Singular extremals on Lie groups

We investigate the space of singular curves associated to a distribution ofk-planes, or, what is the same thing, a nonlinear deterministic control system linear in controls. A singular curve is one for which the associated linearized system is not controllable. If a quadratic positive-definite cost function is introduced, then the corresponding optimal control problem is known as the sub-Riemannian geodesic problem. The original motivation for our work was the question “Are all sub-Riemannian minimizers smooth?” which is equivalent to the question “Are singular minimizers necessarily smooth?” Our main result concerns the singular curves for a class of homogeneous systems whose state spaces are compact Lie groups. We prove that for this class each singular curve lies in a lower-dimensional subgroup within which it is regular and we use this result to prove that all sub-Riemannian minimizers are smooth. A central ingredient of our proof is a symplectic-geometric characterization of singular curves formulated by Hsu. We extend this characterization to nonsmooth singular curves. We find that the symplectic point of view clarifies the situation and simplifies calculations.     

Controllability on classical Lie groups

We consider the problem of accessibility and controllability of certain bilinear systems. These evolve on Lie groups whose Lie algebras are the normal real forms of complex simple Lie algebras. Previous results by other authors were obtained under the assumption that the controlled vector field is strongly regular. Our paper is aimed at weakening this requirement, and involves relating the root structure of elements in a Lie algebra as above to the nodal connection graphs obtained from their standard matrix representations. This is in turn related to a standard irreducibility assumption on the uncontrolled vector field. The abstract results on generation of Lie algebras are of some independent interest. The work of this author was supported in part by a Fulbright grant, while she was visiting the Arizona State University, and by the Centro de Matemática da Universidade de Coimbra/INIC. The work of this author was partially supported by AFOSR Contract No. 85-0224A.     

Hamiltonian point of view of non-Euclidean geometry and elliptic functions

This paper offers a new way of looking at the classical geometries and the theory of elliptic functions through Hamiltonian systems on Lie groups. In particular, the paper shows that: (i) the classical models of non-Euclidean geometries are canonically induced by bi-invariant sub-Riemannian metrices on Lie groups which act by left-actions on the underlying space; (ii) there is a class of canonical variational problems on Lie groups G whose projections on homogeneous spaces G/K generalize Euler's elasticae and include all curves of constant curvature and all -functions of Weierstrass; (iii) complex Lie groups unify non-Euclidean geometries and complex elasticae offer a distinctive look at the elliptic functions.     

An estimation problem in compact Lie groups

In this paper an estimation problem is formulated and solved where the process evolves in a compact, semisimple Lie group. The problem is to find explicitly the conditional probability density of a random initial condition of a Brownian motion in the Lie group given a vector-valued function of the group-valued Brownian motion at some fixed positive time. This function is constant on orbits of the Lie group formed by conjugation. The conditional probability density function has a simple, explicit form using some orbit properties of the Lie group. Many physical problems can be modelled as evolving on compact, semisimple Lie groups, e.g. the motion of a rigid body with a fixed point.     

Constructing algebraic groups from their Lie algebras

A connected algebraic group in characteristic 0$0$ is uniquely determined by its Lie algebra. In this paper algorithms are given for constructing an algebraic group in characteristic 0$0$, given its Lie algebra. In particular, this leads to an algorithm for finding a maximal reductive subgroup and the unipotent radical of an algebraic group.     

Uniform controllable sets of left-invariant vector fields on compact Lie groups

A set of vector fields on a differentiable manifold M is said to be uniformly completely controllable (u.c.c.) if there exists a nonnegative integer N such that evert pair (p, q) of point of M can be joined by a trajectory, or positive orbit, of which involves at most N switches.

In this article we show that if M is a Lie group G and a set of left-invariant vector fields on G, N must be greater than or equal to dim(G)-1. We also construct sets of vector fields which are uniformly completely controllable in dim(G)-1 switches when G is the Lie group of any compact real form of g and g runs over all classical simple Lie algebras over .     

