Kinematic Analysis of a Partially Decoupled 3-DOF Parallel Wrist

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.


Introduction
Recent research focuses on the parallel wrist manipulators with two or three motorized axes [1][2][3][4][5][6][7][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 singularityfree 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][14][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 pantilt 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.

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 The reciprocal product of two screws is equal to the instantaneous work of the wrench to the motion of the body: where the twist $  = [ 2 ; r 2 × 2 +v 2 ] denotes the instantaneous motion of the rigid body.The wrench $  = [f 1 ; r 1 ×f 1 +C 1 0 ] 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.

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 θ , 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] $   ∘ $ , = 0  = 1, 2, . . .,   (4) 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).

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.

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 baseplatform, 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.
Since the universal joint is equvilent to two intersecting revolute joints, three kinematic limbs are also denoted as limb

Limb Kinematics Analysis
A coordinate system - is attached to the base-platform.The point  is chosen as the origin point; the -axis is along the link , the -axis is along the link , and the -axis is perpendicular to both  and  axes.
Nomenclature used in this paper is as listed: (i) []  denotes the posture of link  under the - coordinate system, as a part of the  kinematic limb.
(ii) [a  ]  denotes the axis of the revolute joint   under the - coordinate system, as a part of the  kinematic limb.
(iii)   denotes the rotated angle of the revolute joint   .
(iv)  3 and  6 denote the length of the actuated prismatic joints  3 and  6 , respectively.
And the geometrics of the mechanism are defined as  =  = ,  =  = ,  3 = , and  6 = .And Rot(a i ,   ) denotes the rotation matrix about the direction vector a i by the angle   and can be represented as [22]: where vers   = 1 − cos   ,   = cos   ,   = sin   , and a i = {  ,   ,   }.By using formula (9), the axis of the th kinematic joint of serial limb could be obtained, ) can be represented as The constraint wrench system which the kinematic limb  enforces on the moving-platform  can be calculated by reciprocal screw theory: The constraint wrench $ r  is the force which is across the origin and is parallel to the revolute joint  2 .The constraint wrench $ r  is along with the axis of revolute joint  8  , since the revolute joint  8  is parallel to  2 .Locking the actuation joint  3 , the actuation wrench of limb  can be obtained by (5) as Further, where   denotes the angle velocity of the revolute joints   ( = 1, 2, 4, 12) and v 3 denotes the velocity of the prismatic joint  3 .
The actuation wrench of limb , which is the force along with the axis of the prismatic joint  3 , is obtained by (14) as The constraint wrench system which the kinematic limb  enforces on the moving-platform  can be calculated as The constraint wrench $ r  is the force along the -axis.Locking the actuation joint  6 , the actuation wrench of kinematic limb  can be obtained by The actuation wrench of limb , which is the force along with the axis of the prismatic joint  3 , is obtained by (18) as The constraint wrench system which the kinematic limb  enforces on the moving-platform  and can be calculated as The constraint wrench system consists of three nonlinear forces $ r 1, , $ r 2, , and $ r 3, , which intersect at the origin point.
Locking the actuation joint  9 , the actuation wrench of limb , which is a torque along with the -axis, can be obtained as

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 matrix  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.

The Constraint Singularity
. The constraint wrench system of moving-platform is determined by ( 12), (17), and ( 21), 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, 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.

The Actuation Wrench
Singularity.The actuation wrench system of moving-platform is determined by ( 15), (19), and (22), The forces $   and $   keep nonlinear.The torque $   is nonlinear with both $   and $   .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.

The Limb Singularity. Limb singularities occur at the following configurations:
(i) When  = 0 ∘ or  = 180 ∘ , $ 1 and $ 12 are linear and the actuation joint  3 will be locked by the mechanical system; the rank of twist system of limb  will decrease to 1.
(ii) When  = 0 ∘ or  = 180 ∘ , the actuation joint  6 will be locked by the mechanical system; the rank of twist system of limb  will decrease to 2.
(iii) When the angle of revolute joint  10 is zero, that means the limb  is folded; the rank of twist system of the limb  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.

Kinematics Analysis
6.1.Inverse Kinematics.The output angles and input values of the parallel wrist are the angles  1 ,  8 ,  12 and  6 ,  3 ,  9 , 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.

The Input Value of the Active Prismatic Joints.
Since the angle  is defined as the angle between the links  and ,  is defined as the angle between the links  and ; the relationships between ,  and  1 ,  8 can be observed as  =  1 + /2,  =  8 + /2.
The input value of the active prismatic joints can be obtained from the geometrics of the triangles △ and △,

The Input Value of the Active Revolute Joint.
According to the topology of the parallel wrist, the revolute joints  11 and  10 are perpendicular.Hence, the dot product of the axis vectors of revolute joints  11 and  10 should be zero: Substituting the coordinates of a 11 in ( 11) and a 10 in (20) and into (27), we obtain where cos  12 cos  1 + sin  12 cos  sin  1 ̸ = 0. Finally the inverse kinematics of the mechanism can be obtained from ( 26) and (28),  33), the direct kinematics of the moving-platform  are obtained: 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 around  12 is actuated by prismatic joints  6 ,  3 and the revolute joint  9 .The parallel wrist is a partially decoupled mechanism.

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 where  6min and  6max are, respectively, the minimum and maximum lengths of prismatic joints  6 and where  3min and  3max are, respectively, the minimum and maximum lengths of prismatic joints  3 .When  6 is equal to  6min or  6max ,  3 is equal to  3max or  3min ; 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 (iii) Link Interference.The links of parallel wrist can be approximated by cylinders of diameter .The minimum distance between every two adjacent line segments is   .The constraint on the relative position of all pairs of links can be denoted as 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  and , when the triangle △ degenerates to the coaxial lines.The workspace analysis shows the workspace of the unit length vector associated with the moving-platform  of the modified parallel wrist.
(2) The maximum range of output angle  8 can be (0, 2), which is optimized by on the least interference design methodology introduced in [27].

Decoupled Motion Study
The mathematical results of the cases are shown in Table 1.
The results were developed by using the MATHEMATICA software package.The geometric parameters of the parallel wrist are given as  =  = 200 cm,  =  = 100 cm, and  =  =  = 98 cm.The performance of partially decoupled motion of the 3-DOF parallel wrist is shown in Figure 4. 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 Figure 4(a), the parallel wrist changes from the 1st posture to 2nd posture.
The CAD model of partially decoupled parallel wrist

Figure 1 :
Figure 1: The schematic diagram and design of the partially decoupled 3-DOF parallel wrist.

Figure 4 :
Figure 4: The demonstration of partially decoupled motion.