Euclidean Group Research Articles

On quotients of generalized Euclidean group rings

ABSTRACTLet R = ℤ[C] be the integral group ring of a finite cyclic group C. Dennis et al. [4] proved that R is a generalized Euclidean ring in the sense of Cohn [3], i.e., SLn(R) is generated by the elementary matrices for all n. We prove that every proper quotient of R is also a generalized Euclidean ring.

Logarithmic control, trajectory tracking, and formation for nonholonomic vehicles on Lie group SE(2)

ABSTRACTThis paper studies the problem of set-point stabilisation, trajectory tracking, and formation tracking for unicycle-type vehicles by taking advantage of the exponential coordinates and the other mathematical tools provided by the Lie group setting of the special Euclidean group SE(2). Motivated by recent developments in geometric control approaches for holonomic systems, we first study the stabilisation problem for nonholonomic constrained kinematic systems by improving the logarithmic stabilising control laws in such a way that they can satisfy nonholonomic constraints too. We then extend the control design to the problem of trajectory tracking by proposing auxiliary systems and investigating the conditions for which the adjoint map in the Lie algebra preserves the velocity constraint. This leads to a global control law which is valid for general time-varying as well as non-smooth trajectories. The tracking control law is also applied to the problem of leader-follower formation tracking by constructing nonholonomic virtual leaders in the desired formation. Finally, the kinematic control law is translated to a differential drive robot using the backstepping technique. To demonstrate the effectiveness of the controllers, the numerical results are presented and the performances are compared with the other three controllers in the literature.

Computationally Efficient Rigid-Body Gaussian Process for Motion Dynamics

In this letter, we address the modeling and learning of complex nonlinear rigid-body motions employing Gaussian processes. As the common procedure of using Euler angles in the Gaussian process results in inaccurate predictions for large rotations, we represent the input data by axis-angle pseudovectors for rotations and Euclidean vectors for translation. Our decision in favor of this representation of the special Euclidean group $SE(3)$ is due to its computational efficiency. To allow Gaussian process estimation on a non-Euclidean input domain, such as the space of rigid motions, we generalize the model by introducing novel mean and covariance functions on $SE(3)$ . We prove that those functions fulfill the requirements of Gaussian processes. The proposed approach is validated on simulated and on real human motion data. Our results demonstrate significant benefits of the proposed rigid-body Gaussian process with respect to alternative variants in terms of regression performance and computational efficiency.

Skeleton‐Based Human Action Recognition via Screw Matrices

With the recent advent of low-cost acquisition depth cameras, extracting 3D body skeleton has become relatively easier, which significantly lighten many difficulties in 2D videos including occlusions, shadows and background extraction, etc. Directly perceived features, for example points, lines and planes, can be easily extracted from 3D videos such that we can employ rigid motions to represent skeletal motions in a geometric way. We apply screw matrices, acquired by using rotations and translations in 3D space, to model single and multi-body rigid motion. Since screw matrices are members of the special Euclidean group SE(3), an action can be represented as a point on a Lie group, which is a differentiable manifold. Using Lie-algebraic properties of screw algebra, isomorphic to se(3), the classical algorithms of machine learning in vector space can be expanded to manifold space. We evaluate our approached on three public 3D action datasets: MSR Action3D dataset, UCF Kinect dataset and Florence3D-Action Dataset. The experimental results show that our approaches either match or exceed state-of-the-art skeleton-based human action recognition approaches.

Análisis Cinemático del Manipulador Paralelo 4-PRUR Mediante la Teoría de Tornillos

In this work the kinematics of a parallel manipulator performing Schönflies motion is investigated by means of the theory of screws. As an intermediate step, the displacement analysis is reported in semi-closed form solution based on the coordinates of two points embedded in the moving platform. This strategy allows to employ only one reference frame avoiding the computation of the rotation matrix. The input-output equations of velocity and acceleration are systematically obtained by resorting to reciprocal-screw theory. To this aim, the robot is treated as a six-degrees-of-freedom parallel manipulator incorporating pseudo kinematic pairs connecting the limbs to the fixed platform and one virtual kinematic chain in order to apply without restrictions the Lie algebra se(3) of the Euclidean group SE(3). The singularity analysis is investigated based on the input-output equation of velocity. Numerical examples are included in order to show the application of the method.