Integral control on Lie groups

In this paper, we extend the popular integral control technique to systems evolving on Lie groups. More explicitly, we provide an alternative definition of “integral action” for proportional(–derivative)-controlled systems whose configuration evolves on a nonlinear space, where configuration errors cannot be simply added up to compute a definite integral. We then prove that the proposed integral control allows to cancel the drift induced by a constant bias in both first order (velocity) and second order (torque) control inputs for fully actuated systems evolving on abstract Lie groups. We illustrate the approach by 3-dimensional motion control applications.     

You-Kaveh图像去噪模型扩散系数的改进

本文在You-Kaveh模型的基础上,提出了一个新的扩散系数,得到了一个去噪效果更好的方程,新方程不但能够去除高斯噪声,而且能够很好地去除椒盐噪声。同时,改进了模型中拉普拉斯算子的离散形式,使其包含更多的图像信息,能够更准确地判断图像的特征。采用本文方法处理后的图像,避免了用二阶偏微分方程处理图像常出现的"阶梯"效应;同时,和同类的四阶偏微分方程去噪模型相比,本文方法的处理结果不会出现"斑"点,因此视觉效果更加理想。最后通过实验证明了该方法的有效性。     

Uniform finite generation of compact Lie groups

Consider a compact connected Lie group G and the corresponding Lie algebra . Let {X₁,…,X_m} be a set of generators for the Lie algebra . We prove that G is uniformly finitely generated by {X₁,…,X_m}. This means that every element KG can be expressed as K=e^Xt₁e^Xt₂···e^Xt_l, where the indeterminates X are in the set {X₁,…,X_m}, , and the number l is uniformly bounded. This extends a previous result by F. Lowenthal in that we do not require the connected one dimensional Lie subgroups corresponding to the X_i, i=1,…,m, to be compact. We link the results to the existence of universal logic gates in quantum computing and discuss the impact on bang bang control algorithms for quantum mechanical systems.     

Automatic estimation of asymmetry for gradient-based alignment of noisy images on Lie group

Many parametric image alignment approaches assume equality of the images to register up to motion compensation. In presence of noise this assumption does not hold. In particular, for gradient-based approaches, which rely on the optimization of an error functional with gradient descent methods, the performances depend on the amount of noise in each image. We propose in this paper to use the Asymmetric Composition on Lie groups (ACL) formulation of the alignment problem to improve the robustness in presence of asymmetric levels of noise. The ACL formulation, generalizing state-of-the-art gradient-based image alignment, introduces a parameter to weight the influence of the images during the optimization. Three new methods are presented to estimate this asymmetry parameter: one supervised (MVACL) and two fully automatic (AACL and GACL). Theoretical results and experimental validation show how the new algorithms improve robustness in presence of noise. Finally, we illustrate the interest of the new approaches for object tracking under low-light conditions.     

Differential invariants of a Lie group action: Syzygies on a generating set

Given a group action, known by its infinitesimal generators, we exhibit a complete set of syzygies on a generating set of differential invariants. For that we elaborate on the reinterpretation of Cartan’s moving frame by Fels and Olver [Fels, M., Olver, P.J., 1999. Moving coframes. II. Regularization and theoretical foundations. Acta Appl. Math. 55 (2), 127–208]. This provides constructive tools for exploring algebras of differential invariants.     

Optimization on Lie manifolds and pattern recognition

Several pattern recognition problems can be reduced in a natural way to the problem of optimizing a nonlinear function over a Lie manifold. However, optimization on Lie manifolds involves, in general, a large number of nonlinear equality constraints and is hence one of the hardest optimization problems. We show that exploiting the special structure of Lie manifolds allows one to devise a method for optimizing on Lie manifolds in a computationally efficient manner. The new method relies on the differential geometry of Lie manifolds and the underlying connections between Lie groups and their associated Lie algebras. We describe an application of the new Lie group method to the problem of diagnosing malignancy in the cytological extracts of breast tumors. The diagnosis method that we present has a mean sensitivity of 98.086% and a predictive index of 0.0602, making it the most accurate and reliable diagnostic method reported thus far.     

