Computer Aided Geometric Design of Motion Interpolants

Ge, Q. J.; Ravani, Bahram

doi:10.1115/1.2919447

Cited by 89 publications

(58 citation statements)

References 13 publications

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“…Proof: The first part follows immediately from (7). For the second part, we can use (7) to prove that the eigenvalues of are given in terms of the eigenvalues of by (8) Because is positive definite, it follows that which implies , i.e., is positive definite.…”

Section: B Induced Metric Onmentioning

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: B Induced Metric Onmentioning

confidence: 99%

“…Ge and Ravani [7] used the dual-unit quaternion representation of and subsequently applied Euclidean methods to interpolate in this space. Jütler [8] formulated a more general version of the polynomial interpolation by using dual (instead of dual unit) quaternions to parameterize .…”

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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“…For the Cayley-Rodrigues parameter and unit quaternion trajectories, when p = 0 the path followed by these trajectories is identical to the minimal geodesic, but these curves fail to be unitspeed: the Cayley-Rodrigues parameters warp the time axis according to tan t=2, while for the unit quaternions time warping occurs as a result of projecting a line segment (connecting two points on the three-sphere in 4 ) onto the three-sphere. For p¿1 the unit quaternion trajectory has less distortion than the canonical co-ordinate trajectory, although for other arbitrary initial angular velocities the crossover point varied over the entire range of p. In general, our experimental results suggest that for p between 0 and =2 the canonical co-ordinate trajectory has the least deviation from the optimal trajectory, while for p greater than =2 the unit quaternion trajectory has less deviation.…”

Section: Two-point Interpolationmentioning

confidence: 96%

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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“…Recently, it has been realized that curve design methods in computer aided geometric design (CAGD) provide elegant tools to handle this problem. The methods vary with the different parametrizations of the rigid-body motion such as quaternion [9], dual quaternion [10], and Lie algebra [11]. The rigid-body motion in refs.…”

Section: Introductionmentioning

confidence: 99%

Kinematic generation of ruled surface based on rational motion of point-line

Zhang

Zhu

Ding

et al. 2011

Sci. China Technol. Sci.

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This paper studies representation of rigid combination of a directed line and a reference point on it (here referred to as a "point-line") using dual quaternions. The geometric problem of rational ruled surface design is viewed as the kinematic problem of rational point-line motion design. By using the screw theory in kinematics, mappings from the spaces of lines and point-lines in Euclidean three-dimensional space into the hyperplanes in dual quaternion space are constructed, respectively. The problem of rational point-line motion design is then converted to that of projective Bézier or B-spline image curve design in hyperplane of dual quaternions. This kinematic method can unify the geometric design of ruled surfaces and tool path generation for five-axis numerical control (NC) machining. line space, point-line space, rational motion, dual quaternion hyperplane, ruled surface generation Citation: Zhang X M, Zhu L M, Ding H, et al. Kinematic generation of ruled surface based on rational motion of point-line. Sci China Tech Sci, 2012, 55: 6271,

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Computer Aided Geometric Design of Motion Interpolants

Cited by 89 publications

References 13 publications

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

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

Cubic spline algorithms for orientation interpolation

Kinematic generation of ruled surface based on rational motion of point-line

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