The Kinematic Models of the SINS and Its Errors on the SE(3) Group in the Earth-Centered Inertial Coordinate System.

In this paper, the kinematic models of the Strapdown Inertial Navigation System (SINS) and its errors on the SE(3) group in the Earth-Centered Inertial frame (ECI) are established. On the one hand, with the ECI frame being regarded as the reference, based on the joint representation of attitude and velocity on the SE(3) group, the dynamic of the local geographic coordinate system (n-frame) and the body coordinate system (b-frame) evolve on the differentiable manifold, respectively, and the high-order expansion of the Baker-Campbell-Haussdorff equation compensates for the non-commutative motion errors stimulated by strong maneuverability. On the other hand, the kinematics of the left- and right-invariant errors of the n-frame and the b-frame on the SE(3) group are separately derived, where the errors of the b-frame completely depend on inertial sensor errors, while the errors of the n-frame rely on position errors and velocity errors. In this way, the errors brought by the inconsistency of the reference coordinate system are tackled, and a novel attitude error definition is introduced to separate and decouple the factors affecting the dynamic of the n-frame errors and the b-frame errors for better attitude estimation. Through a turntable experiment and a car-mounted field experiment, the effectiveness of the proposed kinematic models in estimating attitude has been verified, with a remarkable improvement in yaw angle accuracy in the case of large initial misalignment angles, and the models developed have better robustness compared to the traditional SE(3) group-based model.

An Improved Initial Alignment Method Based on SE2(3)/EKF for SINS/GNSS Integrated Navigation System with Large Misalignment Angles.

This paper proposes an improved initial alignment method for a strap-down inertial navigation system/global navigation satellite system (SINS/GNSS) integrated navigation system with large misalignment angles. Its methodology is based on the three-dimensional special Euclidean group and extended Kalman filter (SE2(3)/EKF) and aims to overcome the challenges of achieving fast alignment under large misalignment angles using traditional methods. To accurately characterize the state errors of attitude, velocity, and position, these elements are constructed as elements of a Lie group. The nonlinear error on the Lie group can then be well quantified. Additionally, a group vector mixed error model is developed, taking into account the zero bias errors of gyroscopes and accelerometers. Using this new error definition, a GNSS-assisted SINS dynamic initial alignment algorithm is derived, which is based on the invariance of velocity and position measurements. Simulation experiments demonstrate that the alignment method based on SE2(3)/EKF can achieve a higher accuracy in various scenarios with large misalignment angles, while the attitude error can be rapidly reduced to a lower level.

Dynamics of invariant solutions of the DNA model using Lie symmetry approach.

The utilization of the Lie group method serves to encapsulate a diverse array of wave structures. This method, established as a robust and reliable mathematical technique, is instrumental in deriving precise solutions for nonlinear partial differential equations (NPDEs) across a spectrum of domains. Its applications span various scientific disciplines, including mathematical physics, nonlinear dynamics, oceanography, engineering sciences, and several others. This research focuses specifically on the crucial molecule DNA and its interaction with an external microwave field. The Lie group method is employed to establish a five-dimensional symmetry algebra as the foundational element. Subsequently, similarity reductions are led by a system of one-dimensional subalgebras. Several invariant solutions as well as a spectrum of wave solutions is obtained by solving the resulting reduced ordinary differential equations (ODEs). These solutions govern the longitudinal displacement in DNA, shedding light on the characteristics of DNA as a significant real-world challenge. The interactions of DNA with an external microwave field manifest in various forms, including rational, exponential, trigonometric, hyperbolic, polynomial, and other functions. Mathematica simulations of these solutions confirm that longitudinal displacements in DNA can be expressed as periodic waves, optical dark solitons, singular solutions, exponential forms, and rational forms. This study is novel as it marks the first application of the Lie group method to explore the interaction of DNA molecules.

Influence of magnetic field-dependent viscosity on Casson-based nanofluid boundary layers: A comprehensive analysis using Lie group and spectral quasi-linearization method.

This study examines the effects of magnetic-field-dependent (MFD) viscosity on the boundary layer flow of a non-Newtonian sodium alginate-based Fe3O4 nanofluid over an impermeable stretching surface. The non-Newtonian Casson and homogeneous nanofluid models are utilized to derive the governing flow and heat transfer equations. Applying Lie group transformations to dimensional partial differential equations yields nondimensional ordinary differential equations, which are then numerically solved using the spectral quasi-linearization technique. The analysis primarily focuses on the impacts of the MFD viscosity parameter, nanoparticle volume fraction of Fe3O4, and magnetic parameters on the flow and heat transfer characteristics. The local skin friction and heat transfer rate behaviors influenced by viscosity changes due to the magnetic field are discussed. It is found that MFD viscosity significantly impacts flow and thermal energies, enhancing skin friction coefficients and reducing Nusselt numbers in the boundary layer region.

