RSSR Mechanisms Research Articles

Kinematic geometry of spatial RSSR mechanisms

Two different novel methods to derive the input–output (IO) equation of arbitrary RSSR linkages are described. Both methods share some common elements, i.e., they use the standard Denavit–Hartenberg notation to first describe the linkage as an open kinematic chain, and Study’s kinematic mapping to describe the displacement of the coordinate frame attached to the end-effector of the chain with respect to the relatively non-moving base frame. The kinematic closure equation is obtained in the seven-dimensional projective kinematic mapping image space by equating the eight Study soma coordinates to the identity array. Then two methods are successfully applied to eliminate the intermediate joint angle parameters leading to the degree four biquadratic implicit algebraic IO equation: (a) the linear implicitisation algorithm, which can be applied after rearranging the closure equation such that the linkage can be viewed as two serial RS chains, and (b) numerical elimination theory using pseudowitness sets. Both approaches lead to the same IO equation. The utility of this algebraic form of the IO equation is illustrated with three detailed application examples.

Kinematic Geometry of Spatial Rssr Mechanisms

Kinematic Geometry of Spatial Rssr Mechanisms

A New Coupler Critical Dimensions (CCD) Method for Linkage Mechanisms Mobility Analysis

A new Coupler Critical Dimensions (CCD) approach to define mobility criteria (crank, rocker conditions, or existence) for linkage mechanisms has been presented in this paper. The idea is to analyze the extreme lengths of a mechanism coupler link when the mechanism is on the extreme of its existence or changing its mobility condition. The method leads a set of expressions of the constant mechanism parameters that can be used to define the exact dimensional limits of the coupler link in the mechanism. These expressions present sufficient and necessary dimensional conditions for the mechanism existence and become a turning point to change its mobility from crank to a rocker and vice versa. At the boundaries of the coupler dimensions, the mechanism reaches its the change-point configuration where the mechanism may switch either from one work function to another or from existence to non-existence. The method has been successfully applied to the planar 4R, spatial RSSR, and planar multiloop linkage mechanisms. The obtained results prove the effectiveness and accuracy of the method in defining the limits of the mechanism rotatability conditions or existence in general.

On the Spin Surface of RSSR Mechanisms With Parallel Rotary Axes

Due to Cayley’s theorem the line s∊Σ (=moving system) spanned by the centers of the spherical joints of a revolute-spherical-spherical-revolute linkage generates a surface of degree 8. In the special case of parallel rotary axes of the R-joints the corresponding ruled surface is only of degree 6. Now the point locus of any point X∊Σ\{s} is a surface of order 16 (general case) or of order 12 (special case). Hunt (1978, Kinematic Geometry of Mechanisms, Clarendon, Oxford) suggested that the circularity of this so called spin-surface for the general case is 8 and this was later proved. We demonstrate that the circularity of the spin-surface for the special case is 4 instead of 6 as given in the literature (1994, “The (True) Stewart Platform Has 12 Configurations,” Proceedings of the IEEE International Conference on Robotics and Automation, pp. 2160–2165). As a consequence generalized triangular symmetric simplified manipulators (the three rotary axes need not be coplanar) with two parallel rotary joints can have up to 16 solutions instead of 12 (2006, Parallel Robots, 2nd ed., Springer, New York). We show that this upper bound cannot be improved by constructing an example for which the maximal number of assembly modes is reached. Moreover, we list all parallel manipulators of this type where more than 4×2=8 points are located on the imaginary spherical circle.

Assembly Configurations of Spatial Single-Loop Single-DOF Mechanisms

This paper examines the problem of finding the assembly configurations (ACs), also called circuits, of spatial single-loop single-DOF mechanisms. All six lower pairs are considered: revolute (R), cylindric (C), prismatic (P), helical (H), spheric (S), and planar (E). The RSSR, CSS, RRSC, CCS, and RRSRR mechanisms are covered, along with all variations in which an R joint is replaced with a P or H joint, or an S joint is replaced with an E joint. The RRCRC, RRRCC, and RRRRCR mechanisms are covered, along with all appropriate variations in which an R joint is replaced by a P or H joint. A numerical method is given to find the ACs, and is illustrated with several examples.

