Screw System Theory Research Articles

Kinestatic Analysis of General Parallel Manipulators

Robotic mechanisms in general can be of either serial-chain, parallel-chain, or hybrid (a combination of both parallel and serial chains) geometry. While it can be asserted that kinematic theories and techniques are well established for fully serial-chain manipulators, the same assertion cannot be made when it is considered in the general context. In this article, we present a general procedure for systematic formulation and characterization of the instantaneous kinematics for a robotic mechanism with a general parallel-chain geometry. A kinestatic approach based on screw system theory is adopted in this treatment. The resulting equation is a compact Jacobian matrix of the system which includes attributes from not only the active joints but also the passive constraints. An example has been provided to demonstrate the methodology as well as its theoretical feasibility.

Reflecting on the life of Kenneth H. Hunt and his contributions to Mechanism

Kenneth Henderson Hunt, see Fig. 1, was the Foundation Professor of Engineering, and the inaugural Dean of the Faculty of Engineering at Monash University, Victoria, Australia. He was the Dean of Engineering from 1961 until 1975 and then served as the Chair of Mechanism from 1976 until 1985. He is remembered as the scientist who rediscovered Ball's theory of screws and who made significant contributions to screw system theory which he applied to the kinematics of spatial mechanisms, shaft couplings, and robotics. Kenneth authored two classical books Mechanisms and Motion and Kinematic Geometry of Mechanisms, and co-authored the more modern text Robots and Screw Theory: Applications of Kinematics and Statics to Robotics. He was also one of the founding fathers, contributing to the foundation act with the initial member organizations, of the International Federation for the Theory of Machines and Mechanisms (in 1998 the official name was changed to IFToMM, acronym for the International Federation for the Promotion of Mechanism and Machine Science). The federation was officially born in 1969 and Kenneth served on the first Executive Council and was the founding chairman of the Australian Committee.

Relevance and Transferability for Parallel Mechanisms With Reconfigurable Platforms

The parallel mechanism with a reconfigurable platform retains all advantages of parallel mechanisms and provides additional functions by virtue of the reconfigurable platform, leading to kinematic coupling between limbs that restricts development of the mechanism. This paper aims at dealing with kinematic coupling between limbs by investigating the transferability of limb constraints and their degrees of relevance to the platform constraints based on the geometric model of the mechanism. The paper applies screw-system theory to verifying the degree of relevance between limb constraint wrenches and platform constraint wrenches, and reveals the transferability of limb constraints, to obtain the final resultant wrenches and twists of the end effector. The proposed method is extended to parallel mechanisms with planar n-bar reconfigurable platforms, spherical n-bar reconfigurable platforms, and other spatial reconfigurable platforms and lends itself to a way of studying a parallel mechanism with a reconfigurable platform.

Reconfiguration of the plane-symmetric double-spherical 6R linkage with bifurcation and trifurcation

This study presents a double-spherical 6R overconstrained linkage which is a variant of the Sarrus linkage and investigates its constraint-induced bifurcation and trifurcation. In light of the unique geometry of the double-spherical 6R linkage, which is also a typical plane-symmetric Bricard 6R loop, its parametric constraints are explored. Based on constraint analysis in screw system theory, both design and motion parameters that lead to constraint singularities are revealed and the transitory positions for bifurcation and trifurcation are identified among geometric constraints induced singular positions. The analysis reveals that the presented 6R overconstrained linkage is able to reconfigure its configurations by passing transitory positions and to evolve to distinct motion branches including spherical 4R linkages and serial kinematic chains.

Structural Design and Performance Study of Parallel Metamorphic Mechanism Based on Screw System Theory

A new type of metamorphic kinematic pair is designed according to the traditional kinematic pair, and further a novel parallel metamorphic cutting mechanism is designed. Based on screw system theory the constraints of structure branched chain on the end of the moving platform are analyzed before and after transformation, and then the degrees of freedom of the novel parallel metamorphic cutting mechanism is calculated, and the exact axis that the mechanism rotates across can also be obtained.

Mobility and Singularity Analysis of a Class of Two Degrees of Freedom Rotational Parallel Mechanisms Using a Visual Graphic Approach

In this paper, a visual graphic approach is presented for the mobility and singularity analysis of mechanisms with no helical pair. The presented method is established upon the reciprocal screw system theory. Using the visual graphic approach, the mobility and singularity analysis mainly requires applying a few simple rules and involves into no formula derivation. As a case study, the mobility and singularity analysis is implemented for a class of two degrees of freedom (DOF) rotational parallel mechanisms (RPMs), including the Omni-Wrist III with four limbs and its two derived architectures with three limbs called the T-type and Δ-type RPMs. The Δ-type one is found to has kinematic properties close to the Omni-Wrist III.

