On the Effect of Dual Weights in Computer Aided Design of Rational Motions

Purwar, Anurag; Ge, Q. J.

doi:10.1115/1.1906263

Cited by 32 publications

(21 citation statements)

References 20 publications

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“…, 4, u i is a non-unit quaternion proportional to the unit quaternion u i . Considering that in most applications it is not strictly necessary to operate with unit quaternions (quaternions can actually be treated as a vector of homogeneous coordinates [15]), this is an interesting alternative because it avoids the computation of square roots and divisions. Even if we need to compute the rotation matrix corresponding to a non-unit quaternion, we do not need to previously normalize it.…”

Section: Shepperd-markley Methodsmentioning

confidence: 99%

On Cayley’s Factorization with an Application to the Orthonormalization of Noisy Rotation Matrices

Sarabandi

Pérez-Gracia

Thomas

2019

Adv. Appl. Clifford Algebras

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A real orthogonal matrix representing a rotation in four dimensions can be decomposed into the commutative product of a leftand a right-isoclinic rotation matrix. This operation, known as Cayley's factorization, directly provides the double quaternion representation of rotations in four dimensions. This factorization can be performed without divisions, thus avoiding the common numerical issues attributed to the computation of quaternions from rotation matrices. In this paper, it is shown how this result is particularly useful, when particularized to three dimensions, to re-orthonormalize a noisy rotation matrix by converting it to quaternion form and then obtaining back the corresponding proper rotation matrix. This re-orthonormalization method is commonly implemented using the Shepperd-Markley method, but the method derived here is shown to outperform it by returning results closer to those obtained using the Singular Value Decomposition which are known to be optimal in terms of the Frobenius norm.Mathematics Subject Classification (2010). Primary 99Z99; Secondary 15A66.

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Section: Shepperd-markley Methodsmentioning

confidence: 99%

On Cayley’s Factorization with an Application to the Orthonormalization of Noisy Rotation Matrices

Sarabandi

Pérez-Gracia

Thomas

2019

Adv. Appl. Clifford Algebras

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“…It should be noted that re-parametrization of a rational Bézier motion leaves the path of the motion unchanged, but the "speed" of the resulting motion is in general different from the original motion; see ref. [21]. However, in NC machining the actual feedrates are defined in NC codes, independent of the parametrizations.…”

Section: Ruled Surface Design As Curve Design In Dual Quaternion Hypementioning

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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“…In this context it should also be mentioned that the PSH-map is the analogue of the map in P 5 sending a screw $ to its axis (cf. [17] and [9,Remark 10] [13] and [14].…”

Section: Remarkmentioning

confidence: 99%

Kinematic Interpretation of the Study Quadric’s Ambient Space

Nawratil

2018

Advances in Robot Kinematics 2018

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It is well known that real points of the Study quadric (sliced along a 3-dimensional generator space) correspond to displacements of the Euclidean 3-space. But we still lack of a kinematic meaning for the points of the ambient 7-dimensional projective space P 7 . This paper gives one possible interpretation in terms of displacements of the Euclidean 4-space. From this point of view we also discuss the extended inverse kinematic map, motions corresponding to straight lines in P 7 and linear complexes of SE(3)-displacements. Moreover we present an application of this interpretation in the context of interactive motion design.

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On the Effect of Dual Weights in Computer Aided Design of Rational Motions

Cited by 32 publications

References 20 publications

On Cayley’s Factorization with an Application to the Orthonormalization of Noisy Rotation Matrices

On Cayley’s Factorization with an Application to the Orthonormalization of Noisy Rotation Matrices

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

Kinematic Interpretation of the Study Quadric’s Ambient Space

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