Trajectory Generation for Three-Degree-of-Freedom Cable-Suspended Parallel Robots Based on Analytical Integration of the Dynamic Equations

2016 ◽  
Vol 8 (4) ◽  
Author(s):  
Xiaoling Jiang ◽  
Clément Gosselin
Keyword(s):  
Differential Equations ◽  
Trajectory Generation ◽  
Parallel Robots ◽  
Degree Of Freedom ◽  
Dynamic Equations ◽  
Special Cases ◽  
Linear Differential ◽  
Pass Through ◽  
Cable Forces ◽  
Three Degree Of Freedom

This paper proposes a trajectory generation technique for three degree-of-freedom (3-dof) planar cable-suspended parallel robots. Based on the kinematic and dynamic modeling of the robot, positive constant ratios between cable tensions and cable lengths are assumed. This assumption allows the transformation of the dynamic equations into linear differential equations with constant coefficients for the positioning part, while the orientation equation becomes a pendulum-like differential equation for which accurate solutions can be found in the literature. The integration of the differential equations is shown to yield families of translational trajectories and associated special frequencies. This result generalizes the special cases previously identified in the literature. Combining the results obtained with translational trajectories and rotational trajectories, more general combined motions are analyzed. Examples are given in order to demonstrate the results. Because of the initial assumption on which the proposed method is based, the ratio between cable forces and cable lengths is constant and hence always positive, which ensures that all cables remain under tension. Therefore, the acceleration vector remains in the column space of the Jacobian matrix, which means that the mechanism can smoothly pass through kinematic singularities. The proposed trajectory planning approach can be used to plan dynamic trajectories that extend beyond the static workspace of the mechanism.

Trajectory Generation for Three-Degree-of-Freedom Cable-Suspended Parallel Robots Based on Analytical Integration of the Dynamic Equations

2015 ◽  
Author(s):  
Xiaoling Jiang ◽  
Clément Gosselin
Keyword(s):  
Differential Equations ◽  
Trajectory Generation ◽  
Dynamic Modelling ◽  
Parallel Robots ◽  
Dynamic Equations ◽  
Initial Assumption ◽  
Special Cases ◽  
Linear Differential ◽  
Pass Through ◽  
Cable Forces

This paper proposes a trajectory generation technique for three-dof planar cable-suspended parallel robots. Based on the kinematic and dynamic modelling of the robot, positive constant ratios between cable tensions and cable lengths are assumed. This assumption allows the transformation of the dynamic equations into linear differential equations with constant coefficients for the positioning part, while the orientation equation becomes a pendulum-like differential equation for which accurate solutions can be found in the literature. The integration of the differential equations is shown to yield families of translational trajectories and associated special frequencies. This result generalizes the special cases previously identified in the literature. Combining the results obtained with translational trajectories and rotational trajectories, more general combined motions are analysed. Examples are given in order to demonstrate the results. Because of the initial assumption on which the proposed method is based, the ratio between cable forces and cable lengths is constant and hence always positive, which ensures that all cables remain under tension. Therefore, the acceleration vector remains in the column space of the Jacobian matrix, which means that the mechanism can smoothly pass through kinematic singularities. The proposed trajectory planning approach can be used to plan dynamic trajectories that extend beyond the static workspace of the mechanism.

Singularity robust inverse dynamics of planar 2-RPR parallel manipulators

2004 ◽  
Vol 218 (7) ◽  
pp. 721-730 ◽  
Author(s):  
S Kemal Ider
Keyword(s):  
Parallel Manipulator ◽  
Inverse Dynamics ◽  
Full Rank ◽  
Parallel Manipulators ◽  
Parallel Robots ◽  
Dynamic Equations ◽  
Revolute Joints ◽  
Planar Parallel Manipulator ◽  
Pass Through ◽  
Full Utilization

In planar parallel robots, limitations occur in the functional workspace because of interference of the legs with each other and because of drive singularities where the actuators lose control of the moving platform and the actuator forces grow without bounds. A 2-RPR (revolute, prismatic, revolute joints) planar parallel manipulator with two legs that minimizes the interference of the mechanical components is considered. Avoidance of the drive singularities is in general not desirable since it reduces the functional workspace. An inverse dynamics algorithm with singularity robustness is formulated allowing full utilization of the workspace. It is shown that if the trajectory is planned to satisfy certain conditions related to the consistency of the dynamic equations, the manipulator can pass through the drive singularities while the actuator forces remain stable. Furthermore, for finding the actuator forces in the vicinity of the singular positions a full rank modification of the dynamic equations is developed. A deployment motion is analysed to illustrate the proposed approach.

