Be´zier Curves on Riemannian Manifolds and Lie Groups with Kinematics Applications

Park, F. C.; Ravani, Bahram

doi:10.1115/1.2826114

Cited by 154 publications

(74 citation statements)

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“…This idea was extended by Ge and Ravani [5] and Park and Ravani [6] to spatial motions. The focus in these papers is on the generalization of the notion of interpolation from the Euclidean space to a curved space.…”

Section: Introductionmentioning

confidence: 99%

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An SVD-based projection method for interpolation on SE(3)

Belta

Kumar

2002

IEEE Trans. Robot. Automat.

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This paper develops a method for generating smooth trajectories for a moving rigid body with specified boundary conditions. Our method involves two key steps: 1) the generation of optimal trajectories in GA + (n), a subgroup of the affine group in R n and 2) the projection of the trajectories onto SE(3), the Lie group of rigid body displacements. The overall procedure is invariant with respect to both the local coordinates on the manifold and the choice of the inertial frame. The benefits of the method are threefold. First, it is possible to apply any of the variety of well-known efficient techniques to generate optimal curves on GA + (n). Second, the method yields approximations to optimal solutions for general choices of Riemannian metrics on SE(3). Third, from a computational point of view, the method we propose is less expensive than traditional methods. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This journal article is available at ScholarlyCommons: http://repository.upenn.edu/meam_papers/10 334 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 18, NO. 3, JUNE 2002 An SVD-Based Projection Method for Interpolation on SE(3) Calin Belta, Student Member, IEEE, and Vijay Kumar, Senior Member, IEEE Abstract-This paper develops a method for generating smooth trajectories for a moving rigid body with specified boundary conditions. Our method involves two key steps: 1) the generation of optimal trajectories in + ( ), a subgroup of the affine group in IR and 2) the projection of the trajectories onto (3), the Lie group of rigid body displacements. The overall procedure is invariant with respect to both the local coordinates on the manifold and the choice of the inertial frame. The benefits of the method are threefold. First, it is possible to apply any of the variety of well-known efficient techniques to generate optimal curves on + ( ). Second, the method yields approximations to optimal solutions for general choices of Riemannian metrics on (3). Third, from a computational point of view, the method we propose is less expensive than traditional methods.

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Section: Introductionmentioning

confidence: 99%

“…It is worth noting that all these works (with the exception of [6]) use a particular parameterization of the group and do not discuss the invariance of their methods. In contrast, Noakes et al [13] derived the necessary conditions for cubic splines on general manifolds without using a coordinate chart.…”

Section: Introductionmentioning

confidence: 99%

An SVD-based projection method for interpolation on SE(3)

Belta

Kumar

2002

IEEE Trans. Robot. Automat.

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“…The Bezier curve in Riemannian space is calculated to solve the problem of robotic workspace fitting. 21 By changing the original constant kinetic energy, the optimal smooth trajectories for a set of mobile robots are generated. 22 Generally speaking, all existing works are on rigid body motion with one degree of freedom.…”

Section: Introductionmentioning

confidence: 99%

Optimal three-dimensional biped walking pattern generation based on geodesics

Zhang

Zhou

2017

International Journal of Advanced Robotic Systems

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The innovative three-dimensional humanoid biped gait planning method using geodesics is introduced in this article. In order to control three-dimensional walking, the three-dimensional linear inverted pendulum model is studied in our energy-optimal gait planning based on geodesics. The kinetic energy of the three-dimensional linear inverted pendulum model is calculated at first. Based on this kinetic energy model, the Riemannian metric is defined and the Riemannian surface is further determined by this Riemannian metric. The geodesic is the shortest line between two points on the Riemannian surface. This geodesic is the optimal kinetic energy gait for the center of gravity because the kinetic energy along the geodesic is invariant according to the geometric property of geodesics and the walking is energy-saving. Finally, a simulation experiment using a 12-degree-of-freedom biped robot model is implemented. The gait sequences of the simulated RoboErectus humanoid robot are obtained in the ROS (Robot Operating System) Gazebo environment. The proposed geodesics approach is compared with the traditional sinusoidal interpolation method by analyzing the torque feedback of each joint of both legs, and our geodesics optimization gait planning method for three-dimensional linear inverted pendulum model walking control is verified by the assessment results.

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“…[9][10][11][12][13] The fundamental idea of the geodesic-based trajectory planning method is to replace line or arc segments in a Euclidean space by geodesics in a Riemannian manifold. Because a geodesic curve generally represents (locally) the shortest path between two points in a Riemannian manifold, 14 the geodesic-based method can always yield an optimal trajectory.…”

Section: Introductionmentioning

confidence: 99%

An improved geodesic algorithm for trajectory planning of multi-joint robots

Chen

Tang

2016

International Journal of Advanced Robotic Systems

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The present work proposes an improved geodesic algorithm for the trajectory planning of multi-joint robots. First, all of the joint variables are chosen to set up a generalized local coordinate system for the product of the positional space and the orientational one. Second, by defining a new Riemannian metric that contains both the positional and rational parameters, the traditional geodesic algorithm is improved so that it becomes capable of planning robot trajectories that include both the position and the orientation. To demonstrate the effectiveness of the improvement, trajectories are planned for two typical joint robots: one being a planar 3R and the other a spatial RRPP. It is verified that the improved algorithm can generate not only smooth motions for the joints but also smooth and accurate motions for the end-effector. The improved algorithm applies to multi-joint robots with no more than six degrees of freedom.

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Be´zier Curves on Riemannian Manifolds and Lie Groups with Kinematics Applications

Cited by 154 publications

References 10 publications

An SVD-based projection method for interpolation on SE(3)

An SVD-based projection method for interpolation on SE(3)

Optimal three-dimensional biped walking pattern generation based on geodesics

An improved geodesic algorithm for trajectory planning of multi-joint robots

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