KINEMATIC ANALYSIS OF A NOVEL THREE DEGREE-OF-FREEDOM PLANAR PARALLEL MANIPULATOR

International Journal of Robotics and Automation, Vol. 24, No. 2, 2009 KINEMATIC ANALYSIS OF A NOVEL THREE DEGREE-OF-FREEDOM PLANAR PARALLEL MANIPULATOR B. Li, J. Zhao, X. Yang, and Y. Hu Abstract In this paper, a novel unsymmetrical five four bar Planar Parallel Manipulator (PPM) with three Degree-of-Freedoms (DOFs) is proposed, the new manipulator is composed of a double-layer five-bar part with a four-bar part located in between. Targeting for the application of Chinese dish cooking machine, the detailed kinematics analysis for the new PPM is developed. Different from traditional three-dof PPM, the manipulator has only four assembly modes. In orientation workspace analysis, the workspace map is shown to be an annular zone. The two-point velocity method is used in dexterity analysis to solve the inhomogeneous problem, and the five four bar PPM exhibits good dexterity. By analysis of both Inverse Kinematics Singularity (IKS) and Direct Kinematics Singularity (DKS), the distributions of the singular curves in workspace are obtained, the DKS analysis also reveals the similarity between the DKS and the direct displacement analysis: the related analysis can be decomposed into a four-bar part and a five-bar part respectively. All the analysis presented in the paper can facilitate the dimensional synthesis and trajectory planning for the manipulator. Key Words Parallel manipulator, kinematics, workspace, dexterity, singularity 1. Introduction In recent years, there was a trend towards the analysis and synthesis of the reduced-dof, i.e. less than six, parallel manipulators [1]. Although spatial parallel manipulators deserve more research efforts for their complexity in kinematics, dynamics and control, PPM have also attracted considerable attentions for their simple structure and easy Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, 518055, P.R. China; State Key Laboratory of Robotics and System (HIT), Harbin, 150001, P.R. China; e-mail: libing.sgs@hit.edu.cn, jguozhao@gmail.com, yangxj@hitsz.edu.cn Cognitive Technologies Lab, Shenzhen Institute of Advanced Integration Technology, Chinese Academy of Sciences/Chinese University of Hong Kong, Shenzhen, 518067, P.R. China; e-mail: ying.hu@siat.ac.cn Recommended by Prof. Jingzhou Yang (paper no. 206-3314) implementation. Many three-dof PPMs have been proposed in the literatures, which can be classified into two categories: nonredundant and redundant. A nonredundant three-dof PPM has only three actuators and is conventionally made up of three identical limbs connecting the fixed base to the moving platform. In fact, Merlet [2] enumerated all the seven feasible structures for them. Some of them, especially the 3-RRR, 3-PRR and 3-RPR type, have been studied extensively [3, 4]. A redundantly actuated PPM has more than three actuators. Wu et al. proposed a three-dof PPM with two symmetrical limbs, each of which contains two prismatic actuators [5]. Ebrahimi et al. invented a new kinematically redundant PPM by adding one prismatic actuator to each limb of traditional 3-RRR manipulator [6]. All the above mentioned manipulators are of symmetrical nature. In this paper, a new three-dof PPM with unsymmetrical structure is proposed. The manipulator evolves from the traditional 3-RRR manipulator by relocating the end of one limb to the end of adjacent limb as shown in Fig. 1. In the figure, the end of the second limb is relocated to the end of first limb. Thus the first and second limbs become one limb and it constitutes a five-bar part. If we let it be still, then the remaining part of the manipulator is a four-bar part with the moving platform as one of its link. Therefore, we can call this new manipulator as a five four bar manipulator. A similar method has been used to obtain parallel kinematics machines, such as 6 6, 3 6 and 3 3 connections of a hexapod, but they still have symmetrical structure. To the best of authors knowledge, the proposed unsymmetrical five four bar manipulator is novel. The manipulator is originally designed for an automatic Chinese dish cooking machine to imitate skilled chef s operation, which needs the cooking pan to perform a rather complicated planar motion. The typical Chinese cuisine includes a cooking pan and fire for heating up the in-pan material, there are two main motions for a skilled chef to operate on the cooking pan: one is to move the pan with a planar ellipse trajectory to avoid any cooking material sticking to the pan; the second motion is to make the cooking pan move quickly upward and downward, thus the in-pan material can be turned over so that fire can evenly heat up the double sides of the material, this is the 158