Simultaneous Measurements of Noncommuting Observables: Positive Transformations and Instrumental Lie Groups.

We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function, defined relative to the invariant measure of the instrumental Lie group. This diffusion can be analyzed using Wiener path integration, stochastic differential equations, or a Fokker-Planck-Kolmogorov equation. This way of considering instrument evolution we call the Instrument Manifold Program. We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover that we call the universal instrumental Lie group, which is independent not just of states, but also of Hilbert space. The universal instrument is generically infinite dimensional, in which case the instrument's evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, and it can be analyzed within the differential geometry of the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, position and momentum, and the three components of angular momentum. As these measurements are performed continuously, they converge to strong simultaneous measurements. For a single observable, this results in the standard decay of coherence between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups.

Simulation and design of shaped pulses beyond the piecewise-constant approximation.

Response functions of resonant circuits create ringing artefacts if their input changes rapidly. When physical limits of electromagnetic spectroscopies are explored, this creates two types of problems. Firstly, simulation: the system must be propagated accurately through every response transient, this may be computationally expensive. Secondly, optimal control: circuit response must be taken into account; it may be advantageous to design pulses that are resilient to such distortions. At the root of both problems is the popular piecewise-constant approximation for control sequences in the rotating frame; in magnetic resonance it has persisted since the earliest days and has become entrenched in the commercially available hardware. In this paper, we report an implementation and benchmarks of recent Lie-group methods that can efficiently simulate and optimise smooth control sequences.

Symplectic Foliation Structures of Non-Equilibrium Thermodynamics as Dissipation Model: Application to Metriplectic Nonlinear Lindblad Quantum Master Equation.

The idea of a canonical ensemble from Gibbs has been extended by Jean-Marie Souriau for a symplectic manifold where a Lie group has a Hamiltonian action. A novel symplectic thermodynamics and information geometry known as "Lie group thermodynamics" then explains foliation structures of thermodynamics. We then infer a geometric structure for heat equation from this archetypal model, and we have discovered a pure geometric structure of entropy, which characterizes entropy in coadjoint representation as an invariant Casimir function. The coadjoint orbits form the level sets on the entropy. By using the KKS 2-form in the affine case via Souriau's cocycle, the method also enables the Fisher metric from information geometry for Lie groups. The fact that transverse dynamics to these symplectic leaves is dissipative, whilst dynamics along these symplectic leaves characterize non-dissipative phenomenon, can be used to interpret this Lie group thermodynamics within the context of an open system out of thermodynamics equilibrium. In the following section, we will discuss the dissipative symplectic model of heat and information through the Poisson transverse structure to the symplectic leaf of coadjoint orbits, which is based on the metriplectic bracket, which guarantees conservation of energy and non-decrease of entropy. Baptiste Coquinot recently developed a new foundation theory for dissipative brackets by taking a broad perspective from non-equilibrium thermodynamics. He did this by first considering more natural variables for building the bracket used in metriplectic flow and then by presenting a methodical approach to the development of the theory. By deriving a generic dissipative bracket from fundamental thermodynamic first principles, Baptiste Coquinot demonstrates that brackets for the dissipative part are entirely natural, just as Poisson brackets for the non-dissipative part are canonical for Hamiltonian dynamics. We shall investigate how the theory of dissipative brackets introduced by Paul Dirac for limited Hamiltonian systems relates to transverse structure. We shall investigate an alternative method to the metriplectic method based on Michel Saint Germain's PhD research on the transverse Poisson structure. We will examine an alternative method to the metriplectic method based on the transverse Poisson structure, which Michel Saint-Germain studied for his PhD and was motivated by the key works of Fokko du Cloux. In continuation of Saint-Germain's works, Hervé Sabourin highlights the, for transverse Poisson structures, polynomial nature to nilpotent adjoint orbits and demonstrated that the Casimir functions of the transverse Poisson structure that result from restriction to the Lie-Poisson structure transverse slice are Casimir functions independent of the transverse Poisson structure. He also demonstrated that, on the transverse slice, two polynomial Poisson structures to the symplectic leaf appear that have Casimir functions. The dissipative equation introduced by Lindblad, from the Hamiltonian Liouville equation operating on the quantum density matrix, will be applied to illustrate these previous models. For the Lindblad operator, the dissipative component has been described as the relative entropy gradient and the maximum entropy principle by Öttinger. It has been observed then that the Lindblad equation is a linear approximation of the metriplectic equation.

