Fr45.3. Kinematic Analysis and Performance Evaluation of the 3-PUU Parallel Module of a 3D Printing Manipulator

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1 th International Conference on Control, Automation, Robotics & Vision Marina Bay Sands, Singapore, 10-12th December 2014 (ICARCV 2014) Fr45.3 Kinematic Analysis and Performance Evaluation of the 3-PUU Parallel Module of a 3D Printing Manipulator Song Lu 1 Department of Electromechanical Engineering 1 University of Macau Avenida da Universidade, Taipa, Macao, China robot.slu@gmail.com Yangmin Li 1,2, Tianjin Key Laboratory of the Design and Intelligent Control of the Advanced Mechatronical System 2 Tianjin University of Technology Tianjin , China Corresponding author s ymli@umac.mo Abstract Recently, 3D printing manipulators have attracted extensive attention since they have become promising tools to perform the practical prototyping and distributed manufacturing tasks. To improve the kinematic accuracy, dexterity and efficiency of 3D printing manipulators, the concept of a 6-DOF hybrid manipulator, consisting of a 3-DOF parallel manipulator and a 3-DOF rotational wrist, is proposed in this paper. According to the requirement of 3D printing movements, a three-prismatic-universal-universal (3-PUU) translational parallel manipulator (TPM) is designed. Several kinematic properties of the 3-PUU TPM under study are investigated, including the inverse and forward kinematic problems, workspace determination, and dexterity. Both the inverse kinematics and forward kinematics solutions are derived in closed form, and Jacobian matrix is derived analytically. Moreover, in view of the physical constraints imposed by mechanical joints, the reachable workspace is determined. Finally, the dexterity characteristic of the 3-PUU TPM is evaluated based on the condition number of its Jacobian matrix. Index Terms Three-dimensional printing, hybrid kinematic manipulator, kinematic analysis, performance evaluation. I. INTRODUCTION Three-dimensional printing(3dp), a process for producing 3D solid objects of virtually any shape from a digital model using a layer-by-layer (LBL) process, has received extensive attention in the biomedical field in recent years for its potential applications [1], [2]. The primary advantage of 3D printing technique is its ability to create almost any shape or geometric feature. However, one of the main disadvantages for the use of 3D printers in final product manufacturing is its lack of dimensional accuracy [3]. 3D printing applications such as industrial design, bioprinting, solid repairing, toy modeling, education and training require a particular mechanism structure for various purposes. Design requirements for a mechanism of the targeted 3D printing manipulator are capability of no less than threedegrees-of-freedom motion, capability of high-frequency motion, high position accuracy, high flexibility, and large This research was sponsored by Macao Science and Technology Development Fund (Grant No. 108/2012/A3 and 108/2012/A3), Research Committee of University of Macau (Grant No. MYRG183(Y1-L3)FST11- LYM, MYRG203(Y1-L4)-FST11-LYM). position and orientational workspace. Typically the mechanisms of 3D printing devices are divided into serial structures and parallel structures. Serial mechanisms such as Objet Eden 350 V printer [3], Fab@Home printer [5] and MtoBS [4] have open kinematic structures such that they usually provide large workspace but show relatively low stiffness and accuracy, low nominal load/weight ratio, and heavy structure [6]. However, parallel manipulators have closed kinematic structures such that they provide smaller workspace but show low weight, compact structure, better accuracy, and stiffness [7] [11]. Since the orientational workspace of common parallel manipulators is usually very limited, therefore, in the last two decades, hybrid kinematic manipulators (HKM) which are composed by a combination of serial and parallel kinematic chain architectures have addressed great attention [12] [15]. Hybrid kinematic manipulators have both better performances and advantages which not only combines the better stiffness and more precise position that