On the direct problem singularities of a class of 3-DOF parallel manipulators

Robotica ◽  
2004 ◽  
Vol 22 (4) ◽  
pp. 389-394 ◽  
Author(s):  
R. Di Gregorio
Keyword(s):  
Rigid Body ◽  
Direct Problem ◽  
Parallel Manipulators ◽  
Degree Of Freedom ◽  
Kinematic Pair ◽  
Kinematic Chains ◽  
Position Analysis ◽  
Direct Kinematics ◽  
Relative Motions

The 3-PS structure features one rigid body (platform) connected to another rigid body (base) by means of three kinematic chains (limbs) of type PS (P and S stand for prismatic pair and spherical pair, respectively). All the 3-degree-of-freedom parallel manipulators with three connectivity-5 limbs, each one constituted of one passive (i.e. not actuated) prismatic pair, one passive spherical pair and one actuated kinematic pair of any type, become 3-PS structures when the actuated pairs are locked. Direct kinematics of this class of manipulators is tied to the properties of the 3-PS structure. In particular, the direct position analysis is tied to the assembly modes of the 3-PS structure; whereas the determination of the singularities of the direct instantaneous problem is tied to the determination of the singular geometries of the 3-PS structure, where instantaneous relative motions between platform and base are possible. The solution of these two problems is necessary both for designing the manipulators and for controlling them during motion. This paper deal with the determination of the singular geometries of the 3-PS structure.

Analytic Form Solution of the Direct Position Analysis of a Wide Family of Three-Legged Parallel Manipulators

2005 ◽  
Vol 128 (1) ◽  
pp. 264-271 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Analytic Form ◽  
Parallel Manipulators ◽  
Analytical Form ◽  
Form Solution ◽  
Kinematic Chains ◽  
Position Analysis ◽  
In Series ◽  
The One ◽  
Generic Structures ◽  
Wide Family

A wide family of parallel manipulators (PMs) is the one that groups all PMs with three legs where the legs become kinematic chains constituted of a passive spherical pair (S) in series with either a passive prismatic pair (P) or a passive revolute pair (R) when the actuators are locked. The topologies of the structures generated by these manipulators, when the actuators are locked, are ten. Two out of these topologies are the SR-2PS topology (one SR leg and two PS legs) and the SP-2RS topology (one SP leg and two RS legs). This paper presents two algorithms. The first one determines all the assembly modes of the SR-2PS structures. The second one determines all the assembly modes of the SP-2RS structures. The presented algorithms can be applied without changes to solve, in analytical form, the direct position analysis (DPA) of all the parallel manipulators that generate a SR-2PS structure or a SP-2RS structure when the actuators are locked. In particular, the closure equations of two generic structures, one of type SR-2PS and the other of type SP-2RS, are written. The eliminants of the two systems of equations are determined and the solution procedures are presented. Finally, the proposed procedures are applied to real cases. This work demonstrates that (i) the DPA solutions of any PM that becomes a SR-2PS structure are at most eight, and (ii) the DPA solutions of any PM that becomes a SP-2RS structure are at most sixteen.

Determination of the Workspace of 6-DOF Parallel Manipulators

1989 ◽  
Author(s):  
C. Gosselin
Keyword(s):  
Parallel Manipulator ◽  
Graphical Representation ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Degree Of Freedom ◽  
Geometrical Properties ◽  
Cartesian Space ◽  
Six Degree Of Freedom

Abstract This paper presents an algorithm for the determination of the workspace of parallel manipulators. The method described here, which is based on geometrical properties of the workspace, leads to a simple graphical representation of the regions of the three-dimensional Cartesian space that are attainable by the manipulator with a given orientation of the platform. Moreover, the volume of the workspace can be easily computed by performing an integration on its boundary, which is obtained from the algorithm. Examples are included to illustrate the application of the method to a six-degree-of-freedom fully-parallel manipulator.

Closed-Form Determination of the Location of a Rigid Body by Seven In-Parallel Linear Transducers

1996 ◽  
Author(s):  
Carlo Innocenti
Keyword(s):  
Rigid Body ◽  
Linear Equations ◽  
Parallel Manipulators ◽  
Rigid Bodies ◽  
Body Location ◽  
Position And Orientation ◽  
General Geometry ◽  
Two Bodies

Abstract The paper presents an original analytic procedure for unambiguously determining the relative position and orientation (location) of two rigid bodies based on the readings from seven linear transducers. Each transducer connects two points arbitrarily chosen on the two bodies. The sought-for rigid-body location simply results by solving linear equations. The proposed procedure is suitable for implementation in control of fully-parallel manipulators with general geometry. A numerical example shows application of the reported results to a case study.

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.

Direct Position Analysis in Analytical Form of Parallel Manipulators Generating Structures With Topology SR-2PS

2004 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Analytical Form ◽  
Solution Procedure ◽  
Generic Structure ◽  
Kinematic Chains ◽  
Position Analysis ◽  
In Series ◽  
The One ◽  
Wide Family

A wide family of parallel manipulators (PMs) is the one that groups all the PMs with three legs where the legs become kinematic chains constituted of a passive spherical pair (S) in series with either a passive prismatic pair (P) or a passive revolute pair (R) when the actuators are locked. The topologies of the structures generated by these manipulators, when the actuators are locked, are ten. One out of these topologies is the SR-2PS topology (one SR leg and two PS legs). This paper presents an algorithm that determines all the assembly modes of the structures with topology SR-2PS in analytical form. The presented algorithm can be applied without changes to solve, in analytical form, the direct position analysis of any parallel manipulator which generates a SR-2PS structure when the actuators are locked. In particular, the closure equations of a generic structure with topology SR-2PS are written. The eliminant of this system of equations is determined and the solution procedure is presented. Finally, the proposed procedure is applied to a real case. This work demonstrates that the solutions of the direct position analysis of any parallel manipulator which generates a SR-2PS structure when the actuators are locked are at most eight.

