J. Cent. South Univ. Technol. (2010) 17: 323-331

DOI: 10.1007/s11771-010-0049-8

Dynamic modeling and analysis of 3-RRS parallel manipulator with flexible links

LIU Shan-zeng(刘善增)^{1, 2}, YU Yue-qing(余跃庆)¹, ZHU Zhen-cai(朱真才)²,

SU Li-ying(苏丽颖)¹, LIU Qing-bo(刘庆波)¹

1. College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology,

Beijing 100124, China;

2. School of Mechanical and Electrical Engineering, China University of Mining and Technology,

Xuzhou 221116, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2010

Abstract: The dynamic modeling and solution of the 3-RRS spatial parallel manipulators with flexible links were investigated. Firstly, a new model of spatial flexible beam element was proposed, and the dynamic equations of elements and branches of the parallel manipulator were derived. Secondly, according to the kinematic coupling relationship between the moving platform and flexible links, the kinematic constraints of the flexible parallel manipulator were proposed. Thirdly, using the kinematic constraint equations and dynamic model of the moving platform, the overall system dynamic equations of the parallel manipulator were obtained by assembling the dynamic equations of branches. Furthermore, a few commonly used effective solutions of second-order differential equation system with variable coefficients were discussed. Newmark numerical method was used to solve the dynamic equations of the flexible parallel manipulator. Finally, the dynamic responses of the moving platform and driving torques of the 3-RRS parallel mechanism with flexible links were analyzed through numerical simulation. The results provide important information for analysis of dynamic performance, dynamics optimization design, dynamic simulation and control of the 3-RRS flexible parallel manipulator.

Key words: flexible robot; parallel manipulator; dynamic analysis; kineto-elastodynamics analysis; driving torque

1 Introduction

The parallel manipulator is typically made of two rigid bodies (a base and a moving platform), one movable and the other fixed, connected to each other through at least two independent kinematic chains. Such a manipulator owns the advantages of high mechanical stiffness, wide bandwidth and large load-weight ratio. Therefore, it is very popular in many industrial applications [1-5]. There have been many reports of research on the kinematics of parallel manipulators, but few subjects dynamics have been investigated. Moreover, most researches on the dynamics of parallel manipulators are based on rigid links. In some applications, however, link flexibility needs to be considered [6-8]. When a direct-drive arm for laser cutting, for instance, tracks a curved trajectory at a high speed with an acceleration of over 3G, the minimum tracking error should be less than 0.2 mm. Thus, mechanical vibration has to be controlled to achieve satisfactory performance. Modeling of flexible mechanisms has attracted much attention in recent years. However, most researches reported were on serial mechanisms, i.e., open-loop mechanisms, while less on closed-loop structures, i.e., four-bar linkage mechanisms. Dynamic modeling of parallel mechanisms, characterized by multiple closed-loop chains, has not been studied extensively [7-11]. FATTAH et al [7] presented a method of modeling the dynamics of a 3-DOF spatial parallel manipulator with flexible links. The natural orthogonal complement of constraint matrix was used to derive the minimum number of motion equations and to eliminate the nonworking kinematic constraint forces caused by the kinematic coupling of links. The effect of geometric nonlinearities in elastic deformations has been considered in the formulated model. In Ref.[8], a sub-structuring modeling procedure was introduced for developing a dynamic model of a planar parallel platform. PIRAS et al [9] presented the results of a dynamic finite element analysis on a 3-PRR planar parallel robot with flexible links. Results showed that, for a given high-speed motion, the configuration of the mechanism exerted a significant influence on the nature of resultant elastic vibrations. WANG et al [10] investigated the kinematics and dynamics in a 6-DOF RTS fully parallel manipulator with elastic joints. DU and YU [11] established the standard dynamics formulation including equivalent torque of elastic moment with generalized input according to the principle of virtual work and Lagrange method.