Figure 1. From 3-RRR manipulator to a new manipulator. Figure 2. Proposed novel three-dof PPM. key requirement of Chinese cooking process. Traditional serial manipulator is used in the development of the first prototype, which includes a ball screw for one direction translation, a crank-slider mechanism for the other direction translation and a motor for rotation. The major drawback for this manipulator is that it cannot imitate the chef s dexterous motion. Moreover, the actuators are rather cumbersome because not all of them are mounted on the base. The proposed PPM in this paper is designed to overcome the above shortcomings. Due to the special architecture of the proposed PPM, the five four bar manipulator exhibits different characteristics from the traditional 3-RRR manipulator. First of all, all the three actuators can be installed at the base and thus reduce their inertia. Furthermore, the control process can be significantly simplified as the manipulator can be treated as a combination of five-bar and four-bar mechanisms, precise imitation of the chef s motion can be achieved by trajectory planning. However, the manipulator s stiffness may not be strong enough. For the specific application, some modifications are made to improve the characteristics of the proposed manipulator. As the assembly space is limited, we change the fixed base to form an isosceles triangle; we also use a double-layer five-bar to solve the problem of stiffness reduction. In addition, the moving platform is also modified to avoid mechanical inferences as a cooking container should be attached on it. By 159 the above modifications, a novel manipulator is obtained whose solid model is depicted in Fig. 2(a). The fixed base is connected to the moving platform by a double-layer fivebar part and with a four-bar part lying in between. For this manipulator, a national patent has been applied [7]. Due to the special architecture of proposed manipulator, the five four bar manipulator exhibits different characteristics from the traditional 3-RRR manipulator. This paper is organized as follows: in Section 2, the displacement analysis of the manipulator is performed. Based on the displacement analysis, the workspace, dexterity and singularity, analysis are performed in Section 3 5 respectively. Finally, a conclusion is drawn based on all the analysis. 2. Displacement Analysis 2.1 Detailed Description of the Five Four Bar Manipulator The kinematic structure of the five four bar manipulator is shown in Fig. 2(b). In the manipulator, the five-bar part is O 1 A 1 B 2 A 2 O 2, where B 1 also belongs to it. The four-bar part is O 3 A 3 B 3 B 1. Isosceles O 1 O 2 O 3 defines the fixed base, and B 3 B 1 P forms the moving platform. ΔA 1 B 1 B 2 in the figure as a whole is a rigid link connecting to other parts by three revolute joints. By the Grubler- Kutzbach formula, we can easily conclude the DOF of the

manipulator is three. A base frame OXY with its origin at midpoint of O 1 O 2 is established, its x-axis points along the direction of O 1 O 2 and y-axis along OO 3. Three actuators are all located at O 1, O 2 and O 3 respectively, and α 1, α 2, α 3 are used to denote the actuating angle with respect to x-axis. The orientation of moving platform is denoted by β, which is the angle between B 3 B 1 and the x-axis. Coordinates of P in base frame is used to represent the position of the moving platform. Hence the pose (position and orientation) of the moving platform can be described by p =(x, y, β) T. A body frame O X Y attached to the moving platform is also established with it origin at P, x -axis points along the direction of B 3 B 1. To simplify subsequent analysis, we denote the vector of each link as follows: r 1 = OO 1, r 2 = OO 2, r 3 = OO 3, a 1 = O 1 A 1, a 2 = O 2 A 2, a 3 = O 3 A 3, b 1 = A 1 B 1, b 2 = A 2 B 2, b 3 = A 3 B 3, c 1 = B 1 P, c 2 = B 2 B 1, c 3 = B 3 P, d 1 = A 1 B 2, d 2 = B 3 B 1. The bold face letter is used to represent vectors. Let λ = π B 3 B 1 P, B 2 B 1 A 1 = γ 1, A 1 B 2 B 1 = γ 2. In the rest of this paper, the normal lowercase letters are scalars and can be used to represent the Euclidian norm of vectors, e.g. a 1 = a 1 is the length of link O 1 A 1. We use subscript x and y to denote the coordinates of any point. For example, (B 1x, B 1y ) T represents the coordinates of point B 1 in base frame. The same notation applies to the two components of a vector. For instance, a 1 =(a 1x,a 1y ) T means the vector components in the body frame. 