Geometric Construction of Be´zier Motions

Ge, Q. J.; Ravani, Bahram

doi:10.1115/1.2919446

Cited by 80 publications

(41 citation statements)

References 9 publications

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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%

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%

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

Belta

Kumar

2002

IEEE Trans. Robot. Automat.

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“…Figure 1 shows the six key frames and the additional 10 interpolated frames computed using Bezier interpolation for dual quaternion coordinates [6,7]. Figure 2 presents the same interpolation computed using double quaternion coordinates.…”

Section: Example CC Chain Designmentioning

confidence: 99%

“…We use the Bezier interpolation strategy of Ge and Ravani [6] to generate the sequence of positions under the control of a set of key frames; also see [7]. Table 1 lists the key frames used for these examples, and Figs.…”

Section: The Specified Trajectorymentioning

confidence: 99%

Dimensional Synthesis of Robots using a Double Quaternion Formulation of the Workspace

McCarthy

Ahlers

2000

Robotics Research

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This paper presents a synthesis procedure for a robot that guides an end-effector as close as possible to a user-specified trajectory. The technique maps the goal workspace from the group of spatial displacements, SE(3), to the group of 4 × 4 rotations, SO(4), in order to obtain a bi-invariant metric for the design procedure. Double quaternions are used to provide a convenient parameterization for SO(4). An example is presented that compares designs obtained using dual quaternions to those for double quaternions, for varying locations of the fixed frame, in order to demonstrate the technique.

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“…In this article we present a class of algorithms for e ciently generating C 2 bi-invariant trajectories on SO (3), and compare their performance vis-Â a-vis the minimum angular acceleration curve. The algorithms are based on three popular co-ordinate parametrizations for rotation matrices: the unit quaternion, Cayley-Rodrigues parameter, and canonical co-ordinate representations.…”

Section: Introductionmentioning

confidence: 99%

Cubic spline algorithms for orientation interpolation

Kang

Park

1999

Int. J. Numer. Meth. Engng.

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SUMMARYThis article presents a class of spline algorithms for generating orientation trajectories that approximately minimize angular acceleration. Each algorithm constructs a twice-di erentiable curve on the rotation group SO(3) that interpolates a given ordered set of rotation matrices at speciÿed knot times. Rotation matrices are parametrized, respectively, by the unit quaternion, canonical co-ordinate, and Cayley-Rodrigues representations. All the algorithms share the common feature of (i) being invariant with respect to choice of ÿxed and moving frames (bi-invariant), and (ii) being cubic in the parametrized co-ordinates. We assess the performance of these algorithms by comparing the resulting trajectories with the minimum angular acceleration curve.

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Geometric Construction of Be´zier Motions

Cited by 80 publications

References 9 publications

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

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

Dimensional Synthesis of Robots using a Double Quaternion Formulation of the Workspace

Cubic spline algorithms for orientation interpolation

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