Inverse position analysis, workspace determination and position synthesis of parallel manipulators with 3-RSR topology

Robotica ◽  
2003 ◽  
Vol 21 (6) ◽  
pp. 627-632 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Parallel Manipulators ◽  
End Effector ◽  
New Approach ◽  
Position Analysis ◽  
Starting Point ◽  
Inverse Position Analysis ◽  
The Common ◽  
Three Degree Of Freedom ◽  
Position Synthesis

Manipulators with 3-RSR topology are three-degree-of-freedom parallel manipulators that may be either spherical or mixed-motion manipulators. The inverse position analysis (IPA) and the workspace determination of 3-RSR manipulators are addressed by means of a new approach. The new approach is centered on a particular form of the closure equations called compatibility equations. The compatibility equations contain only the six coordinates (end-effector coordinates) which locates the end-effector pose (position and orientation) with respect to the frame, and the geometric constants of the manipulator. When the manipulator geometry is assigned, the common solutions of the compatibility equations are the end-effector coordinates which identify the end-effector poses belonging to the manipulator workspace. Moreover, they can be the starting point to easily solve the IPA. The presented compatibility equations can be also used to solve the position synthesis of the 3-RSR manipulator. This way of solving the position synthesis will demonstrate that only approximated solutions exist when more than eight end-effector poses are given.

Forward Problem Singularities of Manipulators Which Become PS-2RS or 2PS-RS Structures When the Actuators Are Locked

2003 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Rigid Body ◽  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Geometric Interpretation ◽  
Forward Problem ◽  
End Effector ◽  
Kinematic Chains ◽  
Position And Orientation ◽  
Manipulator Design

The instantaneous forward problem (IFP) singularities of a parallel manipulator (PM) must be determined during the manipulator design and avoided during the manipulator operation, because they are configurations where the end-effector pose (position and orientation) cannot be controlled by acting on the actuators any longer, and the internal loads of some links become infinite. When the actuators are locked, PMs become structures consisting of one rigid body (platform) connected to another rigid body (base) by means of a number of kinematic chains (limbs). The geometries (singular geometries) of these structures where the platform can perform infinitesimal motion correspond to the IFP singularities of the PMs the structures derive from. This paper studies the singular geometries both of the PS-2RS structure and of the 2PS-RS structure. In particular, the singularity conditions of the two structures will be determined. Moreover, the geometric interpretation of their singularity conditions will be provided. Finally, the use of the obtained results in the design of parallel manipulators which become either PS-2RS or 2PS-RS structures, when the actuators are locked, will be illustrated.

Synthesis of RCCC Linkages to Visit Four Given Poses

2015 ◽  
Vol 7 (3) ◽  
Author(s):  
Shaoping Bai ◽  
Jorge Angeles
Keyword(s):  
Rigid Body ◽  
Exact Solutions ◽  
Kinematic Pair ◽  
Approximate Nature ◽  
The Asymmetry ◽  
Robust Formulation

This paper focuses on the problem of synthesis of spatial four-bar linkages of the RCCC type for rigid-body guidance with four given poses, R denoting a revolute, C a cylindrical kinematic pair. While synthesis equations for CC and RC dyads are available in the literature, the synthesis of spatial RCCC four-bar linkages requires special attention, due to its asymmetric topology. We revisit the problem to cope robustly with the asymmetry, namely, the approximate nature of the RC dyad and the infinity of exact solutions of the CC dyad for the number of given poses. Our approach includes a robust formulation of the synthesis of CC dyads, for the determination of axis-congruences. Moreover, we formulate a uniform synthesis equation, which enables us to treat both RC and CC dyads, with properly selected constraints for both cases. Two design examples are included.

On the Direct Kinematics of Spherical Three-Degree-of-Freedom Parallel Manipulators of General Architecture

1994 ◽  
Vol 116 (2) ◽  
pp. 594-598 ◽  
Author(s):  
C. M. Gosselin ◽  
J. Sefrioui ◽  
M. J. Richard
Keyword(s):  
Parallel Manipulators ◽  
Degree Of Freedom ◽  
Real Solutions ◽  
Three Degree Of Freedom ◽  
General Architecture ◽  
Direct Kinematics

In this paper, the direct kinematics of general spherical parallel three-degree-of-freedom manipulators is investigated. A polynomial of degree 8 is obtained to describe this problem and it is shown that this polynomial is minimal since 8 real solutions corresponding to actual configurations have been found for a given set of actuator coordinates and a given architecture. This result completes the study on the direct kinematics of spherical three-degree-of-freedom parallel manipulators undertaken by the authors in a previous paper. An example of an architecture and a set of actuator coordinates which lead to 8 real solutions is presented to illustrate the results.

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