Deformation of the elastic links is taken into account in kineto-elastodynamics (KED) studies of moving mechanisms due to external inertial loads [9, 12]. The flexible deformation of the links plays a significant role in high-speed operation because the links are usually lighter in weight and hence has lower structural stiffness than the heavier links. The finite element code used here is based on the linear theory of KED, because it is assumed that the small-amplitude structural vibrations of the mechanism do not exert a significant effect on its rigid-body motion. Therefore, the influence of elastic deformations on the rigid-body motion is ignored, and the motion equations are solved for prescribed normal motions. However, the influence of the prescribed rigid-body motion on the elastic deformations is accounted for. The aim of this paper is to derive the dynamic equations of a 3-RRS parallel manipulator with flexible links. Axial, transverse (in two planes) and torsional deformation of the links is considered in the mathematical model of the manipulator. Furthermore, the dynamic responses of the movable platform and the driving torques of the flexible parallel mechanism are analyzed.

2 Dynamic equations

A spatial 3-RRS parallel mechanism with revolute actuators is illustrated in Fig.1. The manipulator consists of three identical legs B_iC_iP_i, where i=1, 2, 3, a rigid moving triangular platform P₁P₂P₃, and a fixed platform B₁B₂B₃, assumed rigid as well. Each leg contains two flexible links that are coupled by a revolute joint. The legs are connected to platform P₁P₂P₃ through spherical joints and coupled to the base through revolute joints. This manipulator has three rigid degrees of freedom and three motors located on the fixed platform B₁B₂B₃ to drive the actuated joints. A basal Cartesian coordinate frame designated as the O-XYZ frame is fixed at the center of the base platform with the Z-axis pointing vertically upward and the Y-axis pointing the pin joint 1, B₁. Similarly, a coordinate frame P-X′Y′Z′ is assigned to the center of the upper platform, with the Z′-axis normal to the platform and the Y′-axis pointing the ball joint 1, P₁. The local frame B_i-x′_i1y′_i1z′_i1 (i=1, 2, 3) is fixed on the base platform with its z′_i1-axis pointing vertically upward and its x′_i1-axis being co-linear with the direction of the revolute joint axis, as shown in Fig.1. Moreover, Fig.1 includes two reference coordinate frames B_i-x_i1y_i1z_i1 and C_i-x_i2y_i2z_i2 (i=1, 2, 3) attached to the first and the second links of the ith leg, respectively. The x_i1-axis is co-linear with the direction of the first revolute joint axis, the z_i1-axis points from B_i toward C_i, and the y_i1-axis is perpendicular to both the x_i1-axis and the z_i1-axis. The x_i2-axis is co-linear with the direction of the second revolute joint axis, the z_i2-axis points from C_i toward P_i, and the y_i2-axis is perpendicular to both the x_i2-axis and the z_i2-axis.

Fig.1 Sketch map of spatial 3-RRS parallel manipulator

2.1 Spatial flexible beam element

In order to model the spatial 3-RRS parallel manipulator, a new model of spatial flexible beam element is proposed, as shown in Fig.2, where δ₁-δ₃ and δ₁₀-δ₁₂, δ₄-δ₆ and δ₁₃-δ₁₅, δ₇-δ₉ and δ₁₆-δ₁₈ are the axial or transverse displacements, rotary angles and curvatures at nodes A and B, respectively. It is supposed that a spatial flexible beam element is subject to axial, lateral (in two planes) and torsional deformation. A point in the element will have elastic displacements in x, y, z directions and around x, y, z directions. They can be defined as: W_x(x, t), W_y(x, t), W_z(x, t), ψ_x(x, t), ψ_y(x, t) and ψ_z(x, t). These deflection variables can be described uniquely by a set of interpolation functions and generalized coordinates, and expressed as follows:

                             (1)

                             (2)

Fig.2 Sketch map of spatial flexible beam element

                              (3)

                             (4)

                   (5)

                   (6)

where δ=[δ₁ δ₂ … δ₁₈]^T is the generalized coordinates of the element displacement, as shown in Fig.2, N_A, N_B, N_C and N_D?R^18?1 are the vectors of interpolation polynomials [12-13].

