Euclidean Group Research Articles

Geometric Pseudospectral Method on SE(3) for Rigid-Body Dynamics with Application to Aircraft

General pseudospectral method is extended to the special Euclidean group SE(3) by virtue of equivariant map for rigid-body dynamics of the aircraft. On SE(3), a complete left invariant rigid-body dynamics model of the aircraft in body-fixed frame is established, including configuration model and velocity model. For the left invariance of the configuration model, equivalent Lie algebra equation corresponding to the configuration equation is derived based on the left-trivialized tangent of local coordinate map, and the top eight orders truncated Magnus series expansion with its coefficients of the solution of the equivalent Lie algebra equation are given. A numerical method called geometric pseudospectral method is developed, which, respectively, computes configurations and velocities at the collocation points and the endpoint based on two different collocation strategies. Through numerical tests on a free-floating rigid-body dynamics compared with several same order classical methods in Euclidean space and Lie group, it is found that the proposed method has higher accuracy, satisfying computational efficiency, stable Lie group structural conservativeness. Finally, how to apply the previous discretization scheme to rigid-body dynamics simulation and control of the aircraft is illustrated.

Feedback Classification of Invariant Control Systems on Three-Dimensional Lie Groups

We consider left-invariant control affine systems evolving on Lie groups. In this context, feedback equivalence specializes to detached feedback equivalence. We characterize (local) detached feedback equivalence in a simple algebraic manner. We then classify all (full-rank) systems evolving on three-dimensional Lie groups. A representative is identified for each equivalence class. Systems on the Heisenberg group, the Euclidean group, and the orthogonal group are treated in full, as typical examples. In these three cases, simple algebraic characterizations of the equivalence classes are also exhibited. A few remarks conclude the paper.

Riemannian Means on Special Euclidean Group and Unipotent Matrices Group

Among the noncompact matrix Lie groups, the special Euclidean group and the unipotent matrix group play important roles in both theoretic and applied studies. The Riemannian means of a finite set of the given points on the two matrix groups are investigated, respectively. Based on the left invariant metric on the matrix Lie groups, the geodesic between any two points is gotten. And the sum of the geodesic distances is taken as the cost function, whose minimizer is the Riemannian mean. Moreover, a Riemannian gradient algorithm for computing the Riemannian mean on the special Euclidean group and an iterative formula for that on the unipotent matrix group are proposed, respectively. Finally, several numerical simulations in the 3-dimensional case are given to illustrate our results.

Consensus based formation control laws for systems on Lie groups

This paper designs the consensus based formation control laws for multi-agent systems defined on Lie groups. The specific results are illustrated on SE(2) and SE(3) (the special Euclidean groups of rigid body motions and each of which shares the geometric structure of a Lie group). The paper develops the control laws for the problem concerned under the assumption that the communication topologies of the multi-agent systems are spanning-tree-shaped, ring-shaped and complete, respectively. At last, several examples are included in the numerical simulations to verify the theoretical results.

Eigenvector synchronization, graph rigidity and the molecule problem.

The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. In this paper, we extend the previous work and propose the 3D-As-Synchronized-As-Possible (3D-ASAP) algorithm, for the graph realization problem in ℝ3, given a sparse and noisy set of distance measurements. 3D-ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch, there corresponds an element of the Euclidean group, Euc(3), of rigid transformations in ℝ3, and the goal was to estimate the group elements that will properly align all the patches in a globally consistent way. Furthermore, 3D-ASAP successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose 3D-spectral-partitioning (SP)-ASAP, a faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a pre-processing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably with similar state-of-the-art localization algorithms.

