scholarly journals Stability of Manipulator Configuration Under External Loading

2012 ◽  
Author(s):  
Alexandr Klimchik ◽  
Anatol Pashkevich ◽  
Damien Chablat
Keyword(s):  
Equilibrium Configuration ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Static Equilibrium ◽  
Practical Significance ◽  
External Loading ◽  
Kinematic Chains ◽  
Cartesian Stiffness Matrix ◽  
The Stability ◽  
Cartesian Stiffness

The paper is devoted to the analysis of robotic manipulator behavior under internal and external loadings. The main contributions are in the area of stability analysis of manipulator configurations corresponding to the loaded static equilibrium. In contrast to other works, in addition to usually studied the end-platform behavior with respect to the disturbance forces, the problem of configuration stability for each kinematic chain is considered. The proposed approach extends the classical notion of the stability for the static equilibrium configuration that is completely defined the properties of the Cartesian stiffness matrix only. The advantages and practical significance of the proposed approach are illustrated by several examples that deal with serial kinematic chains and parallel manipulators. It is shown that under the loading the manipulator workspace may include some specific points that are referred to as elastostatic singularities where the chain configurations become unstable.

The differential calculus of screws: Theory, geometrical interpretation, and applications

2009 ◽  
Vol 223 (6) ◽  
pp. 1449-1468 ◽  
Author(s):  
J J Cervantes-Sánchez ◽  
J M Rico-Martínez ◽  
G González-Montiel ◽  
E J González-Galván
Keyword(s):  
Differential Calculus ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Higher Order ◽  
Abstract Concepts ◽  
Serial Manipulators ◽  
Kinematic Chains ◽  
Finite Displacements ◽  
Typical Shape ◽  
Derivatives Of

This article presents a novel and original formula for the higher-order time derivatives, and also for the partial derivatives of screws, which are successively computed in terms of Lie products, thus leading to the automation of the differentiation process. Through the process and, due to the pure geometric nature of the derivation approach, an enlightening physical interpretation of several screw derivatives is accomplished. Important applications for the proposed formula include higher-order kinematic analysis of open and closed kinematic chains and also the kinematic synthesis of serial and parallel manipulators. More specifically, the existence of a natural relationship is shown between the differential calculus of screws and the Lie subalgebras associated with the expected finite displacements of the end effector of an open kinematic chain. In this regard, a simple and comprehensible methodology is obtained, which considerably reduces the abstraction level frequently required when one resorts to more abstract concepts, such as Lie groups or Lie subalgebras; thus keeping the required mathematical background to the extent that is strictly necessary for kinematic purposes. Furthermore, by following the approach proposed in this article, the elements of Lie subalgebra arise in a natural way — due to the corresponding changes in screws through time — and they also have the typical shape of the so-called ordered Lie products that characterize those screws that are compatible with the feasible joint displacements of an arbitrary serial manipulator. Finally, several application examples — involving typical, serial manipulators — are presented in order to prove the feasibility and validity of the proposed method.

A Non-Linear Stiffness Model for Serial and Parallel Manipulators

2017 ◽  
Vol 5 (1) ◽  
pp. 34-62
Author(s):  
Manoj Kumar
Keyword(s):  
Numerical Technique ◽  
Sudden Change ◽  
Parallel Manipulators ◽  
Rigid Bodies ◽  
Stiffness Analysis ◽  
Cartesian Stiffness Matrix ◽  
Stiffness Model ◽  
Non Linear ◽  
Linear Behaviour ◽  
Cartesian Stiffness

The paper presents a methodology to enhance the stiffness analysis of serial and parallel manipulators with passive joints. It directly takes into account the loading influence on the manipulator configuration and, consequently, on its Jacobians and Hessians. The main contributions of this paper are the introduction of a non-linear stiffness model for the manipulators with passive joints, a relevant numerical technique for its linearization and computing of the Cartesian stiffness matrix which allows rank-deficiency. Within the developed technique, the manipulator elements are presented as pseudo-rigid bodies separated by multidimensional virtual springs and perfect passive joints. Simulation examples are presented that deal with parallel manipulators of the Ortholide family and demonstrate the ability of the developed methodology to describe non-linear behaviour of the manipulator structure such as a sudden change of the elastic instability properties (buckling).

Dexterous Workspace of n-PRRR Planar Parallel Manipulators

2012 ◽  
Vol 4 (3) ◽  
Author(s):  
André Gallant ◽  
Roger Boudreau ◽  
Marise Gallant
Keyword(s):  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Divergence Theorem ◽  
End Effector ◽  
Kinematic Chains ◽  
Prismatic Joint ◽  
Planar Surfaces ◽  
Revolute Joints ◽  
Gauss Divergence Theorem ◽  
Planar Parallel Manipulators

In this work, a method is presented to geometrically determine the dexterous workspace boundary of kinematically redundant n-PRRR (n-PRRR indicates that the manipulator consists of n serial kinematic chains that connect the base to the end-effector. Each chain is composed of two actuated (therefore underlined) joints and two passive revolute joints. P indicates a prismatic joint while R indicates a revolute joint.) planar parallel manipulators. The dexterous workspace of each nonredundant RRR kinematic chain is first determined using a four-bar mechanism analogy. The effect of the prismatic actuator is then considered to yield the workspace of each PRRR kinematic chain. The intersection of the dexterous workspaces of all the kinematic chains is then obtained to determine the dexterous workspace of the planar n-PRRR manipulator. The Gauss divergence theorem applied to planar surfaces is implemented to compute the total dexterous workspace area. Finally, two examples are shown to demonstrate applications of the method.