2.2 Inverse Displacement Analysis For the Inverse Displacement Analysis (IDA), p=(x, y, β) T is given, the actuation angles q =(α 1,α 2,α 3 ) T are to be found. Although the manipulator evolves from the 3-RRR PPM, we cannot solve the IDA by substitution of specific value to existing solutions because of the modifications we made. Therefore, the conventional vector loop method is used to solve the problem. In fact, the manipulator can be decomposed into three subchains as shown in Fig. 3. We can formulate three vector loop equations according to the three subchains as follows: OP = r 1 + a 1 + b 1 + c 1 OP = r 2 + a 2 + b 2 + c 2 + c 1 (1) OP = r 3 + a 3 + b 3 + c 3 From which we can obtain the solution to IDA as follows: α i = arctan where: ( f i g i δ i e i e 2 i + f 2 i g2 i e i g i + δ i f i e 2 i + f 2 i g2 i ) (i =1, 2, 3) e 1 =2a 1 [x + r 1 c 1 cos(λ + β)] f 1 =2a 1 [y c 1 sin(λ + β)] g 1 =[x + r 1 c 1 cos(λ + β)] 2 +[y c 1 sin(λ + β)] 2 + a 2 1 b e 2 =2a 2 [x r 2 c 1 cos(λ + β) c 2x ] f 2 =2a 2 [y c 1 sin(λ + β) c 2y ] g 2 =[x r 1 c 1 cos(λ + β) c 2x ] 2 +[y c 1 sin(λ + β) c 2y ] 2 + a 2 2 b 2 2 e 3 =2a 3 [x d 2 cos β c 1 cos(λ + β)] f 3 =2a 2 [y r 3 d 2 sin β c 1 sin(λ + β)] g 3 =[x d 2 cos β c 1 cos(λ + β)] 2 +[y r 3 d 2 sin β c 1 sin(λ + β)] 2 + a 2 3 b 2 3 and δ i = ±1 is branch index, which determines the working modes of the manipulator. Note that c 2x and c 2y in the above equations can be obtained only after α 1 is solved. Therefore, the solution for the second subchain depends on the first subchain. This is the reason that the IDA cannot be obtained from existing 3-RRR formulas. The manipulator has a total of eight working modes as shown in Fig. 4. In the figure + means the value of branch index being 1, while means 1. For example, mode (a) with + + + means δ 1, δ 2 and δ 3 all taking the (2) Figure 3. Three sub chains in the five four bar manipulator. 160

values of 1. For automatic cooking application, the cooking pan connected in the point P of Fig. 2(b) will swing back and forth to avoid any material sticking to the pan during the cooking process, the required workspace is almost a rectangular area located in the right-side of the mechanism, so the configuration of the manipulator is selected to work in the mode of Fig. 4(e). 2.3 Direct Displacement Analysis Direct displacement analysis is to solve the moving platform s pose p =(x, y, β) T given the actuation angles q =(α 1,α 2,α 3 ) T. The vector loop method is used to solve the problem. Obviously, if we have the coordinates of B 1 and B 3, then the pose of the moving platform can be easily obtained. Therefore, the direct displacement analysis is converted to solve B 1 and B 3. In the five-bar part, from loop OO 1 A 1 B 2 O and OO 2 A 2 B 2 O, we can solve for OB 2, and then coordinates of B 1 can be easily obtained. Similarly, in the four-bar part, from loop OO 3 A 3 B 3 O and OB 1 B 3 O, we can solve for OB 3, thus coordinates of B 3 can be obtained. Note that there exists two solutions for both B 1 and B 3, for a given set of actuator angles, the manipulator has four assembly modes. 3. Workspace Analysis The workspace of parallel manipulators can be divided into different categories [2]: the constant orientation workspace(cow), orientation workspace, inclusive orientation workspace, reachable or maximum workspace, total orientation workspace(tow), dexterous workspace etc. For automatic cooking application, only the COW and TOW are studied in this paper. Merlet [8] used efficient geometrical algorithms to determine various workspace for 3-RPR manipulator which could be easily extended to other symmetrical PPMs. However, these algorithms cannot be applied to five four bar PPM with unsymmetrical structure. Thus, in this paper, the usual numerical polar search method will be used to obtain the boundary of different workspace [9]. To facilitate the analysis, without otherwise stated, the manipulator s dimensions are as follows: r 1 = r 2 =0.075, r 3 =0.1, a 1 =0.22, a 2 =0.2, a 3 =0.3, b 1 =0.6, b 2 =0.4, b 3 =0.44, c 1 =0.2, c 2 =0.22, d 1 =0.4, d 2 =0.12, λ = π 4. COW is defined as the set of positions of a reference point on the moving platform when its orientation is fixed. For the five four bar manipulator, we choose P as the reference point, then COW is the translation workspace of P when β is a constant. TOW is defined as the set of positions that the moving platform can be reached for any orientation in a specified range. In this paper, the numerical search method is used to obtain the boundary for TOW. Generally speaking, the rotational range for a revolute joint is [0, 2π], but in practical applications due to the working environment limitation, the actuator can only rotate in a specified range. Thus, it is important to study the COW under actuator constraints. Let us π specify the ranges of the actuator angles as: 3 α 1 4π 3, 161 π 3 α 2 2π 3, π 6 α 3 π 2 and let the manipulator work as mode (e) as shown in Fig. 4. The resulting COW with β = π/3 and TOW with β [0, 2π] are illustrated in Fig. 5. In the figure, the annular area circled by the solid line is the COW, while inside the COW the annular area circled by the dashed line is the TOW. As a matter of fact, the TOW is the intersection of COW for all the β in a specified interval, hence the TOW is always a subset of COW. Dexterous workspace is defined as the positions of a reference point on the moving platform which can be reached for any orientation. It is a special case of TOW as the range of orientation is [0, 2π], hence the dexterous workspace can be obtained by the same searching method for TOW. 4. Dexterity Analysis Condition number of the Jacobian matrix is a prevalent kinematic performance index to evaluate the manipulator s dexterity. Let the actuated joint variables be denoted by a vector q and the center point of the moving platform be described by a vector p by differentiating the displacement constraint equations with respect to time, two Jacobian matrices J p and J q can be obtained as follows: b 1x J p = b 2x (c 2xb 2y c 2y b 2x )a 1x b 2y (c 2xb 2y c 2y b 2x )a 1y b 1x a 1y b 1y a 1x b 1x a 1y b 1y a 1x b 3x b 1y b 3y b 1x c 1y b 1y c 1x (c 1x a 1y c 1y a 1x )(c 2x b 2y c 2y b 2x ) (c 1x b 2y c 1y b 2x ) b 1x a 1y b 1y a 1x b 3x c 3y b 3y c 3x (3) a 1x b 1y a 1y b 1x 0 0 J q = 0 a 2x b 2y a 2y b 2x 0 0 0 a 3x b 3y a 3y b 3x Then the overall Jacobian matrix for the parallel manipulator can be written as J = J 1 p J q. The condition number of traditional Jacobian can only be applied to those manipulators with both only one type of actuation joints and for either positioning or orienting task [10]. If these conditions are not satisfied, the condition number may vary if different units are chosen in the Jacobian. For the five four bar manipulator, although its actuation joints are all revolute joints, its moving platform has mixed translational and rotational motion. Thus, the last row in the direct Jacobian is dimensionless while the first two rows have units of length. To solve the unit inhomogeneous problem, a two points linear velocities method proposed by Gosselin [11] is utilized in this paper. The Jacobian matrix is posture dependent, thus the condition number also varies with posture. To get a performance index that only depends on the manipulator s dimensions, a global conditioning index based on the average condition number over the whole workspace was proposed

Figure 4. Eight working modes of five four bar manipulator. [12 13]. For the five four bar PPM, the global conditioning index can be defined as C = C(β)dβ/λ 2 λ 1, where λ 2 λ 1 C(β) is the global condition number with β [λ 1,λ 2 ]. Let W be the workspace and c(j ) be the condition number, C(β) can be defined as C(β)= W c(j )dw/ W dw. For the application of automatic cooking, we need the five four bar PPM to work as mode Fig. 4(e) in a specified workspace. As mentioned in Section 2.2 the target workspace area is a rectangular workspace which is marked with a black square in Fig. 5 with vertices planar coordinates at [ 0.1, 0.7], [ 0.1, 0.8], [0.2, 0.7] and [0.2, 0.8]. Based on the equation of global condition number, we can get C( π 4 )=1.76. Given β [0, 4π 9 ], by calculating global condition numbers for every β in this interval, the minimum global condition number happens at β =0.83776, which means the manipulator s average dexterity is the best in the specified workspace when β =0.83776. By the equation of global conditioning index and using the same data, we can get C =2.61 indicating the good dexterity of the five four bar PPM. 5. Singularity Analysis There are a variety of classifications for singularities, the most common classification is based on the two Jaco- 162 Figure 5. COW for β = π 3, TOW for β [ 0, π 2 ]. bian matrices. For this classification, three types of singularities are: Inverse Kinematic Singularity (IKS) when det(j q ) = 0, Direct Kinematic Singularity (DKS) when det(j p ) = 0 and combined singularity when both det(j p ) = 0 and det(j q ) = 0. Since the combined singularity can occur only in special architecture, only the IKS and DKS are studied in this paper. For simplicity, we assume the orientation of the moving platform is fixed, i.e. β is a constant.

