A Differential-Geometric Analysis of Singularities of Point Trajectories of Serial and Parallel Manipulators

1998 ◽  
Vol 123 (1) ◽  
pp. 80-89 ◽  
Author(s):  
Ashitava Ghosal ◽  
Bahram Ravani
Keyword(s):  
Degrees Of Freedom ◽  
Parallel Manipulators ◽  
Rate Of Change ◽  
Degree Of Freedom ◽  
Task Space ◽  
Singular Configurations ◽  
Point Trajectories ◽  
Singular Configuration ◽  
Three Degree Of Freedom ◽  
Derivatives Of

In this paper, we present a differential-geometric approach to analyze the singularities of task space point trajectories of two and three-degree-of-freedom serial and parallel manipulators. At non-singular configurations, the first-order, local properties are characterized by metric coefficients, and, geometrically, by the shape and size of a velocity ellipse or an ellipsoid. At singular configurations, the determinant of the matrix of metric coefficients is zero and the velocity ellipsoid degenerates to an ellipse, a line or a point, and the area or the volume of the velocity ellipse or ellipsoid becomes zero. The degeneracies of the velocity ellipsoid or ellipse gives a simple geometric picture of the possible task space velocities at a singular configuration. To study the second-order properties at a singularity, we use the derivatives of the metric coefficients and the rate of change of area or volume. The derivatives are shown to be related to the possible task space accelerations at a singular configuration. In the case of parallel manipulators, singularities may lead to either loss or gain of one or more degrees-of-freedom. For loss of one or more degrees-of-freedom, the possible velocities and accelerations are again obtained from a modified metric and derivatives of the metric coefficients. In the case of a gain of one or more degrees-of-freedom, the possible task space velocities can be pictured as growth to lines, ellipses, and ellipsoids. The theoretical results are illustrated with the help of a general spatial 2R manipulator and a three-degree-of-freedom RPSSPR-SPR parallel manipulator.

Differential Geometric Analysis of Singularities of Point Trajectories of Serial and Parallel Manipulators

1998 ◽  
Author(s):  
Ashitava Ghosal ◽  
Bahram Ravani
Keyword(s):  
Degrees Of Freedom ◽  
Geometric Analysis ◽  
Parallel Manipulators ◽  
Rate Of Change ◽  
Degree Of Freedom ◽  
Local Geometry ◽  
Singular Configurations ◽  
Point Trajectories ◽  
Definition Of ◽  
Three Degree Of Freedom

Abstract In this paper, we present a differential-geometric analysis of singularities of point trajectories of two and three-degree-of-freedom serial and parallel manipulators. At non-singular configurations, the first order local properties are characterized by the metric coefficients, and, geometrically, by the shape and size of a velocity ellipse and ellipsoid for two and three-degree-of-freedom motions respectively. At singular configurations, the definition of a metric is no longer valid and the velocity ellipsoid degenerates to an ellipse, a line or a point, and the area or the volume of the velocity ellipse or ellipsoid becomes zero. The second and higher order properties, such as curvature, are also not defined at a singularity. In this paper, we use the rate of change of the area or volume to characterize the singularities of the point trajectory. For parallel manipulators, singularities may lead to either loss or gain of one or more degrees-of-freedom. For loss of degree of freedom, the ellipsoid degenerates to an ellipse, a line, or a point as in serial manipulators. For a gain of degree-of-freedom the singularities can be pictured as growth to lines, ellipses, and ellipsoids. The method presented gives a clear geometric picture as to the possible directions and magnitude of motion at a singularity and the local geometry near a singularity. The theoretical results are illustrated with the help of a general spatial 2R manipulator and a three-degree-of-freedom RPSSPR-SPR parallel manipulator.

