RR Dyad Research Articles

Approximate Motion Generation of an Epicyclic Gear Train With Noncircular Gears Based on Optimization–Homotopy Algorithm

Abstract This study aims to make approximate motion generation that has multiple solutions to facilitate the selection of the best mechanism parameters. Therefore, an approximate synthesis method for an epicyclic gear train (EGT) with noncircular gears based on optimization–homotopy algorithm is proposed in this paper. First, the multipose synthesis objective function of a simplified model of EGT (planar RR dyad) is established. The approximate motion generation is transformed into the extremum problem of multivariate function, and a variable is introduced to expand the solution space. A homotopy algorithm is used to solve the gradient equation of the objective function to obtain the finite range solution domain. Second, the feasible solution domain of EGT is obtained by identifying the total rotation of revolute joints of the RR dyad, and a mechanism error evaluation index is established. In accordance with actual constraints, the most suitable solution is selected in the feasible solution domain, and the relative angular displacement relationship of the selected RR dyad is used to determine the transmission ratio and pitch curve of noncircular gears to realize the design of an EGT. Lastly, a design example is provided to illustrate the feasibility of the method.

A synthesis method of spatial over-constrained mechanisms based on kinematics of serial manipulators

AbstractThis paper proposes a modular method based on the kinematics of serial manipulators to synthesize over-constrained mechanisms. Because the PPP manipulator has an unlimited work space, its end-effector can be constrained to trace a trajectory identical to those of another open-chain manipulator, including a P joint single link and an RR dyad. In doing so, two open-chain manipulators can be concatenated to form closed-loop mechanisms, including PPPP, PPPRR, or PPCR mechanisms. To design over-constrained mechanisms efficiently, the Denavit–Hartenberg convention is adopted to describe the PPP manipulator kinematically, and the Euler angles are utilized to derive geometric constraints of synthesized over-constrained mechanisms. Next, kinematic equations of the PPP manipulator can be modularized and applicable to analyze different closed-loop mechanisms. At last, by adjusting link lengths, twisted angles, and joint angles of the synthesized PPPRR and PPCR mechanisms to form other over-constrained mechanisms configurationally. The novelty of this research lies in modularizing the over-constrained mechanism into two movable serial manipulators whose end-effectors share identical trajectory and orientation. Thus, defining geometrical constraints of the over-constrained mechanism can be transformed into finding angular parameters describing the orientation of these two serial manipulators such that the end-effector coordinate system of two manipulators can properly be aligned. Angular parameters of the serial manipulators can be easily determined by means of Euler angles, which yields an advantage of easy calculation since it only involves the computation of Euler angles parameters. The presented method can be extended to the kinematic synthesis and analysis of more spatial closed-loop mechanisms.

A novel planar motion generation method based on the synthesis of planetary gear train with noncircular gears

Four-bar linkage has been used in motion generation synthesis for a long time, but there are inevitably some shortcomings such as order defects and poor design flexibility. In this work, a planetary gear train with noncircular gears is proposed to realize planar motion generation synthesis. Compared with the four-bar linkage, it has the advantages of a compact structure, full rotatability, and flexible design. The planetary gear train with noncircular gears is composed of two interrelated parts: a two-stage noncircular gear pair and the RR dyad. The synthesis method is divided mainly into three steps: First, the constraint equation of RR dyad for planar motion generation is established, and a solution region of the double revolute joint (RR) dyads is calculated by kinematic mapping theory. Second, after the full rotatability identification of RR dyads and the determination of each transmission ratio, the centrodes of two-stage noncircular gear pairs, which correspond to the specific RR dyad, are obtained. Finally, the planetary gear train with noncir-cular gears is established to pass the given poses. This method has a universal significance and can be applied to many practical cases.

A Unified Approach to Exact and Approximate Motion Synthesis of Spherical Four-Bar Linkages Via Kinematic Mapping

This paper studies the problem of spherical four-bar motion synthesis from the viewpoint of acquiring circular geometric constraints from a set of prescribed spherical poses. The proposed approach extends our planar four-bar linkage synthesis work to spherical case. Using the image space representation of spherical poses, a quadratic equation with ten linear homogeneous coefficients, which corresponds to a constraint manifold in the image space, can be obtained to represent a spherical RR dyad. Therefore, our approach to synthesizing a spherical four-bar linkage decomposes into two steps. First, find a pencil of general manifolds that best fit the task image points in the least-squares sense, which can be solved using singular value decomposition (SVD), and the singular vectors associated with the smallest singular values are used to form the null-space solution of the pencil of general manifolds; second, additional constraint equations on the resulting solution space are imposed to identify the general manifolds that are qualified to become the constraint manifolds, which can represent the spherical circular constraints and thus their corresponding spherical dyads. After the inverse computation that converts the coefficients of the constraint manifolds to the design parameters of spherical RR dyad, spherical four-bar linkages that best navigate through the set of task poses can be constructed by the obtained dyads. The result is a fast and efficient algorithm that extracts the geometric constraints associated with a spherical motion task, and leads naturally to a unified treatment for both exact and approximate spherical motion synthesis.