Observer’s observables. Residual diffeomorphisms

We investigate the fate of diffeomorphisms when the radial gauge is imposed in canonical general relativity. As shown elsewhere, the radial gauge is closely related to the observer’s observables. These observables are invariant under a large subgroup of diffeomorphisms which results in their usefulness for canonical general relativity. There are, however, some diffeomorphisms, called residual diffeomorphisms, which might be ‘observed’ by the observer as they do not preserve her observables. The present paper is devoted to the analysis of these diffeomorphisms in the case of the spatial and spacetime radial gauges. Although the residual diffeomorphisms do not form a subgroup of all diffeomorphisms, we show that their induced action in the phase space does form a group. We find the generators of the induced transformations and compute the structure functions of the algebras they form. The obtained algebras are deformations of the algebra of the Euclidean group and the algebra of the Poincaré group in the spatial and spacetime case, respectively. In both cases the deformation depends only on the Riemann curvature tensor and in particular vanishes when the space or spacetime is flat.

Variants of stability of discontinuous groups for Euclidean motion groups

Let [Formula: see text] be a Lie group, [Formula: see text] a closed subgroup of [Formula: see text] and [Formula: see text] a discontinuous group for the homogeneous space [Formula: see text]. Given a deformation parameter [Formula: see text], the deformed subgroup [Formula: see text] may fail to act properly discontinuously on [Formula: see text]. To understand this phenomenon in the case when [Formula: see text] stands for an Euclidean motion group [Formula: see text], we compare the notion of stability for discontinuous groups (cf. [T. Kobayashi and S. Nasrin, Deformation of properly discontinuous action of [Formula: see text] on [Formula: see text], Int. J. Math. 17 (2006) 1175–1193]) with its variants. We prove that the defined stability variants hold when [Formula: see text] turns out to be a crystallographic subgroup of [Formula: see text].

Integrated Guidance and Feedback Control of Underactuated Robotics System in SE(3)

An integrated guidance and feedback control scheme for steering an underactuated vehicle through desired waypoints in three-dimensional space, is developed here. The underactuated vehicle is modeled as a rigid body with four control inputs. These control inputs actuate the three degrees of freedom of rotational motion and one degree of freedom of translational motion in a vehicle body-fixed coordinate frame. This actuation model is appropriate for a wide range of underactuated vehicles including spacecraft with internal attitude actuators, vertical take-off and landing (VTOL) aircraft, fixed-wing multirotor unmanned aerial vehicles (UAVs), maneuverable robotic vehicles, etc. The guidance problem is developed on the special Euclidean group of rigid body motions, SE(3), in the framework ofgeometric mechanics, which represents the vehicle dynamics globally on this configuration manifold. The integrated guidance and control algorithm selects the desired trajectory for the translational motion that passes through the given waypoints, and the desired trajectory for the attitude based on the desired thrust direction to achieve the translational motion trajectory. A feedback control law is then obtained to steer the underactuated vehicle towards the desired trajectories in translation and rotation. This integrated guidance and control scheme takes into account known bounds on control inputs and generates a trajectory that is continuous and at least twice differentiable, which can be implemented with continuous and bounded control inputs. The integrated guidance and feedback control scheme is applied to an underactuated quadcopter UAV to autonomously generate a trajectory through a series of given waypoints in SE(3) and track the desired trajectory in finite time. The overall stability analysis of the feedback system is addressed. Discrete time models for the dynamics and control schemes of the UAV are obtained in the form of Lie group variational integrators using the discrete Lagrange-d’Alembert principle. Almost global asymptotic stability of the feedback system over its state space is shown analytically and verified through numerical simulations.

Global Formulation of an Extended Kalman Filter on SE(3) for Geometric Control of a Quadrotor UAV

An extended Kalman filter (EKF) is developed on the special Euclidean group, SE(3) for geometric control of a quadrotor UAV. It is obtained by performing an intrinsic form of linearization on SE(3) to estimate the state of the quadrotor from noisy measurements. The proposed estimator considers all of the coupling effects between rotational and translational dynamics, and it is developed in a coordinate-free fashion. The desirable features of the proposed EKF are illustrated by numerical examples and experimental results for several scenarios. The proposed estimation scheme on SE(3) has been unprecedented and these results can be particularly useful for aggressive maneuvers in GPS denied environments or in situations where parts of onboard sensors fail.

