Full Rotatability Research Articles

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 Novel Kinematically Redundant Planar Parallel Robot Manipulator With Full Rotatability

This paper presents a novel kinematically redundant planar parallel robot manipulator, which has full rotatability. The proposed robot manipulator has an architecture that corresponds to a fundamental truss, meaning that it does not contain internal rigid structures when the actuators are locked. This also implies that its rigidity is not inherited from more general architectures or resulting from the combination of other fundamental structures. The introduced topology is a departure from the standard 3-RPR (or 3-RRR) mechanism on which most kinematically redundant planar parallel robot manipulators are based. The robot manipulator consists of a moving platform that is connected to the base via two RRR legs and connected to a ternary link, which is joined to the base by a passive revolute joint, via two other RRR legs. The resulting robot mechanism is kinematically redundant, being able to avoid the production of singularities and having unlimited rotational capability. The inverse and forward kinematics analyses of this novel robot manipulator are derived using distance-based techniques, and the singularity analysis is performed using a geometric method based on the properties of instantaneous centers of rotation. An example robot mechanism is analyzed numerically and physically tested; and a test trajectory where the end effector completes a full cycle rotation is reported. A link to an online video recording of such a capability, along with the avoidance of singularities and a potential application, is also provided.

Geometric analysis of coupler-link mobility and circuits for planar four-bar linkages

The problem of mobility analysis of planar four-bar linkages is revisited. The mobility is analyzed in terms of the range of coupler angle by a geometric approach, with a special interest on its full rotatability when the linkage is driven by the coupler link. A constraint triangle which characterizes the movement of the coupler link is constructed. By virtue of the constraint triangle, a new formulation on the couple curve rotation is derived, upon which the ranges of coupler angle are determined and the turning points are identified. Moreover, with the range of coupler angle, the number of circuits is determined. Examples are included to demonstrate the mobility analysis with the new approach.

Mobility Criteria of Planar Single-Loop N-Bar Chains With Prismatic Joints

This paper presents the extension of the N-bar rotatability laws to N-bar chains containing prismatic joints. The extension is based on the principle that a prismatic joint may be regarded as a revolute joint located at infinity in the direction normal to the sliding path. The effects of long and short links, full rotatability, linkage classification, and formation of branches and sub-branches are discussed. The extension provides a consistent method to understand all aspects of linkage rotatability disregarding the existence of prismatic joints. The results are demonstrated by several examples.

Full Rotatability and Singularity of Six-Bar and Geared Five-Bar Linkages

Full rotatability identification is a problem frequently encountered in linkage analysis and synthesis. The full rotatability of a linkage refers to a linkage, in which the input may complete a continuous rotation without the possibility of encountering a dead center position. In a complex linkage, the input rotatability of each branch may be different. This paper presents a unified and comprehensive treatment for the full rotatability identification of six-bar and geared five-bar linkages, regardless of the choice of input joints or reference link. A general way to identify all dead center positions and the associated branches is discussed. Special attention and detail discussion is given to the more difficult condition, in which the input is not given through a joint in the four-bar loop or to a gear link. A branch without a dead center position has full rotatability. Using the concept of joint rotation space, the branch of each dead center position, and hence, the branch without a dead center position can be identified. One may find the proposed method to be generally and conceptually straightforward. The treatment covers all linkage inversions.

Stretch rotation and complete mobility identification of Watt six-bar chains

Mobility of linkages refers to the problems concerning branch, full rotatability, singularities, and order of motion. This paper uses the stretch and rotation of a four-bar loop to convert a Watt six-bar linkage to an equivalent simple Stephenson linkage. It shows the mobility of a Watt six-bar linkage is affected by a hidden five-bar chain. The equivalency offers a simple and clear visual explanation on the formation of branches and sub-branches and how Watt and Stephenson linkages differ in mobility. The resulting mobility algorithm requires no stretch rotation. The dead center positions in the second four-bar loop are the branch points. It reveals that although a Watt six-bar linkage may have only up to four-branches, a branch may have up to six sub-branches. The results offer a simple algorithm suitable for automated identification of branch, sub-branch, and full rotatability. The algorithm is valid for Watt linkages with or without prismatic joints and is independent of linkage inversions. Examples are presented for illustration.

On mechanism design optimization for motion generation

A set of Burmester curves can represent an infinite number of planar four-bar motion generator solutions for a given series of prescribed rigid-body poses. Unfortunately, given such a vast number of possible mechanical solutions, it is difficult for designers to arbitrarily select a Burmester curve solution that ensures full link rotatability, produces feasible transmission angles and is as compact as possible. This work presents an algorithm for selecting planar four-bar motion generators with respect to Grashof conditions, transmission angle conditions and having the minimal perimeter value. This algorithm has been codified into MathCAD for enhanced analysis capabilities and ease of use. The example in this work demonstrates the synthesis of a compact planar, four-bar crank rocker motion generator with feasible transmission angles.

