scholarly journals Geometric Interpretation of the Derivatives of Parallel Robots’ Jacobian Matrix With Application to Stiffness Control

2003 ◽  
Vol 125 (1) ◽  
pp. 33-42 ◽  
Author(s):  
N. Simaan ◽  
M. Shoham
Keyword(s):  
Closed Form ◽  
Degrees Of Freedom ◽  
Jacobian Matrix ◽  
Geometric Interpretation ◽  
Parallel Robots ◽  
Redundant Robot ◽  
Moving Platforms ◽  
Stiffness Control ◽  
Derivatives Of ◽  
Direct Kinematics

This paper presents a closed-form formulation and geometrical interpretation of the derivatives of the Jacobian matrix of fully parallel robots with respect to the moving platforms’ position/orientation variables. Similar to the Jacobian matrix, these derivatives are proven to be also groups of lines that together with the lines of the instantaneous direct kinematics matrix govern the singularities of the active stiffness control. This geometric interpretation is utilized in an example of a planar 3 degrees-of-freedom redundant robot to determine its active stiffness control singularity.

Gateway Configuration for Rapid Pick-and-Place Operations of 2-Degrees-of-Freedom Parallel Robots

2020 ◽  
Vol 13 (1) ◽  
Author(s):  
Andrea Martin-Parra ◽  
David Rodriguez-Rosa ◽  
Sergio Juarez-Perez ◽  
Guillermo Rubio-Gomez ◽  
Antonio Gonzalez-Rodriguez ◽  
...  
Keyword(s):  
Energy Consumption ◽  
Degrees Of Freedom ◽  
Jacobian Matrix ◽  
Parallel Robots ◽  
End Effector ◽  
Joint Coordinates ◽  
Pick And Place ◽  
Low Energy Consumption ◽  
Simulation Results ◽  
The Difference

Abstract This article presents a new assembling for 2 degrees-of-freedom (DOFs) parallel robots for executing rapid pick-and-place operations with low energy consumption. A conventional design of 2-DOF parallel robots is based on five-bar mechanisms. Collisions between links are highly possible, restricting the end-effector workspace and/or increasing the trajectory time to avoid collisions. In this article, an alternative assembling for preventing collisions is presented. This novel assembling allows exploring the difference between the four five-bar mechanism configurations for the same position of the end-effector. Some of these configurations yield to lower time and/or lower energy consumption for the same motorization. First, a dynamic model of the robot has been developed using matlab® and simulink® and validated by comparison with the results obtained by adams® software. A robust cascade PD regulator for controlling joint coordinates has been tuned providing a high accurate end-effector positioning. Finally, simulation results of four configurations are presented for executing controlled maneuvers. The obtained results demonstrate that the conventional configuration is the worst one in terms of trajectory time or energy consumption and, conversely, the best one corresponds to an uncommonly used configuration. A workspace map where all configurations provide faster maneuvers has been obtained in terms of Jacobian matrix and mechanism elbows distance. The results presented here allow designing a rapid manipulator for pick-and-place operations.

Fully-Isotropic T1R2-Type Parallel Robots With Three Degrees of Freedom

2005 ◽  
Author(s):  
Grigore Gogu
Keyword(s):  
Degrees Of Freedom ◽  
Jacobian Matrix ◽  
Parallel Manipulators ◽  
Parallel Robots ◽  
Velocity Space ◽  
Linear Transformations ◽  
Matrix Mapping ◽  
Joint Velocity ◽  
First Time ◽  
Three Degrees Of Freedom

The paper presents singularity-free fully-isotropic T1R2-type parallel manipulators (PMs) with three degrees of freedom. The mobile platform has one independent translation (T1) and two rotations (R2). A method is proposed for structural synthesis of fully-isotropic T1R2-type PMs based on the theory of linear transformations. A one-to-one correspondence exists between the actuated joint velocity space and the external velocity space of the moving platform. The Jacobian matrix mapping the two vector spaces of fully-isotropic T1R2-type PMs presented in this paper is the 3x3 identity matrix throughout the entire workspace. The condition number and the determinant of the Jacobian matrix being equal to one, the manipulator performs very well with regard to force and motion transmission capabilities. As far as we are aware, this paper presents for the first time in the literature solutions of singularity-free T1R2-type PMs with decoupled an uncoupled motions, along with the fully-isotropic solutions.

