse function of a polynomial. This is in contrast with Markov Chain Monte Carlo algorithms, which are not known to enjoy rapid mixing on this problem, and only give approximate samples. As a direct application, we use this to sample from the posterior distribution of a rank-1 matrix inference problem in polynomial time.

We introduce Tempered Geodesic Markov Chain Monte Carlo (TG-MCMC) algorithm for initializing pose graph optimization problems, arising in various scenarios such as SFM (structure from motion) or SLAM (simultaneous localization and mapping). TG-MCMC is first of its kind as it unites global non-convex optimization on the spherical manifold of quaternions with posterior sampling, in order to provide both reliable initial poses and uncertainty estimates that are informative about the quality of solutions. We devise theoretical convergence guarantees and extensively evaluate our method on synthetic and real benchmarks. Besides its elegance in formulation and theory, we show that our method is robust to missing data, noise and the estimated uncertainties capture intuitive properties of the data. Papers published at the Neural Information Processing Systems Conference.