Type Synthesis of 3-DOF Translational Parallel Manipulators Based on Screw Theory

2004 ◽  
Vol 126 (1) ◽  
pp. 83-92 ◽  
Author(s):  
Xianwen Kong ◽  
Cle´ment M. Gosselin
Keyword(s):  
Screw Theory ◽  
General Procedure ◽  
Linear Equations ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Type Synthesis ◽  
Displacement Analysis ◽  
Kinematic Chains ◽  
Small Joint ◽  
Actuated Joints

A method is proposed for the type synthesis of 3-DOF (degree-of-freedom) translational parallel manipulators (TPMs) based on screw theory. The wrench systems of a translational parallel kinematic chain (TPKC) and its legs are first analyzed. A general procedure is then proposed for the type synthesis of TPMs. The type synthesis of legs for TPKCs, the type synthesis of TPKCs as well as the selection of actuated joints of TPMs are dealt with in sequence. An approach to derive the full-cycle mobility conditions for legs for TPKCs is proposed based on screw theory and the displacement analysis of serial kinematic chains undergoing small joint motions. In addition to the TPKCs proposed in the literature, TPKCs with inactive joints are synthesized. The phenomenon of dependent joint groups in a TPKC is revealed systematically. The validity condition of actuated joints of TPMs is also proposed. Finally, linear TPMs, which are TPMs whose forward displacement analysis can be performed by solving a set of linear equations, are also revealed.

Symmetry and invariants of kinematic chains and parallel manipulators

Robotica ◽  
2012 ◽  
Vol 31 (1) ◽  
pp. 61-70 ◽  
Author(s):  
Roberto Simoni ◽  
Celso Melchiades Doria ◽  
Daniel Martins
Keyword(s):  
Group Theory ◽  
Automorphism Group ◽  
Kinematic Analysis ◽  
Group Structure ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Formal Definition ◽  
Kinematic Chains ◽  
Number Synthesis ◽  
Definition Of

SUMMARYThis paper presents applications of group theory tools to simplify the analysis of kinematic chains associated with mechanisms and parallel manipulators. For the purpose of this analysis, a kinematic chain is described by its properties, i.e. degrees-of-control, connectivity and redundancy matrices. In number synthesis, kinematic chains are represented by graphs, and thus the symmetry of a kinematic chain is the same as the symmetry of its graph. We present a formal definition of symmetry in kinematic chains based on the automorphism group of its associated graph. The symmetry group of the graph is associated with the graph symmetry. By using the group structure induced by the symmetry of the kinematic chain, we prove that degrees-of-control, connectivity and redundancy are invariants by the action of the automorphism group of the graph. Consequently, it is shown that it is possible to reduce the size of these matrices and thus reduce the complexity of the kinematic analysis of mechanisms and parallel manipulators in early stages of mechanisms design.

On the Stochastic Stability and Observability of Controlled Serial Kinematic Chains

2010 ◽  
Author(s):  
Fabio Bonsignorio
Keyword(s):  
Stochastic Stability ◽  
Stochastic System ◽  
Tangent Space ◽  
Kinematic Chain ◽  
Gaussian Distributions ◽  
Shannon Theory ◽  
Kinematic Chains ◽  
Controlled Systems ◽  
The Stability

In this paper the stability and observability of a controlled serial kinematic chain are analyzed with reference to a characterization of observability and stability for a stochastic system grounded in the application of Shannon theory to controlled systems. This approach was proposed in 2004 by H. Touchette and S. Lloyd. In particular it is analyzed in depth the case in which errors on the joints follow (concentrated) Gaussian distributions. In this case the property of Lie Groups (and related tangent space Lie algebra), studied from G. Chirikjian et al., allow to carry out the study of stochastic serial kinematic chains in a simplified way and to properly identify stability and observability conditions from a Shannon information standpoint.