In terms of the O-XYZ frame, the element nodal deflection vector is defined as

           (7)

whereas, referred to the o-xyz frame, the nodal deformation is expressed as

                                     (8)

where R is the transformation matrix from the global frame O-XYZ to the element frame o-xyz.

Fig.3 shows the finite element model of a B_iC_iP_i branch. A total of two elements are employed, one for each link. The corresponding placement of nodes is shown in Fig.3. There are coincident nodes (point C_i) at the revolute connection between B_iC_i and C_iP_i. Boundary conditions for zero translational displacement and rotation angle are imposed at node C_i (cantilever hypothesis [12, 14] for the link of B_iC_i).

Fig.3 Finite element model of B_iC_iP_i branch: (a) Element coordinates; (b) Global coordinates

Before the dynamic equations of 3-RRS parallel manipulator can be obtained, it is necessary to derive the kinetic and elastic deformation energy of the spatial flexible beam element.

2.2 Kinetic energy

The kinetic energy of the spatial flexible beam element is expressed as

 (9)

where L is the length of the beam element, A is the cross-sectional area, r is the material density, and I_p is the area polar moment of inertia around x-axis, and W_ax(x, t), W_ay(x, t), W_az(x, t) and  are the absolute displacements and angular displacement of a point in the element [15].

Substituting Eq.(9) for Eqs.(1)-(4) and performing the integration, one obtains

                              (10)

where m_e is the element mass matrix [15], and  

2.3 Elastic deformation energy

The elastic deformation energy of the element is composed of axial deformation energy, lateral bending deformation energy and torsional deformation energy. So, the total elastic deformation energy is expressed as follows:

(11)

where E is the elastic modulus, G is the shear modulus, and I_z and I_y are the area inertia moments of the cross-section along z-axis and y-axis.

Substituting Eq.(11) for Eqs.(1)-(4) and performing the integration, one obtains

                              (12)

where k_e is the element stiffness matrix [15], and

2.4 Dynamic model of element

In order to obtain the dynamic equations of parallel manipulator system, the dynamic models of elements should be derived firstly. Then, the equations of the parallel mechanism can be obtained through assembly of the equations of its elements with a standard finite element procedure.

With the nodal displacement δ assumed as the generalized coordinates, Lagrange equation is expressed as

                   (13)

where f_e is the vector for generalized external nodal forces including the inertia forces in rigid-body motion [10].

Substituting Eqs.(10) and (12), Eq.(13) becomes

                             (14)

Substituting Eq.(8) and premultiplying by R^T, the element equations are expressed in terms of the O-XYZ frame as

                            (15)

where the element matrices are

, , .

2.5 Dynamic model of moving platform

The moving platform is considered as a rigid body. Its motion equations are expressed in terms of the P-X′Y′Z′ frame as

                                 (16)

where , m₀ is the mass of the

moving platform, [I] is 3×3 identity matrix, J is the moment of inertia matrix of the moving platform about the P-X′Y′Z′ frame, δ₀=[δ₀₁ δ₀₂ … δ₀₆]^T is the displacement deflection vector of the moving platform about the P-X′Y′Z′ frame induced by the flexible deformation of the links, and F₀ is the vector of generalized external forces including the inertia forces of nominal motion.

Then, the motion equation of the moving platform is expressed in terms of the O-XYZ frame as

                         (17)

where R₀ is the transformation matrix from the global frame O-XYZ to the local frame P-X'Y'Z', U₀∈R^6×1 is the displacement deflection vector of the moving platform about the O-XYZ frame due to the flexible deformation of the flexible links.

2.6 Kinematic constraint equations

The mechanism joints are considered as ideal holonomic connections. It is assumed that the position coordinates of the ball joints at point P_i, where i=1, 2, 3, in the O-XYZ frame are  Thus, the displacement constraint equation between the moving platform P₁P₂P₃ and the legs B_iC_iP_i, where i=1, 2, 3, can be expressed as

         (18)

where is the displacement deflection of point P_i, where i=1, 2, 3, in the O-XYZ frame.