Dynamic Modelling of a Three Degree-of-Freedom Planar Platform-Type Manipulator

1997 ◽  
Author(s):  
Hazem A. Attia ◽  
Maher G. Mohamed
Keyword(s):  
Differential Equations ◽  
Efficient Solution ◽  
Equations Of Motion ◽  
Translational Motion ◽  
Dynamic Modelling ◽  
Degree Of Freedom ◽  
Constraint Forces ◽  
Dynamic Formulation ◽  
System Of Particles ◽  
Three Degree Of Freedom

Abstract In this paper, the dynamic modelling of a planar three degree-of-freedom platform-type manipulator is presented. A kinematic analysis is carried out initially to evaluate the initial coordinates and velocities. The dynamic model of the manipulator is formulated using a two-step transformation. Initially, the dynamic formulation is written in terms of the Cartesian coordinates of a dynamically equivalent system of particles. Since there is no rotational motion associated with a particle, then the differential equations of motion are derived by applying Newton’s second law to study the translational motion of the particles. The constraint forces between the particles are expressed in terms of Lagrange multipliers. Then, the differential equations of motion are written in terms of the relative joint variables. This leads to an efficient solution and integration of the equations of motion. A numerical example is presented and a computer program is developed.

Modeling, Simulation, and Modal Analysis Hydraulic Valve Lifter With Oil Compressibility Effects

1989 ◽  
Author(s):  
T. Hatch ◽  
A. P. Pisano
Keyword(s):  
Differential Equations ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Check Valve ◽  
Linear Differential Equations ◽  
Third Order ◽  
Mode Behavior ◽  
Special Cases ◽  
Linear Differential ◽  
Hydraulic Valve

Abstract A two-degree-of-freedom (2-DOF), analytical model of a hydraulic valve lifter is derived. Special features of the model include the effects of bulk oil compressibility, multi-mode behavior due to plunger check valve modeling, and provision for the inclusion of third and fourth body displacements to aid In the use of the model in extended, multi-DOF systems. It is shown that motion of the lifter plunger and body must satisfy a coupled system of third-order, non-linear differential equations of motion. It is also shown that the special cases of zero oil compressibility and/or 1-DOF motion of lifter plunger can be obtained from the general third-order equations. For the case of zero oil compressibility, using Newtonian fluid assumptions, the equations of motion are shown to reduce to a system of second-order, linear differential equations. The differential equations are numerically integrated in five scenarios designed to test various aspects of the model. A modal analysis of the 2-DOF, compressible model with an external contact spring is performed and is shown to be in excellent agreement with simulation results.

scholarly journals Solvability and Stability of Impulsive Set Dynamic Equations on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
Shihuang Hong ◽  
Jing Gao ◽  
Yingzi Peng
Keyword(s):  
Differential Equations ◽  
Time Scales ◽  
Difference Equations ◽  
Time Scale ◽  
Dynamic Equations ◽  
Stability Of Solutions ◽  
Real Numbers ◽  
Generalized Derivative ◽  
Special Cases ◽  
Existence And Stability

A class of new nonlinear impulsive set dynamic equations is considered based on a new generalized derivative of set-valued functions developed on time scales in this paper. Some novel criteria are established for the existence and stability of solutions of such model. The approaches generalize and incorporate as special cases many known results for set (or fuzzy) differential equations and difference equations when the time scale is the set of the real numbers or the integers, respectively. Finally, some examples show the applicability of our results.

scholarly journals Optimal Kinematic Redundancy Planning for Planar Mobile Cable-Driven Parallel Robots

2018 ◽  
Author(s):  
Tahir Rasheed ◽  
Philip Long ◽  
David Marquez-Gamez ◽  
Stéphane Caro
Keyword(s):  
Parallel Robot ◽  
Parallel Robots ◽  
Degree Of Freedom ◽  
Static Equilibrium ◽  
Kinematic Redundancy ◽  
Pick And Place ◽  
Moving Platform ◽  
Three Degree Of Freedom

