近地小行星仅测角相对导航可观性判据

针对近地小行星仅测角相对导航系统的非线性可观测性分析问题,本文基于非线性系统局部弱可观性理论给出了新的可观测性判据.首先,通过分析非线性系统李导数的共有属性,给出了李导数梯度空间的共有子空间.进一步,利用该共有子空间和非线性系统局部弱可观性定义给出了新的可观性判据.经理论证明,该判据适用于非线性系统任意阶李导数,并且给出的可观性结论可与高阶可观性矩阵秩判据的结论等价.与传统的秩判据相比,该判据具有无需进行高阶李导数运算,无需进行高维矩阵运算的优点.因此该判据适合在轨计算,具有一定的工程价值.最后仿真结果验证了可观性判据的有效性.     

Canonical Decomposition of Steerable Functions

This paper describes a general mathematical formulation for the problem of constructing steerable functions. The formulation is based on Lie group theory and is thus applicable to transformations which are Lie groups, such as, rotation, translation, scaling, and affine transformation. For one-parameter and Abelian multi-parameter Lie transformation groups, a canonical decomposition of all possible steerable functions, derived using the Jordan decomposition of matrices, is developed. It is shown that any steerable function under Lie transformation groups can be described using this decomposition. Finally, a catalog of steerable functions for several common multi-parameter image transformation groups is also provided.     

Lie Algebra and System Identification Techniques for 3D Rigid Motion Estimation and Monocular Tracking

This paper addresses the parameters’ estimation of 2D and 3D transformations. For the estimation we present a method based on system identification theory, we named it the “A-method”. The transformations are considered as elements of the Lie group GL(n) or one of its subgroups. We represent the transformations in terms of their Lie Algebra elements. The Lie algebra approach assures to follow the shortest path or geodesic in the involved Lie group. To prove the potencial of our method, two experiments are presented. The first one is a monocular estimation of 3D rigid motion of an object in the visual space. With this aim, the six parameters of the rigid motion are estimated based on measurements of the six parameters of the affine transformation in the image. Secondly, we present the estimation of the affine or projective transformations involved in monocular region tracking. Jaime Ortegón-Aguilar received his degree in computer sciences at the Universidad Autonoma de Yucatan in Merida, Mexico in 2000. He earned his M.Sc. degree at the Cinvestav in Guadalajara, Mexico in 2002. He received his PhD degree from Cinvestav in 2006. His research interests include image processing, computer vision, robotics and applications of geometric algebra. Eduardo Jose Bayro-Corrochano gained his Ph.D. in Cognitive Computer Science in 1993 from the University of Wales at Cardiff. From 1995 to 1999 he has been Researcher and Lecturer at the Institute for Computer Science, Christian Albrechts University, Kiel, Germany, working on applications of geometric Clifford algebra to cognitive systems. At present is a full professor at CINVESTAV Unidad Guadalajara, México, Department of Electrical Engineering and Computer Science. His current research interest focuses on geometric methods for artificial perception and action systems. It includes geometric neural networks, visually guided robotics, humanoids, color image processing, Lie bivector algebras for early vision and robot maneuvering. He developed the quaternion wavelet transform for quaternion multi-resolution analysis using the phase concept. He is associate editor of Robotics and Journal of Advanced Robotic Systems and member of the editorial board of Journal of Pattern Recognition, Journal of Mathematical Imaging and Vision, Iberoamerican Journal of Computer and Systems and Journal of Theoretical and Numerical Approximation. He is editor and author of the following books: Geometric Computing for Perception Action Systems, E. Bayro-Corrochano, Springer Verlag, 2001; Geometric Algebra with Applications in Science and Engineering, E. Bayro-Corrochano and G. Sobczyk (Eds.), Birkhauser 2001; Handbook of Geometric Computing for Pattern Recognition, Computer Vision, Neurocomputing and Robotics, E. Bayro-Corrochano, Springer Verlag, 2005. He has published over 120 refereed journal, book chapters and conference papers. He is fellow of the IAPR society.     

