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

Authors: Fan  Zhang; Yunping  Zhu; Tomonari  Furukawa; Wanqing  Song
Journal: Journal of Robotics
 (2015)

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## 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.


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## Body

## 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 Section2. 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$1=[s1,s10] and $2=[s2,s20] can be represented as [19](1)$1∘$2=s1·s20+s2·s10.The reciprocal product of two screws is equal to the instantaneous work of the wrench to the motion of the body:(2)$f∘$m=f1·v2+C1·ω2+a12f1·ω2,where the twist $m=[ω2;r2×ω2+v2] denotes the instantaneous motion of the rigid body. The wrench $f=[f1;r1×f1+C01] 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 ofCi number of 1-DOF joints. The instantaneous motion of the moving-platform, $p, can be expressed as a linear combination of Ci twists [20],(3)$p=∑i=1Cjθ˙i,j$i,jforj=1,2,…F,where θ˙i,j denotes the intensity and $i,j represents a unit screw associated with the ith joint of the jth kinematic limb. F denotes the number of limbs of the parallel mechanism.The constraints of thejth kinematic limb, $jr, which are reciprocal to Cj number of twists of jth limb [21](4)$jr∘$i,j=0j=1,2,…,Ciforms a (6-Cj) reciprocal screw system.If we lock the actuated joint ofjth limb, the rank of the reciprocal screw system increases by 1. The additional reciprocal screws, denoted as $ja, are reciprocal to all the passive joint screws of jth limb and impose work on $k,j, the actuated joint of jth limb (the kth joint of jth limb) [21],(5)$ja∘$i,j=0fori=1,2,…,k-1,k+1,…,Cj$ja∘$k,j≠0.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](6)Jc=$1rT$2rT⋮$FrT.The constraint matrix of the parallel mechanism can be formulated as(7)Ja=$1aT$1aT$k,1$2aT$2aT$k,2⋮$FaT$FaT$k,F.Each row of constraint Jacobian matrix represents a constraint imposed by limb. The rows of actuation Jacobian matrix represent the actuation acting by limbs.


## 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$1=[s1,s10] and $2=[s2,s20] can be represented as [19](1)$1∘$2=s1·s20+s2·s10.The reciprocal product of two screws is equal to the instantaneous work of the wrench to the motion of the body:(2)$f∘$m=f1·v2+C1·ω2+a12f1·ω2,where the twist $m=[ω2;r2×ω2+v2] denotes the instantaneous motion of the rigid body. The wrench $f=[f1;r1×f1+C01] 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 ofCi number of 1-DOF joints. The instantaneous motion of the moving-platform, $p, can be expressed as a linear combination of Ci twists [20],(3)$p=∑i=1Cjθ˙i,j$i,jforj=1,2,…F,where θ˙i,j denotes the intensity and $i,j represents a unit screw associated with the ith joint of the jth kinematic limb. F denotes the number of limbs of the parallel mechanism.The constraints of thejth kinematic limb, $jr, which are reciprocal to Cj number of twists of jth limb [21](4)$jr∘$i,j=0j=1,2,…,Ciforms a (6-Cj) reciprocal screw system.If we lock the actuated joint ofjth limb, the rank of the reciprocal screw system increases by 1. The additional reciprocal screws, denoted as $ja, are reciprocal to all the passive joint screws of jth limb and impose work on $k,j, the actuated joint of jth limb (the kth joint of jth limb) [21],(5)$ja∘$i,j=0fori=1,2,…,k-1,k+1,…,Cj$ja∘$k,j≠0.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](6)Jc=$1rT$2rT⋮$FrT.The constraint matrix of the parallel mechanism can be formulated as(7)Ja=$1aT$1aT$k,1$2aT$2aT$k,2⋮$FaT$FaT$k,F.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 Figure1 (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: R stands for revolute joint; RN stands for revolute joint with an axis across the origin O; P stands for prismatic joint; and U stands for universal joint.Figure 1
The schematic diagram and design of the partially decoupled 3-DOF parallel wrist.
(a)
The CAD model of partially decoupled parallel wrist
(b)
The schematic diagram of mechanismSince the universal joint is equvilent to two intersecting revolute joints, three kinematic limbs are also denoted as limbDEFO (R1NR2P3R4R12N), limb GHO (R5P6R7R8NR12N), and limb ABCO (R9NR10NR11N limb), as shown in Figure 1(b). And R1N⊥R2, R2⊥P3, R2//R4//R8N, R5⊥P6, R1//R5//R7, P3⊥P6, R9N⊥R10N, and R10N⊥R11N, while ⊥ denotes lines are perpendicular, and // denotes lines are parallel in geometry. The actuated joints of the parallel wrist are prismatic joints P6, P3 and the revolute joint R9. The moving-platform is attached at the link CO.


