Reciprocal Screw Research Articles

Structure Synthesis of a Class of 3-DOF Rotational Parallel Manipulators

The theory of reciprocal screws is used for the enumeration of the feasible limb structures of a class of 3-degree-of-freedom (DOF) rotational parallel manipulators. Geometric conditions for the moving platform to possess finite spherical motion are described. A table of feasible limbs that can be used for the construction of 3-DOF rotational parallel manipulators is developed.

Force-unconstrained poses for a redundantly-actuated planar parallel manipulator

A study of the effect of including redundant actuators on the existence of force-unconstrained configurations for a planar parallel layout of joints is presented. The 3-branch, revolute-jointed, 8-bar configuration is considered in detail. This layout is referred to as a 3-RRR configuration to denote its three branches and the number of joints per branch. Concepts of reciprocal screw quantities and kinematic geometry are used to determine the feasibility of the existence of force-unconstrained positions. Redundant actuation is discussed from the perspective of eliminating families of degeneracies. For the 3-RRR it is shown that one order of infinity of unconstrained configurations is eliminated for each actuator, beyond three, added.

Identifying multi-DOF-loss velocity degeneracies in kinematically-redundant manipulators

A new methodology for identifying multi-DOF (degree-of-freedom) loss velocity-degenerate (singular) configurations of kinematically-redundant manipulators is presented. The methodology is based on extending an existing single-DOF-loss velocity degeneracy identification methodology [J. Robot. Syst. 17 (11) (2000) 633; Mech. Mach. Theory 36 (3) (2001) 397]. The new methodology uses the properties of reciprocal screws to determine the multi-DOF-loss velocity-degenerate configurations of kinematically-redundant manipulators. A by-product of the multi-DOF-loss methodology is that reciprocal screws for each velocity-degenerate configuration are determined. These reciprocal screws characterize the motion loss for each degenerate configuration. The 2-DOF-loss velocity degeneracies for a 7-jointed spherical-revolute-spherical manipulator are derived to demonstrate the effectiveness of the reciprocity-based methodology for determining multi-DOF-loss velocity degeneracies.

Configuration analysis of the Schatz linkage

This paper investigates different configurations of the Schatz linkage based on the analysis of a reciprocal screw and of the relationship between the reciprocal screw and its stem-screw system, which consists of twists of freedom located at six revolute joints of the linkage. A new method of using cofactors of an augmenting screw is used to obtain the reciprocal screw. The stem-screw system of order 5 of the linkage is developed from the special geometry of the six revolute joints and closed-form displacement solutions are provided based on the stem-screw system. The screw surface representing the trajectory of the reciprocal screw with a chosen pitch is established. This surface is then used to characterize zero-pitch reciprocal screw configurations. The special relationship between the reciprocal screw and the stem-screw system is analysed and used to characterize the constraint wrench and configurations created by the changes of the constraint wrench with a non-zero pitch. Consequently, a set of configurations are presented in conjunction with a ruled surface produced by the progression of the constraint wrench when the linkage drive joint rotates from 5° to 85°.

A Linear Algebraic Procedure in Obtaining Reciprocal Screw Systems

AbstractThis paper presents a linear algebraic procedure in obtaining (6−n) reciprocal screws from an n‐screw system of a robotic manipulator. The procedure starts by formulating the reciprocal relationship between a screw system and its reciprocal screw system in matrix form, and augmenting the screw matrix which is assembled with n screws in the screw system. Vectors of cofactors of the augmented row are then produced and the partitioning of the screw matrix is implemented to generate each of the (6−n) reciprocal screws. The theory developed in this paper provides a novel and simple procedure to obtain reciprocal screws from any screw system. The paper illustrates applications to cases with different screw systems. © 2003 Wiley Periodicals, Inc.