博弈控制论简述

博弈控制论是近年来出现的博弈与控制的交叉学科,虽然它融合了博弈和控制双方的工具和方法,但无论从研究对象,研究方法与现有的结论看,它都不完全从属于经典博弈或经典控制的范畴,因此,是一个具有自身特色的新学术生长点.本文的目的是对这个新方向做一个简述介绍,内容包括:i)基于博弈的控制;ii)基于势博弈的优化;iii)博弈中的状态空间方法;iv)国内几个研究博弈控制论的团队.最后,对博弈控制论的未来做一展望.     

覆盖学习的道路优化算法

在之前的研究中,已经针对李群多连通空间上具有不同类别特征的研究对象,提出了多连通覆盖学习算法,成功地将覆盖学习应用到多连通李群空间.主要针对多连通覆盖学习算法中连通道路的交叉问题,考虑在李群空间上寻找一条测地曲线,使得映射后不同单连通空间上的道路的关联度最小化、同一单连通空间上的道路的关联度最大化,从而实现连通空间上类别判别性能的优化.首先回顾李群连通性质的相关知识;然后,简单介绍了多连通覆盖学习算法,并针对问题给出新的优化算法;最终,通过与经典覆盖学习算法、李群均值算法以及原始算法的比较实验,证明了该优化算法具有更好的分类性能.     

Nonlinear systems possessing linear symmetry

This paper tackles linear symmetries of control systems. Precisely, the symmetry of affine nonlinear systems under the action of a sub‐group of general linear group GL(n,?). First of all, the structure of state space (briefly, ss) symmetry group and its Lie algebra for a given system is investigated. Secondly, the structure of systems, which are ss‐symmetric under rotations, is revealed. Thirdly, a complete classification of ss‐symmetric planar systems is presented. It is shown that for planar systems there are only four classes of systems which are ss‐symmetric with respect to four linear groups. Fourthly, a set of algebraic equations are presented, whose solutions provide the Lie algebra of the largest connected ss‐symmetry group. Finally, some controllability properties of systems with ss‐symmetry group are studied. As an auxiliary tool for computation, the concept and some properties of semi‐tensor product of matrices are included. Copyright © 2006 John Wiley & Sons, Ltd.     

非线性倒立摆起摆和稳定控制研究

该文深入研究了直线倒立摆的摆起和稳定控制。通过两种方法对倒立摆非线性模型进行线性化得到线性化模型。第一种方法是利用微分几何李导数理论，对坐标和输入输出进行变换，然后在局部进行线性化得到一个线性化模型；另一种方法直接在平衡点附近进行线性化得到线性化模型。最后用MATLAB仿真和实物控制分别证明了能量反馈理论和稳定控制理论具有良好的控制效果，且通过Lie理论对系统进行坐标变换得到线性化模型比略去高阶项得到的线性化模型具有更好的鲁棒性和控制效果。     

积分水平集的多目标规划

给出了一种基于积分水平集的求解多目标规划的概念性算法。用数论中确定性的一致分布的数值积分来逼近水平值和水平集,从而得到了实现性算法和算法终止准则。最后给出数值例子证明了算法的有效性。     

Waveform diversity via mutual information

A novel criterion for waveform diversity in radar systems is presented that is based on the information theoretic concept of Shannon mutual information (MI). In general, waveform diversity refers to adaptively changing a transmitted waveform based on the target and interference environment. MI is a measure of the information in a random variable or vector about another random variable or vector. It is shown herein that an MI framework provides the basis for a general and powerful criterion for evaluation of candidate waveforms, as well as design of waveforms. The criterion inherently includes radar system and scenario statistical information. In addition, the criterion reduces to a simple analytical function of a set of scalar parameters for the case of Gaussian-distributed proper random processes. It is shown that those scalar parameters are also the canonical correlations of the random variables in the MI expression that involves the received radar signal at two distinct time instants. This property suggests the possibility of new insights into waveform diversity using MI as a criterion. The concept is presented assuming a SAR system configuration, but can be extended to other radar modes and functions, as well as to other sensing applications.     