Design of a Teaching Robot with Three Degrees of Freedom

2012 ◽  
Vol 619 ◽  
pp. 325-328
Author(s):  
You Jun Huang ◽  
Ze Lun Li ◽  
Zhi Cheng Huang
Keyword(s):  
Degrees Of Freedom ◽  
Low Cost ◽  
Degree Of Freedom ◽  
Good Repeatability ◽  
Main Components ◽  
Opening And Closing ◽  
Three Degree Of Freedom ◽  
Three Degrees Of Freedom ◽  
Application Prospects

A teaching robot with three degree of freedom is designed. The three degrees of freedom are: waist rotation, lifting and stretching of the arm and opening and closing of the gripper. The designs of the main components are: a mobile chassis, parallel rails, horizontal rails and manipulator. The teaching robot designed has the features of low cost, easy to regulation, good repeatability and it has good promotion and application prospects in the field of teaching.

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.

Rudder/Fin Roll Stabilization of the USCG WMEC 901 Class Vessel

1999 ◽  
Vol 36 (03) ◽  
pp. 157-170
Author(s):  
Jerrold N. Sgobbo ◽  
Michael G. Parsons
Keyword(s):  
Degrees Of Freedom ◽  
Multiple Input Multiple Output ◽  
Cross Coupling ◽  
Degree Of Freedom ◽  
Rolling Motion ◽  
Rolling Moment ◽  
Multiple Input ◽  
Input Multiple Output ◽  
Three Degree Of Freedom ◽  
The U.S

The U.S. Coast Guard's 270-ft Medium Endurance Cutter (WMEC) operates with an active fin stabilization system. This system was designed using a one-degree-of-freedom (1-DOF) model in the roll direction. The controller was designed separate from the heading autopilot. The effects of the rudders and their ability to produce a significant rolling moment were also neglected as well as the cross coupling of roll motions into other degrees of freedom. This paper studies the effects of the rudders on the rolling motion of the ship using a three-degree-of-freedom (3-DOF) model. A simple optimal heading autopilot is designed and combined with the existing fin roll controller to investigate the effects of the rudders on the roll motions of this class of vessel. A rudder roll controller and a multiple input-multiple output (MIMO) rudder/fin controller are designed as well. Significant roll reduction can be achieved using the MIMO rudder/fin controller.

Improving Motion Stability of the Plane 3-RPR Parallel Manipulator at Singular Configuration

2018 ◽  
Author(s):  
Yu-Tong Li ◽  
Yu-Xin Wang
Keyword(s):  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Angular Speed ◽  
Motion Stability ◽  
Singular Configurations ◽  
Movable Platform ◽  
Singular Configuration ◽  
Translation Type ◽  
External Forces ◽  
Gerschgorin Circle

Kinematic parameters have significant influences on the motion stability of parallel manipulators at singular configureations. Taking the plane 3-RPR parallel manipulator as an example, the motion stability at different types of singular configurations corresponding to the angular speed and velocity of the movable platform are investigated. At first, the second order of uncoupled dynamics equation for the 3-RPR parallel manipulator is established with the aid of the second class Lagrange approach. According to the Lyapunov first approximate stability criterion, the approximate conditions for the 3-RPR parallel manipulator with a stabile motion at singular configurations are determined based on the Gerschgorin circle theorem. Next, the exact Hurwitz criterion is utilized to study the motion stability and the load capability of the manipulator corresponding to the angular speed and velocity of the movable platform, as well as the directions of the external forces at two kinds of singular configurations: with a gained rotation-type DOF, and with a gained translation-type DOF, respectively. The results show that increasing both the angular speed and the velocity of the mass center of the movable platform can efficiently improve the motion stability of the 3-RPR parallel manipulator at singular configurations.

