Dynamic Modeling and Analysis of a 2PRU-UPR Parallel Robot Based on Screw Theory

This paper proposes the systemic dynamic modeling and analysis of a 2PRU-UPR parallel robot with two rotations and one translation based on screw theory, where P, R and U denote prismatic, revolute and universal joints, respectively. Compared with existing parallel robots having two rotations and one translation, the two actuated prismatic joints of the 2PRU-UPR parallel robot are mounted on a fixed base to reduce the movable mass and improve the dynamic response. First, the inverse kinematics are presented. Next, adopting the screw-based method, the velocity and acceleration of joints and limbs of the 2PRU-UPR parallel robot are analyzed in detail. The actuated forces of the three actuators are then obtained according to the principle of virtual work. Additionally, a numerical simulation is conducted using ADAMS software to investigate the dynamic model of the 2PRU-UPR manipulator and to verify the correctness of the theoretical results. Finally, distributions of the dynamic manipulability ellipsoid index are used to evaluate the dynamic translational and rotational performances of the 2PRU-UPR parallel robot. A prototype based on the dynamic analysis has been fabricated. The dynamic modeling and evaluation provide a basis for the efficient and precise control of the 2PRU-UPR parallel robot in actual machining manipulations. The 2PRU-UPR parallel robot has great potential in machining workpieces with curved surfaces.

Parallel robots have been intensively studied over past decades and used in many applications, from simple pickand-place operations to advanced electronic manufacturing [1]- [3]. Having a closed structure, parallel robots have potential advantages of a high load-carrying capacity, good positioning accuracy and low inertia [4]- [7].
Dynamic modeling plays an important role in the predesign stage of the development of parallel robots and is essential for dynamic performance analysis and parameter optimization. The relations between forces and motions are investigated through modeling [8] [9] and are useful for the development of the control scheme [10], [11]. The dynamic analysis of parallel robots is carried out adopting several methods, mainly the Newton-Euler method [12]- [16], Lagrange method [17]- [21] and the adoption of the virtual work principle [22]- [26]. Additionally, methods such as the The associate editor coordinating the review of this manuscript and approving it for publication was Vlad Diaconita . use of Gibbs-Appell equations [27], [28] and Kane equation [29], [30] are used in the dynamic analysis of parallel robots.
The Newton-Euler method is the most common approach used in analyzing the dynamic model; it is relatively straightforward to implement and can be understood easily. However, the Newton-Euler method [12]- [16] requires many differential algebraic equations to be established, especially when the parallel system is complex and has multiple constraints. The derivation of equations requires much computational time.
In contrast with the Newton-Euler method, the Lagrange method [17]- [21] is based on the kinetic energy and potential energy of the system. The Lagrange method is based on pure mathematics instead of geometry. It is simple to develop the dynamic model with energy and generalized forces. However, a long computational time is required for matrix operations and the derivation of partial differentials. It is noted that the Lagrange method is not good for programming and computation. The use of the virtual work principle [22]- [26] is based on the system's overall virtual energy, which eliminates the derivation of constrained forces/moments. Such an energy approach for the analysis of a parallel robot can be simplified applying screw theory [31]- [35]. The principle of virtual work is suitable for the programming and computation of parallel robots, including complex models [36]- [38].
Adopting screw theory and the virtual work principle, this paper presents the systemic dynamic modeling and analysis of the three-degree-of-freedom (3-DOF) 2PRU-UPR parallel robot with two rotations and one translation (2R1T) [39]. Compared with existing 2R1T parallel robots, such as the parallel module of the Exechon robot [40], the 2UPR-SPR parallel robot with 13 single-DOF joints, the parallel module of the Tricept robot [41], and the 3UPS-UP parallel robot with 21 single-DOF joints, the 2PRU-UPR parallel robot only has 12 single-DOF joints, which is helpful in terms of decreasing the deformation and clearance caused by the joints [39]. Here, P, R, U and S denote prismatic, revolute, universal and spherical joints, respectively. Additionally, two of actuated prismatic joints of the 2PRU-UPR parallel robot are mounted on a fixed base to reduce the movable mass and improve the dynamic response. In industry, the 2PRU-UPR parallel robot has great potential in machining workpieces with curved surfaces. It is therefore necessary to develop a dynamic model that can be used in further research, such as the design of control systems and experiments. Through screw theory, the expressions of the velocity and acceleration of each joint and limb of the 2PRU-UPR parallel robot are established easily, and these expressions are easy to understand and implement. Using the principle of virtual work, the actuated forces of the 2PRU-UPR parallel robot are obtained directly. The correctness of the theoretical results for a given trajectory is verified by comparing with numerical results obtained using ADAMS software. Using a dynamic model with clear physical meaning and a simple expression allows easier analysis of the dynamic performance of the 2PRU-UPR parallel robot. Additionally, the dynamic model can be used to achieve real-time control with high efficiency and precision, which is beneficial for machining workpieces in industry.
In evaluating the dynamic performance of the 2PRU-UPR parallel robot, the present paper employs the dynamic manipulability ellipsoid (DME) [2], [42]- [44] to depict manipulability, namely the ease with which the position and orientation of the manipulator's moving platform can be changed. The distribution of this index of the 2PRU-UPR parallel robot is obtained and discussed. The results of the paper will serve as criteria for the structural optimization and motion control of a prototype of the 2PRU-UPR parallel robot.
The remainder of the paper is organized as follows. Section 2 briefly introduces the basics of screw theory. Section 3 presents the inverse kinematics analysis. Section 4 analyzes the velocity and acceleration of joints and limbs of the 2PRU-UPR parallel robot. Section 5 presents the derivation of actuated forces and verifies the correctness of the theoretical method. Section 6 presents distributions of the DME for different operating heights. Section 7 describes the prototype and hard system of the 2PRU-UPR parallel robot. Section 8 presents conclusions.