条件事件代数在专家系统中的应用研究

条件事件代数是在确保规则概率与条件概率相容的前提下,把布尔代数上的逻辑运算推广到条件事件（规则）集合中的逻辑代数系统,本文介绍了条件事件代数的基本原理和性质,并利用条件事件代数解决了专家系统的规则循环问题,克服了传统谓词逻辑在推理过程中的局限性。     

多智能体系统可控性的图论刻画

研究领航者-跟随者结构多智能体系统的可控性问题.利用松弛的等价分化刻画了系统可控性与信息拓扑结构之间的关系,为系统可控性提供了基于图论的判别方法.基于置换群理论将对称的概念推广到多领航者系统,并证明了3种图论工具（对称性、等价分化和松弛的等价分化）之间的包含关系.仿真结果验证了所提出方法的有效性.     

Screw and Lie group theory in multibody kinematics

After three decades of computational multibody system (MBS) dynamics, current research is centered at the development of compact and user-friendly yet computationally efficient formulations for the analysis of complex MBS. The key to this is a holistic geometric approach to the kinematics modeling observing that the general motion of rigid bodies and the relative motion due to technical joints are screw motions. Moreover, screw theory provides the geometric setting and Lie group theory the analytic foundation for an intuitive and compact MBS modeling. The inherent frame invariance of this modeling approach gives rise to very efficient recursive \(O ( n ) \) algorithms, for which the so-called “spatial operator algebra” is one example, and allows for use of readily available geometric data. In this paper, three variants for describing the configuration of tree-topology MBS in terms of relative coordinates, that is, joint variables, are presented: the standard formulation using body-fixed joint frames, a formulation without joint frames, and a formulation without either joint or body-fixed reference frames. This allows for describing the MBS kinematics without introducing joint reference frames and therewith rendering the use of restrictive modeling convention, such as Denavit–Hartenberg parameters, redundant. Four different definitions of twists are recalled, and the corresponding recursive expressions are derived. The corresponding Jacobians and their factorization are derived. The aim of this paper is to motivate the use of Lie group modeling and to provide a review of different formulations for the kinematics of tree-topology MBS in terms of relative (joint) coordinates from the unifying perspective of screw and Lie group theory.     

活动标架的构造及其在模式识别中的应用研究

基于Mark Fels和Peter J.Olver的活动标架理论,给出了用经典算法和改进的递归算法来构造活动标架和微分不变量的代数构造算法,并以一个李变换群为例演示了两种方法的构造过程。结果证明递归构造方法与经典的Maurer- Cartan方法相比较,不仅能够系统地应用于任意的变换群作用,也不要求一个slice的存在,且对于多参数的变换群来说,其递归构造方式使得相应的活动标架和微分不变量的构造过程更便捷,也容易实现。重要的是,相应的Maurer-Cartan形也被一步步地构造获得。所获得结果不仅是新的,且为微分不变量在签名曲线中的应用研究提供了基础理论支撑。     

A Geometric Theory for Analysis and Synthesis of Sub-6 DoF Parallel Manipulators

Mechanism synthesis is mostly dependent on the designer's experience and intuition and is difficult to automate. This paper aims to develop a rigorous and precise geometric theory for analysis and synthesis of sub-6 DoF (or lower mobility) parallel manipulators. Using Lie subgroups and submanifolds of the special Euclidean group SE(3), we first develop a unified framework for modelling commonly used primitive joints and task spaces. We provide a mathematically rigorous definition of the notion of motion type using conjugacy classes. Then, we introduce a new structure for subchains of parallel manipulators using the product of two subgroups of SE(3) and discuss its realization in terms of the primitive joints. We propose the notion of quotient manipulators that substantially enriches the topologies of serial manipulators. Finally, we present a general procedure for specifying the subchain structures given the desired motion type of a parallel manipulator. The parallel mechanism synthesis problem is thus solved using the realization techniques developed for serial manipulators. Generality of the theory is demonstrated by systematically generating a large class of feasible topologies for (parallel or serial) mechanisms with a desired motion type of either a Lie subgroup or a submanifold.     