On the kinematics of a new parallel mechanism with Schoenflies motion

Robotica ◽  
2015 ◽  
Vol 34 (9) ◽  
pp. 2056-2070 ◽  
Author(s):  
Po-Chih Lee ◽  
Jyh-Jone Lee
Keyword(s):  
Closed Form ◽  
Condition Number ◽  
Kinematic Analysis ◽  
Parallel Mechanism ◽  
Parallel Manipulator ◽  
Matrix Algebra ◽  
Degrees Of Freedom ◽  
Jacobian Matrix ◽  
Pick And Place

SUMMARYThis paper investigates the kinematics of one new isoconstrained parallel manipulator with Schoenflies motion. This new manipulator has four degrees of freedom and two identical limbs, each having the topology of Cylindrical–Revolute–Prismatic–Helical (C–R–P–H). The kinematic equations are derived in closed-form using matrix algebra. The Jacobian matrix is then established and the singularities of the robot are investigated. The reachable workspaces and condition number of the manipulator are further studied. From the kinematic analysis, it can be shown that the manipulator is simple not only for its construction but also for its control. It is hoped that the results of the evaluation of the two-limb parallel mechanism can be useful for possible applications in industry where a pick-and-place motion is required.

Prototyping and Validation of a Fixed-Actuator 3-Leg 6-DOF Robot

2018 ◽  
Author(s):  
Nathan A. Jensen ◽  
Carl A. Nelson
Keyword(s):  
Parallel Mechanism ◽  
Degrees Of Freedom ◽  
Parallel Robot ◽  
Parallel Robots ◽  
Mechanism Analysis ◽  
Joint Angles ◽  
Maximum Extension ◽  
Partial Derivatives ◽  
Expected Performance ◽  
Derivatives Of

While 6-leg, 6 DOF parallel robots offer advantages over serial mechanisms in many applications, they suffer from mobility limitation pertaining to both the maximum extension of links and link interference. The latter of these can be mitigated by a reduction of the number of links in the mechanism. The end-effector’s degrees of freedom are maintained by adding controllable degrees of freedom to the remaining legs. This paper presents a prototype of a previously proposed 3-leg, 6-DOF parallel robot. A measure of its workspace is also shown and compared to that of a similarly sized 6-leg parallel mechanism. Analysis of partial derivatives of Cartesian points with respect to joint angles is also explored to give a metric of expected performance in different regions of workspace.

Structural synthesis of maximally regular T3R2-type parallel robots via theory of linear transformations and evolutionary morphology

Robotica ◽  
2009 ◽  
Vol 27 (1) ◽  
pp. 79-101 ◽  
Author(s):  
G. Gogu
Keyword(s):  
Degrees Of Freedom ◽  
Jacobian Matrix ◽  
Parallel Robots ◽  
Velocity Space ◽  
Evolutionary Morphology ◽  
Structural Synthesis ◽  
Real Time Control ◽  
Linear Transformations ◽  
Time Control ◽  
Moving Platform

SUMMARYThe paper presents structural synthesis of maximally regular T3R2-type parallel robotic manipulators (PMs) with five degrees of freedom. The moving platform has three independent translations (T3) and two rotations (R2). A method is proposed for structural synthesis of maximally regular T3R2-type PMs based on the theory of linear transformations and evolutionary morphology. A one-to-one correspondence exists between the actuated joint velocity space and the external velocity space of the moving platform. The Jacobian matrix mapping the two vector spaces of maximally regular T3R2-type PMs presented in this paper is the 5×5 identity matrix throughout the entire workspace. The condition number and the determinant of the Jacobian matrix being equal to one, the manipulator performs very well with regard to force and motion transmission capabilities. Kinematic analysis of maximally regular parallel robots is trivial and no computation is required for real-time control. This paper presents in a unified approach the structural synthesis of PMs with five degrees of freedom with decoupled and uncoupled motions, along with the maximally regular solutions.

scholarly journals Design Blocks in Simulink to Detection Singularity in the Workspace of Gough-Stewart Robot Manipulator

2019 ◽  
Vol 27 (1) ◽  
pp. 1-15
Author(s):  
Hassan Mohammed Alwan ◽  
Riyadh Ahmed Sarhan
Keyword(s):  
Degrees Of Freedom ◽  
Jacobian Matrix ◽  
Robot Manipulator ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Lagrange Method ◽  
Six Degrees Of Freedom ◽  
Moving Platforms ◽  
Current Input ◽  
Fixed Base

This work deals with Gough-Stewart robot manipulator, which has six degrees of freedom, six actuators, fixed base, and moving platforms. Here, the Jacobian matrix derived to detect the singular point in the workspace for manipulator at determinant of Jacobian matrix equal to zero, then derived the equation of motion from the dynamic analysis by Lagrange method to verify the singular points with Jacobian where the forces increase rapidly at this point. Finally, design blocks in Simulink include the Jacobian matrix and the equations of motion to detection the singularities at any time for current input parameters (X, Y, Z, α, β, γ), where the determinant of the Jacobian equal to zero at maximum forces.

