Kinematic analysis of the 3-CUP parallel mechanism

doi:10.1016/j.rcim.2013.03.002

Robotics and Computer-Integrated Manufacturing

Volume 29, Issue 5, October 2013, Pages 382-395

https://doi.org/10.1016/j.rcim.2013.03.002 Get rights and content

Highlights

•
We present closed form solutions to the position and velocity analysis of the 3-CUP parallel mechanism.
•
Numerical examples and simulations validate the analytical results.
•
The parasitic motion workspace is presented.
•
The singular configurations of the mechanism are analyzed using screw theory.

Abstract

This paper investigates the kinematics of a parallel mechanism that is composed of three identical CUP legs evenly distributed on the fixed base. The platform of the mechanism has three degrees-of-freedom, namely: two rotations and one translation along the axis perpendicular to the base. The paper obtains closed form solutions for the inverse and forward kinematics problems. Furthermore, the Jacobian matrix is determined in order to solve the instantaneous kinematics analysis. It is used for the identification of the singular configurations of the mechanism, which are investigated by applying screw theory. The parasitic motions of the platform are determined by means of a workspace analysis. This paper uses several simulations and numerical examples to prove the accuracy of the analytical results.

Introduction

A parallel mechanism (PM) can be defined as a multi degree-of-freedom (henceforth denoted as DOF) mechanism composed of a moving platform and a base connected by at least two serial kinematic chains in parallel [1]. One of the first applications of this type of mechanisms is believed to be the tire testing machine introduced by Gough and Whitewall [2], followed by the motion simulation platform built by Stewart [3]. Another example of this class of mechanisms is the Delta Robot [4], one of the most commercially successful parallel manipulators, which is commonly used in pick-and-place applications. It is well known in the related literature that parallel mechanisms possess several advantages with respect to their counterpart, i.e. serial manipulators, namely: (i) high stiffness, (ii) overload capacity, and (iii) high acceleration. However, PMs also present some drawbacks, such as: (i) limited workspace, and (ii) kinematical complexity.

Parallel mechanisms can be used in a wide range of applications, such as flight simulators [3], medicine [5], pick-and-place [4], haptics [6], among others [2], [4]. The variety of applications in which PMs can be used, their advantages in comparison with the serial counterpart, along with their kinematical complexity have motivated the research community for the past fifty years to investigate this type of mechanisms. However in order to be more specific, it is important to mention the classification of PMs according to their mobility (number of DOF). A full mobility PM is denoted as a mechanism with 6 DOF. On the other hand, a lower mobility PM has less than 6 DOF, which is the focus of the current trend in the research community. Lower mobility PMs are typically used in more specific tasks instead of general purpose tasks; however there is a certain loss of flexibility due to the reduction in DOF of the manipulator.

The 3-CUP (where C, U, and P denote cylindrical, universal, and prismatic joint, respectively) PM analyzed in this paper consists of a moving platform and a fixed base connected by three identical legs that are evenly distributed on the fixed base. This robotic architecture was proposed by Rodriguez-Leal and Dai [7] based on Artiomimetics. For convenience of the reader, a CAD model of the 3-CUP manipulator is shown in Fig. 1. The mobility of the 3-CUP PM has been studied by Rodriguez-Leal et al. [8] where it was determined that the moving platform has 3 DOF: i.e. one translation axis perpendicular to the fixed base and two rotations about two skew axes. Strictly speaking, the 3-CUP PM is a lower mobility (i.e. less than 6 DOF), 2R1T (namely, 2 rotations and 1 translation) parallel mechanism. The study of kinematics is a very crucial aspect approached by researchers in order to define possible applications of PM [9], [10], which in most cases result in the solution of nonlinear equation systems [11], [12], [13]. This paper investigates four key aspects of the 3-CUP PM: the (i) inverse and forward kinematics position problem, (ii) inverse and forward instantaneous kinematics, (iii) singularity analysis, and (iv) workspace analysis. The aforementioned analyses provide an essential insight into the characteristic of the mechanism and some important features such as the position limits and singular configurations that prevent malfunctioning or breakdown of the manipulator.

The paper is arranged into eight sections. Firstly, Section 2 introduces the notation used in this work, and describes the topology of the PM. The inverse kinematics position problem of the mechanism is presented in Section 3, where its solution is obtained by solving the closed-loop equation for the unknown variables. In Section 4 a geometric solution for the forward kinematics problem is proposed. The instantaneous kinematics analysis is included in Section 5, and it is obtained with the differentiation of the closed loop equation resulting in a simplified Jacobian matrix. The identification of singular configurations is performed using two approaches (i.e. analyzing the determinant of the Jacobian matrix and using screw theory) and is contained in Section 6. Furthermore, a 3D shape of the workspace is obtained using computer code programming in Section 7. Finally, Section 8 presents some conclusions and future research on this 3-CUP PM.

Section snippets

Notation and manipulator description

For convenience of the reader, scalar quantities are represented in this paper by lower case lettering in italic type (e.g. a), vectors by lower case lettering in bold italic type (e.g. d_i), and matrices by upper case lettering in bold upright type (e.g. R). Cartesian coordinate reference frames are symbolized by upper case lettering (e.g. G). Furthermore, upper case lettering in italic type is used for orthogonal axes, centroids, and points fixed in links. The subscript ij denotes the joint j

Inverse kinematic analysis

The inverse kinematics problem consists in determining the actuators strokes (i.e. d_i) in order to configure the end-effector in a desired position and orientation.

