Motion characteristics analysis of a novel spherical two-degree-of-freedom parallel mechanism

: In this paper, a novel two-degree-of-freedom (2DOF) spherical parallel mechanism(SPM) with symmetrical structure is proposed and analyzed. First, the models of forward kinematics and inverse kinematics are established based on D-H parameters, and the Jacobian matrix of the mechanism is obtained and verified. Second, the workspace of the mechanism is obtained according to inverse kinematics and link interference conditions. Next, rotational characteristics analysis shows that the end effector can achieve continuous rotation about an axis located in the mid-plane and passing through the rotation center of the mechanism. Moreover, the rotational characteristics of the mechanism are proved and motion planning is carried out. A numerical example is given to verify the kinematics analysis and motion planning. Finally, some variant mechanisms can be synthesized. This work lays the foundation for the motion control and practical application of this 2DOF SPM.


Introduction
Spherical Parallel Mechanism (SPM) is a special spatial parallel mechanism. Its end effector can rotate freely around the point. The SPM has important application value and has been widely used, such as the azimuth tracking system [1] , the bionic robot [2] , surgical robot [3] , and the medical device [4] .The research about SPM mostly focuses on 2DOF SPM [5] and 3DOF SPM [6] . The theoretical research and practical application of 3DOF SPM are quite mature. For example, theoretical research about the typical 3-RRR 3DOF SPM has been studied in terms of its working space [7] , singularity [8] , dexterity [9] , stiffness [10] , dynamics [11] . And in practical engineering applications, Gosselin proposed the famous agile eye in 1994 [12] , and Houssem Saafi et al. made a research on the haptic medical device [4] , etc. In most cases, the 2DOF SPM can satisfy application requirements, such as pointing mechanisms [13] used in spherical engraving machines, azimuth tracking of satellite antennas, and automatic ground tracking equipment for various aircraft, etc., and some 2DOF artificial wrists sorted out by Bajaj [14] .
The representative 2DOF SPM is the spherical 5R mechanism. Ouerfelliz studied the direct and inverse kinematics [15] , kinematic and dynamic optimization of a general spherical 5R linkage. Cervantes-Sanchez analyzed its workspace and singularity [16] . Zhang et al. had a further analysis of the workspace of spherical 5R mechanism and spherical 2DOF parallel mechanism with actuation redundancy [17] , as well as dynamic analysis [18,19] , trajectory planning [20] and parameter optimization [21] . Yu et al. introduced [22] a simple and visual graphic method for mobility analysis of parallel mechanisms and presented a novel 2DOF rotational parallel mechanism derived from well-known Omni Wrist III. Dong et al. analyzed [23] the kinematics, singularity and workspace of a class of 2DOF rotational parallel manipulators in geometric approach. Chen et al. proposed [24] a new geometric kinematic modeling approach based on the concept of instantaneous single-rotation-angle and used for the 2DOF RPMs with symmetry in a homo-kinetic plane. Kim et al. deformed the spherical 5R mechanism [25] , designed the spatial self-adaptive finger clamp, and conducted constraint analysis, optimization design of the structure and grasping experiment on it. Xu et al. established a theory regarding the type synthesis of the two-rotational-degrees-of-freedom parallel mechanism with two continuous rotational axes systematically [26] . Terence et al. conducted the decoupling design of 5R spherical mechanism and compared it with traditional 5R spherical mechanism in motion characteristics and workspace [27] . Cao et al. obtained a three-rotation, one-translation (3R1T) manipulator for minimally invasive surgery by connecting the revolute pair and the prismatic pair to a 2DOF spherical mechanism [28] , and analyzed its kinematics and singularity. Alamdar et al. introduced a new non-symmetric 5R-SPM and developed a geometrical approach to analyze its configurations and singularities [29] .
In this paper, we propose a novel 2DOF SPM with symmetrical structure and its variant Mechanism. The article is organized as follows: Sec.2 provides a description of the proposed SPM's structure, analysis of its mobility, the models of forward kinematics and inverse kinematics are established, and the Jacobian matrix of the mechanism is obtained and verified. In Sec.3, the workspace of the mechanism is obtained. The rotation characteristics of SMP are analyzed in Sec.4. Sec.5 describes variant mechanism of the 2DOF SPM. Finally, the conclusions are drawn.

Mobility Analysis
The schematic diagram of the 2DOF SPM is shown in Fig. 1, all the revolute axes intersect at one-point O, called the rotation center of the mechanism. The base is connected with the end effector by three spherical serial 3R sub-chains: B1B2B3, B4B5B6 and B7B8B9. There is a special spherical sub-chain consisting of link 9, link 10 and component 11 and connected by two arc prismatic pairs, limits the revolute axes OB2, OB5 and OB8 on a plane, which is defined as the mid-plane of the mechanism. And the spherical 3R sub-chains B7B8B9 and component 11 forming a symmetric double arc slider-rocker mechanism aims at keeping the mid-plane always coplanar with the angular bisector of spherical angle ∠B1B2B3 [30] , ensuring the base and the end effector are symmetric with respect to the mid-plane during the movement of the mechanism.
where d is the order of a mechanism (for the spherical mechanism d=3), n is the number of components including the base, g is the number of kinematic pairs, f i is the freedom of the i-th ·4· kinematic pair. For this mechanism n=11, g=14, and ∑ f i =14. Therefore, the mechanism has 2DOFs.