Mobility Analysis of RSSR Mechanisms by Working Volume

An RSSR mechanism is analyzed by treating the linkage as two manipulators. The mobility region of the input link or the output link is derived from the intersection region of the working volumes generated by two manipulators. It can be found by solving the region bounded by an ellipse and a unit circle. Concise discriminants and straightforward strategy are developed to determine the type for some special RSSR linkages. Grashof’s rule is derived for determining the type of planar four-bar and proved to be the primary criterion for simply skewed four-bar.

On Realization of Spherical Joints In RSSR Mechanisms

A spherical joint must allow adequate mobility between the ball and socket. Unless the specific geometric requirement of the spherical joint, namely the trajectory of the ball stem in the socket, is known, the realization of a synthesized mechanism containing a spherical joint can become an endless trial and error process or even a fruitless attempt. Typical examples can be found in RSSR mechanisms. For a synthesized RSSR mechanism, this paper presents the closed form equation describing the spherical trajectory of the ball in the socket and the design and manufacturing issues of the spherical joints. It offers the classification scheme of spherical joints based on the required ball rotatability in the socket and shows how the necessary socket opening of the spherical joints is affected. The size and location of the socket in a spherical joint must be properly determined to meet the rotatability requirement and in many situations, a conventional spherical joint with a circular socket opening cannot be used. Essential geometric and location information regarding to the use of non-circular socket opening, spherical grooved pair, as well as revolute joints to replace the conventional spherical joints to realize a synthesized RSSR mechanism is presented. The proposed concept and method can be extrapolated to other mechanisms containing spherical joints.

The RSSR mechanisms with partially constant transmission angle

It is required to calculate roots of a quartic equation for finding extremums of the transmission angle of an RSSR mechanism. If three roots of this equation are equal, the transmission angle will remain nearly constant within an interval. By normalizing the mechanism dimensions, the required conditions for these types of mechanisms are discussed.

Mobility analysis of spatial 4- and 5-link mechanisms of the RS class

When a spatial mechanism has many links, the mobility analysis and the input-output relation are very difficult to obtain due to the high degree of the underlying polynomial. For spatial RS-class mechanisms such as RSSR, RRSS, and RRSRR mechanisms, the quadratic input-output relation and the associated mobility condition are developed in this paper. The method is based on the geometric interpretation of the distance function for spatial mechanisms. The method can be applied to mechanisms which have a symmetric relation between two conjugate configurations. Especially, for the spatial 5-link RRSRR mechanism, a maximum of 12 limit positions of the crank are obtained by the proposed method.

The spatial RSSR mechanisms with constant transmission angle

The dimensional conditions for the existence of spatial RSSR mechanisms having constant transmission angle are investigated. For this purpose, the dimensions to make zero coefficients of trigonometric terms in the expression of transmission angle, are obtained. By normalizing these with respect to a1, a design chart is presented.

Mobility Analysis and Type Identification of Four-Link Mechanisms

An alternative derivation of the simple criteria, in terms of linkage parameters, demarking the zones of nonexistence of double-crank and crank-rocker RSSR mechanisms is presented. The existence of a crank is verified by polynomial discriminants of the limit position. Closed from solutions are obtained for type identification for 4R spherical, RCCC, RSSP, and planar 4R mechanisms by treating these as the limiting cases of an RSSR mechanism.

Approximated Grashof-Type Movability Conditions for RSSR Mechanisms With Force Transmission Limitations

Closed-form Grashof-type movability conditions are derived for closed-loop RSSR mechanisms using a geometrical approximation technique. The conditions that ensure the presence of crank-rocker and drag-link type mechanisms are derived with and without force transmission limitations. The force transmission limitations may be specified as a function of the output link angle. The accuracy of the approximated conditions is analyzed. As an example, the conditions are used to synthesize a function generating mechanism with fully rotatable crank and with various force transmission requirements. The developed technique is general, and can be applied to other similar spatial mechanisms. The application of this approach to geometrical synthesis of open-loop chain manipulators is discussed.

