Optimal estimation for the satellite attitude using star tracker measurements
Automatica (Journal of IFAC)
An estimation problem in compact Lie groups
Systems & Control Letters
Fast transforms and sampling for compact groups
Fast transforms and sampling for compact groups
The Motor Extended Kalman Filter: A Geometric Approach for Rigid Motion Estimation
Journal of Mathematical Imaging and Vision
Robot Motion Planning and Control
Robot Motion Planning and Control
A Mathematical Introduction to Robotic Manipulation
A Mathematical Introduction to Robotic Manipulation
Nonholonomic Motion Planning
Computing MAP trajectories by representing, propagating and combining PDFs over groups
ICCV '03 Proceedings of the Ninth IEEE International Conference on Computer Vision - Volume 2
Nonholonomic Modeling of Needle Steering
International Journal of Robotics Research
Rotation Recovery from Spherical Images without Correspondences
IEEE Transactions on Pattern Analysis and Machine Intelligence
Error propagation on the Euclidean group with applications to manipulator kinematics
IEEE Transactions on Robotics
The Path-of-probability Algorithm for Steering and Feedback Control of Flexible Needles
International Journal of Robotics Research
Three-dimensional Motion Planning Algorithms for Steerable Needles Using Inverse Kinematics
International Journal of Robotics Research
Registration of 2D Points Using Geometric Algebra and Tensor Voting
Journal of Mathematical Imaging and Vision
International Journal of Robotics Research
Articulated motion segmentation of point clouds by group-valued regularization
EG 3DOR'12 Proceedings of the 5th Eurographics conference on 3D Object Retrieval
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A nonholonomic system subjected to external noise from the environment, or internal noise in its own actuators, will evolve in a stochastic manner described by an ensemble of trajectories. This ensemble of trajectories is equivalent to the solution of a Fokker–Planck equation that typically evolves on a Lie group. If the most likely state of such a system is to be estimated, and plans for subsequent motions from the current state are to be made so as to move the system to a desired state with high probability, then modeling how the probability density of the system evolves is critical. Methods for solving Fokker-Planck equations that evolve on Lie groups then become important. Such equations can be solved using the operational properties of group Fourier transforms in which irreducible unitary representation (IUR) matrices play a critical role. Therefore, we develop a simple approach for the numerical approximation of all the IUR matrices for two of the groups of most interest in robotics: the rotation group in three-dimensional space, SO(3), and the Euclidean motion group of the plane, SE(2). This approach uses the exponential mapping from the Lie algebras of these groups, and takes advantage of the sparse nature of the Lie algebra representation matrices. Other techniques for density estimation on groups are also explored. The computed densities are applied in the context of probabilistic path planning for kinematic cart in the plane and flexible needle steering in three-dimensional space. In these examples the injection of artificial noise into the computational models (rather than noise in the actual physical systems) serves as a tool to search the configuration spaces and plan paths. Finally, we illustrate how density estimation problems arise in the characterization of physical noise in orientational sensors such as gyroscopes.