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Gregory S. Chirikjian - One of the best experts on this subject based on the ideXlab platform.
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online ultrasound sensor calibration using gradient descent on the Euclidean Group
International Conference on Robotics and Automation, 2014Co-Authors: Martin Kendal Ackerman, Alexis Cheng, Emad M Boctor, Gregory S. ChirikjianAbstract:Ultrasound imaging can be an advantageous imaging modality for image guided surgery. When using ultrasound imaging (or any imaging modality), calibration is important when more advanced forms of guidance, such as augmented reality systems, are used. There are many different methods of calibration, but the goal of each is to recover the rigid body transformation relating the pose of the probe to the ultrasound image frame. This paper presents a unified algorithm that can solve the ultrasound calibration problem for various calibration methodologies. The algorithm uses gradient descent optimization on the Euclidean Group. It can be used in real time, also serving as a way to update the calibration parameters on-line. We also show how filtering, based on the theory of invariants, can further improve the online results. Focusing on two specific calibration methodologies, the AX = XB problem and the BX−1 p problem, we demonstrate the efficacy of the algorithm in both simulation and experimentation.
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ICRA - Online ultrasound sensor calibration using gradient descent on the Euclidean Group
2014 IEEE International Conference on Robotics and Automation (ICRA), 2014Co-Authors: Martin Kendal Ackerman, Alexis Cheng, Emad M Boctor, Gregory S. ChirikjianAbstract:Ultrasound imaging can be an advantageous imaging modality for image guided surgery. When using ultrasound imaging (or any imaging modality), calibration is important when more advanced forms of guidance, such as augmented reality systems, are used. There are many different methods of calibration, but the goal of each is to recover the rigid body transformation relating the pose of the probe to the ultrasound image frame. This paper presents a unified algorithm that can solve the ultrasound calibration problem for various calibration methodologies. The algorithm uses gradient descent optimization on the Euclidean Group. It can be used in real time, also serving as a way to update the calibration parameters on-line. We also show how filtering, based on the theory of invariants, can further improve the online results. Focusing on two specific calibration methodologies, the AX = XB problem and the BX−1 p problem, we demonstrate the efficacy of the algorithm in both simulation and experimentation.
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Conformational Statistics of Dna and Diffusion Equations on The Euclidean Group
Mathematics of DNA Structure Function and Interactions, 2009Co-Authors: Gregory S. ChirikjianAbstract:Semi-flexible (or wormlike) polymer chains such as DNA possess bending and torsional stiffness. Given a semi-flexible polymer structure that is subjected to Brownian motion forcing, the distribution of relative positions and orientations visited by the distal end of the chain relative to its proximal end provides important information about the molecule that can be linked to experimental observations. This probability density of end-to-end position and orientation can be obtained by solving a Fokker-Planck equation that describes a diffusion process on the Euclidean motion Group. In this paper, methods for solving this diffusion equation are reviewed. The techniques presented are valid for chains of up to several persistence lengths in open environments, where the effects of excluded volume can be neglected.
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Error propagation on the Euclidean Group with applications to manipulator kinematics
IEEE Transactions on Robotics, 2006Co-Authors: Yunfeng Wang, Gregory S. ChirikjianAbstract:Error propagation on the Euclidean motion Group arises in a number of areas such as errors that accumulate from the base to the distal end of manipulators. We address error propagation in rigid-body poses in a coordinate-free way, and explain how this differs from other approaches proposed in the literature. In this paper, we show that errors propagate by convolution on the Euclidean motion Group, SE(3). When local errors are small, they can be described well as distributions on the Lie algebra se(3). We show how the concept of a highly concentrated Gaussian distribution on SE(3) is equivalent to one on se(3). We also develop closure relations for these distributions under convolution on SE(3). Numerical examples illustrate how convolution is a valuable tool for computing the propagation of both small and large errors
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Synthesis of binary manipulators using the Fourier transform on the Euclidean Group
Journal of Mechanical Design, 1999Co-Authors: Alexander B Kyatkin, Gregory S. ChirikjianAbstract:In this paper we apply the Fourier transform on the Euclidean motion Group to solve problems in kinematic design of binary manipulators. In recent papers it has been shown that the workspace of a binary manipulator can be viewed as a function on the motion Group, and it can be generated as a generalized convolution product. The new contribution of this paper is the numerical solution of mathematical inverse problems associated with the design of binary manipulators. We suggest an anzatz function which approximates the manipulator's density in analytical form and has few free fitting parameters. Using the anzatz functions and Fourier methods on the motion Group, linear and non-linear inverse problems (i.e., problems of finding the manipulator's parameters which produce the total desired workspace density) are solved.
