Use of dual numbers in kinematical analysis of spatial mechanisms. Part II: applying the method for the generalised Cardan mechanism

Abstract Complex dual numbers w̌1=x1+iy1+εu1+iεv1 which form a commutative ring are for the first time introduced in this paper. Arithmetic operations and functions of complex dual numbers are defined. Complex dual numbers are used to solve dual polynomial equations. It is shown that the singularities of a dual input-output displacement polynomial equation of a mechanism correspond to its singularity positions. This new method of identifying singularities provides clear physical insight into the geometry of the singular configurations of a mechanism, which is illustrated through analysis of special configurations of the RCCC spatial mechanism. Numerical solutions for dual polynomial equations and complex dual numbers are conveniently implemented in the CH language environment for analysis of the RCCC spatial mechanism. Like the dual number, the complex dual number is a useful mathematical tool for analytical and numerical treatment of spatial mechanisms.

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Computations of Dual Numbers in the Extended Finite Dual Plane

19th Design Automation Conference: Volume 2 — Design Optimization; Geometric Modeling and Tolerance Analysis; Mechanism Synthesis and Analysis; Decomposition and Design Optimization ◽

10.1115/detc1993-0378 ◽

1993 ◽

Author(s):

Harry H. Cheng

Keyword(s):

Programming Language ◽

Data Type ◽

Complex Numbers ◽

Dual Numbers ◽

Linguistic Features ◽

Mathematical Functions ◽

Dual Plane ◽

Spatial Mechanisms ◽

Computational Aspects ◽

Dual Functions

Abstract The numerical computational aspects of dual numbers in the CH programming language are presented in this paper. Dual is a built-in data type in CH. Dual numbers and dual metanumbers are described in the extended dual plane and extended finite dual plane. The arithmetic and relational operations, and built-in mathematical functions are defined for both dual numbers and dual metanumbers of DualZero, DualInf, and DualNaN. Due to polymorphism, the syntax of dual arithmetic and relational operations, and built-in dual functions are the same as those for real and complex numbers in the CH programming language. The linguistic features and handling of user’s dual functions and dual arrays, as well as applications of dual numbers in robotics and spatial mechanisms in the CH programming language can be found in (Cheng, 1993c).

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Computer-Aided Displacement Analysis of Spatial Mechanisms

20th Design Automation Conference: Volume 1 — Dynamic Mechanical Systems; Geometric Modeling and Features; Concurrent Engineering ◽

10.1115/detc1994-0052 ◽

1994 ◽

Author(s):

Sean Thompson ◽

Harry H. Cheng

Keyword(s):

Programming Languages ◽

Design Automation ◽

Data Type ◽

Complex Numbers ◽

Dual Numbers ◽

Displacement Analysis ◽

Spatial Mechanism ◽

Programming Paradigm ◽

Spatial Mechanisms ◽

Programming Techniques

Abstract Recently, Cheng (1993) introduced the CH programming language. CH is designed to be a superset of ANSI C with all programming features of FORTRAN. Many programming features in CH are specifically designed and implemented for design automation. Handling dual number as a basic built-in data type in the language is one example. Formulas with dual numbers can be translated into CH programming statements as easily as formulas with real and complex numbers. In this paper we will show that both formulation and programming with dual numbers are remarkably simple for analysis of complicated spatial mechanisms within the programming paradigm of CH. With computational capabilities for dual formulas in mind, formulas for analysis of spatial mechanisms are derived differently from those intended for implementation in computer programming languages without dual data type. We will demonstrate some formulation and programming techniques in the programming paradigm of CH through a displacement analysis of the RCRCR five-link spatial mechanism. A CH program that can obtain both numerical and graphical results for complete displacement analysis of the RCRCR mechanism will be presented.

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Singularity Analysis of Spatial Mechanisms Using Dual Polynomials and Complex Dual Numbers

Journal of Mechanical Design ◽

10.1115/1.2829444 ◽

1999 ◽

Vol 121 (2) ◽

pp. 200-205 ◽

Cited By ~ 8

Author(s):

H. H. Cheng ◽

S. Thompson

Keyword(s):

Numerical Solutions ◽

Polynomial Equation ◽

Singularity Analysis ◽

Polynomial Equations ◽

Dual Numbers ◽

Output Displacement ◽

Spatial Mechanism ◽

Spatial Mechanisms ◽

Physical Insight ◽

Language Environment

Complex dual numbers wˇ = x + iy + εu + iεv which form a commutative ring are introduced in this paper to solve dual polynomial equations numerically. It is shown that the singularities of a dual input-output displacement polynomial equation of a mechanism correspond to its singularity positions. This new method of identifying singularities provides clear physical insight into the geometry of the singular configurations of a mechanism, which is illustrated through analysis of special configurations of the RCCC spatial mechanism. Numerical solutions for dual polynomial equations and complex dual numbers are conveniently implemented in the CH language environment for analysis of the RCCC spatial mechanism.

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