## 4. Limb Kinematics Analysis

A coordinate systemO-XYZ is attached to the base-platform. The point O is chosen as the origin point; the x-axis is along the link DE, the z-axis is along the link AO, and the y-axis is perpendicular to both x and z axes.Nomenclature used in this paper is as listed:(i)
[X]k denotes the posture of link X under the O-XYZ coordinate system, as a part of the k kinematic limb.(ii)
[ai]k denotes the axis of the revolute joint Ri under the O-XYZ coordinate system, as a part of the k kinematic limb.(iii)
θi denotes the rotated angle of the revolute joint Ri.(iv)
L3 and L6 denote the length of the actuated prismatic joints P3 and P6, respectively.And the geometrics of the mechanism are defined asOE=OG=a, OF=OH=b, L3=EF, and L6=GH. And(8)β=∠EOF=arccos⁡a2+b2-L322ab,α=∠GOH=arccos⁡a2+b2-L622ab,ψ=∠OEF=arccos⁡a2+L32-b22aL3,δ=∠OGH=arccos⁡a2+L62-b22aL6.R
o
t
(
a
i
,
θ
i
) denotes the rotation matrix about the direction vector ai by the angle θi and can be represented as [22]:(9)Rotai,θi=axaxversθi+cθiayaxversθi-azsθiazaxversθi+aysθiaxayversθi+azsθiayayversθi+cθiazayversθi-axsθiaxazversθi-aysθiayazversθi+axsθiazazversθi+cθi,where versθi=1-cos⁡θi, cθi=cos⁡θi, sθi=sin⁡θi, and ai={ax,ay,az}.By using formula (9), the axis of the jth kinematic joint of serial limb could be obtained,(10)aj=Rotaj-1,θi-1⋯Rota2,θ2Rota1,θ1.


### 4.1.DEFO Limb

The twist system of kinematic limbDEFO (R1NR2P3R4R12N) can be represented as (11)$1=1,0,0;0,0,0$2=0,cos⁡θ1,sin⁡θ1;0,-sin⁡θ1,cos⁡θ1$3=0,0,0;cos⁡ψ,-sin⁡θ1sin⁡ψ,cos⁡θ1sin⁡ψ$4=0,cos⁡θ1,sin⁡θ1;sin⁡β,-cos⁡βsin⁡θ1,cos⁡βcos⁡θ1$12=-cos⁡β,-sin⁡βsin⁡θ1,sin⁡βcos⁡θ1;0,0,0.The constraint wrench system which the kinematic limbDEFO enforces on the moving-platform CO can be calculated by reciprocal screw theory: (12)$DEFOr=0,cos⁡θ1,sin⁡θ1;0,0,0.The constraint wrench$DEFOr is the force which is across the origin and is parallel to the revolute joint R2. The constraint wrench $DEFOr is along with the axis of revolute joint R8N, since the revolute joint R8N is parallel to R2.Locking the actuation jointP3, the actuation wrench of limb DEFO can be obtained by (5) as(13)$DEFOa·$3≠0$DEFOa·$i=0i=1,2,4,12.Further,(14)$DEFOa·v3≠0$DEFOa·ωi=0i=1,2,4,12,where ωi denotes the angle velocity of the revolute joints Ri(i=1,2,4,12) and v3 denotes the velocity of the prismatic joint P3.The actuation wrench of limbDEFO, which is the force along with the axis of the prismatic joint P3, is obtained by (14) as(15)$DEFOa=cos⁡ψ,-sin⁡θ1sin⁡ψ,cos⁡θ1sin⁡ψ;0,0,0.


### 4.2.GHO Limb

The twist system of kinematic limbGHO can be represented as(16)$5=1,0,0;0,1,0$6=0,0,0;0,sin⁡δ,cos⁡δ$7=1,0,0;0,cos⁡α,sin⁡α$8=0,sin⁡α,-cos⁡α;0,0,0$12=sin⁡θ8,cos⁡αcos⁡θ8,sin⁡αcos⁡θ8;0,0,0.The constraint wrench system which the kinematic limbGHO enforces on the moving-platform CO can be calculated as(17)$GHOr=1,0,0;0,0,0.The constraint wrench$GHOr is the force along the x-axis.Locking the actuation jointP6, the actuation wrench of kinematic limb GHO can be obtained by(18)$DEFOa·v6≠0$DEFOa·ωi=0i=5,7,8,12.The actuation wrench of limbGHO, which is the force along with the axis of the prismatic joint P3, is obtained by (18) as(19)$GHOa=0,sin⁡δ,cos⁡δ;0,0,0.