Finite‐Time Optimal Tracking Control for Dynamic Systems on Lie Groups

This paper investigates the problem of finite‐time optimal tracking control for dynamic systems on Lie groups for the situation when the tracking time and/or the cost functions need to be considered. The specific results are illustrated on SE(3) (the specific Euclidean groups of rigid body motions). The tracking time is given according to task requirements in advance. By using Pontryagin's maximum principle (PMP) on Lie groups and the backstepping method, a finite‐time optimal tracking control law is designed to track a desired reference trajectory at the given time. Simultaneously, the corresponding cost functions are guaranteed to be optimal. Compared with existing results of optimal control on Lie groups, it is noteworthy that we consider the finite‐time tracking control for dynamic systems rather than kinematic systems. Furthermore, the obtained optimal control law is described by explicit formulations, which is significant for practical applications.     

Application of Lie Algebras to Visual Servoing

A novel approach to visual servoing is presented, which takes advantage of the structure of the Lie algebra of affine transformations. The aim of this project is to use feedback from a visual sensor to guide a robot arm to a target position. The target position is learned using the principle of teaching by showing in which the supervisor places the robot in the correct target position and the system captures the necessary information to be able to return to that position. The sensor is placed in the end effector of the robot, the camera-in-hand approach, and thus provides direct feedback of the robot motion relative to the target scene via observed transformations of the scene. These scene transformations are obtained by measuring the affine deformations of a target planar contour (under the weak perspective assumption), captured by use of an active contour, or snake. Deformations of the snake are constrained using the Lie groups of affine and projective transformations. Properties of the Lie algebra of affine transformations are exploited to provide a novel method for integrating observed deformations of the target contour. These can be compensated with appropriate robot motion using a non-linear control structure. The local differential representation of contour deformations is extended to allow accurate integration of an extended series of small perturbations. This differs from existing approaches by virtue of the properties of the Lie algebra representation which implicitly embeds knowledge of the three-dimensional world within a two-dimensional image-based system. These techniques have been implemented using a video camera to control a 5 DoF robot arm. Experiments with this implementation are presented, together with a discussion of the results.     

Discrete‐time invariant extended Kalman filter on matrix Lie groups

In this article, we derive symmetry preserving discrete‐time invariant extended Kalman filters (IEKF) on matrix Lie groups. These Kalman filters offer an advantage over classical extended Kalman filters as the error dynamics for such filters are independent of the group configuration which, in turn, provides a uniform estimate of the region of convergence. In contrast to existing techniques in the literature, the discrete‐time IEKF is derived using minimal tools from differential geometry which simplifies the derivation and the representation of IEKF. In our technique, the linearized error dynamics is defined on the Lie algebra directly using variational approaches, unlike conventional approaches where the error dynamics is translated to an Euclidean space using the logarithm map before its linearization. Moreover, the Kalman gains and its associated difference Riccati equations are derived in operator spaces by setting a discrete‐time optimal control problem and solving it with discrete‐time Pontryagin's maximum principle. The proposed discrete‐time IEKF is implemented for the attitude dynamics of the rigid body, which is a benchmark problem in control. It is observed from the numerical studies that the IEKF is computationally less intensive and provides better performance than the classical extended Kalman filter.     

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

This 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.     

On a new method for finding generalized equivalence transformations for differential equations involving arbitrary functions

A new efficient method for finding generalized equivalence transformations for a class of differential equation systems via its related extended classical symmetries is presented. This technique can be further adapted to find the equivalence transformations for the mathematical model. It applies to classes of differential systems whose arbitrary functions involve all equations’ independent variables. As a consequence, any symbolic manipulation program designed to find classical Lie symmetries can also be used to determine generalized equivalence transformations and equivalence transformations, respectively, without any modification of the program. The method has been implemented as the maple routine gendefget and is based on the maple package desolv(by Carminati and Vu). The nonlinear stationary heat conduction parameter identification problem is considered as an example.