Kinematic design of the RSSR-SC spatial linkage based on rotatability conditions

This paper studies the relationship between the physical dimensions of the RSSR-SC spatial linkage and the range of motion of its input link. The study results in a quartic equation that defines any configuration of the linkage for given values of the input link angle. Obtained from this quartic equation, we use three polynomial discriminants in order to find the corresponding rotatability condition for all cases available for the motion of the input link. Examples for each case are provided. Moreover, during the design process, a branching analysis is also conducted, since the main goal of this work is to design the RSSR-SC linkage in such a way that its input link has full rotatability without the undesirable branching problem.

A unified approach for the investigation of branch and circuit defects

In the first part of this paper, a graphical method is proposed which unifies the resolution of branch and circuit defects, and full or partial input link rotatability in graphical mechanism synthesis. Linkages with partial rotatability of the input link can have one single range of motion of the input link or two disjointed ranges of motion of the input link. Circuit defect will appear only if the input link has two disjointed ranges of motion and if the design positions belong to different ranges. Starting from the follower link, the graphical method develops elliptical and hyperbolic boundaries of regions, in a fixed reference system, where the choice of the ground pin of the driving (input) link either would yield a mechanism free of branch and circuit defects, or most likely would yield potential-circuit-defective-mechanisms (PCDM). Then, the analytical method is presented and it involves the development of the algebraic curves, in a reference system connected to the coupler, which represents the boundaries of the PCDM regions. The analytical method enables the unification of the PCDM regions with the branch defect region (due to Filemon) and also those for the design of mechanisms with transmission angle within prescribed limits (due to Gupta). Both graphical and analytical methods are developed and illustrated with examples for the planar four-bar mechanisms.

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.

Mobility criteria of geared five-bar linkages

This article proposes simple Grashof type criteria to identify the full rotatability of geared five-bar linkages and provides a systematic approach to study the effects of the five-bar loop and gear constraint in the mobility of a geared five-bar linkage. A fully rotatable geared linkage must have a C-C (crank-crank) type host five-bar linkage. All fully rotatable geared linkages are grouped into A, B, and C types according to the joint rotation space of their host linkages. In type A linkages, a gear constraint may have any gear ratio and phase angle unless a change-point position exists. In type B linkages, the gear ratio may be a unit or other values but the phase angle must be properly chosen to achieve full rotatability. In type C linkages, the two terminal gears must rotate with the same speed in the same direction relative to the center link between them and have a proper phase angle. Any geared linkage which does not belong to one of these three types does not have full rotatability. A table summarizing the required phase angle conditions for a fully rotatable geared linkage with a unit gear ratio is also presented.

Rotatability Laws for N-Bar Kinematic Chains and Their Proof

This paper presents a thorough proof and adds valuable insight into the theorems [1] concerning the mobility of planar N-bar linkages. The paper includes four parts: the assemblability, the revolvability between two links, the full rotatability, and the classification. The proof is based on the assemblability condition of N-bar linkages and the revolvability condition of an angle. Examples are also provided to further clarify the use of these theorems. In these theorems, Grashof’s Law becomes a very special case.

Mobility Criteria of Single-Loop N-Bar Linkages

Single-closed-loop N-bar mobility criteria are presented. These criteria provide a simple and effective method (1) to determine the full rotatability of any single closed-loop linkages, (2) to predict the revolvability between two adjacent links, (3) to explain and identify the existence of singular positions, (4) to classify linkages, and (5) to identify the difference of singular positions between linkages of different types. These criteria are fundamental tools toward a better understanding on the mobility of more complicated linkages.

Rotatability Considerations for Spherical Four-Bar Linkages With Applications to Robot Wrist Design

The following result for spherical four-bar mechanisms is proved: Let all links of a spherical four-bar linkage be in the range [0, π]. Consider two linkages—one primary (p) and another secondary (s) according to the condition αip + αfp < αis + αfs where αs = π − αp. Then the input link is fully rotatable iff αip + αfp < αc + α0, |αip − αfp| > |αc − α0| and αip + αfp + αc + α0 < 2π. An alternate form of this result also is developed. Applications of these results in determining full or partial rotatability of grippers in spherical robot wrists are discussed.

Synthesis of fully rotatable R-S-S-R linkages

Algebraic-geometrical techniques to design fully rotatable R-S-S-R linkages with transmission angle control are presented. The concept of upper bounding Cos μ where μ is the transmission angle, is used to obtain feasible regions for linkages with full input-link rotatability. The proposed feasible regions have algebraic boundaries. This concept was developed earlier for the precision synthesis of planar four-bar linkage and it has now been extended to the precision synthesis of R-S-S-R linkage.