### 4.3.ABCO Limb

The twist system of kinematic limbABCO can be represented as(20)$9=0,0,-1;0,0,0$10=a10,b10,c10;0,0,0$11=a11,b11,c11;0,0,0.The constraint wrench system which the kinematic limbABCO enforces on the moving-platform CO and can be calculated as(21)$1,ABCOr=1,0,0;0,0,0$2,ABCOr=0,1,0;0,0,0$3,ABCOr=0,0,1;0,0,0.The constraint wrench system consists of three nonlinear forces$1,ABCOr, $2,ABCOr, and $3,ABCOr, which intersect at the origin point.Locking the actuation jointR9, the actuation wrench of limb ABCO, which is a torque along with the z-axis, can be obtained as(22)$ABCOa=0,0,0;0,0,1.


## 4.1.DEFO Limb

The twist system of kinematic limbDEFO (R1NR2P3R4R12N) can be represented as (11)$1=1,0,0;0,0,0$2=0,cos⁡θ1,sin⁡θ1;0,-sin⁡θ1,cos⁡θ1$3=0,0,0;cos⁡ψ,-sin⁡θ1sin⁡ψ,cos⁡θ1sin⁡ψ$4=0,cos⁡θ1,sin⁡θ1;sin⁡β,-cos⁡βsin⁡θ1,cos⁡βcos⁡θ1$12=-cos⁡β,-sin⁡βsin⁡θ1,sin⁡βcos⁡θ1;0,0,0.The constraint wrench system which the kinematic limbDEFO enforces on the moving-platform CO can be calculated by reciprocal screw theory: (12)$DEFOr=0,cos⁡θ1,sin⁡θ1;0,0,0.The constraint wrench$DEFOr is the force which is across the origin and is parallel to the revolute joint R2. The constraint wrench $DEFOr is along with the axis of revolute joint R8N, since the revolute joint R8N is parallel to R2.Locking the actuation jointP3, the actuation wrench of limb DEFO can be obtained by (5) as(13)$DEFOa·$3≠0$DEFOa·$i=0i=1,2,4,12.Further,(14)$DEFOa·v3≠0$DEFOa·ωi=0i=1,2,4,12,where ωi denotes the angle velocity of the revolute joints Ri(i=1,2,4,12) and v3 denotes the velocity of the prismatic joint P3.The actuation wrench of limbDEFO, which is the force along with the axis of the prismatic joint P3, is obtained by (14) as(15)$DEFOa=cos⁡ψ,-sin⁡θ1sin⁡ψ,cos⁡θ1sin⁡ψ;0,0,0.


## 4.2.GHO Limb

The twist system of kinematic limbGHO can be represented as(16)$5=1,0,0;0,1,0$6=0,0,0;0,sin⁡δ,cos⁡δ$7=1,0,0;0,cos⁡α,sin⁡α$8=0,sin⁡α,-cos⁡α;0,0,0$12=sin⁡θ8,cos⁡αcos⁡θ8,sin⁡αcos⁡θ8;0,0,0.The constraint wrench system which the kinematic limbGHO enforces on the moving-platform CO can be calculated as(17)$GHOr=1,0,0;0,0,0.The constraint wrench$GHOr is the force along the x-axis.Locking the actuation jointP6, the actuation wrench of kinematic limb GHO can be obtained by(18)$DEFOa·v6≠0$DEFOa·ωi=0i=5,7,8,12.The actuation wrench of limbGHO, which is the force along with the axis of the prismatic joint P3, is obtained by (18) as(19)$GHOa=0,sin⁡δ,cos⁡δ;0,0,0.


## 4.3.ABCO Limb

The twist system of kinematic limbABCO can be represented as(20)$9=0,0,-1;0,0,0$10=a10,b10,c10;0,0,0$11=a11,b11,c11;0,0,0.The constraint wrench system which the kinematic limbABCO enforces on the moving-platform CO and can be calculated as(21)$1,ABCOr=1,0,0;0,0,0$2,ABCOr=0,1,0;0,0,0$3,ABCOr=0,0,1;0,0,0.The constraint wrench system consists of three nonlinear forces$1,ABCOr, $2,ABCOr, and $3,ABCOr, which intersect at the origin point.Locking the actuation jointR9, the actuation wrench of limb ABCO, which is a torque along with the z-axis, can be obtained as(22)$ABCOa=0,0,0;0,0,1.