On the elimination of branching in the synthesis of spatial single-loop mechanisms with lower pairs

SUMMARYA single-loop spatial mechanism kinematically becomes an open robot, if we separate the grounded joint of the input link which may then be considered as the end effector of the robot. Any position of the end-effector within the workspace of such an open robot can be reached via a number of different configurations of the links. These configurations are called “branches” of the open robot for that particular position of the end effector.If the open robot is now stretched to a limiting position by a force exerted on the end effector, all the possible branches of the mechanism approach each other. When they become coincident, they form the “limiting configuration”. Any two related branches are at opposite sides of the limiting configuration. From the relationship between the links in th elimiting configuration and in related branches, conditions for aviodance of branching of the original closed-loop mechanism can be obtained. This is necessary in order to assure that a set of consistent relative displacements are specified for the open robot to move displacements are specified for the open robot to move toward the desired end-effector position without jumping from one branch to another. As for the closed-loop mechanism, open robot branching aviodance ensures that a desired sequence of positions of a particular floating link in the loop will be generated without changing the branch of the link configuration.In this paper, the above approach is applied to RSSR, RRSC, RRSRR, RRRRRRR and RPCRRR spatial closed-loop motion-generator mechanisms and the corresponding conditions for aviodance of branching in the synthesis of the mechanisms are derived.

The analytical synthesis of spring elements in spatial mechanisms

This paper deals with an algebraic method for dynamic synthesis of spatial mechanisms. By combining the extension of Burmester's Theorem and the concept of energy equilibrium, the relationship between the spring constraint and the geometric constraints in spatial mechanism is analyzed. Furthermore, this paper develops the equations for synthesis of RCCC, RSSR and RRRSR mechanisms. Finally an approximation, considering spring free length, is developed.

Mechanism Link Rotatability and Limit Position Analysis Using Polynomial Discriminants

A theory is proposed for algebraically determining the limit positions of single-degree-of-freedom mechanisms. The absence of limit positions indicates that the link being considered is a fully rotating crank. This theory is applied in the present paper to the RSSR and RRSS spatial mechanisms. Conditions for spatial mechanisms analogous to Grashof’s law should be attainable using this theory. A numerical example is given to illustrate the theory.

A note on the synthesis and dissimilar function-cognates of the RSSR and RCR 0CR mechanisms

Function-cognates are those mechanisms that provide exactly the same input-output displacement relationships as the original mechanism. Dissimilar cognates are cognates having a different structure from that of the original mechanism. It is shown that there are no RSSR mechanisms function-cognate to a given planar four-bar or spherical four-bar. On the other hand, formulae for the infinity of RCR 0CR cognates of the planar four-bar are given. It is further shown that for a given RCR 0CR mechanism (or for a given general trigonometric implicit relationship linear in cosines and sines of input and output rotations individually), there is always one and only one RSSR mechanism giving the same relationship. The latter is therefore also the case for an eight-precision point synthesis of the RSSR mechanism.

Mechanical Error Analysis of Spatial Linkages

In this paper the effect on the output error of function-generating spatial linkages due to tolerances on the links and clearances at the hinges is analyzed. Tolerances on link lengths are assumed to be normally distributed. For the clearance error in spherical, prismatic, and revolute joints, uniform probability density function is assumed. However, the models developed and the analysis procedure proposed can be used for any other probability density function including the mean and variance values derived from experimental information. A synthesis procedure to allocate tolerances and clearances on different members of linkages to restrict the output error within specified units is developed. Results of analysis and synthesis are given for two example problems involving RRSS and RSSR mechanisms.

Complete Force Balancing of Spatial Linkages

A new method permits the complete force balancing of spatial linkages such as the RSSR and RSSP mechanisms. Operator equations are written describing the location of the center of mass for the given mechanism. The coefficients of the time-dependent terms are set to zero by properly locating the mass centers of selected links. As a result, the location of the total center of mass is invariant and there is no resultant shaking force.