Active Joint Stiffness Regulation to Achieve Isotropic Compliance in the Euclidean Space

This paper is dedicated to the relationship between the external force applied on a point of a robot end-effector and its consequent displacement in static conditions. Both the force and the displacement are herein considered in the Euclidean space E(3). This fact represents a significant simplification of the approach, since it avoids some problems related to the absence of a natural positive definite metric on the Special Euclidean Group SE(3). On the other hand, such restriction allows the method to find closed-form solutions to a large class of problems in robot statics. The peculiar goal of this investigation consists of setting up a procedure which guarantees at least one pose at which any force applied (in E(3)) to an end-effector point is always parallel to its consequent displacement (also in E(3)). This property, which will be referred to as isotropic compliance in E(3), makes the robot tip static behavior uniform with respect to all directions, namely, isotropic, although not homogeneous, since it holds only in some poses. Achieving isotropic compliance in E(3) is a task more general than the classical problem of finding a pose with unit condition number, which does not include the case of different elements in the diagonal joint stiffness matrix. For this reason, the object of the present investigation could not be furtherer simplified to the classical kinetostatic problem in terms of the jacobian matrix alone. The paper reveals how the force–displacement parallelism can be achieved by using a method based on a simple proportional-derivative (PD) controller strategy. The method can be applied when the passive and active stiffness act, on the joints, either in parallel or in series, and the magnitude of the displacement response can be chosen by imposing appropriate values for the overall joints compliance. Results show that for the three analyzed examples, namely, the RR, RRP, and RRR manipulators, with arbitrary lengths of the links, there is, at least, one pose for which the sought property is achieved.

Moving Frames and Conservation Laws for Euclidean Invariant Lagrangians

In recent work, the authors show the mathematical structure behind both the Euler–Lagrange system and the set of conservation laws, in terms of the differential invariants of the group action and a moving frame. In this paper, the authors demonstrate that the knowledge of this structure allows to find the first integrals of the Euler–Lagrange equations, and subsequently, to solve by quadratures, variational problems that are invariant under the special Euclidean groups SE(2) and SE(3).

Nonlinear Robust Tracking Control of a Quadrotor UAV on SE(3)

AbstractThis paper provides nonlinear tracking control systems for a quadrotor unmanned aerial vehicle (UAV) that are robust to bounded uncertainties. A mathematical model of a quadrotor UAV is defined on the special Euclidean group, and nonlinear output‐tracking controllers are developed to follow (i) an attitude command, and (ii) a position command for the vehicle center of mass. The controlled system has the desirable properties that the tracking errors are uniformly ultimately bounded, and the size of the ultimate bound can be reduced arbitrarily by control system parameters. Numerical examples illustrating complex maneuvers are provided.

Sensor network localization by eigenvector synchronization over the euclidean group

We present a new approach to localization of sensors from noisy measurements of a subset of their Euclidean distances. Our algorithm starts by finding, embedding, and aligning uniquely realizable subsets of neighboring sensors called patches. In the noise-free case, each patch agrees with its global positioning up to an unknown rigid motion of translation, rotation, and possibly reflection. The reflections and rotations are estimated using the recently developed eigenvector synchronization algorithm, while the translations are estimated by solving an overdetermined linear system. The algorithm is scalable as the number of nodes increases and can be implemented in a distributed fashion. Extensive numerical experiments show that it compares favorably to other existing algorithms in terms of robustness to noise, sparse connectivity, and running time. While our approach is applicable to higher dimensions, in the current article, we focus on the two-dimensional case.

Unitary cocycle representations of the Galilean line group: Quantum mechanical principle of equivalence

We present a formalism of Galilean quantum mechanics in non-inertial reference frames and discuss its implications for the equivalence principle. This extension of quantum mechanics rests on the Galilean line group, the semidirect product of the real line and the group of analytic functions from the real line to the Euclidean group in three dimensions. This group provides transformations between all inertial and non-inertial reference frames and contains the Galilei group as a subgroup. We construct a certain class of unitary representations of the Galilean line group and show that these representations determine the structure of quantum mechanics in non-inertial reference frames. Our representations of the Galilean line group contain the usual unitary projective representations of the Galilei group, but have a more intricate cocycle structure. The transformation formula for the Hamiltonian under the Galilean line group shows that in a non-inertial reference frame it acquires a fictitious potential energy term that is proportional to the inertial mass, suggesting the equivalence of inertial mass and gravitational mass in quantum mechanics.

