Enumeration of parallel manipulators

Robotica ◽  
2009 ◽  
Vol 27 (4) ◽  
pp. 589-597 ◽  
Author(s):  
Roberto Simoni ◽  
Andrea Piga Carboni ◽  
Daniel Martins
Keyword(s):  
Group Theory ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Internal Symmetry ◽  
New Method ◽  
Chain Mechanism ◽  
End Effector ◽  
Group Of Automorphisms ◽  
Vertex Graph

SUMMARYIn this paper, we present a new method of enumeration of parallel manipulators with one end-effector. The method consists of enumerating all the manipulators possible with one end-effector that a single kinematic chain can originate. A very useful simplification for kinematic chain, mechanism and manipulator enumeration is their representation through graphs. The method is based on group theory where abstract structures are used to capture the internal symmetry of a structure in the form of automorphisms of a group. The concept used is orbits of the group of automorphisms of a colored vertex graph. The theory and some examples are presented to illustrate the method.

Enumeration of kinematic chains and mechanisms

2008 ◽  
Vol 223 (4) ◽  
pp. 1017-1024 ◽  
Author(s):  
R Simoni ◽  
A P Carboni ◽  
D Martins
Keyword(s):  
Group Theory ◽  
Kinematic Chain ◽  
Equivalence Classes ◽  
New Method ◽  
Complete List ◽  
Planar Mechanisms ◽  
Group Of Automorphisms ◽  
Kinematic Chains ◽  
New Concepts

A problem still unsolved in kinematics is the enumeration of a complete list of kinematic chains and mechanisms without isomorphisms and without degenerate chains that operate in any screw system. In this paper, a method for the enumeration of kinematic chains without isomorphisms and degenerate chains for all screw systems and a new method of enumeration of kinematic chain inversions (i.e. mechanisms) based on group theory techniques are presented. New concepts of the group theory are introduced and a new method of enumeration of inversions is presented. Kinematic inversions are related to the symmetries of the chain which can be identified analysing the corresponding graph. The symmetry of a graph can be identified for the group of automorphisms of the graph and its orbits provides sets of vertices (links) that are in the same equivalence classes, i.e. they have the same properties of symmetry. The main definitions of group theory and examples of application of the new method of enumeration of inversions are presented. New results are obtained and divided in two classes: original results in non-planar screw systems (λ≠3) and results in agreement with a previously published list for planar kinematic chains and planar mechanisms (inversions). Two tables (1 and 3) are provided, which are up-to-date lists of kinematic chains and mechanisms for several screw systems.

Dexterous Workspace of n-PRRR Planar Parallel Manipulators

2012 ◽  
Vol 4 (3) ◽  
Author(s):  
André Gallant ◽  
Roger Boudreau ◽  
Marise Gallant
Keyword(s):  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Divergence Theorem ◽  
End Effector ◽  
Kinematic Chains ◽  
Prismatic Joint ◽  
Planar Surfaces ◽  
Revolute Joints ◽  
Gauss Divergence Theorem ◽  
Planar Parallel Manipulators

In this work, a method is presented to geometrically determine the dexterous workspace boundary of kinematically redundant n-PRRR (n-PRRR indicates that the manipulator consists of n serial kinematic chains that connect the base to the end-effector. Each chain is composed of two actuated (therefore underlined) joints and two passive revolute joints. P indicates a prismatic joint while R indicates a revolute joint.) planar parallel manipulators. The dexterous workspace of each nonredundant RRR kinematic chain is first determined using a four-bar mechanism analogy. The effect of the prismatic actuator is then considered to yield the workspace of each PRRR kinematic chain. The intersection of the dexterous workspaces of all the kinematic chains is then obtained to determine the dexterous workspace of the planar n-PRRR manipulator. The Gauss divergence theorem applied to planar surfaces is implemented to compute the total dexterous workspace area. Finally, two examples are shown to demonstrate applications of the method.

