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Journal of Bionic Engineering 2009年第4期6卷 350-356页

作者： Wilfried Konrad Christian Apeltauer Jrg Frauendiener Wilhelm Barthlott Anita Roth-Nebelsick Institute for Geosciences University of Tubingen D-72016 Tubingen Germany Department of Mathematics and Statistics University of Otago Dunedin 9054 New Zealand Centre of Mathematics for Applications University of Oslo NO-0317 Oslo Norway Nees Institute for Biodiversity of Plants University of Bonn D-53115 Bonn Germany State Museum of Natural History Stuttgart Rosenstein 1 D-70191 Stuttgart Germany

We describe a few mathematical tools which allow to investigate whether air-water interfaces exist(under prescribed conditions)and are mechanically stable and temporally *** terms of physics,air-water interfaces are governed by the Young-Laplace *** they are surfaces of constant mean curvature which represent solutions of a nonlinear elliptic partial differential *** explicit solutions of this equation can be obtained only in very special cases,it is -under moderately special circumstances-possible to establish the existence of a solution without actually solving the differential *** also derive criteria for mechanical stability and temporal persistence of an air *** we calculate the lifetime of a non-persistent air ***,we apply these tools to two examples which exhibit the symmetries of 2D *** examples can be viewed as abstractions of the biological model represented by the aquatic fern Salvinia.

关键词： interfaces air layers differential geometry stability persistence Salvinia

A unified dynamic model for locomotion and manipulation of a snake-like robot based on differential geometry

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Science China(Information Sciences) 2011年第2期54卷 318-333页

作者： WANG ZhiFeng1,3, MA ShuGen1,2, LI Bin1 & WANG YueChao1 1State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China 2Department of Robotics, Ritsumeikan University, Shiga-ken 525-8577, Japan 3Graduate University of Chinese Academy of Sciences, Beijing 100039, China State Key Laboratory of Robotics Shenyang Institute of Automation Chinese Academy of Sciences Shenyang China Graduate University of Chinese Academy of Sciences Beijing China Department of Robotics Ritsumeikan University Shiga-ken Japan

A snake-like robot, whose body is a seried-wound articulated mechanism, can move in various environments. In addition, when one end is fixed on a base, the robot can manipulate objects. A method of dynamic modeling for locomotion and manipulation of the snake-like robot is developed in order to unify the dynamic equations of two states. The transformation from locomotion to manipulation is a mechanism reconfiguration, that is, the robot in locomotion has not a fixed base, but it in manipulation has one. First, a virtual structure method unifies the two states in mechanism (e.g., an embedding in the configuration space); second, the product-of-exponentials formula describes the kinematics; third, the dynamics of locomotion and manipulation are established in a Riemannian manifold; finally, based on the analysis of the dynamic model, the dynamics of manipulation can be directly degenerated from those of locomotion, and this degeneration relation is proved through using the Gauss equations. In the differential geometry formulation, this method realizes the unification of the dynamics of locomotion and manipulation. According to a geometrical point of view, the unified dynamic model for locomotion and manipulation is considered as a submanifold problem endowed with geometric meaning. In addition, the unified model offers an insight into the dynamics of the snake-like robot beyond the dynamic model separately established for locomotion or manipulation.

关键词： snake-like robot dynamics unified model differential geometry locomotion manipulation

Coordinated standoff tracking of moving targets using differential geometry

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Journal of Zhejiang University-Science C(Computers and Electronics) 2014年第4期15卷 284-292页

作者： Zhi-qiang SONG Hua-xiong LI Chun-lin CHEN Xian-zhong ZHOU Feng XU Department of Control and System Engineering School of Management and Engineering Nanjing University Department of Information Suzhou Institute of Trade and Commerce No. 207 Research Institute China North Industries Group Corporation

This research is concerned with coordinated standoff tracking, and a guidance law against a moving target is proposed by using differential geometry. We first present the geometry between the unmanned aircraft(UA) and the target to obtain the convergent solution of standoff tracking when the speed ratio of the UA to the target is larger than one. Then, the convergent solution is used to guide the UA onto the standoff tracking geometry. We propose an improved guidance law by adding a derivative term to the relevant algorithm. To keep the phase angle difference of multiple UAs, we add a second derivative term to the relevant control law. Simulations are done to demonstrate the feasibility and performance of the proposed approach. The proposed algorithm can achieve coordinated control of multiple UAs with its simplicity and stability in terms of the standoff distance and phase angle difference.