## 5. Singularity Analysis

The Jacobian matrix of parallel mechanism can be represented as the actuation Jacobian matrix and constraint Jacobian matrix [21]. Thus the singularities occur: (i) when the actuation Jacobian matrix is singular, while the constraint Jacobian matrix is invertible; (ii) when the constraint Jacobian matrix is singular; (iii) when the kinematic limb is singular [21, 23].When the matrixA is singular, that means the row vectors of the matrix are linear. In this way, the singular problem of Jacobian matrix can be transformed into judging the ranks of the wrench system and twist system of limbs.


### 5.1. The Constraint Singularity

The constraint wrench system of moving-platform is determined by (12), (17), and (21),(23)$GHOr=1,0,0;0,0,0$DEFOr=0,cos⁡θ1,sin⁡θ1;0,0,0$1,ABCOr=1,0,0;0,0,0$2,ABCOr=0,1,0;0,0,0$3,ABCOr=0,0,1;0,0,0.It is easy to see that the rank of the constraint wrench systems keeps 3, which means the constraint Jacobian matrix will be nonsingular. This parallel wrist has no constrain singularities across the workspace.The DOF of the parallel wrist can be verified by these constraints. Based on the reciprocal screw theory, the twist system of the moving-platform can be calculated,(24)$1m=1,0,0;0,0,0$2m=0,1,0;0,0,0$3m=0,0,1;0,0,0.As seen from (24), the parallel wrist has three successive rotational DOFs, which guarantee the moving-platform of the parallel wrist can achieve the spherical motion around the origin.


### 5.2. The Actuation Wrench Singularity

The actuation wrench system of moving-platform is determined by (15), (19), and (22),(25)$DEFOa=cos⁡ψ,-sin⁡θ1sin⁡ψ,cos⁡θ1sin⁡ψ;0,0,0$GHOa=0,sin⁡δ,cos⁡δ;0,0,0$ABCOa=0,0,0;0,0,1.The forces$DEFOa and $GHOa keep nonlinear. The torque $aABCO is nonlinear with both $DEFOa and $GHOa. Since the rank of actuation wrench system does not decrease, the actuation Jacobian matrix will be invertible. The parallel wrist has no actuation singularities across the workspace.


### 5.3. The Limb Singularity

Limb singularities occur at the following configurations:(i)
Whenβ=0° or β=180°, $1 and $12 are linear and the actuation joint P3 will be locked by the mechanical system; the rank of twist system of limb DEFO will decrease to 1.(ii)
Whenα=0° or α=180°, the actuation joint P6 will be locked by the mechanical system; the rank of twist system of limb GHO will decrease to 2.(iii)
When the angle of revolute jointR10 is zero, that means the limb ABCO is folded; the rank of twist system of the limb ABCO will decrease to 2.Under these configurations, the DOFs of the kinematic limb will degenerate, the parallel mechanism will fail to meet the condition of three rotational DOFs, and the mechanism will be locked.


## 5.1. The Constraint Singularity

The constraint wrench system of moving-platform is determined by (12), (17), and (21),(23)$GHOr=1,0,0;0,0,0$DEFOr=0,cos⁡θ1,sin⁡θ1;0,0,0$1,ABCOr=1,0,0;0,0,0$2,ABCOr=0,1,0;0,0,0$3,ABCOr=0,0,1;0,0,0.It is easy to see that the rank of the constraint wrench systems keeps 3, which means the constraint Jacobian matrix will be nonsingular. This parallel wrist has no constrain singularities across the workspace.The DOF of the parallel wrist can be verified by these constraints. Based on the reciprocal screw theory, the twist system of the moving-platform can be calculated,(24)$1m=1,0,0;0,0,0$2m=0,1,0;0,0,0$3m=0,0,1;0,0,0.As seen from (24), the parallel wrist has three successive rotational DOFs, which guarantee the moving-platform of the parallel wrist can achieve the spherical motion around the origin.


