Singularity-free Workspace Research Articles

Unified kinematics and optimal design of a 3rRPS metamorphic parallel mechanism with a reconfigurable revolute joint

This paper introduces a new metamorphic parallel mechanism based on a reconfigurable revolute (rR) joint and the mechanism consists of three rRPS (rR joint-prismatic joint-spherical joint) limbs. Reconfiguration principle of the rR joint and the rRPS limb is explained, based on which the 3rRPS metamorphic parallel mechanism can be reconfigurable between two working motion types, pure rotation (3R) motion and one translation and two rotation (1T2R) motion. Using the limb geometric constraint model, analytical forward kinematics is solved in a unified way for both motion types. Reciprocal screw based Jacobian is obtained for singularity analysis which is then used for singularity-free workspace analysis. Based on those, maximum singularity-free workspace and kinematics performance based criteria are applied in optimizing basic mechanism parameters considering input and passive joint limitations. A unified objective function with variable design priorities represented by function weights of the two topologies is proposed and examples are illustrated. The introduced new parallel mechanism covers the two very useful motion types, 3R and 1T2R, while the model in the paper provides basis of modeling and optimal design for further applications.

Maximal singularity-free orientation workspace over a position region of Gough–Stewart platform

Maximizing the singularity-free workspace of parallel manipulators is highly desirable in a context of robot design. So far, no work has been found to address the maximal singularity-free orientation workspace over a position region. In practice, this type of workspace is interesting because a mechanism often works in a range of positions. This work focuses on the Gough–Stewart platform. An optimal position at which the mechanism holds the maximal singularity-free orientation workspace is determined. This optimal position lies on the line which is perpendicular to the base and passes the centroid of the base. Considering the symmetry, a parallelepiped with centre at the determined optimal position could be an interesting working position region for the Gough–Stewart platform. Two algorithms are presented to compute the maximal singularity-free orientation workspace over such an interesting position region. An example is provided for demonstration.

Optimal dimensional synthesis of a symmetrical five-bar planar upper-extremity neuromotor device

Individuals with hemiplegia suffer from impaired arm movements that appear as a marked change in arm stiffness. A quantitative measure of arm stiffness would characterize rehabilitation therapy effectively, while little mechanism is designed to implement the function. A symmetrical five-bar linkage consisting of two revolute joints and three prismatic joints is presented. Inverse kinematics and forward kinematics are obtained first. Then inverse singularities and direct singularities of the mechanism are gained. The global stiffness index is defined based on the results of kinematics analysis. Finally, optimal dimensional synthesis of the mechanism in terms of maximum stiffness is conducted by genetic algorithms. The calculation results show that with the length of both the two linkage a=830 mm, the interacting angle of the two guides 2δ=4.48 radian, and the maximum range of displacement of the two carriers d max=940 mm, the mechanism achieves highest rigidity and its workspace is singularity-free, which covers the human left and right arm range of motion. The proposed novel mechanism featuring high rigidity and a singularity-free workspace can provide rehabilitation training, also solve the problem of quantitative measure of arm stiffness.

Constraint Conditions Determination for Singularity-free Workspace of Central Symmetric Parallel Robots

The kinematic map between actuators and end-effector in parallel robot is one of the main constraints for its workspace. With a geometric analysis on the map, this paper reveals the internal characteristics of the central symmetric parallel robot workspace, and presents a series of topology based constraint conditions for the numerical calculation in singularity-free workspace optimization. We use homeomorphism theorems and finite distortion principles to introduce these constraints, and explain how the singularity-free workspace can be considered as a simple connected set under these conditions. Finally, we determine the constraint conditions for two typical central symmetric parallel robots: the 3- RPR structure planar parallel robot and the 6-SPS structure Stewart parallel robot. And the constraint conditions are applied in the singularity-free workspace optimization respectively.