关键词： Unmanned aircraft Standoff tracking differential geometry Coordinated control

Rotational Hypersurfaces with Constant Gauss-Kronecker Curvature

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Chinese Annals of Mathematics,Series B 2022年第3期43卷 343-358页

作者： Yuhang LIU Yunchu DAI Beijing International Center for Mathematical Research Peking UniversityBeijing 100871China Department of Mathematics Stanford UniversityCalifornia 94305USA

The authors study rotational hypersurfaces with constant Gauss-Kronecker curvature in R^(n).They solve the ODE associated with the generating curve of such hypersurface using integral expressions and obtain several geometric properties of such *** particular,they discover a class of non-compact rotational hypersurfaces with constant and negative Gauss-Kronecker curvature and finite volume,which can be seen as the higher-dimensional generalization of the pseudo-sphere.

关键词： differential geometry Gauss-Kronecker curvature Ordinary differential equation

Optimal State Estimation and Fault Diagnosis for a Class of Nonlinear Systems

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IEEE/CAA Journal of Automatica Sinica 2020年第2期7卷 517-526页

作者： Hamed Kazemi Alireza Yazdizadeh Department of Electrical Engineering Shahid Beheshti UniversityTehran ***Iran

This study proposes a scheme for state estimation and,consequently,fault diagnosis in nonlinear ***,an optimal nonlinear observer is designed for nonlinear systems subject to an actuator or plant *** utilizing Lyapunov's direct method,the observer is proved to be optimal with respect to a performance function,including the magnitude of the observer gain and the convergence *** observer gain is obtained by using approximation of Hamilton-Jacobi-Bellman(HJB)*** approximation is determined via an online trained neural network(NN).Next a class of affine nonlinear systems is considered which is subject to unknown disturbances in addition to fault *** this case,for each fault the original system is transformed to a new form in which the proposed optimal observer can be applied for state estimation and fault detection and isolation(FDI).Simulation results of a singlelink flexible joint robot(SLFJR)electric drive system show the effectiveness of the proposed methodology.

关键词： differential geometry fault detection and isolation(FDI) fault diagnosis neural network(NN) nonlinear observer and filter design optimal state estimation

Computing Open-Loop Optimal Control of the q-Profile in Ramp-Up Tokamak Plasmas Using the Minimal-Surface Theory

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Plasma Science and Technology 2013年第5期15卷 403-410页

作者：许超欧勇盛 Eugenio SCHUSTER 于欣 The State Key Laboratory of Industrial Control Technology Zhejiang University Shenzhen Institutes of Advanced Technology Chinese Academy of Sciences Department of Mechanical Engineering & Mechanics Lehigh University Ningbo Institute of Technology Zhejiang University

The q-profile control problem in the ramp-up phase of plasma discharges is consid- ered in this work. The magnetic diffusion partial differential equation （PDE） models the dynamics of the poloidal magnetic flux profile, which is used in this work to formulate a PDE-constrained op-timization problem under a quasi-static assumption. The minimum surface theory and constrained numeric optimization are then applied to achieve suboptimal solutions. Since the transient dy- namics is pre-given by the minimum surface theory, then this method can dramatically accelerate the solution process. In order to be robust under external uncertainties in real implementations, PID （proportional-integral-derivative） controllers are used to force the actuators to follow the computational input trajectories. It has the potential to implement in real-time for long time discharges by combining this method with the magnetic equilibrium update.

