Academic literature on the topic 'Riemannian geometric framework'

In this article we survey some recent developments in optimal robot design, and collect some of the differential geometric approaches into a general mathematical framework for robot design. The geometric framework permits a set of coordinate-free definitions of robot performance that can be optimized for designing both open- and closed-chain robotic mechanisms. In particular, workspace volume is precisely defined by regarding the rigid body motions as a Riemannian manifold, and various features of actuators, as well as inertial characteristics of the robot, can be captured by the suitable selection of a Riemannian metric in configuration space. The integral functional of harmonic mapping theory also provides a simple and elegant global description of dexterity.

In this article we survey some recent developments in optimal robot design, and collect some of the differential geometric approaches into a general mathematical framework for robot design. The geometric framework permits a set of coordinate-free definitions of robot performance that can be optimized for designing both open- and closed-chain robotic mechanisms. In particular, workspace volume is precisely defined by regarding the rigid body motions as a Riemannian manifold, and various features of actuators, as well as inertial characteristics of the robot, can be captured by the suitable selection of a Riemannian metric in configuration space. The integral functional of harmonic mapping theory also provides a simple and elegant global description of dexterity.

The aim of this paper is to provide the geometrical structure of a gravitational field that includes the addition of dark matter in the framework of a Riemannian and a Riemann–Sasaki spacetime. By means of the classical Riemannian geometric methods we arrive at modified geodesic equations, tidal forces, and Einstein and Raychaudhuri equations to account for extra dark gravity. We further examine an application of this approach in cosmology. Moreover, a possible extension of this model on the tangent bundle is studied in order to examine the behavior of dark matter in a unified geometric model of gravity with more degrees of freedom. Particular emphasis shall be laid on the problem of the geodesic motion under the influence of dark matter.

Geometric Quantum Mechanics is a mathematical framework that shows how quantum theory may be expressed in terms of Hamiltonian phase-space dynamics. The states are points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is specified by the Schrödinger equation in this framework. The quest to express the uncertainty principle in geometrical language has recently become the focus of significant research in geometric quantum mechanics. One has demonstrated that the Robertson–Schrödinger uncertainty principle, which is a stronger version of the uncertainty relation, can be defined in terms of symplectic form and Riemannian metric. On the basis of this formulation, we study the dynamical behavior of the uncertainty relation for the spin 1 system in this work. We show that under Hamiltonian flow, the Robertson–Schrödinger uncertainty principles are not invariant. This is because, unlike the symplectic area, the Riemannian metric is not invariant under Hamiltonian flow throughout the evolution process.

Symmetric positive defined (SPD) matrix has attracted increasing research focus in image/video analysis, which merits in capturing the Riemannian geometry in its structured 2D feature representation. However, computation in the vector space on SPD matrices cannot capture the geometric properties, which corrupts the classification performance. To this end, Riemannian based deep network has become a promising solution for SPD matrix classification, because of its excellence in performing non-linear learning over SPD matrix. Besides, Riemannian metric learning typically adopts a kNN classifier that cannot be extended to large-scale datasets, which limits its application in many time-efficient scenarios. In this paper, we propose a Bag-of-Matrix-Summarization (BoMS) method to be combined with Riemannian network, which handles the above issues towards highly efficient and scalable SPD feature representation. Our key innovation lies in the idea of summarizing data in a Riemannian geometric space instead of the vector space. First, the whole training set is compressed with a small number of matrix features to ensure high scalability. Second, given such a compressed set, a constant-length vector representation is extracted by efficiently measuring the distribution variations between the summarized data and the latent feature of the Riemannian network. Finally, the proposed BoMS descriptor is integrated into the Riemannian network, upon which the whole framework is end-to-end trained via matrix back-propagation. Experiments on four different classification tasks demonstrate the superior performance of the proposed method over the state-of-the-art methods.