## 5.2. The Actuation Wrench Singularity

The actuation wrench system of moving-platform is determined by (15), (19), and (22),(25)$DEFOa=cos⁡ψ,-sin⁡θ1sin⁡ψ,cos⁡θ1sin⁡ψ;0,0,0$GHOa=0,sin⁡δ,cos⁡δ;0,0,0$ABCOa=0,0,0;0,0,1.The forces$DEFOa and $GHOa keep nonlinear. The torque $aABCO is nonlinear with both $DEFOa and $GHOa. Since the rank of actuation wrench system does not decrease, the actuation Jacobian matrix will be invertible. The parallel wrist has no actuation singularities across the workspace.


## 5.3. The Limb Singularity

Limb singularities occur at the following configurations:(i)
Whenβ=0° or β=180°, $1 and $12 are linear and the actuation joint P3 will be locked by the mechanical system; the rank of twist system of limb DEFO will decrease to 1.(ii)
Whenα=0° or α=180°, the actuation joint P6 will be locked by the mechanical system; the rank of twist system of limb GHO will decrease to 2.(iii)
When the angle of revolute jointR10 is zero, that means the limb ABCO is folded; the rank of twist system of the limb ABCO will decrease to 2.Under these configurations, the DOFs of the kinematic limb will degenerate, the parallel mechanism will fail to meet the condition of three rotational DOFs, and the mechanism will be locked.


## 6. Kinematics Analysis

### 6.1. Inverse Kinematics

The output angles and input values of the parallel wrist are the anglesθ1,θ8,θ12 and L6,L3,θ9, respectively. Usually the inverse kinematics of parallel mechanism can be accomplished by the rotation matrix [24] or kinematics modeling [25]. However, for the partially decoupled parallel wrist, the inverse kinematics can be further derived to two linear and one nonlinear kinematic equations.


#### 6.1.1. The Input Value of the Active Prismatic Joints

Since the angleα is defined as the angle between the links OG and OH, β is defined as the angle between the links OE and OF; the relationships between α,β and θ1, θ8 can be observed as α=θ1+π/2, β=θ8+π/2.The input value of the active prismatic joints can be obtained from the geometrics of the triangles △EOF and △GOH,(26)L3=a2+b2-2abcos⁡βL6=a2+b2-2abcos⁡α.


#### 6.1.2. The Input Value of the Active Revolute Joint

According to the topology of the parallel wrist, the revolute jointsR11 and R10 are perpendicular. Hence, the dot product of the axis vectors of revolute joints R11 and R10 should be zero:(27)a11DEFO·a10ABCO=0.Substituting the coordinates ofa11 in (11) and a10 in (20) and into (27), we obtain(28)θ9=arctan⁡sin⁡θ12sin⁡βcos⁡θ12cos⁡θ1+sin⁡θ12cos⁡βsin⁡θ1,where cos⁡θ12cos⁡θ1+sin⁡θ12cos⁡βsin⁡θ1≠0.Finally the inverse kinematics of the mechanism can be obtained from (26) and (28),(29)L3=a2+b2+2absin⁡θ8L6=a2+b2+2absin⁡θ1θ9=arctan⁡sin⁡θ12sin⁡βcos⁡θ12cos⁡θ1+sin⁡θ12cos⁡βsin⁡θ1.


### 6.2. Direct Kinematics

The direct kinematic problem was solved by establishing the kinematics constraint equations of the same link under different kinematic limbs. The first two output angles were derived by formulating the kinematics of the linkOF; then the third output angle was obtained by the kinematics of the moving-platform CO.


#### 6.2.1. The Two Decoupled Output Angles of Moving-Platform

The posture of revolute jointR12 associated with the different kinematic limbs should be the same. Thus, the vector of the axis of R12 in (11) and (16) should be the equivalent, obtaining(30)a12DEFO=a12GHO.Further we yields(31)-cos⁡β-sin⁡βsin⁡θ1sin⁡βcos⁡θ1=sin⁡θ8cos⁡αcos⁡θ8sin⁡acos⁡θ8.The output angle of the moving-platform,θ1 and θ8, can be calculated as(32)θ1=α-π2θ8=β-π2.


#### 6.2.2. The Third Coupled Output Angle of Moving-Platform

Based on the derived equation (27), the output angle θ12 can be obtained:(33)θ12=arctan⁡cos⁡θ1sin⁡θ9cos⁡θ9sin⁡β-cos⁡βsin⁡θ1sin⁡θ9while cos⁡θ9sin⁡β-cos⁡βsin⁡θ1sin⁡θ9≠0.Combining (32) and (33), the direct kinematics of the moving-platform CO are obtained:(34)θ1=α-π2(35)θ8=β-π2(36)θ12=arctan⁡cos⁡θ1sin⁡θ9cos⁡θ9sin⁡β-cos⁡βsin⁡θ1sin⁡θ9.Among the three DOFs of parallel wrist, two rotational DOFs are independent DOFs, and each DOF is controlled by a single actuated joint. The third rotational DOF aroundR12 is actuated by prismatic joints P6, P3 and the revolute joint R9. The parallel wrist is a partially decoupled mechanism.


