New Complex-Number Form of the Cubic of Stationary Curvature in a Computer-Oriented Treatment of Planar Path-Curvature Theory for Higher-Pair Rolling Contact

1982 ◽  
Vol 104 (1) ◽  
pp. 233-238 ◽  
Author(s):  
G. N. Sandor ◽  
A. G. Erdman ◽  
L. Hunt ◽  
E. Raghavacharyulu
Keyword(s):  
Complex Number ◽  
Rate Of Change ◽  
Rolling Contact ◽  
Planar Mechanisms ◽  
Digital Computation ◽  
Complex Arithmetic ◽  
Moving Plane ◽  
Curvature Theory ◽  
Path Curvature ◽  
Number Form

New complex number forms of the Euler-Savary Equation (ESE) for higher-pair rolling contact planar mechanisms were derived in a former paper by the authors. The present work, based on the former, deals with the derivation of the cubic of stationary curvature (CSC) in complex-vector form, suitable for digital computation. The CSC or Burmester’s circlepoint curve and its conjugate, the centerpoint curve for four infinitesimally close positions of the moving plane requires taking into account not only the curvature but also the rate of change of curvature of the rolling centrodes in the immediate vicinity of the position considered. The analytical procedure based on the theory developed in the present paper, when programmed for digital computation using complex arithmetic, takes care of the algebraic signs automatically, without the need for observing traditional sign conventions. The analysis is applicable to both higher-pair and lower-pair planar mechanisms. An example using the complex-number approach illustrates this.

New Complex-Number Forms of the Euler-Savary Equation in a Computer-Oriented Treatment of Planar Path-Curvature Theory for Higher-Pair Rolling Contact

1982 ◽  
Vol 104 (1) ◽  
pp. 227-232 ◽  
Author(s):  
G. N. Sandor ◽  
A. G. Erdman ◽  
L. Hunt ◽  
E. Raghavacharyulu
Keyword(s):  
Complex Number ◽  
Rolling Contact ◽  
Radius Of Curvature ◽  
Planar Mechanisms ◽  
Kinematic Synthesis ◽  
Synthesis Of Mechanisms ◽  
Curvature Theory ◽  
Path Curvature ◽  
Planar Linkages ◽  
Contact Mechanism

It is well known from the theory of Kinematic Synthesis of planar mechanisms that the Euler-Savary Equation (ESE) gives the radius of curvature and the center of curvature of the path traced by a point in a planar rolling-contact mechanism. It can also be applied in planar linkages for which equivalent roll-curve mechanisms can be found. Typical example: the curvature of the coupler curve of a four-bar mechanism. Early works in the synthesis of mechanisms concerned themselves with deriving the ESE by means of combined graphical and algebraic techniques, using certain sign conventions. These sign conventions often become sources of error. In this paper new complex-number forms of the Euler-Savary Equation are derived and are presented in a computer-oriented format. The results are useful in the application of path-curvature theory to higher-pair rolling contact mechanisms, such as cams, gears, etc., as well as linkages, once the key parameters of an equivalent rolling-contact mechanism are known. The complex-number technique has the advantage of eliminating the need for the traditional sign conventions and is suitable for digital computation. An example is presented to illustrate this.

Singular Solutions in Burmester Theory

1973 ◽  
Vol 95 (2) ◽  
pp. 572-576 ◽  
Author(s):  
R. E. Kaufman
Keyword(s):  
Complex Number ◽  
Singular Solutions ◽  
Design Technique ◽  
Planar Mechanisms ◽  
Point Position ◽  
Proper Interpretation ◽  
General Synthesis ◽  
Number Development

A unified complex number development of four position planar finite position theory is presented. This formulation shows that Burmester circlepoint-centerpoint theory specializes to include slider points, concurrency points, poles, and point position reduction by proper interpretation of the trivial roots of the general synthesis equations. Thus a single design technique can be used for the multiposition synthesis of most pin or slider-jointed planar mechanisms. Four position function, path, or motion generating linkages can all be designed in this manner.

