Singularity analysis of a planar parallel mechanism with revolute joints based on a geometric approach (original) (raw)
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A novel criterion for singularity analysis of parallel mechanisms
Mechanism and Machine Theory, 2019
A novel criterion for singularity analysis of parallel robots is presented. It relies on screw theory, the 3-dimensional Kennedy theorem, and the singular properties of minimal parallel robots. A parallel robot is minimal if in any generic configuration, activating any leg/limb causes a motion in all its joints and links. For any link of the robot, a pair of legs is removed. In the resulting 2 degrees-of-freedom mechanism, all possible instantaneous screw axes belong to a cylindroid. A center axis of this cylindroid is determined. This algorithm is performed for three different pairs of legs. The position is singular, if the instantaneous screw axis of the chosen link crosses and is perpendicular to three center axes of the cylindroids. This criterion is applied to a 6/6 Stewart Platform and validated on a 3/6 Stewart Platform using results known in the literature. It is also applied to two-platform minimal parallel robots and verified through the Jacobian; hence demonstrating its general applicability to minimal robots. Since any parallel robot is decomposable into minimal robots, the criterion applies to all constrained parallel mechanisms.
Kinematics and singularity analyses of a 4-dof parallel manipulator using screw theory
Mechanism and Machine Theory, 2006
In this work, the kinematics and singularity analyses of a four degrees of freedom parallel manipulator are investigated using the theory of screws. As an initial step, the forward position analysis is carried out and forward position equations are obtained in a closed form, thanks to the simplicity of the architecture of the proposed mechanism. Afterwards, simple and compact expressions are derived for the velocity and reduced acceleration states of the moving platform w.r.t. the fixed coordinate frame of the manipulator, both in screw form, through each limb, and as six-dimensional vectors. The Klein form, a symmetrical bilinear form of the Lie algebra e(3), plays a central role in the present work. A numerical example is provided to demonstrate the efficacy of screw theory in efficiently analysing the kinematics and singularity of the parallel manipulator. The numerical results from the kinematic analysis are verified with results produced with a commercially available dynamic and kinematic simulation program.
Singularity Analysis of Closed-loop Mechanisms and Parallel Manipulators
Singularity analysis is the study of gain or loss in degrees of freedom of a mechanism at a particular configuration. Singularity analysis is related to the degeneracy of Jacobian matrices relating configuration variables with task space variables. In this paper, we present an improved Jacobian formulation which can be used to identify not only the gain or loss of degrees of freedom, but, in addition, can be used to determine if the gain results in redundant degree of freedom. The approach and its advantages are illustrated with several examples.
IEEE Access
In this contribution, the Jacobian analysis of a four-legged six-degrees-of-freedom decoupled parallel manipulator is carried out through the screw theory. As an intermediate step, for the sake of completeness the inverse/forward displacement analysis as well as the computation of the workspace of the robot are achieved by taking advantage of the decoupled orientation and position of the moving platform. Afterward, the input/output equation of velocity of the parallel robot is obtained by harnessing of the properties of reciprocal screw systems. Once the Jacobian matrices are identified and investigated, the analysis of singularities for the robot manipulator emerges as a natural application of the Jacobian analysis. Numerical examples are included with the purpose to show the practicality and versatility of the method of kinematic analysis. Furthermore, the numerical results obtained by means of the theory of screws are successfully verified with the aid of commercially available software like ADAMS.
Singularity analysis of planar parallel manipulators
Mechanism and Machine Theory, 1995
With regard to planar parallel-manipulators, a general classification of singularities into three groups is introduced. The classification scheme relies on the properties of the Jacobian matrices of the manipulator at hand. The Jacobian matrices of two classes of planar parallel manipulators are derived and the three types of singularities are identified for them. The first class contains 20 manipulators constructed with three different combinations of legs of the PRR, PPR, RRR and RPR types, P and R representing prismatic and revolute pairs, respectively. The second class consists of 4 manipulators constructed with three legs of the PRP and RRP types. Finally, one example for each class is included. Contrary to earlier claims, we show that the third type of singularity is not necessarily architecturedependent.
This paper introduces a new metamorphic parallel mechanism consisting of four reconfigurable rTPS limbs. Based on the reconfigurability of the reconfigurable Hooke (rT) joint, the rTPS limb has two phases while in one phase the limb has no constraint to the platform, in the other it constrains the spherical joint center to lie on a plane. This results in the mechanism to have ability of reconfiguration between different topologies with variable mobility. Geometric constraint equations of the platform rotation matrix and translation vector are set up based on the point-plane constraint, which reveals the bifurcated motion property in the topology with mobility 2 and the geometric condition with mobility change in altering to other mechanism topologies. Following this, a unified kinematics limb modeling is proposed considering the difference between the two phases of the reconfigurable rTPS limb. This is further applied for the mechanism modeling and both the inverse and forward kinematics is analytically solved by combining phases of the four limbs covering all the mechanism topologies. Based on these, a unified singularity modeling is proposed by defining the geometric constraint forces and actuation forces in the Jacobian matrix with their change in the variable topologies in terms of constraint screws. Analysis of workspace with singularity distribution is carried out using this model and corresponding singularity loci are obtained with special singular configurations illustrated.
Singularity Analysis of a 3-PRRR Kinematically Redundant Planar Parallel Manipulator
Journal of Mechanical Engineering, 2010
Finding Singular configurations (singularities) is one of the mandatory steps during the design and control of mechanisms. Because, in these configurations, the instantaneous kinematics is locally undetermined that causes serious problems both to static behavior and to motion control of the mechanism. This paper addresses the problem of determining singularities of a 3-PRRR kinematically redundant planar parallel manipulator by use of an analytic technique. The technique leads to an input -output relationship that can be used to find all types of singularities occurring in this type of manipulators.
Jacobian Analysis of Limited DOF Parallel Manipulator using Wrench and Reciprocal Screw Principle
This paper presents a new methodology of formulating Jacobian matrix for limited degree of freedom (DOF) parallel kinematic machine (PKM), which is a very important tool to relate the end-effector velocity with the joint rate velocity. Even if it is believed by many researchers that Jacobian matrix is critical to generating the trajectories of the prescribed geometry in the end-effector space, it was cumbersome to formulate in simple and descriptive form by partial differentiation. In this work screw mathematics is used to formulate the Jacobian matrix in simple and integrated form under a unified framework and it is proved that the resulted Jacobian matrix is 6 6 which provides clear information about the architecture and singularity condition of the manipulator. Obtaining Jacobian matrix in unambiguous way is very crucial step to formulate and solve velocity, acceleration and motor torque equations with less computational burden. The 3PRS parallel kinematic machine is selected as an example to demonstrate the methodology. The numerical solution is obtained using MATLAB. IJERTV3IS040580 c a J J J , c a J J J .The generalized Jacobian
Kinematic Analysis of Planar Parallel Mechanism
The paper deals with kinematic model of parallel mechanism equipped with elastic members. The main aim is to determine the workspace of particular point of the mechanism in order to designing of whole mechanism for the future. Whole mechanisms should consist of several same segments which creates concept of serial mechanism. Since there is investigated parallel mechanism, at first has to be solved inverse kinematic model in order to obtain generalized variables, which are necessary for direct kinematic model. In the paper Jacobian matrix is derived for investigated mechanism. Then the algorithm for inverse kinematic model is described. The inverse kinematic model uses damped least squares method which appears as suitable method for our purposes. The results of inverse kinematic solution is consequently used for direct kinematic model, which is described by homogeneous transformation matrices. The planar parallel mechanism is simulated in software Matlab and the results are expressed in the graphs as well as our approach is discussed in the conclusion.