This paper establishes a systematic procedure to determine the instantaneous invariants for a member in a planar mechanism and the curvature ratios of the path of a point fixed in the member. Closed-form expressions are derived for the instantaneous invariants and the curvature ratios as continuous functions of the input variable of the mechanism. The methods that are proposed in this paper can be applied to the design of planar mechanisms in general. For a given mechanism, the configuration defined by the input variable at which the member achieves the optimal approximation of a prescribed rigid motion can be determined. Alternatively, a point fixed in a member can be selected such that it will generate a trajectory matching a given curve with high precision. For purposes of illustration, the paper details the analytical procedure for a simple epicyclic gear train and a four-bar mechanism. The instantaneous invariants and the curvature ratios are generated for the complete operating cycle of each mechanism.

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.

This paper describes how Fuzzy Set Theory and Fuzzy Numbers can be applied to the synthesis of planar and spatial mechanisms. The main advantage of using Fuzzy Set Theory over other methods of mechanism synthesis is that uncertainties in positions, rotations and link lengths can be quantified using the concept of membership functions. A membership function is an explicit representation of the deviation of the value of a parameter from its specified value. Numerical examples include the synthesis of a four-bar path generating mechanism with uncertain prescribed timing as well as uncertain path points and the RSSR spatial function generating mechanism. This work is expected to be a significant contribution to optimal mechanism design.

An aspect of dual-number coordinate-transformation matrices is used to establish iterative methods for determining the rotational and translational displacements in the kinematic analysis of complex spatial mechanisms.

The formulation of design procedures for rigid-body mechanisms has benefited from the application of type-synthesis techniques. Therefore, with modifications to allow for inclusions of compliance, type synthesis is seen as a useful tool in the design of compliant mechanisms. Previous efforts have developed methods that result in a large number of possible design solutions to a given problem. This paper deals primarily with the development of a simplified compliant-mechanism type-synthesis methodology that limits the number of design solutions considered. The techniques are derived by modifying existing compliant mechanism type-synthesis techniques to yield a simpler model with greater pragmatic value.

This paper reports analysis equations for the computerized third-order analysis of dyads. Such equations are an extension of those reported by C.H. Suh and C.W. Radcliffe. Examples of tracing cubic of stationary curvature and quartic of derivative curvature for multiloop linkages are herein described.

The instantaneous screw axis is used in the generation of the swept volume of a moving object. The envelope theory is used to determine the boundary surfaces of the swept volume. Specifically, the envelope surfaces generated by a plane polygon, cylindrical and spherical surfaces are presented. Furthermore, the ruled surfaces generated by edges of the moving object are discussed.

The paper presents a new method for the dimensional synthesis of the spatial guidance linkage that features the guided body connected to the base by the interposition of five rods having spherical joints at both extremities. The linkage is required to index the guided body through seven arbitrarily-chosen rigid-body positions. Core of the proposed method is an original algebraic elimination procedure that allows five unknowns to be dropped from a set of six second-order algebraic equations in six unknowns. As a result, a final univariate polynomial equation of twentieth order is obtained whose twenty roots, in the complex domain, represent as many possible placements for a connecting rod. A numerical example is reported.

Type synthesis of rigid-link mechanisms provides a means to determine mechanism topologies before considering link dimensions. The formulation of design procedures for rigid-body mechanisms has benefited from the application of type synthesis techniques. Therefore, type synthesis is seen as a useful tool in the development of design procedures for compliant mechanisms as well. The focus of this paper is to propose and exemplify a design procedure for compliant mechanisms that employs the type synthesis techniques developed for compliant mechanisms.

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.

In this article, a methodology, which yields (in closed-form) the functional relation between “any” two joint variables of a one degree-of-freedom planar mechanism, is developed. For instance, the transmission angle and crank-rotatibility synthesis algorithms (Soylu, 1993; Soylu and Kanberoğlu, 1993) require such a generic input-output equation. The equation is obtained in an optimal manner which minimizes the computational effort associated with it. The tools of theory of elimination and symbolic manipulation are also used in the developed method.