关键词： advanced plasma operations current profile dynamics optimal control theory,minimal surface equation differential geometry

EQUIVALENT NORMAL CURVATURE APPROACH MILLING MODEL OF MACHINING FREEFORM SURFACES

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Chinese Journal of Mechanical Engineering 2008年第3期21卷 52-57页

作者： YI Xianzhong MA Weiguo QIHaiying YAN Zesheng GAO Deli School of Mechanical Engineenng Yangtze University Jingzhou 434023 China Key Laboratory for Thermal Science and Power Engineering of Ministry of Education Tsinghua University Beijing 100084 China Tianjin Pipe（Group） Corporation Ltd. Tianjin 300301 China Key Laboratory for Petroleum Engineenng of Ministry of Education China University of Petroleum Beijing 102249 China

A new milling methodology with the equivalent normal curvature milling model machining freeform surfaces is proposed based on the normal curvature theorems on differential geometry. Moreover, a specialized whirlwind milling tool and a 5-axis CNC horizontal milling machine are introduced. This new milling model can efficiently enlarge the material removal volume at the tip of the whirlwind milling tool and improve the producing capacity. The machining strategy of this model is to regulate the orientation of the whirlwind milling tool relatively to the principal directions of the workpiece surface at the point of contact, so as to create a full match with collision avoidance between the workpiece surface and the symmetric rotational surface of the milling tool. The practical results show that this new milling model is an effective method in machining complex three- dimensional surfaces. This model has a good improvement on finishing machining time and scallop height in machining the freeform surfaces over other milling processes. Some actual examples for manufacturing the freeform surfaces with this new model are given.

关键词： Equivalent normal curvature milling model Whirlwind milling method Freeform surfaces 5-axis CNC machine differential geometry

Sampling Streaming Data Along Geodesic

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Chinese Journal of Electronics 2015年第2期24卷 251-257页

作者： LI Wenping YANG Jing ZHANG Jianpei College of Computer Science and Technology Harbin Engineering University College of Mathematics Physics and Information Engineering Jiaxing University

This paper proposes an approach to sample data stream based on differential geometry. Our aim is to take advantage of information of discarded data and support stream to generate different number of transactions during different periods. To this end, we establish a novel data stream model represented by a surface, within which time is quantified and probability, value and time,viewed as one united body, could be calculated simultaneously. We pro ject data stream onto a surface of the model and replace points which have the shortest geodesic distance with their mid-point. To the best of our knowledge,this is the first work on introducing differential geometry as a sampling trick. Experimental results show that our approach is effective.

关键词： Data stream Sampling Data stream model differential geometry Geodesic

Smooth shapes with spherical topology:Beyond traditional modeling,efficient deformation,and interaction

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Computational Visual Media 2017年第3期3卷 199-215页

作者： D.Schmitter P.García-Amorena M.Unser Biomedical Imaging Group cole Polytechnique Fédérale de Lausanne(EPFL)1015 LausanneSwitzerland

Existing shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but non-interpolating schemes such as subdivision or NURBS. Both polygon models and subdivision methods require a large number of parameters to model smooth *** need fewer parameters but have a complicated rational expression and non-uniform shifts in their formulation. We present a new method to construct deformable closed surfaces, which includes exact spheres, by combining the best of two worlds: a smooth, interpolating model with a continuously varying tangent plane and well-defined curvature at every point on the surface. Our formulation is considerably simpler than NURBS and requires fewer parameters than polygon meshes. We demonstrate the generality of our method with applications including intuitive user-interactive shape modeling,continuous surface deformation, shape morphing,reconstruction of shapes from parameterized point clouds, and fast iterative shape optimization for image segmentation. Comparisons with discrete methods and non-interpolating approaches highlight the advantages of our framework.

关键词： shape modeling spherical topology parametric surfaces splines differential geometry

Geometrical description of denormalized thermodynamic manifold

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Chinese Physics B 2009年第9期18卷 3790-3794页

作者：吴利萍孙华飞曹丽梅 Department of Mathematics Beijing Institute of Technology

In view of differential geometry, the state space of thermodynamic parameters is investigated. Here the geometrical structures of the denormalized thermodynamic manifold are considered. The relation of their geometric... 详细信息

关键词： thermodynamic differential geometry geometrical metric