Inverse Velocity and Singularity Analysis of Low‐DOF Serial Manipulators

AbstractThis paper presents a method for exact inverse velocity analysis of low‐DOF (degrees of freedom n<6) serial manipulators. For a low‐DOF serial manipulator, the number of independently controllable variables in the Cartesian space is equal to the number of joint variables in the joint space, and the remaining 6−n variables are linearly dependent on these independent variables. This paper employs the theory of reciprocal screws to determine a mapping between the independent velocity components in the Cartesian space and the joint rates in the joint space. It is shown that singular conditions of a low‐DOF manipulator depend on choice of independent variables. A 5‐DOF and a 4‐DOF manipulator are analyzed, and a numerical example in which the end effector of a 4‐DOF manipulator is commanded to follow a straight line is used to demonstrate the methodology. © 2003 Wiley Periodicals, Inc.

Analytic Formulation of Reciprocal Screws and Its Application to Nonredundant Robot Manipulators

One of the most important and widely used concepts in the kinematic analysis of robot manipulators is the reciprocal screw. However, there are no general expressions and easy methods to obtain the reciprocal screw in an analytic manner. This paper suggests an analytic formulation of the reciprocal screws of arbitrarily aligned screw systems. Since the reciprocal screws obtained in this paper are represented by the direction vectors and the position vectors of the given screws, we can analyze the relation between the reciprocal screw system and the given screw system easily. With the results, to find a reciprocal screw is to solve an algebraic equation of the corresponding system of screws. In order to show the usefulness of the result, several examples related to the robot manipulator are provided. For a nonredundant serial manipulator, the pseudo inverse of the Jacobian matrix is shown to be equivalent to the wrench matrix obtained by the reciprocity. For a parallel manipulator, a leg is isolated to obtain an independent part from the manipulator and is analyzed analytically. The proposed method can be applied to any arbitrarily aligned screw system.

Feasible Motion Solutions for Serial Manipulators at Singular Configurations

When a serial manipulator is at a singular configuration, the Jacobian matrix will lose its full rank causing the manipulator to lose one or more degrees of freedom. This paper presents a novel approach to model the manipulator kinematics and solve for feasible motions of a manipulator at singular configurations such that the precise path tracking of a manipulator at such configurations is possible. The joint screw linear dependency is determined by using known line varieties so that not only the singular configurations of a manipulator can be identified but also the dependent joint screws can be determined. Feasible motions in Cartesian space are identified by using the theory of reciprocal screws and the resulting equations of constraint. The manipulator first-order kinematics is then modeled by isolating the linearly dependent columns and rows of the Jacobian matrix such that the mapping between the feasible motions in Cartesian space and the joint space motions can be uniquely determined. Finally, a numerical example is used to demonstrate the feasibility of the approach. The simulation results show that a PUMA-type robot can successfully track a path that is singular at all times.

Structure Synthesis of a Class of 4-DoF and 5-DoF Parallel Manipulators with Identical Limb Structures

In this paper, a systematic approach is developed for the structural synthesis of a class of four-degrees-of-freedom (4-DoF) and 5-DoF overconstrained parallel manipulators with identical serial limbs. The theory of screws and reciprocal screws is employed for the analysis of the geometric conditions that must be met by the limbs of such parallel manipulators. Limb structures that can be used for constructing 4-DoF or 5-DoF parallel manipulators are enumerated according to the reciprocity of the twist and wrench systems. The assembly conditions of a parallel manipulator built by using identical C- or F-limbs are discussed. Several 4-DoF and 5-DoF parallel manipulators are sketched as examples.

Null–space construction using cofactors from a screw–algebra context

This paper develops an approach for constructing the null space N ( A ) of a linear system of homogeneous equations using the cofactors of an augmented coefficient matrix A . The relationship between the row space R ( A T ) and null space is exploited by introducing an augmenting vector which is linearly independent of the row space and dependent on the null space. The resultant null space is shown to be a vector of cofactors of the augmenting row of the coefficient matrix and is invariant. This provides a straightforward solution to a linear system of homogeneous equations without going through Gauss-Seidel elimination. The approach is derived from a onedimensional null space and is extended to a multidimensional one by partitioning the coefficient matrix and consequently constructing a set of ( n − m ) null–space vectors based on cofactors. Examples are given and accuracy is compared with Gauss–Seidel elimination. The approach is further used in a screw–algebra context with a simple procedure to obtain a system of reciprocal screws representing a set of constraint wrenches from a set of twists of freedom, in the form of a linear system of homogeneous equations in R 6 . The paper provides rigorous proofs and applications in both linear algebra and advanced kinematics.