Figure 6. IKS curves. Figure 7. DKS configurations. 5.1 IKS Curve In (3), we have obtained the expression for J q. From det(j q ) = 0, we know that the first two links in each subchain are collinear, i.e., O i A i and A i B i are collinear. This is the case when different branches of the inverse displacement meet. Hence, this type of singularity often occurs at the workspace boundary. The singularity curves for β = π/3 are shown in Fig. 6. All the curves for the three subchains are plotted in Fig. 6(a), where the dash dotted line is the curve for the first chain, the dashed line is for the third and the solid line is the coupler curve for the second. The COW for β = π/3 is added to all the IKS curves in Fig. 6(b), where the boundary of COW is the bold solid line. From the figure, the boundary of COW is made up of portions of IKS curves. Thus, we can alternatively get the COW from IKS analysis. 5.2 DKS Curve The DKS is much more difficult than IKS, as one can see from the complex structure of J p in (3). Many works on DKS just list a few special DKS configurations which 163 can be viewed geometrically [14]. To obtain the DKS curve for the five four bar PPM, the analytical method is very complicated and sometimes is not available. In fact, numerical method is sufficient to get the DKS curves. Therefore, by using Matlab simulation, the DKS curves for all the eight working modes are obtained. Actually in the expressions of DKS, when δ 3 takes the same sign it means that different branches of direct displacement meet for the four-bar part, as shown in Fig. 7(a), the singularity happens when A 3 B 3 and B 3 B 1 are collinear. If we disregard the DKS curve for the fourbar part, then there will be the cases of the first two branch indices δ 1 and δ 2 taking the same sign, in this case the corresponding configuration for the manipulator is shown in Fig. 7(b). In the figure, A 1 and A 2 are coincident, this case can happen only when b 2 = d 1. Note that in the configuration shown in Fig. 7(b) the manipulator can perform self-motion, i.e. the manipulator can undergo continuous motion if all the actuators are locked. Indeed, under this condition, it is a four-bar mechanism with A 3 and A 1 or A 2 as its joints attach to the fixed base. Based on the simulation results, we can find that the DKS curves can be divided into two parts: the four-bar

part and the five-bar part. For the working modes having the same sign of δ 3, the four-bar part of the DKS curves is the same. Similarly, if δ 1 and δ 2 have the same sign, then the five-bar part of the DKS curves is the same. This result corresponds to the four-bar part and five-bar part in direct displacement analysis. Apart from the two singular configurations in Fig. 7, there is another type of singularity. The singular configuration happens when A 1 B 2 and A 2 B 2 are collinear, which means two branches of direct displacement for the five-bar part meet. This can not happen if d 1 + b 2 >a 1 + a 2 + r 1 + r 2 is satisfied. 6. Conclusion In this paper, a novel unsymmetrical five four bar PPM with three-dofs is proposed, the manipulator is composed a double-layer five-bar part with a four-bar part in between. Due to the special architecture, all the three subchains are considered in solving the kinematics of the manipulator. The vector loop method is utilized to solve the displacement problem, different from traditional three-dof PPM, the manipulator has only four assembly modes. In workspace analysis, the COW is shown to be an annular zone, the numerical search method is used to analyse the TOW of the manipulator. The two points velocity method which can better solve the inhomogeneous problem is used in dexterity analysis, in this way homogeneous Jacobian matrix is generated. The five four bar PPM also has good dexterity as can be seen from the resulting dexterity plot. In singularity analysis, both the IKS and DKS curves are depicted. The results for the IKS show that the boundary of COW is a part of IKS curve. Examination of the DKS curve reveals the similarity between the DKS and the