Singularity Analysis of Serial Robot-Manipulators

1996 ◽  
Vol 118 (4) ◽  
pp. 520-525 ◽  
Author(s):  
A. Karger
Keyword(s):  
Degrees Of Freedom ◽  
Robot Manipulator ◽  
Robot Manipulators ◽  
Singularity Analysis ◽  
Singular Set ◽  
Singular Configurations ◽  
Special Cases ◽  
Serial Robot ◽  
Singular Configuration ◽  
Actual Configuration

This paper is devoted to the description of the set of all singular configurations of serial robot-manipulators. For 6 degrees of freedom serial robot-manipulators we have developed a theory which allows to describe higher order singularities. By using Lie algebra properties of the screw space we give an algorithm, which determines the degree of a singularity from the knowledge of the actual configuration of axes of the robot-manipulator only. The local shape of the singular set in a neighbourhood of a singular configuration can be determined as well. We also solve the problem of escapement from a singular configuration. For serial robot-manipulators with the number of degrees of freedom different from six we show that up to certain exceptions singular configurations can be avoided by a small change of the motion of the end-effector. We also give an algorithm which allows to determine equations of the singular set for any serial robot-manipulator. We discuss some special cases and give examples of singular sets including PUMA 560.

scholarly journals Feedback Linearisation for Nonlinear Vibration Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
S. Jiffri ◽  
P. Paoletti ◽  
J. E. Cooper ◽  
J. E. Mottershead
Keyword(s):  
Degrees Of Freedom ◽  
Nonlinear Problems ◽  
Second Order ◽  
Degree Of Freedom ◽  
Sensors And Actuators ◽  
Order Matrix ◽  
Problem Class ◽  
Aeroelastic Model ◽  
Matrix Differential ◽  
Three Degree Of Freedom

Feedback linearisation is a well-known technique in the controls community but has not been widely taken up in the vibrations community. It has the advantage of linearising nonlinear system models, thereby enabling the avoidance of the complicated mathematics associated with nonlinear problems. A particular and common class of problems is considered, where the nonlinearity is present in a system parameter and a formulation in terms of the usual second-order matrix differential equation is presented. The classical texts all cast the feedback linearisation problem in first-order form, requiring repeated differentiation of the output, usually presented in the Lie algebra notation. This becomes unnecessary when using second-order matrix equations of the problem class considered herein. Analysis is presented for the general multidegree of freedom system for those cases when a full set of sensors and actuators is available at every degree of freedom and when the number of sensors and actuators is fewer than the number of degrees of freedom. Adaptive feedback linearisation is used to address the problem of nonlinearity that is not known precisely. The theory is illustrated by means of a three-degree-of-freedom nonlinear aeroelastic model, with results demonstrating the effectiveness of the method in suppressing flutter.

Inertial Vibration Damping Control for a Flexible Base Manipulator

2002 ◽  
Author(s):  
Lynnane E. George ◽  
Wayne J. Book
Keyword(s):  
Performance Index ◽  
Degrees Of Freedom ◽  
Position Control ◽  
Joint Space ◽  
Vibration Damping ◽  
Degree Of Freedom ◽  
Inertia Effects ◽  
Inertial Interaction ◽  
Flexible Base ◽  
Three Degree Of Freedom

A rigid (micro) robot mounted serially to the tip of a long, flexible (macro) manipulator is often used to increase reach capability, but flexibility in the macromanipulator can interfere with positioning accuracy. A rigid manipulator attached to a flexible but unactuated base was used to study a scheme to achieve positioning of the micromanipulator combined with enhanced vibration damping of the base. Inertial interaction forces and torques acting between the robot and its base were modeled and studied to determine how to use them to damp the vibration. One issue is that there are locations in the workspace where the rigid robot loses its ability to create interactions in one or more degrees of freedom. These “inertial singularities” are functions of the rigid robot’s joint variables. A performance index was developed to predict the ability of the rigid robot to damp vibration and will help ensure the robot is operating in joint space configurations favorable for inertial damping. When the performance index is used along with the appropriate choice of feedback gains, the inertia effects, or those directly due to accelerating the robot’s links, have the greatest influence on the interactions. By commanding the robot link’s accelerations out of phase with the base velocity, vibration energy will be removed from the system. This signal is then added to the rigid robot’s position control signal. Simulations of a rigid three degree of freedom anthropomorphic robot mounted on a flexible base were developed and show the effectiveness of the control scheme. In addition, experimental results demonstrating two degree of freedom vibration damping are included.

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