Complete real solution of the five-orientation motion generation problem for a spherical four-bar linkage

For a spherical four-bar linkage, the maximum number of the spherical RR dyad (R: revolute joint) of five-orientation motion generation can be at most 6. However, complete real solution of this problem has seldom been studied. In order to obtain six real RR dyads, based on Strum’s theorem, the relationships between the design parameters are derived from a 6th-degree univariate polynomial equation that is deduced from the constraint equations of the spherical RR dyad by using Dixon resultant method. Moreover, the Grashof condition and the circuit defect condition are taken into account. Given the relationships between the design parameters and the aforementioned two conditions, two objective functions are constructed and optimized by the adaptive genetic algorithm(AGA). Two examples with six real spherical RR dyads are obtained by optimization, and the results verify the feasibility of the proposed method. The paper provides a method to synthesize the complete real solution of the five-orientation motion generation, which is also applicable to the problem that deduces to a univariate polynomial equation and requires the generation of as many as real roots.

Synthesis of Planar Mechanisms for Pick and Place Tasks With Guiding Positions

A novel dimensional synthesis technique for solving the mixed exact and approximate motion synthesis problem for planar RR kinematic chains is presented. The methodology uses an analytic representation of the planar RR dyad's rigid body constraint equation in combination with an algebraic geometry formulation of the exact synthesis for three prescribed positions to yield designs that exactly reach the prescribed pick and place positions while approximating an arbitrary number of guiding positions. The result is a dimensional synthesis technique for mixed exact and approximate motion generation for planar RR dyads. A solution dyad may be directly implemented as a 2R open chain or two solution dyads may be combined to form a planar 4R closed chain, also known as a planar four-bar mechanism. The synthesis algorithm utilizes only algebraic geometry and does not require the use of a numerical optimization algorithm or a metric on elements of SE(2); the group of planar displacements. Two implementations of the synthesis algorithm are presented; computational and graphical construction. Moreover, the kinematic inversion of the algorithm is also included. Two examples that demonstrate the synthesis technique are included.

Orientation Order Analysis of Spherical Four-Bar Mechanisms

In this paper, we present a novel methodology for orientation order analysis of spherical RR dyads. The methodology is a spherical generalization of the recent works of Myszka, Murray, and Schmiedeler for assessing position order of planar RR dyads. The objective of the methodology is to determine if a prescribed fixed axis location for a spherical RR dyad will result in the dyad guiding a moving rigid-body through a set of finitely separated spherical orientations in the desired order (e.g., 1, 2, 3, 4, etc.). The planar propeller methodology of Myszka, Murray, and Schmiedeler is extended to yield a spherical hoop methodology. The result is a useful tool to determine if a given spherical RR dyad will guide a moving body through a set of prescribed orientations in the desired order. Finally, we demonstrate the utility of the hoop method for order analysis of spherical RR dyads in two case studies.

Assessing Position Order in Rigid Body Guidance: An Intuitive Approach to Fixed Pivot Selection

Several established methods determine if an RR dyad will pass through a set of finitely separated positions in order. The new method presented herein utilizes only the displacement poles in the fixed frame to assess whether a selected fixed pivot location will yield an ordered dyad solution. A line passing through the selected fixed pivot is rotated one-half revolution about the fixed pivot, in a manner similar to a propeller with infinitely long blades, to sweep the entire plane. Order is established by tracking the sequence of displacement poles intersected. With four or five positions, fixed pivot locations corresponding to dyads having any specified order are readily found. Five-position problems can be directly evaluated to determine if any ordered solutions exist. Additionally, degenerate four-position cases for which the set of fixed pivots corresponding to ordered dyads that collapse to a single point on the center point curve can be identified.

Determining Burmester Points from the Analysis of a Planar Platform

This paper presents a technique for determining the Burmester points of five finitely separated positions in a plane. A Burmester point is the location of the fixed pivot of an RR dyad that can reach each of the five positions. These Burmester points are obtained from a “compatibility platform” constructed from selected relative poles associated with the five specified displacements. We show that the relative position pole of two assembly configurations of this platform is a fixed pivot compatible with the five positions. An example originally presented by Sandor and Erdman is used to verify the calculations.