This paper develops information geometric representations for nonlinear filters in continuous time. The posterior distribution associated with an abstract nonlinear filtering problem is shown to satisfy a stochastic differential equation on a Hilbert information manifold. This supports the Fisher metric as a pseudo-Riemannian metric. Flows of Shannon information are shown to be connected with the quadratic variation of the process of posterior distributions in this metric. Apart from providing a suitable setting in which to study such information-theoretic properties, the Hilbert manifold has an appropriate topology from the point of view of multi-objective filter approximations. A general class of finite-dimensional exponential filters is shown to fit within this framework, and an intrinsic evolution equation, involving Amari's -1-covariant derivative, is developed for such filters. Three example systems, one of infinite dimension, are developed in detail.

A geometric framework for metrics of maximal acceleration which is applicable to large proper accelerations is discussed, including a theory of connections associated with the geometry of maximal acceleration. In such a framework, it is shown that the uniform bound on the proper maximal acceleration implies a uniform bound for certain bilinear combinations of the Riemannian curvature components in the domain of the spacetime where curvature is finite.

The structural network of the brain, or structural connectome, can be represented by fiber bundles generated by a variety of tractography methods. While such methods give qualitative insights into brain structure, there is controversy over whether they can provide quantitative information, especially at the population level. In order to enable population-level statistical analysis of the structural connectome, we propose representing a connectome as a Riemannian metric, which is a point on an infinite-dimensional manifold. We equip this manifold with the Ebin metric, a natural metric structure for this space, to get a Riemannian manifold along with its associated geometric properties. We then use this Riemannian framework to apply object-oriented statistical analysis to define an atlas as the Fréchet mean of a population of Riemannian metrics. This formulation ties into the existing framework for diffeomorphic construction of image atlases, allowing us to construct a multimodal atlas by simultaneously integrating complementary white matter structure details from DWMRI and cortical details from T1-weighted MRI. We illustrate our framework with 2D data examples of connectome registration and atlas formation. Finally, we build an example 3D multimodal atlas using T1 images and connectomes derived from diffusion tensors estimated from a subset of subjects from the Human Connectome Project.

We present an embedding of the Tsallis entropy into the three-dimensional Heisenberg group, in order to understand the meaning of generalized independence as encoded in the Tsallis entropy composition property. We infer that the Tsallis entropy composition induces fractal properties on the underlying Euclidean space. Using a theorem of Milnor/Wolf/Tits/Gromov, we justify why the underlying configuration/phase space of systems described by the Tsallis entropy has polynomial growth for both discrete and Riemannian cases. We provide a geometric framework that elucidates Abe's formula for the Tsallis entropy, in terms the Pansu derivative of a map between sub-Riemannian spaces.