Jochen Trumpf - One of the best experts on this subject based on the ideXlab platform.
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discrete update pose filter on the special Euclidean Group se 3
Conference on Decision and Control, 2019Co-Authors: Mohammad Zamani, Jochen TrumpfAbstract:This paper proposes two variants of the Geometric Approximate Minimum Energy (GAME) filter on the Special Euclidean Group SE(3) in the case that exteroceptive measurements are obtained in discrete time. Continuous-discrete versions of the GAME filter are provided that near-continuously predict pose and its covariance using high frequency interoceptive measurements and then update these estimates utilizing low frequency exteroceptive measurements obtained in discrete time. The two variants of the proposed filter are differentiated in their derivation due to the choice of affine connection used on SE(3). The proposed discrete update filters are derived based on first principles of deterministic minimum-energy filtering extended for discrete time measurements and derived directly on SE(3). The performance of the proposed filters is demonstrated and compared in simulations with a short discussion of practical implications of the choice of affine connection.
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CDC - Discrete update pose filter on the special Euclidean Group SE(3)
2019 IEEE 58th Conference on Decision and Control (CDC), 2019Co-Authors: Mohammad Zamani, Jochen TrumpfAbstract:This paper proposes two variants of the Geometric Approximate Minimum Energy (GAME) filter on the Special Euclidean Group SE(3) in the case that exteroceptive measurements are obtained in discrete time. Continuous-discrete versions of the GAME filter are provided that near-continuously predict pose and its covariance using high frequency interoceptive measurements and then update these estimates utilizing low frequency exteroceptive measurements obtained in discrete time. The two variants of the proposed filter are differentiated in their derivation due to the choice of affine connection used on SE(3). The proposed discrete update filters are derived based on first principles of deterministic minimum-energy filtering extended for discrete time measurements and derived directly on SE(3). The performance of the proposed filters is demonstrated and compared in simulations with a short discussion of practical implications of the choice of affine connection.
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gradient like observer design on the special Euclidean Group se 3 with system outputs on the real projective space
arXiv: Optimization and Control, 2015Co-Authors: Tarek Hamel, Robert Mahony, Jochen TrumpfAbstract:A nonlinear observer on the Special Euclidean Group $\mathrm{SE(3)}$ for full pose estimation, that takes the system outputs on the real projective space directly as inputs, is proposed. The observer derivation is based on a recent advanced theory on nonlinear observer design. A key advantage with respect to existing pose observers on $\mathrm{SE(3)}$ is that we can now incorporate in a unique observer different types of measurements such as vectorial measurements of known inertial vectors and position measurements of known feature points. The proposed observer is extended allowing for the compensation of unknown constant bias present in the velocity measurements. Rigorous stability analyses are equally provided. Excellent performance of the proposed observers are shown by means of simulations.
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CDC - Gradient-like observer design on the Special Euclidean Group SE(3) with system outputs on the real projective space
2015 54th IEEE Conference on Decision and Control (CDC), 2015Co-Authors: Minhduc Hua, Tarek Hamel, Robert Mahony, Jochen TrumpfAbstract:A nonlinear observer on the Special Euclidean Group SE(3) for full pose estimation, that takes the system outputs on the real projective space directly as inputs, is proposed. The observer derivation is based on a recent advanced theory on nonlinear observer design. A key advantage with respect to existing pose observers on SE(3) is that we can now incorporate in a unique observer different types of measurements such as vectorial measurements of known inertial vectors and position measurements of known feature points. The proposed observer is extended allowing for the compensation of unknown constant bias present in the velocity measurements. Rigorous stability analyses are equally provided. Excellent performance of the proposed observers are shown by means of simulations.