Coriolis acceleration analysis of planar mechanisms by complex-number algebra

1982 ◽  
Vol 17 (6) ◽  
pp. 405-414 ◽  
Author(s):  
George N Sandor ◽  
E Raghavacharyulu ◽  
Arthur G Erdman
Keyword(s):  
Complex Number ◽  
Planar Mechanisms ◽  
Coriolis Acceleration ◽  
Acceleration Analysis

Order Rectification for Complex Number Based Burmester Curves

1991 ◽  
Vol 113 (3) ◽  
pp. 239-247 ◽  
Author(s):  
T. R. Chase ◽  
W. E. Fang
Keyword(s):  
Complex Number ◽  
Planar Mechanisms

A new solution to the order rectification problem for a driving dyad of planar mechanisms is presented. The method identifies sections of both Burmester curves where the driving link rotates in a single direction when passing through four precision positions in sequence. The new solution describes desirable regions of the curves in terms of the complex number parameters used to generate the curves, providing a complex number equivalent to available pole based order rectification procedures. The new solution is stated in a summary form that is readily codifiable. An example is presented. The theory underlying the new solution is then developed in detail.

Dynamic Characteristics of Spatial Mechanisms

1969 ◽  
Vol 91 (1) ◽  
pp. 228-233 ◽  
Author(s):  
E. J. Givens ◽  
J. C. Wolford
Keyword(s):  
Linkage Analysis ◽  
Dynamic Characteristics ◽  
Energy Method ◽  
Planar Mechanisms ◽  
Matrix Methods ◽  
Digital Computation ◽  
Spatial Mechanisms ◽  
Spatial Linkages

An energy method given by Quinnfor determining the dynamic characteristics of planar mechanisms under the action of displacement-related forces is extended to spatial linkages. These linkages may, in addition, be subjected to time-related forces or to velocity-related damping forces. Recently developed matrix methods are used in the linkage analysis resulting in a method well suited to digital computation.

The Application of Curvature Theory to the Trajectory Generation Problem of Robot Manipulators

1992 ◽  
Vol 114 (4) ◽  
pp. 677-680 ◽  
Author(s):  
M. M. Stanisˇic´ ◽  
K. Lodi ◽  
G. R. Pennock
Keyword(s):  
Robot Manipulators ◽  
Trajectory Generation ◽  
Rate Of Change ◽  
Speed Ratio ◽  
Coordination Problem ◽  
Geometric Problem ◽  
Curvature Theory ◽  
Instantaneous Speed ◽  
Generation Problem

This paper illustrates a new application of planar curvature theory to the geometric problem of trajectory generation by a two-link manipulator. The theory yields the instantaneous speed ratio, and the rate of change of the speed ratio, which correspond to the geometry of a desired point trajectory. Separate from the purely geometric speed ratio problem (i.e., the coordination problem) is the time based problem of controlling the joint rates in order to move with the specified path variables.

A Polynomial Formulation of Inverse Kinematics of Rolling Contact

2015 ◽  
Vol 7 (4) ◽  
Author(s):  
Lei Cui ◽  
Jian S. Dai
Keyword(s):  
Degrees Of Freedom ◽  
Contact Point ◽  
Rolling Contact ◽  
Moving Object ◽  
Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Univariate Polynomial ◽  
Curvature Theory ◽  
Robotic Devices ◽  
Polynomial Formulation

Rolling contact has been used by robotic devices to drive between configurations. The degrees of freedom (DOFs) of rolling contact pairs can be one, two, or three, depending on the geometry of the objects. This paper aimed to derive three kinematic inputs required for the moving object to follow a trajectory described by its velocity profile when the moving object has three rotational DOFs and thus can rotate about any axis through the contact point with respect to the fixed object. We obtained three contact equations in the form of a system of three nonlinear algebraic equations by applying the curvature theory in differential geometry and simplified the three nonlinear algebraic equations to a univariate polynomial of degree six. Differing from the existing solution that requires solving a system of nonlinear ordinary differential equations, this polynomial is suitable for fast and accurate numerical root approximations. The contact equations further revealed the two essential parts of the spin velocity: The induced spin velocity governed by the geometry and the compensatory spin velocity provided externally to realize the desired spin velocity.