Sur l'homologie des groupes unitaires à coefficients polynomiaux

AbstractLetAbe a ring with anti-involution andFa nice functor (tensor or symmetric power, for example) from finitely-generated projectiveA-modules to abelian groups. We show that the homology of the hyperbolic unitary groupsUn,n(A) with coefficients inF(A2n) can be expressed stably (i.e. after taking the colimit overn) by the homology of these groups with untwisted coefficients and functor homology groups that we can compute in suitable cases (for example, whenAis a field of characteristic 0 or a ring without ℤ-torsion andFa tensor power). This extends the result whereAis a finite field, which was dealt with previously by C. Vespa and the author (Ann. Sci. ENS, 2010).The proof begins by relating, without any assumption onF, our homology groups to the homology of a category of hermitian spaces with coefficients twisted byF. Then, whenFis polynomial, we establish — following a method due to Scorichenko — an isomorphism between this homology and the homology of another category of (possibly degenerate) hermitian spaces, which is computable (in good cases) by standard methods of homological algebra in functor categories (using adjunctions, Künneth formula…). We give some examples.Finally, we deal with the analogous problem for non-hyperbolic unitary groups in some special cases, for example euclidean orthogonal groupsOn(A) (the ringAbeing here commutative). The isomorphism between functor homology and group homology with twisted coefficients does not hold in full generality; nevertheless we succeed to get it whenAis a field or, for example, a subring of ℚ containing ℤ[1/2]. The method, which is similar to that in the previous case, uses a general result of symmetrisation in functor homology proved at the beginning of the article.

Global space-time symmetries of quantized Euclidean and Minkowski superspaces

Abstract Starting with assumptions both simple and natural from “physical” point of view we present a direct construction of the transformations preserving wide class of (anti)commutation relations which describe Euclidean/Minkowski superspace quantizations. These generalized transformations act on deformed superspaces as the ordinary ones do on undeformed spaces but they depend on non(anti)commuting parameters satisfying some consistent (anti)commutation relations. Once the coalgebraic structure compatible with the algebraic one is introduced in the set of transformations we deal with quantum symmetry supergroup. This is the case for intensively studied so called $ N = \frac{1}{2} $ supersymmetry as well as its three parameter extension. The resulting symmetry transformations — supersymmetric extension of θ — Euclidean group can be regarded as global counterpart of appropriately twisted Euclidean superalgebra that has been shown to preserve $ N = \frac{1}{2} $ supersymmetry.

INFINITE DIMENSIONAL STATIONARY RANDOM FIELDS OVER A LOCALLY COMPACT ABELIAN GROUP

In this paper we consider spectral representation of infinite dimensional stationary random fields over an abelian locally compact (or LCA) group, and then extend the results of earlier authors who consider wide sense Markov random fields over a Euclidean group ℝn to the general LCA group context and obtain minimality properties. We also indicate possibilities of some extensions of these results to certain nonstationary classes.

Robust 3D visual tracking using particle filtering on the special Euclidean group: A combined approach of keypoint and edge features

We present a 3D model-based visual tracking approach using edge and keypoint features in a particle filtering framework. Recently, particle-filtering-based approaches have been proposed to integrate multiple pose hypotheses and have shown good performance, but most of the work has made an assumption that an initial pose is given. To ameliorate this limitation, we employ keypoint features for initialization of the filter. Given 2D–3D keypoint correspondences, we randomly choose a set of minimum correspondences to calculate a set of possible pose hypotheses. Based on the inlier ratio of correspondences, the set of poses are drawn to initialize particles. After the initialization, edge points are employed to estimate inter-frame motions. While we follow a standard edge-based tracking, we perform a refinement process to improve the edge correspondences between sampled model edge points and image edge points. For better tracking performance, we employ a first-order autoregressive state dynamics, which propagates particles more effectively than Gaussian random walk models. The proposed system re-initializes particles by itself when the tracked object goes out of the field of view or is occluded. The robustness and accuracy of our approach is demonstrated using comparative experiments on synthetic and real image sequences.

