Closure to “Discussion of ‘Kinematic Synthesis of Plane Curves’” (1967, ASME J. Eng. Ind., 89, pp. 175–176)

1967 ◽  
Vol 89 (1) ◽  
pp. 176-176
Author(s):  
D. W. Lewis ◽  
C. K. Gyory
Keyword(s):  
Plane Curves ◽  
Kinematic Synthesis

Kinematic Synthesis of Plane Curves

1967 ◽  
Vol 89 (1) ◽  
pp. 173-175 ◽  
Author(s):  
D. W. Lewis ◽  
C. K. Gyory
Keyword(s):  
Least Squares ◽  
Plane Curves ◽  
Kinematic Synthesis ◽  
Point Curve ◽  
Damped Least Squares ◽  
Four Bar Linkage

The coupler point curve of a plane mechanism is a curve that may be described by a series of paired coordinates. An extension of the method of “damped least squares” provides a means for successive adjustment of the parameters which define a particular type mechanism. Repetitive application of this process will result in a convergence toward an optimum approximation to the desired curve as described by the series of paired coordinates. The method has been applied to a four-bar linkage as an example of application.

Evaluating Plane Curves As Constraints for Locomotion on Uneven Terrain

2020 ◽  
Author(s):  
Chang Liu ◽  
Mark Plecnik
Keyword(s):  
Physical Mechanism ◽  
Mechanical Design ◽  
Plane Curves ◽  
Kinematic Synthesis ◽  
Single Degree Of Freedom ◽  
Mechanical Advantage ◽  
Work Related ◽  
Uneven Terrain ◽  
Physical Mechanisms ◽  
Single Degree

Abstract This paper focuses on preliminary work related to the discovery of single degree-of-freedom mechanism paths useful for dynamic locomotion tasks. The objective is to bridge a gap between kinematic specifications and emerging dynamic behaviors. This is accomplished by formulating a set of ordinary differential equations that includes essential mechanism characteristics (path traced, mechanical advantage) but excludes all physical mechanism parameters (topology, link lengths). The dynamics represent a rotation constrained body propelled by a foot that is attached to that body by a user-defined path. The foot is powered by a series-elastic actuator acting through a mechanical advantage function that is defined across the length of the path. Through this framework, a range of user-defined paths were tested for effective locomotion on flat and complex terrains. Foot paths and mechanical advantage functions exist outside of any mechanical design, with the goal to discover paradigms worth instantiating into physical mechanisms, a task reserved for kinematic synthesis. This work would empower existing kinematic synthesis techniques to achieve dynamic requirements. In other words, kinematic requirements are transformed from an end to a means. Their dynamic utility would be evaluated by the framework presented in this paper rather than pursued by themselves.

On the Existence of a Cycloidal Burmester Theory in Planar Kinematics

1964 ◽  
Vol 31 (4) ◽  
pp. 694-699 ◽  
Author(s):  
George N. Sandor
Keyword(s):  
Plane Curves ◽  
Motion Generation ◽  
Kinematic Synthesis ◽  
Multiple Generation ◽  
Special Cases ◽  
Moving Plane ◽  
Motion Generator ◽  
Linkage Synthesis

One of the basic theories of kinematic synthesis, namely, Burmester’s classical centerpoint-circlepoint theory, is shown to be one of several special cases of a broader, more general new theory, involving points of the moving plane whose several corresponding positions lie on cycloidal curves. These curves may be generated by “cycloidal cranks.” Such “cycloidpoints,” centers of their generating circles (“circlepoints”) and base circles (“centerpoints”) are proposed to be called “Burmester point trios” (BPT’s). In case of 6 prescribed arbitrary positions, such BPT’s appear to lie, respectively, on three higher plane curves proposed to be called “cycloidpoint,” “circlepoint” and “centerpoint curves,” or, collectively, “generalized Burmester curves.” In the case of hypocycloidal cranks with “Cardanic” proportions, the hypocycloids become ellipses. For 7 prescribed positions, the number of BPT’s is finite. Application to linkage synthesis for motion generation with prescribed order and timing is presented and cognate-motion generator linkages, based on multiple generation of cycloidal curves, are shown to exist. Analytical derivations are outlined for the equations of the “generalized Burmester curves,” and possible further specializations and generalizations are indicated.

scholarly journals Conjectures on Stably Newton Degenerate Singularities

2021 ◽  
Author(s):  
Jan Stevens
Keyword(s):  
Characteristic Zero ◽  
Arbitrary Number ◽  
Milnor Number ◽  
Plane Curves ◽  
Newton Diagram ◽  
Finite Characteristic ◽  
Vanishing Cycles

AbstractWe discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

On an area-preserving inverse curvature flow of convex closed plane curves

2021 ◽  
Vol 280 (8) ◽  
pp. 108931
Author(s):  
Laiyuan Gao ◽  
Shengliang Pan ◽  
Dong-Ho Tsai
Keyword(s):  
Curvature Flow ◽  
Plane Curves ◽  
Area Preserving ◽  
Closed Plane
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