II. SCREW THEORY BASICS
In a Cartesian coordinate system, the position and direction of a line are represented by a vector S. r is a position vector at a given point expressed as Equation (1) represents a straight line using homogeneous coordinates (S; S 0 ) T to show the position and direction of the line in space. There are six components. The straight line is represented by a screw with the form where h = S · S 0 (S · S) is the pitch of the screw. This screw is a line vector and represents a revolute pair or a binding force when the pitch is zero. A screw with an infinite pitch is expressed as S= (0; S) T , where S is a direction vector. This screw represents a prismatic pair or a binding couple.
In modeling based on screw theory, the velocity of the rigid body is denoted V O , the primary component is denoted where When the representation point changes from O to P, only the dual part of the twist changes: where r P/O is the position vector from point O to point P.
From the definition of the screw, S= S S 0 T , the calcu-

III. INVERSE ANALYSIS OF THE 2PRU-UPR PARALLEL ROBOT
To simplify the progress of the dynamic model of the parallel robot, a simplified computer-aided-design model is created as shown in Fig. 1. Here, the moving platform is connected to the fixed base through three kinematic limbs. The first limb, A 1 B 1 , and the second limb, A 2 B 2 , are identical PRU kinematic chains. The axis of the R joint is perpendicular to the axis of the P joint, the first axis of the U joint is parallel to the axis of the R joint, and the second revolute axis of the U joint is connected to the moving platform. The third limb, A 3 B 3 , is a UPR kinematic chain. The first revolute axis of the U joint is perpendicular to the axis of the P joint. The second revolute axis of the U joint is always parallel to the second axes of the U joints of the first two limbs. As shown in Fig. 1, the fixed frame O − xyz is attached to the fixed base, and the x − axis and y − axis pass through points B 3 and B 1 , respectively. A moving frame o − uvw is attached to the moving platform with u − axis along oA 3 and v − axis along oA 1 . The local coordinate frame of each limb, as shown in Fig. 1, is established to describe inertial characteristics. The local coordinate frames B i − x ij y ij z ij are attached to the kinematic links of the 2PRU-UPR parallel robot. The dimensional parameters of the 2PRU-UPR parallel robot are defined as To simplify the dynamic analysis of the 2PRU-UPR parallel robot, η = z α β T is chosen as the vector of independent generalized coordinates in this paper. The dependent coordinates must satisfy the constraints where γ denotes the rotation angle around the z-axis and the term tan denotes the tangent function. As shown in Fig. 1, According to [38], the inverse kinematics of the 2PRU-UPR parallel robot are written as