Kinematics and dynamics of a rigid body in nonholonomic coordinates

In this paper we consider dynamic equations of nonholomic systes in terms of pseudo-coordinates which appear in the general theory of rigid body. Especially, the rotator kinematics and dynamics in nonholonomic coordinates is presented. Besides, using the appropriate methods of the differential geometry and theory of Lie groups and Lie algebras a simplification of motion equations in the tangent space is obtained.     

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.     

一种新的李群分类器在手写体数字中的应用

李群是变换空间的一种基本表示理论。目前针对李群数据所设计的分类器较少,对多分类的效果也不是很好。以手写体数字的应用为背景,引入了支持向量机分类算法来处理李群数据。由于李群数据具有矩阵表现的形式,设计了一种矩阵高斯核函数,使得支持向量机能够处理矩阵数据。仿真结果表明,支持向量机方法在李群数据上具有很好的性能。     

基于李代数的变形目标跟踪

动态几何变形是图像跟踪技术面临的突出难题之一. 本文提出基于李代数的变形目标跟踪方法, 用Gabor特征表征目标, 以仿射李群建立目标几何变形, 利用李代数和李群之间的指数映射将参数的最优化求解从欧氏空间转至光滑流形, 实现了对变形目标的稳定跟踪.从物理层面分析了目标跟踪过程中的参数几何变换的实质, 从理论上对在光滑流形上进行迭代求解的优点进行了详细分析, 并对其收敛性做出了证明.图像序列跟踪测试的对比实验表明, 本文方法较现有基于欧氏空间的算法在收敛速度、跟踪稳定性和精确性方面有显著提高.     

Screw and Lie group theory in multibody dynamics

Screw and Lie group theory allows for user-friendly modeling of multibody systems (MBS), and at the same they give rise to computationally efficient recursive algorithms. The inherent frame invariance of such formulations allows to use arbitrary reference frames within the kinematics modeling (rather than obeying modeling conventions such as the Denavit–Hartenberg convention) and to avoid introduction of joint frames. The computational efficiency is owed to a representation of twists, accelerations, and wrenches that minimizes the computational effort. This can be directly carried over to dynamics formulations. In this paper, recursive \(O ( n ) \) Newton–Euler algorithms are derived for the four most frequently used representations of twists, and their specific features are discussed. These formulations are related to the corresponding algorithms that were presented in the literature. Two forms of MBS motion equations are derived in closed form using the Lie group formulation: the so-called Euler–Jourdain or “projection” equations, of which Kane’s equations are a special case, and the Lagrange equations. The recursive kinematics formulations are readily extended to higher orders in order to compute derivatives of the motions equations. To this end, recursive formulations for the acceleration and jerk are derived. It is briefly discussed how this can be employed for derivation of the linearized motion equations and their time derivatives. The geometric modeling allows for direct application of Lie group integration methods, which is briefly discussed.     

A Group-Theoretic Construction with Spatiotemporal Wavelets for the Analysis of Rotational Motion

This paper presents a group-theoretic approach for the analysis of rotational motion in image sequences. This method relies on Lie algebras, Lie groups and Lie group representations to provide not only the continuous wavelets but also the related tools of harmonic analysis. This approach can be referred to research works presented in J.S. Byrnes et al. (Wavelets and their Applications, Kluwer Academic Publishers, 1994) who strongly influenced this topic. For the purpose of modeling motion transformations, this paper introduces the concepts of Lie algebras and Lie groups as the actual mathematical foundations of all the observable kinematics embedded in spatio-temporal signals. Rotational motion analysis focuses on the estimation of angular velocity and angular accelerations embedded in image sequences. Rotational motion is usually carried on a trajectory, the complete problem at hand consists in estimating not only the angular velocity and its temporal derivatives but also the position, the translational velocity and its temporal derivatives along the carrier trajectory. The paper starts with the usual affine and Galilei groups and proceeds by successive extensions and sections to the rotational group. The theory of group representations is central to provide families of continuous wavelets, special functions, PDE's, ODE's and integral transforms as new mathematical tools of motion analysis in image sequence to perform optimal and selective detection, estimations, tracking, and reconstructions. This paper defines rotational wavelets and proposes a structured approach to perform estimation and tracking in image sequences which fits to Kalman filters. Simulations on real digital image sequences are also presented with tracking and estimation.