Unified Kinematic Analysis of General Planar Parallel Manipulators (original) (raw)
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Direct kinematics of planar parallel manipulators
1996
Abstract We address the problem of finding all the solutions of the direct kinematics for every possible architectures of planar fully parallel manipulators. We show that for this problem all the possible kinematic chains can be reduced to a set of three basic chains and we explain how to calculate the solutions of the forward kinematics for all the combinations of these basic chains and consequently for all the possible architectures of planar parallel robots
The direct kinematics of planar parallel manipulators: Special architectures and number of solutions
Mechanism and Machine Theory, 1994
We address the problem of finding all the solutions of the direct kinematics for every possible architectures of planar fully parallel manipulators. We show that for this problem all the possible kinematic chains can be reduced to a set of three basic chains and we explain how to calculate the solutions of the forward kinematics for all the combinations of these basic chains and consequently for all the possible architectures of planar parallel robots.
2005
Based on a proven exact method to solve the forward kinematics problem , this article is the second one investigating the forward kinematics problem formulation specifically applied to planar parallel manipulators. This part focuses on the displacement based equation systems. The majority of planar tripods can modeled by the 3-RPR parallel manipulator which is a tripod constituted by a fixed base and a triangular mobile platform attached to three kinematics chains with linear (prismatic) actuators located between two revolute joints. In order to implement the algebraic method, the parallel manipulator kinematics shall be formulated as polynomial equation systems where the number of equations are equal or exceeding the unknown numbers. Three geometric formulation were derived to model the difficult forward kinematics problem . The selected algebraic proven method is implementing Gröbner bases from which it constructs an equivalent univariate system. Then, the real roots isolation is performed using this last system. Each real solution exactly corresponds to one manipulator assembly mode. The forward kinematics problem resolution of the planar 3-RPR parallel manipulator outputs 6 complex solutions which become a proven real solution number upper bound. In several typical examples, the resolution performances (computation times and memory usage) are compared. It was then possible to compare the modelings and to reject one. Moreover, the number of real solutions was obtained and the corresponding postures drawn. The algebraic method is exact and produces certified results.
Advanced Robotics, 2006
Based on a proven exact method which solves the forward kinematics problem (FKP) this article investigates the FKP formulation specifically applied to planar parallel manipulators. It focuses on the displacement-based equation systems. The majority of planar tripods can modeled by the 3-RPR parallel manipulator, which is a tripod constituted by a fixed base and a triangular mobile platform attached to three kinematics chains with linear (prismatic) actuators located between two revolute joints. In order to implement the algebraic method, the parallel manipulator kinematics are formulated as polynomial equation systems where the number of equations is equal to or exceeds the number of unknowns. Three geometrical formulations are derived to model the difficult FKP. The selected proven algebraic method uses Gröbner bases from which it constructs an equivalent univariate system. Then, the real roots are isolated using this last system. Each real solution exactly corresponds to one manipulator assembly mode, which is also called a manipulator posture. The FKP resolution of the planar 3-RPR parallel manipulator outputs six complex solutions which become a proven real solution number upper bound. In several typical examples, the resolution performances (computation times and memory usage) are given. It is then possible to compare the models and to reject one. Moreover, a number of real solutions are obtained and the corresponding postures drawn. The algebraic method is exact and produces certified results.