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Minimum-Energy Pose Filtering on the Special Euclidean Group
2012Co-Authors: Mohammad Zamani, Jochen Trumpf, Robert MahonyAbstract:Obtaining a robust estimate for the pose ( attitude and position) of a rigid body moving in three dimensional space using noisy vectorial measurements is a challenging problem. The underlying geometry of pose space, the special Euclidean Group SE(3), makes this problem highly nonlinear and sensitive to measurement noise. According to a recent survey [9], most attitude estimation applications in robotics are currently tackled using extended Kalman filter (EKF) based methods, cf. [4, 1, 19]. However, implementing these methods using linearization and sampling techniques that do not respect the underlying geometry of the system’s state space may cause convergence and stability issues, cf. [8]. State of the art EKFtype methods such as the multiplicative extended Kalman filter (MEKF) [15] and the invariant extended Kalman filter (IEKF) [6] try to compensate by applying modifications to the EKF equations in order to preserve the geometric structure of the estimates. A recent body of work on the design of nonlinear observers, cf. [18, 5, 20, 11, 12], directly exploits the geometric structure of attitude and pose to achieve guaranteed stability and convergence of estimates. However, these observers mainly use constant gains that need to be pre-tuned depending on the application. In recent work [16, 13, 14] the authors have designed nonlinear observers with time varying gains (i.e. nonlinear filters) that are posed directly on the geometric spaces of the unit circle S1 and the rotation Group SO(3). The first two works [16, 13] are heuristic and yield bounds on the distance to optimality of their algorithms. The latter work [14] is based on a systematic deterministic minimumenergy filtering approach due to Mortensen [17]. Although Mortensen’s approach is for systems defined on a Euclidean space, the authors in [14] extend it to a system defined on S1. Krener [10] proved that minimum-energy filters achieve exponentially fast convergence under some conditions including the uniform ob-
John J. Leonard - One of the best experts on this subject based on the ideXlab platform.
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se sync a certifiably correct algorithm for synchronization over the special Euclidean Group
The International Journal of Robotics Research, 2019Co-Authors: David M. Rosen, Luca Carlone, Afonso S. Bandeira, John J. LeonardAbstract:Many geometric estimation problems naturally take the form of synchronization over the special Euclidean Group: estimate the values of a set of unknown poses \(x_{1},\ldots ,x_n \in \text {SE}(d)\) given noisy measurements of a subset of their pairwise relative transforms \(x_{i}^{-1}x_{j}\). Examples of this class include the foundational problems of pose-graph simultaneous localization and mapping (SLAM) (in robotics) and camera motion estimation (in computer vision), among others. This problem is typically formulated as a nonconvex maximum-likelihood estimation that is computationally hard to solve in general. Nevertheless, in this paper we present an algorithm that is able to effciently recover certifiably globally optimal solutions of the special Euclidean synchronization problem in a non-adversarial noise regime. The crux of our approach is the development of a semidefinite relaxation of the maximum-likelihood estimation whose minimizer provides the exact MLE so long as the magnitude of the noise corrupting the available measurements falls below a certain critical threshold; furthermore, whenever exactness obtains, it is possible to verify this fact a posteriori, thereby certifying the optimality of the recovered estimate. We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction effciently. We combine this fast optimization approach with a simple rounding procedure to produce our algorithm, SE-Sync. Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering globally optimal solutions when the available measurements are corrupted by noise up to an order of magnitude greater than that typically encountered in robotics and computer vision applications, and does so more than an order of magnitude faster than the Gauss-Newton-based approach that forms the basis of current state-of-the-art techniques.
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SE-Sync: a certifiably correct algorithm for synchronization over the special Euclidean Group
The International Journal of Robotics Research, 2018Co-Authors: David M. Rosen, Luca Carlone, Afonso S. Bandeira, John J. LeonardAbstract:Many important geometric estimation problems naturally take the form of synchronization over the special Euclidean Group: estimate the values of a set of unknown Group elements x1,…,xn∈SE(d) given ...