Establishment of the coordinate systems
As shown in Fig. 2, a global coordinate system O-x 0 y 0 z 0 is located at the rotation center O with the x 0 -axis passing through point Q, the midpoint of arc link B 1 B 2 , the z 0 -axis is perpendicular to the plane where the arc link B 1 B 2 lies on, and y 0 -axis is defined by right hand rule.
As shown in Fig. 2, a global coordinate system O-x0y0z0 is located at the rotation center O with the x 0 -axis passing through point Q, the midpoint of arc link B 1 B 2 , the z 0 -axis is perpendicular to the plane where the arc link B 1 B 2 lies on, and y 0 -axis is defined by right hand rule. the parameter θ ij , where ij = 21, 32, 43, 54, 65, 61, 74, 87, 81, represent the angle between the two plane that the two adjacent links lying on. Looking at the rotation center along the revolute axis, the positive direction is counterclockwise. Due to the characteristics of the spherical mechanism that each revolute axis intersects at the rotation center O, the parameter α i and d ij equals zero, where ij =21, 32, 43, 54, 65, 61, 74, 87, 81. The i-th local coordinate systems are also located at the rotation center O. The x i -axis along with each revolute axis, where x 1 coincides with x 9 , x 2 coincides with x 10 , x 3 coincides with x 7 , x 5 coincides with x 11 , x 8 coincides with x 12 . The z i -axis is perpendicular to the plane where the i-th link lying on and the y i -axis is defined by right hand rule. Because this mechanism has two DOFs, the configuration can be represented by two angles φ and γ, where φ represents the angle between the vector OP and x 0 -axis, and γ represents the angle between the mid-perpendicular plane of the end effector and the plane O-x 0 z 0 . Designate point P as the output reference point of the mechanism, which is the middle point of arc link B 3 B 4 . And the driving parameters of the mechanism are θ 21 and θ 61 .
In the inverse kinematics, the driving parameters θ 21 and θ 61 can be solved when the configuration parameters φ and γ of the end effector are given.

Description of the configuration
Suppose each link moves on a spherical surface with a radius R, and the position of outputs reference point P can be described by angle φ and ω: cos sin sin sin cos where ω is the angle between the plane OPQ and the plane O-x 0 z 0, which also represents the angle between the projection of vector OP on the plane O-y 0 z 0 and the positive direction of z 0 -axis. The relationship between γ and ω can be derived from spherical triangle PQM and MNQ. According to the characteristics of the spherical mechanism and the knowledge of spherical trigonometry [31] , the relevant parameters are expressed in Fig. 3 for a clear observation. According to the spherical triangular sine theorem, from the spherical triangle M'NQ shown in Fig.  3(b) it can be derived that: Similarly, it is available in a spherical triangle PQM shown in Fig. 3(a): In the two formulas above, ∠M=180°-γ, ∠M'=∠M/2, ∠m=φ, ∠m'=∠m/2, ∠q=∠n, and ∠ N=90°.

Solutions of coordinates with configuration parameters
The trajectory of the point B 2 in the global coordinate system O-x 0 y 0 z 0 is determined by a spherical surface and a plane. As shown in Fig. 4, the radius of the spherical surface is OB 1 and the center is O. The plane is vertical to vector OB 1 and passing through the line B 2 B 2 '. The trajectory equation is:  (cos 1) sin sin sin cos 0 x y z R

Solutions of coordinates with driving parameters
The coordinates of B 2 and B 5 can also be derived by D-H link parameters. [32] between the adjacent local i-th and (i-1)th coordinate system, which is the coordinate transformation from i-th link to (i-1)-th link, it can be obtained by the following equation: 1 Rot( , )Trans(0, 0, )Trans( , 0, 0)Rot( , ) -1 i i T is an inverse transformation matrix between the adjacent local i-th and (i-1)th coordinate system, which is the coordinate transformation from (i-1)-th link to i-th link, and is the transpose matrix of Then, it can be derived that: The coordinates of revolute joints B 2 and B 5 in the global coordinate system O-x 0 y 0 z 0 can be obtained from Eq. (8) and Eq. (9): [ ]

Forward Kinematics of the SPM
Given the driving parameters θ 21 and θ 61 , the solution of the configuration parameters φ and γ can be figured out, that is, the forward kinematics of the spherical mechanism. And the normal vector of the median plane is obtained by Eqs. (10) and (11): x R r y s z t k − = = = , the coordinate of the outputs reference point P can be obtained from the symmetrical characteristic of the mechanism: It can be obtained by the spherical triangular cosine theorem from the spherical triangle M'NQ in Fig. 3(b) that: According to Fig. 3(a), the configuration parameters can be obtained by Eqs. (2), (16) and (17): arccos 1 =180 2 arccos(sin(arctan ) cos( arccos )) 2

Jacobian Matrix Analysis
By taking the derivative of Eq. (2) with respect to time, the following equation can be obtained: sin 0 cos sin sin cos cos cos sin sin From the symmetrical characteristic of the mechanism, we can know that: It can be derived by Eqs. (10), (11) and (19) 6, when i=2 and 6, j=2 and 5 respectively).