Formation tracking control for multiple rigid bodies on matrix Lie groups: A system decomposition approach

SummaryThis paper addresses the finite‐time formation tracking control problem for multiple rigid bodies whose dynamics are defined on the matrix Lie groups, including the special Euclidean group SE(2),SE(3) as specific cases. The reference trajectory in a form of rotational and translational motions is generated offline in advance as the virtual leader for tracking. Moreover, the formation time is specified according to the task, and the desired formation shape is given a priori with respect to the virtual leader. By virtue of the system decomposition approach, intrinsic formation tracking laws are derived for arbitrary initial velocities for the rigid bodies. The tracking controllers are intrinsic meaning that the dynamics and controllers are developed in the body‐fixed frame without a global reference frame. Moreover, based on the geometric convex combination on SE(3), the result is extended to the distributed case where only the neighboring agents' state are used. Two numerical simulations on SE(2) and SE(3) are given respectively to illustrate the validity and robustness of the proposed controllers. Copyright © 2017 John Wiley & Sons, Ltd.

Hyperbolic Pseudoinverses for Kinematics in the Euclidean Group

The kinematics of a robot manipulator are described in terms of the mapping connecting its joint space and the 6-dimensional Euclidean group of motions $SE(3)$. The associated Jacobian matrices map into its Lie algebra $\mathfrak{se}(3)$, the space of twists describing infinitesimal motion of a rigid body. Control methods generally require knowledge of an inverse for the Jacobian. However, for an arm with fewer or greater than six actuated joints or at singularities of the kinematic mapping, this breaks down. The Moore--Penrose pseudoinverse has frequently been used as a surrogate but is not invariant under change of coordinates. Since the Euclidean Lie algebra carries a pencil of invariant bilinear forms that are indefinite, a family of alternative hyperbolic pseudoinverses is available. Generalized Gram matrices and the classification of screw systems are used to determine conditions for their existence. The existence or otherwise of these pseudoinverses also relates to a classical problem addressed by ...

Dynamics of Meandering Spiral Waves with Weak Lattice Perturbations

Re-entrant spiral waves are observed in many different situations in nature, perhaps most importantly in excitable electrophysiological tissue where they are believed to be responsible for pathological conditions such as cardiac arrhythmias, epileptic seizures, and hallucinations. Mathematically, spiral waves occur as solutions to systems of reaction-diffusion partial differential equations (RDPDEs), which are frequently used as models for electrophysiological phenomena. Because of the invariance of these RDPDEs with respect to the Euclidean group $SE(2)$ of planar translations and rotations, much progress has been made in understanding the dynamics and bifurcations of spiral waves using the theory of group-equivariant dynamical systems. In reality, however, Euclidean symmetry is at best an approximation. Inhomogeneities and anisotropy in the medium of propagation of the waves break the Euclidean symmetry and can lead to such phenomena as anchoring and drifting. In this paper, we study the effects on quas...

Some Discussions about the Error Functions on SO(3) and SE(3) for the Guidance of a UAV Using the Screw Algebra Theory

In this paper a new error function designed on 3-dimensional special Euclidean group SE(3) is proposed for the guidance of a UAV (Unmanned Aerial Vehicle). In the beginning, a detailed 6-DOF (Degree of Freedom) aircraft model is formulated including 12 nonlinear differential equations. Secondly the definitions of the adjoint representations are presented to establish the relationships of the Lie groups SO(3) and SE(3) and their Lie algebras so(3) and se(3). After that the general situation of the differential equations with matrices belonging to SO(3) and SE(3) is presented. According to these equations the features of the error function on SO(3) are discussed. Then an error function on SE(3) is devised which creates a new way of error functions constructing. In the simulation a trajectory tracking example is given with a target trajectory being a curve of elliptic cylinder helix. The result shows that a better tracking performance is obtained with the new devised error function.

Wigner functions for the pair angle and orbital angular momentum

The problem of constructing physically and mathematically well-defined Wigner functions for the canonical pair angle and angular momentum is solved. While a key element for the construction of Wigner functions for the planar phase space (q,p) in R^2 is the Heisenberg-Weyl group, the corresponding group for the cylindrical phase space (theta,p) in S^1 x R is the Euclidean group E(2) of the plane and its unitary representations. Here the angle theta is replaced by the pair (cos theta, sin theta) which corresponds uniquely to the points on the unit circle. The main structural properties of the Wigner functions for the planar and the cylindrical phase spaces are strikingly similar. A crucial role plays the sinc function which provides the interpolation for the discontinuous quantized angular momenta in terms of the continuous classical ones, in accordance with the famous Whittaker cardinal function, well-known from interpolation and sampling theories. The quantum mechanical marginal distributions for angle (continuous) and angular momentum (discontinuous) are as usual uniquely obtained by appropriate integrations of the (theta,p) Wigner function. Among the examples discussed is an elementary system of simple "cat" states.