parallel mechanism owns, but also the bigger orientational workspace and better decoupling that serial mechanism owns, moreover, its displacement calculation, singularity analysis, workspace evaluation, statics analysis, motion planning and control are thus significantly simplified [16]. Some successful applicable examples of hybrid manipulators are: Tricept robot from Spain PKMtricept SL [17], [18], M-3iA/6A robot from Japan FANUC Corporation [19], Exechon machine from German DS-Technology Company [20], Galileo Sphere robot from Italian Motor Power Company [21]. Combining serial and parallel kinematic chains in these manipulators gives the possibility to have the advantages of both architectures and reduce their drawbacks. Particularly, a hybrid manipulator can have high accuracy comparable with a parallel manipulator and large workspace comparable with a serial manipulator. Therefore, hybrid kinematic manipulators have great potential applications in 3D printing fields, because of their fine characteristics of position accuracy, dynamics, stiffness, flexibility, etc. As for the parallel or hybrid manipulator, kinematic performance evaluation is an important issue in the development of a parallel manipulator. Kinematic performance evaluation based on Jacobian matrix is an open and challenging problem /14/$ IEEE 1847

2 that is attracting an increasing amount of efforts [22], [23]. Various performance indices, in terms of workspace [24], dexterity [25], [26], velocity/force transmission [27], accuracy [28], and stiffness [29], etc., were proposed to characterize properties of parallel manipulators. Among them, the dexterity of a manipulator is defined through the condition number of its Jacobian matrix by Salisbury and Craig [30]. The physical meaning of dexterity is the ability of the manipulator to arbitrarily change its position and orientation, or apply forces and torques in arbitrary directions. Gosselin applied the dexterity based on the condition number of Jacobian matrix to optimize the design of planar and spatial parallel manipulators [25], [26]. Mechanism optimization, dynamics and control for parallel manipulators are hot and tough research topics in recent years [31] [33], we will investigate them in next step of work. The remainder of this paper is organized as follows. Firstly, in Section II, the conceptual design of a new 6- DOF HKM manipulator to meet the requirements of 3D printing motion is proposed, where the parallel part is the key of the whole kinematic analysis. Kinematics analysis of the 3-PUU parallel manipulator module, including inverse position, forward position, and inverse velocity, are carried out. Afterwards, the reachable position workspace of the parallel manipulator module is generated in Section III. Moreover, kinematic dexterity performance of the 3-PUU TPM based on local condition index is evaluated in Sections IV. Conclusions are provided in the last section. three dimensional translation and rotation, and lay down successive layers of liquid material to build the model from a series of cross sections. Additionally, the whole printing process can be monitored by a high resolution CCD camera which provides reliable image processing. The overview of the 6-DOF hybrid 3D printing manipulator that we synthesize is shown in Fig.2. The parallel-serial mechanism consists of a three-prismatic-universal-universal (3-PUU) parallel mechanism for translational motion and a spatial 3-DOF rotational wrist for orientational motion, namely, the spatial 3-DOF rotational wrist is mounted on the moving platform of the 3-PUU parallel manipulator. The microsyringe printer head filled with printing material is attached to the 3-DOF rotational wrist. The design realizes a large workspace of orientational motion in a compact volume of the manipulator and decoupled translation and rotation, so that the end-effectors position is only controlled by the three legs firstly while the orientation is then determined by the 3-DOF rotational wrist. II. MECHANISM ARCHITECTURE AND KINEMATIC ANALYSIS A. Architecture description Fig. 2. CAD model of the 3-DOF parallel manipulator for 3D printing Fig. 1. Schematic of the 3D printing