The Experts below are selected from a list of 89226 Experts worldwide ranked by ideXlab platform
Jeanfrancois Le Gall - One of the best experts on this subject based on the ideXlab platform.
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Geodesics in large planar maps and in the brownian map
Acta Mathematica, 2010Co-Authors: Jeanfrancois Le GallAbstract:We study Geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe Geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the root by more than one geodesic. The set S is dense in the Brownian map and homeomorphic to a non-compact real tree. Furthermore, for every x in S, the number of distinct Geodesics from x to the root is equal to the number of connected components of S\{x}. In particular, points of the Brownian map can be connected to the root by at most three distinct Geodesics. Our results have applications to the behavior of Geodesics in large planar maps.
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Geodesics in large planar maps and in the brownian map
arXiv: Probability, 2008Co-Authors: Jeanfrancois Le GallAbstract:We study Geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe Geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the root by more than one geodesic. The set S is dense in the Brownian map and homeomorphic to a non-compact real tree. Furthermore, for every x in S, the number of distinct Geodesics from x to the root is equal to the number of connected components of the complement of {x} in S. In particular, points of the Brownian map can be connected to the root by at most three distinct Geodesics. Our results have applications to the behavior of Geodesics in large planar maps.
Bahram Mashhoon - One of the best experts on this subject based on the ideXlab platform.
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the generalized jacobi equation
Classical and Quantum Gravity, 2002Co-Authors: Carmen Chicone, Bahram MashhoonAbstract:The Jacobi equation in pseudo-Riemannian geometry determines the linearized geodesic flow. The linearization ignores the relative velocity of the Geodesics. The generalized Jacobi equation takes the relative velocity into account; that is, when the Geodesics are neighbouring but their relative velocity is arbitrary the corresponding geodesic deviation equation is the generalized Jacobi equation. The Hamiltonian structure of this nonlinear equation is analysed in this paper. The tidal accelerations for test particles in the field of a plane gravitational wave and the exterior field of a rotating mass are investigated. In the latter case, the existence of an attractor of uniform relative radial motion with speed 2−1/2c ≈ 0.7c is pointed out. The astrophysical implication of this result for the terminal speed of a relativistic jet is briefly explored.
Simona-luiza Druta - One of the best experts on this subject based on the ideXlab platform.
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Geodesicity and isoclinity properties for the tangent bundle of the Heisenberg manifold with Sasaki metric
Turkish Journal of Mathematics, 2012Co-Authors: Simona-luiza DrutaAbstract:We prove that the horizontal and vertical distributions of the tangent bundle with the Sasaki metric are isocline, the distributions given by the kernels of the horizontal and vertical lifts of the contact form w on the Heisenberg manifold (H3,g) to (TH3,gS) are not totally geodesic, and the distributions FH=L(E1H,E2H) and FV=L(E1V,E2V) are totally geodesic, but they are not isocline. We obtain that the horizontal and natural lifts of the curves from the Heisenberg manifold (H3,g), are Geodesics on the tangent bundle endowed with the Sasaki metric (TH3,gs), if and only if the curves considered on the base manifold are Geodesics. Then, we get two particular examples of Geodesics on (TH3,gs), which are not horizontal or natural lifts of Geodesics from the base manifold (H3,g).
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Geodesicity and Isoclinity Properties for the Tangent Bundle of the Heisenberg Manifold with Sasaki Metric
arXiv: Differential Geometry, 2010Co-Authors: Simona-luiza DrutaAbstract:We prove that the horizontal and vertical distributions of the tangent bundle with the Sasaki metric are isocline, the distributions given by the kernels of the horizontal and vertical lifts of the contact form $\omega$ from the Heisenberg manifold $(H_3,g)$ to $(TH_3,g^S)$ are not totally geodesic, and the distributions $F^H=L(E_1^H,E_2^H)$ and $F^V=L(E_1^V,E_2^V)$ are totally geodesic, but they are not isocline. We obtain that the horizontal and natural lifts of the curves from the Heisenberg manifold $(H_3,g)$, are Geodesics in the tangent bundle endowed with the Sasaki metric $(TH_3,g^s)$, if and only if the curves considered on the base manifold are Geodesics. Then, we get two particular examples of Geodesics from $(TH_3,g^s)$, which are not horizontal or natural lifts of Geodesics from the base manifold $(H_3,g)$.
Liandong Zhang - One of the best experts on this subject based on the ideXlab platform.
