A novel and efficient kinematic synthesis approach of the spherical 4R function generator for five and six precision points

Cervantes-Sánchez, J. Jesús; Gracia, Luis; Rico-Martínez, José M.; Medellín-Castillo, Hugo I.; González-Galván, Emilio J.

doi:10.1016/j.mechmachtheory.2009.05.006

Cited by 10 publications

(14 citation statements)

References 8 publications

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“…The three functions and the corresponding set of precision points used are shown in Table 1. The first two are taken from [6], which is the only work that gives numerical examples for the problem at hand. However, the method reported in [6] introduces algorithmic singularities at the parameter values ψ 0 = ±π/2 and φ 0 = ±π/2 due to the transformations made between design parameters and design coefficients.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…The first two are taken from [6], which is the only work that gives numerical examples for the problem at hand. However, the method reported in [6] introduces algorithmic singularities at the parameter values ψ 0 = ±π/2 and φ 0 = ±π/2 due to the transformations made between design parameters and design coefficients. , respectively, which can be readily identified as spurious roots because two and one contours, out of four, do not pass through these points.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Innocenti and Parenti-Castelli [3] and Gosselin et al [4] obtained a univariate eighth-degree characteristic polynomial for the direct displacement problem of spherical parallel robots. Moreover, Alizade and Kilit [5] and Cervantes-Sánchez et al [6] synthesized the fourbar spherical function generator (SFG) with five precision points based on a cubic polynomial. Similarly, Dukkipati [7] and Cervantes-Sánchez et al [6] designed the SFG with six precision points via a 16th-degree and a 10th-degree polynomial, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, Alizade and Kilit [5] and Cervantes-Sánchez et al [6] synthesized the fourbar spherical function generator (SFG) with five precision points based on a cubic polynomial. Similarly, Dukkipati [7] and Cervantes-Sánchez et al [6] designed the SFG with six precision points via a 16th-degree and a 10th-degree polynomial, respectively. However, the classical univariate-polynomial approach for computational kinematics has significant drawbacks, such as algorithmic singularities and ill-conditioning, which are illustrated in Section 2 with several examples.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Robustness to algorithmic singularities and sensitivity in computational kinematics

Gracia

Ángeles

2011

Proceedings of the Institution of Mechanical Engineers, Part C:

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A robust approach to computational kinematics intended to cope with algorithmic singularities is introduced in this paper. The approach is based on the reduction of the original system of equations to a subsystem of bivariateequations, as opposed to the multivariate-polynomial reduction leading to the characteristic univariate-polynomial. The effectiveness of the approach is illustrated for the exact function-generation synthesis of planar, spherical, and spatial four-bar linkages. Some numerical examples are provided for the case of the spherical four-bar function generator with six precision points to show the benefits of the proposed method with respect to methods reported in the literature.

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Section: Numerical Examplesmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Robustness to algorithmic singularities and sensitivity in computational kinematics

Gracia

Ángeles

2011

Proceedings of the Institution of Mechanical Engineers, Part C:

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“…Computational methods for the exact kinematic synthesis of a spherical 4R mechanism for function generation synthesis problem are studied by Cervantes-Sánchez et al [29]. Function synthesis with spherical 4R mechanisms for the five precision points is studied analytically by Alizade and Kilit [30].…”

Section: Introductionmentioning

confidence: 99%

Function generation synthesis of spherical 5R mechanism with regional spacing and Chebyshev approximation

Kiper

Bilgincan

2015

Mechanism and Machine Theory

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The Chebyshev approximation is well known to be applicable for the approximation of single input-single output functions by means of a function generator mechanism. The approximation method may be also applied to multi-input functions, although until recently, it was not used for function generation with multi-degrees-of-freedom mechanisms. In a recent study, the authors applied the approximation method to a two-degrees-of-freedom mechanism for the first time, however the selection and iteration of the design points at which the errors were minimized was not satisfactory. In this study, an alternative method of selection and iteration for these design points is introduced and the corresponding spacing is called the "regional spacing". As a case study for the application of the approximation of multi-input functions, a spherical 5R mechanism is used to generate a two input-single-output function. The input joints of the mechanism are selected as one of the fixed joints and the moving mid-joint, whereas the remaining fixed joint represents the output. The synthesis problem is analytically formulated and presented in polynomial form for five and six unknown parameters. The synthesis problem for five unknown parameters is illustrated as a numerical example. Regional spacing is used for the selection and iteration of design points for the synthesis. The Chebyshev approximation along with the Remez algorithm is utilized to find the unknown construction parameters and the error of the function. The design points and the coefficients of the approximation polynomial are determined by numerical iteration using six moving points. At each iteration step, the design points are relocated at the extremum error points in their respective regions. Iterations are repeated until the magnitudes of the extremum point errors are approximately equal. Finally, the construction parameters of the mechanism are determined and the variation of the percentage error between the desired and generated function values is obtained.

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Synthesis of a special RPSPR spatial linkage function generator for six precision points

Cervantes-Sánchez

Gracia

Alba-Ruiz

et al. 2011

Mechanism and Machine Theory

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A novel and efficient kinematic synthesis approach of the spherical 4R function generator for five and six precision points

Cited by 10 publications

References 8 publications

Robustness to algorithmic singularities and sensitivity in computational kinematics

Robustness to algorithmic singularities and sensitivity in computational kinematics

Function generation synthesis of spherical 5R mechanism with regional spacing and Chebyshev approximation

Synthesis of a special RPSPR spatial linkage function generator for six precision points

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