Research on the Necessity of Lie Group Strapdown Inertial Integrated Navigation Error Model Based on Euler Angle.

In response to the lack of specific demonstration and analysis of the research on the necessity of the Lie group strapdown inertial integrated navigation error model based on the Euler angle, two common integrated navigation systems, strapdown inertial navigation system/global navigation satellite system (SINS/GNSS) and strapdown inertial navigation system/doppler velocity log (SINS/DVL), are used as subjects, and the piecewise constant system (PWCS) matrix, based on the Lie group error model, is established. From three aspects of variance estimation, the observability and performance of the system with large misalignment angles for low, medium, and high accuracy levels, traditional error model, Lie group left error model, and right error model are compared. The necessity of research on Lie group error model is analyzed quantitatively and qualitatively. The experimental results show that Lie group error model has better stability of variance estimation, estimation accuracy, and observability than traditional error model, as well as higher practical value.

Equivariant Oka theory: survey of recent progress.

We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group G acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle E of homogeneous spaces for a group bundle G , all over a reduced Stein space X with compatible actions of a reductive complex group on E, G , and X. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov's Oka principle based on a notion of a G-manifold being G-Oka.

A Novel Distribution for Representation of 6D Pose Uncertainty.

The 6D Pose estimation is a crux in many applications, such as visual perception, autonomous navigation, and spacecraft motion. For robotic grasping, the cluttered and self-occlusion scenarios bring new challenges to the this field. Currently, society uses CNNs to solve this problem. The CNN models will suffer high uncertainty caused by the environmental factors and the object itself. These models usually maintain a Gaussian distribution, which is not suitable for the underlying manifold structure of the pose. Many works decouple rotation from the translation and quantify rotational uncertainty. Only a few works pay attention to the uncertainty of the 6D pose. This work proposes a distribution that can capture the uncertainty of the 6D pose parameterized by the dual quaternions, meanwhile, the proposed distribution takes the periodic nature of the underlying structure into account. The presented results include the normalization constant computation and parameter estimation techniques of the distribution. This work shows the benefits of the proposed distribution, which provides a more realistic explanation for the uncertainty in the 6D pose and eliminates the drawback inherited from the planar rigid motion.

Lie-Group Theoretic Approach to Shape-Sensing Using FBG-Sensorized Needles Including Double-Layer Tissue and S-Shape Insertions.

Flexible bevel-tipped needles are often used for needle insertion in minimally-invasive surgical techniques due to their ability to be steered in cluttered environments. Shapesensing enables physicians to determine the location of needles intra-operatively without requiring radiation of the patient, enabling accurate needle placement. In this paper, we validate a theoretical method for flexible needle shape-sensing that allows for complex curvatures, extending upon a previous sensor-based model. This model combines curvature measurements from fiber Bragg grating (FBG) sensors and the mechanics of an inextensible elastic rod to determine and predict the 3D needle shape during insertion. We evaluate the model's shape sensing capabilities in C- and S-shape insertions in single-layer isotropic tissue, and C-shape insertions in two-layer isotropic tissue. Experiments on a four-active area, FBG-sensorized needle were performed in varying tissue stiffnesses and insertion scenarios under stereo vision to provide the 3D ground truth needle shape. The results validate a viable 3D needle shape-sensing model accounting for complex curvatures in flexible needles with mean needle shape sensing root-mean-square errors of 0.160 ± 0.055 mm over 650 needle insertions.

Lie symmetries of Benjamin-Ono equation.

Lie Symmetry analysis is often used to exploit the conservative laws of nature and solve or at least reduce the order of differential equation. One dimension internal waves are best described by Benjamin-Ono equation which is a nonlinear partial integro-differential equation. Present article focuses on the Lie symmetry analysis of this equation because of its importance. Lie symmetry analysis of this equation has been done but there are still some gaps and errors in the recent work. We claim that the symmetry algebra is of five dimensional. We reduce the model and solve it. We give its solution and analyze them graphically.