scholarly journals Position and Singularity Analysis of a Class of Planar Parallel Manipulators with a Reconfigurable End-Effector

Machines ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 7
Author(s):  
Tommaso Marchi ◽  
Giovanni Mottola ◽  
Josep M. Porta ◽  
Federico Thomas ◽  
Marco Carricato
Keyword(s):  
Inverse Kinematics ◽  
Kinematic Chain ◽  
Experimental Tests ◽  
Parallel Manipulators ◽  
Singularity Analysis ◽  
Parallel Robots ◽  
End Effector ◽  
Revolute Joints ◽  
A Chain ◽  
And Control

Parallel robots with configurable platforms are a class of robots in which the end-effector has an inner mobility, so that its overall shape can be reconfigured: in most cases, the end-effector is thus a closed-loop kinematic chain composed of rigid links. These robots have a greater flexibility in their motion and control with respect to rigid-platform parallel architectures, but their kinematics is more challenging to analyze. In our work, we consider n-RRR planar configurable robots, in which the end-effector is a chain composed of n links and revolute joints, and is controlled by n rotary actuators located on the base of the mechanism. In particular, we study the geometrical design of such robots and their direct and inverse kinematics for n=4, n=5 and n=6; we employ the bilateration method, which can simplify the kinematic analysis and allows us to generalize the approach and the results obtained for the 3-RRR mechanism to n-RRR robots (with n>3). Then, we study the singularity configurations of these robot architectures. Finally, we present the results from experimental tests that have been performed on a 5–RRR robot prototype.

Symmetry and invariants of kinematic chains and parallel manipulators

Robotica ◽  
2012 ◽  
Vol 31 (1) ◽  
pp. 61-70 ◽  
Author(s):  
Roberto Simoni ◽  
Celso Melchiades Doria ◽  
Daniel Martins
Keyword(s):  
Group Theory ◽  
Automorphism Group ◽  
Kinematic Analysis ◽  
Group Structure ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Formal Definition ◽  
Kinematic Chains ◽  
Number Synthesis ◽  
Definition Of

SUMMARYThis paper presents applications of group theory tools to simplify the analysis of kinematic chains associated with mechanisms and parallel manipulators. For the purpose of this analysis, a kinematic chain is described by its properties, i.e. degrees-of-control, connectivity and redundancy matrices. In number synthesis, kinematic chains are represented by graphs, and thus the symmetry of a kinematic chain is the same as the symmetry of its graph. We present a formal definition of symmetry in kinematic chains based on the automorphism group of its associated graph. The symmetry group of the graph is associated with the graph symmetry. By using the group structure induced by the symmetry of the kinematic chain, we prove that degrees-of-control, connectivity and redundancy are invariants by the action of the automorphism group of the graph. Consequently, it is shown that it is possible to reduce the size of these matrices and thus reduce the complexity of the kinematic analysis of mechanisms and parallel manipulators in early stages of mechanisms design.

Selecting Kinematic Structures of Parallel Manipulators Using Symmetry and Connectivity

2015 ◽  
Author(s):  
Roberto Simoni ◽  
Henrique Simas ◽  
Daniel Martins
Keyword(s):  
Conceptual Design ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Connectivity Matrix ◽  
Early Stages

This paper presents an application of symmetry and connectivity to select kinematic structures of parallel manipulators. One kinematic chain can originate several mechanisms and each mechanism can originate several parallel manipulators and, in early stages of conceptual design, it is difficult to decide what is the most promising one. Hunt [1] introduced the concept of connectivity and, since then, the connectivity has been used as an important parameter to select the most appropriated parallel manipulators to develop determined task. However, it is difficult to analyze non isomorphic parallel manipulators from the connectivity matrix. In this sense, in this paper, we apply symmetry to reduce the set of parallel manipulators to a manageable few with the desired connectivity. As a result, all promising parallel manipulators originating from a kinematic chain can be analyzed without isomorphisms.

