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.

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

1998 ◽  
Vol 120 (2) ◽  
pp. 293-298 ◽  
Author(s):  
C. Innocenti
Keyword(s):  
Rigid Body ◽  
Linear Equations ◽  
Parallel Manipulators ◽  
Rigid Bodies ◽  
Body Location ◽  
Position And Orientation ◽  
General Geometry ◽  
Two Bodies

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 Actual Configuration of the General Stewart Platform Using Only One Additional Sensor

1999 ◽  
Vol 121 (1) ◽  
pp. 21-25 ◽  
Author(s):  
V. Parenti-Castelli ◽  
R. Di Gregorio
Keyword(s):  
Stewart Platform ◽  
Rigid Bodies ◽  
Displacement Sensor ◽  
Leg Length ◽  
Displacement Sensors ◽  
On Line ◽  
Actual Configuration ◽  
General Geometry

This paper presents a procedure for the determination of the actual configuration of the general geometry Stewart platform (GSP), a fully-parallel manipulator that features two rigid bodies connected to each other via spherical pairs by six controlled-length legs. The six leg length measurements, provided by the displacement sensors incorporated in the leg hardware equipment, do not make it possible to uniquely find the GSP configuration because several configurations are possible for a given set of leg lengths. Several extra sensors in addition to those incorporated in the leg equipment have been proposed in the literature in order to obtain a one-to-one correspondence between the measurements and the actual GSP configuration. The proposed procedure makes use of only one additional displacement sensor and relies upon the analytical results available in the literature for a particular type of Stewart platform. The procedure, which uniquely defines the actual configuration of the GSP, is not intended for on-line implementation. Three different algorithms are proposed for computation and their efficiency compared. A case study is reported that confirms the effectiveness of the procedure.

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.

Velocity and Acceleration Analysis of Contact Between Three-Dimensional Rigid Bodies

1996 ◽  
Vol 63 (4) ◽  
pp. 974-984 ◽  
Author(s):  
N. Sankar ◽  
V. Kumar ◽  
Xiaoping Yun
Keyword(s):  
Rigid Body ◽  
Relative Motion ◽  
Three Dimensional ◽  
Rigid Bodies ◽  
Local Surface ◽  
Contact Points ◽  
Pure Rolling ◽  
Acceleration Analysis ◽  
Rigid Body Motions ◽  
Two Bodies

During manipulation and locomotion tasks encountered in robotics, it is often necessary to control the relative motion between two contacting rigid bodies. In this paper we obtain the equations relating the motion of the contact points on the pair of contacting bodies to the rigid-body motions of the two bodies. The equations are developed up to the second order. The velocity and acceleration constraints for contact, for rolling, and for pure rolling are derived. These equations depend on the local surface properties of each contacting body. Several examples are presented to illustrate the nature of the equations.

Inverse Kinematics of Serial-Chain Manipulators

1996 ◽  
Vol 118 (3) ◽  
pp. 396-404 ◽  
Author(s):  
Hong-You Lee ◽  
Charles F. Reinholtz
Keyword(s):  
Inverse Kinematics ◽  
Minimum Degree ◽  
Complete Solution ◽  
Linear Equations ◽  
End Effector ◽  
Serial Chain ◽  
Inverse Kinematics Problem ◽  
Univariate Polynomials ◽  
General Geometry

This paper proposes a unified method for the complete solution of the inverse kinematics problem of serial-chain manipulators. This method reduces the inverse kinematics problem for any 6 degree-of-freedom serial-chain manipulator to a single univariate polynomial of minimum degree from the fewest possible closure equations. It is shown that the univariate polynomials of 16th degree for the 6R, 5R-P and 4R-C manipulators with general geometry can be derived from 14, 10 and 6 closure equations, respectively, while the 8th and 4th degree polynomials for all the 4R-2P, 3R-P-C, 2R-2C, 3R-E and 3R-S manipulators can be derived from only 2 closure equations. All the remaining joint variables follow from linear equations once the roots of the univariate polynomials are found. This method works equally well for manipulators with special geometry. The minimal properties may provide a basis for a deeper understanding of manipulator geometry, and at the same time, facilitate the determination of all possible configurations of a manipulator with respect to a given end-effector position, the determination of the workspace and its subspaces with the different number of configurations, and the identification of singularity positions of the end-effector. This paper also clarifies the relationship between the three known solutions of the general 6R manipulator as originating from a single set of 14 equations by the first author.

