Designing Planar Mechanisms Using a Bi-Invariant Metric in the Image Space of SO (3)

In this paper we present a technique for using a bi-invariant metric in the image space of spherical displacements for designing planar mechanisms for n (> 5) position rigid body guidance. The goal is to perform the dimensional synthesis of the mechanism such that the distance between the position and orientation of the guided body to each of the n goal positions is minimized. Rather than measure these distances in the plane, we introduce an approximating sphere and identify rotations which are equivalent to the planar displacements to a specified tolerance. We then measure distances between the rigid body and the goal positions using a bi-invariant metric on the image space of SO(3). The optimal linkage is obtained by minimizing this distance over all of the n goal positions. The paper proceeds as follows. First, we approximate planar rigid body displacements with spherical displacements and show that the error induced by such an approximation is of order 1/R2, where R is the radius of the approximating sphere. Second, we use a bi-invariant metric in the image space of spherical displacements to synthesize an optimal spherical 4R mechanism. Finally, we identify the planar 4R mechanism associated with the optimal spherical solution. The result is a planar 4R mechanism that has been optimized for n position rigid body guidance using an approximate bi-invariant metric with an error dependent only upon the radius of the approximating sphere. Numerical results for ten position synthesis of a planar 4R mechanism are presented.

Synthesis of Defect Free Six Bar Linkages for Body Guidance Through up to Six Finitely Separated Positions

In the synthesis of watt I six bar linkage, for finitely separated design positions, or in higher order design, constraint equations become highly nonlinear and transcendental. This paper presents a method to decouple the synthesis problem to the synthesis of two path generator 4-bar linkages. Based on this decoupled system, an explicit design methodology is developed, enabling a three, four, five or six body guidance position Watt I linkage to be designed while the designer has choice of some body pivots and ground pivots. Numerical procedures for higher number of positions are also discussed. The methodology allows the designer to obtain an entire set of solutions to a particular design problem. As a spin off from this work, a methodology is also presented to obtain complete solution sets of four bar path generators capable of passing through up to seven precision points, with a procedure that can be eventually extended to eight and nine path points. Design considerations such as branching and transmission angles are also considered.

Analysis and Synthesis of Overconstrained Mechanism

This paper presents a new systematic method for dealing with overconstrained mechanisms, and describes how the method was used to discover new overconstrained mechanisms and correct errors in several previously published overconstraint conditions. With this one method we are able to verify all previously known overconstrained mechanisms. In addition, this method yields the input-output equations of any single-loop overconstrained mechanism. For all new and corrected overconstrained mechanisms, numerical examples of input-output curves are presented.

Analysis and Synthesis of Linear Transmission Mechanisms: Coupled Mechanisms and Mechanisms Wth Prismatic Joints

The characteristics of linear transmission mechanisms are studied. Using the characteristics, the kinematic and synthesis of linear transmission mechanisms are expanded. First, the synthesis of mechanisms with prismatic joints in the equivalent open-loop chain is developed. Then the kinematics and synthesis of mechanisms with coupled joint motion are also derived. Two coupled mechanisms are used as examples to demonstrate the application potential in the industry.

A Method for Generating Geared Links From Linkages

A concept of generating geared links by inverting general linkages is proposed, which is based on the topological graph, and applies substitution principle of mechanisms. It makes it possible to apply the methods, data and computer programs of general linkages to geared links.

Five Position Synthesis of Spatial CC Dyads

This paper presents a new technique for determining the fixed axes of spatial CC dyads for rigid body guidance through five finitely separated positions. A CC dyad is a kinematic chain consisting of a floating link connected by a cylindric joint to a crank which in turn is connected to ground by a second cylindric joint. The lines that can be axes of the fixed joint are shown to be obtained from a “compatibility platform” constructed from selected relative screw axes associated with the five specified displacements. We show that the screw axis of the displacement of this platform is a fixed axis of a CC dyad compatible with the five positions. Roth’s original example is presented to verify the calculations. The specialization of this procedure to planar and spherical five position synthesis is also presented.

Bézier Curves on Riemannian Manifolds and Lie Groups With Kinematics Applications

In this article we generalize the concept of Bézier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau’s algorithm for constructing Bézier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their algebraic group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Bézier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Bezier sense) using this recursive algorithm. We apply this algorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body.

A Method for Type Synthesis of Mechanisms Using Phase Diagrams

A method for synthesizing the types of spatial as well as planar mechanisms is expressed in this paper by using the concept of phase diagram in metallurgy. The concept represented as a type synthesis technique is applied to (a) planar mechanisms with n degrees of freedom and simple loop, (b) spatial mechanisms with single degree of freedom and simple loop, to enumerate all the possible mechanisms with physically realizable kinematic pairs. Based on the technique described, a set of new reciprocating mechanisms is generated as a practical application.

Performance Quality, Sensitivity, and Tolerance Specification of Mechanisms

By employing Taguchi’s concept to mechanism synthesis, this paper presents the theory and technique to identify a robust design, which is the least sensitive to the tolerances, for mechanisms and to determine the tolerance specification for the best performance and manufacturability. The method is demonstrated in finite and infinitesimal position synthesis. The sensitivity Jacobian is first introduced to relate the performance tolerances and the dimensional tolerances. The Rayleigh quotient of the sensitivity Jacobian, which is equivalent to Taguchi’s signal to noise ratio, is then used to define the performance quality and a sensitivity index is introduced to measure the sensitivity of the performance quality to the dimensional tolerances for the whole system. The ideal tolerance specification is obtained in closed form. It shows how the tolerance specification affects the performance quality and that the performance quality can be significantly improved by tightening a key tolerance while loosening the others. The theory is general and the technique is readily adaptable to almost any form and type of mechanical system, including multiple-loop linkages and mechanical assemblies or even structures.

A Direct Rotatability Criterion for Spherical Four-Bar Linkages

The necessary and sufficient conditions for the full input rotatability in a spherical four bar linkage are proved. The direct criterion is: for all twist angles α in the range [0, π], the excess (deficit) of the sum of the frame and input twist angles over (from) π should, in absolute value, be greater than that for the coupler and follower twist angles; the difference between the follower and input twist angles, in absolute value, should be greater than that for the coupler and follower twist angles. Application of the direct criterion to full rotatability of other links are discussed and some variations in the form of the criterion are developed.