A Statically Balanced Gough/Stewart-Type Platform: Conception, Design, and Simulation

2009 ◽  
Vol 1 (3) ◽  
Author(s):  
Marco Carricato ◽  
Clément Gosselin
Keyword(s):  
Sufficient Conditions ◽  
Parallel Manipulators ◽  
Point Of View ◽  
Theoretical Point ◽  
Constant Force ◽  
Parallel Mechanisms ◽  
End Effector ◽  
Neutral Equilibrium ◽  
The Difference ◽  
Elastic Elements

Gravity compensation of spatial parallel manipulators is a relatively recent topic of investigation. Perfect balancing has been accomplished, so far, only for parallel mechanisms in which the weight of the moving platform is sustained by legs comprising purely rotational joints. Indeed, balancing of parallel mechanisms with translational actuators, which are among the most common ones, has been traditionally thought possible only by resorting to additional legs containing no prismatic joints between the base and the end-effector. This paper presents the conceptual and mechanical designs of a balanced Gough/Stewart-type manipulator, in which the weight of the platform is entirely sustained by the legs comprising the extensible jacks. By the integrated action of both elastic elements and counterweights, each leg is statically balanced and it generates, at its tip, a constant force contributing to maintaining the end-effector in equilibrium in any admissible configuration. If no elastic elements are used, the resulting manipulator is balanced with respect to the shaking force too. The performance of a study prototype is simulated via a model in both static and dynamic conditions, in order to prove the feasibility of the proposed design. The effects of imperfect balancing, due to the difference between the payload inertial characteristics and the theoretical/nominal ones, are investigated. Under a theoretical point of view, formal and novel derivations are provided of the necessary and sufficient conditions allowing (i) a body arbitrarily rotating in space to rest in neutral equilibrium under the action of general constant-force generators, (ii) a body pivoting about a universal joint and acted upon by a number of zero-free-length springs to exhibit constant potential energy, and (iii) a leg of a Gough/Stewart-type manipulator to operate as a constant-force generator.

A New Measurement Device for Kinematic Calibration of Parallel Manipulators

2004 ◽  
Author(s):  
Abdul Rauf ◽  
Sung-Gaun Kim ◽  
Jeha Ryu
Keyword(s):  
Parallel Manipulators ◽  
Measurement Procedure ◽  
Initial Guess ◽  
Measurement Noise ◽  
Kinematic Calibration ◽  
Kinematic Parameters ◽  
End Effector ◽  
Experimental Procedure ◽  
New Device ◽  
Rotation Component

Kinematic calibration is a process that estimates the actual values of geometric parameters to minimize the error in absolute positioning. Measuring all the components of Cartesian posture assure identification of all parameters. However, measuring all components, particularly the orientation, can be difficult and expensive. On the other hand, with partial pose measurements, experimental procedure is simpler. However, all parameters may not be identifiable. This paper proposes a new device that can be used to identify all kinematic parameters with partial pose measurements. Study is performed for a 6 DOF (degree-of-freedom) fully parallel Hexa Slide manipulator. The device, however, is general and can be used for other parallel manipulators. The proposed device consists of a link with U joints on both sides and is equipped with a rotary sensor and a biaxial inclinometer. When attached between the base and the mobile platform, the device restricts the end-effector’s motion to 5 DOF and measures two position components and one rotation component of the end-effector. Numerical analyses of the identification Jacobian reveal that all parameters are identifiable. Computer simulations show that the identification is robust for the errors in the initial guess and the measurement noise. Intrinsic inaccuracies of the device can significantly deteriorate the calibration results. A measurement procedure is proposed and cost functions are discussed to prevent propagation of the inaccuracies to the calibration results.