## 6.1. Inverse Kinematics

The output angles and input values of the parallel wrist are the anglesθ1,θ8,θ12 and L6,L3,θ9, respectively. Usually the inverse kinematics of parallel mechanism can be accomplished by the rotation matrix [24] or kinematics modeling [25]. However, for the partially decoupled parallel wrist, the inverse kinematics can be further derived to two linear and one nonlinear kinematic equations.


### 6.1.1. The Input Value of the Active Prismatic Joints

Since the angleα is defined as the angle between the links OG and OH, β is defined as the angle between the links OE and OF; the relationships between α,β and θ1, θ8 can be observed as α=θ1+π/2, β=θ8+π/2.The input value of the active prismatic joints can be obtained from the geometrics of the triangles △EOF and △GOH,(26)L3=a2+b2-2abcos⁡βL6=a2+b2-2abcos⁡α.


### 6.1.2. The Input Value of the Active Revolute Joint

According to the topology of the parallel wrist, the revolute jointsR11 and R10 are perpendicular. Hence, the dot product of the axis vectors of revolute joints R11 and R10 should be zero:(27)a11DEFO·a10ABCO=0.Substituting the coordinates ofa11 in (11) and a10 in (20) and into (27), we obtain(28)θ9=arctan⁡sin⁡θ12sin⁡βcos⁡θ12cos⁡θ1+sin⁡θ12cos⁡βsin⁡θ1,where cos⁡θ12cos⁡θ1+sin⁡θ12cos⁡βsin⁡θ1≠0.Finally the inverse kinematics of the mechanism can be obtained from (26) and (28),(29)L3=a2+b2+2absin⁡θ8L6=a2+b2+2absin⁡θ1θ9=arctan⁡sin⁡θ12sin⁡βcos⁡θ12cos⁡θ1+sin⁡θ12cos⁡βsin⁡θ1.


## 6.1.1. The Input Value of the Active Prismatic Joints

Since the angleα is defined as the angle between the links OG and OH, β is defined as the angle between the links OE and OF; the relationships between α,β and θ1, θ8 can be observed as α=θ1+π/2, β=θ8+π/2.The input value of the active prismatic joints can be obtained from the geometrics of the triangles △EOF and △GOH,(26)L3=a2+b2-2abcos⁡βL6=a2+b2-2abcos⁡α.


## 6.1.2. The Input Value of the Active Revolute Joint

According to the topology of the parallel wrist, the revolute jointsR11 and R10 are perpendicular. Hence, the dot product of the axis vectors of revolute joints R11 and R10 should be zero:(27)a11DEFO·a10ABCO=0.Substituting the coordinates ofa11 in (11) and a10 in (20) and into (27), we obtain(28)θ9=arctan⁡sin⁡θ12sin⁡βcos⁡θ12cos⁡θ1+sin⁡θ12cos⁡βsin⁡θ1,where cos⁡θ12cos⁡θ1+sin⁡θ12cos⁡βsin⁡θ1≠0.Finally the inverse kinematics of the mechanism can be obtained from (26) and (28),(29)L3=a2+b2+2absin⁡θ8L6=a2+b2+2absin⁡θ1θ9=arctan⁡sin⁡θ12sin⁡βcos⁡θ12cos⁡θ1+sin⁡θ12cos⁡βsin⁡θ1.


## 6.2. Direct Kinematics

The direct kinematic problem was solved by establishing the kinematics constraint equations of the same link under different kinematic limbs. The first two output angles were derived by formulating the kinematics of the linkOF; then the third output angle was obtained by the kinematics of the moving-platform CO.


### 6.2.1. The Two Decoupled Output Angles of Moving-Platform

The posture of revolute jointR12 associated with the different kinematic limbs should be the same. Thus, the vector of the axis of R12 in (11) and (16) should be the equivalent, obtaining(30)a12DEFO=a12GHO.Further we yields(31)-cos⁡β-sin⁡βsin⁡θ1sin⁡βcos⁡θ1=sin⁡θ8cos⁡αcos⁡θ8sin⁡acos⁡θ8.The output angle of the moving-platform,θ1 and θ8, can be calculated as(32)θ1=α-π2θ8=β-π2.