IV. VELOCITY AND ACCELERATION ANALYSIS OF THE 2PRU-UPR PARALLEL ROBOT
The generalized velocity of the moving platform is obtained by taking derivatives of (7): The linear velocity and rotational velocity of the moving platform are therefore where v and ω are the linear velocity and rotational velocity of a given point P on the moving platform and K p is the mapping between the velocity of the moving platform and the generalized coordinates. K p is expressed as The linear velocity and rotational velocity are differentiated to obtain the linear acceleration and rotational acceleration of the moving platform: Here, the first limb (i.e., the PRU limb) is taken as an example to verify the proposed method. With respect to the fixed frame O − xyz, the twist system of the first limb in the fixed frame O − xyz is written as The corresponding twist system of the first limb is rewritten as a 6 × 1 matrix and the velocity of the rigid body is written as In the case of the 2PRU-UPR parallel robot, the linear expression of the moving platform is The linear velocity and rotational velocity of the moving platform are expressed according to (5) as Equation (16) is written in matrix form as where Using (13), the Jacobian matrix of the first limb is obtained as The velocity of the moving platform, relative to the fixed link, was solved as (17). The Jacobian matrix of the mechanism is reversible if the mechanism does not have motion redundancy. The rotational velocity i ω i+1 is obtained by inversing the Jacobian matrix J i : We define a unit screw S (i) unit , which is rewritten as a 6 × 1 matrix. According to the structural parameters of the 2PRU-UPR parallel robot, the unit screw S (i) unit intersects all screw vectors except the prismatic pair of the ith limb, such that the reciprocal product of the unit screw S (i) unit and all the screws of the ith limb except the prismatic pair screw is zero. This is expressed as where j S j+1 denotes all screws of the ith limb except the prismatic pair screw. The Klein form is applied to both sides of (20). The relationship between the drive velocity of each limb and the velocity of the moving platform is written aṡ where Here, J i ω is a joint velocity matrix for the ith limb. The drive velocity of the ith limb is obtained using (22) and the velocity of the moving platform. Rearranging (22) gives We define a coefficient G j k ω k+1 for the relation between the velocity of active joint j and the rotational velocity of passive joint k and obtain Substituting (24) into (23) yields Equation (25) shows that the joint rotational velocity of the corresponding limb is derived from the driving velocity and coefficient G The vector of the centroid of link C m is denoted, with respect to its own coordinate system, as r o . With respect to the base coordinate system, the centroid of link C m is expressed as r n = R Li r o . The centroid velocity screw of the component is (27) We define the centroid partial screw of a component as We denote the acceleration of the rigid body by A o , denote the primary component by ϑ(A O ) =ω, and express the dual component as D( whereω is the angular velocity and a 0 − ω × v O is the linear acceleration of the rigid body at point O. Applying the theory of primary kinematics, the acceleration of the given point P is obtained as

VOLUME 8, 2020
A reference coordinate system is attached to the fixed link, such that the acceleration of the moving part m relative to the fixed link is derived as where The reciprocal product of two screws S 1 S 2 was introduced in (6). When the mechanism does not have motion redundancy, the Jacobian matrix of the mechanism is reversible, and (31) is rewritten as The corresponding value of screw S L based on (32) is a 6 × 1 matrix. According to (29), the acceleration of the rigid body is According to (30), the acceleration of the link in the 2PRU-UPR parallel robot is The centroid acceleration of the moving platform and sliders is The centroid acceleration of the links is