This paper describes a new type of constant-velocity coupling capable of producing solid joint angles greater than 180 degrees. Analytical expressions describing the range of motion are developed based on the kinematic closure equations of an equivalent spatial mechanism. This mathematical description is then used as a tool in the kinematic design and analysis of a constant-velocity coupling capable of producing a 240 degree solid angle output motion.

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.

This paper presents a new viewpoint about structural composition of planar kinematic chains: single-opened-chains, which is composed of binary links, are regarded as basic structural units of mechanisms. The constraint characteristics (the constraint factors, Δ j ) of single opened chains and the constraint characteristics (the coupled degree κ and the κ -algorithm) of mechanical networks are presented. Thus a kinematic chain with v independent loops is regarded to be composed of one basic link and v single-opened-chains in regular sequence. Based on the ordered single-opened-chains and the topological constraints characteristics of mechanisms, a powerful new method for structural synthesis of planar kinematic chains with plane and nonplane linear graphs has been developed. This permits the development of a highly efficient and completely automatic program for the computer-generated enumeration of structural types of mechanisms. The method is illustrated by applying to the case of kinematic chains with up to six independent loops on a personal computer. The ordered single-opened-chains and the topological characteristics are used for setting up a new unified model for structics, kinematics and dynamics of planar mechanisms.

A dual iterative method for displacement analysis of a spatial mechanism is presented in this paper. The algorithm and formulation based upon 3 × 3 dual transformation matrices are succinct. They can be easily implemented in the C H programming language. The algorithm has been numerically verified by dual iterative displacement analysis of an RCCC four-link spatial mechanism.

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.

In this paper we present an algebraic formulation for the constraint manifold of the spatial CC dyad in the image space of spatial displacements. We then present a technique, utilizing the constraint manifold, for performing the dimensional synthesis of mechanisms for rigid body guidance through n positions. Finally, we present numerical results for 10 prescribed positions for the coupler of a spatial 4 C mechanism.

Approximate kinematic equations are developed for multiple-input small-crank mechanisms derived from a basic planar four-bar linkage. Application of a method in which an output link is presumed to comprise a mean and a perturbational motions, the vector loop approach, and a recursive formulation facilitates derivation of the approximate kinematic equations. For a general n-input mechanism, two sets of constraint equations are established: a set of nonlinear recursive equations relating the mean link orientations and a set of linear recursive equations for the purturbational motions. The recursive procedure is demonstrated by developing approximate kinematic equations for four-bar based two-, three- and four-input small-crank mechanisms.

The advancing requirements of instrumentation in astronomy has led to the development of telescopes with segmented active surfaces. In such telescopes, movable segments are controlled together to maintain the correct figure of the reflecting surface. This paper investigates telescope active surfaces by representing each movable segment as a 3-degree of freedom (DoF) parallel manipulator of the type described by Lee and Shah (1988a, 1988b). The workspace of this manipulator is examined for two types of actuator mounting geometries. In the first case, pin joint connections are placed at the base and ball joints at the end-effector. In the second case, the joints are reversed. Following a general investigation, the results are specialized to reflect the behavior of small motions, as active surfaces require only small adjustments. Finally, the effects of the kinematics on the performance of an example surface are determined. The analysis reveals that placing the ball joint connections at the base provides a passive correction of uncontrolled states in the surface.

In this paper, two different procedures are proposed for the optimum design of drag-link planar and spherical four-bar linkages. First, we present a new graphical synthesis method based on 3-dimensional parametric plots of three commonly-used performance indices, namely, the design error, the structural error, and the transmission quality. The plots reveal that minimum values of the design and structural errors occur at a common point in a subspace that is defined by mobility constraints. Second, we propose a numerical procedure based on a constrained nonlinear least-square scheme that aims at optimizing the design error and transmission quality of the linkage. As shown by numerical examples, the graphical method provides a general picture of the problem under study and can be used to evaluate the goodness of the synthesized linkages.