A Recurrent Neural-Network-Based Real-Time Dynamic Model for Soft Continuum Manipulators.

This paper introduces and validates a real-time dynamic predictive model based on a neural network approach for soft continuum manipulators. The presented model provides a real-time prediction framework using neural-network-based strategies and continuum mechanics principles. A time-space integration scheme is employed to discretize the continuous dynamics and decouple the dynamic equations for translation and rotation for each node of a soft continuum manipulator. Then the resulting architecture is used to develop distributed prediction algorithms using recurrent neural networks. The proposed RNN-based parallel predictive scheme does not rely on computationally intensive algorithms; therefore, it is useful in real-time applications. Furthermore, simulations are shown to illustrate the approach performance on soft continuum elastica, and the approach is also validated through an experiment on a magnetically-actuated soft continuum manipulator. The results demonstrate that the presented model can outperform classical modeling approaches such as the Cosserat rod model while also shows possibilities for being used in practice.

Rate of Entropy Production in Stochastic Mechanical Systems.

Entropy production in stochastic mechanical systems is examined here with strict bounds on its rate. Stochastic mechanical systems include pure diffusions in Euclidean space or on Lie groups, as well as systems evolving on phase space for which the fluctuation-dissipation theorem applies, i.e., return-to-equilibrium processes. Two separate ways for ensembles of such mechanical systems forced by noise to reach equilibrium are examined here. First, a restorative potential and damping can be applied, leading to a classical return-to-equilibrium process wherein energy taken out by damping can balance the energy going in from the noise. Second, the process evolves on a compact configuration space (such as random walks on spheres, torsion angles in chain molecules, and rotational Brownian motion) lead to long-time solutions that are constant over the configuration space, regardless of whether or not damping and random forcing balance. This is a kind of potential-free equilibrium distribution resulting from topological constraints. Inertial and noninertial (kinematic) systems are considered. These systems can consist of unconstrained particles or more complex systems with constraints, such as rigid-bodies or linkages. These more complicated systems evolve on Lie groups and model phenomena such as rotational Brownian motion and nonholonomic robotic systems. In all cases, it is shown that the rate of entropy production is closely related to the appropriate concept of Fisher information matrix of the probability density defined by the Fokker-Planck equation. Classical results from information theory are then repurposed to provide computable bounds on the rate of entropy production in stochastic mechanical systems.

Estimating Lower Limb Kinematics Using a Lie Group Constrained Extended Kalman Filter with a Reduced Wearable IMU Count and Distance Measurements.

Tracking the kinematics of human movement usually requires the use of equipment that constrains the user within a room (e.g., optical motion capture systems), or requires the use of a conspicuous body-worn measurement system (e.g., inertial measurement units (IMUs) attached to each body segment). This paper presents a novel Lie group constrained extended Kalman filter to estimate lower limb kinematics using IMU and inter-IMU distance measurements in a reduced sensor count configuration. The algorithm iterates through the prediction (kinematic equations), measurement (pelvis height assumption/inter-IMU distance measurements, zero velocity update for feet/ankles, flat-floor assumption for feet/ankles, and covariance limiter), and constraint update (formulation of hinged knee joints and ball-and-socket hip joints). The knee and hip joint angle root-mean-square errors in the sagittal plane for straight walking were 7.6±2.6∘ and 6.6±2.7∘, respectively, while the correlation coefficients were 0.95±0.03 and 0.87±0.16, respectively. Furthermore, experiments using simulated inter-IMU distance measurements show that performance improved substantially for dynamic movements, even at large noise levels (σ=0.2 m). However, further validation is recommended with actual distance measurement sensors, such as ultra-wideband ranging sensors.

Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics.

In this paper, we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau's symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equivariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such as momentum maps, Casimir functions, coadjoint orbits, and Lie-Poisson brackets with cocycles, as unifying structures appearing in various applications of this framework to information geometry and machine learning. For instance, we discuss the expression of the Fisher metric in presence of equivariance and we exploit the property of the entropy of the Souriau model as a Casimir function to apply a geometric model for energy preserving entropy production. We illustrate this framework with several examples including multivariate Gaussian probability densities, and the Bogoliubov-Kubo-Mori metric as a quantum version of the Fisher metric for quantum information on coadjoint orbits. We exploit this geometric setting and Lie group equivariance to present symplectic and multisymplectic variational Lie group integration schemes for some of the equations associated with Souriau symplectic and polysymplectic models, such as the Lie-Poisson equation with cocycle.

