Journal of Robotics

Volume 2015, Article ID 790414, 9 pages

http://dx.doi.org/10.1155/2015/790414

## Kinematic Analysis of a Partially Decoupled 3-DOF Parallel Wrist

^{1}Shanghai University of Engineering Science, Shanghai 24060, China^{2}Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24060, USA

Received 15 July 2015; Revised 7 October 2015; Accepted 22 October 2015

Academic Editor: Shahram Payandeh

Copyright © 2015 Fan Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A unique spherical parallel wrist with three partially decoupled rotational degrees of freedom (DOFs) is introduced in this paper. The mechanism has the significant advantages of few singularities and simple partially decoupled kinematics. A modified parallel wrist is optimized to have the least link interference workspace. Finally, the decoupled motion is studied in detail to exhibit the kinematic performance of the mechanism.

#### 1. Introduction

Recent research focuses on the parallel wrist manipulators with two or three motorized axes [1–8]. The parallel wrist manipulators could be an effective tool for wrist, shoulder, and ankle instrument in medical robots, tracking mechanism, and advanced manufacturing mechanisms. To achieve high performance, including low inertia and high accuracy, remains challenging in the above mentioned fields, which gives rise to the study of the structural synthesis and optimization for the 3-DOF parallel wrist manipulators.

The coupled parallel wrist manipulators refer to the mechanisms whose input-output kinematics are nonlinear [9]. Existing researches revealed the problems about coupled parallel wrists, such as the singularities and the workspace problems. “Agile eye” is one of the most well-known parallel wrists, whose workspace is flawed with six singularity curves which correspond to self-motions of the moving-platform [10]. The Omni-Wrist III mechanism has a singularity-free workspace; however, the workspace is bounded by a spatial curved surface due to its variable center-of-rotation characteristic [11]. A family of 2-DOF coupled rotational parallel manipulators with an equal-diameter spherical pure rotation (ESPR) are proposed, to improve the workspace problem of Omni-Wrist [2]. Other coupled parallel wrists like the 3-RRUR [7], the 2-DOF 5R spherical parallel manipulator [5], the 3-UPU pure rotational parallel mechanism [6], the 2-DOF solar tracking mechanism [1], and the planar-spherical overconstrained mechanisms [12] are also reported to have the singularities or workspace problems.

The parallel wrists with diagonal matrix and triangular matrix are known as the partially decoupled parallel wrist (or the uncoupled parallel wrist) and the decoupled parallel wrist (or the fully decoupled parallel wrist), respectively. These mechanisms are reported to have fewer singularities and simple kinematics [13–15]. Carricato and Parenti-Castelli proposed a decoupled 2-DOF parallel wrist by optimizing the topology of mechanisms [14]. Hervé synthesized a family of 2-DOF parallel wrists which can achieve uncoupled pan-tilt motion by group theory method [15]. Zeng and Huang established the type synthesis method of the rotational decoupled parallel mechanism and proposed a novel 2-DOF decoupled parallel rotational wrist [16].

Though 2-DOF partially decoupled and decoupled parallel wrists have been well-studied, the methodology for kinematic analysis of the 3-DOF partially decoupled and decoupled parallel wrist still requires further investigations. Lubin et al. formulated the topology condition for the synthesis of the partially decoupled spherical parallel mechanism and obtained a novel 3-DOF partially decoupled spherical parallel mechanism [13]. Gogu presented a family of decoupled and partially decoupled 3-DOF parallel wrist [9]. Kuo and Dai presented a fully decoupled remote center-of-motion parallel manipulator which can achieve 3-DOF spherical motion and a translational motion [17]; later they also proposed a variant of the Agile Eye with fully decoupled structure [18]. The methodology for the kinemaic analysis of partially decoupled wrists still needs further research to formulate a general method for 3-DOF spherical parallel wrists.

This paper proposed a unique partially decoupled 3-DOF parallel wrist. The fundamental of screw theory is briefly introduced in Section 2. The geometry of the parallel wrist is introduced in Section 3. And the link analysis is presented by using the reciprocal screw theory in Section 4. The singularity analysis is presented in Section 5. The kinematics analysis is presented in Section 6. And the workspace analysis and decoupled motion study are discussed in Sections 7 and 8. Finally, discussion and conclusion are addressed in Section 9.

#### 2. The Fundamental of Screw Theory

##### 2.1. The Reciprocal Screw Theory

A screw is six-dimensional in a homogeneous coordinate, which can completely describe the direction and position of vector in three-dimensional space. The reciprocal product of two screws and can be represented as [19]

The reciprocal product of two screws is equal to the instantaneous work of the wrench to the motion of the body:where the twist denotes the instantaneous motion of the rigid body. The wrench denotes the wrench imposed on the rigid body.

If the reciprocal product is zero, the wrench denotes the constraint of the mechanical system to the instantaneous motion of rigid body. The geometrical conditions of the constraint and the instantaneous motion are as follows:(i)If the constraint is a force, the force is perpendicular to the translational motion and is coplanar (intersecting, parallel, or coaxial) with the revolute motion.(ii)If the constraint is a torque, the torque is perpendicular to the rotational motion.

##### 2.2. The Constraint and Actuation Wrench of the Limb

The serial kinematic limb may be considered as a serial chain of number of 1-DOF joints. The instantaneous motion of the moving-platform, , can be expressed as a linear combination of twists [20],where denotes the intensity and represents a unit screw associated with the th joint of the th kinematic limb. denotes the number of limbs of the parallel mechanism.

The constraints of the th kinematic limb, , which are reciprocal to number of twists of th limb [21]forms a (6-) reciprocal screw system.

If we lock the actuated joint of th limb, the rank of the reciprocal screw system increases by 1. The additional reciprocal screws, denoted as , are reciprocal to all the passive joint screws of th limb and impose work on , the actuated joint of th limb (the th joint of th limb) [21],

Thus for a given mechanism, the constraint and actuated wrench could be calculated by (4) and (5).

##### 2.3. The Jacobian Matrix in Screw Formulation

The actuation matrix of the parallel mechanism can be formulated as [21]

The constraint matrix of the parallel mechanism can be formulated as

Each row of constraint Jacobian matrix represents a constraint imposed by limb. The rows of actuation Jacobian matrix represent the actuation acting by limbs.

#### 3. The Geometry of a Parallel Wrist

The general geometry of the partially decoupled 3-DOF parallel wrist of is shown in Figure 1 (Figure 1(b) is kinematically equivalent to Figure 1(a)). The mechanism consists of a base-platform, a moving-platform, and three kinematic limbs. The nomenclature for the kinematic joint is as follows: stands for revolute joint; stands for revolute joint with an axis across the origin ; stands for prismatic joint; and stands for universal joint.