Computer-aided displacement analysis of spatial mechanisms using the CH programming language

Cheng, Harry H.; Thompson, Sean

doi:10.1016/0965-9978(95)00074-7

Cited by 9 publications

(5 citation statements)

References 9 publications

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“…Let the unit pure dual octonions  =  + 𝜀  and  =  + 𝜀  be an element of P 6 ⊆ S 6 , then the unit dual octonion Proof. Any pure dual octonion can be written as a dual vector as in (10). Also, according to Theorem 5, since every element of P 6 corresponds to two intersecting perpendicular lines, the unit pure dual octonion  =  + 𝜀  represents the intersecting perpendicular directed lines d  and d  while the unit pure dual octonion  =  + 𝜀  represents the intersecting perpendicular directed lines d  and d  .…”

Section: Discussionmentioning

confidence: 99%

“…E. Study used dual numbers and dual vectors in his work on the geometry and kinematics of lines [8]. The application of dual numbers has increased recently in a number of disciplines, including kinematics, dynamics, robotics, computer-aided geometric design, mechanism design and rigid body modelling, group theory, and field theory [9][10][11][12][13][14]. As a result, dual numbers are an important number system with many uses in mathematics and physics, especially in mechanics and theoretical kinematics.…”

Section: Introductionmentioning

confidence: 99%

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Dual octonions and rigid body kinematics

Çakır,

Bektaş,

Yılmaz

2024

Math Methods in App Sciences

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In this paper, we first give the basic information about octonions and present the Euclidean rotation matrix formed by an octonion in seven‐dimensional Euclidean space. Next, we define and introduce the ‐module and dual vectors using dual numbers. Then, we provide the transformation that maps the points on the unit dual sphere one‐to‐one with the directed lines in . We also define a subset of the unit dual sphere, demonstrating that each element of this subset corresponds to two intersecting perpendicular directed lines in seven‐dimensional Euclidean space. Following that, we introduce dual octonions with their basic algebraic properties and examine rigid body (screw) motions in seven‐dimensional Euclidean space using dual octonions. Finally, we define an operator and express that this operator transforms two perpendicular intersecting directed lines in seven‐dimensional Euclidean space into two perpendicular intersecting directed lines.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dual octonions and rigid body kinematics

Çakır,

Bektaş,

Yılmaz

2024

Math Methods in App Sciences

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“…More accurate approximations can be obtained by progressively increasing the sampling resolution, as in [7], that, after plotting the input-output function by initially sampling the input variable by steps of 1 • , the interval bounds are refined until the third decimal position to give the values 69.350 • < θ 1 < 410.471 • for one assembly mode, and 148.788 • < θ 1 < 308.299 • for the second, in full accordance with the results presented here (see Table 2). The limitation of this approach is that it can only find the bounding values within the precision of the discretization used initially and, in the extreme case of a mechanism with isolated solutions or very short solution intervals, they may be completely missed.…”

Section: Introductionmentioning

confidence: 99%

Solution intervals for variables in spatial RCRCR linkages

Celaya

2019

Mechanism and Machine Theory

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An analytic method to compute the solution intervals for the input variables of spatial RCRCR linkages and their inversions is presented. The input-output equation is formulated as the intersection of a single ellipse with a parameterized family of ellipses, both related with the possible values that certain dual angles determined by the configuration of the mechanism can take. Bounds for the angles of the input pairs of the RCRCR and RRCRC inversions are found by imposing the tangency of two ellipses, what reduces to analyzing the discriminant of a fourth degree polynomial. The bounds for the input pair of the RCRRC inversion is found as the intersection of a single ellipse with the envelope of the parameterized family of ellipses. The method provides the bounds of each of the assembly modes of the mechanism as well as the local extrema that may exist for the input variable.

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“…The computation of the dual stiffness matrix has been implemented using Ch programming language. 15,16 The stiffness matrix A linear elastic system can be modeled as a rigid body connected to the ground by springs arranged in parallel. In such a system, the stiffness matrix K ½ is a linear map which relates wrench and twist.…”

Section: Introductionmentioning

confidence: 99%

A model for computing the dual stiffness matrix of the human knee joint

Enea

Pennestrı̀

Valentini

2013

Proceedings of the Institution of Mechanical Engineers, Part K:

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This paper discusses an application of dual algebra to the analysis of human knee joint stiffness matrix. According to the proposed methodology, the general stiffness matrix characterizing the elasticity properties of the knee ligaments is reduced to a dual diagonal stiffness matrix. For this purpose, the similarity transform was extended to the field of dual numbers. The theoretical framework proposed seems particularly useful when comparing stiffness matrices obtained from different experimental setup and within different Cartesian coordinate systems.

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Computer-aided displacement analysis of spatial mechanisms using the CH programming language

Cited by 9 publications

References 9 publications

Dual octonions and rigid body kinematics

Dual octonions and rigid body kinematics

Solution intervals for variables in spatial RCRCR linkages

A model for computing the dual stiffness matrix of the human knee joint

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