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A Certifiably Correct Algorithm for Synchronization over the Special Euclidean Group
arXiv: Robotics, 2016Co-Authors: David M. Rosen, Luca Carlone, Afonso S. Bandeira, John J. LeonardAbstract:Many geometric estimation problems take the form of synchronization over the special Euclidean Group: estimate the values of a set of poses given noisy measurements of a subset of their pairwise relative transforms. This problem is typically formulated as a maximum-likelihood estimation that requires solving a nonconvex nonlinear program, which is computationally intractable in general. Nevertheless, in this paper we present an algorithm that is able to efficiently recover certifiably globally optimal solutions of this estimation problem in a non-adversarial noise regime. The crux of our approach is the development of a semidefinite relaxation of the maximum-likelihood estimation whose minimizer provides the exact MLE so long as the magnitude of the noise corrupting the available measurements falls below a certain critical threshold; furthermore, whenever exactness obtains, it is possible to verify this fact a posteriori, thereby certifying the optimality of the recovered estimate. We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction efficiently. We combine this fast optimization approach with a simple rounding procedure to produce our algorithm, SE-Sync. Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering globally optimal solutions when the available measurements are corrupted by noise up to an order of magnitude greater than that typically encountered in robotics applications, and does so at a computational cost that scales comparably with that of direct Newton-type local search techniques.
Robert Mahony - One of the best experts on this subject based on the ideXlab platform.
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output regulation on the special Euclidean Group se 3
Conference on Decision and Control, 2016Co-Authors: Simone De Marco, Lorenzo Marconi, Tarek Hamel, Robert MahonyAbstract:This paper addresses the output regulation problem of rigid bodies whose kinematic configuration space lies on the Special Euclidean Group SE(3). Reference trajectories to be tracked are generated by an autonomous system, referred to as exosystem, defined on the Special Euclidean Group as well. Only partial relative pose measurements associated to the “natural” linear left Group action on SE(3) along with the pose and the velocity of the controlled body are available. The proposed control action embeds a copy of the exosystem kinematics properly updated by means of relative information error in the same spirit of internal model principle.
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CDC - Output regulation on the Special Euclidean Group SE(3)
2016 IEEE 55th Conference on Decision and Control (CDC), 2016Co-Authors: Simone De Marco, Lorenzo Marconi, Tarek Hamel, Robert MahonyAbstract:This paper addresses the output regulation problem of rigid bodies whose kinematic configuration space lies on the Special Euclidean Group SE(3). Reference trajectories to be tracked are generated by an autonomous system, referred to as exosystem, defined on the Special Euclidean Group as well. Only partial relative pose measurements associated to the “natural” linear left Group action on SE(3) along with the pose and the velocity of the controlled body are available. The proposed control action embeds a copy of the exosystem kinematics properly updated by means of relative information error in the same spirit of internal model principle.
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gradient like observer design on the special Euclidean Group se 3 with system outputs on the real projective space
arXiv: Optimization and Control, 2015Co-Authors: Tarek Hamel, Robert Mahony, Jochen TrumpfAbstract:A nonlinear observer on the Special Euclidean Group $\mathrm{SE(3)}$ for full pose estimation, that takes the system outputs on the real projective space directly as inputs, is proposed. The observer derivation is based on a recent advanced theory on nonlinear observer design. A key advantage with respect to existing pose observers on $\mathrm{SE(3)}$ is that we can now incorporate in a unique observer different types of measurements such as vectorial measurements of known inertial vectors and position measurements of known feature points. The proposed observer is extended allowing for the compensation of unknown constant bias present in the velocity measurements. Rigorous stability analyses are equally provided. Excellent performance of the proposed observers are shown by means of simulations.
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CDC - Gradient-like observer design on the Special Euclidean Group SE(3) with system outputs on the real projective space
2015 54th IEEE Conference on Decision and Control (CDC), 2015Co-Authors: Minhduc Hua, Tarek Hamel, Robert Mahony, Jochen TrumpfAbstract:A nonlinear observer on the Special Euclidean Group SE(3) for full pose estimation, that takes the system outputs on the real projective space directly as inputs, is proposed. The observer derivation is based on a recent advanced theory on nonlinear observer design. A key advantage with respect to existing pose observers on SE(3) is that we can now incorporate in a unique observer different types of measurements such as vectorial measurements of known inertial vectors and position measurements of known feature points. The proposed observer is extended allowing for the compensation of unknown constant bias present in the velocity measurements. Rigorous stability analyses are equally provided. Excellent performance of the proposed observers are shown by means of simulations.
