3R Chain Research Articles

Structural–parametric synthesis of single-loop 6R and 7R mechanisms using factorization of motion polynomials and its application in grippers

Abstract In this study, the so-called quasi-spiral motion pattern of a particular 3R (revolute) chain, which could track a quasi-spiral curve, is discovered for application in envelope gripping. Naturally, a novel perspective on the synthesis of 1-DOF (degree of freedom) single-loop 6R and 7R mechanisms is presented to implement the quasi-spiral motion pattern. In these processes, the method of factorization of the motion polynomial is used to synthesize a series of single-loop mechanisms with the motion to generate quasi-spiral curves. By employing the proposed method, numerous innovative 6R and 7R mechanisms have been developed to construct dual layers of grippers based on specific network regulations. Taking a novel 6R mechanism as an example, the forward solution of the mechanism is determined, and the relationship between different rotation angles is obtained. Then, displaying the changes in each angle during the grasping process. Finally, the novel 6R and 7R mechanisms are employed as a unit to determine the networking method and assemble new types of grippers. Grippers formed by the aforementioned 1-DOF mechanisms exhibit the characteristics of envelope grasping.

Mixed Position and Twist Space Synthesis of 3R Chains

Mixed-position kinematic synthesis is used to not only reach a certain number of precision positions, but also impose certain instantaneous motion conditions at those positions. In the traditional approach, one end-effector twist is defined at each precision position in order to achieve better guidance of the end-effector along a desired trajectory. For one-degree-of-freedom linkages, that suffices to fully specify the trajectory locally. However, for systems with a higher number of degrees of freedom, such as robotic systems, it is possible to specify a complete higher-dimensional subspace of potential twists at particular positions. In this work, we focus on the 3R serial chain. We study the three-dimensional subspaces of twists that can be defined and set the mixed-position equations to synthesize the chain. The number and type of twist systems that a chain can generate depend on the topology of the chain; we find that the spatial 3R chain can generate seven different fully defined twist systems. Finally, examples of synthesis with several fully defined and partially defined twist spaces are presented. We show that it is possible to synthesize 3R chains for feasible subspaces of different types. This allows a complete definition of potential motions at particular positions, which could be used for the design of precise interaction with contact surfaces.

Solution region synthesis method of planar one-dof six-bar linkages for specifying 4-positions of a 3R open chain

This paper addresses the synthesis problem of one-dof six-bar linkage motion generation for four precision points. A novel and convenient synthesis approach is introduced. With our method, a 3R chain that guides the movement of its end-effector through a set of four specified task positions is given. We then attach two RR links to this 3R chain to obtain a six-bar linkage. Motion generation can produce an infinite number of solutions for four desired positions. It is difficult for designers to synthesize defect-free linkages and synthesize the best linkage through rapidly and precisely selecting curve points that satisfy every design condition. To solve this problem, a solution region method is given in this paper. With this method an infinite number of mechanism solutions can be expressed on a coordinate plane. In addition, the paper still puts forward a distinguishing method for testing circuit and branch defects for Watt-I six-bar linkages. The feasible mechanism solution regions without any defects are obtained by using the solution region method.

A Design System for Six-Bar Linkages Integrated With a Solid Modeler

This paper presents a design system for planar and spherical six-bar linkages, which is integrated with a solid modeler. The user specifies a backbone 3R chain in five task configurations in the sketch mode of the solid modeler and executes the design system. Two RR constraints are computed, which constrain the 3R chain to a single degree-of-freedom six-bar linkage. There are six ways that these constraints can be added to the 3R serial chain to yield as many as 63 different linkages in case of planar six-bar linkages and 165 in case of spherical six-bar linkages. The performance of each candidate is analyzed, and those that meet the required task are presented to the designer for selection. The design algorithm is run iteratively with random variations applied to the task configurations within user-specified tolerance zones, to increase the number of candidate designs. The output is a solid model of the six-bar linkage. Examples are presented, which demonstrate the effectiveness of this strategy for both planar and spherical linkages.