scholarly journals Path-planning of a hybrid parallel robot using stiffness and workspace for foot rehabilitation

2018 ◽  
Vol 10 (1) ◽  
pp. 168781401775415 ◽  
Author(s):  
Alireza Rastegarpanah ◽  
Hamid Rakhodaei ◽  
Mozafar Saadat ◽  
Mohammad Rastegarpanah ◽  
Naresh Marturi ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Parallel Robot ◽  
Parallel Robots ◽  
Degree Of Freedom ◽  
Position Vector ◽  
Parallel Mechanisms ◽  
Moving Platforms ◽  
Maximum Stiffness ◽  
In Series ◽  
Applied Forces

Stiffness is one of the important parameters for estimating the performance of hybrid parallel robots as it is not constant throughout its workspace. The aim of this study is to provide an optimum path based on maximum stiffness within the workspace of a 9-degree-of-freedom hybrid parallel mechanism configuration, which includes nine linear actuators connecting one stationary and two moving platforms in series. The proposed robot is designed for ankle rehabilitation, where accurate and precise movement of lower extremities is required. The design takes advantage of two important characteristics of parallel robots: stiffness and workspace. The proposed methodology to determine the stiffness of hybrid robot in three single axes is based on calculation of position vector of each actuator in any particular pose, by considering the inverse kinematics of the system, in order to obtain the magnitude and direction of the applied forces. The results obtained from the workspace calculations have been compared with those of two standard parallel mechanisms including a 6-degree-of-freedom hexapod and a tripod with 3 degrees of freedom. The stiffness of the robot has been calculated in simulation and then compared with those of a developed prototype hybrid model in two different case studies.

Constraint Singularities of Force Transmission in Nonredundant Parallel Robots With Less Than Six Degrees of Freedom

2003 ◽  
Vol 125 (3) ◽  
pp. 557-563 ◽  
Author(s):  
Matteo Zoppi ◽  
Luca E. Bruzzone ◽  
Rezia M. Molfino ◽  
Rinaldo C. Michelini
Keyword(s):  
Degrees Of Freedom ◽  
Jacobian Matrix ◽  
Parallel Robots ◽  
Numerical Results ◽  
Structural Behavior ◽  
Force Transmission ◽  
Constraint Force ◽  
Six Degrees Of Freedom ◽  
Local Properties ◽  
Delta Robot

The analysis of the workspace singularities is one of the fundamental aspects in the design of parallel robots. The architecture singularities are generally studied analysing the local properties of the Jacobian matrix. Nevertheless, for limited-DOF parallel robots, there is a category of singularities (constraint or constructive singularities), relating to the constraint force transmission, which are not described by this matrix. This paper deals with a general approach to the analysis of these singularities, used in the synthesis of a Linear Delta robot to suitably modify its geometry, remarkably improving the structural behavior. Details and numerical results are provided.

scholarly journals Closed-form time derivatives of the equations of motion of rigid body systems

2021 ◽  
Author(s):  
Andreas Müller ◽  
Shivesh Kumar
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Formulation ◽  
Dynamics Of Rigid Body ◽  
Control Forces ◽  
And Control ◽  
Derivatives Of ◽  
Insight Into

AbstractDerivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

Design and analysis of CICABOT: a novel translational parallel manipulator based on two 5-bar mechanisms

Robotica ◽  
2011 ◽  
Vol 30 (3) ◽  
pp. 449-456 ◽  
Author(s):  
M. F. Ruiz-Torres ◽  
E. Castillo-Castaneda ◽  
J. A. Briones-Leon
Keyword(s):  
Parallel Manipulator ◽  
Jacobian Matrix ◽  
Geometric Analysis ◽  
Vector Equation ◽  
Prismatic Joints ◽  
Hollow Cylinders ◽  
Moving Platform ◽  
Direct Kinematics

SUMMARYThis work presents the CICABOT, a novel 3-DOF translational parallel manipulator (TPM) with large workspace. The manipulator consists of two 5-bar mechanisms connected by two prismatic joints; the moving platform is on the union of these prismatic joints; each 5-bar mechanism has two legs. The mobility of the proposed mechanism, based on Gogu approach, is also presented. The inverse and direct kinematics are solved from geometric analysis. The manipulator's Jacobian is developed from the vector equation of the robot legs; the singularities can be easily derived from Jacobian matrix. The manipulator workspace is determined from analysis of a 5-bar mechanism; the resulting workspace is the intersection of two hollow cylinders that is much larger than other TPM with similar dimensions.

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