direct displacement problem: they can all be decomposed into a four-bar part and a five-bar part. All the analysis presented in this paper can facilitate the dimensional synthesis and trajectory planning for the new manipulator. The results of the kinematics analysis for the novel PPM in this paper show that the proposed five four bar PPM can better meet the requirements of the Chinese dish cooking machine. The selected work mode for this particular application can provide a desired workspace with good dexterity throughout the effective workspace zone. Acknowledgements This work is supported by Natural Science Foundation of China (Project No: 60875060) and State Key Laboratory of Robotics and System (HIT) (Project No: SKLRS200719). The work is also partly supported by 863 Hi-Tech Research and Development Program of China (Project No: 2006AA040205). References [1] Q.C. Li, Z. Huang, & J.M. Herve, Type synthesis of 3R2T 5- DOF parallel mechanisms using the Lie group of displacements, IEEE Transaction of Robotics and Automation, 20(2), 2004, 173 180. [2] J.P. Merlet, Parallel robots, Second Edition (Netherlands: Springer, 2006), 54 68. 164 [3] I.A. Bonev, D. Zlatanov, & C.M. Gosselin, Singularity analysis of 3-DOF planar parallel mechanisms via screw theory, ASME Journal of Mechanical Design, 125(3), 2003, 573 581. [4] C.M. Gosselin, S. Lemieux, & J.P. Merlet, A new architecture of planar three-degree-of-freedom parallel manipulator, Proc. IEEE Int. Conf. Robotics and Automation, Minneapolis, MN, 1996, 3738 3743. [5] J. Wu, J.S. Wang, & L.P. Wang, Optimal kinematic design and application of a redundantly actuated 3DOF planar parallel manipulator, ASME Journal of Mechanical Design, 130(5), 2008, 054503. [6] I. Ebrahimi, J.A. Carretero, & R. Boudreau, 3-PRRR redundant planar parallel manipulator: inverse displacement, workspace, and singularity analyses, Mechanism and Machine Theory, 42(8), 2007, 1007 1016. [7] B. Li, H.J. Yu, & J.G. Zhao, A Novel Three DOF Planar Parallel Manipulator, P. R. China Patent Pending, 2007, Registration No. 2007100730733. [8] J.P. Merlet, C.M. Gosselin, & N. Mouly, Workspace of planar parallel manipulators, Mechanism and Machine Theory, 33(1/2), 1998, 7 20. [9] Z. Huang, Y.S. Zhao, & T.S. Zhao, Advanced Spatial Mechanism Chinese Edition, (China: Advanced Education Press, 2006). [10] L.W. Tsai, Robot Analysis: The Mechanics of Serial and Parallel Manipulators (USA: John Wiley & Sons, 1999). [11] C.M. Gosselin, Dexterity indices for planar and spatial robotic manipulator, Proc. IEEE Int. Conf. Robotics and Automation, Cincinnati, USA, 1990, 650 655. [12] C.M. Gosselin & J. Angeles, A global performance index for the kinematic optimization of robotic manipulators, ASME Journal of Mechanical Design, 113(3), 1991, 220 226. [13] K.H. Hunt, Kinematic Geometry of Mechanisms (New York: Oxford University Press, 1978). [14] I.A. Bonev, Geometric analysis of parallel mechanisms, doctoral disseration, Laval University, Quebec, Canada, 2002. Biographies robust design, etc. Bing Li is a Professor and currently Head of the Department of Mechanical Engineering and Automation, Shenzhen Graduate School, Harbin Institute of Technology of China. He received his Ph.D. from Department of Mechanical Engineering of the Hong Kong Polytechnic University. His research interests include parallel manipulators, parallel kinematics machine, sheet metal assembly, Jianguo Zhao is a graduate student from the Department of Mechanical Engineering and Automation, Shenzhen Graduate School, Harbin Institute of Technology of China. His research interests focus on parallel manipulators, etc.

Xiaojun Yang is an Assistant Professor of Department of Mechanical Engineering and Automation, Shenzhen Graduate School, Harbin Institute of Technology of China. His research interests focus on the design and control of the parallel mechanisms. Ying Hu is an Associate Research Professor of Cognitive Technologies Lab, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences. Her research interests focus on parallel robot, medical assistant robot, etc. 165