system with a parallel manipulator Most 3D Printing processes are layer-based with only three-dimensional motions in x, y and z axes, with major limitations that they can only print vertical or horizontal surfaces. To be able to print smooth surfaces in multiple directions and from multiple angles, as shown in Fig.1, a 3D printing system consisting of a 6-DOF hybrid manipulator is proposed. The 3D printable models are created with a CAD package, and the 3D printing system is capable of carrying out an additive process under computer control. After reading the design from 3D printable file (STL file), the controller drives a 6-DOF hybrid manipulator to implement The schematic diagram of the 3-PUU parallel manipulator is described in Fig.3, which consists of a moving platform, a fixed base and three limbs with identical kinematic structure. Each limb connects the fixed base to the moving platform by a prismatic (P) joint followed by two universal (U) joints in sequence, where the P joint is implemented by a lead screw actuation system driven by a servomotor. The linear guideways are perpendicular to the ground and arranged into axial symmetrical structure. The reference frame O xyz, attached to the base, and the moving frame O x y z, attached to the platform, are located with O and O at the center of the equilateral triangle ΔB 1 B 2 B 3 and ΔA 1 A 2 A 3. Let the x-axis direct along OB 1 and be parallel to x axis, and the z and z axes be normal to ΔB 1 B 2 B 3 and ΔA 1 A 2 A 3, respectively. The moving platform possesses only a translational motion, therefore, define r =( x y z ) T as the position vector of the moving platform with respect to the fixed frame O xyz. 1848

3 Fig. 3. Schematic representation of the 3-PUU TPM B. Inverse Position Analysis Given the position of the moving platform, inverse position analysis of the 3-PUU TPM involves the determination of the linear displacements of sliders. As shown in Fig.3, the position vector of r =(x y z ) T of point O can be expressed by where r = b i + q i e + d i + lw i a i, (i= 1, 2, 3) (1) a i =[ s cos β i s sin β i 0 ] T (2) b i =[ S cos β i S sin β i 0 ] T (3) d i =[ d cos β i d sin β i 0 ] T (4) e =[ ] T (5) where s, S, q i, β i, l, e, w i, d i, a i and b i denote the radius of the moving platform, the radius of the fixed base, the linear displacement of slider, the angular of point B i in the coordinate O xyz, the length of the strut, the unit vector along the lead screw, the vector along the strut, the vector from a lead screw to the center point of universal joint C i, the position vector of point A i in the coordinate O x y z and the position vector of point B i in the coordinate O xyz, respectively. Taking the norm of both sides of Eq.(1), yields q i = r T e (r T e) 2 r + a i b i d i 2 + l 2 (6) and w i = r+a i b i d i q i e (7) l C. Forward Position Analysis The forward displacement analysis of this parallel manipulator consists of determining the position vector r of the moving platform for a given set of inputs, q i (i =1, 2, 3). Referring to Fig.3 and rearranging Eq.(1), we obtain r (b i + d i a i ) q i e = r r 0i q i e = lw i (8) where r 0i =(b i + d i a i ) = ( (S d s)cosβ i = ( ) T x 0i y 0i z 0i ( S + d + s)sinβ i 0 ) T (9) Taking Euclidean norm on both sides of Eq.(4) gives q 2 1 2q 1 z +(x x 01 ) 2 +(y y 01 ) 2 + z 2 l 2 =0 (10) q 2 2 2q 2 z +(x x 02 ) 2 +(y y 02 ) 2 + z 2 l 2 =0 (11) q 2 3 2q 3 z +(x x 03 ) 2 +(y y 03 ) 2 + z 2 l 2 =0 (12) Subtracting Eq.(10) from Eq.(11), we obtain (q2 2 q1) 2 2(q 2 q 1 )z +(x 2 02 x 2 01) 2(x 02 x 01 )x +(y02 2 y01) 2 2(y 02 y 01 )y =0 (13) Subtracting Eq.(12) from Eq.(11), we obtain (q2 2 q3) 2 2(q 2 q 3 )z +(x 2 02 x 2 03) 2(x 02 x 03 )x +(y02 2 y03) 2 2(y 02 y 03 )y =0 (14) Substituting Eq.(9) into Eq.(13) leads to (q 2 2 q 2 1) 2(q 2 q 1 )z +(S d s)(3x + 3y) =0 (15) Substituting Eq.(9) into Eq.(14) leads to (q 2 2 q 2 3) 2(q 2 q 3 )z +(S d s)( 3y) =0 (16) Solving Eq.(15) and Eq.(16), we obtain x = (q2 1 q3) 2 2(q 1 q 3 )z (S d s) (17) y = (q2 2 q 2 3)+2(q 2 q 3 )z 3(S d s) (18) Substituting Eq.(17) and Eq.(18) into Eq.(11) leads to where G 1 z 2 + G 2 z + G 3 =0 (19) G 1 = 4(q 1 q 3 ) (q 2 q 3 ) 2 (S d s) 2 +1 (20) G 2 = 4(q2 1 q3)(q 2 1 q 3 ) 4 3 (q2 2 q3)(q 2 2 q 3 ) (S d s) 2 2q 1 (21) G 3 = (q2 1 q3) (q2 2 q3) 2 2 (S d s) 2 +(S d s) 2 + q1 2 l 2 (22) According to the actual assembly situation, the position of moving platform in z direction is given by z = G 2 + G 2 2 4G 1 G 3 (23) 2G 1 Substituting Eq.(20) into Eq.(17) and Eq.(18), we obtain the solutions of x and y. And apparently, it can be seen that there exist only solution for forward position analysis in this type 3PUU parallel manipulator structure. 1849