A New Algorithm Based on Two Extra-Sensors for Real-Time Computation of the Actual Configuration of the Generalized Stewart-Gough Manipulator

1998 ◽  
Author(s):  
V. Parenti-Castelli ◽  
R. Di Gregorio
Keyword(s):  
Real Time ◽  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Computational Burden ◽  
The Real ◽  
Position Analysis ◽  
Extra Information ◽  
Actual Configuration ◽  
General Geometry

Abstract It is well known that the direct position analysis of fully-parallel manipulators provides more than one solution, i.e., more than one configuration of the mechanism is possible for a given set of the actuated variables of motion. Extra information is, thus, necessary to find the actual configuration of the manipulator. This paper presents a new algorithm for the real-time computation of the actual configuration of the generalized Stewart-Gough manipulator, also known as 6-6 fully-parallel manipulator with general geometry. The proposed algorithm makes use of two extra rotary sensors in addition to the six normally implemented in the servosystems of the manipulator. A one-to-one correspondence between the sensor measurements and the manipulator configuration is provided. With respect to other algorithms recently presented in the literature, the proposed method greatly reduces the computational burden. Finally a case study shows the effectiveness of the proposed procedure.

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.

Paradoxes in Rigid-Body Kinematics of Structures

1998 ◽  
Vol 65 (1) ◽  
pp. 218-222
Author(s):  
L. Mentrasti
Keyword(s):  
Rigid Body ◽  
Kinematic Analysis ◽  
Sufficient Conditions ◽  
Rigid Bodies ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Kinematic Chains ◽  
Straight Line ◽  
Two Bodies

The paper discusses two paradoxes appearing in the kinematic analysis of interconnected rigid bodies: there are structures that formally satisfy the classical First and Second Theorem on kinematic chains, but do not have any motion. This can arise when some centers of instantaneous rotation (CIR) relevant to two bodies coincide with each other (first kind paradox) or when the CIRs relevant to three bodies lie on a straight line (second kind paradox). In these cases two sets of new theorems on the CIRs can be applied, pointing out sufficient conditions for the nonexistence of a rigid-body motion. The question is clarified by applying the presented theory to several examples.

Analytic Form Solution of the Direct Position Analysis of the SP-2RS Architectures

2004 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Polynomial Equation ◽  
Analytic Form ◽  
Parallel Manipulators ◽  
Equation System ◽  
Rigid Bodies ◽  
Form Solution ◽  
Position Analysis ◽  
Non Linear ◽  
Non Linear System

When the actuators are locked, parallel manipulators (PMs) become parallel structures, that are structures constituted by two rigid bodies (platform and base) connected by a number of kinematic chains (limbs) with only passive kinematic pairs. A set of PMs is the one collecting the manipulators (SP-2RS architectures) which become structures with one limb of type SP and two limbs of type RS (P, R and S stand for prismatic pair, revolute pair and spherical pair respectively). The analytic determination of the assembly modes of the SP-2RS structures (i.e. the solution in analytic form of the direct position analysis of the SP-2RS architectures) has not been presented in the literature yet. This paper presents the solution in analytic form of the DPA of the SP-2RS architectures. In particular, the closure equation system of a generic SP-2RS structure is written in the form of three non-linear equations in three unknowns. The solution of the non-linear system is reduced to the determination of the roots of a sixteenth-degree univariate polynomial equation plus a simple back substitution procedure. The proposed solution algorithm is applied to a real case. The result of this study is that the solutions of the direct position analysis of all the SP-2RS architectures are at most sixteen and can be analytically determined through the proposed algorithm.

Properties of the SX-YS-ZS Structures and Singularity Determination in Parallel Manipulators Which Generate Those Structures

2004 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Rigid Body ◽  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Geometric Interpretation ◽  
Forward Problem ◽  
Unified Approach ◽  
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 (legs). 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. In this paper, the singular geometries of the structures with topology SX-YS-ZS (S stands for spherical pair, whereas X, Y and Z stand for three generic one-dof pair which may be or may not be of the same type) are studied with a unified approach. The presented approach leads to obtain an analytic condition which allows all the singular geometries of these structures to be determined. Moreover, the geometric interpretation of the found singularity condition and the exhaustive enumeration of the types of singular geometries is provided. Finally, the use of the presented results in the design of the manipulators which become one structure with topology SX-YS-ZS when the actuators are locked is discussed.

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