### 6.2.2. The Third Coupled Output Angle of Moving-Platform

Based on the derived equation (27), the output angle θ12 can be obtained:(33)θ12=arctan⁡cos⁡θ1sin⁡θ9cos⁡θ9sin⁡β-cos⁡βsin⁡θ1sin⁡θ9while cos⁡θ9sin⁡β-cos⁡βsin⁡θ1sin⁡θ9≠0.Combining (32) and (33), the direct kinematics of the moving-platform CO are obtained:(34)θ1=α-π2(35)θ8=β-π2(36)θ12=arctan⁡cos⁡θ1sin⁡θ9cos⁡θ9sin⁡β-cos⁡βsin⁡θ1sin⁡θ9.Among the three DOFs of parallel wrist, two rotational DOFs are independent DOFs, and each DOF is controlled by a single actuated joint. The third rotational DOF aroundR12 is actuated by prismatic joints P6, P3 and the revolute joint R9. The parallel wrist is a partially decoupled mechanism.


## 6.2.1. The Two Decoupled Output Angles of Moving-Platform

The posture of revolute jointR12 associated with the different kinematic limbs should be the same. Thus, the vector of the axis of R12 in (11) and (16) should be the equivalent, obtaining(30)a12DEFO=a12GHO.Further we yields(31)-cos⁡β-sin⁡βsin⁡θ1sin⁡βcos⁡θ1=sin⁡θ8cos⁡αcos⁡θ8sin⁡acos⁡θ8.The output angle of the moving-platform,θ1 and θ8, can be calculated as(32)θ1=α-π2θ8=β-π2.


## 6.2.2. The Third Coupled Output Angle of Moving-Platform

Based on the derived equation (27), the output angle θ12 can be obtained:(33)θ12=arctan⁡cos⁡θ1sin⁡θ9cos⁡θ9sin⁡β-cos⁡βsin⁡θ1sin⁡θ9while cos⁡θ9sin⁡β-cos⁡βsin⁡θ1sin⁡θ9≠0.Combining (32) and (33), the direct kinematics of the moving-platform CO are obtained:(34)θ1=α-π2(35)θ8=β-π2(36)θ12=arctan⁡cos⁡θ1sin⁡θ9cos⁡θ9sin⁡β-cos⁡βsin⁡θ1sin⁡θ9.Among the three DOFs of parallel wrist, two rotational DOFs are independent DOFs, and each DOF is controlled by a single actuated joint. The third rotational DOF aroundR12 is actuated by prismatic joints P6, P3 and the revolute joint R9. The parallel wrist is a partially decoupled mechanism.


## 7. Workspace Analysis

The wrist mechanism has three rotational DOFs, so the workspace is defined as all reachable positions of the platform about a fixed point in 3D space. There are three main mechanical constraints that limit the workspace of the parallel wrist: (i) the actuators’ stroke, (ii) the range of the passive joints, and (iii) the link interference. By applying the inverse kinematics of parallel wrist, we judge the boundaries of the orientation workspace.(i) The Actuators’ Stroke. Consider(37)L6min≤L6≤L6max,L3min≤L3≤L3max,where L6min and L6max are, respectively, the minimum and maximum lengths of prismatic joints L6 and where L3min and L3max are, respectively, the minimum and maximum lengths of prismatic joints L3. When L6 is equal to L6min or L6max, L3 is equal to L3max or L3min; the node can be written into the array which forms the boundary of the orientation workspace.(ii) Range of the Passive Joints. The universal joints in the mechanism are replaced by two intersecting joints, and the range of the passive revolute joints could be considered to be (0,2π]. Thus during the calculation of the workspace, the range of the passive joints is ignored.(iii) Link Interference. The links of parallel wrist can be approximated by cylinders of diameter D. The minimum distance between every two adjacent line segments is Di. The constraint on the relative position of all pairs of links can be denoted as(38)Di≥D.The minimum distance between two line segments can be calculated by the multistep algorithm introduced in [26].Based on the least interference design methodology introduced in [27], the parallel wrist is optimized to have the maximum link interference-free workspace, as shown in Figure 2. There is only quite little link interference between links OF and FE, when the triangle △EOF degenerates to the coaxial lines.Figure 2
The modified parallel wrist.The workspace analysis shows the workspace of the unit length vector associated with the moving-platformOC of the modified parallel wrist.From Figure3, we can derive the following:(1)
The output angleθ1 of the parallel wrist is reachable within (-π/2,π/2), when output angle θ8 is equal to any angle within (-π/2,π/2).(2)
The maximum range of output angleθ8 can be (0,2π), which is optimized by on the least interference design methodology introduced in [27].Figure 3
The workspace of the parallel wrist.