4 D. Inverse Velocity Analysis Taking the derivative of Eq.(1) with respect to time, the velocity equations of the moving platform are described as v = q i e + lω i w i, (i =1, 2, 3) (24) where v, q i and ω i denote the velocity of moving platform, velocity of ith slider and angular velocity of ith strut, respectively. Taking dot multiplying both sides of the expression in Eq.24 with the vector w i gives the relation between a single strut velocity and the angular velocity of the moving platform as q i = wt i wi T (25) ev Writing (25) for i =1, 2, 3, and rearranging in a matrix form gives q = J 1 q J x Ẋ = JẊ (26) configurations associated with these solutions are depicted consecutively in Fig.4. (a) J = J q = [ w T 1 w T 2 w T 3 w1 T e w2 T e ] T (27) w3 T e w1 T e w2 T e 0 (28) 0 0 w3 T e J x = [ w T 1 w T 2 w T 3 ] T (29) where q = ( q 1 q 2 q 3 ) T, Ẋ = ( ẋ ẏ ż ) T, J, Jx, and J q denote the vector that contains the slider velocities, the velocity vector of the moving platform, the Jacobian matrix of the 3-PUU TPM, the direct kinematic Jacobian matrix, and the inverse kinematic Jacobian matrix, respectively. TABLE I ARCHITECTURE PARAMETERS OF THE 3-PUU TPM (b) S s l d β 1 β 2 β 3 (m) (m) (m) (m) (rad) (rad) (rad) π/3 π/3 E. Numerical Examples To show the efficiency of the kinematic models, the architecture parameters of the 3-PUU TPM are presented in Table I. For the forward position analysis, given three different sets of inputs ( q 1 q 2 q 3 ), output results ( x y z ) can be calculated as shown in Table II, and their corresponding TABLE II FORWARD POSITION ANALYSISOFTHE3-PUU TPM Inputs(m) Outputs(m) q 1 q 2 q 3 x y z case(a) case(b) case(c) Fig. 4. (c) Different configurations for forward position III. WORKSPACE ANALYSIS The workspace of a mechanism is defined as the set of all end effector configurations which can be reached by some choice of joints coordinates. Comparing with their serial counterparts, parallel manipulators have relatively small workspace, thus the workspace of a parallel manipulator is one of the utmost important aspects which reflect its working capacity and kinematic performance. 1850

5 A. Physical Constraints 1) the stroke limits of sliders. q min q i q max (30) where q min and q max represent the minimum boundary and the maximum boundary of slider respectively. 2) the motion height limit of moving platform. z h b h w (31) where h b and h w represent the height of the post and rotational wrist, respectively. 3) the revolute angle limits of u joint. The cone angles of u joint are as follows. ( ) n T φ max φ i = arcsin i1 w i φ max n i1 (32) ( ) n T ϕ max ϕ i = arcsin i2 w i ϕ max n i2 (33) where n i1 = (sinβ i, cos β i, 0) T and n i2 = ( cos β i,sinβ i, 0) T denote the normal vectors of the radial plane and the tangential plane, respectively. B. Simulation TABLE III THE CONSTRAINTS PARAMETERS OF THE 3-PUU TPM q min q max h b h w φ max ϕ max (m) (m) (m) (m) (rad) (rad) π/6 π/3 The constraints parameters of a 3PUU parallel manipulator are given in Tab.III. The reachable workspace of the manipulator generated by Matlab program is shown in Fig. 5. It is observed that the reachable position workspace is 120 symmetrical about the motion directions of three actuators. In the whole range of the workspace, the cross-sectional shape is close to an equilateral triangle. IV. KINEMATIC PERFORMANCE EVALUATION One important issue in the design of robot manipulators is evaluating their kinematic performance based on their Jacobian matrix. Contrary to serial manipulators, the Jacobian matrix of a parallel manipulator transforms the platform velocity vector into the actuators velocity vector or transforms the actuator force