3-RPR Manipulators Research Articles

Singularity distance computations for 3-RPR manipulators using intrinsic metrics

Avoiding singularities is a crucial task in robotics and path planning. This paper proposes a novel algorithm for detecting the closest singularity to a given pose for nine interpretations of the 3-RPR manipulator. The algorithm utilizes intrinsic metrics based on the framework's total elastic strain energy density, employing the physical concept of Green-Lagrange strain. The constrained optimization problem for detecting the closest singular configuration with respect to these metrics is solved globally using tools from numerical algebraic geometry implemented in the software package Bertini. The effectiveness of the proposed algorithm is demonstrated on a 3-RPR manipulator executing a one-parametric motion. Additionally, the obtained intrinsic singularity distances are compared with extrinsic metrics. Finally, the paper illustrates the advantage of employing a well-defined metric for identifying the closest singularities in comparison with the existing methods in the literature, highlighting its application in design optimization.

Singularities of a planar 3-RPR parallel manipulator with joint clearance

SUMMARYIf the joint clearances of the joints of a manipulator are considered, an unconstrained motion of the end-effector can be computed. This is true for all poses of the manipulator, even with all actuators locked.This paper presents how this unconstrained motion can be determined for a planar 3-RPR manipulator. The singularities are then studied. It is shown that when clearances are considered, the singularity curves normally found in the workspace of such a manipulator become singular zones. These zones can be significant and greatly reduce the usable workspace of a manipulator. Since a prescribed configuration that would not, in theory, corresponds to a singular pose can become singular due to the unconstrained motion, the results of this paper are relevant to manipulator design and trajectory planning.

Dynamic characteristics of a 3-RPR planar parallel manipulator with flexible intermediate links

SUMMARYThis paper presents the dynamic modeling of a 3-RPR planar parallel manipulator with three flexible intermediate links in order to investigate the effects of the intermediate links flexibility on the undesired vibrations of the end-effector. For this purpose, the intermediate links are modeled as Euler--Bernoulli beams with two types of fixed-pinned and fixed-free boundary conditions based on the assumed mode method (AMM). The equations of motion of the 3-RPR manipulator are formulated using the augmented Lagrange multipliers method in the form of differential algebraic equations (DAEs) by incorporating the elastic and rigid coordinates in the set of generalized coordinates. After defining the initial conditions and imposing external forces to the manipulator, the equations are then solved numerically using the Modified Extended Backward-Differentiation Formula Implicit (MEBDFI) approach. Comparison of the simulation results for two different boundary conditions shows clearly the effects of flexibility of the intermediate links on the vibration of the end-effector trajectory. Results of this work can be used for the dynamic modeling of other manipulators or to design a controller for reducing the undesired vibrations.

A unified approach to the accuracy analysis of planar parallel manipulators both with input uncertainties and joint clearance

This paper presents a unified approach to predict the accuracy performance of the general planar parallel manipulators (PPMs) both due to the input uncertainties and the joint clearance. Based on the theory of envelope, a geometric method is employed to uniformly construct the indeterminate influences of these two error sources on the pose (position and orientation) deviation of the manipulators. According to the generalized kinematic mapping of constrained plane motions, the end-effector's exact output error bound for a specified configuration can then be obtained as an accurate and complete description for the manipulator's accuracy performance, from which not only the maximal position and orientation errors but also their coupling relationship can be derived. The planar 3-RPR manipulator is intensively studied as a numerical example and several numerical simulations are provided to demonstrate the correctness and effectiveness the proposed approach.

Cusp Points in the Parameter Space of Degenerate 3-RPR Planar Parallel Manipulators

This paper investigates the conditions in the design parameter space for the existence and distribution of the cusp locus for planar parallel manipulators. Cusp points make possible nonsingular assembly-mode changing motion, which increases the maximum singularity-free workspace. An accurate algorithm for the determination is proposed amending some imprecisions done by previous existing algorithms. This is combined with methods of cylindric algebraic decomposition, Gröbner bases, and discriminant varieties in order to partition the parameter space into cells with constant number of cusp points. These algorithms will allow us to classify a family of degenerate 3-RPR manipulators.

Force optimization of kinematically-redundant planar parallel manipulators following a desired trajectory

In this work, an optimization-based methodology for resolving the generalized forces for kinematically-redundant planar parallel manipulators following a desired trajectory is presented. The proposed methodology assumes that the manipulator is performing a task that is slow enough to allow kinetostatic analysis to be used. Two test trajectories were used to show the effectiveness of the proposed methodology. The results for a kinematically-redundant 3-PRPR manipulator were compared against the results for a non-redundant 3-RPR manipulator. The results show that the redundant manipulator has improved force capabilities compared to the non-redundant manipulator. In particular, the redundant manipulator is able to pass through singular configurations with feasible generalized forces, something the non-redundant manipulator cannot do.

A Simple Proof That Generic 3-RPR Manipulators Have Two Aspects

Avoiding singularities in the workspace of a parallel robot is an important issue. The case of 3-RPR planar robots is an important subject of theoretical studies. We study the singularities of planar 3-RPR robots by using a new parameterization of the singular locus in a modified workspace. This approach enables us to give a simple alternative proof of a result recently proved by Husty: the complement of the singular locus in the workspace of a generic 3-RPR manipulator has two connected components (called aspects); we also give a procedure to design a singularity-free path connecting any two points in the same aspect. The parameterization introduced in this paper, due to its simple geometric properties, proves to be useful for the study of the singularities of 3-RPR robots.