## 8. Decoupled Motion Study

The mathematical results of the cases are shown in Table1. The results were developed by using the MATHEMATICA software package. The geometric parameters of the parallel wrist are given as OE=OG=200 cm, OF=OH=100 cm, and OA=OB=OC=98 cm.Table 1
The cases of direct kinematics.
Posture
Input
Output
L
3 (cm)
L
6 (cm)
θ
9 (°)
θ
1 (°)
θ
8 (°)
θ
12 (°)
1
148
220
30
45.23
−44.62
41.84
2
148
170
20
−31.84
−44.62
22.15
3
148
260
20
26.10
−44.62
22.30
4
258
260
20
26.10
24.46
11.23
5
258
260
45
26.10
24.46
24.26The performance of partially decoupled motion of the 3-DOF parallel wrist is shown in Figure4. Four videos are attached as supplement materials (see Supplementary Material available online at http://dx.doi.org/10.1155/2015/790414):(1)
As shown in Figure4(a), the parallel wrist changes from the 1st posture to 2nd posture.The parallel wrist changes from the 1st posture (L3 = 148 cm, L6 = 220 cm, and θ9 = 30°) to 2nd posture (L3 = 148 cm, L6 = 170 cm, and θ9 = 20°). The output angle θ8 stays at −44.62°. In Figure 4(a), the 1st posture is shown in dotted line, and the 2nd posture is shown in real line.(2)
As shown in Figure4(b), the parallel wrist changes from the 2nd posture to the 3rd posture.The parallel wrist changes from the 2nd posture (L3 = 148 cm, L6 = 170 cm, and θ9 = 20°) to 3rd posture (L3 = 148 cm, L6 = 260 cm, and θ9 = 20°). In Figure 4(b), the 2nd posture is shown in dotted line, and the 3rd posture is shown in real line.(3)
As shown in Figure4(c), the parallel wrist changes from the 3rd posture to the 4th posture.The parallel wrist changes from the 3rd posture (L3 = 148 cm, L6 = 260 cm, and θ9 = 20°) to 4th posture (L3 = 258 cm, L6 = 260 cm, and θ9 = 20°). In Figure 4(c), the 3rd posture is shown in dotted line, and the 4th posture is shown in real line.(4)
As shown in Figure4(d), the parallel wrist changes from the 4th posture to the 5th posture.The parallel wrist changes from the 4th posture (L3 = 258 cm, L6 = 260 cm, and θ9 = 20°) to 5th posture (L3 = 258 cm, L6 = 260 cm, and θ9 = 45°). In Figure 4(d), the 4th posture is shown in dotted line, and the 5th posture is shown in real line.(5)
As shown in Figures4(a) and 4(b), output angle θ8 is controlled by actuated joint P3.(6)
As shown in Figures4(c) and 4(d), output angle θ1 is controlled by actuated joint P6.(7)
As shown in Figures4(b) and 4(c), output angle θ12 is controlled by actuated joints P3, P6, and R9.Figure 4
The demonstration of partially decoupled motion.
(a)
The posture whenL3 = 148 cm, L6 = 220~170 cm, and θ9 = 30~20°
(b)
The posture whenL3 = 148 cm, L6 = 170~260 cm, and θ9 = 20°
(c)
The posture whenL3 = 148~258 cm, L6 = 260 cm, and θ9 = 20°
(d)
The posture whenL3 = 258 cm, L6 = 260, and θ9 = 20~45°


## 9. Conclusion and Discussion

In this paper, a 3-DOF spherical parallel wrist was introduced. Then link analysis is presented by using the reciprocal screw theory. As a result, the singularity analysis is deduced. The kinematics analysis and workspace analysis are presented. The decoupled motion analysis shows that the two rotational DOFs of the mechanism can be controlled by a single actuation, respectively, while the third rotational DOF is controlled by three actuated joints. Thus, the third DOF is a weak-coupling with the other two rotational DOFs.The mechanism has linear kinematics, so it has the potential for application as the ankle or wrist of the robot. The kinematics of the parallel wrist is simple and easy to control, which ensures that the wrist will achieve high precision motion.



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*Source: 790414-2015-11-23.xml*