vector into the platform force vector, and thus any measure of dexterity and manipulability may be expressed in terms of properties of Jacobian matrix. A. Dexterity The dexterity of a manipulator can be thought as the ability of the manipulator to arbitrarily change its position and orientation, or apply forces and torques in arbitrary directions. It is defined on the basis of the condition number of the Jacobian matrix cond(j) = J J 1 (34) Mathematically, if 2-norm of the matrix is chosen for, the condition number is given by cond 2 (J) = σ max (35) σ min where σ max (J) and σ min (J) represent the maximum and minimum singular values of Jacobian matrix J, respectively. As a measure of dexterity, the condition number ranges in value from one (isotropy) to infinity (singularity) and measures the degree of ill-conditioning of the Jacobian matrix, i.e., nearness of the singularity. If the condition number of the manipulator Jacobian matrix is close to one in certain configurations, the manipulator is identified to be in isotropic configurations which mean that the manipulator has completely same mapping magnification in all directions. On the contrary, if it is close to infinity, the manipulator is regarded to be in the vicinity of singular configurations, thus the condition number is to be kept as small as possible. Since the kinematic property of the 3-PUU TPM prototype does not change along the z axis, a local dexterity index (LDI) on a specific z plane rather than a global dexterity index in the whole workspace is properly chosen, LDI can be defined as 1 LDI = cond 2 (J) = σ min (36) σ max which varies in the range [0 1], when LDI=0, it indicates that the parallel manipulators therein singular configuration, and LDI=1 indicates an isotropy configuration. Fig. 5. (c) Reachable workspace of the 3-PUU TPM B. Simulation In this section, evaluation of dexterity of the 3PUU TPM is carried out by investigating the distribution of kinematic performance indices with respect to configuration. The distribution of LDI is given in Fig. 6 when the { end-effector of the 3-PUU TPM is in the circle (x, y, z) } x 2 + y 2 =0.14m,z =0.6m. It is shown that the dexterity of the 3-PUU TPM is maximum when the endeffector is on point I (x =0,y =0,z =0.6m), and it is 1851

6 Fig. 6. (c) Distribution of dexterity - LDI minimum when the end-effector is on the edge of the circle. Therefore, the 3-PUU TPM has the best dexterity when the end-effector is on point I, and it has the worst dexterity when the end-effector is on the edge of the circle. Moreover, the distribution of dexterity is symmetrical to the plane y =0. V. CONCLUSION The kinematics analysis, workspace determination and performance evaluation of the 3-PUU parallel module for a 3D printing manipulator is carried out in this paper. Forward position, inverse position, inverse velocity and inverse acceleration are analytically analyzed. It is noteworthy that the solution for forward position analysis can be derived an in closed form. Moreover, in view of the physical constraints imposed by sliders, revolute joints and structure limitations, the reachable workspace which is 120 symmetrical about the motion directions of three actuators is determined. The cross-sectional shape of the reachable workspace is close to an equilateral triangle. Considering the kinematic property of the 3-PUU TPM prototype does not change along the z axis, a local dexterity as the kinematic performance index is taken. The best dexterity and worst dexterity of 3-PUU TPM occur when the end-effector is on point I and on the edge of the circle, respectively. REFERENCES [1] J. Jung, H. Kang H, T. Kang, J. Park, J. park and D. Cho, Projection image-generation algorithm for fabrication of a complex structure using projection-based microstereolithography, Int. J. Prec. Eng. Manuf., vol.13, no.3, pp , [2] B. Derby, Printing and prototyping of tissues and scaffolds, Science, vol.338, no.6109, pp , [3] C. Cajal, J. Santolaria, J. Velazquez, S. Aguado and J. Albajez, Volumetric error compensation technique for 3D printers, Procedia Eng., vol.63, pp , [4] F. Pati, J. Shim, J. Lee and D. 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