A General Geometric Index for Solving the Forward Kinematics of Planar Parallel Manipulators
2021 9th RSI International Conference on Robotics and Mechatronics (ICRoM), 2021
In this paper, a new geometrical method is proposed to solve the forward kinematics of planar parallel manipulators exhibiting two translational and one rotational DOF which leads to high-order system polynomial expressions. The main idea has arisen from a geometrical interpretation of the intersection among the vertex space of each kinematic chain. From the type synthesis performed for planar parallel manipulators with identical kinematic structure, it has been revealed that the number of every possible configuration for this type of manipulators is eighteen, which the solution of the forward kinematic problem can be expressed using a univariate polynomial expression. However, among these eighteen configuration only six of them lead to high order polynomials and therefore do not have closedform solutions. The proposed method in this paper is based on an index representing the intersection of the circles which are extracted from the vertex space of each kinematic limb of the manipulator. In this approach, the orientation angle of the moving platform is selected from an interval; and based on the selected orientation angle, the circles obtained from the vertex space of each limb are determined. The forward kinematic solution happens where all the circles intersect at one common point where this geometrical phenomenon is expressed using the proposed index. In order to illustrate the performance of the proposed approach, three examples are solved. It should be noted that the foregoing method succeeded even in solving examples such as manipulators with six real solutions, manipulators with degenerate answers, which are reported in the literature as special cases.
2005
Based on a proven exact method to solve the forward kinematics problem , this article investigates the forward kinematics problem formulation specifically applied to planar parallel manipulators. This first part focuses on the position-based equation systems. The majority of planar tripods can modeled by the 3-RPR parallel manipulator. This manipulator is a tripod constituted by a fixed base and a triangular mobile platform attached to three kinematics chains with linear (prismatic) actuators located between two revolute joints. In order to implement algebraic methods, the parallel manipulator kinematics shall be formulated as polynomial equations systems where the number of equations are equal to the unknown numbers. Two kinematics position-based formulation were derived to model the difficult forward kinematics problem . The selected algebraic proven method is implementing Gröbner bases from which it constructs an equivalent univariate system. Then, the real roots isolation is performed using this last system. Each real solution exactly corresponds to one manipulator assembly mode. The forward kinematics problem resolution of the planar 3-RPR parallel manipulator outputs 6 complex solutions which become a proven real solution number upper bound. In several typical examples, the resolution performances (computation times and memory usage) are compared and the number of real solutions were obtained and the corresponding postures drawn. The algebraic method is exact and produces certified results.
On the kinematic constraint surfaces of general three-legged planar robot platforms
Mechanism and Machine Theory, 2003
The variants of general three-legged planar robot platforms are enumerated and classified. Constraint surfaces corresponding to individual platform legs in the kinematic mapping image space are classified and parametrized. The parametric equations are free from representational singularities. The entire set consists of hyperboloids of one sheet and hyperbolic paraboloids. This result corrects an oversight in the body of literature. These surfaces have important applications to the kinematic analysis of planar three-legged robot platforms, hence appropriate attention should be given to their classification.
Kinematic Analysis of Parallel Manipulators
Parallel Kinematic Machines, 1999
In the world of robotics, the parallel kinematic manipulators have proved their unique potentials like rigid structural capacities, better dynamic behaviours, and excellent spatial positional accuracy for many applications. One of the prime focus areas of the paper is to test and to establish a simplified novel approach for the Inverse Kinematics problem solution for a 3-PSU Parallel Kinematic Manipulator platform. Additionally, the methodology for simulation of Inverse Kinematics has been checked and verified. The obtained solution is further used for positional simulation of virtual prototype of 3-PSU PKM using Pro/Mechanism module. Further, these derived data is used for simulation of the Rigid Dynamics Solution for PKM (considering inertial forces acting on actuator) is also simulated in case of circular contouring.
A planar quaternion approach to the kinematic synthesis of a parallel manipulator
Robotica, 1997
In this paper we present a technique for designing planar parallel manipulators with platforms capable of reaching any number of desired poses. The manipulator consists of a platform connected to ground by RPR chains. The set of positions and orientations available to the end-effector of a general RPR chain is mapped into the space of planar quaternions to obtain a quadratic manifold. The coefficients of this constraint manifold are functions of the locations of the base and platform R joints and the distance between them. Evaluating the constraint manifold at each desired pose and defining the limits on the extension of the P joint yields a set of equations. Solutions of these equations determine chains that contain the desired poses as part of their workspaces. Parallel manipulators that can reach the prescribed workspace are assembled from these chains. An example shows the determination of three RPR chains that form a manipulator able to reach a prescribed workspace.