A Synthesis Method for Placing Workpieces in RPR Planar Robotic Workcells in Which Large Rotations Are Required at the Workpiece

1996 ◽  
Author(s):  
K. D. Chaney ◽  
J. K. Davidson
Keyword(s):  
Reference Frame ◽  
Synthesis Method ◽  
Analytical Representation ◽  
New Method ◽  
Two Dimensional ◽  
End Effector ◽  
Radial Location ◽  
Practical Applications ◽  
Full Turn ◽  
Large Rotations

Abstract A new method is developed for determining both a satisfactory location of a workpiece and a suitable mounting-angle of the tool for planar RPR robots that can provide dexterous workspace. The method is an analytical representation of the geometry of the robot and the task, and is particularly well suited to applications in which the task requires large rotations of the end-effector. It is determined that, when the task requires that the end-effector rotate a full turn at just two locations and when the first or third joint in the robot is rotatable by one turn, then the radial location of the workpiece is fixed in the workcell but its angular location is not fixed. When the mounting-angle of the tool is also a variable, the method accommodates tasks in which the tool must rotate a full turn at three locations on the workpiece. The results are presented as coordinates of points in a two-dimensional Cartesian reference frame attached to the workcell. Consequently, a technician or an engineer can determine the location for the workpiece by laying out these coordinates directly in the workcell. Example problems illustrate the method. Practical applications include welding and deposition of adhesives.

Dynamic Modeling of a Parallel Manipulator With Three Translational Degrees of Freedom

1998 ◽  
Author(s):  
Richard Stamper ◽  
Lung-Wen Tsai
Keyword(s):  
Parallel Manipulator ◽  
Degrees Of Freedom ◽  
Close Agreement ◽  
Jacobian Matrix ◽  
Parallel Manipulators ◽  
Kinematic Structure ◽  
End Effector ◽  
Moving Platform ◽  
Euler Approach ◽  
Computationally Intensive

Abstract The dynamics of a parallel manipulator with three translational degrees of freedom are considered. Two models are developed to characterize the dynamics of the manipulator. The first is a traditional Lagrangian based model, and is presented to provide a basis of comparison for the second approach. The second model is based on a simplified Newton-Euler formulation. This method takes advantage of the kinematic structure of this type of parallel manipulator that allows the actuators to be mounted directly on the base. Accordingly, the dynamics of the manipulator is dominated by the mass of the moving platform, end-effector, and payload rather than the mass of the actuators. This paper suggests a new method to approach the dynamics of parallel manipulators that takes advantage of this characteristic. Using this method the forces that define the motion of moving platform are mapped to the actuators using the Jacobian matrix, allowing a simplified Newton-Euler approach to be applied. This second method offers the advantage of characterizing the dynamics of the manipulator nearly as well as the Lagrangian approach while being less computationally intensive. A numerical example is presented to illustrate the close agreement between the two models.

Determination of the Singularity Loci of Spherical Three-Degree-of-Freedom Parallel Manipularors

1992 ◽  
Author(s):  
Clément M. Gosselin ◽  
Jaouad Sefrioui
Keyword(s):  
Graphical Representation ◽  
Jacobian Matrix ◽  
Parallel Manipulators ◽  
Degree Of Freedom ◽  
End Effector ◽  
Cartesian Space ◽  
Singularity Locus ◽  
Three Degree Of Freedom ◽  
Analytical Expressions

Abstract In this paper, an algorithm for the determination of the singularity loci of spherical three-degree-of-freedom parallel manipulators with prismatic atuators is presented. These singularity loci, which are obtained as curves or surfaces in the Cartesian space, are of great interest in the context of kinematic design. Indeed, it has been shown elsewhere that parallel manipulators lead to a special type of singularity which is located inside the Cartesian workspace and for which the end-effector becomes uncontrollable. It is therfore important to be able to identify the configurations associated with theses singularities. The algorithm presented is based on analytical expressions of the determinant of a Jacobian matrix, a quantity that is known to vanish in the singular configurations. A general spherical three-degree-of-freedom parallel manipulator with prismatic actuators is first studied. Then, several particular designs are investigated. For each case, an analytical expression of the singularity locus is derived. A graphical representation in the Cartesian space is then obtained.

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