Motion Generation of Planar Six- and Eight-Bar Slider Mechanisms as Constrained Robotic Systems

In this paper, we formulated a systematic design methodology for the design of planar six- and eight-bar slider mechanisms for motion generation applications that require more complex motion than the slider–crank mechanism. We show how two RR dyads can be synthesized and attached to planar PRR and PRR–3R chain for the dimensional synthesis of planar six- and eight-bar slider mechanisms, respectively. The results are 15 different types of one degree-of-freedom planar six- and eight-bar linkages with a prismatic joint at its base. We demonstrate the design process with the design of a multifunctional wheelchair that could transform its structure between a self-propelled wheelchair and a walking guide meant for outpatient rehabilitation purpose.

Geometric characterization of the workspace of non-orthogonal rotation axes

In this article, a novel characterization of the workspace of 3R chains with non-orthogonal, intersecting axes is derived by describing the set of singular orientations as two tori that separate two-solvable from non-solvable orientations within $SO(3)$. Therefore, the tori provide the boundary of the workspace of the axes' constellation. The derived characterization generalizes a recent result obtained by Piovan and Bullo. It is based on a specific, novel representation of rotations, called unit ball representation, which allows to interpret the workspace characterization with ease. In an appendix, tools for dealing with angles and rotations are introduced and the equivalence of unit quaternion representation and unit ball representation is described.

Kinematics of Hex-Piderix - A Six-Legged Robot - Using Screw Theory

In this paper the kinematic analysis of a six-legged robot, hereafter named Hex-piderix, is carried out. A three revolute (3R) chain has been chosen for each limb in order to mimic the leg structure of an insect. The rotation matrix, with unitary vectors, and the Denavit-Hartenberg (D-H) conventions are used to find the pose of the thorax. The problem of inverse position is solved by geometrical analysis. The direct and inverse infinitesimal kinematics are obtained by the reciprocal screw theory, considering a suction cup attached to each leg and modelling it as a UP linkage. A numerical example of the thorax pose was made by solving the equations obtained from the direct position analysis. The equations of the inverse position analysis were solved to obtain the angles of the joints. Finally, the velocity values of the thorax obtained from the infinitesimal kinematics were validated by simulating the movements of Hex-piderix using specialized software.

Structural Synthesis of 6 Bar Mechanisms as Mechanically Constrained 3R Chains

In this paper, we consider the planar robot formed by 3R chain. To mechanically constrain the relative movement of the joints so that the end-effector reaches a specified set of task positions, two additional links are added to the 3R chain. Graphical representation is presented for enumeration process. Reverse transformation technique is developed to reconstruct 6-bar mechanisms from their corresponding graphs. Structural synthesis process yields designs for seven different forms of a six-bar linkage the Watt I and Stephenson I, II, and III six-bar linkages.

21st Century Kinematics: Synthesis, Compliance, and Tensegrity

As we move into the second decade of the 21st century, we can identify three research trends that we can expect to persist into the future. They are the analysis and synthesis of (i) spatial mechanisms and robotic systems, (ii) compliant linkage systems, and (iii) tensegrity and cable-driven systems. In each case, we find that researchers are formulating and solving polynomial systems of total degrees that dwarf those associated with major kinematics problems of the previous century.

The synthesis of six-bar linkages as constrained planar 3R chains

Abstract In this paper, we constrain a planar serial chain formed by three revolute joints, so that it becomes a one degree-of-freedom six-bar linkage. Our focus is on the various ways that two RR chains can be attached to constrain the links of the 3R chain, and ensure that the end-effector passes through a set of five specified task positions. This synthesis process yields designs for the Watt I and Stephenson I, II, and III six-bar linkages. This approach does not apply to the Watt II because its floating link is not connected to the ground frame by a 3R chain. We demonstrate the synthesis process with the design of a six-bar steering linkage intended for use on a gravity racer.