Instantaneous Invariants and Curvature Analysis of a Planar Four-Link Mechanism

1994 ◽  
Vol 116 (4) ◽  
pp. 1173-1176 ◽  
Author(s):  
An Tzu Yang ◽  
G. R. Pennock ◽  
Lih-Min Hsia
Keyword(s):  
Rigid Body ◽  
Fourth Order ◽  
Arbitrary Point ◽  
Canonical System ◽  
Planar Mechanisms ◽  
Operating Cycle ◽  
Curvature Analysis ◽  
Crank Angle ◽  
Moving Plane ◽  
Derivatives Of

This paper shows that the canonical system and the instantaneous invariants for a moving plane, which is connected to the fixed plane by a revolute-revolute crank, are functions of the derivatives of the crank angle. Then closed-form expressions are derived for the curvature ratios of the path generated by an arbitrary point fixed in the moving plane, in terms of the coordinates of the point and the instantaneous invariants of the plane. For illustrative purposes, numerical results are presented for the instantaneous invariants (up to the fourth-order) of the coupler of a specified crank-rocker mechanism, as a function of the input angle. In addition, the paper shows the variation in the first and second curvature ratios of an arbitrary coupler curve during the complete operating cycle of the mechanism. The authors hope that, based on the results presented here, a variety of useful tools for the kinematic design of planar mechanisms, with a rotary input, will be developed for plane rigid body guidance as well as curve generation.

Kinematic Synthesis of Adjustable Mechanisms—Part 1: Function Generation

1973 ◽  
Vol 95 (2) ◽  
pp. 417-422 ◽  
Author(s):  
Joseph F. McGovern ◽  
George N. Sandor
Keyword(s):  
Complex Number ◽  
Relative Orientation ◽  
Complex Numbers ◽  
Input Output ◽  
Kinematic Synthesis ◽  
Digital Computation ◽  
Closed Form Solutions ◽  
Function Generation ◽  
Planar Linkages ◽  
Adjustable Mechanisms

One of the major advantages of linkages over cams is the ease with which they can be adjusted to modify their output. Incorporating an adjustment into the design of a linkage makes it possible to make a selection from two, three, or more different outputs by simply making the adjustment. Adjustable mechanisms make it possible to use the same hardware for different input-output relationships. The adjustment considered in this paper is the changing of a fixed pivot location. This paper presents a method of synthesizing adjustable planar linkages for function generation with finitely separated precision points and higher order synthesis involving prescribed velocities, accelerations, and higher accelerations. The method of synthesis is analytical with closed form solutions and utilizes complex numbers. The method is programmed for automatic digital computation. The linkages considered are a four-bar, a geared five-bar, and a geared six-bar mechanism. Examples include adjustable mechanisms which have been successfully synthesized with the method developed here. Future extensions of the complex number method to include adjustment by changing the length of a link and by changing of the relative orientation of the gears in geared linkages are outlined.

Triad Synthesis for up to Five Design Positions With Application to the Design of Arbitrary Planar Mechanisms

1987 ◽  
Vol 109 (4) ◽  
pp. 426-434 ◽  
Author(s):  
T. R. Chase ◽  
A. G. Erdman ◽  
D. R. Riley
Keyword(s):  
Complex Number ◽  
Path Generation ◽  
Dimensional Synthesis ◽  
Planar Mechanisms ◽  
Relative Precision ◽  
Synthesis Techniques ◽  
New Synthesis ◽  
Actual Mechanism ◽  
Synthesis Tool

A new synthesis tool, the triad, is introduced to enable simplified synthesis of very complex planar mechanisms. The triad is a connected string of three vectors representing jointed rigid links of an actual mechanism. The triad is used as a tool to model an original mechanism topology with a set of simpler components. Each triad is then used to generate a set of “relative precision positions” which, in turn, enables the dimensional synthesis of each triad with well-established motion and path generation techniques for simple four-bar linkages. Two independent derivations of the relative precision positions are provided. All common triad geometries amenable to simple dyad synthesis techniques are presented. The triad geometries summarized here may be applied to two, three, four, and five precision position problems using graphical, algebraic, or complex number formulations of Burmester theory. Examples are provided.

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