Geometric Design of Linkages

McCarthy, J. Michael; Soh, Gim Song

doi:10.1007/978-1-4419-7892-9

Cited by 289 publications

(275 citation statements)

References 83 publications

(145 reference statements)

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“…and Y(.) are rotation matrices about z and y axes, respectively [34] and S and C represent the sine and cosine functions, respectively.…”

Section: Fig 1 Spherical Five-bar Mechanismmentioning

confidence: 99%

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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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“…and Y(.) are rotation matrices about z and y axes, respectively [34] and S and C represent the sine and cosine functions, respectively.…”

Section: Fig 1 Spherical Five-bar Mechanismmentioning

confidence: 99%

“…Motion generation synthesis is studied with least square approximation method for spherical linkages by Alizade et al [26]. A general theory for synthesis of spherical mechanisms is given in [34]. The reader is suggested to read Alizade et al [26,30,31] for a more detailed review of design of spherical mechanisms.…”

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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“…Some researchers use the deficient term as a substitute to under-actuated, however what we mean by a deficient manipulator is a manipulator with less degrees-of-freedom (dof) than the task-space dimension. Although analytical synthesis methods for single dof mechanisms are widely studied [3,4], mostly optimization methods are utilized for determining link length dimensions of multi-dof mechanisms (ex. see [5]).…”

Section: Introductionmentioning

confidence: 99%

Function Generation Synthesis with a 2-DoF Overconstrained Double-Spherical 7R Mechanism Using the Method of Decomposition and Least Squares Approximation

Kiper

Bağdadioğlu

2014

New Trends in Mechanism and Machine Science

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Abstract. This study addresses the approximate function generation synthesis with an overconstrained two degrees-of-freedom double spherical 7R mechanism using least squares approximation with equal spacing of the design points on the input domain. The 7R mechanism is a constructed by combining a spherical 5R mechanism with a spherical 4R mechanism with distant centers and a common moving link and then removing the common link. This construction allows the analysis and synthesis of the resulting single-loop mechanism by decomposing it into fictitious 5R and 4R loops. The two inputs to the mechanism are provided in the 5R loop and the output is in the 4R loop. The fictitious output of the 5R loop is an input to the 4R loop this intermediate variable is used to also decompose the function to be generated. This decomposition provides the designer extra freedom in synthesis and enables decreasing the error of approximation. A case study is presented at the end of the study where the 7R design is compared with an equivalent spherical 5R mechanism; hence the advantage of the 7R mechanism is demonstrated.

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“…This generalized problem finds a practical application in the design of articulated serial chains. For example, the TS chain shown in Figure 1 has a gimbal joint (T-joint) at its shoulder and a ball joint (S-joint) at the wrist of its gripper, and no elbow joint, (McCarthy 2000 [15]). Thus, the wrist center P moves on a sphere about the center B of the gimbal joint which we call the reachable surface of the chain.…”

Section: Introductionmentioning

confidence: 99%

“…His result was a graphical solution to a set of five quadratic equations in five unknown paramters. Today, these equations are relatively easy to solve numerically, see Suh and Radcliffe (1978)[32], Sandor and Erdman (1984) [26], or McCarthy (2000) [15]. Chen and Roth (1967)[4] introduced a generalization of this problem which seeks points and lines in a moving body that take positions on surfaces associated with articulated chains used to build robot manipulators.…”

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confidence: 99%

The Computation of Reachable Surfaces for a Specified Set of Spatial Displacements

McCarthy

2005

Handbook of Geometric Computing

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In this chapter, we consider the geometry of configurations of points generated by arbitrary sets of positions of a rigid body. The goal is to find those points in the moving body that lie on specific algebraic surfaces. The result is a set of algebraic equations that can be remarkably complex.The problem originates with Burmester's determination (Burmester 1886[3]) of points in a body that lie on a circle for an arbitrary set of five planar positions. He used these Burmester points to design a linkage that guided the body through the specified positions. His result was a graphical solution to a set of five quadratic equations in five unknown paramters. Today, these equations are relatively easy to solve numerically, see Suh and Radcliffe (1978)[32], Sandor and Erdman (1984) [26], or McCarthy (2000) [15]. Chen and Roth (1967)[4] introduced a generalization of this problem which seeks points and lines in a moving body that take positions on surfaces associated with articulated chains used to build robot manipulators. See Craig (1989) [5] or Tsai (1999)[34] for a discussion of serial chain robots. Our focus is two-jointed chains that supports a spherical wrist. The center of this wrist traces a surface which is said to be "reachable" by the chain. Considering the various ways of assembling these articulated chains, we obtain seven reachable algebraic surfaces, and the problem reduces to computing the dimensions of these chains from a set of polynomial equations.It is interesting how quickly the complexity of the problem increases with the number of dimensional parameters and the degree of the surface. The total degree of the polynomial systems that we consider range from 32 for the simplest to over 4 million for the most complex. The equations have significant internal structure, so it is possible to use a linear product decomposition (Bernshtein 1975[2] and Morgan et al. (1995)[17]) to provide a better bound on the number of solutions, which range from 10 to over 800,000. Where possible we have used polynomial elimination to verify the number of roots (Neilsen and Roth 1995[18] and Husty 1996[6]), but in the more complex examples

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Geometric Design of Linkages

Cited by 289 publications

References 83 publications

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

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

Function Generation Synthesis with a 2-DoF Overconstrained Double-Spherical 7R Mechanism Using the Method of Decomposition and Least Squares Approximation

The Computation of Reachable Surfaces for a Specified Set of Spatial Displacements

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