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Minimum-Energy Pose Filtering on the Special Euclidean Group
2012Co-Authors: Mohammad Zamani, Jochen Trumpf, Robert MahonyAbstract:Obtaining a robust estimate for the pose ( attitude and position) of a rigid body moving in three dimensional space using noisy vectorial measurements is a challenging problem. The underlying geometry of pose space, the special Euclidean Group SE(3), makes this problem highly nonlinear and sensitive to measurement noise. According to a recent survey [9], most attitude estimation applications in robotics are currently tackled using extended Kalman filter (EKF) based methods, cf. [4, 1, 19]. However, implementing these methods using linearization and sampling techniques that do not respect the underlying geometry of the system’s state space may cause convergence and stability issues, cf. [8]. State of the art EKFtype methods such as the multiplicative extended Kalman filter (MEKF) [15] and the invariant extended Kalman filter (IEKF) [6] try to compensate by applying modifications to the EKF equations in order to preserve the geometric structure of the estimates. A recent body of work on the design of nonlinear observers, cf. [18, 5, 20, 11, 12], directly exploits the geometric structure of attitude and pose to achieve guaranteed stability and convergence of estimates. However, these observers mainly use constant gains that need to be pre-tuned depending on the application. In recent work [16, 13, 14] the authors have designed nonlinear observers with time varying gains (i.e. nonlinear filters) that are posed directly on the geometric spaces of the unit circle S1 and the rotation Group SO(3). The first two works [16, 13] are heuristic and yield bounds on the distance to optimality of their algorithms. The latter work [14] is based on a systematic deterministic minimumenergy filtering approach due to Mortensen [17]. Although Mortensen’s approach is for systems defined on a Euclidean space, the authors in [14] extend it to a system defined on S1. Krener [10] proved that minimum-energy filters achieve exponentially fast convergence under some conditions including the uniform ob-
Venu Madhav Govindu - One of the best experts on this subject based on the ideXlab platform.
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efficient and robust registration on the 3d special Euclidean Group
International Conference on Computer Vision, 2019Co-Authors: Uttaran Bhattacharya, Venu Madhav GovinduAbstract:We present a robust, fast and accurate method for registration of 3D scans. Using correspondences, our method optimizes a robust cost function on the intrinsic representation of rigid motions, i.e., the Special Euclidean Group SE(3). We exploit the geometric properties of Lie Groups as well as the robustness afforded by an iteratively reweighted least squares optimization. We also generalize our approach to a joint multiview method that simultaneously solves for the registration of a set of scans. Our approach significantly outperforms the state-of-the-art robust 3D registration method based on a line process in terms of both speed and accuracy. We show that this line process method is a special case of our principled geometric solution. Finally, we also present scenarios where global registration based on feature correspondences fails but multiview ICP based on our robust motion estimation is successful.
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Efficient and Robust Registration on the 3D Special Euclidean Group
arXiv: Computer Vision and Pattern Recognition, 2019Co-Authors: Uttaran Bhattacharya, Venu Madhav GovinduAbstract:We present an accurate, robust and fast method for registration of 3D scans. Our motion estimation optimizes a robust cost function on the intrinsic representation of rigid motions, i.e., the Special Euclidean Group $\mathbb{SE}(3)$. We exploit the geometric properties of Lie Groups as well as the robustness afforded by an iteratively reweighted least squares optimization. We also generalize our approach to a joint multiview method that simultaneously solves for the registration of a set of scans. We demonstrate the efficacy of our approach by thorough experimental validation. Our approach significantly outperforms the state-of-the-art robust 3D registration method based on a line process in terms of both speed and accuracy. We also show that this line process method is a special case of our principled geometric solution. Finally, we also present scenarios where global registration based on feature correspondences fails but multiview ICP based on our robust motion estimation is successful.
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ICCV - Efficient and Robust Registration on the 3D Special Euclidean Group
2019 IEEE CVF International Conference on Computer Vision (ICCV), 2019Co-Authors: Uttaran Bhattacharya, Venu Madhav GovinduAbstract:We present a robust, fast and accurate method for registration of 3D scans. Using correspondences, our method optimizes a robust cost function on the intrinsic representation of rigid motions, i.e., the Special Euclidean Group SE(3). We exploit the geometric properties of Lie Groups as well as the robustness afforded by an iteratively reweighted least squares optimization. We also generalize our approach to a joint multiview method that simultaneously solves for the registration of a set of scans. Our approach significantly outperforms the state-of-the-art robust 3D registration method based on a line process in terms of both speed and accuracy. We show that this line process method is a special case of our principled geometric solution. Finally, we also present scenarios where global registration based on feature correspondences fails but multiview ICP based on our robust motion estimation is successful.