Lévy processes through time shift on oscillator Weyl algebra

We extend the Lie{algebra time shift technique, introduced in (2), from the usual Weyl algebra (associated to the additive group of a Hilbert space H) to the generalized Weyl algebra (oscillator algebra), associated to the semi{direct product of the additive group H with the unitary group on H (the Euclidean group of H, in the terminology of (14)). While in the usual Weyl algebra the possible quantum extensions of the time shift are essentially reduced to isomorphic copies of the Wiener process, in the case of the oscilla- tor algebra a larger class of Levy process arises.Our main result is the proof of the fact that the generators of the quantum Markov semigroups, associated to these time shifts, share with the quantum extensions of the Laplacian the important property that the generalized Weyl operators are eigenoperators for them and the corresponding eigenvalues are explicitly computed in terms of the Levy{Khintchin factor of the underlying classical Levy process.

P-adic renormalization group solutions and the euclidean renormalization group conjectures

We discuss algebraic similarity of the Wilson’s renormalization groups in the Euclidean and p-adic spaces. Automodel Hamiltonians have identical form in both cases in the framework of perturbation theory. Fermionic p-adic model has exact renormalization group solution which generates a list of non-trivial conjectures for the Euclidean case.

Passivity-Based Pose Synchronization in Three Dimensions

This paper addresses passivity-based pose synchronization in the Special Euclidean group <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SE</i> (3) . We first develop a passivity-based distributed velocity input law to achieve pose synchronization. We next show that the pose synchronizes exponentially fast under an assumption on the initial states of the bodies, with an exponential convergence rate given by algebraic connectivity of interconnection graphs. We also prove pose synchronization in the presence of communication delays and topology switches. We moreover give further extensions of the present law, where desirable velocities and collision avoidance are taken into account. Finally, the effectiveness of the present inputs is demonstrated through numerical simulations and experiments on a planar (2D) test bed.

Integration of the EPDiff equation by particle methods

The purpose of this paper is to apply particle methods to the numerical solution of the EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. This behavior allows for the description of many rich physical applications, but also introduces difficult numerical challenges. We present a particle method for the EPDiff equation that is well-suited for this class of solutions and for simulating collisions between wavefronts. Discretization by means of the particle method is shown to preserve the basic Hamiltonian, the weak and variational structure of the original problem, and to respect the conservation laws associated with symmetry under the Euclidean group. Numerical results illustrate that the particle method has superior features in both one and two dimensions, and can also be effectively implemented when the initial data of interest lies on a submanifold.

Invariant Histograms

We introduce and study a Euclidean-invariant distance histogram function for curves. For a sufficiently regular plane curve, we prove that the cumulative distance histograms based on discretizing the curve by either uniformly spaced or randomly chosen sample points converge to our histogram function. We argue that the histogram function serves as a simple, noise-resistant shape classifier for regular curves under the Euclidean group of rigid motions. Extensions of the underlying ideas to higher-dimensional submanifolds, as well as to area histogram functions invariant under the group of planar area-preserving affine transformations, are discussed.

Using Lie group symmetries for fast corrective motion planning

In this paper we develop an algorithmic framework allowing for fast and elegant path correction exploiting Lie group symmetries and operating without the need for explicit control strategies such as cross-track regulation. These systems occur across the gamut of robotics, notably in locomotion, be it ground, underwater, airborne, or surgical domains. Instead of reintegrating an entire trajectory, the method selectively alters small key segments of an initial trajectory in a consistent way so as to transform it via symmetry operations. The algorithm is formulated for arbitrary Lie groups and applied in the context of the special Euclidean group and subgroups thereof. A sampling-based motion planner is developed that uses this method to create paths for underactuated systems with differential constraints. It is also shown how the path correction method acts as a controller within a feedback control loop for real-time path correction. These approaches are demonstrated for ground vehicles in the plane and for flexible bevel tip needle steering in space. The results show that using symmetry-based path correction for motion planning provides a prudent and simple, yet computationally tractable, integrated planning and control strategy.