We consider the problem of designing trajectory tracking feedback control laws for La- grangian mechanical systems in a Riemannian geometric framework. Classical nonlinear control techniques that rely on Euclidean parameterizations of nonlinear confguration manifolds, severely restrict the region of operation of the system due to singularities of local coordinate charts. The primary focus of our study is to develop a generic, con- structive and intrinsic (coordinate independent) procedure for global control design such that the closed loop operational envelop is signifcantly enhanced. An important class of systems where the proposed control design techniques have been applied are underactu- ated unmanned aerial vehicles (UAV). Such systems are physically capable of executing aggressive and global (unrestricted) maneuvers as a result of enhanced mechanical de- sign and actuation technology. However, developing autonomous controllers such that the closed loop system can execute such maneuvers is indeed a formidable problem. Part 1: Global control on Riemannian manifolds (chapter 3 and 4): In the frst part of the thesis, we consider simple Lagrangian mechanical control sys- tems evolving on compact Riemannian manifolds, whose coordinate independent Euler- Lagrange equations of motion are established through the Levi-Civita connection corre- sponding to the kinetic-energy metric tensor. When the system is fully actuated), using the Riemannian connection structure, we develop a generic and constructive trajectory tracking feedback control law based on integrator back-stepping where the confguration error is the gradient of the squared geodesic distance between the confguration of the system and the reference trajectory, and the velocity error is the di erence between the velocity of the system and the parallel translation of the velocity of the reference trajec- tory, along a minimal geodesic connecting the confguration of the system and the point on the reference trajectory. The control law is appended with a feed-forward term which is the covariant derivative of the distance-gradient and the parallel-translation term. We demonstrate that this control law achieves asymptotically stable tracking when the confguration of the system is within injectivity radius of the point on the reference tra- jectory. The primary reason is that the control law does not encounter the cut locus, where it is no longer well defned, and around which it is no longer smooth. We then use our study of the compact Riemannian cut locus (which is the primary topological obstruction in global control design) in chapter 2 where we establish certain structural and dynamical properties, and thereby show that the control gains can be chosen large enough such that the confguration of the system does not intersect the cut locus of the point on the reference trajectory for all positive time, provided it starts away from the cut locus (arbitrarily close to it) initially. We thereby extend the region of stability of the above control law to an arbitrarily large domain of the tangent bundle. We then append the control law with a dynamic feedback in order to achieve globally exponentially stable tracking. We now restrict our attention to compact Lie groups which are naturally equipped with a bi-invariant metric structure, which enables us in constructing an elegant and computationally simple version of the generic control law on Riemannian manifolds. Exploiting the isometry of the group action, we convert the global tracking problem to a local tracking problem within the injectivity radius, and a global stabilization problem. Unlike the generic Riemannian case, we show that the components of the control law can be easily computed using only the Lie group structure (i.e. the group actions, exponential and logarithm map). We fnally study the problem of under-actuated differentially constrained mechanical systems where the velocity is constrained to a regular distribution. The equations of motion are established through a metric-compatible connection called the 'constrained connection', which is not necessarily torsion free if the differential constraint is non- integrable (i.e. non-holonomic). We extend the control design techniques previously established, to achieve global output tracking of such systems. Part 2: Application to under-actuated aerial vehicles on SE(3) (chapter 5 and 6): In the second part of the thesis, we apply the geometric control design techniques to two under-actuated aerial vehicles; multi-rotors and thrust vectored vertical take-o and landing (VTOL) aircraft, whose confguration evolves on the Lie group SE(3). In our study of multi-rotors, we frst design a globally-exponentially stable controller for tracking the position and relative heading angle of a quadrotor, which is considered as a rigid body subjected to a force along the body z axis and three torques about the body axes. The equations of motion can be written as a cascade of two subsystems, one which is a fully actuated rotational subsystem on SO(3), the output of which is the input to a translational subsystem on R3. We use the previously described control design on R3 using the bi-invariant metric (from the cannonical Ad-invariant inner product on the Lie algebra) and cascade this control with a saturated thrust feedback control on R3 in order to achieve global asymptotically stable tracking at an exponential rate. This design is then augmented with a fault tolerant strategy, which ensures that the controller continues to track the position of the center of mass of the quadrotor in spite of a rotor failure. This is achieved by relinquishing control of the heading angle, and designing a reduced-attitude control law which tracks the orientation of the thrust axis on S2. We use the global output tracking control design discussed in part 1 to achieve this. The reduced attitude control law is then cascaded with the saturated thrust feedback control on R3 as in the previous case. We then study the bi-spinner problem which is a rigid body with only two fxed co-axial rotors. Such a vehicle is severely under-actuated and therefore global tracking control is indeed a formidable problem. We propose a multi-scale geometric controller under the assumption that the angular velocity of the bispinner about the thrust-axis is signifcantly higher than the other two angular velocity components. We design a control law which globally tracks position trajectories with only two functioning rotors. In our study of thrust vectored VTOL UAVs, we consider an axis-symmetric aerial vehicle subjected to a terminally applied vectored thrust and torque about the axis of symmetry. Typical examples of such vehicles are thrust-propelled rockets, submersible torpedos, tail-sitter drones etc. The diffculty in control design for such a problem is that we can no longer write the equations of motion as a cascade system as we did in the case of multi rotors. The reason is that the control inputs which produce torques for the rotations in SO(3) also generate forces which result in translations in R3. Further, this coupling results in an unstable inverse input-output system (non-minimumphase), which renders the control design problem formidable. In order to resolve this difficulty, we frst impose a non-integrable differential constraint on the system, such that the constrained system admits a differentially at output i.e. The Huygens center-of-oscillation. We reformulate the translational dynamics with respect to this point to convert the tracking control problem into one which involves a cascade system as in the previous case, and apply the reduced attitude and saturated thrust feedback law to achieve global tracking of the center of mass, when the differential constraint is satisfed. This control law is augmented with another component which ensures that the differential constraint is asymptotically stabilized which ensures global asymptotic tracking for all initial conditions in the tangent bundle. Another important factor that the control design adresses is the constraint on the thrust of the vehicle to be strictly bounded above zero. We provide simulation results which illustrate the effectiveness, robustness and global tracking performance of the proposed controllers.