Jacobian Analysis of Limited-DOF Parallel Manipulators

This paper presents a methodology for the Jacobian analysis of limited degrees-of-freedom (DOF) parallel manipulators. A limited-DOF parallel manipulator is a spatial parallel manipulator which has less than six degrees-of-freedom. It is shown that a 6×6 Jacobian matrix, which provides information about both architecture and constraint singularities, can be derived by making use of the theory of reciprocal screws. The 3-UPU and 3-RPS parallel manipulators are used as examples to demonstrate the methodology.

General Methodology for Type Synthesis of Symmetrical Lower-Mobility Parallel Manipulators and Several Novel Manipulators

In the area of parallel robots, the use of lower-mobility parallel manipulators for many tasks requiring fewer than six degrees of freedom (DoF) has drawn a lot of interest. This paper treats one fundamental problem in the study of lower-mobility parallel manipulators: type synthesis. Using reciprocal screw theory, we define the mechanism constraint system and limb constraint system. We then investigate the relations between the mechanism constraint system and the limb constraint system under different geometrical conditions. Three tables describing the relations are presented. Based on the three tables, we propose a constraint-synthesis method for type synthesis of symmetrical lower-mobility parallel manipulators. The method used in the paper to construct the kinematic structure of the limb of lower-mobility parallel manipulators with prescribed DoF is simple and systematic. Examples are included to illustrate the application of the method and some novel lower-mobility parallel manipulators with 3-, 4- and 5-DoF are synthesized.

Geometric Analysis of Overconstrained Parallel Manipulators with Three and Four Degrees of Freedom.

This paper investigates the constraint characteristics of parallel manipulators with three and four degrees of freedom. An analytical method of using equivalent screw groups is developed in the study based on the linear independence of screw systems and reciprocal screw systems. A procedure for determining overconstraints in spatial parallel mechanisms with fewer six degrees of freedom is given. With this method, the constraint characteristics of four parallel mechanisms with three or four degrees of freedom, i. e. 3-RRC, 3-CPR, 3-URU/SPS and 4-UPU mechanisms, are revealed. The constraint characteristics of three types of parallelogram linkage unites are then discovered, which gives the corresponding equivalent representations in terms of serial mechanisms. The approach is further applied to the geometric analysis of parallel mechanisms with hybrid limbs including 3-R(4U)RPP and 3-R(4S)RRRP mechanisms.

Interrelationship between screw systems and corresponding reciprocal systems and applications

This paper discusses the relationship between a screw system and its corresponding reciprocal system based upon set theory, proposes a theorem, makes a proposition with corollaries to specify the relationship between the two screw systems and characterises all cases in screw systems. The theory is then applied to the one-, two-, and three-systems and their corresponding reciprocal five-, four-and three-systems, and used to predict reciprocal screws and the subsequent relationship with the existing screw system. A further application is given by selecting screws to design a system such that a certain intersection is desirable between the system and its reciprocal system. As a result, a method of generating a one-system of reciprocal screw from a five-system and of generating a reciprocal system from a co-reciprocal system is given.

Reciprocity-based resolution of velocity degeneracies (singularities) for redundant manipulators

A methodology for determining the velocity-degenerate (singular) configurations of joint-redundant manipulators is presented. The developed methodology uses the properties of reciprocal screws to determine all degenerate configurations resulting in a single degree-of-freedom (DOF) motion loss. Six joint screws ($ sub 1 ,$ sub 2 ,…,$ sub 6 ) are chosen to form one six-joint sub-group Jacobian, J sub . Setting the determinant of J sub to zero ( | J sub |=0 ) allows all conditions (say n in total) that cause the six-joint sub-group to become velocity degenerate to be identified. Reciprocal screws, W recip 1 , W recip 2 ,…, W recip n , can be found for each of the n velocity-degeneracy conditions. The reciprocal screw, W recip i , is reciprocal to the six joints that comprise J sub when the ith J sub degeneracy condition is true, but will not necessarily be reciprocal to the redundant joints $ r 1 ,$ r 2 ,…,$ r k−6 (where k is the total number of joints and k>6). Taking reciprocal products of W recip i and each redundant joint $ r 1 ,$ r 2 ,…,$ r k−6 and setting the results to zero yields all additional conditions necessary for W recip i to be simultaneously reciprocal to all of the redundant joints. This procedure is repeated for all of the reciprocal screws. The method identifies all unique sets of conditions that result in the redundant-manipulator joints $ sub 1 ,$ sub 2 ,…,$ sub 6 ,$ r 1 ,$ r 2 ,…,$ r k−6 becoming degenerate. A by-product of the methodology is that reciprocal screws for all degenerate configurations are derived. Examples are presented to demonstrate the effectiveness of the methodology at determining the velocity-degenerate configurations for joint-redundant manipulators.