Mobile Cable-Driven Parallel Robots (MCDPRs) are special type of Reconfigurable Cable Driven Parallel Robots (RCDPRs) with the ability of undergoing an autonomous change in their geometric architecture. MCDPRs consists of a classical Cable-Driven Parallel Robot (CDPR) carried by multiple Mobile Bases (MBs). Generally MCDPRs are kinematically redundant due to the additional mobilities generated by the motion of the MBs. As a consequence, this paper introduces a methodology that aims to determine the best kinematic redundancy scheme of Planar MCDPRs (PMCDPRs) with one degree of kinematic redundancy for pick-and-place operations. This paper also discusses the Static Equilibrium (SE) constraints of the PMCDPR MBs that are needed to be respected during the task. A case study of a PMCDPR with two MBs, four cables and a three degree-of-freedom (DoF) Moving Platform (MP) is considered.

Tracking Controllers for Uncertain Systems: Application to a Manutec R3 Robot

1989 ◽  
Vol 111 (4) ◽  
pp. 609-618 ◽  
Author(s):  
Martin Corless
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
General Class ◽  
Uncertain Systems ◽  
Mechanical Systems ◽  
Degree Of Freedom ◽  
Tracking Controllers ◽  
Asymptotic Tracking ◽  
Three Degree Of Freedom

We consider a class of uncertain dynamical systems described by ordinary differential equations and characterized by certain structural conditions and known bounding functions. For a feasible class of desired state motions we present a class of controllers which assure asymptotic tracking to within any desired degree of accuracy. The results are applied to a general class of mechanical systems and are illustrated by a simple example and by application to a three degree-of-freedom model of a Manutec r3 robot.

Modeling, Simulation, and Modal Analysis of a Hydraulic Valve Lifter With Oil Compressibility Effects

1991 ◽  
Vol 113 (1) ◽  
pp. 46-54 ◽  
Author(s):  
C. T. Hatch ◽  
A. P. Pisano
Keyword(s):  
Differential Equations ◽  
Modal Analysis ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Check Valve ◽  
Third Order ◽  
Bulk Oil ◽  
Special Cases ◽  
Linear Differential ◽  
Hydraulic Valve

A two-degree-of-freedom (2-DOF), analytical model of a hydraulic valve lifter is derived. Special features of the model include the effects of bulk oil compressibility, multimode behavior due to plunger check valve modeling, and provision for the inclusion of third and fourth body displacements to aid in the use of the model in extended, multi-DOF systems. It is shown that motion of the lifter plunger and body must satisfy a coupled system of third-order, nonlinear differential equations of motion. It is also shown that the special cases of zero oil compressibility and/or 1-DOF motion of lifter plunger can be obtained from the general third-order equations. For the case of zero oil compressibility, using Newtonian fluid assumptions, the equations of motion are shown to reduce to a system of second-order, linear differential equations. The differential equations are numerically integrated in five scenarios designed to test various aspects of the model. A modal analysis of the 2-DOF, compressible model with an external contact spring is performed and is shown to be in excellent agreement with simulation results.

Kinematic Isotropic Configuration of Spatial Cable-Driven Parallel Robots

2011 ◽  
Vol 1 (4) ◽  
pp. 61-86
Author(s):  
Hamoon Hadian ◽  
Abbas Fattah
Keyword(s):  
Parallel Manipulators ◽  
Parallel Robot ◽  
Numerical Approach ◽  
Parallel Robots ◽  
Degree Of Freedom ◽  
Tension Index ◽  
Symbolic Method ◽  
Three Degree Of Freedom ◽  
Cable Robots

In this paper, the authors study the kinematic isotropic configuration of spatial cable-driven parallel robots by means of four different methods, namely, (i) symbolic method, (ii) geometric workspace, (iii) numerical workspace and global tension index (GTI), and (iv) numerical approach. The authors apply the mentioned techniques to two types of spatial cable-driven parallel manipulators to obtain their isotropic postures. These are a 6-6 cable-suspended parallel robot and a novel restricted three-degree-of-freedom cable-driven parallel robot. Eventually, the results of isotropic conditions of both cable robots are compared to show their applications.

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