Frame-invariance in finite element formulations of geometrically exact rods

This article is concerned with finite element implementations of the threedimensional geometrically exact rod. The special attention is paid to identifying the condition that ensures the frame invariance of the resulting discrete approximations. From the perspective of symmetry, this requirement is equivalent to the commutativity of the employed interpolation operator I with the action of the special Euclidean group SE(3), or I is SE(3)-equivariant. This geometric criterion helps to clarify several subtle issues about the interpolation of finite rotation. It leads us to reexamine the finite element formulation first proposed by Simo in his work on energy-momentum conserving algorithms. That formulation is often mistakenly regarded as non-objective. However, we show that the obtained approximation is invariant under the superposed rigid body motions, and as a corollary, the objectivity of the continuum model is preserved. The key of this proof comes from the observation that since the numerical quadrature is used to compute the integrals, by storing the rotation field and its derivative at the Gauss points, the equivariant conditions can be relaxed only at these points. Several numerical examples are presented to confirm the theoretical results and demonstrate the performance of this algorithm.

Nonlinear Observer for 3D Rigid Body Motion Estimation Using Doppler Measurements

This work presents a nonlinear observer that estimates the translational and rotational motion of a rigid body based on measurements of configuration (pose), angular velocity, and radial velocity as well as modeled forces and torques. The radial velocity measurements are provided by a single direction Doppler sensor. Using a conveniently defined Lyapunov function, a nonlinear observer on the Special Euclidean Group (SE(3)) is derived and almost globally stability is guaranteed. The resulting estimation error is almost globally exponentially convergent for sufficiently rich motion. Numerical simulation results are presented that illustrate the performance of the proposed solution.

Optimal control on special Euclidean group via natural gradient algorithm

Considering the optimal control problem about the control system of the special Euclidean group whose output only depends on its input is meaningful in practical applications. The optimal control considered here is described as the output matrix is as close as possible to the target matrix by adjusting the system input. The geodesic distance is adopted as the measure of the difference between the output matrix and the target matrix, and the trajectory of the control input obtained in the process is achieved. Furthermore, some numerical simulations are shown to illustrate our outcomes based on the natural gradient descent algorithm for optimizing the control system of the special Euclidean group.

Strapdown Inertial Navigation Algorithms Based on Lie Group

This paper presents a framework for a strapdown Inertial Navigation System (INS) algorithm design by using Lie group and Lie algebra. The general way to solve Lie group differential equations is introduced. Investigations reveal that this general Lie group method provides a simpler unified way to solve differential equations involving direction cosine matrix, quaternion and dual quaternion, which are widely used in INS algorithm design. Furthermore, we also present a new INS algorithm based on the Special Euclidean group se(3) under the guidelines of Lie group method. Analyses show that se(3) algorithm has the same accuracy as a dual quaternion algorithm, this is also justified by numerical simulations. Though the se(3) algorithm has no improvements in accuracy, the general Lie group method used in the design process shows its brevity and uniformity.

Design and Implementation of a Low Cost Mini Quadrotor for Vision Based Maneuvers in GPS Denied Environments

This paper presents the design and implementation detail of an advanced mini quadrotor system, including the low cost commercial-off-the-shelf (COTS) electronics and advanced control algorithm. The proposed quadrotor has a gross takeoff weight of 758[Formula: see text]g and 360[Formula: see text]mm frame diagonal size. It is capable of semi-autonomous maneuver in GPS denied environments, solely relying on onboard sensors and computers. A globally defined quadrotor model is formularized, and a nonlinear velocity tracking controller is implemented on the special Euclidean group SE(3). An optical flow and ultrasonic-based onboard downward-facing camera is used as the primary sensor to provide velocity and altitude measurement feedback for the controller. The control and sensor fusion algorithm is developed under Arduino compatible open source electronics.

Global swarming while preserving connectivity via Lagrange–Poincarè equations

In this paper, we exploit symmetry properties of multi-agent robot systems to design control laws that preserve connectivity while swarming. We start by designing a connectivity control law for agents with configuration spaces R3 and SO(3) that is invariant under the action of the special Euclidean group SE(3) and the special orthogonal group SO(3), respectively. Therefore, the dynamics of such multi-agent systems is amenable to be reduced by these group actions. We then utilize the Lagrange–Poincaré equations that split the Euler–Lagrange equations for the multi-agent system into horizontal and vertical parts. The invariance of the connectivity controller implies that its control effort has zero vertical component. We then use the resulting vertical equations of motion to design a control law that asymptotically stabilizes the centroid and the orientation of the swarm at a desired pose.