Now, the kinematic constraint equations allow us to assemble the equations of all the flexible links to obtain the governing motion equations of the parallel manipulator.

2.7 Dynamic model of parallel manipulator

If Eqs.(15) and (17) are expanded to system size, all the element equations with kinematic constraint equations are combined, and proportional damping is incorporated, then the complete dynamic equation of the system is expressed as

                        (19)

where M, C and K are the total mass, damping and stiffness matrices of the system, respectively, F is the generalized external force vector of the system including the inertia forces of rigid-body motion, and U is the vector of the system’s generalized coordinates in terms of O-XYZ frame. All of them are time-dependent. Rayleigh damping has been introduced into Eq.(19), with the following formula of the damping matrix C as linear combination of the mass and stiffness matrices [12-14, 16]:

                             (20)

where l₁ is the mass damping coefficient, and l₂ is the stiffness damping coefficient. The magnitude of the Rayleigh coefficients is largely determined by the energy dissipation characteristics of the structural materials.

3 Solutions of dynamic equations

Mathematically, Eq.(19) represents a second-order linear differential equation system with variable coefficients and, in principle, the solutions to the equations can be obtained with the standard procedures of solving differential equations with constant coefficients. However, the procedures proposed for solving the general system of differential equations will become very expensive if the order of the matrices is large. In practical finite element analysis, we are therefore mainly interested in two methods of solution: direct integration and mode superposition. A few commonly used effective methods of direct integration are central difference, Houbolt, Wilson θ, and Newmark methods. In the solution the time span under consideration, T, is subdivided into n equal time intervals Δt (i.e. Δt=T/n), and in the employed integration scheme, an approximate solution at times 0, Δt, 2Δt, …, t, t+Δt, T is established. Owing to its least numerical errors, the Newmark method appears to be the most effective [16]. In the following discussion we will seek the solution of Eq.(19) using the Newmark method.

The Newmark integration scheme can be interpreted as the extension of the linear acceleration method. The following assumptions are used [16]:

             (21)

   (22)

where 0≤m₁≤1 and 0≤2m₂≤1, and m₁ and m₂ are parameters that can be determined by integral precision and stability. The Newmark integration is unconditionally stable provided that m₁≥0.5 and m₂≥0.25(m₁+0.5)².

In addition to Eqs.(21) and (22), in order to obtain displacements, velocities and accelerations at time t+Δt, dynamic Eq.(19) at time t+Δt is also considered:

              (23)

Solving Eq.(19) for  in terms of , and then substituting  into Eq.(21), we obtain equations for  and , each in terms of the unknown displacement  only. These two relations for  and  are substituted into Eq.(23) to calculate  at time t+Δt. After that, with Eqs.(21) and (22),  and  at time t+Δt can also be calculated.

4 Numerical simulation

The distance between revolute joint center B_i and point O is expressed as OB_i=R, where i=1, 2, 3. The distance between ball joint center P_i and point P is expressed as PP_i=r, where i=1, 2, 3. The lengths of the links B_iC_i and C_iP_i are l_i1 and l_i2, respectively, where i=1, 2, 3. The position coordinate of point P in the O-XYZ frame is denoted by (x_P, y_P, z_P)^T, and the orientation coordinate of the moving platform by means of Z-Y-X Euler angles is denoted by (α, β, γ). The inertia moments of the moving platform about  and Z′-axis are denoted by J_X′, J_Y′, and J_Z′, respectively.

The parameters of the 3-RRS parallel mechanism are specified as: each link is made of steel, mass density r=7.80×10³ kg/m³, elastic modulus E=211 GPa, shear modulus G=80 GPa, l_i1=l_i2=0.15 m (i=1, 2, 3), the cross-section is 1.5 mm×5 mm, m₀=0.152 kg, J_X′=  1.06×10^{-2 kg?m2}, J_Y′=2.74×10^{-4 kg?m2}, J_Z′=1.09×10^{-2 kg?m2}, r=0.10 m, R=0.12 m, l₁=2.0×10^-3, l₂=3.0×10^-4, ?t=0.01 s, m₁=0.528 0, m₂=0.264 2, and T=1 s. In this simulation, the moving platform is set to move on a desired trajectory given as

                 (24)

where s(t) is a time function that takes values between 0 and 1. Suppose that a cycloidal motion is selected for time function:

, 0≤t≤T                  (25)

where T is the traveling time.