Singularity-free Workspace of Parallel Robots

Singularity-free Workspace of Parallel Robots

Unified Singularity Modeling and Reconfiguration of 3rTPS Metamorphic Parallel Mechanisms with Parallel Constraint Screws

This paper presents a unified singularity modeling and reconfiguration analysis of variable topologies of a class of metamorphic parallel mechanisms with parallel constraint screws. The new parallel mechanisms consist of three reconfigurable rT PS limbs that have two working phases stemming from the reconfigurable Hooke (rT) joint. While one phase has full mobility, the other supplies a constraint force to the platform. Based on these, the platform constraint screw systems show that the new metamorphic parallel mechanisms have four topologies by altering the limb phases with mobility change among 1R2T (one rotation with two translations), 2R2T, and 3R2T and mobility 6. Geometric conditions of the mechanism design are investigated with some special topologies illustrated considering the limb arrangement. Following this and the actuation scheme analysis, a unified Jacobian matrix is formed using screw theory to include the change between geometric constraints and actuation constraints in the topology reconfiguration. Various singular configurations are identified by analyzing screw dependency in the Jacobian matrix. The work in this paper provides basis for singularity-free workspace analysis and optimal design of the class of metamorphic parallel mechanisms with parallel constraint screws which shows simple geometric constraints with potential simple kinematics and dynamics properties.

Determination of the maximal singularity-free workspace of 3-DOF parallel mechanisms with a constructive geometric approach

This paper proposes a novel approach to obtain the maximal singularity-free regions of planar parallel mechanisms, which is based on a constructive geometric reasoning. The proposed approach consists of two algorithms. First, the borders of the singularity-free region associated with an arbitrary starting point of the moving platform are obtained. Then, the second algorithm finds the center of the maximal singularity-free circle, which is obtained using the so-called offset curve algorithm. The procedure is applied to a 3-PRR planar parallel mechanism as an example and the obtained results illustrate graphically the effectiveness of the proposed algorithm. The proposed approach can be directly applied to obtain the maximal singularity-free circle of similar parallel mechanisms, which is not the case for other approaches proposed in the literature that are limited to a given parallel mechanism, namely, the 3-RPR planar parallel mechanism. Moreover, as the main feature of the proposed approach, it can be implemented both in a CAD system and in a computer algebra system where non-convex and reentrant curves can be considered.

Tri-pyramid Robot: Design and kinematic analysis of a 3-DOF translational parallel manipulator

3-DOF translational parallel manipulators have been developed in many different forms, but they still have respective disadvantages in different applications. To overcome their disadvantages, the structure and constraint design of a 3-DOF translational parallel manipulator is presented and named the Tri-pyramid Robot. In the constraint design of the presented manipulator, a conical displacement subset is defined based on displacement group theory. A triangular pyramidal constraint is presented and applied in the constraint designs between the manipulator@?s subchains. The structural properties including the decoupled motions, overconstraint elimination, singularity free workspace, fixed actuators and isotropic configuration are analyzed and compared to existing structures. The Tri-pyramid Robot is constrained and realized by a minimal number of 1-DOF joints. The kinematic position solutions, workspace with variation of structural parameters, Jacobian matrix, isotropic and dexterity analysis are performed and evaluated in the numerical simulations.

Determining the maximal singularity-free circle or sphere of parallel mechanisms using interval analysis

SUMMARYThis paper proposes a systematic algorithm based on the concept of interval analysis to obtain the maximal singularity-free circle or sphere within the workspace of parallel mechanisms. As case studies the 3-RPR planar and 6-UPS parallel mechanisms are considered to illustrate the relevance of the algorithm for 2D and 3D workspaces. To this end, the main algorithm is divided into four sub-algorithms, which eases the understanding of the main approach and leads to a more effective and robust algorithm to solve the problem. The first step is introduced to obtain the constant-orientation workspace and then the singularity locus. The main purpose is to obtain the maximal singularity-free workspace for an initial guess. Eventually, the general maximal singularity-free workspace is obtained. The main contribution of the paper is the proposition of a systematic algorithm to obtain the maximal singularity-free circle/sphere in the workspace of parallel mechanisms. The combination of a maximal singularity-free circle or sphere with the workspace analysis by taking into account the stroke of actuators, as additional constraint to the latter problem, is considered. Moreover, the center point of the circle/sphere is not restrained to a prescribed point.