To be able to describe the other fundamental interactions, beyond quantum electrodynamics (QED), weak and strong interactions, it is necessary to generalize the concept of gauge symmetry to non-Abelian groups. Therefore, in this chapter, a quantum field theory (QFT)-invariant under local, that is, space-time-dependent, transformations of matrix representations of a general compact Lie groups are constructed. Inspired by the Abelian example, the geometric concept of parallel transport is introduced, a concept discussed more extensively later in the framework of Riemannian manifolds. All the required mathematical quantities for gauge theories then appear naturally. Gauge theories are quantized in the temporal gauge. The equivalence with covariant gauges is then established. Some formal properties of the quantized theory, like the Becchi–Rouet–Stora–Tyutin (BRST) symmetry, are derived. Feynman rules of perturbation theory are derived, the regularization of perturbation theory is discussed, a somewhat non-trivial problem. Some general properties of the non-Abelian Higgs mechanism are described.

This chapter has two purposes; to describe a few elements of differential geometry that are required in different places in this work, and to provide, for completeness, a short introduction to general relativity (GR) and the problem of its quantization. A few concepts related to reparametrization (more accurately, diffeomorphism) of Riemannian manifolds, like parallel transport, affine connection, or curvature, are recalled. To define fermions on Riemannian manifolds, additional mathematical objects are required, the vielbein and the spin connection. Einstein–Hilbert's action for classical gravity GR is defined and the field equations derived. Some formal aspects of the quantization of GR, following the lines of the quantization of non-Abelian gauge theories, are described. Because GR is not renormalizable in four dimensions (even in its extended forms like supersymmetric gravity), at present time, a reasonable assumption is that GR is the low-energy, large-distance remnant of a more complete theory that probably no longer has the form of a quantum field theory (QFT) (strings, non-commutative geometry?). In the terminology of critical phenomena, GR belongs to the class of irrelevant interactions: due to the presence of the massless graviton, GR can be compared with an interacting theory of Goldstone modes at low temperature, in the ordered phase. The scale of this new physics seems to be of the order of 10¹⁹ GeV (Planck's mass). Still, because the equations of GR follow from varying Einstein–Hilbert action, some regularized form is expected to be relevant to quantum gravity. In the framework of GR, the presence of a cosmological constant, generated by the quantum vacuum energy, is expected, but it is extremely difficult to account for its extremely small, measured value.

Abstract We present a Riemannian geometric framework for variational approaches to geometric design. Optimal surface design is regarded as a special case of the more general problem of finding a minimum distortion mapping between Riemannian manifolds. This geometric approach emphasizes the coordinate-invariant aspects of the problem, and engineering constraints are naturally embedded by selecting a suitable metric in the physical space. In this context we also present an engineering application of the theory of harmonic maps.