Velocity degeneracy determination for the kinematically redundant CSA/ISE STEAR testbed manipulator

The Canadian Space Agency (CSA)/International Submarine Engineering (ISE) STEAR Testbed Manipulator (STM) is a ground-based manipulator similar to the arms of the Special Purpose Dextrous Manipulator (SPDM) being built for the International Space Station (ISS). This work presents the determination of the velocity-degenerate (singular) configurations for the STM, a system comprised of two seven-jointed redundant manipulators. The two “arms” of the STM are mirror images of one another. The analysis presented is for the STM-1 or “right” arm, but it applies to the STM-2 or “left” arm since they are mirror images and have similar dimensions and joint ranges. A methodology based on reciprocity of screws is used to identify all configurations causing one degree-of-freedom (DOF) instantaneous motion loss in the STM. It is shown that there are five families of two-condition configurations resulting in a single DOF loss for manipulators kinematically similar to the STM-1. Reciprocal screws characterizing the lost motion DOF for each of the degenerate configurations are derived. Analysis of the velocity degeneracies shows that for the STM-1 dimensions and joint limits, only four of the five velocity-degenerate configurations are feasible. © 2000 John Wiley & Sons, Inc.

On equivalent motions in spatial linkages

A common feature of spatial linkages is the restriction, perhaps multiply, of a form of motion to containment within portion of the chain while the gross configuration emulates a simpler linkage, such as the four-slider loop. We investigate this phenomenon by examining a group of related overconstrained linkages. In doing so we make use of displacement–closure equations, screw motors and the notion of the reciprocal screw.

Analysis and Synthesis of Schatz Six-Revolute Mechanisms.

Based on geometrical intuition and computer graphics, we present the formation of two kinds of Schatz six-revolute mechanisms and two types of algebraic surfaces on which the corresponding reciprocal screw axes lie under any possible configurations. Then, we derive the general kinematic closed-form solutions and show the absence of stationary configuration of both mechanisms using four-by-four matrix and differential algebra. Moreover, the parametric formulations of coupler-point motion provide a complete higher-order analysis of the coupler curve. In practical application, this mechanism is further used for the dimensional synthesis, path and motion generation synthesis, by the nonlinear programming optimization method. At last, two numerical examples are taken to illustrate the design algorithm.

Extension of Usable Workspace of Rotational Axes in Robot Planning

Singularity of a robot manipulator is one of the obstacles that influences its capabilities. This paper discusses constrained and allowable rotational motion resulting from lost translational freedom when the robot is singular. A convenient method and simple and clear expression to determine the allowable rotational axes and the subspace that they form, under Jacobian singularity, is analyzed and presented. Different configurations, reciprocal screws, and subspaces of allowable-rotational-axes are derived in a case study involving a classic robot. The result is useful in applications involving robot path planning in task space as it extends the usable workspace of rotational axes.

Study on the kinematic characteristics of 3 DOF in-parallel actuated platform mechanisms

This paper presents a simple method based on the screw theory for determination of the possible motion characteristics for constrained 3 degree-of-freedom (DOF) parallel mechanisms when the linear displacements of the platform are restricted. The theory is significant for understanding novel constrained multi-DOF mechanisms, their inverse kinematics and robot path planning. The paper firstly discusses theoretically the relationship between the reciprocal screws and the motions of the mechanisms, then presents the analysis of the kinematic characteristics of several 3 DOF in-parallel actuated mechanisms.