Screw Theory Based Methodology for the Deterministic Type Synthesis of Flexure Mechanisms

Flexure mechanism synthesis, however, is still a comparably difficult task. This paper aims at exploring a simple but systematic type synthesis methodology for general flexure mechanisms. The applied mathematical tool is reciprocal screw system theory in geometric form, and the proposed approach is an improvement of freedom and constraint topology (FACT), which is based on the FACT approach, combining with other methods including equivalent compliance mapping, set operation on building blocks, etc. As a result, it enables the type synthesis of flexure mechanisms simple, complete, and effective. What is more significant is that the proposed approach makes the unified type synthesis of both constraint-based and kinematics-based flexure mechanisms available. That is also the new contribution to the flexure de-sign.

Type synthesis of a class of spatial lower-mobility parallel mechanisms with orthogonal arrangement based on Lie group enumeration

Type synthesis of lower-mobility parallel mechanisms (PMs) has drawn extensive interests, particularly two main approaches were established by using the reciprocal screw system theory and Lie group theory, respectively. Although every above approach provides a universal framework for structural design of general lower-mobility PMs, type synthesis is still a comparably difficult task for the PMs with particular geometry or required to fulfill some specified tasks. This paper aims at exploring a simple and effective synthesis method for lower-mobility parallel mechanisms with orthogonal arrangement (OPMs), and the applied mathematical tool is established in the displacement group theory. For this purpose, the concept of the Cartesian DOF-characteristic matrix, originated from canonical displacement subgroup and displacement submanifold, is proposed. A new approach based on combination of the atlas of Cartesian DOF-characteristic matrix and displacement group-theoretic method is addressed for both exhaustive classification and type synthesis of OPMs. Type synthesis for some representatives of 3-DOF OPMs verifies effectiveness of the proposed approach.

Numeration and type synthesis of 3-DOF orthogonal translational parallel manipulators

Low-degree-of-freedom (Low-DOF) parallel manipulators (PMs) have drawn extensive interests, particularly in type synthesis in which two main formal approaches were established by using the reciprocal screw system theory and Lie group theory. This paper aims at numeration and type synthesis of orthogonal translational parallel manipulators (OTPMs) by resorting to an integration of the group-based method and graphical representation of the topology. For this purpose, the concept of Cartesian DOF-characteristic matrix, originated from displacement subgroup and displacement submanifold, is proposed. A new approach based on the combination of the atlas of Cartesian DOF-characteristic matrix and the displacement group-theoretic method is addressed for both exhaustive classification and type synthesis of OTPMs. The proposed approach is prone to construct an orthogonal structure and is easy to realize the complete classification and exhaustive enumeration of this class of low-DOF PMs.

Kinematic design of large displacement precision XY positioning stage by using cross strip flexure joints and over-constrained mechanism

Flexures are widely used in precision machines since they offer frictionless, particle-free, and low maintenance operation, and they provide extremely high resolution. A large displacement precision XY positioning stage is designed by using cross strip flexure joints. An over-constrained mechanism is used to incorporate symmetry to cancel out the effects of center shifting in large-motion flexures. Advanced kinematic techniques such as screw system theory are used to achieve a good kinematic design. Existing flexure-based translation stages usually have motion range to size ratios of less than 0.01 as compared to 0.25 or higher in this research. It is believed that large-motion flexure-based XY stages can be a cost-effective solution for semiconductor applications, particularly the ones that operate in vacuum.

Mobility in Metamorphic Mechanisms of Foldable/Erectable Kinds

This paper looks at a 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 kinematic 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.

Kinestatic Analysis of General Parallel Manipulators

Robotic mechanisms in general can be of either serial-chain, parallel-chain, or hybrid (a combination of both parallel and serial chains) geometry. While it can be asserted that kinematic theories and techniques are well established for fully serial-chain manipulators, the same assertion cannot be made when it is considered in the general context. In this article, we present a general procedure for systematic formulation and characterization of the instantaneous kinematics for a robotic mechanism with a general parallel-chain geometry. A kinestatic approach based on screw system theory is adopted in this treatment. The resulting equation is a compact Jacobian matrix of the system which includes attributes from not only the active joints but also the passive constraints. An example has been provided to demonstrate the methodology as well as its theoretical feasibility.

Workspace of planar cooperating robots with rolling contacts

In multifingered grippers and multiarm manipulation systems it is possible to manipulate a grasped object by rolling the object along the contacting links. This affords greater versatility, a potentially larger workspace and the ability to adapt to external loads without regrasping. The reachable workspace, or the set of points that can be reached by a reference point on the object, is an important measure of performance. The functional workspace, a subspace of the reachable workspace in which the object can be held in a force-closed grasp, is also an important design consideration. In this paper, the reachable and functional workspaces and methods to determine their boundaries are studied. The basic idea is to determine extreme configurations using screw system theory and obtain the locus of the reference point as the system is moved through these configurations. Although this procedure has been used in an earlier paper, the modeling of rolling contacts and consideration of joint limits require a different treatment. Examples of planar grasps with two manipulators are used to illustrate the application of the theory.