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Kuka youBot arm shortest path planning based on Geodesics
2013 IEEE International Conference on Robotics and Biomimetics (ROBIO), 2013Co-Authors: Liandong Zhang, Changjiu ZhouAbstract:The state-of-the-art Kuka youBot is an open-source robot platform. In order to improve youBot arm manipulation performance, a novel robotic trajectory planning method based on Geodesics is used for Kuka youBot arm shortest path trajectory planning in this paper. Geodesic is the necessary condition of the shortest length between two points on the Riemannian surface in which the covariant derivative of the geodesic's tangent vector is zero. The Riemannian metric is constructed according to the distance metric by arc length of the youBot arm trajectory to achieve shortest path. Once the Riemannian metric is obtained, the corresponding Riemannian surface is solely determined. Then the geodesic equations on this surface can be determined and calculated. For the given initial conditions of the trajectory, the geodesic equations can be solved and the results are the optimal trajectory of the youBot arm in the joint space for the given metric. The planned trajectories in the joint space can also be mapped into the workspace. A simple trajectory planning example on Kuka youBot arm from camera pose ready point to object grasping point is given to demonstrate the feasibility of the proposed approach.
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Optimal energy gait planning for humanoid robot using Geodesics
2010 IEEE Conference on Robotics Automation and Mechatronics, 2010Co-Authors: Liandong Zhang, Changjiu Zhou, Peijie Zhang, Zhiwei Song, Yue Pik KongAbstract:A novel gait planning method using Geodesics for humanoid robot is given in this paper. Both center of gravity (COG) and the exact Single Support Phase (SSP) are studied in our energy optimal gait planning based on Geodesics. The kinetic energy of a 2-dimensional inverted pendulum is obtained at first. We regard the kinetic energy as the Riemannian metric and the geodesic on this metric is studied and this is the shortest line between two points on the Riemannian surface. This geodesic is the optimal kinetic energy gait for the COG because the kinetic energy along geodesic is invariant according to the geometric property of Geodesics and the walking is stable and no impact. Then the walking in Single Support Phase is studied and the energy optimal gait for the swing leg is obtained using our Geodesics method. Finally, experiments using traditional joint angles interpolating method and using our Geodesics optimization method are carried out respectively and the corresponding currents of the joint motors are recorded. With the currents comparing results, the feasibility of this new gait planning method is verified.
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Robot Optimal Trajectory Planning Based on Geodesics
2007 IEEE International Conference on Control and Automation, 2007Co-Authors: Liandong Zhang, Changjiu ZhouAbstract:Geometric characteristics of Geodesics in the Riemannian surface are used to make robotic optimal trajectory planning in this paper. Distance length and kinetic energy are regarded as Riemannian metrics respectively, and the Riemannian surfaces are determined by the corresponding metrics, and they represent the robotics kinematics and dynamics respectively. The geodesies on the Riemannian surface are calculated and are regarded as the optimal trajectory. Geodesic is the necessary condition of the shortest length between two points on the Riemannian surface and the covariant derivative of the geodesic's tangent vector is zero. When to implement optimal trajectory planning with arc length as the Riemannian metric, geodesic makes the shortest length between two points. The end-effector's velocity is invariant along the geodesic and the acceleration is zero. So the motion is very smooth. When system's kinetic energy as the Riemannian metric, the geodesic between two points on the kinetic surface makes the kinetic energy remain invariant. Computer calculation and simulation verify that the method based on geodesic is good at trajectory planning especially when the trajectory is linear or certain index should be minimized.
Carmen Chicone - One of the best experts on this subject based on the ideXlab platform.
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the generalized jacobi equation
Classical and Quantum Gravity, 2002Co-Authors: Carmen Chicone, Bahram MashhoonAbstract:The Jacobi equation in pseudo-Riemannian geometry determines the linearized geodesic flow. The linearization ignores the relative velocity of the Geodesics. The generalized Jacobi equation takes the relative velocity into account; that is, when the Geodesics are neighbouring but their relative velocity is arbitrary the corresponding geodesic deviation equation is the generalized Jacobi equation. The Hamiltonian structure of this nonlinear equation is analysed in this paper. The tidal accelerations for test particles in the field of a plane gravitational wave and the exterior field of a rotating mass are investigated. In the latter case, the existence of an attractor of uniform relative radial motion with speed 2−1/2c ≈ 0.7c is pointed out. The astrophysical implication of this result for the terminal speed of a relativistic jet is briefly explored.