V. FORCE ANALYSIS AND NUMERICAL SIMULATION
We use 0 V n Cm and 0 A n Cm to denote the velocity and acceleration of a centroid C m of a rigid body whose mass is m, respectively. We assume a force F n I ,Cm acting on the center of mass C m : where I n = 0 R n I • n 0 R n T is the inertia of the link with respect to the base coordinate system.
Denoting the acceleration due to gravity by g, the force F n G,Cm is written as Assuming that an external force f n E and torque τ n E act at the center of mass of the link, the external force F n E,Cm is written as In the dynamic equations of the mechanism, the force of each link must be based on the same coordinate system. The vector of force F n acting on a link is written as The instantaneous power ω n acting on the mass center of the link is expressed as In the case of the 2PRU-UPR parallel robot, the force screw acting on the moving platform and the sliders is The force screw acting on the link is Here, m l , m s and m p are the masses of the corresponding links, sliders, and moving platform, respectively. Parameter g denotes gravity. Parameters I s , I l and I P denote the moments of inertia of the corresponding links, sliders and moving platform, respectively. The driving force is calculated using (41) and the principle of virtual work: The principle of virtual work is that for any virtual displacement, the sum of virtual work done by external forces and internal forces of a multi-rigid body system under the effect of an external drive is zero. The virtual velocity δq i is substituted into (45) to give For an arbitrary choice of the virtual velocity δq i , the total virtual work is zero. δW = 0 if and only if and the drive can be obtained. Equation (47) solves out the drive of the 2PRU-UPR parallel robot. The drive of the mechanism is derived through the given mechanism model parameters and the track of the moving platform. We import the model into ADAMS software for simulation. Finally, the simulation results of ADAMS are verified with the calculated drive values.
In the numerical calculation, the lengths of the first and second limbs are l 1 = l 2 = 0.259m, the dimensions of the moving platform are a 1 = a 2 = 0.074m, and the distance of the sub-center U of the third limb from the origin of the fixed coordinate system is b 3 = 0.148m. Other structural parameters are listed in Table 1. The trajectories are tested for the dynamic model α = arctan 0.015sin(πt) 0.15+0.03πt where t is time. The duration of simulation is set at 0.6s. VOLUME 8, 2020 Given the trajectory (48) of the moving platform, the driving forces of prismatic joints are calculated from the above dynamic analysis. Figure 2 shows that the driving forces of the 2PRU-UPR parallel robot obtained using the proposed method and ADAMS are almost the same. The proposed theoretical dynamic model can therefore be regarded as an alternative for the accurate dynamic modeling of the 2PRU-UPR parallel robot.

VI. DYNAMIC PERFORMANCE ANALYSIS OF THE 2PRU-UPR PARALLEL ROBOT
This section uses the DME method [2], [42]- [44] to evaluate the dynamic performance of the 2PRU-UPR parallel robot. By ignoring the effects of velocity and gravity on the actuated forces, this method develops the mapping relationship between the acceleration of the moving platform and actuated forces, from which the dynamic performance of the 2PRU-UPR parallel robot can be evaluated. Additionally, by separating the Jacobian matrix, the DME method can be adopted to analyze the accelerations of translation and rotation of the moving platform under a driving force for prismatic joints. This section also derives distributions of DME indices in the workspace for different operational heights. These dynamic analyses will be useful for the control and experiments of the 2PRU-UPR parallel robot in future work.
The system equation of motion of the parallel manipulator is + C(q,q)q + G(q).
In this paper, the dynamic model of the 2PRU-UPR parallel robot is rewritten as  prismatic joints, the loads applied to the moving platform and the related torques.
For the 2PRU-UPR parallel robot, we rewrite (50) in the form where F d is the driving force applied to the sliders of the parallel robot. The 3 × 3 matrix J d is related to the system's generalized coordinates; i.e.,ḋ = J dq . The vectoṙ v e = ω T Pv T P T denotes the rotational and translational accelerations of the end-effector of the parallel mechanism. The 6 × 3 matrix J e = J P = [ J T P,R J T P,T ] T is the Jacobian of the parallel mechanism's end-effector.
According to (51), a dynamic manipulability ellipsoid is obtained to describe the performance of the moving platform in terms of acceleration in various directions as whereã =v e −J eq is the acceleration of the moving platform.
inertia matrix of the parallel mechanism, andJ = J e J −1 d is a 6 × 3 matrix. A separate dynamic matrix associated with the rotation and translation of the manipulator is derived from (52). By replacing the corresponding parts of the Jacobian matrix with separate parts, two dynamics manipulability ellipsoids are obtained for the evaluation of the performances of rotational and translational accelerations: whereã R andã T are related to the rotational and translational aspects ofã, respectively. Accordingly, the separate Jacobians are given byJ R = J P,R J −1 d andJ T = J P,T J −1 d , both of which are 3 × 3 matrices.
It is found from (7) that coordinate y is related to coordinate z. Combined with the mobility characteristic of the VOLUME 8, 2020 2PRU-UPR parallel robot, the ellipsoids described by (53) can be understood as a two-dimensional subspace of a threedimensional rotational space and a two-dimensional subspace of a three-dimensional translational space. The rotational and translational accelerations are therefore limited to two different planes. On this basis, bothJ T are symmetric semidefinite matrices with rank 2, and each has two positive eigenvalues and one zero eigenvalue. Using these nonzero eigenvalues, the condition number of the 3 × 3 matrixMJ + R is used to evaluate the isotropic characteristics of rotation in the dynamic performance of manipulators: where σ R,1 and σ R,2 are nonzero singular values ofMJ + R and σ R,1 ≤ σ R,2 . Similarly, the translation in the dynamic performance of manipulators is where σ T ,1 and σ T ,2 are nonzero singular values ofMJ + T and σ T ,1 ≤ σ T ,2 .
Kinematics analysis reveals that parameters α, β and z affect the dynamic manipulability. Hence, in the performance analysis, the DME is analyzed using α, β and z. The dynamic index of translation is affected by the parameter z, which increases from 0.1 to 0.22 m, as shown in Fig. 3(a)-(d). The dynamic index of rotation is also affected by parameter z, as shown in Fig. 4(a)-(d). Figure 3(a)-(d) shows the DME for the manipulator at its highest position. In this case, the manipulator is asymmetrical around the α axis, which can be explained in that the translation is only associated with the first and second limbs. The third limb contributes nothing to the translation. The model is symmetrical along the α axis but asymmetrical along the β axis. With z increasing from 0.1 to 0.22 m, the range of the DME index in the direction of the β axis increases and the distribution moves along the β axis. The reason may be that with z increasing from 0.1 to 0.22 m, the angles of the third limb and moving platform are smaller when the moving platform is parallel to the fixed base. Figure 3(a)-(d) also shows that when α > 30 • , α < −30 • or β > 30 • , the translational dynamic performance is poor owing the 2PRU-UPR parallel robot being near the singular configurations in these regions. Additionally, the maximum value of the translational performance increases from 0.45 to 0.7 with z increasing from 0.1 to 0.22 m, which means that a better range for translational operations is around z = 0.22m. Figure 4(a)-(d) shows the manipulator at two higher positions with symmetry along the α axis. The two higher positions are achieved as a result of the positions of the limbs. Rotation around the α axis is only associated with the first and second limbs while rotation around the β axis is only associated with the third limb. As the model is symmetrical along the α axis and asymmetrical along the β axis, the dynamic index of rotation is symmetrical along the α axis and asymmetrical along the β axis. With an increase in z from 0.1 to 0.22 m, the distance between the manipulator and driving pairs lengthens, leading to a smaller dynamic index for rotation. Figure 4(a)-(d) also shows that the rotational dynamic performance is poor when α > 5 • or α < −5 • . In contrast with the distributions of translational dynamic performances, the maximum value of the index of rotational performance decreases from 0.9 to 0.45 with z increasing from 0.1 to 0.22 m, which means that the better region for industrial application (i.e., the better range for rotational operations) is around z = 0.1m. Figures 3 and 4 show that it is necessary to select different operational ranges according to the actual needs of rotation and translation in future applications.