A Compact Inverse Velocity Solution for Redundant Robots

A compact inverse rate kinematics solution for serial-chain, redundant, n-jointed robots is presented. The serial-chain inverse rate kinematics problem is dual to the inverse statics problem in in-parallel systems such as cooperating robots, in which the task of distributing the load wrench between multiple subsystems is typically underspecified. Therefore, methods for obtaining the wrench distribution in an in-parallel system can be applied to the underconstrained inverse kinematics problem for serial-chain manipulators. In particular, a solution that minimizes a weighted-norm of the joint rates vector is derived in this article. The advantages of this method are the ability to accommodate any type of joint and the availability of analytical, closed-form solutions for the joint rates. Finally, a geometric interpretation of the solution is provided using screw system theory.

Statics of In-Parallel Manipulator Systems

In-parallel systems can have some active and some passive joints (i.e., joints with and without joint actuators). The questions that we address in this paper are: how to choose the active and passive joints, how to know whether a set of active joints is sufficient, redundant, or minimal to control the end-effector, and whether there exists a unique choice of active actuators. Our approach to study this problem is based on instantaneous properties of series chains as derived from the theory of screw systems. In this framework, we identify those screws that correspond to the joint actuators of the component series chains. Once the correspondence is made, the sought after actuator properties are deduced from the vector space properties of these screws.

Instantaneous Kinematics of Parallel-Chain Robotic Mechanisms

This paper addresses the instantaneous kinematics of robotic manipulators which have an in-parallel scheme of actuation. Hybrid geometries which require both serial and parallel actuation are also considered. Multifingered grippers, walking vehicles, and multiarm manipulation systems in addition to robot arms with a parallel structure can be included in this broad category. The direct and inverse kinematics (and statics) of these devices is discussed with particular attention to applications in control. An analytical method based on screw system theory for obtaining transformation equations between joint and end-effector coordinates is described. Special configurations in which the end-effector gains or loses a degree of freedom, which are also known as geometric singularities, are an important consideration in this study. This is because the number of special configurations or singularities in the workspace is far more for in-parallel manipulators than that for serial manipulators. The special configurations for a planar dual-arm manipulation system, which can be kinematically modeled as a 5-R linkage, are discussed in some detail as an example.

Orthogonal spaces and screw systems

The classification of screw systems is systematically reformulated in terms of the theory of orthogonal spaces. The Lie algebra of the Euclidean group is isomorphic to the algebra of infinitesimal screws. The orthogonal space structure of the Lie algebra can thus be used for translating the theory of screw systems into a theory of subspaces of the Lie algebra of the Euclidean group. The reciprocal product, also known as the Klein form, is the leading criterion for the classification of screw systems. Furthermore, the Killing form provides a secondary criterion for a finer classification.

Structural Kinematics of In-Parallel-Actuated Robot-Arms

An alternative is here put forward to counterbalance the present-day preoccupation with series-actuated robot-arms. A systematic study of robots and manipulators, now concentrating on “in-parallel” actuator-arrangements, reveals many geometries applicable either to entire robot-arms or to parts of otherwise series-actuated arms. No survey of this kind is possible without drawing heavily on the theory of screw systems. Having established a means of enumerating possible geometries, screw theory is again invoked to highlight, in broad terms, the patterns considered to be most promising. Criteria for avoiding undesirable robot-arm-configurations are touched upon, and certain aspects of the performance of in-parallel-actuated robot-arms are compared and contrasted with those of series-actuated arms. Within the bounds here set (thought to be realistic) the survey on its own is intended to be exhaustive, but many details remain to be investigated. This paper aims to do no more than edge open the door a little further towards in-parallel actuation in robot-arms; others may then consider that further study could be productive.

Intrinsic formulation of problems of geometry and kinematics of mechanisms

It is possible to analyse any mechanism in a way which is independent of a particular choice of a system of axes. This intrinsic method establishes a bridge between the theory of displacements and the theory of screw systems.

On relative freedom between links in kinematic chains with cross-jointing

In a short series of even individually important papers, fundamental tools have been developed for the determination of the mobility of a linkage and the relative freedom between two particular members. It has become evident that, to complete the basic set of analytical procedures, a direct algorithm is required for determining the freedom between links which are separated by cross-jointing. This paper provides such a method which, further, incorporates quantitative relationships and embraces earlier results, particularly the series and parallel laws of screw system theory.