Singularity-free workspace analysis of general 6-UPS parallel mechanisms via convex optimization

This paper investigates the singularity-free zones in the workspace of general 6-UPS parallel mechanisms. The emphasis is placed on obtaining the maximum volume ellipsoid or sphere in the singularity-free subregions, while taking into account the actuator strokes. The proposed algorithms are based on the convex optimization and have several advantages over the existing methods such as Lagrange multipliers approach, which makes it appropriate for other applicable optimization problems. For determining the maximal singularity-free zones, in order to find the maximum volume ellipsoid, a judicious iterative procedure, referred to as Improved Lower Bound Semidefinite Programming, is proposed. Additionally, different reformulations of the problem are proposed in order to solve other interesting problems in the robotics community such as obtaining the maximum volume sphere. Furthermore, an approach based on the sum of squares method is proposed to solve this problem for a general Gough–Stewart platform with any arbitrary geometric parameter which is conducive to a polynomial optimization problem. The computational time for the proposed algorithms are considerably low compared to other methods proposed in the literature for obtaining the singularity-free workspace which opens an avenue to use them as systematic algorithms for on-line purposes.

Parallel singularity-free design with actuation redundancy: A case study of three different types of 3-degree-of-freedom parallel mechanisms with redundant actuation

Typical parallel mechanisms suffer from parallel singularity due to kinematic coupling of multichains. This paper investigates how to remove parallel singularities by using redundant actuations. First, actuation wrenches and constraint wrenches forming the full direct kinematic Jacobian matrix are derived. After briefly addressing conditions for their constraint singularities, Grassmann–Cayley algebra is employed to identify parallel singularities. Then, employing Grassmann line geometry, the locations and the minimum number of redundant actuators are identified for the parallel mechanisms to have parallel singularity-free workspace. Three different types of 3-degree-of-freedom parallel mechanisms such as planar, spherical, and spatial parallel mechanisms are given as exemplary devices.

Performance enhancement of a three-degree-of-freedom parallel tool head via actuation redundancy

This paper presents a novel type of three degree-of-freedom (DOF) redundant parallel tool head (PTH) which is developed by introducing actuation redundancy to an originally designed non-redundant 3-DOF PTH. To diminish the physical constraints imposed by spherical joints, a modified spherical joint with large tilting capacity is introduced. The two types of PTHs, the redundant form and the non-redundant one, are then fully compared with each other in terms of singularity distribution, workspace shape, kinematic performance and stiffness behavior. The comparison results show that the redundant PTH has several notable advantages over the non-redundant one, including enlarged singularity-free workspace, improved dexterity performance and higher stiffness.

Design and Experimental Study of a 3-DOF Parallel Swivel Head

Traditional parallel mechanisms (PMs) are usually characterized by small tilting capability. To overcome this problem, a 3-DOF parallel swivel head (PSH) with large tilting capacity is proposed in this paper. The proposed PSH, which is structurally developed from a conventional 3-PRS parallel mechanism (PM), can achieve a large tilting capability by means of structural improvements. Firstly, a modified spherical joint with a maximum tilting angle of ±120o is devised to diminish the physical restrictions on the orientation workspace. Secondly, a UPS typed leg is introduced for the sake of singularity elimination. The superiority of the proposed PSH is theoretically proved by investigations of singularity-free orientation workspace and then is experimentally validated using a prototype fabricated. The theoretical and experimental results illustrate that the proposed PSH has a large tilting capacity and thus can be used as swivel head for a hybrid machine tool which is designed to be capable of realizing both horizontal and vertical machining.

Six-DOF parallel manipulators with maximal singularity-free joint space or workspace

SUMMARYSingularity-free workspace is a very important criterion for the design of manipulators, especially for parallel manipulators which are well known for their limited workspace and complex singularities. This paper studies geometric parameters and dexterity measures that affect the size of a singularity-free joint space and proposes methods for the development of 6-DOF Stewart–Gough parallel manipulators that have better singularity-free joint space. With a local dexterity measure as the objective function, a systematic method is employed to search for the design with a maximal singularity-free joint space. The related workspaces are also investigated. It is shown that the workspace is not proportional to the size of the joint space and that manipulators with a larger singularity-free workspace usually have relatively poor dexterity.