VII. PROTOTYPING
A prototype of the 2PRU-UPR parallel robot has been built, as shown in Fig. 5. An exploded diagram of the UPR limb is shown in Fig. 6. Each limb comprises a support frame, linear units and links connected by revolute joints, as shown in the figure. A linear unit includes a servo motor (ECMA-C10401FS), a link, ball screws (R16-5T3-FSI), couplings, a linear guide (MGN9C-2-R315-Z0-H-M), a support, a bearing shaft and a sliding block. The key link parameters were introduced in Section 5. To guarantee the workspace and avoid a singularity, the displacement of the prismatic pairs is limited within a certain range d i ∈ 75mm, 310mm , and the screw length and lead are set at 400 and 5 mm, respectively. To prevent the sliding block from breaking, two optoelectronic switches are installed near the two ends of the screw. When the sliding block is detected by the optoelectronic switch, the control system stops the movement of the corresponding servo motor. Experiments on the dynamics, accuracy and kinematic calibration will be conducted in future work.

VIII. CONCLUSION
The dynamic modeling and performance analysis of a 3-DOF 2PRU-UPR parallel robot were presented. On the basis of inverse kinematics, expressions of the velocity and acceleration of joints and limbs of the 2PRU-UPR parallel robot were easily derived using screw theory, and the actuated forces were then obtained directly using the principle of virtual work. The proposed modeling method was verified in numerical simulation against dynamics modeling using ADAMS software. The DME was used to evaluate the dynamic translational and rotational performances of the 2PRU-UPR parallel robot, and the effects of the operating height on the distribution of the DME were discussed in detail, providing criteria for further precise control and high-speed machining. A prototype suitable for the machining of curved surfaces was built.