Verification of Kinematic Analysis
Verify the correctness of Jacobian matrix and the forward kinematics by comparing the numerical solution obtained by Eqs. (22) and (23) and the measurement obtained by 3D model, when giving two tiny value as inputs [33] . Four sets of data under two general configurations are given, as shown in Table 1. Then, the correctness of inverse kinematic model is verified in the same way, which means the correctness of the kinematics analysis of the 2DOF SPM.

Workspace Analysis
Due to the interference of the mechanism, the reference point P of the effector can't reach every point on the spherical surface. As shown in Fig. 1, suppose the width of each link of the mechanism is 8mm, the effective radius is R=200mm, that is, OP=OQ=200mm, and α 1 =α 4 =60°, α 2 =α 3 =α 5 =α 6 =40°, α 7 =α 8 =50°.  configuration of the mechanism To avoid interference, considering the width of the links, assume that the angle between the rotation axes OB 1 and OB 3 and the angle between OB 4 and OB 6 are not less than 10°. The workspace of the mechanism in Fig. 6 can be obtained according to the inverse kinematics and the interference condition. The specific limited configuration and corresponding position parameters of the mechanism are shown in Table 2.

Equivalent Rotation Characteristics
The end effector of the 2DOF SPM can realize continuous rotation around the axis that passes through the rotation center and lies on the mid-plane during the moving process. Moreover, the 2DOF SPM also has the following motion properties: The end effector can move from the initial position to the final position around a fixed axis no matter how the revolute axis changes in the motion process, called equivalent rotation of the mechanism. As the simplified motion model shown in Fig. 7(a), the end effector moves from position I to position II, and the mid-planes at the initial and final positions are v 1 and v 2 , the symmetric points of Q about the mid-plane are P 1 and P 2 , respectively. The line l is the intersection line of the two mid-planes, and the axis of the equivalent rotation [34] . For a clearly obversion, a plane s 3 is set, which passes through line OP 1 and is perpendicular to line l, as shown in Fig. 7

Motion Planning of the Equivalent Rotation
As shown in Fig. 7(a), the two parameters φ 1 and γ 1 of the initial configuration of the mechanism and the two parameters φ 2 and γ 2 of the final configuration are given. The coordinates of output reference point can be obtained by Eq. (2). The equation of axis l, which is the intersection line of the two mid-planes, can be obtained by Eq. (15). The equation of plane s 3 , which is passing through lines QP 1 and QP 2 , can be obtained according to the structural characteristics. And the coordinates of the point S can be obtained by the equations of axis l and plane s 3 .
Then the rotated angle of the output reference point P can be derived that: (24) where SP1=P1-S and SP2=P2-S.

SP SP SP SP
The direction vector l=[l x l y l z ] T and the rotation angle θ of the end effector rotating around the axis l are already obtained, and the rotation matrix R (θ) can be expressed by: The vector QP2 can be expressed as: The coordinates of point P 2 can be obtained by the Eq. (26), and the other parameters of the mechanism can be obtained by the inverse kinematics described in Section 2.2. Thereby, the driving parameters θ 21 and θ 61 of the rotation process can be obtained. It provides the basis for the motion planning of the spherical mechanism.

Variant Mechanisms of the 2DOF SPM
Based on 3DOF planar sub-chain, a group of variant 2DOF SPMs with the same characteristics are synthesized, providing more possibilities for practical application.
In the middle of this mechanism, there are two arc prismatic joints which connect links 9, 10 and 11. Its function is to keep the lines OB 2 , OB 5 and OB 8 on the same mid-plane. We can use a 3DOF planar sub-chain to provide the same constraints. In this way, we can get a group of variant 2DOF SPM without arc prismatic joints. Compared with the variation mechanism, the mechanism with arc prismatic joints really needs higher precision to ensure the motion performance of the mechanism. But its disadvantage is taking more space than the original mechanism.
There are seven different configurations of the 3DOF planar sub-chain can be obtained:  [RPP], in which R represents revolute joint and P represents prismatic joint [35] , and several 2DOF SPMs based on them are shown in Fig. 9. Fig. 9. Several 2DOF SPMs based on 3DOF planar sub-chain

Conclusions
A novel 2DOF Spherical Parallel Mechanism (SPM) is proposed. The SPM can realize continuous rotation around any line on the mid-plane which passes through the rotation center of the spherical mechanism, and the rotational axis can be fixed during the rotation process, which means any form of motion of the mechanism can be transformed to a rotation with a fixed axis.
The forward and inverse kinematics of the mechanism are solved based on D-H parameters and analytical geometry. The inverse Jacobian matrix of the 2DOF SPM is obtained by taking the derivative of the constraint equation, and its workspace is analyzed by considering the interference condition of the links. The correctness of the kinematics and motion planning of the mechanism is verified by the motion examples presented.
A group of variant 2DOF SPMs are constructed based on 3DOF planar sub-chain, that can provide more possibilities for practical application.