Fig.4 illustrates the displacement errors of the moving platform in the global frame O-XYZ, wherein, ε_X, ε_Y and ε_Z refer to the position errors in directions X, Y and Z, respectively; ε_γ, ε_β and ε_α refer to the orientation errors around X-, Y- and Z-axis, respectively. With these preconditions, it is found that the maximum position error is -7.2 mm in direction Z, and the maximum orientation error is -0.055 rad around Y-axis. Hence, we should not ignore the influence of elastic links, especially when the moving platform moves at a relatively high speed. It is necessary for us to analyze the displacement errors of the flexible parallel manipulator.

Fig.4 Displacement errors of moving platform: (a) Position error in direction X; (b) Position error in direction Y; (c) Position error in direction Z; (d) Orientation error around X-axis; (e) Orientation error around Y-axis; (f) Orientation error around Z-axis

Figs.5 and 6 show the results of the velocity and acceleration simulation of the moving platform, respectively. v_X, v_Y and v_Z represent the velocities in directions X, Y and Z, and ω_γ, ω_β and ω_α refer to the angular velocities around X-, Y- and Z-axis, respectively. a_X, a_Y and a_Z refer to the accelerations in X, Y and Z directions, and a_γ, a_β and a_α refer to the angular accelerations around X-, Y- and Z-axis, respectively. From these figures, it is observed that the motion of the moving platform is complicated. Therefore, in order to make the manipulator to carry out desired motions actually, there is necessity to develop the motion control algorithms for the flexible parallel manipulator.

Fig.5 Velocities and angular velocities of moving platform: (a) Velocity in direction X; (b) Velocity in direction Y; (c) Velocity in direction Z; (d) Angular velocity around X-axis; (e) Angular velocity around Y-axis; (f) Angular velocity around Z-axis

Fig.6 Acceleration and angular acceleration of moving platform: (a) Acceleration in X direction; (b) Acceleration in Y direction;(c) Acceleration in Z direction; (d) Angular acceleration around X-axis; (e) Angular acceleration around Y-axis; (f) Angular acceleration around Z-axis

Fig.7 shows the results of driving torque simulation when the traveling time T equals 1 s for the 3-RRS parallel manipulator, in which τ₁, τ₂ and τ₃ refer to the driving torques of the driving links B₁C₁, B₂C₂ and B₃C₃, respectively. The value analysis of the driving torque is shown in Table 1.

Fig.7 Time-histories of driving torques: (a) Driving torque τ₁; (b) Driving torque τ₂; (c) Driving torque τ₃

Table 1 Value analysis of driving torques

5 Conclusions

(1) A new spatial beam element is presented and the dynamic equations of a 3-RRS parallel manipulator with flexible links are derived.

(2) The dynamic responses of the moving platform and driving torques of the 3-RRS parallel mechanism with flexible links are analyzed through numerical simulation. The results show that the link flexibility exerts a significant effect on displacement errors, velocities, accelerations and driving torques of the parallel manipulator.

(3) A method for the dynamic analysis, simulation, optimization design and control of the 3-RRS flexible parallel mechanisms is established.

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Foundation item: Projects(50875002, 60705036) supported by the National Natural Science Foundation of China; Project(3062004) supported by Beijing Natural Science Foundation, China; Project(20070104) supported by the Key Laboratory of Complex Systems and Intelligence Science, Institute of Automation, Chinese Academy of Sciences; Project(2009AA04Z415) supported by the National High-Tech Research and Development Program of China

Received date: 2009-06-15; Accepted date: 2009-09-30

Corresponding author: YU Yue-qing, Professor; Tel: +86-10-67391702; E-mail: yqyu@bjut.edu.cn

(Edited by YANG You-ping)