Multi-Stage Meta-Learning for Few-Shot with Lie Group Network Constraint.

Deep learning has achieved many successes in different fields but can sometimes encounter an overfitting problem when there are insufficient amounts of labeled samples. In solving the problem of learning with limited training data, meta-learning is proposed to remember some common knowledge by leveraging a large number of similar few-shot tasks and learning how to adapt a base-learner to a new task for which only a few labeled samples are available. Current meta-learning approaches typically uses Shallow Neural Networks (SNNs) to avoid overfitting, thus wasting much information in adapting to a new task. Moreover, the Euclidean space-based gradient descent in existing meta-learning approaches always lead to an inaccurate update of meta-learners, which poses a challenge to meta-learning models in extracting features from samples and updating network parameters. In this paper, we propose a novel meta-learning model called Multi-Stage Meta-Learning (MSML) to post the bottleneck during the adapting process. The proposed method constrains a network to Stiefel manifold so that a meta-learner could perform a more stable gradient descent in limited steps so that the adapting process can be accelerated. An experiment on the mini-ImageNet demonstrates that the proposed method reached a better accuracy under 5-way 1-shot and 5-way 5-shot conditions.

Lie Group Statistics and Lie Group Machine Learning Based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation.

In 1969, Jean-Marie Souriau introduced a "Lie Groups Thermodynamics" in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau's model considers the statistical mechanics of dynamic systems in their "space of evolution" associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (e.g., Galileo's group in classical physics). Souriau Lie Group Thermodynamics was also addressed 30 years after Souriau by R.F. Streater in the framework of Quantum Physics by Information Geometry for some Lie algebras, but only in the case of null cohomology. Souriau method could then be applied on Lie groups to define a covariant maximum entropy density by Kirillov representation theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau's moment map. For this case, the coadjoint action on moment map is equivariant. For non-null cohomology, we give the case of Lie group SE(2). Finally, we will propose a new geometric definition of Entropy that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping.

Self-similarity in fate and transport of contaminants in groundwater.

Similarity within non-linear geophysical processes under different space and time has been investigated in literature to understand and model the sophisticated nature of these processes. This study deals with the self-similarity of the governing processes of fate and transport of contaminants in groundwater, which include advection, dispersion, sorption, and degradation, either by chemical reaction or microbiological interaction. As such, self-similarity conditions of three-dimensional advective-dispersive-reactive transport (ADR) equation with various initial and boundary conditions are obtained by employing one-parameter Lie group of point scaling transformations. The strength of the proposed scaling approach lies in its ability to handle initial and boundary conditions of the process together with the governing equation. Then, utilizing reported conditions in Borden Field experiment domain, numerical simulations are performed to demonstrate examples of self-similarity when first-order degradation rate constant is zero and nonzero under three sorption conditions. The results of the numerical examples showed the effectiveness of the derived self-similarity conditions and the strength of the proposed approach in handling initial and boundary conditions of the governing transport process. It is possible to obtain both smaller and larger size self-similar (scaled-up or scaled-down) domains, which result in faster and slower transport processes, respectively. The obtained self-similarity criteria can offer further understanding of ADR transport phenomena and can provide spatial, temporal, and economical flexibility in constructing scaled models of field conditions governed by three-dimensional ADR transport process.

Representation and Characterization of Nonstationary Processes by Dilation Operators and Induced Shape Space Manifolds.

We proposed in this work the introduction of a new vision of stochastic processes through geometry induced by dilation. The dilation matrices of a given process are obtained by a composition of rotation matrices built in with respect to partial correlation coefficients. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, only one dilation matrix is obtained and it corresponds therefore to Naimark dilation. When the process is nonstationary, a set of dilation matrices is obtained. They correspond to Kolmogorov decomposition. In this work, the nonstationary class of periodically correlated processes was of interest. The underlying periodicity of correlation coefficients is then transmitted to the set of dilation matrices. Because this set lives on the Lie group of rotation matrices, we can see them as points of a closed curve on the Lie group. Geometrical aspects can then be investigated through the shape of the obtained curves, and to give a complete insight into the space of curves, a metric and the derived geodesic equations are provided. The general results are adapted to the more specific case where the base manifold is the Lie group of rotation matrices, and because the metric in the space of curve naturally extends to the space of shapes; this enables a comparison between curves' shapes and allows then the classification of random processes' measures.