The Determination of Equilibrium Configurations of Spring-Restrained Mechanisms Using (4 × 4) Matrix Methods

1967 ◽  
Vol 89 (1) ◽  
pp. 87-93 ◽  
Author(s):  
D. F. Livermore
Keyword(s):  
Kinematic Chain ◽  
External Loading ◽  
Matrix Methods ◽  
Kinematic Chains ◽  
Equilibrium Configurations ◽  
Force Systems ◽  
Velocity Information ◽  
Equilibrium Analyses ◽  
External Loads

A spring-restrained, multiple-loop, multiple-degree-of-freedom kinematic chain will normally have one or more stable equilibrium configurations when steady external loads are applied to it. The “kinematic equivalent” of a vehicle and its suspension linkages is a common example of such a system. Changes in external loading due to cornering, braking, and so on, can produce important changes in the equilibrium configuration of the suspension. This paper presents a general method for determining the equilibrium configurations of spring-restrained, kinematic chains under the action of steady external loading. The iterative (4 × 4) matrix method of displacement analysis, previously developed for single-loop chains, is extended to complex chains and is used to determine the displacement and velocity information required for equilibrium analyses. The final results are general computer programs which will determine displacement and/or equilibrium configurations for simple or complex mechanism systems wherein the applied force systems may be considered conservative.

Dexterous Workspace of N-PRRR Planar Parallel Manipulators

2010 ◽  
Author(s):  
Andre´ Gallant ◽  
Roger Boudreau ◽  
Marise Gallant
Keyword(s):  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Divergence Theorem ◽  
Kinematic Chains ◽  
Planar Surfaces ◽  
Gauss Divergence Theorem ◽  
Planar Parallel Manipulators ◽  
Workspace Boundary

In this work, a method is presented to geometrically determine the dexterous workspace boundary of kinematically redundant n-PRRR planar parallel manipulators. The dexterous workspace of each non-redundant RRR kinematic chain is first determined using a four-bar mechanism analogy. The effect of the prismatic actuator is then considered to yield the workspace of each PRRR kinematic chain. The intersection of the dexterous workspaces of all the kinematic chains is then obtained to determine the dexterous workspace of the planar n-PRRR manipulator. The Gauss Divergence Theorem applied to planar surfaces is implemented to compute the total dexterous workspace area. Finally, two examples are shown to demonstrate applications of the method.

A geometrical formulation for the workspace of parallel manipulators

Robotica ◽  
2021 ◽  
pp. 1-11
Author(s):  
Matteo Russo ◽  
Marco Ceccarelli
Keyword(s):  
Parallel Mechanism ◽  
Parallel Manipulator ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Analytical Models ◽  
Kinematic Chains ◽  
Motion Constraints ◽  
Geometrical Formulation ◽  
Geometric Formulation ◽  
Algebraic Formulation

Abstract In study this paper, a geometric formulation is proposed to describe the workspace of parallel manipulators by using a recursive approach as an extension of volume generation for solids of revolution. In this approach, the workspace volume and boundary for each limb of the parallel manipulator is obtained with an algebraic formulation derived from the kinematic chain of the limb and the motion constraints on its joints. Then, the overall workspace of the mechanism can be determined as the intersection of the limb workspaces. The workspace of different kinematic chains is discussed and classified according to its external shape. An algebraic formulation for the inclusion of obstacles in the computation is also proposed. Both analytical models and numerical simulations are reported with their advantages and limitations. An example on a 3-SPR parallel mechanism illustrates the feasibility of the formulation and its efficiency.

Research Regarding Improvement of Torsional Stiffness for Planetary Speed Reducers Used in the Actuation of Industrial Robots

2015 ◽  
Vol 809-810 ◽  
pp. 718-723
Author(s):  
Paul Alin Butunoi ◽  
Gheorghe Stan ◽  
Ana Lăcrămioara Ungureanu
Keyword(s):  
Power Distribution ◽  
Kinematic Chain ◽  
Industrial Robots ◽  
Torsional Stiffness ◽  
Computation Method ◽  
Planetary Gears ◽  
Kinematic Chains ◽  
Speed Reducer ◽  
The Stability ◽  
Kinematic Diagram

Planetary speed reducers are used in the actuation of the revolute kinematic joints thanks to their specific advantages. However, in order to achieve a good positioning accuracy for the kinematic chains, these must have reduced backlash and high torsional stiffness. Therefore the deformations of the elements composing the mechanical structure under the action of gearing forces should be as low as possible, thus leading to backlash reduction. This also becomes important since the presence of backlash in the structure of the planetary speed reducer affects the stability of the overall kinematic chain. In this paper, a novel computation method for bearing deflections is proposed, starting from the kinematic diagram of the planetary speed reducer. Having known the kinematic diagram, it becomes possible to compute the gearing forces, keeping in mind the non-uniform power distribution on the planetary gears. The bearing reactions were computed in two cases: when the planetary gears are free to spin on the planetary carrier and when the planetary gears are fixed on the planet carrier, being supported at the extremities. Based on the computed values for bearing reactions, the corresponding deflections are determined. Once the values of the bearing reactions and deflections had been determined, the load-deflection and stiffness-deflection diagrams were elaborated for ball and roller bearings, allowing the determination of the bearing type that ensures low deflections and high stiffness. Based on the results, some recommendations are made, allowing the reduction of bearing deflections, and increasing the torsional stiffness.

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