Denoting G (X, Y, Z) and H (U, V, W) as the global and platform reference frames that are located on the base and moving platform centroids, respectively (see Fig. 1), consider the loop closure equation for leg i defined in the G frame as (see Fig. 2) $p + {}^{H}b_{i} = a_{i} + d_{i}$ where $p = [\begin{matrix} p_{x} \\ p_{y} \\ p_{z} \end{matrix}], b_{i} = {}^{G}R_{H} \cdot R (W, χ_{i}) \cdot {[b_{i}, 0, 0]}^{T}, a = [\begin{matrix} a C ψ_{i} \\ a S ψ_{i} \\ 0 \end{matrix}]$ and ${}^{G}R_{H} = [\begin{matrix} C γ C β & C γ S β S α - S \end{matrix}$

Forward kinematic analysis

The forward kinematics problem, also referred as direct kinematics problem, consists on determining the position and orientation of the moving platform knowing the magnitude of the input strokes. In this particular case, the strokes d_i are known and p_x, p_y, p_z, α, β and γ are to be determined. This work approaches the forward kinematics problem by analyzing the orientation and the position of the platform independently.

Since the d_i strokes magnitudes are given, therefore the three points in

Instantaneous kinematics analysis

The instantaneous kinematics analysis, which is also known in literature as velocity analysis, is approached in this paper by differentiating the closed-loop Eq. (2), namely: $[\begin{matrix} {\dot{p}}_{x} \\ {\dot{p}}_{y} \\ {\dot{p}}_{z} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ {\dot{d}}_{i} \end{matrix}] - \frac{d}{d t} (b_{i} [\begin{matrix} u_{x} C χ_{i} + v_{x} S χ_{i} \\ u_{y} C χ_{i} + v_{y} S χ_{i} \\ u_{z} C χ_{i} + v_{z} S χ_{i} \end{matrix}])$ where ${[\begin{matrix} {\dot{p}}_{x}, & {\dot{p}}_{y}, & {\dot{p}}_{z} \end{matrix}]}^{T} = {[\begin{matrix} ν_{x}, & ν_{y}, & ν_{z} \end{matrix}]}^{T}$

Substituting χ₁=0°, χ₂=120°, and χ₃=240° and Eq. (15b) into Eq. (15a) results in the following nine equations: $ν_{x} = b_{1} (C γ S β) ω_{y} + b_{1} (C β S γ) ω_{z} - (C β S γ) {\dot{b}}_{1} ν_{y} = b_{1} (S γ S β) ω_{y} - b_{1} (C β C γ) ω_{z} - (C β S γ) {\dot{b}}_{1} ν_{z} = b_{1} (C β) ω_{y} + S β {\dot{b}}_{1} + {\dot{d}}_{1} 2 ν_{x} + b_{2} ((\sqrt{3} C α C γ S β + S α S γ) ω_{x} + C γ (\sqrt{3} C β S α + S$

Singularity analysis

This work follows two methods in order to perform a comprehensive analysis of the singular configurations of the 3-CUP parallel mechanism. First, the determinant of the Jacobian matrix is analyzed in order to identify the singular configurations of the manipulator [16]. Furthermore, the motion and constraint screw sets are investigated in the resulting singular configurations in order to determine the instantaneous mobility of the mechanism [17]. The determinant of the Jacobian matrix from Eq.

Workspace analysis

The approach followed in this paper for having an insight into the possible orientations reached by the manipulator, considers the unit vector that is normal to the platform from Eq. (9). The components of this unit vector change according to the platform orientation and is calculated by using all the possible combinations of strokes magnitudes. The plot of the orientation workspace is shown in Fig. 9, which using the dimensional parameters a=433.013 mm, d_min=655 mm and d_max=1155 mm; result in the

Conclusions

This paper presented the kinematics of the 3-CUP parallel mechanism. The inverse kinematics problem was solved by investigating the closed-loop equation. Unique closed-form solutions for the active strokes, passive strokes and parasitic motions were obtained. The forward kinematics problem was solved by first investigating the orientation of the platform, since the α and β angles were known, the positioning of the platform was determined using the closed-form solutions obtained in the inverse

Acknowledgments

This work has been conducted under the financial support of CONACYT during the MSc study of the first author at Tecnologico de Monterrey, Mexico.

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Kinematic analysis of the 3-CUP parallel mechanism

Highlights

Abstract

Introduction

Section snippets

Notation and manipulator description

Inverse kinematic analysis

Forward kinematic analysis

Instantaneous kinematics analysis

Singularity analysis

Workspace analysis

Conclusions

Acknowledgments

Mechanism and Machine Theory

Mechanism and Machine Theory

Parallel robots

A platform with six degrees of freedom

Proceedings of the Institute of Mechanical Engineers

Surgical bedside master console for neurosurgical robotic system

International Journal for Computer Assisted Radiology and Surgery

Shade, a new 3-DOF haptic device

IEEE Transactions on Robotics and Automation