Planning Singularity-Free Paths on Closed-Chain Manipulators

This paper provides an algorithm for computing singularity-free paths on closed-chain manipulators. Given two nonsingular configurations of the manipulator, the method attempts to connect them through a path that maintains a minimum clearance with respect to the singularity locus at all points, which guarantees the controllability of the manipulator everywhere along the path. The method can be applied to nonredundant manipulators of general architecture, and it is resolution complete. It always returns a path whenever one exists at a given resolution or determines path nonexistence otherwise. The strategy relies on defining a smooth manifold that maintains a one-to-one correspondence with the singularity-free C-space of the manipulator, and on using a higher dimensional continuation technique to explore this manifold systematically from one configuration, until the second configuration is found. If desired, the method can also be used to compute an exhaustive atlas of the whole singularity-free component reachable from a given configuration, which is useful to rapidly resolve subsequent planning queries within such component, or to visualize the singularity-free workspace of any of the manipulator coordinates. Examples are included that demonstrate the performance of the method on illustrative situations.

Cusp Points in the Parameter Space of Degenerate 3-RPR Planar Parallel Manipulators

This paper investigates the conditions in the design parameter space for the existence and distribution of the cusp locus for planar parallel manipulators. Cusp points make possible nonsingular assembly-mode changing motion, which increases the maximum singularity-free workspace. An accurate algorithm for the determination is proposed amending some imprecisions done by previous existing algorithms. This is combined with methods of cylindric algebraic decomposition, Gröbner bases, and discriminant varieties in order to partition the parameter space into cells with constant number of cusp points. These algorithms will allow us to classify a family of degenerate 3-RPR manipulators.

Influence of kinematic redundancy on the singularity-free workspace of parallel kinematic machines

In this paper the effect of kinematic redundancy in order to reduce the singularity loci of the direct kinematics and to increase the operational, i.e., singularityfree, workspace is demonstrated. The proposed approach consists of additional prismatic actuators allowing one or more base joints to move linearly. As a result, a selective reconfiguration can be performed in order to avoid singular configurations. Exemplarily, kinematically redundant schemes of four structures, the 3RRR, the 3RPR, the 6UPS, and the 6RUS, are considered. The relationship between the redundancy and the operational workspace is studied and several analysis examples demonstrate the effectiveness of the proposed concept. Furthermore, the additional benefit of an increasing number of redundant actuators is discussed.

Kinematic Analysis of a Planar Mechanism Driven by Two Orthogonal Layout Actuators

In this paper, a planar five-bar mechanism driven by two orthogonal layout actuators is proposed. The inverse and forward kinematics can be described in closed form. The Jacobi matrix, singularity and workspace of the mechanism are investigated. In addition, charts of some performance indices, such as dexterity, velocity, payload capability and stiffness index, are plotted. The results show that the mechanism has a flatter singularity-free workspace, which is usually our research object. On the other hand, Dexterity, Payload capability and stiffness index of the mechanism are basically the same distribution on the workspace, which are axis symmetric. But, the trend of velocity index is opposite. The proposed mechanism can be applied to the field of machine tools or used as the mobile base for a spatial manipulator. The results of the paper are very useful for the design and application of the new mechanism

A new approach to determine the maximal singularity-free zone of 3-RPR planar parallel manipulator

SUMMARYThis paper presents a new algorithm to find the singularity-free cylindrical workspace of parallel manipulators. Because of the limited workspace of parallel manipulators cluttered with different types of singularities, a simple and robust technique to determine continuous singularity-free zones in the workspace of parallel manipulators is required. Here, the largest singularity-free cylinder within the workspace for any prescribed orientation ranging around a reference orientation angle of moving platform is determined. To this end, Particle Swarm Optimization is utilized to find the closest point on the singularity surface to the axis of the cylinder. By implementing the algorithm on 3-RPR planar parallel manipulator, the results show that this algorithm improves the efficiency and leads to significantly larger singularity-free workspace than those reported earlier.

Kinematics of a star-triangle planar parallel manipulator

This paper addresses kinematics of a 3 degree-of-freedom (DOF) planar parallel manipulator called Star-Triangle manipulator. The manipulator has good accuracy and a relatively large singularity-free workspace. First, position analysis of the manipulator is implemented indicating that both the inverse and forward position problems of the manipulator have only one solution. Then, velocity and singularity analyses of the manipulator are carried out and its isotropic configurations are identified using two Jacobian matrices. Based on the isotropic conditions, the workspace of manipulator is determined geometrically and it is shown that all points in the workspace are free of singularities. At last, kinematic accuracy of the manipulator is investigated on the workspace through a kinematic conditioning index (KCI).