Polynomial Solutions to the Displacement Analysis of 10-Link 1-DOF Mechanisms

This paper presents closed-form polynomial solutions to the displacement analysis problem of planar 10-link mechanisms with 1 degree-of-freedom (DOF). Using the successive elimination procedure presented herein, the input-output (I/O) polynomials as well as the number of assembly configurations for five mechanisms resulting from two 10-link kinematic chains are presented. It is shown that the displacement analysis problems for all five mechanisms can be reduced to a univariate polynomial devoid of any extraneous roots. This univariate polynomial corresponds to the I/O polynomial of the mechanism. In addition, one of the examples also illustrates how trigonometric manipulations in conjunction with tangent half-angle substitutions can lead to non-trivial extraneous roots in the solution process. Theoretical conditions for identifying and eliminating these extraneous roots are also presented.

Structure Types Synthesis of Multiloop Spatial Kinematic Chains With General Variable Constraints

A systematic procedure is presented for the structure type synthesis of multiloop spatial kinematic chains with general variable constraints in this paper. The parameters and the structure types of the contracted graphs and the branch chains used to synthesize such kinematic chains are given for kinematic chains with up to four independent loops. The assignments for the constraints values of all the loops in a kinematic chain are discussed. Using these as the basis, the structure types of the multiloop spatial kinematic chains with hybrid constraints could be synthesized.

Mobility in Metamorphic Mechanisms of Foldable/Erectable Kinds

This paper looks at class of mechanisms that change structure when erected or folded. The class includes a variety of artefacts and decorative gifts and boxes comprised of flat card creased to enable the folding or unfolding of a structure. Such a structure admits kinematics study in keeping with theory of mechanisms when the creases are treated as hinges joining card and paper panels treated as links. New horizons have been brought up in the use and mechanised manufacture of mechanisms of this kind. Here typical types are described in terms of their fundamental parts and their equivalent mechanisms. Screw system theory is brought into the analysis of mechanisms of these kinds, particularly those containing multiple loops. Different geometry and system combinations are used for the study of mobility and kinematics making use of the result from the equivalent screw systems.

A Proposed Categorization Method and Category Domain Inequality Development Procedure for Ground Joints of Five Bar Linkages

This paper presents a method for the development of sets of symbolic inequalities in terms of link lengths for the prediction of the rotation capabilities of ground joints of single-loop five-bar linkages. The inequalities are obtained from the combination of the loop equation of a five-bar linkage and its derivatives and the application of simple logic operations. The rotation capabilities of ground joints are divided into three categories: the incomplete-rotation ground joints, the conditioned complete-rotation ground joints, and the unconditioned complete-rotation ground joints. The derived sets of inequalities define the domain, in a five-dimension space of the five link lengths, for each of the rotation categories. In this paper, the definition of each category is clearly described and the derivations of sets of inequalities are explained in details. A computer program was constructed to examine the completeness and correctness of the categorization method and to analyze the given five-bar linkages to determine the appropriate categories for their ground joints.

An Introduction to Burmester Field Theory

This paper introduces the Burmester field for motion-generation dyads with four design positions. The Burmester field is the region swept by a Burmester curve as one or more of the design positions varies. The Burmester field’s geometric features are shown to be related to the poles. The most significant feature are anchor poles that remain stationary as the design positions are changed. The envelope of the Burmester field found with the variation of a single design parameter is presented. This work demonstrates that the envelope of the Burmester field consists of segments of Burmester curves and segments found using envelope theory. A number of examples are presented and discussed.

Orthonormal Isotropic Vector Bases

Orthonormal bases of isotropic vectors for indefinite square matrices are proposed and solved. A necessary and sufficient condition is that the matrix must have zero trace. A recursive algorithm is presented for computer applications. The isotropic vectors of 3 × 3 matrices are solved explicitly. Deviatoric stresses in continuum mechanics, the existence of isotropic vectors (particularly in screw space), and stiffness synthesis by springs are shown to be related to the isotropic vector problem.

Numerical Solution for Manipulator Forward Dynamics BVPs

A new algorithm is presented for iterative solution of systems of nonlinear ordinary differential equations (ODEs) with any order for multibody dynamics and control problems. The collocation technique (based on the explicit fixed-point iteration scheme) may be used for solving both initial value problems (IVPs) and boundary value problems (BVPs). The BVP is solved by first transforming it into the IVP. If the Lipschitz constant is large and the algorithm diverges in a single (‘long’) domain, the domain is partitioned into a number of subdomains and the local solutions of the corresponding BVPs are matched either locally (in parallel) or globally. The technique is general and may be applied to general systems of ODEs in any field. As an illustration, the forward dynamics problem of a manipulator is solved as an IVP and then as a BVP.

Linkage Synthesis of a Four-Bar Mechanism for N Precision Points Using Simulated Annealing

This paper presents an alternative way of linkage synthesis by using a computational intelligence technique: Simulated Annealing. The technique allows to define n precision points of a desired path to be followed by a four-bar linkage (path generation problem). The synthesis problem is transformed into an optimization one in order to use the Simulated Annealing algorithm. With this approach, a path can be better specified since the user will be able to provide more “samples” than the usual limited number of five allowed by the classical methods. Several examples are shown to demonstrate the advantages of this alternative synthesis technique.

A Computer Implementation of the Position Solution of Planar Linkages Using Vector Loop Decomposition

A general closed-form approach to the solution of loop equations of planar n-bar linkages is presented. Each loop of a set of canonical independent loops is decomposed to a set of vectors. Several common combinations of revolute and prismatic joints are defined. By evaluating the types of joints at each end of a vector, the magnitude and direction of the vector are determined to be known constants or unknown variables. This leads to an identification of the number of unknowns and the distribution of unknowns in the loop. This identification allows the unknowns to be found by matching the situation to one of the unique, closed-form cases for a solvable loop. A computer software application has been developed and is analyzed for efficiency.

Analytical Dynamics of Unrooted Multibody-Systems With Symmetries

The objectives of this paper are to (i) exploit the structure of Euler-Liouville equations for multibody systems and separate the external and internal aspects of motion, (ii) specialize these equations to systems with special mass and geometric properties such as holonomoids and orthotropoids, (iii) apply the results to special orthotropoids, the spheroidal linkages of Wohlhart, and write their equations of motion in a simple and elegant manner.